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On the degree of strong approximation of almost periodic
functions in the Stepanov sense
Włodzimierz Łenski and Bogdan Szal
University of Zielona Góra
Faculty of Mathematics, Computer Science and Econometrics
65-516 Zielona Góra, ul. Szafrana 4a, Poland
W.Lenski@wmie.uz.zgora.pl, B.Szal @wmie.uz.zgora.pl
( )
Abstract
Considering the class of almost periodic functions in the Stepanov sense we
extend and generalize the results of the first author [11]. as well as
the results of L. Leindler [7] and P. Chandra [4, 5] .
Key words: Degree of strong approximation, Almost periodic
functions, Strong approximation, Special sequences.
2000 Mathematics Subject Classification: 42A24, 41A25.
1 Introduction
Let $S^{p}\;\left(1<p\leq\infty\right)$ be the class of all almost
periodic functions in the Stepanov sense $\left(1<p<\infty\right)$ or
uniformly almost periodic $\left(p=\infty\right)$ with the norm
$$\|f\|_{S^{p}}:=\left\{\begin{array}[]{c}\sup\limits_{u}\left\{\frac{1}{\pi}%
\int_{u}^{u+\pi}\mid f(t)\mid^{p}dt\right\}^{1/p}\text{ \ \ when \ \ }1<p<%
\infty,\\
\sup\limits_{u}\mid f(u)\mid\text{ \ \ when \ \ }p=\infty.\end{array}\right.$$
Suppose that the Fourier series of $f\in S^{p}$ has the form
$$Sf\left(x\right)=\sum_{\nu=-\infty}^{\infty}A_{\nu}\left(f\right)e^{i\lambda_{%
\nu}x},\text{ \ \ where \ }A_{\nu}\left(f\right)=\lim_{L\rightarrow\infty}%
\frac{1}{L}\int_{0}^{L}f(t)e^{-i\lambda_{\nu}t}dt,$$
with the partial sums
$$S_{\gamma_{k}}f\left(x\right)=\sum_{\left|\lambda_{\nu}\right|\leq\gamma_{k}}A%
_{\nu}\left(f\right)e^{i\lambda_{\nu}x}$$
and that $0=\lambda_{0}<\lambda_{\nu}<\lambda_{\nu+1}$ if $\nu\in\mathbb{N}=\left\{1,2,3...\right\},$ $\underset{v\rightarrow\infty}{\lim}\lambda_{\nu}=\infty,$ $\lambda_{-\nu}=-\lambda_{\nu,}$ $\left|A_{\nu}\right|+\left|A_{-\nu}\right|>0.$ Let $\Omega_{\alpha,p}$, with some fixed positive $\alpha$ , be the set of functions of class $S^{p\text{ }}$ whose Fourier exponents satisfy the condition
$$\lambda_{\nu+1}-\lambda_{\nu}\geq\alpha\text{ \ \ }\left(\nu\in\mathbb{N}%
\right).$$
In case $f\in\Omega_{\alpha,p}$
$$S_{\lambda_{k}}f\left(x\right)=\int_{0}^{\infty}\left\{f\left(x+t\right)+f%
\left(x-t\right)\right\}\Psi_{\lambda_{k},\lambda_{k}+\alpha}\left(t\right)dt,$$
where
$$\Psi_{\lambda,\eta}\left(t\right)=\frac{2\sin\frac{\left(\eta-\lambda\right)t}%
{2}\sin\frac{\left(\eta+\lambda\right)t}{2}}{\pi\left(\eta-\lambda\right)t^{2}%
}\text{ \ \ }\left(0<\lambda<\eta,\text{
\ }\left|t\right|>0\right).$$
Let $A:=\left(a_{nk}\right)$ $\left(k,n=0,1,...\right)$ be a lower
triangular infinite matrix of real numbers satisfying the following
condition:
$$\text{ }a_{nk}\geq 0\text{ }\left(k,n=0,1,...\right),\text{ }a_{nk}=0\text{
}\left(k>n\right)\text{ and }\sum\limits_{k=0}^{n}a_{nk}=1.$$
(1)
Let us consider the strong mean
$$H_{n,A,\gamma}^{q}f\left(x\right)=\left\{\sum_{k=0}^{n}a_{n,k}\left|S_{\gamma_%
{k}}f\left(x\right)-f\left(x\right)\right|^{q}\right\}^{1/q}\text{ \ \ \ }%
\left(q>0\right)\text{.}$$
(2)
As measures of approximation by the quantity (2), we use the best
approximation of $f$ by entire functions $g_{\sigma}$ of exponential type $\sigma$ bounded on the real axis, shortly $g_{\sigma}\in B_{\sigma}$ and
the moduli of continuity of $\ f$ defined by the formulas
$$E_{\sigma}(f)_{S^{p}}=\inf_{g_{\sigma}}\left\|f-g_{\sigma}\right\|_{S^{p}},$$
$$\omega f\left(\delta\right)_{S^{p}}=\sup_{\left|t\right|\leq\delta}\left\|f%
\left(\cdot+t\right)-f\left(\cdot\right)\right\|_{S^{p}\text{ }}$$
and
$$w_{x}f(\delta)_{p}:=\left\{\frac{1}{\delta}\int_{0}^{\delta}\left|\varphi_{x}%
\left(t\right)\right|^{p}dt\right\}^{1/p}\text{ with }1<p<\infty,$$
where $\varphi_{x}\left(t\right):=f\left(x+t\right)+f\left(x-t\right)-2f\left(x\right)$, respectively.
A sequence $c:=\left(c_{n}\right)$ of nonnegative numbers tending to zero
is called the Rest Bounded Variation Sequence, or briefly $c\in RBVS$, if it
has the property
$$\sum\limits_{k=m}^{\infty}\left|c_{n}-c_{n+1}\right|\leq K\left(c\right)c_{m}$$
(3)
for all natural numbers $m$, where $K\left(c\right)$ is a constant
depending only on $c$.
A sequence $c:=\left(c_{n}\right)$ of nonnegative numbers will be called
the Head Bounded Variation Sequence, or briefly $c\in HBVS$, if it has the
property
$$\sum\limits_{k=0}^{m-1}\left|c_{n}-c_{n+1}\right|\leq K\left(c\right)c_{m}$$
(4)
for all natural numbers $m$, or only for all $m\leq N$ if the sequence $c$
has only finite nonzero terms and the last nonzero terms is $c_{N}$.
Therefore we assume that the sequence $\left(K\left(\alpha_{n}\right)\right)_{n=0}^{\infty}$ is bounded, that is, that there exists a constant $K$ such that
$$0\leq K\left(\alpha_{n}\right)\leq K$$
holds for all $n$, where $K\left(\alpha_{n}\right)$ denote the sequence
of constants appearing in the inequalities (3) or (4) for the
sequence $\alpha_{n}:=\left(a_{nk}\right)_{k=0}^{\infty}$.Now we can
give the conditions to be used later on. We assume that for all $n$ and $0\leq m\leq n$
$$\sum\limits_{k=m}^{\infty}\left|a_{nk}-a_{nk+1}\right|\leq Ka_{nm}$$
(5)
and
$$\sum\limits_{k=0}^{m-1}\left|a_{nk}-a_{nk+1}\right|\leq Ka_{nm}$$
(6)
hold if $\alpha_{n}:=\left(a_{nk}\right)_{k=0}^{\infty}$ belongs to $RBVS$ or $HBVS$, respectively.
The $C$-norm of the deviation $\left|\sum_{k=0}^{n}a_{n,k}\left[S_{k}f\left(x\right)-f\left(x\right)\right]%
\right|,$ with the
partial sums $S_{k}f$ of classical trigonometric Fourier series, was
estimated by P. Chandra [4] [5] for monotonic sequences $\left(a_{nk}\right)$ and by L. Leindler [7] for the sequences of bounded
variation. These results were generalized by W. Łenski [11] who
considered the strong means $H_{n,A}^{q\text{ }},$ also in classical case,
and the functions belonging to the $L^{p}$. In present paper we shall
considered the almost periodic functions from the Stepanov class giving
similarly estimations for the strong means $H_{n,A}^{q\text{ }}$ in
individual points and in norms.
We shall write $I_{1}\ll I_{2}$ if there exists a positive constant $C$ such
that $I_{1}\leq CI_{2}$.
2 Main results
Let us consider a function $w_{x}$ of modulus of continuity type on the
interval $[0,+\infty),$ i.e. a nondecreasing continuous function having the
following properties: $w_{x}\left(0\right)=0,$ $w_{x}\left(\delta_{1}+\delta_{2}\right)\leq w_{x}\left(\delta_{1}\right)+w_{x}%
\left(\delta_{2}\right)$ for any $\delta_{1},\delta_{2}\geq 0$ with $x$ such
that the set
$$\displaystyle\Omega_{\alpha,p}\left(w_{x}\right)$$
$$\displaystyle=$$
$$\displaystyle\left\{f\in\Omega_{\alpha,p}:\left[\frac{1}{\delta}\int_{0}^{%
\delta}\left|\varphi_{x}\left(t\right)-\varphi_{x}\left(t\pm\gamma\right)%
\right|^{p}dt\right]^{1/p}\ll w_{x}\left(\gamma\right)\right.$$
$$\displaystyle\left.\text{\ and \ }w_{x}f\left(\delta\right)_{p}\ll w_{x}\left(%
\delta\right)\text{ \ , \ where \ }\gamma,\delta>0\right\}$$
is nonempty. It is clear that $\Omega_{\alpha,p}\left(w_{x}\right)\subseteq\Omega_{\alpha,p^{\prime}}\left(w_%
{x}\right),$ for $p^{\prime}\leq p<\infty.$
Our main results are the following:
Theorem 1
Let (1) and (6) hold. Suppose $w_{x}$ is such that
$$\left\{u^{\frac{p}{q}}\int\limits_{u}^{\pi}\frac{\left(w_{x}\left(t\right)%
\right)^{p}}{t^{1+\frac{p}{q}}}dt\right\}^{\frac{1}{p}}=O\left(uH_{x}\left(u%
\right)\right)\text{ \ \ as \ \ }u\rightarrow 0^{+},$$
(7)
where $H_{x}\left(u\right)\geq 0$, $1<p\leq q$ and
$$\int\limits_{0}^{t}H_{x}\left(u\right)du=O\left(tH_{x}\left(t\right)\right)%
\text{ \ \ as \ \ }t\rightarrow 0^{+}.$$
(8)
If $f\in\Omega_{\alpha,p}\left(w_{x}\right),$ then
$$H_{n,A,\gamma}^{q}f\left(x\right)=O\left(a_{nn}H_{x}\left(a_{nn}\right)+\left%
\{\sum_{k=0}^{n}a_{n,k}\left(E_{\alpha k/2}\left(f\right)_{S^{p}}\right)^{q}%
\right\}^{1/q}\right),$$
(9)
where $q$ is such that $1<q\left(q-1\right)^{-1}\leq p\leq q.$
Theorem 2
Let (1), (5), (7) and (8) hold. If $f\in\Omega_{\alpha,p}\left(w_{x}\right)$, then
$$H_{n,A,\gamma}^{q}f\left(x\right)=O\left(a_{n0}H_{x}\left(a_{n0}\right)+\left%
\{\sum_{k=0}^{n}a_{n,k}\left(E_{\alpha k/2}\left(f\right)_{S^{p}}\right)^{q}%
\right\}^{1/q}\right),$$
(10)
where $q$ is such that $1<q\left(q-1\right)^{-1}\leq p\leq q.$
Consequently, we can immediately derive the results on norm approximation.
Theorem 3
Let (1) and (6) hold. Suppose $\omega f\left(\cdot\right)_{S^{\widetilde{p}}}$ is such that
$$\left\{u^{\frac{p}{q}}\int\limits_{u}^{\pi}\frac{\left(\omega f\left(t\right)_%
{S^{\widetilde{p}}}\right)^{p}}{t^{1+\frac{p}{q}}}dt\right\}^{\frac{1}{p}}=O%
\left(uH\left(u\right)\right)\text{ \ \ as \ \ }u\rightarrow 0^{+}$$
(11)
holds, with $1<p\leq q\leq\widetilde{p}$ , where additionally $H$ $\left(\geq 0\right)$ instead of $H_{x}$ satisfies the condition (8). If $f\in\Omega_{\alpha,\widetilde{p}}$, then
$$\left\|H_{n,A,\gamma}^{q^{\prime}}f\left(\cdot\right)\right\|_{S^{\widetilde{p%
}}}=O\left(a_{nn}H_{x}\left(a_{nn}\right)\right),$$
with $q^{\prime}\in(0,q]$, where $q$ is such that $1<q\left(q-1\right)^{-1}\leq p\leq q.$
Theorem 4
Let (1) and (5) hold. Suppose $\omega f\left(\cdot\right)_{S^{\widetilde{p}}}$ is such that (11) holds, with $1<p\leq q\leq\widetilde{p}$ , where additionally $H$ $\left(\geq 0\right)$ instead of $H_{x}$ satisfies the condition (8). If $f\in\Omega_{\alpha,\widetilde{p}}$, then
$$\left\|H_{n,A,\gamma}^{q^{\prime}}f\left(\cdot\right)\right\|_{S^{\widetilde{p%
}}}=O\left(a_{n0}H_{x}\left(a_{n0}\right)\right),$$
with $q^{\prime}\in(0,q]$, where $q$ is such that $1<q\left(q-1\right)^{-1}\leq p\leq q.$
Remark 1
Analyzing our proofs and dividing the integral in the formula
$$\left\{\sum_{k=0}^{n}a_{n,k}\left|\int_{0}^{\infty}\varphi_{x}\left(t\right)%
\Psi_{k+\kappa}\left(t\right)dt\right|^{q}\right\}^{1/q}$$
into parts with $\frac{\pi}{n+1}$ instead of $a_{n,n}$ or $a_{n,0}$ we can
obtain the next series of theorems analogously as in [11].
3 Lemmas
To prove our theorems we need the following lemmas.
Lemma 1
[11] If (7) and (8) hold, then
$$\int\limits_{0}^{u}\frac{w_{x}f\left(t\right)}{t}dt=O\left(uH_{x}\left(u\right%
)\right)\text{ \ }\left(u\rightarrow 0_{+}\right).$$
(12)
Lemma 2
[16, Theorem 5.20 II, Ch. XII] Suppose that $1<q\left(q-1\right)^{-1}\leq p\leq q$ and $\xi=\frac{1}{p}+\frac{1}{q}-1$. If $\left|t^{-\xi}g\left(t\right)\right|\in L^{p,}$ then
$$\left\{\frac{\left|a_{0}\left(g\right)\right|^{q}}{2}+\sum\limits_{k=0}^{%
\infty}\left(\left|a_{k}\left(g\right)\right|^{q}+\left|b_{k}\left(g\right)%
\right|^{q}\right)\right\}^{\frac{1}{q}}\ll\left\{\int\limits_{-\pi}^{\pi}%
\left|t^{-\xi}g\left(t\right)\right|^{p}dt\right\}^{\frac{1}{p}}.$$
(13)
4 Proofs of the Results
4.1 Proof of Theorem 1
In the proof we will use the following function $\Phi_{x}f\left(\delta,\nu\right)=\frac{1}{\delta}\int_{\nu}^{\nu+\delta}%
\varphi_{x}\left(u\right)du,$ with $\delta=\delta_{n}=\frac{\pi}{n+1}$ and its estimate
from [10, Lemma 1, p.218]
$$\left|\Phi_{x}f\left(\xi_{1},\xi_{2}\right)\right|\leq w_{x}\left(\xi_{1}%
\right)+w_{x}\left(\xi_{2}\right)$$
(14)
for $f\in\Omega_{\alpha,p}\left(w_{x}\right)$ and any $\xi_{1},\xi_{2}>0$.
Since, for $n=0$ our estimate is evident we consider $n>0$, only.
Denote by $S_{k}^{\ast}f$ the sums of the form
$$S_{\frac{\alpha k}{2}}f\left(x\right)=\sum_{\left|\lambda_{\nu}\right|\leq%
\frac{\alpha k}{2}}A_{\nu}\left(f\right)e^{i\lambda_{\nu}x}$$
such that the interval $\left(\frac{\alpha k}{2},\frac{\alpha\left(k+1\right)}{2}\right)$ does not contain any $\lambda_{\nu}.$ Applying
Lemma 1.10.2 of [9] we easily verify that
$$S_{k}^{\ast}f\left(x\right)-f\left(x\right)=\int_{0}^{\infty}\varphi_{x}\left(%
t\right)\Psi_{k}\left(t\right)dt,$$
where $\varphi_{x}\left(t\right):=f\left(x+t\right)+f\left(x-t\right)-2f\left(x\right)$ and $\Psi_{k}\left(t\right)=\Psi_{\frac{\alpha k}{2},\frac{\alpha\left(k+1\right)}{%
2}}\left(t\right),$ i.e.
$$\Psi_{k}\left(t\right)=\frac{4\sin\frac{\alpha t}{4}\sin\frac{\alpha\left(2k+1%
\right)t}{4}}{\alpha\pi t^{2}}$$
(see also [3], p.41). Evidently, if the interval $\left(\frac{\alpha k}{2},\frac{\alpha\left(k+1\right)}{2}\right)$ contains a Fourier
exponent $\lambda_{\nu},$ then
$$S_{\frac{\alpha k}{2}}f\left(x\right)=S_{k+1}^{\ast}f\left(x\right)-\left(A_{%
\nu}\left(f\right)e^{i\lambda_{\nu}x}+A_{-\nu}\left(f\right)e^{-i\lambda_{\nu}%
x}\right).$$
Since (see [1, p.78] and [2, p. 7])
$$\left\{\sum_{\nu=-\infty}^{\infty}\left|A_{\nu}\left(f\right)\right|^{q}\right%
\}^{1/q}\leq\left\|f\right\|_{B^{p}}\text{ \ \ and \ \ }\left\|f\right\|_{B^{p%
}}\leq\left\|f\right\|_{S^{p}}\text{,}$$
where $\left\|\cdot\right\|_{B^{p,}}$ with $p\geq 1,$ is the
Besicovitch norm, so we have
$$\left|A_{\pm\nu}\left(f\right)\right|=\left|A_{\pm\nu}\left(f-g_{\alpha\mu/2}%
\right)\right|\leq\left\|f-g_{\alpha\mu/2}\right\|_{S^{p}}=E_{\alpha\mu/2}%
\left(f\right)_{S^{p}},$$
for some $g_{\alpha\mu/2}\in B_{\alpha\mu/2},$ with $\alpha k/2<\alpha\mu/2<\lambda_{\nu}.$ Therefore, the deviation
$$\left\{\sum_{k=0}^{n}a_{n,k}\left|S_{\frac{\alpha k}{2}}f\left(x\right)-f\left%
(x\right)\right|^{q}\right\}^{1/q}$$
can be estimated from above by
$$\left\{\sum_{k=0}^{n}a_{n,k}\left|\int_{0}^{\infty}\varphi_{x}\left(t\right)%
\Psi_{k+\kappa}\left(t\right)dt\right|^{q}\right\}^{1/q}+\left\{\sum_{k=0}^{n}%
a_{n,k}\left(E_{\alpha k/2}\left(f\right)_{S^{p}}\right)^{q}\right\}^{1/q},$$
where $\kappa$ equals $0$ or $1$. Applying the Minkowski inequality we
obtain
$$\left\{\sum_{k=0}^{n}a_{n,k}\left|\int_{0}^{\infty}\varphi_{x}\left(t\right)%
\Psi_{k+\kappa}\left(t\right)dt\right|^{q}\right\}^{1/q}$$
$$=\left\{\sum_{k=0}^{n}a_{n,k}\left|\left(\int\limits_{0}^{\frac{2\pi}{\alpha}a%
_{n,n}}+\int\limits_{\frac{2\pi}{\alpha}a_{n,n}}^{\frac{2\pi}{\alpha}}+\int%
\limits_{\frac{2\pi}{\alpha}}^{\infty}\right)\varphi_{x}\left(t\right)\Psi_{k+%
\kappa}\left(t\right)dt\right|^{q}\right\}^{1/q}$$
$$\leq\left\{\sum_{k=0}^{n}a_{n,k}\left|I_{1}(k)\right|^{q}\right\}^{1/q}+\left%
\{\sum_{k=0}^{n}a_{n,k}\left|I_{2}(k)\right|^{q}\right\}^{1/q}+\left\{\sum_{k=%
0}^{n}a_{n,k}\left|I_{3}(k)\right|^{q}\right\}^{1/q}.$$
By (1), integrating by parts, we obtain
$$\left\{\sum_{k=0}^{n}a_{n,k}\left|I_{1}(k)\right|^{q}\right\}^{1/q}\leq\left\{%
\sum_{k=0}^{n}a_{n,k}\left|\frac{4}{\alpha\pi}\int\limits_{0}^{\frac{2\pi}{%
\alpha}a_{n,n}}\varphi_{x}\left(t\right)\frac{\sin\frac{\alpha t}{4}}{t^{2}}%
\sin\frac{\alpha t}{4}(2k+2\kappa+1)dt\right|^{q}\right\}^{1/q}$$
$$\leq\frac{1}{\pi}\int\limits_{0}^{\frac{2\pi}{\alpha}a_{n,n}}\frac{\left|%
\varphi_{x}\left(t\right)\right|}{t}dt=\frac{1}{\pi}\int\limits_{0}^{\frac{2%
\pi}{\alpha}a_{n,n}}\frac{1}{t}\left(\frac{d}{dt}\int\limits_{0}^{t}\left|%
\varphi_{x}\left(s\right)\right|ds\right)dt$$
$$=\frac{1}{\pi}\left[\frac{1}{t}\int\limits_{0}^{t}\left|\varphi_{x}\left(s%
\right)\right|ds\right]_{t=0}^{t=\frac{2\pi}{\alpha}a_{n,n}}+\frac{1}{\pi}\int%
\limits_{0}^{\frac{2\pi}{\alpha}a_{n,n}}\frac{1}{t^{2}}\left(\int\limits_{0}^{%
t}\left|\varphi_{x}\left(s\right)\right|ds\right)dt$$
$$=\frac{1}{\pi}w_{x}f\left(\frac{2\pi}{\alpha}a_{n,n}\right)_{1}+\frac{1}{\pi}%
\int\limits_{0}^{\frac{2\pi}{\alpha}a_{n,n}}\frac{1}{t}w_{x}f\left(t\right)_{1%
}dt$$
$$\ll w_{x}f\left(a_{n,n}\right)_{1}+\int\limits_{0}^{\frac{2\pi}{\alpha}a_{n,n}%
}\frac{1}{t}w_{x}f\left(t\right)_{1}dt$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{\pi}w_{x}f\left(\frac{2\pi}{\alpha}a_{n,n}\right)_{1}+%
\frac{1}{\pi}\int\limits_{0}^{\frac{2\pi}{\alpha}a_{n,n}}\frac{1}{t}w_{x}f%
\left(t\right)_{1}dt.$$
(15)
$$\displaystyle\ll$$
$$\displaystyle w_{x}f\left(a_{n,n}\right)_{1}+\int\limits_{0}^{\frac{2\pi}{%
\alpha}a_{n,n}}\frac{1}{t}w_{x}f\left(t\right)_{1}dt.$$
It is clear that $w_{x}f\left(\delta\right)_{1}/\delta$ is nondecreasing
with respect to $\delta>0$ and $w_{x}f\left(\delta\right)_{1}\leq w_{x}f\left(\delta\right)_{p}$ for $p\geq 1$. Using these properties we
have
$$\left\{\sum_{k=0}^{n}a_{n,k}\left|I_{1}(k)\right|^{q}\right\}^{1/q}\ll a_{n,n}%
\int\limits_{a_{n,n}}^{\pi}\frac{w_{x}f\left(t\right)_{1}}{t^{2}}+\int\limits_%
{0}^{a_{n,n}}\frac{1}{t}w_{x}f\left(\frac{2\pi}{\alpha}t\right)_{1}dt$$
$$\ll\left\{a_{n,n}\int\limits_{a_{n,n}}^{\pi}\frac{\left(w_{x}f\left(t\right)_{%
1}\right)^{p}}{t^{2}}\right\}^{\frac{1}{p}}+\int\limits_{0}^{a_{n,n}}\frac{1}{%
t}w_{x}f\left(t\right)_{1}dt.$$
Since $f\in\Omega_{\alpha,p}\left(w_{x}\right)$ and (8) holds,
Lemma 1 and (7) give
$$\left\{\sum_{k=0}^{n}a_{n,k}\left|I_{1}(k)\right|^{q}\right\}^{1/q}=O\left(a_{%
nn}H_{x}\left(a_{nn}\right)\right).$$
If (6) holds, then
$$a_{n,\mu}-a_{n,m}\leq\left|a_{n,\mu}-a_{n,m}\right|\leq\sum\limits_{k=\mu}^{m-%
1}\left|a_{n,k}-a_{n,k+1}\right|\leq Ka_{n,m}$$
for any $m\geq\mu\geq 0.$ Hence we have
$$a_{n,\mu}\leq\left(K+1\right)a_{n,m}.$$
(16)
From this, we get
$$\displaystyle\left\{\sum_{k=0}^{n}a_{n,k}\left|I_{2}(k)\right|^{q}\right\}^{1/q}$$
$$\displaystyle\leq$$
$$\displaystyle\left\{\left(K+1\right)a_{n,n}\right\}^{\frac{1}{q}}\left\{\sum%
\limits_{k=0}^{n}\left|\frac{4}{\alpha\pi}\int\limits_{\frac{2\pi}{\alpha}a_{n%
,n}}^{\frac{2\pi}{\alpha}}\frac{\varphi_{x}\left(t\right)\sin\frac{\alpha t}{4%
}}{t^{2}}\sin\frac{\alpha t}{4}(2k+2\kappa+1)dt\right|^{q}\right\}^{\frac{1}{q}}$$
$$\ll\frac{8}{\alpha^{2}}\left(a_{nn}\right)^{\frac{1}{q}}\left\{\sum\limits_{k=%
0}^{n}\left|\frac{\alpha}{2\pi}\int\limits_{\frac{2\pi}{\alpha}a_{n,n}}^{\frac%
{2\pi}{\alpha}}\frac{\varphi_{x}\left(t\right)\sin\frac{\alpha t}{4}}{t^{2}}%
\sin\frac{\alpha t}{4}(2\kappa+1)\cos\frac{\alpha kt}{2}dt\right|^{q}\right\}^%
{\frac{1}{q}}$$
$$+\frac{8}{\alpha^{2}}\left(a_{nn}\right)^{\frac{1}{q}}\left\{\sum\limits_{k=0}%
^{n}\left|\frac{\alpha}{2\pi}\int\limits_{\frac{2\pi}{\alpha}a_{n,n}}^{\frac{2%
\pi}{\alpha}}\frac{\varphi_{x}\left(t\right)\sin\frac{\alpha t}{4}}{t^{2}}\cos%
\frac{\alpha t}{4}(2\kappa+1)\sin\frac{\alpha kt}{2}dt\right|^{q}\right\}^{%
\frac{1}{q}}.$$
Using inequality (13), we have
$$\left\{\sum_{k=0}^{n}a_{n,k}\left|I_{2}(k)\right|^{q}\right\}^{1/q}\ll\left(a_%
{nn}\right)^{\frac{1}{q}}\left\{\int\limits_{\frac{2\pi}{\alpha}a_{n,n}}^{%
\frac{2\pi}{\alpha}}\frac{\left|\varphi_{x}\left(t\right)\right|^{p}}{t^{1+%
\frac{p}{q}}}\right\}^{\frac{1}{p}}.$$
Integrating by parts, we obtain
$$\left\{\sum_{k=0}^{n}a_{n,k}\left|I_{2}(k)\right|^{q}\right\}^{1/q}=\left(a_{%
nn}\right)^{\frac{1}{q}}\left\{\left[\frac{1}{t^{1+\frac{p}{q}}}\int\limits_{0%
}^{t}\left|\varphi_{x}\left(t\right)\right|^{p}ds\right]_{t=\frac{2\pi}{\alpha%
}a_{n,n}}^{t=\frac{2\pi}{\alpha}}\right.$$
$$\left.\left(1+\frac{p}{q}\right)\int\limits_{\frac{2\pi}{\alpha}a_{n,n}}^{%
\frac{2\pi}{\alpha}}\frac{1}{t^{2+\frac{p}{q}}}\left(\int\limits_{0}^{t}\left|%
\varphi_{x}\left(t\right)\right|^{p}ds\right)dt\right\}^{\frac{1}{p}}$$
$$=\left(a_{nn}\right)^{\frac{1}{q}}\left\{\left[\frac{1}{t^{\frac{p}{q}}}\left(%
w_{x}f\left(t\right)_{p}\right)^{p}\right]_{t=\frac{2\pi}{\alpha}a_{n,n}}^{t=%
\frac{2\pi}{\alpha}}+\left(1+\frac{p}{q}\right)\int\limits_{\frac{2\pi}{\alpha%
}a_{n,n}}^{\frac{2\pi}{\alpha}}\frac{1}{t^{1+\frac{p}{q}}}\left(w_{x}f\left(t%
\right)_{p}\right)^{p}dt\right\}^{\frac{1}{p}}$$
$$\ll\left(a_{nn}\right)^{\frac{1}{q}}\left\{\left(w_{x}f\left(\frac{2\pi}{%
\alpha}\right)_{p}\right)^{p}+\int\limits_{\frac{2\pi}{\alpha}a_{n,n}}^{\frac{%
2\pi}{\alpha}}\frac{1}{t^{1+\frac{p}{q}}}\left(w_{x}f\left(t\right)_{p}\right)%
^{p}dt\right\}^{\frac{1}{p}}.$$
(17)
Since $f\in\Omega_{\alpha,p}\left(w_{x}\right)$, (7) gives
$$\left\{\sum_{k=0}^{n}a_{n,k}\left|I_{2}(k)\right|^{q}\right\}^{1/q}\ll\left(a_%
{nn}\right)^{\frac{1}{q}}\left\{\left(w_{x}\left(\pi\right)\right)^{p}+\int%
\limits_{a_{nn}}^{\pi}\frac{1}{t^{1+\frac{p}{q}}}\left(w_{x}\left(t\right)%
\right)^{p}dt\right\}^{\frac{1}{p}}$$
$$\ll\left\{\left(a_{nn}\right)^{\frac{p}{q}}\int\limits_{a_{nn}}^{\pi}\frac{%
\left(w_{x}\left(t\right)\right)^{p}}{t^{1+\frac{p}{q}}}dt\right\}^{\frac{1}{p%
}}=O\left(a_{nn}H_{x}\left(a_{nn}\right)\right).$$
For the third term we obtain
$$\left\{\sum_{k=0}^{n}a_{n,k}\left|I_{3}(k)\right|^{q}\right\}^{1/q}\leq$$
$$\leq\left\{\sum_{k=0}^{n}a_{n,k}\left|\sum_{\mu=1}^{\infty}\int\limits_{\frac{%
2\pi}{\alpha}\mu}^{\frac{2\pi}{\alpha}\left(\mu+1\right)}\left[\varphi_{x}%
\left(t\right)-\Phi_{x}f\left(\delta_{k},t\right)\right]\Psi_{k+\kappa}\left(t%
\right)dt\right|{}^{q}\right\}^{1/q}$$
$$+\left\{\sum_{k=0}^{n}a_{n,k}\left|\sum_{\mu=1}^{\infty}\int\limits_{\frac{2%
\pi}{\alpha}\mu}^{\frac{2\pi}{\alpha}\left(\mu+1\right)}\Phi_{x}f\left(\delta_%
{k},t\right)\Psi_{k+\kappa}\left(t\right)dt\right|^{q}\right\}^{1/q}$$
$$=\left\{\sum_{k=0}^{n}a_{n,k}\left|I_{31}(k)\right|^{q}\right\}^{1/q}+\left\{%
\sum_{k=0}^{n}a_{n,k}\left|I_{32}(k)\right|^{q}\right\}^{1/q}$$
and
$$\left|I_{31}(k)\right|\leq\frac{4}{\alpha\pi}\sum_{\mu=1}^{\infty}\int\limits_%
{\frac{2\pi}{\alpha}\mu}^{\frac{2\pi}{\alpha}\left(\mu+1\right)}\left|\varphi_%
{x}\left(t\right)-\Phi_{x}f\left(\delta_{k},t\right)\right|t^{-2}dt$$
$$\leq\frac{4}{\alpha\pi}\sum_{\mu=1}^{\infty}\int\limits_{\frac{2\pi}{\alpha}%
\mu}^{\frac{2\pi}{\alpha}\left(\mu+1\right)}\left[\frac{1}{\delta_{k}t^{2}}%
\int_{0}^{\delta_{k}}\left|\varphi_{x}\left(t\right)-\varphi_{x}\left(t+u%
\right)\right|du\right]dt$$
$$=\frac{4}{\alpha\pi}\frac{1}{\delta_{k}}\int\limits_{0}^{\delta_{k}}\sum_{\mu=%
1}^{\infty}\left\{\int\limits_{\frac{2\pi}{\alpha}\mu}^{\frac{2\pi}{\alpha}%
\left(\mu+1\right)}\frac{1}{t^{2}}\left|\varphi_{x}\left(t\right)-\varphi_{x}%
\left(t+u\right)\right|dt\right\}du$$
$$=\frac{4}{\alpha\pi}\frac{1}{\delta_{k}}\int\limits_{0}^{\delta_{k}}\sum_{\mu=%
1}^{\infty}\left\{\left[\frac{1}{t^{2}}\int_{0}^{t}\left|\varphi_{x}\left(s%
\right)-\varphi_{x}\left(s+u\right)\right|ds\right]_{t=\frac{2\pi}{\alpha}\mu}%
^{t=\frac{2\pi}{\alpha}\left(\mu+1\right)}\right.$$
$$+\left.2\int\limits_{\frac{2\pi}{\alpha}\mu}^{\frac{2\pi}{\alpha}\left(\mu+1%
\right)}\left[\frac{1}{t^{3}}\int_{0}^{t}\left|\varphi_{x}\left(s\right)-%
\varphi_{x}\left(s+u\right)\right|ds\right]dt\right\}du$$
$$\ll\left|\frac{1}{\delta_{k}}\int\limits_{0}^{\delta_{k}}\sum_{\mu=1}^{\infty}%
\left\{\frac{1}{[\frac{2\pi}{\alpha}\left(\mu+1\right)]^{2}}\int\limits_{0}^{%
\frac{2\pi}{\alpha}\left(\mu+1\right)}\left|\varphi_{x}\left(s\right)-\varphi_%
{x}\left(s+u\right)\right|ds\right.\right.$$
$$\left.-\left.\frac{1}{[\frac{2\pi}{\alpha}\mu]^{2}}\int\limits_{0}^{\frac{2\pi%
}{\alpha}\mu}\left|\varphi_{x}\left(s\right)-\varphi_{x}\left(s+u\right)\right%
|ds\right\}du\right|$$
$$+\frac{1}{\delta_{k}}\int\limits_{0}^{\delta_{k}}\sum_{\mu=1}^{\infty}\left\{%
\int\limits_{\frac{2\pi}{\alpha}\mu}^{\frac{2\pi}{\alpha}\left(\mu+1\right)}%
\left[\frac{1}{t^{3}}\int_{0}^{t}\left|\varphi_{x}\left(s\right)-\varphi_{x}%
\left(s+u\right)\right|ds\right]dt\right\}du.$$
Since $f\in\Omega_{\alpha,p}\left(w_{x}\right)$, thus for any $x$
$$\displaystyle\lim_{\zeta\rightarrow\infty}\frac{1}{\zeta^{2}}\int\limits_{0}^{%
\zeta}\left|\varphi_{x}\left(s\right)-\varphi_{x}\left(s+u\right)\right|ds$$
$$\displaystyle\leq$$
$$\displaystyle\lim_{\zeta\rightarrow\infty}\frac{1}{\zeta}w_{x}\left(u\right)%
\leq\lim_{\zeta\rightarrow\infty}\frac{1}{\zeta}w_{x}\left(\delta_{k}\right)$$
$$\displaystyle\leq$$
$$\displaystyle\lim_{\zeta\rightarrow\infty}\frac{1}{\zeta}w_{x}\left(\pi\right)%
=0,$$
and therefore
$$\left|I_{31}(k)\right|\leq\frac{1}{\delta_{k}}\int\limits_{0}^{\delta_{k}}%
\frac{\alpha}{2\pi}\left[\frac{\alpha}{2\pi}\int_{0}^{2\pi/\alpha}\left|%
\varphi_{x}\left(s\right)-\varphi_{x}\left(s+u\right)\right|ds\right]du$$
$$+\frac{1}{\delta_{k}}\int\limits_{0}^{\delta_{k}}w_{x}\left(u\right)du\sum_{%
\mu=1}^{\infty}\left\{\int\limits_{\frac{2\pi}{\alpha}\mu}^{\frac{2\pi}{\alpha%
}\left(\mu+1\right)}\frac{1}{t^{2}}dt\right\}$$
$$\ll\frac{1}{\delta_{k}}\int\limits_{0}^{\delta_{k}}w_{x}\left(u\right)du+w_{x}%
\left(\delta_{k}\right)\sum_{\mu=1}^{\infty}\frac{1}{\frac{2\pi}{\alpha}\mu^{2%
}}\ll w_{x}\left(\delta_{k}\right).$$
Next, we will estimate the term $\left|I_{32}(k)\right|.$ So,
$$I_{32}(k)=\frac{2}{\alpha\pi}\sum_{\mu=1}^{\infty}\int\limits_{\frac{2\pi}{%
\alpha}\mu}^{\frac{2\pi}{\alpha}\left(\mu+1\right)}\frac{\Phi_{x}f\left(\delta%
_{k},t\right)}{t^{2}}\frac{d}{dt}\left(-\frac{\cos\frac{\alpha t\left(k+\kappa%
\right)}{2}}{\frac{\alpha\left(k+\kappa\right)}{2}}+\frac{\cos\frac{\alpha t%
\left(k+\kappa+1\right)}{2}}{\frac{\alpha\left(k+\kappa+1\right)}{2}}\right)dt$$
$$=\frac{2}{\alpha\pi}\sum_{\mu=1}^{\infty}\left[\frac{\Phi_{x}f\left(\delta_{k}%
,t\right)}{t^{2}}\left(-\frac{\cos\frac{\alpha t\left(k+\kappa\right)}{2}}{%
\frac{\alpha\left(k+\kappa\right)}{2}}+\frac{\cos\frac{\alpha t\left(k+\kappa+%
1\right)}{2}}{\frac{\alpha\left(k+\kappa+1\right)}{2}}\right)\right]_{t=\frac{%
2\pi}{\alpha}\mu}^{t=\frac{2\pi}{\alpha}\left(\mu+1\right)}$$
$$\displaystyle+\frac{2}{\alpha\pi}\sum_{\mu=1}^{\infty}\int\limits_{\frac{2\pi}%
{\alpha}\mu}^{\frac{2\pi}{\alpha}\left(\mu+1\right)}\frac{d}{dt}\left(\frac{%
\Phi_{x}f\left(\delta_{k},t\right)}{t^{2}}\right)\left(\frac{\cos\frac{\alpha t%
\left(k+\kappa\right)}{2}}{\frac{\alpha\left(k+\kappa\right)}{2}}-\frac{\cos%
\frac{\alpha t\left(k+\kappa+1\right)}{2}}{\frac{\alpha\left(k+\kappa+1\right)%
}{2}}\right)dt$$
$$\displaystyle=$$
$$\displaystyle I_{321}\left(k\right)+I_{322}\left(k\right)$$
Since $f\in\Omega_{\alpha,p}\left(w_{x}\right)$, thus for any $x$
(using (14))
$$\lim_{\zeta\rightarrow\infty}\left|\frac{\Phi_{x}f\left(\delta_{k},\frac{2\pi}%
{\alpha}\zeta\right)}{\left[\frac{2\pi}{\alpha}\zeta\right]^{2}}\left(-\frac{%
\cos\left[\pi\zeta(k+\kappa)\right]}{\frac{\alpha\left(k+\kappa\right)}{2}}+%
\frac{\cos\left[\pi\zeta\left(k+\kappa+1\right)\right]}{\frac{\alpha\left(k+%
\kappa+1\right)}{2}}\right)\right|$$
$$\leq\lim_{\zeta\rightarrow\infty}\frac{w_{x}\left(\delta_{k}\right)+w_{x}\left%
(\frac{2\pi}{\alpha}\zeta\right)}{2\pi^{2}\zeta^{2}k}\ll\lim_{\zeta\rightarrow%
\infty}\frac{w_{x}\left(\delta_{k}\right)+\zeta w_{x}\left(\frac{2\pi}{\alpha}%
\right)}{\zeta^{2}k}\ll w_{x}\left(\pi\right)\lim_{\zeta\rightarrow\infty}%
\frac{1+\zeta}{\zeta^{2}}=0,$$
and therefore
$$I_{321}\left(k\right)=\frac{2}{\alpha\pi}\sum_{\mu=1}^{\infty}\left[\frac{\Phi%
_{x}f\left(\delta_{k},\frac{2\pi}{\alpha}\left(\mu+1\right)\right)}{\left[%
\frac{2\pi}{\alpha}\left(\mu+1\right)\right]^{2}}\left(-\frac{\cos\left[\pi%
\left(\mu+1\right)\left(k+\kappa\right)\right]}{\frac{\alpha\left(k+\kappa%
\right)}{2}}\right.\right.$$
$$+\left.\frac{\cos\left[\pi\left(\mu+1\right)\left(k+\kappa+1\right)\right]}{%
\frac{\alpha\left(k+\kappa+1\right)}{2}}\right)$$
$$-\left.\frac{\Phi_{x}f\left(\delta_{k},\frac{2\pi}{\alpha}\mu\right)}{\left[%
\frac{2\pi}{\alpha}\mu\right]^{2}}\left(-\frac{\cos\left[\pi\mu(k+\kappa)%
\right]}{\frac{\alpha\left(k+\kappa\right)}{2}}+\frac{\cos\left[\pi\mu\left(k+%
\kappa+1\right)\right]}{\frac{\alpha\left(k+\kappa+1\right)}{2}}\right)\right]$$
$$=-\frac{2}{\alpha\pi}\frac{\Phi_{x}f\left(\delta_{k},2\pi/\alpha\right)}{\left%
[2\pi/\alpha\right]^{2}}\left(-\frac{\left(-1\right)^{\left(k+\kappa\right)}}{%
\frac{\alpha\left(k+\kappa\right)}{2}}+\frac{\left(-1\right)^{\left(k+\kappa+1%
\right)}}{\frac{\alpha\left(k+\kappa+1\right)}{2}}\right)$$
$$=-\frac{1}{\pi^{3}}\Phi_{x}f\left(\delta_{k},2\pi/\alpha\right)\left(-1\right)%
^{\left(k+\kappa+1\right)}\left(\frac{1}{k+\kappa+1}+\frac{1}{k+\kappa}\right).$$
Using (14), we get
$$\left|I_{321}\left(k\right)\right|\ll\frac{1}{\pi^{3}}\frac{2}{k+1}\left|\Phi_%
{x}f\left(\delta_{k},2\pi/\alpha\right)\right|\leq\frac{2}{\pi^{3}\left(k+1%
\right)}\left(w_{x}\left(\delta_{k}\right)+w_{x}\left(2\pi/\alpha\right)\right).$$
Similarly
$$I_{322}\left(k\right)=\frac{2}{\alpha\pi}\sum_{\mu=1}^{\infty}\int\limits_{%
\frac{2\pi}{\alpha}\mu}^{\frac{2\pi}{\alpha}\left(\mu+1\right)}\left(\frac{%
\frac{d}{dt}\Phi_{x}f\left(\delta_{k},t\right)}{t^{2}}-\frac{2\Phi_{x}f\left(%
\delta_{k},t\right)}{t^{3}}\right)$$
$$\cdot\left(\frac{\cos\frac{\alpha t\left(k+\kappa\right)}{2}}{\frac{\alpha%
\left(k+\kappa\right)}{2}}-\frac{\cos\frac{\alpha t\left(k+\kappa+1\right)}{2}%
}{\frac{\alpha\left(k+\kappa+1\right)}{2}}\right)dt$$
and
$$\left|I_{322}\left(k\right)\right|\ll\frac{8}{\alpha^{2}\left(k+1\right)\pi}%
\sum_{\mu=1}^{\infty}\left[\int\limits_{\frac{2\pi}{\alpha}\mu}^{\frac{2\pi}{%
\alpha}\left(\mu+1\right)}\frac{\left|\varphi_{x}\left(t+\delta_{k}\right)-%
\varphi_{x}\left(t\right)\right|}{\delta_{k}t^{2}}dt\right.$$
$$+\left.2\int\limits_{\frac{2\pi}{\alpha}\mu}^{\frac{2\pi}{\alpha}\left(\mu+1%
\right)}\frac{\left|\Phi_{x}f\left(\delta_{k},t\right)\right|}{t^{3}}dt\right]$$
$$\leq\frac{8}{\alpha^{2}\left(k+1\right)\pi\delta_{k}}\sum_{\mu=1}^{\infty}\int%
\limits_{\frac{2\pi}{\alpha}\mu}^{\frac{2\pi}{\alpha}\left(\mu+1\right)}\frac{%
\left|\varphi_{x}\left(t+\delta_{k}\right)-\varphi_{x}\left(t\right)\right|}{t%
^{2}}dt$$
$$+\frac{16}{\alpha^{2}\left(k+1\right)\pi}\sum_{\mu=1}^{\infty}\int\limits_{%
\frac{2\pi}{\alpha}\mu}^{\frac{2\pi}{\alpha}\left(\mu+1\right)}\frac{w_{x}%
\left(\delta_{k}\right)+w_{x}\left(t\right)}{t^{3}}dt$$
$$\ll\frac{1}{\left(k+1\right)\delta_{k}}w_{x}\left(\delta_{k}\right)+\frac{1}{k%
+1}\sum_{\mu=1}^{\infty}\left[\left(w_{x}\left(\delta_{k}\right)+w_{x}\left(%
\frac{2\pi\left(\mu+1\right)}{\alpha}\right)\right)\frac{\alpha^{2}}{4\pi^{2}%
\mu^{3}}\right]$$
$$\ll w_{x}\left(\delta_{k}\right)+\frac{1}{k+1}\left[w_{x}\left(\delta_{k}%
\right)\sum_{\mu=1}^{\infty}\frac{1}{\mu^{3}}+\sum_{\mu=1}^{\infty}\frac{w_{x}%
\left(\frac{2\pi\left(\mu+1\right)}{\alpha}\right)}{\mu^{3}}\right]$$
$$\ll w_{x}\left(\delta_{k}\right)+\frac{1}{k+1}\left(w_{x}\left(\delta{}_{k}%
\right)+w_{x}\left(\frac{4\pi}{\alpha}\right)\sum_{\mu=1}^{\infty}\frac{\mu+1}%
{\mu^{3}}\right)$$
$$\ll w_{x}\left(\delta_{k}\right)+\frac{1}{k+1}\left(w_{x}\left(\delta_{k}%
\right)+w_{x}\left(\frac{4\pi}{\alpha}\right)\right).$$
Therefore
$$\left|I_{3}\left(k\right)\right|\ll w_{x}\left(\delta_{k}\right)+\frac{1}{k+1}%
\left(w_{x}\left(\delta_{k}\right)+w_{x}\left(\frac{2\pi}{\alpha}\right)+w_{x}%
\left(\frac{4\pi}{\alpha}\right)\right)$$
and thus
$$\left\{\sum_{k=0}^{n}a_{n,k}\left|I_{3}(k)\right|^{q}\right\}^{1/q}\ll\left\{%
\sum_{k=0}^{n}a_{n,k}\left(w_{x}\left(\frac{\pi}{k+1}\right)+\frac{1}{k+1}w_{x%
}\left(\frac{\pi}{\alpha}\right)\right)^{q}\right\}^{1/q}$$
$$\ll\left\{\sum_{k=0}^{n}a_{n,k}\left(w_{x}\left(\frac{\pi}{k+1}\right)\right)^%
{q}\right\}^{1/q}.$$
From (16) we obtain
$$\sum_{k=0}^{n}a_{n,k}\left(w_{x}\left(\frac{\pi}{k+1}\right)\right)^{q}\leq%
\sum\limits_{k=0}^{\left[\frac{1}{\left(K+1\right)a_{n,n}}\right]-1}a_{nk}%
\left(w_{x}\left(\frac{\pi}{k+1}\right)\right)^{q}$$
$$+\sum\limits_{k=\left[\frac{1}{\left(K+1\right)a_{n,n}}\right]-1}^{n}a_{nk}%
\left(w_{x}\left(\frac{\pi}{k+1}\right)\right)^{q}.$$
Using (1), (16) and the monotonicity of the function $w_{x}$,
from (7) and (12), we get
$$\sum_{k=0}^{n}a_{n,k}\left(w_{x}\left(\frac{\pi}{k+1}\right)\right)^{q}\leq%
\left(K+1\right)a_{n,n}\sum\limits_{k=0}^{\left[\frac{1}{\left(K+1\right)a_{n,%
n}}\right]-1}\left(w_{x}\left(\frac{\pi}{k+1}\right)\right)^{q}$$
$$+\left(w_{x}\left(\pi\left(K+1\right)a_{n,n}\right)\right)^{q}\sum\limits_{k=%
\left[\frac{1}{\left(K+1\right)a_{n,n}}\right]-1}^{n}a_{nk}$$
$$\ll a_{n,n}\int\limits_{1}^{\frac{1}{\left(K+1\right)a_{n,n}}}\left(w_{x}\left%
(\frac{\pi}{t}\right)\right)^{q}dt+\left(w_{x}\left(a_{n,n}\right)\right)^{q}%
\ll a_{n,n}\int\limits_{a_{n,n}}^{\pi}\frac{\left(w_{x}\left(u\right)\right)^{%
q}}{u^{2}}du+\left(w_{x}\left(a_{n,n}\right)\right)^{q}$$
$$\leq a_{n,n}\int\limits_{a_{n,n}}^{\pi}\frac{\left(w_{x}\left(u\right)\right)^%
{q}}{u^{1+p/q+1-p/q}}du+\left(4w_{x}\left(\frac{a_{n,n}}{2}\right)\right)^{q}$$
$$\leq\left(a_{n,n}\right)^{p/q}\int\limits_{a_{n,n}}^{\pi}\frac{\left(w_{x}%
\left(u\right)\right)^{q}}{u^{1+p/q}}du+\left(8\int\limits_{\frac{a_{n,n}}{2}}%
^{a_{n,n}}\frac{w_{x}\left(u\right)}{u}du\right)^{q}$$
$$\ll\left(a_{n,n}\right)^{p/q}\int\limits_{a_{n,n}}^{\pi}\frac{\left(w_{x}\left%
(u\right)\right)^{q}}{u^{1+p/q}}du+\left(\int\limits_{0}^{a_{n,n}}\frac{w_{x}%
\left(u\right)}{u}du\right)^{q}\ll\left(a_{n,n}H_{x}\left(a_{n,n}\right)\right%
)^{q}.$$
Summing up we obtain that (9) is proved and the proof is complete.
4.2 Proof of Theorem 2
Under the notation of the before proof we can write
$$\left\{\sum_{k=0}^{n}a_{n,k}\left|\int_{0}^{\infty}\varphi_{x}\left(t\right)%
\Psi_{k+\kappa}\left(t\right)dt\right|^{q}\right\}^{1/q}$$
$$=\left\{\sum_{k=0}^{n}a_{n,k}\left|\left(\int\limits_{0}^{\frac{2\pi}{\alpha}a%
_{n,0}}+\int\limits_{\frac{2\pi}{\alpha}a_{n,0}}^{\frac{2\pi}{\alpha}}+\int%
\limits_{\frac{2\pi}{\alpha}}^{\infty}\right)\varphi_{x}\left(t\right)\Psi_{k+%
\kappa}\left(t\right)dt\right|^{q}\right\}^{1/q}$$
$$\leq\left\{\sum_{k=0}^{n}a_{n,k}\left|J_{1}(k)\right|^{q}\right\}^{1/q}+\left%
\{\sum_{k=0}^{n}a_{n,k}\left|J_{2}(k)\right|^{q}\right\}^{1/q}+\left\{\sum_{k=%
0}^{n}a_{n,k}\left|J_{3}(k)\right|^{q}\right\}^{1/q},$$
using the Minkowski inequality. Applying the property of the class $RBVS$
instead of the property of $HBVS$ our proof will be similar to the proof of
Theorem 1.
References
[1]
A. Avantaggiati, G. Bruno and B. Iannacci, The Hausdorff-Young
theorem for almost periodic functions and some applications, Nonlinear
analysis, Theory, Methods and Applications, Vol. 25, No. 1, (1995), pp.
61-87.
[2]
A. D. Bailey, Almost Everywhere Convergence of Dyadic Partial
Sums of Fourier Series for Almost Periodic Functions, Master of Philosophy,
A thesis submitted to School of Mathematics of The University of Birmingham
for the degree of Master of Philosophy, September 2008.
[3]
A. S. Besicovitch, Almost periodic functions, Cambridge, 1932.
[4]
P. Chandra, On the degree of approximation of a class of
functions by means of Fourier series, Acta Mat. Hung., 52 (1988), 199-205.
[5]
P. Chandra, A note on the degree of approximation of continuous
functions, Acta Math. Hungar., 62 (1993), 21-23.
[6]
L. Leindler, On the degree of approximation of continuous
functions, Acta Math. Hungar., 104 (2004), 105-113.
[7]
L Leindler, On the uniform convergence and boundedness of a
certain class of sine series, Analysis Math., 27 (2001), 279-285.
[8]
L. Leindler, A new extension of monotone sequence and its
application, J. Inequal. Pure and Appl. Math., 7(1) (2006), Art. 39, 7 pp.
[9]
B. L. Levitan, Almost periodic functions, Gos. Izdat.
Tekh-Teoret. Liter., Moscov 1953 (in Russian).
[10]
W. Łenski, Pointwise strong and very strong approximation of
Fourier series, Acta Math. Hung., 115(3), (2007), p.215-233.
[11]
W. Łenski, On the degree of strong approximation of integrable
functions, J. Inequal. Pure and Appl. Math., 8(3) (2007), Art. 70.
[12]
P. Pych-Taberska, Approximation properties of the partial sums
of Fourier series of almost periodic functions, Studia Math. XCVI
(1990),91-103.
[13]
S. Tikhonov, Trigonometric series with general monotone
coefficients, J. Math. Anal. Appl., 326(1) (2007), 721-735.
[14]
S. Tikhonov, On uniform convergence of trigonometric series.
Mat. Zametki, 81(2) (2007), 304-310, translation in Math. Notes, 81(2)
(2007), 268-274.
[15]
S. Tikhonov, Best approximation and moduli of smoothness:
Computation and equivalence theorems, J. Approx. Theory, 153 (2008), 19-39.
[16]
A. Zygmund, Trigonometric Series, Cambridge, 2002. |
Morita equivalences on Brauer algebras and BMW algebras of simply-laced types
Shoumin Liu111The author is funded by the NSFC (Grant No. 11601275, Youth Program).
Abstract
The Morita equivalences of classical Brauer algebras and classical Birman-Murakami-Wenzl algebras have been well studied.
Here we study the Morita equivalence problems on these two kinds of algebras of simply-laced type, especially for them with the generic parameters.
We show that Brauer algebras and Birman-Murakami-Wenzl algebras of simply-laced type are Morita equivalent to
the direct sums of some group algebras of Coxeter groups and some Hecke algebras of some Coxeter groups, respectively.
1 Introduction
In [2], when the author study the invariant theory of orthogonal groups, the Brauer algebras are defined
as a class of diagram algebras, which becomes the most classical examples in Schur-weyl duality. If we regard some horizontal strands
in the diagram algebras as roots of Coxeter groups of type ${\rm A}$, it is natural to define the Brauer algebras of other types associated to
other Dynkin diagrams. Cohen, Frenks, and Wales define the Brauer algebras of simply-laced types in [6], and describe
some properties of these algebras. The Birman-Murakami-Wenzl algebras ($BMW$ in short) which is defined in [1] and [22],
can be considered as a quantum version of
classical Brauer algebras. Analogously, the $BMW$ algebras can be extended to other simply-laced types in [8].
The Morita equivalences ([21]) and Quasi-heredity ([3]) are important properties of associative algebras.
As cellular algebras([14], [15], [16]),
these properties of classical Brauer algebras and $BMW$ algebras, even some related algebras, are well studied in many papers,
such as König and Xi ([16], [17],[18],[29]),
Rui and Si([23], [24], [25], [26], [27]). Their results are based on studying the bilinear forms
for defining their cellular structures.
Therefore, it is natural to ask the Morita equivalences and quasi-heredity on the Brauer algebras and $BMW$ algebras of simply-laced types, especial
for type ${\rm D}_{n}$ and type ${\rm E}_{n}$ ($n=6,7,8$). Our paper will focus on these algebras with generic parameters, and is sketched as the following.
In Section 2, we first recall two equivalent definitions of cellular algebra from [15] and [16],
and introduce some basic properties of cellular algebras, especially about Morita equivalence.
In section 3, we recall the definition of Brauer algebras (${\rm Br}(Q,k)$) of simply-laced types and some results from
[6]. In section 4, we prove the Morita equivalence and quasi-heredity of ${\rm Br}(Q,k)$ with some
conditions on ground field $k$ and generic parameter $\delta$. In section 5, by analyzing the structure of ${\rm Br}({\rm D}_{n},k)$,
we show some results about the semi-simplicity of ${\rm Br}({\rm D}_{n},k)$ with $\delta$ evaluated. In section 6, similar to Section 4, we present
the Morita equivalence and quasi-heredity on $BMW$ algebras of simply-laced types.
2 Cellular algebra
We first recall the definition of cellular algebra from [14] and [15].
Definition 2.1.
An associative
algebra ${A}$ over a commutative ring $R$ is cellular if there is a quadruple
$(\Lambda,T,C,*)$ satisfying the following three conditions.
(C1)
$\Lambda$ is a finite partially ordered set. Associated to each
$\lambda\in\Lambda$, there is a finite set $T(\lambda)$. Also, $C$ is an
injective map
$$\coprod_{\lambda\in\Lambda}T(\lambda)\times T(\lambda)\rightarrow{A}$$
whose image is an $R$-basis of ${A}$.
(C2)
The map $*:{A}\rightarrow{A}$ is an
$R$-linear anti-involution such that
$C(x,y)^{*}=C(y,x)$ whenever $x,y\in T(\lambda)$ for some $\lambda\in\Lambda$.
(C3)
If $\lambda\in\Lambda$ and $x,y\in T(\lambda)$, then, for any
element $a\in{A}$,
$$aC(x,y)\equiv\sum_{u\in T(\lambda)}r_{a}(u,x)C(u,y)\ \ \ {\rm mod}\ {A}_{<%
\lambda},$$
where $r_{a}(u,x)\in R$ is independent of $y$
and where ${A}_{<\lambda}$ is the $R$-submodule of ${A}$ spanned by $\{C(x^{\prime},y^{\prime})\mid x^{\prime},y^{\prime}\in T(\mu)\mbox{ for }\mu<\lambda\}$.
Such a quadruple $(\Lambda,T,C,*)$ is called a cell datum for
${A}$.
There is also an equivalent definition due to König and Xi in [16].
Definition 2.2.
Let $A$ be $R$-algebra. Assume there is an anti-automorphism $i$ on $A$ with $i^{2}=id$. A two sided ideal $J$ in
$A$ is called cellular if and only if $i(J)=J$ and there exists a left ideal $\Delta\subset J$ such that $\Delta$ has finite rank and
there is an isomorphism of $A$-bimodules $\alpha:J\simeq\Delta\otimes_{R}i(\Delta)$ making the following diagram commutative:
\xymatrix
J\ar[rr]^α\ar[d]_i & & Δ⊗_R i(Δ) \ar[d]^x⊗y→i(y)⊗i(x)
J \ar[rr]^α & & Δ⊗_R i(Δ)
The algebra $A$ is called cellular if there is a vector space decomposition $A=J^{\prime}_{1}\oplus\cdots\oplus J^{\prime}_{n}$ with
$i(J^{\prime}_{j})=J^{\prime}_{j}$ for each $j$ and such that setting $J_{j}=\oplus_{k=1}^{j}J^{\prime}_{j}$ gives a chain of two sided ideals of
$A$ such that for each $j$ the quotient $J^{\prime}_{j}=J_{j}/J_{j-1}$ is a cellular ideal of $A/J_{j-1}$.
Also recall definitions of iterated inflations from [16].
Given an $R$-algebra $B$, a finitely generated free $R$-module $V$, and a bilinear form
$\varphi:V\otimes_{R}V\longrightarrow B$ with values in $B$, we define an associative algebra (possibly without unit)
$A(B,V,\varphi)$ as follows: as an $R$-module, $A(B,V,\varphi)$ equals $V\otimes_{R}V\otimes_{R}B$. The multiplication is defined on basis
element as follows:
$$\displaystyle(a\otimes b\otimes x)(c\otimes d\otimes y):=a\otimes d\otimes x%
\varphi(b,c)y.$$
Assume that there is an involution $i$ on $B$. Assume, moreover, that $i(\varphi(v,w))=\varphi(w,v)$.
If we can extend this involution $i$ to $A(B,V,\varphi)$ by defining $i(a\otimes b\otimes x)=b\otimes a\otimes i(x)$. Then
We call $A(B,V,\varphi)$ is an inflation of $B$ along $V$.
Let $B$ be an inflated algebra (possible without unit) and $C$ be an algebra with unit. We define an algebra structure in such a way that
$B$ is a two-sided ideal and $A/B=C$. We require that $B$ is an ideal, the multiplication is associative, and
that there exists a unit element of $A$ which maps onto the unit of the quotient $C$. The necessary conditions are outlined in
[16, Section 3]. Then we call $A$ an inflation of $C$ along $B$, or iterated inflation of $C$ along $B$.
We present Proposition 3.5 of [16] below.
Proposition 2.3.
An inflation of a cellular algebra is cellular again. In particular, an iterated inflation of $n$ copies of
$R$ is cellular, with a cell chain of length $n$ as in Definition 2.2.
More precisely, the second statement has the following meaning. Start with $C$ a full matrix ring over $R$ and $B$ an inflation of $R$ along a
free $R$-module, and form a new $A$ which is an inflation of the old $A$ along the new $B$, and continue this operation. Then after
$n$ steps we have produced a cellular algebra $A$ with a cell chain of length $n$.
We also have Theorem 4.1 from [16] as follows.
Theorem 2.4.
Any cellular algebra over $R$ is the iterated inflation of finitely many copies of $R$. Conversely, any iterated inflation of
finitely many copies of $R$ is cellular.
Let $A$ be cellular(with identity) which can be realized as an iterated inflation of cellular algebras $B_{l}$ along vector spaces $V_{l}$
for $l=1,\ldots,n.$ This implies that as a vector space
$$\displaystyle A=\oplus_{l=1}^{n}V_{l}\otimes V_{l}\otimes B_{l},$$
and $A$ is cellular with a chain of two sided ideals ${0}=J_{0}\subset J_{1}\cdots\subset J_{n}=A$, which can be refined to a cell chain, and each
quotient $J_{l}/J_{l-1}$ equals $V_{l}\otimes V_{l}\otimes B_{l}$ as an algebra without unit. The involution $i$ of $A$,is defined through the involution $i_{l}$ of the algebra $B_{l}$ where
$i(a\otimes b\otimes x)=b\otimes a\otimes j_{l}(x)$. The multiplication rule of a layer $V_{l}\oplus V_{l}\oplus B_{l}$ is indicated by
$$\displaystyle(a\otimes b\otimes x)(c\otimes d\otimes y):=a\otimes d\otimes x%
\varphi(b,c)y+\text{lower terms}.$$
Here lower terms refers to element in lower layers $V_{h}\otimes V_{h}\otimes B_{h}$ for $h<l$.
Let $1_{B_{l}}$ be the identity of the algebra $B_{l}$.
We recall [27, Theorem 2.6] about the Morita equivalence of celluar algebra.
Theorem 2.5.
Let $R$ be a field. Suppose that $A$ is an iterated inflation of $R$-algebras $B_{1}$, $B_{2}$, $\cdots$, $B_{n}$, where each inflation
is along $R$-vector space $V_{i}$, $1\leq i\leq n$. For each $i$, let $\varphi_{i}:V_{i}\otimes V_{i}\rightarrow B_{i}$ be the bilinear form with respect to
each inflation. If $\varphi_{i}$ is non-singular for all $i$, then
$$A\overset{Morita}{\sim}\oplus_{i=1}^{n}B_{i}.$$
In this paper, we will focus on the quasi-heredity on some algebras, then we recall the definition of quasi-heredity algebra from [3].
Definition 2.6.
Let $k$ be any associative ring, and $A$ be a $k$-algebra. An ideal $J$ in $A$ is called a hereditary ideal if $J$ is idempotent, $J(rad(A))J=0$. and J is a projective left(or, right) $A$-module; the algebra $A$ is called a heredity algebra.
The algebra $A$ is called quasi-hereditary provided there is a finite chain $0=J_{0}\subset J_{1}\subset\dots\subset J_{n}=A$ of ideals in $A$ such that
$J_{j}/J_{j-1}$ is a hereditary ideal in $A/J_{j-1}$ for all $j$. Such a chain is then called a heredity ideal of the quasi-hereditary algebra $A$.
3 Brauer algebras of simply-laced type
We recall the definition of simply-laced Brauer algebra from [6].
Definition 3.1.
Let $Q$ be a graph. The Brauer monoid ${\rm BrM}(Q)$ is the monoid
generated by the symbols $R_{i}$ and $E_{i}$, for each node $i$ of $Q$ and $\delta$,
$\delta^{-1}$ subject to the following relation, where
$\sim$ denotes adjacency between nodes of $Q$.
$$\delta\delta^{-1}=1$$
(3.1)
$$R_{i}^{2}=1$$
(3.2)
$$R_{i}E_{i}=E_{i}R_{i}=E_{i}$$
(3.3)
$$E_{i}^{2}=\delta E_{i}$$
(3.4)
$$R_{i}R_{j}=R_{j}R_{i},\,\,\mbox{for}\,\it{i\nsim j}$$
(3.5)
$$E_{i}R_{j}=R_{j}E_{i},\,\,\mbox{for}\,\it{i\nsim j}$$
(3.6)
$$E_{i}E_{j}=E_{j}E_{i},\,\,\mbox{for}\,\it{i\nsim j}$$
(3.7)
$$R_{i}R_{j}R_{i}=R_{j}R_{i}R_{j},\,\,\mbox{for}\,\it{i\sim j}$$
(3.8)
$$R_{j}R_{i}E_{j}=E_{i}E_{j},\,\,\mbox{for}\,\it{i\sim j}$$
(3.9)
$$R_{i}E_{j}R_{i}=R_{j}E_{i}R_{j},\,\,\mbox{for}\,\it{i\sim j}$$
(3.10)
The Brauer algebra ${\rm Br}(Q)$ is the the free $\mathbb{Z}$-algebra for Brauer monoid ${\rm BrM}(Q)$.
We denote ${\rm Br}(Q,k)={\rm Br}(Q)\otimes_{\mathbb{Z}}k$, where $k$ is an arbitrary ring.
The Brauer algebras ${\rm Br}(Q)$ has been well studied in [6], where the basis and ranks
of finite types are given. Usually we call $R_{i}$s Coxeter generators, and $E_{i}$s Temperley-Lieb generators([28]).
Let $Q$ be a spherical Coxeter diagram of simply laced type, i.e., its connected components are of type
${\rm A}$, ${\rm D}$, ${\rm E}$ as listed in Table 1. This section is to summarize some results in [5].
When $Q$ is ${\rm A}_{n}$, ${\rm D}_{n}$, ${\rm E}_{6}$, ${\rm E}_{7}$, or ${\rm E}_{8}$, we denote it as $Q\in{\rm ADE}$.
Let $(W,T)$ be the Coxeter system of type $Q$ with $T=\{R_{1},\ldots,R_{n}\}$ associated to the diagram of
$Q$ in Table 1.
Let $\Phi$ be the root system of type $Q$, let $\Phi^{+}$ be its positive root system, and let $\alpha_{i}$ be the simple root
associated to the node $i$ of $Q$. We are interested in sets $B$ of mutually commuting reflections, which has a bijective correspondence with sets of
mutually orthogonal roots of $\Phi^{+}$, since each reflection in $W$ is uniquely determined by a positive root and vice versa.
Remark 3.2.
The action of $w\in W$ on $B$ is given by conjugation in case $B$ is described by reflections and given by
$w\{\beta_{1},\ldots,\beta_{p}\}=\Phi^{+}\cap\{\pm w\beta_{1},\ldots,\pm w\beta%
_{p}\}$, in case $B$ is described by positive roots.
For example, $R_{4}R_{1}R_{2}R_{1}\{\alpha_{1}+\alpha_{2},\alpha_{4}\}=\{\alpha_{1}+\alpha_{%
2},\alpha_{4}\}$, where $Q={\rm A}_{4}$.
For $\alpha$, $\beta\in\Phi$, we write $\alpha\sim\beta$ to denote $|(\alpha,\beta)|=1$. Thus, for $i$ and $j$ nodes of
$Q$, we have $\alpha_{i}\sim\alpha_{j}$ if and only if $i\sim j$.
Definition 3.3.
Let $\mathfrak{B}$ be a $W$-orbit of sets of mutually orthogonal positive roots. We say that
$\mathfrak{B}$ is an admissible orbit if for each $B\in\mathfrak{B}$, and $i$, $j\in Q$ with $i\not\sim j$
and $\gamma$, $\gamma-\alpha_{i}+\alpha_{j}\in B$ we have $r_{i}B=r_{j}B$, and each element in $\mathfrak{B}$ is called an admissible root set.
This is the definition from [5], and there is another equivalent definition in [6]. We also state it here.
Definition 3.4.
Let $B\subset\Phi^{+}$ be a mutually orthogonal root set. If for all $\gamma_{1}$, $\gamma_{2}$, $\gamma_{3}\in B$
and $\gamma\in\Phi^{+}$, with $(\gamma,\gamma_{i})=1$, for
$i=1$, $2$, $3$, we have $2\gamma+\gamma_{1}+\gamma_{2}+\gamma_{3}\in B$, then $B$ is called an admissible root set.
By these two definitions, it follows that the intersection of two admissible root sets are admissible.
It can be checked by definition that the intersection of two admissible sets are still admissible. Hence
for a given set $X$ of mutually orthogonal positive roots, the unique smallest admissible set containing
$X$ is called the admissible closure of $X$, and denoted as $X^{\rm cl}$ (or $\overline{X}$). Up to
the action of the corresponding Weyl groups,
all admissible root sets of type ${\rm A}_{n}$, ${\rm D}_{n}$, ${\rm E}_{6}$, ${\rm E}_{7}$, ${\rm E}_{8}$ have appeared in [6], [7] and [11],
and are listed
in Table 2. In the table, the
set
$Y(t)^{*}$ consists of all $\alpha^{*}$ for $\alpha\in Y(t)$, where $\alpha^{*}$ is the unique
positive root orthogonal to $\alpha$ and all other positive roots orthogonal
to $\alpha$ for type ${\rm D}_{n}$ with $n>4$. For type ${\rm D}_{n}$, if we considier the root systems are realized in $\mathbb{R}^{n}$,
with $\alpha_{1}=\epsilon_{2}-\epsilon_{1}$, $\alpha_{2}=\epsilon_{2}+\epsilon_{1}$, $\alpha_{i}=\epsilon_{i}-\epsilon_{i-1}$, for $3\leq i\leq n$,
then $\Phi^{+}=\{\epsilon_{j}\pm\epsilon_{i}\}_{1\leq i<j\leq n}$, then $(\epsilon_{j}\pm\epsilon_{i})^{*}=\epsilon_{j}\mp\epsilon_{i}$. For ${\rm D}_{4}$,
the $t$ can be $0$, $1$, $2$, $3$, which means the number of nods in the coclique.
When $t=2$, although in the Dynkin diagram $\{\alpha_{1},\alpha_{2}\}$ and $\{\alpha_{1},\alpha_{4}\}$ are symmetric, they are in the
different orbits under the Weyl group’s actions. Then the admissible root sets for ${\rm D}_{4}$ can be written as the $W({\rm D}_{4})$’s
orbits of $\emptyset$, $\{\alpha_{3}\}$, $\{\alpha_{1},\alpha_{2}\}$, $\{\alpha_{1},\alpha_{4}\}$, and $\{\alpha_{1},\alpha_{2},\alpha_{4},\alpha_{1}+\alpha_{2}+\alpha_{4}+2\alpha_{3%
}\}.$
Example 3.5.
If $Q={\rm D}_{4}$, the root set $\{\alpha_{1},\alpha_{2},\alpha_{4}\}$ is mutually orthogonal but not admissible,
and its admissible closure is $\{\alpha_{1},\alpha_{2},\alpha_{4},\alpha_{1}+\alpha_{2}+2\alpha_{3}+\alpha_{4}\}$.
Definition 3.6.
Let $\mathcal{A}$ denote the collection of all admissible subsets of $\Phi$ consisting of
mutually orthogonal positive roots.
Members of $\mathcal{A}$ are called admissible sets.
Now we consider the actions of $R_{i}$ on an admissible $W$-orbit $\mathfrak{B}$. When $R_{i}B\neq B$, We say that
$R_{i}$ lowers $B$ if there is a root $\beta\in B$ of minimal height among those moved by $R_{i}$ that
satisfies $\beta-\alpha_{i}\in\Phi^{+}$ or $R_{i}B<B$.
We say that $R_{i}$ raises $B$ if there is a root $\beta\in B$ of minimal height among
those moved by $R_{i}$ that satisfies $\beta+\alpha_{i}\in\Phi^{+}$ or $R_{i}B>B$. By this
we can set an partial order on $\mathfrak{B}=WB$. The poset $(\mathfrak{B},<)$ with this minimal ordering is called the monoidal poset (with respect to $W$) on $\mathfrak{B}$ (so $\mathfrak{B}$ should be admissible for the poset to be monoidal). If $\mathfrak{B}$ just consists of sets
of a single root, the order is determined by the canonical height function on roots. There is an important conclusion in [5], stated below.
This theorem plays a crucial role in obtaining a basis for Brauer algebra of simply laced type in [6].
Theorem 3.7.
There is a unique maximal element in $\mathfrak{B}$.
For any $\beta\in\Phi^{+}$ and $i\in\{1,\ldots,n\}$, there exists a $w\in W$ such that $\beta=w\alpha_{i}$. Then $R_{\beta}:=wR_{i}w^{-1}$ and
$E_{\beta}:=wE_{i}w^{-1}$ are well defined (this is well known from Coxeter
group theory for $R_{\beta}$;
see [6, Lemma 4.2] for $E_{\beta}$). If
$\beta,\gamma\in\Phi^{+}$ are mutually orthogonal, then $E_{\beta}$ and
$E_{\gamma}$ commute (see [6, Lemma 4.3]). Hence, for $B\in\mathcal{A}$, we
define the product
$$\displaystyle E_{B}$$
$$\displaystyle=$$
$$\displaystyle\prod_{\beta\in B}E_{\beta},$$
(3.11)
which is a quasi-idempotent, and the normalized version
$$\displaystyle{\hat{E}}_{B}$$
$$\displaystyle=$$
$$\displaystyle\delta^{-|B|}E_{B},$$
(3.12)
which is an idempotent element of the Brauer monoid.
For a mutually orthogonal root subset $X\subset\Phi^{+}$, we have
$$\displaystyle E_{X^{\rm cl}}=\delta^{|X^{\rm cl}\setminus X|}E_{X}.$$
(3.13)
Let $C_{X}=\{i\in Q\mid\alpha_{i}\perp X\}$ and
let $W(C_{X})$ be the subgroup generated by the
generators of nodes in $C_{X}$.
The subgroup $W(C_{X})$ is called the centralizer of $X$.
The normalizer of $X$, denoted by $N_{X}$ can be defined as
$$N_{X}=\{w\in W\mid E_{X}w=wE_{X}\}.$$
We let $D_{X}$ denote a set of right coset representatives for $N_{X}$ in $W$.
In [6, Definition 3.2], an action of the Brauer monoid
${\rm BrM}(Q)$ on the collection $\mathcal{A}$ of admissible root sets
in $\Phi^{+}$ was indicated below, where $Q\in{\rm ADE}$.
Definition 3.8.
There is an action of the Brauer monoid ${\rm BrM}(Q)$ on the collection
$\mathcal{A}$. The generators $R_{i}$ $(i=1,\ldots,n)$ act by the natural action of
Coxeter group elements on its positive root sets as in Remark 3.2,
and the element $\delta$ acts as the identity,
and the action of $E_{i}$ $(i=1,\ldots,n)$ is defined by
$$E_{i}B:=\begin{cases}B&\text{if}\ \alpha_{i}\in B,\\
(B\cup\{\alpha_{i}\})^{\rm cl}&\text{if}\ \alpha_{i}\perp B,\\
R_{\beta}R_{i}B&\text{if}\ \beta\in B\setminus\alpha_{i}^{\perp}.\end{cases}$$
(3.14)
We will refer to this action as the admissible set action.
This monoid action plays an important role in getting a basis of ${\rm BrM}(Q)$ in [6]. For the basis, we state one conclusion from [6, Proposition 4.9] below.
Proposition 3.9.
Each element of the Brauer monoid ${\rm BrM}(Q)$ can be written in the form
$$\delta^{k}uE_{X}zv,$$
where
$X$ is the highest element from one $W$-orbit in $\mathcal{A}$, $u$, $v^{-1}\in D_{X}$, $z\in W(C_{X})$, and $k\in\mathbb{Z}$.
4 Morita equivalence on ${\rm Br}(Q,k)$ with generic parameter $\delta$
Here we suppose that $k$ is a field.
To satisfy the cellular condition of corresponding Hecke algebra in [13] and
good prime property in [20], and also [6, Table 3], we suppose the
following for the characteristic of $k$(${\rm Char}(k)$).
$$\begin{cases}{\rm Char}(k)\quad\text{no condition},\quad\text{when}\quad Q={%
\rm A}_{n},\\
{\rm Char}(k)\neq 2,\quad\text{when}\quad Q={\rm D}_{n},\\
{\rm Char}(k)\neq 2,3,\quad\text{when}\quad Q={\rm E}_{6},{\rm E}_{7},\\
{\rm Char}(k)\neq 2,3,5\quad\text{when}\quad Q={\rm E}_{8}.\end{cases}$$
(4.1)
Recall the representation $\rho_{\mathfrak{B}}$ of ${\rm BrM}(Q)$ from [6, Lemma 3.4].
Let $V_{\mathfrak{B}}$ be the free right $k[\delta^{\pm 1}][W(C_{\mathfrak{B}})]$ with basis
$\xi_{B}$ for $B\in\mathfrak{B}$, where $W(C_{\mathfrak{B}})=W(C_{X})$ with $X$ being the highest element of $\mathfrak{B}$.
we define $R_{i}V_{B}=V_{R_{i}B}h_{B,i},$
where $h_{B,i}$ is defined in [5, Definition 2].
The action of $E_{i}$ $(i=1,\ldots,n)$ on $V_{\mathfrak{B}}$ is defined by
$$E_{i}\xi_{B}:=\begin{cases}\xi_{B}&\text{if}\ \alpha_{i}\in B,\\
0&\text{if}\ \alpha_{i}\perp B,\\
R_{\beta}R_{i}\xi_{B}&\text{if}\ \beta\in B\setminus\alpha_{i}^{\perp}.\end{cases}$$
(4.2)
Theorem 4.1.
Let $Q\in{\rm ADE}$.
(i)
The associative algebra ${\rm Br}(Q,k)$ is free over $k[\delta^{\pm 1}]$ and of rank as given in the [6, Table 2],
and the algebra is semisimple when tensored with $k(\delta)$.
(ii)
For each irreducible representation $\tau$ of $W(C_{\mathfrak{B}})$, we denote ${\rm dim}(\tau)$ is the dimension of the representation $\tau$.
The algebra ${\rm Br}(Q,k)\otimes_{\mathbb{Z}[\delta,\delta^{-1}]}k(\delta)$ is a direct sum of matrix algebras of size $|\mathfrak{B}|\cdot{\rm dim}(\tau)$ for
$(B,\tau)$ running over all pairs of $W$-orbits $\mathfrak{B}$ in $\mathcal{A}$ and any irreducible representation $\tau$ of $W(C_{\mathfrak{B}})$.
Proof.
The first half of $(i)$ follows from [6, Theorem 1.1]. Checking [6, Table 3],
the number ${\rm Char}(k)$ is a good prime for every $W(C_{\mathfrak{B}})$ with all $W$-orbits $\mathfrak{B}$ in $\mathcal{A}$ for type $Q$, which implies that
$k(\delta)[W(C_{\mathfrak{B}})]$ is split semisimple.
Then the similar argument for proving [6, Theorem 1.1]
can be applied to the second half of $(i)$. The conclusion of $(ii)$ holds for the similar arguments in [6, Corollary 5.6]
and semisimplicity of $k(\delta)[W(C_{\mathfrak{B}})]$ of all $W$-orbits $\mathfrak{B}$ in $\mathcal{A}$ for type $Q$.
∎
Theorem 4.2.
For the algebra ${\rm Br}(Q,k)\otimes_{\mathbb{Z}[\delta,\delta^{-1}]}k(\delta)$, we have
${\rm Br}(Q,k)\otimes_{\mathbb{Z}[\delta,\delta^{-1}]}k(\delta)\overset{morita}%
{\sim}\oplus k(\delta)[W(C_{\mathfrak{B}})],$
where $\mathfrak{B}$ runs over all the $W$-orbits in $\mathcal{A}$, and $W(C_{\emptyset})=W$ when $\mathfrak{B}=\emptyset$.
Furthermore, the algebra ${\rm Br}(Q,k)\otimes_{\mathbb{Z}[\delta,\delta^{-1}]}k(\delta)$ is quasi-hereditary.
Proof.
From the proof of the cellularity theorem [6, Theorem 1.2] in [6, Section 6],
we have that ${\rm Br}(Q,k)\otimes_{\mathbb{Z}[\delta,\delta^{-1}]}k(\delta)$ is an iterated inflation
of $k(\delta)[W(C_{\mathfrak{B}})]$, where $\mathfrak{B}$ runs over all the $W$-orbits in $\mathcal{A}$, including $\mathfrak{B}=\emptyset$ and $W(C_{\emptyset})=W.$
It follows that
$${\rm Br}(Q,k)\otimes_{\mathbb{Z}[\delta,\delta^{-1}]}k(\delta)\cong\bigoplus_{%
\mathfrak{B}}V_{\mathfrak{B}}\otimes V_{\mathfrak{B}}\otimes k(\delta)[W(C_{%
\mathfrak{B}})].$$
By Theorem 4.1, the algebra ${\rm Br}(Q,k)\otimes_{\mathbb{Z}[\delta,\delta^{-1}]}k(\delta)$ is semisimple, therefore
the bilinear forms for defining the iterated inflation structure of ${\rm Br}(Q,k)\otimes_{\mathbb{Z}[\delta,\delta^{-1}]}k(\delta)$
$$\varphi_{\mathfrak{B}}:V_{\mathfrak{B}}\otimes V_{\mathfrak{B}}\rightarrow k(%
\delta)[W(C_{\mathfrak{B}})]$$
are non-singular by [15, Theorem 3.8]. Hence the theorem holds for Theorem 2.5.
The bilinear forms is non-singular, then the algebra ${\rm Br}(Q,k)\otimes_{\mathbb{Z}[\delta,\delta^{-1}]}k(\delta)$ is quasi-hereditary follows from [15, Remark 3.10].
∎
Remark 4.3.
If we take $\{v_{B}\}_{B\in\mathfrak{B}}$ as the basis of $V_{\mathfrak{B}}$, we see that $\varphi_{\mathfrak{B}}$ is a matrix over $k(\delta)[W(C_{\mathfrak{B}})]$ with coefficients of
polynomial of variable $\delta$. By Theorem 4.2, when we consider this algebra with evaluating $\delta$ in $k$, there are only finite
values of $\delta$ so that $\varphi_{\mathfrak{B}}$ fails to be non-singular. So there are only finite $\delta$s in $k$ so that ${\rm Br}(Q,k)\otimes_{\mathbb{Z}[\delta,\delta^{-1}]}k(\delta)$
fails to be semisimple.
Remark 4.4.
Theorem 4.2 is a generalization of [18, Theorem 7.3], which says that the
classical Brauer algebra
$$\mathscr{B}_{n}(\delta)\overset{Morita}{\sim}\bigoplus_{i=1}^{\left\lfloor n/2%
\right\rfloor}k\mathfrak{S}_{n-2i},$$
where the classical Brauer algebra is considered as our Brauer algebra of type ${\rm A}_{n-1}$.
Remark 4.5.
Let $A={\rm Br}(Q,k)\otimes_{\mathbb{Z}[\delta,\delta^{-1}]}k(\delta)$. Since $A$ is quasi-hereditary, we have the following.
(i)
The algebra $A$ has finite global dimension bounded by $2\sum_{\mathfrak{B}}W(C_{\mathfrak{B}})-2$,
where $\mathfrak{B}$ runs over all the $W$-orbits in $\mathcal{A}$, including $\mathfrak{B}=\emptyset$ and $W(C_{\emptyset})=W$([12]).
(ii)
The Cartan matrix of $A$ has determinant $1$([17, Theorem 3.1]).
(iii)
Any cell chain of $A$ is a hereditary Chain ([17, Theorem 3.1]).
5 Semisimplicity of ${\rm Br}({\rm D}_{n},k)$ with $\delta$ specialized
Now we focus on type ${\rm D}_{n}$, and the parameter $\delta$ is specialized in $k$.
Let $\Lambda$ be the cells used to prove the celluarity in [6, Section 6].
The underlying set $\Lambda$ is defined as the union of $\Lambda_{1}$ and
$\Lambda_{2}$, where $\Lambda_{1}=\{t\}_{t=0}^{[\frac{n}{2}]}$
and $\Lambda_{2}=\{(t,\theta)\}_{t=1}^{[\frac{n+1}{2}]}$.
The partial
order on $\Lambda$ is given by
•
$\lambda_{1}>\lambda_{2}$ if
$\lambda_{1}=t_{1}<t_{2}=\lambda_{2}\in\Lambda_{1}$,
•
$\lambda_{1}>\lambda_{2}$ if
$\lambda_{1}=(t_{1},\theta)$, $\lambda_{2}=(t_{2},\theta)\in\Lambda_{2}$ and
$t_{1}<t_{2}$,
•
$\lambda_{1}>\lambda_{2}$ if $\lambda_{1}=t_{1}\in\Lambda_{1}$ and
$\lambda_{2}=(t_{2},\theta)\in\Lambda_{2}$ and $t_{1}\leq t_{2}$.
It can be
illustrated by the following Hasse diagram, where $a>b$ is equivalent to the
existence of a directed path from $a$ to $b$.
longgggg
\xymatrix
0\ar[dr]\ar[r]&1\ar[r]\ar[d] & 2 \ar[d]\ar[r] &3\ar[d]\ar[r]&⋯\ar[d]
& (1,θ) \ar[r] & (2,θ)\ar[r]&(3,θ)\ar[r]&⋯
And we see that the order on $\Lambda$ is partially ordered, but not totally ordered.
As we have seen in the Table 2, There is a class of $Y(t)\cup Y(t)^{*}$,
for $1\leq t\leq\frac{n}{2}$.
The $\Lambda_{2}$ is corresponding to the orbits of those $Y(t)\cup Y(t)^{*}$.
By the [6, Section 1] or by the diagram representation of Brauer algebra of type
${\rm D}_{n}$ in [7], we see that that the structure and the bilinear forms associated to these cells
are the same as ${\rm Br}({\rm A}_{n-1},k)$, except the $\delta$ in ${\rm Br}({\rm A}_{n-1},k)$ is replaced by
$\delta^{2}$, because we replaced each generator of $E_{i}$ in ${\rm Br}({\rm A}_{n-1},k)$ by $E_{i}E_{i}^{*}$ in ${\rm Br}({\rm D}_{n},k)$.
We keep the notation from [23] and [24], Let
$$\mathbb{Z}(n)=\{i\in\mathbb{Z}\mid 4-2n\leq i\leq n-2\}\setminus\{i\in\mathbb{%
Z}\mid 4-2n\leq i\leq 3-n,2\nmid i\}.$$
By [23, Theorem 1.2, Theorem 1.3] and our analysis about the structure of ${\rm Br}({\rm D}_{n},k)$, we have the followings.
Theorem 5.1.
Let $k=\mathbb{C}$. The algebra ${\rm Br}({\rm D}_{n},k)$ is not semisimple, when
(i)
the parameter $\delta^{2}\in\mathbb{Z}(n)$ and $\delta\neq 0$, or
(ii)
the parameter $\delta=0$ and $n\notin\{1,3,5\}$.
Theorem 5.2.
Let $k$ be a field with characteristic $e>0$. The algebra ${\rm Br}({\rm D}_{n},k)$ is not semisimple, when
(i)
the parameter $\delta^{2}\in\mathbb{Z}(n)$, $\delta\neq 0$ and $e\nmid n!$, or
(ii)
the parameter $\delta=0$ and $n\notin\{1,3,5\}$ and $e\nmid n!$.
6 The Morita equivalence on BMW algebras of simply laced types
The Birman-Murakami-Wenzl (BMW in short) algebras are first introduced in [1] and [22],
which can be considered as type ${\rm A}$ in [4],
where the authors extended them to all simply laced types. We present the definition in the below.
Definition 6.1.
Let $Q$ be a simply laced Coxeter diagram of rank $n$. The Birman-Murakami-Wenzl algebra of type
$Q$ is the algebra, denoted by ${\rm B}(Q)$, with ground field $\mathbb{Q}(l,\delta)$, where $l$ and $\delta$ are
transcendental and algebraically independent over $\mathbb{Q}$, whose presentation is given
on generators $g_{i}$ and $e_{i}$ ($i=1$,$2$,$\ldots$, $n$) by the following relations
$$\displaystyle g_{i}g_{j}$$
$$\displaystyle=$$
$$\displaystyle g_{j}g_{i}\qquad\quad\qquad\quad\quad\mbox{for}\ i\nsim j$$
(6.1)
$$\displaystyle g_{i}g_{j}g_{i}$$
$$\displaystyle=$$
$$\displaystyle g_{j}g_{i}g_{j}\qquad\quad\qquad\,\kern-0.4pt\quad\mbox{for}\ {i%
\sim j}$$
(6.2)
$$\displaystyle me_{i}$$
$$\displaystyle=$$
$$\displaystyle l(g_{i}^{2}+mg_{i}-1)\qquad\kern 0.5pt\mbox{for}\,\mbox{any}\ i$$
(6.3)
$$\displaystyle g_{i}e_{i}$$
$$\displaystyle=$$
$$\displaystyle l^{-1}e_{i}\qquad\quad\qquad\kern 0.5pt\quad\quad\mbox{for}\,%
\mbox{any}\ i$$
(6.4)
$$\displaystyle e_{i}g_{j}e_{i}$$
$$\displaystyle=$$
$$\displaystyle le_{i}\qquad\quad\quad\qquad\,\kern-0.4pt\quad\quad\mbox{for}\ {%
i\sim j}$$
(6.5)
where $m=(l-l^{-1})/(1-\delta).$
Remark 6.2.
It is known there is a natural homomorphism of rings from the BMW algebra of type $Q$ to the Brauer algebra of the same type induced on the generators
by $g_{i}\mapsto R_{i}$ and $e_{i}\mapsto E_{i}$ with $l=1$. In [4], [8], [11], it is proved that
the rewritten form in Proposition 3.9 gives a basis of the BMW algebra by changing $R_{i}$ to $g_{i}$ and $E_{i}$ to $e_{i}$.
Similar to the Theorem 4.1, we have the following conclusion for the algebra ${\rm B}(Q)$.
Theorem 6.3.
Let $Q\in{\rm ADE}$.
(i)
The associative algebra ${\rm B}(Q)$ is semisimple.
(ii)
For the algebra ${\rm B}(Q)$, we have
${\rm B}(Q)\overset{morita}{\sim}\oplus\mathcal{H}(C_{\mathfrak{B}}),$
where $\mathfrak{B}$ runs over all the $W$-orbits in $\mathcal{A}$, $\mathcal{H}(C_{\mathfrak{B}})$ is the Hecke algebra of type $C_{\mathfrak{B}}$.
Furthermore, the algebra ${\rm B}(Q)$ is quasi-hereditary.
Proof.
For (i), when $Q$ is of type ${\rm D}_{n}$, $4\leq n$, or ${\rm E}_{n}$, $n=6$, $7$, $8$, the proofs are given in [8, Theorem 1.1],
and [11, Theorem 1], respectively. For type ${\rm A}_{n}$, by modifying our parameters with the parameters in [25] and [26], we can see for the generic
parameters, and the algebra ${\rm B}({\rm A}_{n})$ is semisimple because of [25, Theorem 4.3] and [26, Theorem 2.17].
For (ii), the prooof of the case for $Q={\rm A}_{n}$ can be found in [26, Theorem 2.17]. For $Q$ being other types ,we apply the the argument of the
proof of Theorem 4.2. By (i), the algebra ${\rm B}(Q)$ is semisimple, therefore
the bilinear forms for defining the iterated inflation structure of ${\rm B}(Q)$
$$\varphi_{\mathfrak{B}}^{{}^{\prime}}:V_{\mathfrak{B}}\otimes V_{\mathfrak{B}}%
\rightarrow\mathcal{H}(C_{\mathfrak{B}})$$
which is defined in [8, Section 8] for type ${\rm D}_{n}$ and in the proof of [11, Theorem 8],
are non-singular by [15, Theorem 3.8]. Hence the Morita Equivalence in (ii) holds for Theorem 2.5.
The bilinear forms is non-singular, then the algebra ${\rm B}(Q)$ is quasi-hereditary follows from [15, Remark 3.10].
∎
Remark 6.4.
If we evaluate the parameters of ${\rm Br}(Q,k)$ and ${\rm B}(Q)$, we have to compute the Gram determinants of
the bilinear forms defining their cellular structures as in [25], to judge the semi-simplicity, Morita equivalence and quasi-heredity
about them. Some further work is needed to complete parameter problems about this.
There are also Brauer algebras of multiply-laced type which has already defined and studied in [9],[10], and [19],
and we can explore these properties about them.
References
[1]
J.S. Birman and H. Wenzl, Braids, link polynomials and a new algebra,
Trans. Amer. Math. Soc. 313 (1989), 249–273.
[2]
R. Brauer,
On algebras which are connected with the semisimple continous groups,
Annals of Mathematics, 38 (1937), 857–872.
[3]
E. Cline, B. Parshall and L. Scott, Finite dimensional algebras and highest weight categories, J. reine angew. Math.
391 (1988), 85-99.
[4]
A.M. Cohen, Dié A.H. Gijsbers and D.B. Wales,
BMW algebras of simply laced type,
Journal of Algebra, 286 (2005),107–153.
[5]
A.M. Cohen, Dié A.H. Gijsbers and D.B. Wales,
A poset connected to Artin monoids of simply laced type,
Journal of Combinatorial Theory, Series A 113 (2006) 1646–1666.
[6]
A.M. Cohen, B. Frenk and D.B. Wales, Brauer algebras
of simply laced type, Israel Journal of Mathematics, 173 (2009)
335–365.
[7]
A.M. Cohen, Dié A.H. Gijsbers and D.B. Wales,
Tangle and Brauer diagram algebras of type D${}_{n}$, Journal of Knot theory and its ramifications,
Volume 18. Number 4. April 2009, 447-483.
[8]
A.M. Cohen, Dié A.H. Gijsbers and D.B. Wales,
The BMW Algebras of type D${}_{n}$,
Communication in Algebra, 42 (2014), 22–55.
[9]
A.M. Cohen, S. Liu and S. Yu, Brauer algebras of type C,
Journal of Pure and Applied Algebra, 216 (2012), 407–426.
[10]
A.M. Cohen, S. Liu, Brauer algebras of type B,
Forum Mathematicum, 27 (2015),1163–1202.
[11]
A.M. Cohen and D.B. Wales, The Birman-Murakami-Wenzl algebras of type ${\rm E}_{n}$,
Transformation Groups, 16 (2011),
681–715.
[12]
V. Dlab, C. Ringel, Quasi-hereditary algebras, Illinois J.Math. 33, 1989, 280–291.
[13]
M. Geck, Hecke algebras of finite type are cellular, Inventiones Mathematicae,
169 (2007), 501–517.
[14]
J.J. Graham, Modular representations
of Hecke algebras and related algebras, Ph. D. thesis, University of
Sydney (1995).
[15]
J.J. Graham and G.I. Lehrer, Cellular algebras, Inventiones Mathematicae
123 (1996),
1–44.
[16]
S. König and C. Xi, Cellular algebras: inflations and Morita equivalences,
J. London Math. Soc.
123 (1996),
1–44.
[17]
S. König and C. Xi, When is a celluar algebra quasi-hereditary? ,
Math. Ann,
315 (1999),
281–293.
[18]
S. König and C. Xi, A characteristic free approach to Brauer algebras, Trans. Amer. Math. Soc.
(4) 353 (2001), 1489–1505.
[19]
S. Liu, Brauer algebra of multiply laced Weyl type,
Indagationes Mathematicae, 26 (2015), 526–546.
[20]
G. Lusztig, Characters of reductive groups over a finite field. Ann. Math. Studies, vol.
107. Princeton University Press, Princeton, NJ (1984).
[21]
K. Morita,Duality for modules and its applications to the theory of rings with minimum condition,
Science reports of the Tokyo Kyoiku Daigaku. Section A. 6 (1958), 83–142.
[22]
J. Murakami, The Kauffman polynomial of links and representation theory,
Osaka J. Math., 24 (1987), 745–758.
[23]
H. Rui, A criterion on the semisimple Brauer algebras, Journal of combinatorial theory, Series A,
111 (2005), 78–88.
[24]
H. Rui, M. Si, A criterion on the semisimple Brauer algebras II, Journal of combinatorial theory, Series A,
113 (2006), 1199–1203.
[25]
H. Rui, M. Si, Gram determinants and semisimplicity criteria
for Birman-Murakami-Wenzl algebra, J. reine angew. Math,
631 (2009), 153–179.
[26]
H. Rui, M. Si, Singular parameters for the Birman-Murakami-Wenzl algebra, Journal of pure and applied algebra,
216 (2012), 1295–1305.
[27]
M. Si, Morita equivalence for cyclotomic $BMW$ algebras, J. Algebra.
423 (2015), 573–591.
[28]
H.N.V. Temperley and E. Lieb, Relation between
percolation and colouring problems and other graph theoretical problems
associated with regular planar lattices: some exact results for the
percolation problems, Proc. Royal. Soc.London Ser A. 322 (1971) 251–288.
[29]
C. C. Xi, On the quasi-heredity of Birman-Wenzl algebras, Adv. Math.
154 (2000), 280–298.
Shoumin Liu
Email: s.liu@sdu.edu.cn
School of Mathematics, Shandong University
Shanda Nanlu 27, Jinan,
Shandong Province, China
Postcode: 250100 |
Neutrino Oscillations in a Supersymmetric
SO(10) Model with Type-III See-Saw Mechanism
Takeshi Fukuyama${}^{\dagger}$, Amon Ilakovac${}^{\ddagger}$,
Tatsuru Kikuchi${}^{\dagger}$ and Koichi Matsuda${}^{\star}$
${}^{\dagger}$
Department of Physics, Ritsumeikan University
Kusatsu, Shiga, 525-8577 Japan
E-mail: ,
${}^{\ddagger}$
University of Zagreb, Department of Physics
P.O. Box 331, Bijenička cesta 32, HR-10002 Zagreb, Croatia
E-mail:
${}^{\star}$
Department of Physics, Osaka University
Toyonaka, Osaka, 560-0043, Japan
E-mail:
fukuyama@se.ritsumei.ac.jp
rp009979@se.ritsumei.ac.jp
ailakov@rosalind.phy.hr
matsuda@het.phys.sci.osaka-u.ac.jp
Abstract:
The neutrino oscillations are studied in the framework of
the minimal supersymmetric SO(10) model with Type-III see-saw
mechanism by additionally introducing a number of SO(10) singlet
neutrinos. The light Majorana neutrino mass matrix is given by
a combination of those of the singlet neutrinos and the $SU(2)_{L}$ active
neutrinos. The minimal SO(10) model gives an unambiguous Dirac
neutrino mass matrix, which enables us to predict the masses and
the other parameters for the singlet neutrinos.
These predicted masses take the values accessible and testable
by near future collider experiments under the reasonable assumptions.
More comprehensive calculations on these parameters are also given.
Neutrino Physics, Beyond Standard Model, GUT
††preprint:
1 Introduction
As pointed out in [1], we can construct,
within the context of the standard model (SM), an operator
which gives rise to the neutrino masses as
$${\cal L}_{\rm eff}=\frac{1}{\Lambda}\,(\ell_{L}H)^{T}C^{-1}(\ell_{L}H)\;.$$
(1)
Here $\ell$, $H$ are the lepton doublet and the Higgs doublet,
$C$ is the charge conjugation operator and $\Lambda$ is the scale
in which something new physics appears.
In the usual see-saw mechanism (type-I see-saw mechanism) [2],
the scale parameter $\Lambda$ is interpreted as the energy scale at which
the right-handed neutrinos become active.
In this paper, we explore the other possibility of type-III seesaw,
introducing a set of singlet into the minimal supersymmetric
standard model (MSSM).
The motivations of this are as follows. One comes from the theoretical
reason that string inspired $E_{6}$ models include SO(10) singlets
as a matter content.
The other does from the empirical reason that many indicate reduced
coupling of neutrinos to the $Z^{0}$-boson in the framework of the SM
or the SM with right-handed neutrinos [3, 4].
2 Type-III see-saw mechanism
We begin with reviewing the essential concept of
the type-III see-saw mechanism proposed in the reference
[5, 6, 7, 8, 9].
You can find a detailed study in [10].
In this model, in addition to the usual $SU(2)_{L}$ singlet
$N=\nu_{R}$, we add a new SO(10) singlet neutrino “$S$”,
which has a positive lepton number (+1),
$${\cal L}_{Y}=\int d^{2}\theta\left(Y_{\nu}~{}\overline{N}\ell_{L}\,H_{u}+Y_{s}%
~{}\overline{N}S_{L}\,H_{s}+\mu_{s}~{}S_{L}^{T}C^{-1}S_{L}\right)+h.c.\;,$$
(2)
where $H_{u}$ and $H_{s}$ are the $SU(2)_{L}$ doublet and singlet
chiral superfields, respectively.
This Lagrangian is written in a matrix form in the base with
$\{\nu_{L},~{}N,~{}S_{L}\}$ as follows:
$$\left(\begin{array}[]{ccc}0&m_{D}^{T}&0\\
m_{D}&0&M_{D}^{T}\\
0&M_{D}&\mu_{s}\end{array}\right)\;.$$
(3)
After the spontaneous symmetry breaking, they give masses to
the neutrinos as
$$\displaystyle m_{D}$$
$$\displaystyle=$$
$$\displaystyle Y_{\nu}\left<H_{u}^{0}\right>~{},~{}~{}~{}M_{D}\ =\ Y_{s}\left<H%
_{s}\right>~{}.$$
(4)
Note that the $\mu_{s}$ term in the above breaks an originally existing
global $U(1)_{L}$ and $U(1)_{\cal R}$ symmetries. Thus we can naturally
expect it as a small value compared with the electroweak scale
even around the keV scale,
according to the following reason:
when the $\mu_{s}$ term is arisen from the VEV of a singlet
$\mu_{s}=\lambda\left<S^{\prime}\right>$, there appears a pseudo-NG boson,
called Majoron $J=\Im S^{\prime}$ associated with the spontaneously broken
$U(1)_{L}$ symmetry. Then the keV scale lepton number violation may lead
to an interesting signature in the neutrinoless double beta decay
[11] or becomes a possible candidate for
the cold dark matter [12].
By integrating out $\nu_{R}$,
$$\frac{\partial{{\cal L}}}{\partial{N}}=m_{D}\nu_{L}+M_{D}S_{L}=0\;,$$
(5)
we obtain
$$S_{L}=-\left(\frac{m_{D}}{M_{D}}\right)\nu_{L}\;.$$
(6)
This means that the light neutrino mass eigenstate is a linear combination
of two states $\nu_{L}$ and $S_{L}$ with the mixing angle
$\epsilon\cong m_{D}/M_{D}$:
$$\nu_{\rm light}=\nu_{L}-\epsilon~{}S_{L}\;.$$
(7)
Such an extra mixing term is interesting when we try to explain
the “NuTeV anomaly” through the heavy singlet neutrino contributions
to the neutrino–nucleon scatterings [3, 4].
Putting Eq. (6) into Eq. (9),
we get the effective light neutrino mass matrix as
$$M_{\nu}=\mu_{s}\left(\frac{m_{D}^{T}m_{D}}{M_{D}^{2}}\right)\;.$$
(8)
In general, adding three singlet neutrinos $\{S_{1},S_{2},S_{3}\}$, the
effective light neutrino mass matrix can be written in the matrix form as
$$M_{\nu}=\left(M_{D}^{-1}m_{D}\right)^{T}\mu_{s}\left(M_{D}^{-1}m_{D}\right)\;.$$
(9)
This matrix is diagonalised by Maki-Nakagawa-Sakata (MNS)
mixing matrix $U$ as
$$U^{T}M_{\nu}~{}U={\rm diag}(m_{1},m_{2},m_{3})\;.$$
(10)
An important fact is that the new physics scale has also the
“see-saw structure” as
$$\Lambda\cong\frac{M_{D}^{2}}{\mu_{s}}\;.$$
(11)
Hence this mechanism is sometimes called as “double see-saw” mechanism.
It’s not the actual see-saw type but the inverse see-saw form, because
the small lepton number violating (
/
$L$
) scale $\mu_{s}$
would indicate the large scale.
Now we consider the general three generation cases.
For simplicity, we assume that all $S_{i}$ have a common $\mu_{s}$ term.
Then the light neutrino matrix is written as
$$\mu_{s}\,U^{T}m_{D}^{T}(M_{D}M_{D}^{T})^{-1}m_{D}U={\rm diag}(m_{1},m_{2},m_{3%
})\;,$$
(12)
that is,
$$M_{D}M_{D}^{T}=m_{D}U{\rm diag}\left(\frac{\mu_{s}}{m_{1}},\frac{\mu_{s}}{m_{2%
}},\frac{\mu_{s}}{m_{3}}\right)U^{T}m_{D}^{T}\;.$$
(13)
This symmetric combination can be diagonalised by a single unitary
matrix ${\cal U}$
$${\cal U}^{T}~{}M_{D}M_{D}^{T}~{}{\cal U}={\rm diag}(M_{D1}^{2},M_{D2}^{2},M_{D%
3}^{2})\;.$$
(14)
Here we note that $\cal U$ includes three mixing angles
$\theta^{\prime}_{1}$, $\theta^{\prime}_{2}$, $\theta^{\prime}_{3}$ and six phases
($\delta$, $\zeta_{2}^{L}$, $\zeta_{3}^{L}$,
$\zeta_{1}^{R}$, $\zeta_{2}^{R}$, $\zeta_{3}^{R}$)
$${\cal U}=\left(\begin{array}[]{ccc}1&0&0\\
0&e^{i\zeta_{2}^{L}}&0\\
0&0&e^{i\zeta_{3}^{L}}\\
\end{array}\right)\left(\begin{array}[]{ccc}c_{3}c_{1}&c_{3}s_{1}&s_{3}e^{-i%
\delta}\\
-c_{2}s_{1}-s_{2}c_{1}s_{3}e^{i\delta}&c_{2}c_{1}-s_{2}s_{1}s_{3}e^{i\delta}&s%
_{2}c_{3}\\
s_{2}s_{1}-c_{2}c_{1}s_{3}e^{i\delta}&-s_{2}c_{1}-c_{2}s_{1}s_{3}e^{i\delta}&c%
_{2}c_{3}\\
\end{array}\right)\left(\begin{array}[]{ccc}e^{i\zeta_{1}^{R}}&0&0\\
0&e^{i\zeta_{2}^{R}}&0\\
0&0&e^{i\zeta_{3}^{R}}\\
\end{array}\right),$$
(15)
where $s_{i}:=\sin\theta^{\prime}_{i}$, $c_{i}:=\cos\theta^{\prime}_{i}$.
You should not confuse these mixing angles with those of the MNS
mixing matrix $U$ appearing in Eq. (30).
From this expression, we can obtain a prediction about masses and mixings
for the heavier Dirac mass matrix $M_{D}$ by giving some informations about
the light neutrino masses and mixings and the lighter Dirac mass matrix $m_{D}$.
3 Fermion masses in an SO(10) Model with a singlet
In order to make a prediction on the second Dirac neutrino mass matrix
$M_{D}$, we need an information for the Yukawa couplings of $Y_{\nu}$.
In this paper, we make the minimal SO(10) model extend to add
a number of singlet, which preserves a precise information for $m_{D}$.
We begin with a review of the minimal SUSY SO(10) model proposed in
[13] and recently analysed in detail in references
[14, 15, 16, 17, 18, 19, 20, 21].
Even when we concentrate our discussion on the issue of how to reproduce
the realistic fermion mass matrices in the SO(10) model, there are lots of
possibilities of the introduction of Higgs multiplets.
The minimal supersymmetric SO(10) model includes only one 10
and one $\overline{\bf 126}$ Higgs multiplets in Yukawa couplings
with 16 matter multiplets. Here, in addition to it,
we introduce a number of SO(10) singlet chiral superfields ${\bf 1}$
as new matter multiplets
111The singlet matter multiplet may have it’s origin in some $E_{6}$
representations ${\bf 27}$ or ${\bf 78}$ which are decomposed under the SO(10)
subgroup as ${\bf 27}={\bf 16+10+1}$,
${\bf 78}={\bf 45+16+\overline{16}+1}$.
In such a case, the superpotential given in Eq. (16) may be generated
from the following $E_{6}$ invariant superpotential:
$W_{Y}=Y_{1}^{ij}{\bf 27}_{i}{\bf 27}_{j}{\bf 27}_{H}+Y_{2}^{ij}{\bf 27}_{i}{%
\bf 27}_{j}{\bf 351}^{\prime}_{H}+Y_{3}^{ij}{\bf 27}_{i}{\bf 78}_{j}\overline{%
{\bf 27}}_{H}$.
.
This additional singlet can provide a type-III see-saw mechanism as
described in the previous section.
In order to avoid a large triplet VEV for $\overline{\bf 126}_{H}$
unnecessary in type-III see-saw model,
we use a $U(1)_{\cal R}$ symmetry. The corresponding $U(1)_{\cal R}$
charges are listed in Table 1.
Then the relevant superpotential can be written as
$$W_{Y}=Y_{10}^{ij}{\bf 16}_{i}{\bf 16}_{j}{\bf 10}_{H}+Y_{126}^{ij}{\bf 16}_{i}%
{\bf 16}_{j}\overline{{\bf 126}}_{H}+Y_{s}^{ij}{\bf 16}_{i}{\bf 1}_{j}%
\overline{{\bf 16}}_{H}+\mu_{s}{\bf 1}_{i}^{2}\;.$$
(16)
At low energy after the GUT symmetry breaking, the superpotential leads to
$$\displaystyle W$$
$$\displaystyle=$$
$$\displaystyle\left(Y_{10}^{ij}H_{10}^{u}+Y_{126}^{ij}H_{126}^{u}\right)u^{c}_{%
i}q_{j}+\left(Y_{10}^{ij}H_{10}^{d}+Y_{126}^{ij}H_{126}^{d}\right)d^{c}_{i}q_{j}$$
(17)
$$\displaystyle+$$
$$\displaystyle\left(Y_{10}^{ij}H_{10}^{u}-3Y_{126}^{ij}H_{126}^{u}\right)N_{i}{%
\ell}_{j}+\left(Y_{10}^{ij}H_{10}^{d}-3Y_{126}^{ij}H_{126}^{d}\right)e^{c}_{i}%
{\ell}_{j}$$
$$\displaystyle+$$
$$\displaystyle Y_{s}^{ij}N_{i}S_{j}H_{s}+\mu_{s}S_{i}^{2}\;,$$
where $H_{10}$ and $H_{126}$ correspond to the Higgs doublets in ${\bf 10}_{H}$
and $\overline{{\bf 126}}_{H}$. That is, we have two pairs of Higgs doublets.
In order to keep the successful gauge coupling unification,
we suppose that one pair of Higgs doublets
(a linear combination of $H_{10}^{u,d}$ and $H_{126}^{u,d}$)
is light while the other pair is heavy ($\simeq M_{\rm GUT}$).
The light Higgs doublets are identified
as the MSSM Higgs doublets ($H_{u}$ and $H_{d}$) and given by
$$H_{u}\ =\ \widetilde{\alpha}_{u}~{}H_{10}^{u}+\widetilde{\beta}_{u}~{}H_{126}^%
{u}\;;\quad H_{d}\ =\ \widetilde{\alpha}_{d}~{}H_{10}^{d}+\widetilde{\beta}_{d%
}~{}H_{126}^{d}\;,$$
(18)
where $\widetilde{\alpha}_{u,d}$ and $\widetilde{\beta}_{u,d}$ denote
elements of the unitary matrix which rotate the flavour basis in the original
model into the SUSY mass eigenstates. Omitting the heavy Higgs mass
eigenstates, the low energy superpotential is described by only the light
Higgs doublets $H_{u}$ and $H_{d}$ such that
$$\displaystyle W_{Y}$$
$$\displaystyle=$$
$$\displaystyle\left(\alpha^{u}Y_{10}^{ij}+\beta^{u}Y_{126}^{ij}\right)u^{c}_{i}%
q_{j}H_{u}\ +\ \left(\alpha^{d}Y_{10}^{ij}+\beta^{d}Y_{126}^{ij}\right)d^{c}_{%
i}q_{j}H_{d}$$
(19)
$$\displaystyle+$$
$$\displaystyle\left(\alpha^{u}Y_{10}^{ij}-3\beta^{u}Y_{126}^{ij}\right)N_{i}%
\ell_{j}H_{u}\ +\ \left(\alpha^{d}Y_{10}^{ij}-3\beta^{d}Y_{126}^{ij}\right)e_{%
i}^{c}\ell_{j}H_{d}$$
$$\displaystyle+$$
$$\displaystyle Y_{s}^{ij}N_{i}S_{j}H_{s}\ +\ \mu_{s}S_{i}^{2}\;,$$
where the formulas of the inverse unitary transformation of Eq. (18),
$H_{10}^{u,d}=\alpha^{u,d}H_{u,d}+\cdots$ and
$H_{126}^{u,d}=\beta^{u,d}H_{u,d}+\cdots$, have been used.
Providing the Higgs VEV’s, $\langle H_{u}\rangle=v\sin\beta$ and
$\langle H_{d}\rangle=v\cos\beta$ with $v\simeq 174$ [GeV],
the Dirac mass matrices can be read off as
$$\displaystyle M_{u}$$
$$\displaystyle=$$
$$\displaystyle c_{10}M_{10}+c_{126}M_{126},$$
$$\displaystyle M_{d}$$
$$\displaystyle=$$
$$\displaystyle M_{10}+M_{126},$$
$$\displaystyle m_{D}$$
$$\displaystyle=$$
$$\displaystyle c_{10}M_{10}-3c_{126}M_{126},$$
$$\displaystyle M_{e}$$
$$\displaystyle=$$
$$\displaystyle M_{10}-3M_{126},$$
(20)
where $M_{u}$, $M_{d}$, $m_{D}$ and $M_{e}$ denote up-type quark, down-type quark,
Dirac neutrino and charged-lepton mass matrices, respectively. Note that
all the quark and lepton mass matrices are characterised by only two basic
mass matrices, $M_{10}$ and $M_{126}$, and four complex coefficients
$c_{10}$ and $c_{126}$. In addition to the above mass matrices the above
model indicates the mass matrices,
$$\displaystyle M_{R}$$
$$\displaystyle=$$
$$\displaystyle c_{R}~{}M_{126}\;,$$
$$\displaystyle M_{L}$$
$$\displaystyle=$$
$$\displaystyle c_{L}~{}M_{126}\;,$$
(21)
together with $M_{D}$ given in Eq. (4).
$c_{R}$ and $c_{L}$ correspond to the VEV’s of
$({\bf 10},{\bf 1},{\bf 3})\subset{\overline{\bf 126}}$
and $({\overline{\bf 10}},{\bf 3},{\bf 1})\subset{\overline{\bf 126}}$,
respectively [22]. If $M_{R}$, $M_{L}$, $M_{D}$ terms dominate,
they are called Type-I, Type-II, and Type-III see-saw, respectively.
In this paper, we consider the case $c_{R}\ =\ c_{L}\ =\ 0$, Type-III.
Here $c_{R}=0$ means that the theory does not pass the Pati-Salam phase
and is broken to the standard model directly.
The mass matrix formulas in Eq. (20) leads to the GUT
relation among the quark and lepton mass matrices,
$$\displaystyle M_{e}=c_{d}\left(M_{d}+\kappa M_{u}\right)\;,$$
(22)
where
$$\displaystyle c_{d}$$
$$\displaystyle=$$
$$\displaystyle-\frac{3c_{10}+c_{126}}{c_{10}-c_{126}},$$
(23)
$$\displaystyle\kappa$$
$$\displaystyle=$$
$$\displaystyle-\frac{4}{3c_{10}+c_{126}}.$$
(24)
Without loss of generality, we can take the basis where $M_{u}$ is real
and diagonal, $M_{u}=D_{u}$. Since $M_{d}$ is the symmetric matrix, it is
described as $M_{d}=V_{\mathrm{CKM}}^{*}\,D_{d}\,V_{\mathrm{CKM}}^{\dagger}$
by using the CKM matrix $V_{\mathrm{CKM}}$ and the real diagonal mass matrix
$D_{d}$.
Considering the basis-independent quantities, $\mathrm{tr}[M_{e}^{\dagger}M_{e}]$,
$\mathrm{tr}[(M_{e}^{\dagger}M_{e})^{2}]$ and $\mathrm{det}[M_{e}^{\dagger}M_{e}]$,
and eliminating $|c_{d}|$, we obtain two independent equations,
$$\displaystyle\left(\frac{\mathrm{tr}[\widetilde{M_{e}}^{\dagger}\widetilde{M_{%
e}}]}{m_{e}^{2}+m_{\mu}^{2}+m_{\tau}^{2}}\right)^{2}$$
$$\displaystyle=$$
$$\displaystyle\frac{\mathrm{tr}[(\widetilde{M_{e}}^{\dagger}\widetilde{M_{e}})^%
{2}]}{m_{e}^{4}+m_{\mu}^{4}+m_{\tau}^{4}},$$
(25)
$$\displaystyle\left(\frac{\mathrm{tr}[\widetilde{M_{e}}^{\dagger}\widetilde{M_{%
e}}]}{m_{e}^{2}+m_{\mu}^{2}+m_{\tau}^{2}}\right)^{3}$$
$$\displaystyle=$$
$$\displaystyle\frac{\mathrm{det}[\widetilde{M_{e}}^{\dagger}\widetilde{M_{e}}]}%
{m_{e}^{2}\;m_{\mu}^{2}\;m_{\tau}^{2}},$$
(26)
where $\widetilde{M_{e}}\equiv V_{\mathrm{CKM}}^{*}\,D_{d}\,V_{\mathrm{CKM}}^{\dagger%
}+\kappa D_{u}$.
With input data of six quark masses, three angles and one CP-phase in the
CKM matrix and three charged-lepton masses, we can solve the above equations
and determine $\kappa$ and $|c_{d}|$, but one parameter, the phase of $c_{d}$,
is left undetermined [14, 15, 16].
With input data of six quark masses, three angles and one CP-phase
in the CKM matrix and three charged lepton masses,
we solve the above equations and determine $\kappa$.
The original basic mass matrices, $M_{10}$ and $M_{126}$, are described by
$$\displaystyle M_{10}$$
$$\displaystyle=$$
$$\displaystyle\frac{3+|c_{d}|e^{i\sigma}}{4}V_{\mathrm{CKM}}^{*}\,D_{d}\,V_{%
\mathrm{CKM}}^{\dagger}+\frac{|c_{d}|e^{i\sigma}\kappa}{4}D_{u},$$
(27)
$$\displaystyle M_{126}$$
$$\displaystyle=$$
$$\displaystyle\frac{1-|c_{d}|e^{i\sigma}}{4}V_{\mathrm{CKM}}^{*}\,D_{d}\,V_{%
\mathrm{CKM}}^{\dagger}-\frac{|c_{d}|e^{i\sigma}\kappa}{4}D_{u},$$
(28)
as the functions of $\sigma$, the phase of $c_{d}$, with the solutions
$|c_{d}|$ and $\kappa$ determined by the GUT relation.
Now let us solve the GUT relation and determine $|c_{d}|$ and $\kappa$.
Since the GUT relation of Eq. (22) is valid only at the GUT
scale, we first evolve the data at the weak scale to the corresponding
quantities at the GUT scale with given $\tan\beta$ according to the
renormalization group equations (RGE’s) and use them as input data at the
GUT scale. Note that it is non-trivial to find the solution of the GUT
relation since the number of the free parameters (fourteen) is almost the
same as the number of inputs (thirteen). The solution of the GUT relation
exists only if we take appropriate input parameters. Taking the experimental
data at the $M_{Z}$ scale [23], we get the following values
for charged fermion masses and the CKM matrix at the GUT scale, $M_{\rm GUT}$
with $\tan\beta=10$:
$$\displaystyle m_{u}=0.000980\;,\;\;m_{c}=0.285\;,\;\;m_{t}=113,$$
$$\displaystyle m_{d}=0.00135\;,\;\;m_{s}=0.0201\;,\;\;m_{b}=0.996,$$
$$\displaystyle m_{e}=0.000326\;,\;\;m_{\mu}=0.0687\;,\;\;m_{\tau}=1.17,$$
and
$$\displaystyle V_{\mathrm{CKM}}(M_{G})=\left(\begin{array}[]{ccc}0.975&0.222&-0%
.000940-0.00289i\\
-0.222-0.000129i&0.974+0.000124i&0.0347\\
0.00864-0.00282i&-0.0337-0.000647i&0.999\end{array}\right)$$
in the standard parameterisation. The signs of the input fermion masses
have been chosen to be $(m_{u},m_{c},m_{t})=(+,-,+)$ and
$(m_{d},m_{s},m_{b})=(-,-,+)$.
By using these outputs at the GUT scale as input parameters,
we can solve Eqs. (25) and (26) and find a solution:
$$\displaystyle\kappa=-0.0103+0.000606i\;,$$
$$\displaystyle|c_{d}|=6.32\;.$$
(29)
Once these parameters, $|c_{d}|$ and $\kappa$, are determined, we can describe
all the fermion mass matrices as a functions of $\sigma$ from the mass matrix
formulas of Eqs. (20), (27) and (28).
Thus in the minimal SO(10) model we have almost unambiguous Dirac neutrino
mass matrix $m_{D}$ and, therefore, we can obtain the informations on $M_{D}$
from the neutrino experiments via
$M_{\nu}=(M_{D}^{-1}m_{D})^{T}\mu_{s}(M_{D}^{-1}m_{D})$ as in Eq. (9).
Now we proceed to the numerical calculation of $M_{D}$ from
the well-confirmed neutrino oscillation data. The MNS mixing matrix $U$
in the standard parametrization is
$$U=\left(\begin{array}[]{ccc}c_{13}c_{12}&c_{13}s_{12}e^{i\varphi_{2}}&s_{13}e^%
{i(\varphi_{1}-\delta)}\\
(-c_{23}s_{12}-s_{23}c_{12}s_{13}e^{i\delta})e^{-i\varphi_{2}}&c_{23}c_{12}-s_%
{23}s_{12}s_{13}e^{i\delta}&s_{23}c_{13}e^{i(\varphi_{1}-\varphi_{2})}\\
(s_{23}s_{12}-c_{23}c_{12}s_{13}e^{i\delta})e^{-i\varphi_{1}}&(-s_{23}c_{12}-c%
_{23}s_{12}s_{13}e^{i\delta})e^{-i(\varphi_{1}-\varphi_{2})}&c_{23}c_{13}\\
\end{array}\right)\;,$$
(30)
where $s_{ij}:=\sin\theta_{ij}$, $c_{ij}:=\cos\theta_{ij}$
and $\delta$, $\varphi_{1}$, $\varphi_{2}$ are the Dirac phase and
the Majorana phases, respectively.
Recent KamLAND data tells us that
222Our convention is $\Delta m_{ij}^{2}=m_{i}^{2}-m_{j}^{2}$.
$$\displaystyle\Delta m^{2}_{\oplus}$$
$$\displaystyle=$$
$$\displaystyle\Delta m^{2}_{32}\ =\ 2.1\times 10^{-3}\;\;{\mathrm{eV}}^{2}\;,$$
$$\displaystyle\sin^{2}\theta_{\oplus}$$
$$\displaystyle=$$
$$\displaystyle 0.5\;,$$
$$\displaystyle\Delta m^{2}_{\odot}$$
$$\displaystyle=$$
$$\displaystyle\left|\Delta m^{2}_{21}\right|\ =\ 8.3\times 10^{-5}\;\;{\mathrm{%
eV}}^{2}\;,$$
$$\displaystyle\sin^{2}\theta_{\odot}$$
$$\displaystyle=$$
$$\displaystyle 0.28\;,$$
$$\displaystyle|U_{e3}|^{2}$$
$$\displaystyle<$$
$$\displaystyle 0.061\;.$$
(31)
For simplicity we take $U_{e3}=0$. Note that we can take both signs
of $\Delta m^{2}_{21}$, $\Delta m^{2}_{21}>0$ or $\Delta m^{2}_{21}<0$.
The former is called normal hierarchy, the latter is called inverted
hierarchy. Here we adopt the former case, and take the lightest neutrino
mass eigenvalue as $m_{\ell}=10^{-3}\;{\rm[eV]}$.
Then the mass eigenvalues are written as
$$\displaystyle m_{1}$$
$$\displaystyle=$$
$$\displaystyle m_{\ell}\;,$$
$$\displaystyle m_{2}$$
$$\displaystyle=$$
$$\displaystyle\sqrt{m_{\ell}^{2}+\Delta m^{2}_{\oplus}}\;,$$
$$\displaystyle m_{3}$$
$$\displaystyle=$$
$$\displaystyle\sqrt{m_{\ell}^{2}+\Delta m^{2}_{\oplus}+\Delta m^{2}_{\odot}}\;.$$
(32)
For the light Dirac neutrino mass matrix $m_{D}$, we input the SO(10)
predicted one as was done in the previous section. However, unlike
the case of minimal SO(10) GUT model, we can not fix $\sigma$.
So we can obtain the heavy Dirac neutrino mass matrix $M_{D}$ as
a function of $\mu_{s}$ and the three undetermined parameters,
$\sigma$, two Majorana phases $\varphi_{1}$ and $\varphi_{2}$.
For example, for fixed $\mu_{s}=1~{}{\rm[keV]}$
(For the implication of this value, see the remarks below Eq. (4).)
and $\varphi_{1}=\varphi_{2}=0$,
we get a prediction for the mass spectra of $M_{D}$.
The dependences on the parameters $\sigma$ and $U_{e3}$
for fixed $\sigma=\pi$ are depicted in Fig. 1.
These values are allowed by the present experiments [24]
and are accessible and testable by the Large Hadron Collider (LHC)
at CERN, in which we are able to discover new particles with masses up to
$\lesssim 7$ [TeV] [25].
Of course, these values depend on the ambiguous assumptions taken above.
We may take another strategy adopted in [21].
As shown in the paper [21],
we repeat the substitution of the normally-distributed random numbers
which give the experimental values [26]:
$$\displaystyle\left|{m_{u}\left({2~{}{\rm{GeV}}}\right)}\right|$$
$$\displaystyle=$$
$$\displaystyle 2.9\pm 0.6~{}{\rm{[MeV]}},\quad\left|{m_{d}\left({2~{}{\rm{GeV}}%
}\right)}\right|=5.2\pm 0.9~{}{\rm{[MeV]}},$$
(33)
$$\displaystyle\left|{m_{s}\left({2~{}{\rm{GeV}}}\right)}\right|$$
$$\displaystyle=$$
$$\displaystyle 99\pm 16~{}{\rm{[MeV]}},\quad\left|{m_{c}\left({m_{c}}\right)}%
\right|=1.0-1.4~{}{\rm{[GeV]}},$$
(34)
$$\displaystyle m_{b}\left({m_{b}}\right)$$
$$\displaystyle=$$
$$\displaystyle 4.0-4.5~{}{\rm{[GeV]}},\quad m_{t}^{{\rm{direct}}}=174.3\pm 5.1~%
{}{\rm{[GeV]}},$$
(35)
$$\displaystyle\left|{m_{e}^{{\rm{pole}}}}\right|$$
$$\displaystyle=$$
$$\displaystyle{\rm{0.510998902}}\pm{\rm{0.000000021~{}[MeV]}},$$
(36)
$$\displaystyle\left|{m_{\mu}^{{\rm{pole}}}}\right|$$
$$\displaystyle=$$
$$\displaystyle{\rm{105.658357}}\pm{\rm{0.00005,}}\quad m_{\tau}^{{\rm{pole}}}={%
\rm{1776.99}}\pm{\rm{0.29~{}[MeV]}},$$
(37)
$$\displaystyle\sin\theta_{12}$$
$$\displaystyle=$$
$$\displaystyle 0.2229\pm 0.0022,\quad\sin\theta_{23}=0.0412\pm 0.0020,$$
(38)
$$\displaystyle\sin\theta_{13}$$
$$\displaystyle=$$
$$\displaystyle 0.0036\pm 0.0007,\quad\delta=\left({59\pm 13}\right)^{\circ}$$
(39)
for the quark and charged lepton masses and the CKM mixing and
the Dirac phase parameters 10,000 times.
On the other hand, about the remaining parameters,
we assume Eq. (31), $m_{1}=10^{-3}$ [eV]
and $U_{e3}=0$ at the GUT scale,
and Majorana phases and $\sigma$ move from $0$ to $2\pi$
in 8 equal intervals.
Namely, we scan the possible ranges of undetermined parameters $\sigma$,
$\varphi_{1}$, $\varphi_{2}$ and plotted the three masses of $M_{D}$,
the three mixing angles
and five phases of ${\cal U}$ which diagonalises the mass matrix $M_{D}$
in the basis where $M_{e}$ is real diagonal in Figs. 2–5.
Here we calculated the distributions for sixteen sets of possible
combinations of mass signatures of up-type and down-type quarks.
Figure 1 corresponds to the blue solid line
of Figure 2 with $\mu_{s}=1$ [KeV].
Finally, it is remarkable to say that the see-saw mechanism itself
(or the types of it) can never been proofed and all the models should
take care of all the types of the see-saw mechanism including
the alternatives to it [27, 28].
The test of all these models is due to the applications to the other
phenomelogical consequences, for example, the lepton flavour violating
processes and so on [29, 30].
4 Summary
In this paper, we have constructed an SO(10) model in which the smallness
of the neutrino masses are explained in terms of the type-III see-saw
mechanism. To evaluate the parameters related to the singlet neutrinos,
we have used the minimal SUSY SO(10) model. This model can simultaneously
accommodate all the observed quark-lepton mass matrix data with appropriately
fixed free parameters. Especially, the neutrino-Dirac-Yukawa coupling
matrix are completely determined. Using this Yukawa coupling matrix,
we have calculated the masses and mixings for the not-so-heavy singlet
neutrinos. The obtained ranges of the mass of $M_{D}$ is interesting since
they are testable by a forthcoming LHC experiment.
Acknowledgments.
The work of T.F. is supported in part by the Grant-in-Aid for
Scientific Research from the Ministry of Education, Science and Culture
of Japan (#16540269). He is also grateful to Professors D. Chang and
K. Cheung for their hospitality at NCTS. The work of T.K. and K.M.
are supported by the Research Fellowship of the Japan Society
for the Promotion of Science (#7336 and #3700).
The work of A.I. is supported by the Ministry of Science and
Technology of Republic of Croatia under contract (#0119261).
References
[1]
S. Weinberg,
Phys. Rev. Lett. 43, 1566 (1979).
[2]
T. Yanagida, in Proceedings of the workshop
on the Unified Theory and Baryon Number in the Universe,
edited by O. Sawada and A. Sugamoto (KEK, Tsukuba, 1979);
M. Gell-Mann, P. Ramond, and R. Slansky,
in Supergravity, edited by D. Freedman and P. van Niewenhuizen
(north-Holland, Amsterdam 1979);
R. N. Mohapatra and G. Senjanović,
Phys. Rev. Lett. 44, 912 (1980).
[3]
G. P. Zeller et al. [NuTeV Collaboration],
Phys. Rev. Lett. 88, 091802 (2002)
[Erratum-ibid. 90, 239902 (2003)]
[arXiv:hep-ex/0110059];
The theoretical explanations of this anomaly by including
heavy singlet neutrinos are given in [4].
[4]
T. Takeuchi,
[arXiv:hep-ph/0209109];
W. Loinaz, N. Okamura, T. Takeuchi and L. C. R. Wijewardhana,
Phys. Rev. D 67, 073012 (2003)
[arXiv:hep-ph/0210193];
W. Loinaz, N. Okamura, S. Rayyan, T. Takeuchi and L. C. R. Wijewardhana,
Phys. Rev. D 70, 113004 (2004)
[arXiv:hep-ph/0403306].
[5]
E. Witten,
Nucl. Phys. B 268, 79 (1986).
[6]
R. N. Mohapatra,
Phys. Rev. Lett. 56 (1986) 561.
[7]
R. N. Mohapatra and J. W. F. Valle,
Phys. Rev. D 34, 1642 (1986).
[8]
J. W. F. Valle, in
NUCLEAR BETA DECAYS AND NEUTRINO: proceedings.
Edited by T. Kotani, H. Ejiri, E. Takasugi.
Singapore, World Scientific, 1986. 542p.
[9]
S. M. Barr,
Phys. Rev. Lett. 92, 101601 (2004)
[arXiv:hep-ph/0309152].
[10]
I. Melo,
[arXiv:hep-ph/9612488].
[11]
Z. G. Berezhiani, A. Y. Smirnov and J. W. F. Valle,
Phys. Lett. B 291, 99 (1992)
[arXiv:hep-ph/9207209].
[12]
V. Berezinsky and J. W. F. Valle,
Phys. Lett. B 318, 360 (1993)
[arXiv:hep-ph/9309214].
[13]
K. S. Babu and R. N. Mohapatra,
Phys. Rev. Lett. 70, 2845 (1993)
[arXiv:hep-ph/9209215].
[14]
K. Matsuda, Y. Koide and T. Fukuyama,
Phys. Rev. D 64, 053015 (2001)
[arXiv:hep-ph/0010026].
[15]
K. Matsuda, Y. Koide, T. Fukuyama and H. Nishiura,
Phys. Rev. D 65, 033008 (2002)
[Erratum-ibid. D 65, 079904 (2002)]
[arXiv:hep-ph/0108202];
[16]
T. Fukuyama and N. Okada,
JHEP 0211, 011 (2002)
[arXiv:hep-ph/0205066].
[17]
H. S. Goh, R. N. Mohapatra and S. P. Ng,
Phys. Lett. B 570, 215 (2003)
[arXiv:hep-ph/0303055].
[18]
H. S. Goh, R. N. Mohapatra and S. P. Ng,
Phys. Rev. D 68, 115008 (2003)
[arXiv:hep-ph/0308197].
[19]
B. Bajc, G. Senjanović and F. Vissani,
Phys. Rev. Lett. 90, 051802 (2003)
[arXiv:hep-ph/0210207].
[20]
B. Dutta, Y. Mimura and R. N. Mohapatra,
Phys. Rev. D 69, 115014 (2004)
[arXiv:hep-ph/0402113].
[21]
K. Matsuda,
Phys. Rev. D 69, 113006 (2004)
[arXiv:hep-ph/0401154].
[22]
K. Matsuda, T. Fukuyama and H. Nishiura,
Phys. Rev. D 61, 053001 (2000)
[arXiv:hep-ph/9906433].
[23]
H. Fusaoka and Y. Koide,
Phys. Rev. D 57, 3986 (1998)
[arXiv:hep-ph/9712201].
[24]
P. Achard et al. [L3 Collaboration],
Phys. Lett. B 517, 67 (2001)
[arXiv:hep-ex/0107014].
[25]
S. N. Gninenko, M. M. Kirsanov, N. V. Krasnikov and V. A. Matveev,
[arXiv:hep-ph/0301140].
[26]
S. Eidelman et al. [Particle Data Group],
Phys. Lett. B 592, 1 (2004).
[27]
H. Murayama,
Nucl. Phys. Proc. Suppl. 137, 206 (2004)
[arXiv:hep-ph/0410140].
[28]
A. Y. Smirnov,
[arXiv:hep-ph/0411194].
[29]
F. Deppisch and J. W. F. Valle,
[arXiv:hep-ph/0406040].
[30]
A. Ilakovac and A. Pilaftsis,
Nucl. Phys. B 437, 491 (1995)
[arXiv:hep-ph/9403398]. |
Magnetic dipole transitions in the OH $A\,^{2}\Sigma^{+}\leftarrow X\,^{2}\Pi$ system
Moritz Kirste
Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, 14195 Berlin, Germany
Xingan Wang
Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, 14195 Berlin, Germany
Gerard Meijer
Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, 14195 Berlin, Germany
Koos B. Gubbels
Institute of Theoretical Physics, University of Cologne, Zülpicher Str. 77, 50937 Cologne, Germany
Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, 14195 Berlin, Germany
Radboud University Nijmegen, Institute for Molecules and Materials, Heijendaalseweg 135, 6525 AJ Nijmegen, The Netherlands
Ad van der Avoird
Radboud University Nijmegen, Institute for Molecules and Materials, Heijendaalseweg 135, 6525 AJ Nijmegen, The Netherlands
Gerrit C. Groenenboom
Radboud University Nijmegen, Institute for Molecules and Materials, Heijendaalseweg 135, 6525 AJ Nijmegen, The Netherlands
Sebastiaan Y.T. van de Meerakker
Radboud University Nijmegen, Institute for Molecules and Materials, Heijendaalseweg 135, 6525 AJ Nijmegen, The Netherlands
Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, 14195 Berlin, Germany
(July 19, 2012)
Abstract
We report on the observation of magnetic dipole allowed transitions in the well-characterized $A\,^{2}\Sigma^{+}-X\,^{2}\Pi$ band system of the OH radical. A Stark decelerator in combination with microwave Rabi spectroscopy is used to control the populations in selected hyperfine levels of both $\Lambda$-doublet components of the $X\,^{2}\Pi_{3/2},v=0,J=3/2$ ground state. Theoretical calculations presented in this paper predict that the magnetic dipole transitions in the $\nu^{\prime}=1\leftarrow\nu=0$ band are weaker than the electric dipole transitions by a factor of $2.58\times 10^{3}$ only, i.e., much less than commonly believed. Our experimental data confirm this prediction.
pacs:
The hydroxyl radical (OH) plays a central role in many areas of chemistry and physics, and is one of the most extensively studied molecular species to date. In 1950, Meinel discovered that emission from vibrationally excited OH radicals in the Earth’s atmosphere is responsible for the infrared night-time air glow Meinel:1950 . Detection of the 18 cm absorption lines in the radio spectrum of Cassiopeia A by Weinreb et al. in 1963 revealed the presence of OH in interstellar space Weinreb Nature 1963 . Shortly after, the OH radical was identified as the first molecule to form astrophysical (mega)masers Weaver Nature 1965 ; Baan Astrophysical Journal 1982 . Since then, a wealth of spectroscopic investigations has been carried out in the microwave, infrared, and ultraviolet part of the spectrum, unraveling the electronic, vibrational, rotational, and hyperfine structure of the OH radical.
The OH (${}^{2}\Pi$) radical (together with the similar NO (${}^{2}\Pi$) radical) has also been established as the paradigm for molecular collisions studies. Interest in these open-shell radical species stems from their importance in combustion and atmospheric environments, as well as from their complex rotational structure that exhibits spin-orbit and $\Lambda$-doublet splittings. Ingenious methods have been developed to select OH (${}^{2}\Pi$) radicals in a single rotational (sub)level, to orient them in space ter Meuler Phys. Rev. Lett. 1976 ; Hain Chem. Phys. Lett. 1996 , and to tune their velocity Gilijamse Science 2006 OH+Xe . These methods have allowed collision experiments of transient species at the fully state-resolved level, and have contributed enormously to our present understanding of how intermolecular potentials govern molecular collision dynamics.
Recently, the OH radical has emerged as a benchmark molecule in the rapidly developing field of cold molecules Meerakker Chemical Reviews . The OH radical was one of the first molecular species to be slowed down Bochinski Phys. Rev. Lett. 2003 and to be confined in traps Meerakker Phys. Rev. Lett. 2005 . In the near future, comparison of high-resolution spectroscopic data on cold OH radicals in the laboratory with interstellar megamaser observations may reveal a possible time variation of fundamental constants Hudson Phys. Rev. Lett. 2006 .
In the vast majority of experiments, ground state OH radicals are detected via laser induced fluorescence (LIF) after optical excitation on electric dipole allowed (EDA) transitions of the $A\,^{2}\Sigma^{+}\leftarrow X\,^{2}\Pi$ band using a pulsed dye laser. An important property of the $A-X$ band is that it allows one to selectively probe the population of individual $\Lambda$-doublet components of opposite parity within a rotational state. Although the $\Lambda$-doublet splittings are typically much smaller than the bandwidth of pulsed dye lasers, the measurement of populations in selected $\Lambda$-doublet components is facilitated by the parity selection rules of EDA transitions and the large energy splitting between levels of opposite parity in the $A\,^{2}\Sigma^{+}$ state (see inset to Figure 1). Similar schemes are used to probe $\Lambda$-doublet component resolved populations in other ${}^{2}\Pi$ molecules such as NO, CH, and SH.
Extreme care, however, must be taken when using this approach. In recent experiments in our laboratory, molecular beams of OH with an almost perfect quantum state purity were produced via the Stark deceleration technique. In these experiments, $\geq 99.999$ % of OH radicals in the ${}^{2}\Pi_{3/2},J=3/2$ rotational ground state reside in the upper $\Lambda$-doublet component of $f$ symmetry; the lower $\Lambda$-doublet component of $e$ symmetry is effectively depopulated in the Stark-deceleration process. When the populations in the $e$ and $f$ components were probed using LIF via the $A\leftarrow X$ transition, however, the apparent population in the $e$ state appeared at least one order of magnitude too large. A spectroscopic re-investigation using a laser with a much narrower bandwidth revealed that magnetic dipole allowed (MDA) transitions were responsible for this effect Kirste Science 2012 .
Magnetic dipole allowed transitions have rarely been observed in laser excitation spectra of heteronuclear molecules Yang Journal of Molecular Spectroscopy 2010 . Their existence is generally neglected in quantitative measurements of state populations, potentially leading to a significant misinterpretation of detector signals. In homonuclear molecules, MDA transitions between electronic states are well known to result in “forbidden” band systems that violate the rigorous selection rules for electric dipole transitions. The most famous example is the atmospheric oxygen band, which appears in the red part of the solar spectrum. In contrast, MDA transitions in heteronuclear molecules mostly exist as weak satellite lines parallel to strong EDA transitions. The general rule of thumb is that MDA transitions are about a factor $10^{5}$ weaker than the corresponding EDA transitions Herzberg . Already in the 1920’s, weak satellite lines in the $A-X$ emission band of OH were observed that appeared to correspond to transitions to the “wrong” $\Lambda$-doublet component Dieke:Nature115:194 ; Watson:Nature117:157 ; Mulliken:PhysRev32:388 ; Jack:PRSA120:222 . These lines were tentatively attributed to the MDA transitions by Van Vleck in 1934 Vleck:AJ80:161 , but received little attention ever since.
Here, we present a detailed analysis of MDA transitions in the $A\,^{2}\Sigma^{+}\leftarrow X\,^{2}\Pi$ band of OH. We show that the satellite MDA transitions are surprisingly strong, and only three orders of magnitude weaker than the main EDA transitions. In our experiment we use a Stark-decelerator to produce packets of OH radicals that reside exclusively in the upper $\Lambda$-doublet component of $f$ symmetry. A controlled fraction of the population is transferred to the lower component of $e$ symmetry by using a microwave field. The MDA and EDA $A\,^{2}\Sigma^{+},v=1\leftarrow X\,^{2}\Pi,v=0$ transitions originating from the $f$ and $e$ level, respectively, are spectroscopically resolved using a narrowband pulsed dye laser. The observed ratio of the signal intensities agrees well with theoretical calculations for the EDA and MDA transition strengths.
The relevant energy levels and electronic transitions are shown in the inset to Figure 1. The electronic ground state of OH has a $X\,^{2}\Pi$ configuration. Each rotational level, labeled by $J$, splits into two $\Lambda$-doublet components which are separated by 0.055 cm${}^{-1}$ for the $J=3/2$ rotational ground state. The upper and lower components have $+$ and $-$ parity, and are indicated by the spectroscopic labels $f$ and $e$, respectively. Each of the $\Lambda$-doublet components of the $J=3/2$ state is split into $F=1$ and $F=2$ hyperfine levels. The four resulting levels are referred to hereafter as $\left|X,f,+,F=2\right\rangle$, $\left|X,f,+,F=1\right\rangle$, $\left|X,e,-,F=2\right\rangle$ and $\left|X,e,-,F=1\right\rangle$.
The first electronically excited state of OH has a $A\,^{2}\Sigma^{+}$ configuration. In our experiments, only the $N=0,J=1/2$ rotational ground state of $+$ parity is of relevance. This state is split into two hyperfine states $F=0$ and $F=1$ that are separated by 0.026 cm${}^{-1}$, and are referred to hereafter as $\left|A,+,F=0\right\rangle$ and $\left|A,+,F=1\right\rangle$. The EDA ($P_{1}(1)$) and MDA ($P^{\prime}_{1}(1)$) $A-X$ transitions couple the $\left|X,e,-\right\rangle$ and $\left|X,f,+\right\rangle$ states to the $\left|A,+\right\rangle$ states following the parity changing and parity conserving selection rules for EDA and MDA transitions, respectively.
Our experimental setup is schematically shown in Figure 1. A packet of OH ($X\,^{2}\Pi_{3/2},v=0,J=3/2,f$) radicals with a velocity of 448 m/s is produced by passing a molecular beam of OH through a 2.6 meter long Stark decelerator Scharfenberg PRA 2009 newdec . The Stark decelerator efficiently deflects molecules in the $\left|X,e,-\right\rangle$ states. A phase angle $\phi_{0}=50^{\circ}$ is used to ensure that the OH radicals that exit the decelerator reside exclusively in the $\left|X,f,+,F=2\right\rangle$ state. The end of the Stark decelerator is electrically shielded to prevent any electric stray fields to penetrate into the interaction area.
A controlled fraction of the OH radicals is transferred into the $\left|X,e,-,F=1\right\rangle$ state by inducing the $\left|X,f,+,F=2\right\rangle\rightarrow\left|X,e,-,F=1\right\rangle$ transition at 1.72 GHz with a microwave pulse. For this purpose a 90 mm long microwave antenna is installed 38 mm downstream from the decelerator and perpendicular to the molecular beam axis. No frequency-matched microwave resonator was used. The microwaves are reflected by the vacuum chamber walls filling the whole vacuum chamber, and we assume the microwaves to be unpolarized. The microwave duration and power can be controlled via a microwave switch and attenuator, respectively. The magnetic field in the interaction region is controlled by three copper coils with a diameter of 31 cm each, that are mounted 30 cm from the interaction area. One coil is positioned above the interaction area, one at the side and one at the end.
Two lasers are used to detect the OH radicals via LIF using the 1-0 band of the OH $A\,^{2}\Sigma^{+}\leftarrow X\,^{2}\Pi_{3/2}$ transition around 282 nm. The first laser, a pulsed dye laser (PDL) with a bandwidth of 1.8 GHz, is used to probe the population in the $\left|X,e,-\right\rangle$ state via the EDA $P_{1}(1)$ transition. The second laser, a pulsed dye amplifier (PDA) seeded by a single mode ring dye laser, has a bandwidth of 120 MHz and is used to separate the $P_{1}(1)$ and $P^{\prime}_{1}(1)$ transitions. The power of the PDL and PDA lasers are adjusted to ensure that the transitions are induced under saturated and unsaturated conditions, respectively, and both lasers are linearly polarized in the $z$ direction (see Figure 1 for the definition of the coordinate system). The off-resonant fluorescence is imaged into a photomultiplier tube (PMT).
In the presence of a magnetic field, the $F=1$ and $F=2$ hyperfine states split into 3 and 5 $M_{F}$ Zeeman sublevels, respectively, that are readily resolved in the microwave spectrum. This is illustrated in Figure 2(a) that shows the $\left|X,f,+,F=2\right\rangle\rightarrow\left|X,e,-,F=1\right\rangle$ spectrum around 1.72 GHz, recorded with the broadband PDL system. In the black spectrum no currents are applied to the coils, and nine transitions can be identified corresponding to the nine allowed $\left|X,f,+,F=2,M_{F}\right\rangle\rightarrow\left|X,e,-,F^{\prime}=1,M^{%
\prime}_{F}\right\rangle$ transitions that are split by the Earth’s magnetic field.
For an unambiguous interpretation of the EDA and MDA $A-X$ transitions, and to measure their relative strengths, it is convenient to choose the laser polarization direction parallel to the space quantization axis. The Earth’s magnetic field, however, is not suitable for this, as the direction of the magnetic field vector is in general not parallel to the laser polarization axis. We therefore follow the approach to first compensate the Earth’s magnetic field by applying currents to the three coils, and then to apply a controlled magnetic field that is parallel to the $z$ axis, i.e., the laser polarization axis. The red curve in Fig. 2(a), shows the microwave spectrum that is recorded when currents of 2.10 A, 1.60 A and 0.35 A are passed through the top, side and end coils, respectively. It is seen that in this configuration the Earth’s magnetic field is compensated and the nine lines merge into one. An additional magnetic field in the $z$ direction can be added by changing the current in the top coil, while keeping the current in the other coils constant. We have chosen to reverse the current in the top coil to generate a magnetic field with a magnitude that is twice as large as the $z$-component of the Earth’s magnetic field.
A controlled fraction of the population in each of the $\left|X,f,+,F=2,M_{F}\right\rangle$ levels can be transferred to an individual $M_{F}$ component of the $\left|X,e,-,F=1\right\rangle$ level by applying a microwave pulse with a controlled pulse duration and power. In Figure 2(b), the fluorescence intensity is shown that is measured for five different microwave transitions as a function of the microwave pulse duration. Clear Rabi oscillations are observed, with different Rabi frequencies for each transition due to the differences in transition strength and the unpolarized microwave radiation. These Rabi oscillations were measured for all nine transitions shown in Figure 2(a), and for each transition it was observed that the maxima of the oscillations yield equal signal intensity. We thus conclude that the OH radicals that exit the Stark decelerator are equally distributed over the five $M_{F}$ levels of the $\left|X,f,+,F=2\right\rangle$ state before the microwave field is applied.
Three different microwave transitions are induced that transfer population from the $\left|X,f,+,F=2,M_{F}=0\right\rangle$ into the $M_{F}=1$, $M_{F}=0$ and $M_{F}=-1$ levels of the $\left|X,e,-,F=1\right\rangle$ state, respectively. These transitions are indicated by the red, black and blue arrows in the inset in Figure 3. For each transition, the microwave pulse duration and power was carefully chosen to transfer (2.5$\pm$1)% of all molecules from the $\left|X,f,+,F=2,M_{F}=0\right\rangle$ level. Since this $M_{F}=0$ level contains one fifth of all $F=2$ molecules, 99.5$\pm$0.2% of the OH radicals remain in the $\left|X,f,+,F=2\right\rangle$ state, in all three cases. The error (2$\sigma$) is given by the statistical spread of the Rabi oscillations.
The EDA $P_{1}(1)$ and MDA $P^{\prime}_{1}(1)$ $A-X$ transitions are then investigated in these three cases by probing the populations in the $\left|X,e,-\right\rangle$ and $\left|X,f,+\right\rangle$ states with the narrowband PDA system. This laser can spectroscopically resolve the $\Lambda$-doublet splitting in the $\left|X\right\rangle$ state and the hyperfine splitting in the $\left|A\right\rangle$ state, but not the hyperfine splittings in both $\left|X\right\rangle$ states or any Zeeman splittings. For parallel laser polarization and magnetic field direction, both the EDA and the MDA transitions obey the hyperfine selection rule $\Delta F$=0,$\pm$1. The EDA transition has the additional selection rule $\Delta M_{F}$=0 (with $\Delta F\neq 0$ for $M_{F}=0$), while MDA transitions can only couple states with $\Delta M_{F}$=$\pm$1. As indicated in Figure 3, there are thus six MDA transitions and only one EDA transition for each case.
In Figure 3 the MDA $P^{\prime}_{1}(1)$ and the EDA $P_{1}(1)$ transitions are shown that are recorded in the three cases. The MDA $P^{\prime}_{1}(1)$ transitions appear at the same position and with equal intensity in all spectra. Depending on the $M_{F}$ level that is populated in the $\left|X,e,-,F=1\right\rangle$ state, the EDA $P_{1}(1)$ transition either couples to the $\left|A,+,F=0\right\rangle$ (for $M_{F}=0$) or the $\left|A,+,F=1\right\rangle$ state (for $M_{F}=\pm 1$). These transitions are clearly resolved in the spectra. The former transition appears four times more intense than the latter two transitions that are of equal intensity, as is expected theoretically supplement . The $\Lambda$-doublet splitting is also recognized.
Having observed the MDA transition one might wonder about the presence of electric quadrupole allowed (EQA) transitions. For parallel laser polarization and magnetic field direction, an EQA transition can couple states with $\Delta F$=$\pm$2, $\Delta M_{F}=\pm 1$. In the experiment no EQA $\left|A,+,F=0\right\rangle\leftarrow\left|X,f,+,F=2\right\rangle$ transition was observed, indicating that EQA transitions in the OH (A-X) band are at least two orders of magnitude weaker than MDA transitions. This finding is supported by the theoretical estimate of the EQA transition strength supplement .
The relative strength of the $A-X$ MDA and the EDA transitions can be deduced from the measured spectra, and compared to theory. The strengths of the transitions are calculated from the magnitude of the two transition dipole moments, given by $\mu_{\rm el/mag}=\left|\left\langle A\,^{2}\Sigma^{+},v=1\right|\bm{\hat{\mu}}%
_{\rm el/mag}\left|X\,^{2}\Pi,v=0\right\rangle\right|$ supplement . We find $\mu_{\rm el}=0.0525$ a.u. and $\mu_{\rm mag}=0.142$ a.u. for the electric and magnetic transition dipole moments, respectively, so that $\frac{1}{\alpha^{2}}\cdot\mu_{\rm el}^{2}/\mu_{\rm mag}^{2}=2.58\cdot 10^{3}$. Here, $\alpha$ is the fine-structure constant accounting for the relative strength of the magnetic field compared to the electric field of the laser. Magnetic dipole transitions in the OH ($A-X$) band are thus only three orders of magnitude weaker than electric dipole transitions.
Taking into account the experimental initial distribution of molecules over the quantum states, as well as the direction of the laser polarization and the magnetic field, we find a theoretical ratio of 25.8 for the fluorescence intensities of the EDA $\left|A,+,F=0\right\rangle\leftarrow\left|X,e,-,F=1,M_{F}=0\right\rangle$ transition and the combined six MDA $\left|A,+,F=1\right\rangle\leftarrow\left|X,f,+,F=2\right\rangle$ transitions supplement . The uncertainty in this ratio is estimated to be about 10% supplement . This value agrees well with the experimental value of (18$\pm$8), obtained by comparing the strong central with the left peak in Figure 3. The experimental error is mainly given by the statistical error of the population transfer in the microwave field from the $\left|X,f,+,F=2,M_{F}=0\right\rangle$ to the $\left|X,e,-,F=1,M_{F}\right\rangle$ levels.
In this work we reported on the direct measurement of magnetic dipole transitions in laser excitation spectra of the OH $A\,^{2}\Sigma^{+},v=1\leftarrow X\,^{2}\Pi_{3/2},v=0$ band. These satellite transitions appear only three orders of magnitude weaker than the corresponding main electric dipole transitions, and can potentially lead to a misinterpretation of detector signals when the $\Lambda$-doublet-resolved state populations in OH ($X\,^{2}\Pi$) are measured. This finding may seem of limited significance in some experiments; in experiments in which large differences in $\Lambda$-doublet populations are expected it may be essential. In particular in state-of-the-art molecular beam experiments with unprecedented state purity and precision, magnetic dipole transitions should be carefully considered.
The authors are grateful to Samuel Meek, Nicolas Vanhaecke, Janneke Blokland, Boris Sartakov and Christian Schewe for the support in the realization of the experiment and the analysis of the data. K.B.G. and G.M. acknowledge support from the ERC-2009-AdG under grant agreement 247142-MolChip. K.B.G. and A.v.d.A acknowledge support from the Alexander von Humboldt Foundation. S.Y.T.v.d.M. acknowledges support from the Netherlands Organisation for Scientific Research (NWO) via a VIDI grant. Supplementary information accompanies this paper.
References
(1)
A.B. Meinel, Astrophys. J. 111, 555 (1950).
(2)
S. Weinreb, A.H. Barret, M.L. Meeks, and J.C. Henry, Nature 200, 829 (1963).
(3)
H. Weaver, D.R.W. Williams, N.H. Dieter, and W.T. Lum, Nature 208, 29 (1965).
(4)
W.A. Baan and P.A.D. Wood, Astrophys. J. 260, 49 (1982).
(5)
J.J. ter Meulen, W.L. Meerts, G.W.M. van Mierlo, and A. Dymanus, Phys. Rev. Lett. 36, 1031 (1976).
(6)
T.D. Hain, M.A. Weibel, K.M. Backstrand, P.E. Pope, and T.J. Curtiss, Chem. Phys. Lett. 262, 723 (1996).
(7)
J.J. Gilijamse, S. Hoekstra, S.Y.T. van de Meerakker, G.C. Groenenboom, and G. Meijer, Science 313, 1617 (2006).
(8)
S.Y.T. van de Meerakker, H.L. Bethlem, N. Vanhaecke, and G. Meijer, Chem. Rev., published online, (2012).
(9)
J.R. Bochinski, E.R. Hudson, H.J. Lewandowski, G. Meijer, and J. Ye, Phys. Rev. Lett. 91, 243001 (2003).
(10)
S.Y.T. van de Meerakker, P.H.M. Smeets, N. Vanhaecke, R.T. Jongma, and G. Meijer, Phys. Rev. Lett. 94, 023004 (2005).
(11)
E.R. Hudson, H.J. Lewandowski, B.C. Sawyer, and J. Ye, Phys. Rev. Lett. 96, 143004 (2006).
(12)
M. Kirste, X. Wang, C.H. Schewe, G. Meijer, K. Liu, A. van de Avoird, L.M.C. Janssen, K.B. Gubbels, G.C. Groenenboom, and S.Y.T. van de Meerakker, submitted (2012).
(13)
X. Yang and P.J. Dagdigian, J. Mol. Spectrosc. 198, 189 (1999).
(14)
G. Herzberg, Molecular Spectra and Molecular Structure: Spectra of Diatomic Molecules (Van Nostrand Reinhold, New York, 1950).
(15)
G.H. Dieke, Nature 115, 194 (1925).
(16)
W.W. Watson, Nature 117, 157 (1926).
(17)
R.S. Mulliken, Phys. Rev. 32, 388 (1928).
(18)
D. Jack, Proc. Royal Soc. A 120, 222 (1928).
(19)
J.H. van Vleck, Astrophys. J. 80, 161 (1934).
(20)
L. Scharfenberg, H. Haak, G. Meijer, and S.Y.T van de Meerakker, Phys. Rev. A. 79, 023410 (2009).
(21)
See supplementary material for a detailed description of the theoretical calculation. |
Fast Inter-Prediction based on Decision Trees for AV1 encoding
Abstract
The AOMedia Video 1 (AV1) standard can achieve considerable compression efficiency thanks to the usage of many advanced tools and improvements, such as advanced inter-prediction modes. However, these come at the cost of high computational complexity of encoder, which may limit the benefits of the standard in practical applications. This paper shows that not all sequences benefit from using all such modes, which indicates that a number of encoder optimisations can be introduced to speed up AV1 encoding. A method based on decision trees is proposed to selectively decide whether to test all inter modes. Appropriate features are extracted and used to perform the decision for each block. Experimental results show that the proposed method can reduce the encoding time on average by 43.4% with limited impact on the coding efficiency.
Fast Inter-Prediction based on Decision Trees for AV1 encoding
Jieon Kim*, Saverio Blasi$${}^{\circ}$$, Andre Seixas Dias$${}^{\circ}$$, Marta Mrak$${}^{\circ}$$ and Ebroul Izquierdo*
* Queen Mary University of London (UK)
$${}^{\circ}$$BBC R&D, London (UK)
Index Terms—
AV1, machine learning, decision trees
1 Introduction
The Alliance for Open Media (AOMedia) [1] recently finalised the development of the AOMedia Video 1 (AV1) specification. AOMedia was founded in 2015 as a consortium of over 30 partners from the semiconductor industry, video on demand providers and web browser developers, with the specific objective of creating open, royalty-free multimedia delivery solutions. AV1 is the first outcome of such initiative, and it was built using the VP9 standard specification developed by Google [2] as a base. Similarly to its predecessor, AV1 follows the typical hybrid block-based approach commonly used in many video coding standards.
Thanks to many new optimisations, algorithms and techniques, AV1 has significantly better performance in terms of higher quality with considerable bit rate saving compared to VP9 [3]. This is mostly due to the fact that AV1 adopts a number of new technical contributions, such as enhanced directional intra-prediction, extended reference frames, dynamic spatial and temporal motion vector referencing, overlapped block motion compensation, extended transform kernels, and many others [4]. While these large number of tools and encoder options contribute to the compression efficiency of the standard, encoder implementations are required to select the best configuration for each portion of the sequence being encoded. This comes at the cost of considerable additional computational complexity, which may limit the benefits of using the standard in practical applications [5]. Therefore, algorithms to reduce the encoder run time with limited effects on the standard coding efficiency would be highly beneficial.
In this paper, a method to reduce the complexity of an AV1 encoder based on early termination of inter-prediction is presented. The method is based on machine learning techniques in order to reduce the number of options to test at the encoder side. The rest of the paper is organised as follow. Section 2 briefly presents state-of-the-art encoder speed-up techniques which make use of machine learning. Section 3 provides an overview of AV1 inter-prediction, as well as the motivation of the proposed method. Then, Section 4 presents the proposed binary tree based inter mode decision algorithm. Experimental results and analysis are presented in Section 5. Finally, conclusions are drawn in Section 6.
2 State of the art
Due to the fact that AV1 was recently finalised, limited work is found in the literature related to reducing the complexity of AV1. A paper was presented focusing on predicting the optimal block size of AV1 encoding based on Bayesian inference [6]. In addition, some work was proposed to speed up AV1 encoding in multi-rate configurations, exploiting information obtained in one representation to speed up the encoding of the other representations [7].
AV1 follows a similar architecture to standards developed by the ITU-T VCEG and/or ISO/IEC MPEG, and as such, it is relevant to briefly present some of the speed-up tools based on machine learning that were proposed in this context. Shen et al. [8] proposed an early termination algorithm for transform block size determination in the High Efficiency Video Coding (HEVC) standard, where the Bayesian decision theory was applied to map the variance of the residual coefficients to the block size. In [9] the decision trees generated by data mining tools were utilised to predict the size of HEVC blocks. Furthermore, a method was proposed to limit the number of block sizes to test in HEVC based on exploiting the size of neighbouring blocks [10]. Similarly, a method to select the optimal motion vector precision was proposed [11], based on local features, such as the behaviour of the residual error samples, and global features, such as the amount of edges in the pictures. These methods were proposed in the context of different codecs and may not be applicable to apply directly to the AV1 coding structure.
3 Motivation
In AV1, an inter-predicted block can be encoded with either Single Reference Frame Prediction Mode (SRFPM), in which case one single reference frame with a corresponding motion vector is used for the prediction, or with Compound Reference Frame Prediction Mode (CRFPM), where two reference frames (with two corresponding motion vectors) are used. Up to seven reference frames can be used by either mode, referred to as LAST_FRAME, LAST2_FRAME, LAST3_FRAME, GOLDEN_FRAME, BWDREF_FRAME, ALTREF_FRAME, and ALTREF2_FRAME.
More details on this selection can be found in the literature [4].
Four motion vector candidates are used in SRFPM, which are NEARESTMV, NEARMV, NEWMV and GLOBALMV. NEARESTMV and NEARMV modes employ previously coded motion vectors extracted from spatial neighbours, as shown in Fig. 1. In addition, NEWMV mode performs block based motion estimation to generate a new motion vector for the current block, while GLOBALMV mode performs frame based motion estimation to generate a single motion vector candidate for the whole frame. Eight candidates are used in CRFPM, obtained by combining some of the candidates in SRFPM to perform bi-directional inter-prediction.
In conventional AV1 encoder implementations, the encoder can select among all these different options, the best option for the current block. Typical implementations base these decisions on rate-distortion optimisation techniques, in which the options are compared based on a cost that takes into account the number of bits needed to encode the block, and the corresponding distortion. Clearly, exhaustively searching among all these options can lead to significant encoder complexity.
On the other hand, it is likely that different types of content may benefit from different coding modes. If the encoder could identify which options to use without performing a brute-force search, considerable time savings could be obtained with limited impact on the compression efficiency. Hence, a statistical analysis was performed to analyse which modes are mostly used in specific sequences, as shown in Fig. 2. The figure presents the average occurrence probability of SRFPM, CRFPM and intra-prediction in blocks extracted from inter frames in AV1 software (Mar, 2018 version), where four different QPs (32, 43, 55 and 63) considered. As can be seen, most inter-predicted blocks are encoded mainly with SRFPM. For example, around 91% and 60% of inter-blocks are encoded with SRFPM in sequences RaceHorses and BQTerrace respectively.
Furthermore, the complexity and coding efficiency of using CRFPM were analysed. The AV1 encoder was modified to prevent testing and selection of CRFPM. This modified encoder was compared with an anchor, namely a conventional AV1 encoder that can select CRFPM. The compression performance of the modified encoder with respect to the anchor was measured in terms of the well-known BD-BR metric, a measure of the difference in rate required to encode at the same objective quality with respect to the anchor at different quality points, in percentage [12]. Complexity was measured in terms of the difference in encoding time, calculated as:
$$TS[\%]=(1-\frac{T(C\textsubscript{modified})}{T(C\textsubscript{anchor})})%
\cdot 100,$$
(1)
where $T(C\textsubscript{anchor})$ and $T(C\textsubscript{modified})$ are the total encoding times required by the anchor and the modified encoder, respectively. Results of this test are presented in Table 1.
As can be seen, avoiding testing CRFPM modes reduce complexity by average 61.4% in terms of encoding time. On the other hand, forcing the encoder to simply remove CRFPM modes can have a detrimental effect on encoding performance in some cases, with up to $12.7\%$ efficiency losses in the case of the BQTerrace sequence. In this paper, a method to selectively predict blocks in which CRFPM would be needed is presented using data mining. The proposed method can be considered as a binary-class classification task which is applied to each block to decide whether it should be predicted using SRFPM (Class 0) or using either CRFPM or SRFPM (Class 1).
4 Inter-coding early termination based on decision trees
Decision trees are simple yet an effective tool to learn the relationship between a set of features and the ground truth. A decision tree is a hierarchical structure consisting of a group of decision nodes and terminal leaves, where each node corresponds to a specific test on a single feature, and the terminal leaves provide a classification for the ground truth. The work in this paper made use of a well-known open-source implementation [13] for training the decision trees.
Different from more complex machine learning solutions, the application of trained decision trees has the advantage of being very simple to implement, leading to little additional complexity. This is crucial from the problem at hand of reducing the complexity of an AV1 encoder implementation. Using more complex solutions such as methods based on support vector machines or deep learning may lead to better classification results, but this comes at the cost of high complexity of applying the method itself during encoding. Given the decision needs to be taken for each inter-predicted block during encoding, such complexity would have a detrimental impact on encoding time, compromising the effectiveness of the algorithm.
As with all machine learning techniques, the selection of sequences used for the training is crucial to ensure that the method can generalise well. To this aim, the first 20 frames of a set of well-known video sequences used in the development of MPEG standards [14] was used, as shown Table 2. Motion activity, texture and resolution are different for the selected test sequences.
Also important to the accuracy of a machine learning algorithm is the selection of features to use for the classification. Ideally the features should be highly correlated to the âground truthâ, which in the case under examination is whether a block is encoded using SRFPM or CRFPM, but not very correlated with each other. This minimises the inclusion of unnecessary data and inter-feature correlations. To this end, many features were extracted from each block in the training sequences. Each feature was then classified in terms of its Gini impurity with respect to the ground truth. The Gini impurity is a measurement of the likelihood of making an incorrect classification of a new instance of a random variable, and it is calculated as follows:
$$G=\sum_{i=0}^{N-1}[{{{P(i)}\cdot{(1-P(i))}}]},$$
(2)
where $P(i)$ is the probability of block i being encoded with SRFPM, and i indicates blocks in the data training set.
Following this process, four features were selected, which aims at 80% accuracy of the prediction in Class 0. Table 3 shows the selected features. Two features are taken from each of the adjacent encoded blocks, on the left and on the top of the current block. These features are denoted as ’f1’ and ’f2’ for the left block, and ’f3’ and ’f4’ for the top block. They have been selected because there is a high correlation in coded information between the current block and its neighbours.
Given that the sequences in the training set are of different length and different resolutions, using all blocks in each sequence would lead to an unbalanced number of training samples from each sequence. Sequences at high resolutions may therefore have a higher impact on the training, which is not ideal to ensure the training can generalise well. Therefore, a fixed number of training samples is used per sequence. Moreover, in order to balance the training samples between the two classes of the ground truth, the number of training samples M for each sequence is calculated as
$$\displaystyle M=\left\{\begin{array}[]{@{}cc}p\textsubscript{0}\cdot N,&p%
\textsubscript{0}<0.5\\
(1-p\textsubscript{0})\cdot N,&p\textsubscript{0}\geq 0.5,\end{array}\right.$$
where $p\textsubscript{0}$ represents the accuracy of SRFPM prediction, and $N$ corresponds to the number of training samples in each class. Hence, the data set is balanced with 50% of blocks classified as being predicted using SRFPM, and 50% using CRFPM.
The flowchart of the proposed method is illustrated in Fig. 3. A binary classifier (Classifier A) is firstly used to make a decision per block. If the block is classified as Class 0, which corresponds to the block being predicted using SRFPM, then CRFPM modes are not tested for the current block. Conversely, if the block is classified in Class 1, both SRFPM and CRFPM modes are tested as in conventional AV1 encoders.
5 Experimental Results
The method was tested to evaluate its performance using test sequences that are not part of the training used to develop the method. In order to validate the performance of the proposed method, an encoder was developed using the reference AV1 software (Sep 5, 2018 version) encoder as a basis [15]. The unmodified reference software encoder was also used as anchor for measuring performance. Seven different video sequences [16] were used to evaluate the performance. Each sequence was encoded at four different quality points (obtained using –cq-level=32, 43, 55, 63), to validate the method under different conditions. The performance of the proposed method was measured in terms of BD-BR and encoder time savings. Table 4 shows that the proposed method reduces encoding time on average by 43.4%, with a maximum time saving of 53.0% and a minimum of 33.7%. As can be seen, the method achieves considerably better coding efficiency than the modified encoder which skips testing of CRFPM modes altogether.
The AV1 reference software allows encoding to be performed using a variety of so called ”Speed presets”, namely encoder configurations which limit certain options and tools in order to reduce the complexity. When doing so, the coding efficiency decreases due to the fact the codec is limited in the number of options it can select. On the other hand, the proposed method selects whether to test or not the CRFPM modes on a block-by-block basis based on features of the block, and as such it has a limited impact on coding efficiency. Moreover, the tool can be used on top of existing AV1 Speed presets and still provide additional speed-ups, showing that the method does not overlap with existing complexity reduction schemes. In order to validate these claims, the method was compared with AV1 Speed preset $2$, and it was also tested on top of an AV1 encoder using Speed preset $2$.
Results of these tests are presented in Table 5. In all cases, the unmodified AV1 encoder was used as anchor. As can be seen, using AV1 Speed preset $2$ on its own usually provides lower complexity reductions for higher efficiency losses than using the proposed method (as in Table 4), showing that the Speed preset is less capable of adapting to content-dependent features. Moreover, using the proposed method on top of Speed preset $2$ can still provide considerable complexity reductions, showing that the method is almost orthogonal to the Speed preset. Average $58\%$ and up to $64\%$ complexity reduction can be obtained under these conditions.
6 Conclusions
This paper presented a fast algorithm for AV1 inter prediction based on decision trees, based on the observation that not all sequences benefit from using CRFPM modes. A decision tree was trained based on $7$ features extracted while encoding each block. A decision is performed whether to skip testing of CRFPM modes, or whether to instead test all modes exhaustively. Experimental results show that the encoding time can be reduced on average by 43.4%, with negligible impact on coding efficiency. Future work could focus on using more features and using different classifier to further speed up AV1 encoder implementations.
References
[1]
“Aom - alliance for open media,” http://aomedia.org/.
[2]
D. Mukherjee, J. Bankoski, A. Grange, J. Han, J. Koleszar, P. Wilkins, Y. Xu,
and R.S. Bultje,
“The latest open-source video codec vp9 - an overview and
preliminary results,”
in Picture Coding Symposium (PCS), 2013.
[3]
T. Laude, Y.G. Adhisantoso, J .Voges, M. Munderloh, and J. Ostermann,
“A comparison of JEM and AV1 with HEVC: Coding Tools, Coding
Efficiency and Complexity,”
in Picture Coding Symposium (PCS), June 2018.
[4]
Y. Chen et al.,
“An Overview of Core Coding Tools in the AV1 Video Codec,”
in Picture Coding Symposium (PCS), June 2018.
[5]
A. S. Dias, S. Blasi, F. Rivera, E. Izquierdo, and M. Mrak,
“An overview of recent video coding developments in MPEG and
AOMedia,”
in International Broadcasting Convention (IBC), September 2018.
[6]
B. Guo, Y. Han, and J. Wen,
“Fast block structure determination in av1-based multiple
resolutions video encoding,”
in 2018 IEEE International Conference on Multimedia and Expo
(ICME), July 2018, pp. 1–6.
[7]
B. Guo, X. Chen, J. Gu, Y. Han, and J. Wen,
“A bayesian approach to block structure inference in av1-based
multi-rate video encoding,”
in 2018 Data Compression Conference, March 2018, pp. 383–392.
[8]
L. Shen, Z. Zhang, X. Zhang, P. An, and Z. Liu,
“Fast tu size decision algorithm for hevc encoders using bayesian
theorem detection,”
in Signal Process. Image Commun., March 2015, vol. 32, pp.
121–128.
[9]
G. Correa, P. A. Assuncao, L. V. Agostini, and L. A. da Silva Cruz,
“Fast HEVC encoding decisions using data mining,”
in IEEE Trans. Circuits Syst. Video Technol., Apr. 2015,
vol. 25, pp. 660–673.
[10]
I. Zupancic, S. G. Blasi, E. Peixoto, and E. Izquierdo,
“Inter-prediction optimizations for video coding using adaptive
coding unit visiting order,”
IEEE Transactions on Multimedia, vol. 18, no. 9, pp.
1677–1690, Sept 2016.
[11]
S. G. Blasi, I. Zupancic, E. Izquierdo, and E. Peixoto,
“Adaptive precision motion estimation for hevc coding,”
in 2015 Picture Coding Symposium (PCS), May 2015, pp. 144–148.
[12]
G. Bjontegaard,
“Calculation of average PSNR differences between RD-Curves,”
in ITU-T SG16 Q.6 Document, VCEG-M33, April 2001.
[13]
“Decision tree classification tool for video coding optimisation problems,”
https://github.com/bbc/cu_split.
[14]
F. Bossen,
“Common test conditions and software reference configurations,”
in JCTVC-L1100, October 2012.
[15]
“Av1 source code in the alliance for open media git repository,”
https://aomedia.googlesource.com/aom/.
[16]
“objective-1-fast test set in awcy,”
https://people.xiph.org/%7etdaede/sets/objective-1-fast/. |
Characterization and Modeling of Silicon-on-Insulator Lateral Bipolar Junction Transistors at Liquid Helium Temperature
Yuanke Zhang
Yuefeng Chen
Yifang Zhang
Jun Xu
Chao Luo
and Guoping Guo
This work was supported by the National Natural Science Foundation of China (No. 12034018), Innovation Program for Quantum Science and Technology (No. 2021ZD0302300). (Yuanke Zhang and Yuefeng Chen contributed equally to this work.) (Corresponding author: Chao Luo, e-mail: lc0121@ustc.edu.cn)The authors are with University of Science and Technology of China (USTC), Hefei 230026, Anhui, China, and also with CAS Key Lab of Quantum Information, Hefei 230026, Anhui, China.
Abstract
Conventional silicon bipolars are not suitable for low-temperature operation due to the deterioration of current gain ($\beta$). In this paper, we characterize lateral bipolar junction transistors (LBJTs) fabricated on silicon-on-insulator (SOI) wafers down to liquid helium temperature (4 K). The positive SOI substrate bias could greatly increase the collector current and have a negligible effect on the base current, which significantly alleviates $\beta$ degradation at low temperatures. We present a physical-based compact LBJT model for 4 K simulation, in which the collector current ($\textit{I}_{\textbf{C}}$) consists of the tunneling current and the additional current component near the buried oxide (BOX)/silicon interface caused by the substrate modulation effect. This model is able to fit the Gummel characteristics of LBJTs very well and has promising applications in amplifier circuits simulation for silicon-based qubits signals.
{IEEEkeywords}
Cryogenic, lateral bipolar junction transistors, silicon-on-insulator, characterization, modeling, tunneling, substract modulation
1 Introduction
\IEEEPARstart
Cryogenic electronics has a promising application for deep aerospace exploration, neutrino physics experiments, infrared focal plane array surfaces, etc., and has been studied to design and implement the manipulation and readout circuits of quantum bits (qubits) in recent years[1, 2, 3, 4, 6, 7, 8, 9, 10, 5]. Bipolar junction transistors (BJTs) with high current gain ($\beta$) have been widely used as low-noise local signal amplifiers and can be a potential candidate for spin readout devices of semiconductor qubits[11, 12, 13, 14, 15]. Therefore, heterojunction bipolar transistors are widely studied due to their useful amplification performance in a wide temperature range even down to millikelvin[11, 12, 13]. However, large-scale quantum computing requires the integration of a large number of qubits and circuits on a single chip. In order to be compatible with the fabrication process of silicon-based qubits, silicon homojunction BJTs remain the most promising candidate. Unfortunately, due to the carrier freeze-out in the base region and the narrowing of the bandgap associated with the emitter, $\beta$ degrades severely with decreasing temperature in conventional silicon bipolars[16, 17, 18, 19].
To overcome this problem, fabricating the homojunction BJT laterally on a silicon-on-insulator substrate provides a promising solution[15, 20, 21, 22]. With a voltage applied to the SOI substrate, an additional current component can be generated near the buried oxide (BOX)/silicon interface. In n-p-n type symmetric LBJTs, a positive SOI substrate voltage ($V_{\rm BOX}$) could significantly increase the collector current ($I_{\rm C}$) with almost no change in the base current ($I_{\rm B}$), and thus increase the current gain. Moreover, previous studies have shown that the modulation effect of $V_{\rm BOX}$ remains effective at low temperatures and the signal-to-noise ratio (SNR) gain can also be ameliorated by adjusting $V_{\rm BOX}$[15], which demonstrates the potential application of LBJTs in amplifying weak electronic signals generated at cryogenic temperatures. In order to design cryogenic circuits based on LBJTs, an accurate compact simulation model is necessary. However, compact modeling of LBJTs at low temperatures remains unexplored.
In this article, the low-temperature characteristics of LBJTs fabricated on SOI wafers ranging from 300 K to 4 K are presented. For the first time, a physical-based LBJT compact model is proposed for 4 K simulation. The collector current is consist of the tunneling current and the additional drift-diffusion current component caused by the positive SOI substrate bias. The model calculation results show very good agreement with the measurement data of LBJTs, especially the modulation effect of $V_{\rm BOX}$.
This article is organized as follows. In Section II, we provide a description of the device structure and the cryogenic measurement setup. Section III describes characterization of the devices from 300 K to 4 K and discusses the cryogenic behaviors. In Section IV, we present a physics-based LBJT model for 4 K simulation, and finally, we conclude this article in Section V.
2 Experimental Details
The schematic of an n-p-n type LBJT fabricated on SOI wafers is shown in Fig. 1(a). Two different sizes of LBJTs are tested in this paper: emitter length ($L_{\rm E}$)-base width ($W_{\rm B}$)-emitter wing widths ($W_{\rm E}$)-collector wing widths ($W_{\rm C}$) = 10-0.1-0.2-0.2 and 5-0.1-0.15-0.2 $\mu$m. More detailed fabrication information can be referred to elsewhere[21]. The measurement setup is shown in Fig. 1(b)-(c). The diced sample chips are bonded to the chip carriers using aluminum (Al) wires [Fig. 1(b)] and the electrical characteristic measurement is performed by a Keysight B1500A semiconductor analyzer. The low-temperature environment is provided by liquid nitrogen (77 K)/helium (4 K) dewar [Fig. 1(c)]. A dip-stick with a rhodium-iron resistance thermometer is placed at different heights inside the dewar to reach temperatures between 300 K and 77 K/4 K and it is pre-placed for 15 minutes at each temperature to ensure measurement environment stability. Differential $\beta$ = d$I_{\rm C}$/d$I_{\rm B}$ is used in this paper.
3 Characterization
The Gummel characteristics of the LBJTs measured under $V_{\rm BOX}$ = 0 V and $V_{\rm BOX}$ = 12 V at various temperatures are shown in Fig. 2(a) and (b), respectively. Throughout the article, both $I_{\rm C}$ and $I_{\rm B}$ are normalized by emitter length ($L_{\rm E}$). The slope of $I_{\rm C}$ increases with decreasing temperature due to a $kT/q$ dependence. It should be noted that $I_{\rm C}$-$V_{\rm BE}$ curves measured at 20 K and 4 K essentially overlap under $V_{\rm BOX}$ = 0 V [Fig. 2(a)] and the overlap disappears with a positive $V_{\rm BOX}$ = 12 V [Fig. 2(b)]. This phenomenon can be attributed to two different current transport mechanisms of LBJTs: the E-C tunneling current inside the LBJT and the drift-diffusion current near the BOX/silicon interface. At low temperatures, the potential barrier in the base region prevents the injection of electrons, and $I_{\rm C}$ is mainly composed of the E-C tunneling current[11, 15, 24]. Due to the saturation of electron temperature, $I_{\rm C}$ is almost independent of temperature when $T$$\leq$17 K[24], and thus the overlapping phenomenon occurs. Differently, an additional depletion region near the BOX/silicon interface is generated under a positive $V_{\rm BOX}$ and thus the drift-diffusion current between the collector and emitter is enhanced. As the drift-diffusion current is temperature dependent, the overlap disappears under $V_{\rm BOX}$ = 12 V, as shown in Fig. 2(b).
$I_{\rm C}$ and $I_{\rm B}$ versus $V_{\rm BE}$ of LBJTs with two different sizes under $V_{\rm BOX}$ = 0$\sim$12 V are shown in Fig. 3. Due to the additional drift-diffusion current regulated by the $V_{\rm BOX}$, $I_{\rm C}$ is significantly enhanced with increasing $V_{\rm BOX}$ under medium $V_{\rm BE}$ values. With further increase of $V_{\rm BE}$ ($V_{\rm BE}$$\textgreater$0.8 V at 300 K and $V_{\rm BE}$$\textgreater$1.1 V at 4 K), the current transport inside the LBJTs (i.e. the traditional BJT transport at 300 K and tunneling at 4 K) plays a dominant role and the influence of $V_{\rm BOX}$ is not noticeable anymore. Moreover, due to the shorter poly-Si lines and lower base resistance, the LBJT with smaller $L_{\rm E}$ delivers a higher $I_{\rm C}$[21]. Surprisingly, $I_{\rm B}$ is negligibly affected by $V_{\rm BOX}$. The transport of $I_{\rm B}$ is mainly concentrated on the upper surface of LBJTs, hence $V_{\rm BOX}$ can hardly affect the injection barrier of holes from the base to the emitter. The increased $I_{\rm C}$ and the unaffected $I_{\rm B}$ under the modulation effect of $V_{\rm BOX}$ imply an improvement in $\beta$, as shown in Fig. 4(a) and (b). Under $V_{\rm BOX}$ = 12 V, $\beta$ (at $I_{\rm B}$ = 1 nA/$\mu$m) is improved by $\sim$10 times and $\sim$10${}^{3}$ times at 300 K and 4 K, respectively.
In addition, $\beta$ versus $I_{\rm B}$ at different temperatures under $V_{\rm BOX}$ = 0 V and 12 V are shown in Fig. 4(c)-(d). As expected, $\beta$ deteriorates with decreasing temperature and is significantly improved by the positive SOI substrate bias. The curves at 4 K and 20 K are very similar in Fig. 4(c), which can be attributed to the overlap of $I_{\rm C}$ discussed above. Due to the reduction of the diffusion coefficient ($D_{\rm B}$), $\beta$ under high injection conditions ($I_{\rm B}$$\textgreater$10${}^{-6}$ A/$\mu$m) reduces with decreasing temperature at each temperature, as shown in Fig. 4(c) and (d).
4 Compact Modeling
As $I_{\rm B}$ is composed of diode currents from the base-emitter (B-E) and base-collector (B-C) junctions, its mechanism remains consistent from room temperature to low temperatures. The Gummel-Poon (GP) bipolar model [25] is used to describe $I_{\rm B}$ characteristics of LBJT in this work. With $V_{\rm BE}$ ranging from 0 to 1.5 V and $V_{\rm CE}$ remaining 1 V, the B-C junction is always in reverse bias or weak bias. Therefore, the current of the B-C junction is negligible and $I_{\rm B}$ can be written in the form[26]
$$I_{\rm B}=A_{\rm E}\frac{qn_{i}^{2}}{G_{\rm E}}\exp\left(\frac{qV_{\rm BE}}{kT}\right)$$
(1)
where $A_{\rm E}$ is the area of the emitter-base junction, $n_{i}$ is the intrinsic carrier density, and $G_{\rm E}$ is the emitter Gummel number, which is inversely proportional to the diffusion coefficient $D_{\rm B}$. However, $D_{\rm B}$ and $n_{i}$ reduce dramatically with decreasing temperature. At 4 K, $n_{i}\approx$ 10${}^{-678}$ cm${}^{-3}$, which lies outside the range of IEEE double-precision arithmetic (10${}^{-308}$$\sim$10${}^{308}$)[27], thus resulting in the parameter $I_{\rm SE}$ (B-E leakage saturation current) in the GP model being too small for computers to calculate. Therefore, we modify Eq. (1) as a summation of the diffusion current and the recombination current
$$\begin{split}I_{\rm{B}}&=\frac{I_{\rm{S}}}{B_{\rm{f}}}\{[\exp(S_{\rm diff}(V_{\rm BE}-V_{\rm diff}))-1]\\
&+I_{\rm{SE}}[\exp(S_{\rm RE}(V_{\rm BE}-V_{\rm RE}))-1]\}\cdot f_{\rm fermi}\end{split}$$
(2)
where $S_{\rm diff}$ and $S_{\rm RE}$ are slope parameters of $I_{\rm B}$-$V_{\rm BE}$ in semi-logarithmic scale. $V_{\rm diff}$ and $V_{\rm RE}$ are the voltages corresponding to diffusion and recombination conductance exceeding the minimum conductance across each nonlinear device (GMIN) in SPICE[28]. $I_{\rm S}$, $I_{\rm SE}$, and $B_{\rm f}$ represent the modified saturation current coefficient, the B-E leakage saturation current coefficient, and the ideal forward maximum gain, respectively. $f_{\rm fermi}$ is used to guarantee a zero current at a zero $V_{\rm BE}$[30, 31]. Moreover, the base parasitic resistance $R_{\rm B}$ is also taken into account to precisely calculate $I_{\rm B}$ in the large injection region and the Newton-Raphson iteration[29] is used for solving the current and voltage of the intrinsic base.
As we discussed in Sec. III, $I_{\rm C}$ is mainly composed of the E-C tunneling current at low temperatures. Assuming that the potential barrier in the base region is parabolic in shape, the tunneling current $I_{\rm T\_tunl}$ is given by[32]
$$I_{\rm T\_tunl}=A_{1}\sqrt{v_{b}}\left[\frac{\exp\left(a_{1}v_{e}/\sqrt{v_{b}}\right)-1}{a_{1}v_{e}/\sqrt{v_{b}}}-1\right]\exp\left(-a_{1}\sqrt{v_{b}}\right)$$
(3)
when $v_{b}\geq v_{e}$. And for $v_{b}<v_{e}$ condition, $I_{\rm T\_tunl}$ is given by
$$\begin{split}I_{\rm T\_tunl}&=A_{1}\sqrt{v_{b}}\left[\left[\exp\left(a_{1}\sqrt{v_{b}}\right)-1\right]\left(1-\frac{v_{b}}{v_{e}}\right)\right.\\
&\left.+\frac{\exp\left(a_{1}\sqrt{v_{b}}\right)-1}{a_{1}v_{e}/\sqrt{v_{b}}}-\frac{v_{b}}{v_{e}}\right]\exp\left(-a_{1}\sqrt{v_{b}}\right)\end{split}$$
(4)
where $v_{b}=1-V_{\rm BE}/V_{\rm DEi}$ and $v_{e}=\Delta W_{\rm E}/qV_{\rm DEi}$. $A_{1}$, $a_{1}$, and $V_{\rm DEi}$ present the current density prefactor, exponent factor, and the built-in voltage of the internal BE junction, respectively. $\Delta W_{\rm E}$ is the parameter related to the height of the potential barrier. It is worth noting that the Fermi distribution function $f(\rm E)$ at $T$ = 0 K is used in the solution of $I_{\rm T\_tunl}$. At 0 K, $f(\rm E)$ is a step function, thus such concise Eq. (3) and (4) are obtained. Although the difference in $f(\rm E)$ may lead to some deviations, it is acceptable at the target application temperature of our model, i.e., 20-100 mK (integration with qubits) or 1-4 K (integration with qubits controller).
As $V_{\rm BE}$ increases further, the emitter carrier energy will approach or even exceed the potential barrier height. The tunneling current $I_{\rm T\_tunl}$ will tend to level off or even decrease, and the hot carrier transmission current ($I_{\rm T\_hc}$) plays a dominant role, which is given by
$$I_{\rm T\_hc}=A_{1}a_{1}\frac{v_{e}}{2}\left(1-\frac{v_{b}}{v_{e}}\right)^{2}$$
(5)
where the parameter definitions are the same as Eq. (3) and (4). Fig. 5(a) shows the contribution of $I_{\rm T\_tunl}$ and $I_{\rm T\_hc}$ in our model, in which $A_{1}$ = 8$\times$10${}^{-5}$ A/$\mu$m, $a_{1}$ = 300, $V_{\rm DEi}$ = 1.7 V, and $\Delta W_{\rm E}$ = 0.601 eV. In addition, in order to accurately describe the $I_{\rm C}$ behavior at low $V_{\rm BE}$ values, the trap-assisted tunneling current $I_{\rm T\_ta}$[33, 34] is also taken into account in this model and simply described by an exponential function [see $I_{\rm T\_ta}$ in Fig.5 (b)]. When there is a limited distribution of traps in the bandgap of the base, carriers can tunnel from the emitter to the collector with the assistance of the traps. Therefore, the total tunneling current can be given by $I_{\rm T}$=$I_{\rm T\_tunl}$+$I_{\rm T\_hc}$+$I_{\rm T\_ta}$ and a qualitative illustration of the tunneling mechanisms under different $V_{\rm BE}$ values is shown in Fig. 5(c).
When a positive $V_{\rm BOX}$ is applied, the base region near the BOX is partially depleted and an additional drift-diffusion current is generated near the BOX/silicon interface, as qualitatively illustrated in Fig. 5(d). In this case, a symmetrical LBJT can be viewed as an upside-down MOSFET. The emitter and collector correspond to the source (S) and drain (D), BOX corresponds to the gate (G) oxide, and the base corresponds to the silicon substrate (sub). Differently, the gate voltage ($V_{\rm G}$) in MOSFETs is applied to a poly-Si gate rather than to the bulk Si in LBJTs. To modify this deviation, we introduce an effective gate voltage $V_{\rm G\_eff}=a_{2}V_{\rm BOX}$ in the model, and applying $V_{\rm BE}$ is equivalent to the modulation of $V_{\rm sub}$ in MOSFETs. To calculate this MOSEFT-like drift-diffusion current $I_{\rm MOS}$, the overdrive voltage $V_{\rm ov}$ = $V_{\rm G}-V_{\rm TH}$ in commercial MOSFET model [35] is replaced by
$$V_{\rm ov}=V_{\rm G\_eff}-V_{\rm TH}-\gamma\left(\sqrt{2\phi_{B}-V_{BE}}-\sqrt{2\phi_{B}}\right)$$
(6)
where $V_{\rm TH}$ the threshold voltage, $\phi_{B}$ is the bulk Fermi potential, and $\gamma$ is the body effect parameter in MOSFETs. The effect of the emitter parasitic resistance $R_{\rm E}$ is also calculated by the Newton-Raphson iterative method[29]. Therefore, the total $I_{\rm C}$ can be given by
$$I_{\rm C}=I_{\rm T}+I_{\rm MOS}$$
(7)
and take the $I_{\rm C}$-$V_{\rm BE}$ characteristics of the LBJT with $L_{\rm E}$-$W_{\rm B}$-$W_{\rm E}$-$W_{\rm C}$ = 5-0.1-0.15-0.2 $\mu$m for example, the contribution of each current component in $I_{\rm C}$ is shown in Fig. 5(b). The parameter-fitting results of the proposed compact model for $L_{\rm E}$-$W_{\rm B}$-$W_{\rm E}$-$W_{\rm C}$ = 10-0.1-0.2-0.2 and 5-0.1-0.15-0.2 $\mu$m LBJTs at 4 K are shown in Fig. 6(a) and (b), respectively. Good matching of the measurement and calculation results is obtained in both devices and the proposed model is ready to use for LBJT-contained cryogenic circuit design.
5 Conclusion
In this article, we present the characterization and modeling of LBJTs fabricated on SOI wafers at liquid helium temperature. At low temperatures, $I_{\rm C}$ is mainly composed of the E-C tunneling current and the MOSEFT-like drift-diffusion current generated by positive SOI substrate bias. Based on the modeling of the two current components above, a physical-based LBJT compact model is proposed for 4 K simulation and it shows good fitting results with the measurement data. The proposed model can be used to design and simulate the LBJT-contained cryogenic circuits for local quantum signal amplification.
Acknowledgement
The device fabrication was done by Prof. Zhen Zhang’s group in the Ångström Microstructure Laboratory (MSL) at Uppsala University. Dr. Qitao Hu, Dr. Si Chen, Prof. Zhen Zhang are acknowledged for the device design and fabrication, and the technical staff of MSL are acknowledged for their process support.
References
[1]
A. Ruffino, T.-Y Yang, J. Michniewicz, Y. Peng, E. Charbon, and M. F. Gonzalez-Zalba, “A cryo-CMOS chip that integrates silicon quantum dots and multiplexed dispersive readout electronics”, Nat. Electron., vol. 5, Jan. 2022, pp. 53-59, doi: 10.1038/s41928-021-00687-6.
[2]
E. Charbon, F. Sebastiano, A. Vladimirescu, H. Homulle, S. Visser, L. Song, and R. M. Incande, “Cryo-CMOS for quantum computing,” in IEDM Tech. Dig., Dec. 2016, pp. 13–15, doi: 10.1109/IEDM.2016.7838410.
[3]
M. H. Devoret and R. J. Schoelkopf, “Amplifying quantum signals with the single-electron transistor,” Nature, vol. 406, no. 6799, pp. 1039–1046, Aug. 2000, doi:10.1038/35023253.
[4]
F. A. Zwanenburg, A. S. Dzurak, A. Morello, M. Y. Simmons, L. C. L. Hollenberg, G. Klimeck, S. Rogge, S. N. Coppersmith, and M. A. Eriksson, “Silicon quantum electronics,” Rev. Modern Phys., vol. 85, no. 3, pp. 961–1019, Jul. 2013, doi: 10.1103/RevModPhys.85.961.
[5]
E. Gutiérrez-D, J. Deen, and C. Claeys, Low Temperature Electronics: Physics, Devices, Circuits, and Applications. San Diego, CA, USA: Academic, Oct. 2001, doi: 10.1016/B978-0-12-310675-9.X5000-2.
[6]
B. Patra, R. M. Incandela, J. P. G. van Dijk, H. A. R. Homulle, L. Song, M. Shahmohammadi, R. B. Staszewski, A. Vladimirescu, M. Babaie, F. Sebastiano, and E. Charbon, “Cryo-CMOS circuits and systems for quantum computing applications,” IEEE J. Solid-State Circuits, vol. 53, no. 1, pp. 309–321, Jan. 2018, doi: 2018. 10.1109/JSSC.2017.2737549.
[7]
J. R. Hoff, G. W. Deptuch, Guoying Wu, and Ping Gui, “Cryogenic Lifetime Studies of 130 nm and 65 nm nMOS Transistors for High-Energy Physics Experiments,” IEEE Trans. Nucl. Sci., vol. 62, no. 3, pp. 1255–1261, 2015, doi: 10.1109/TNS.2015.2433793.
[8]
B. Patra, “CMOS circuits and systems for cryogenic control of silicon quantum processors,” 2021. [Online]. Available: https://doi.org/10.4233/uuid:cea59727-fda2-41e1-ba87-9404ef22202d.
[9]
Y. Zhang et al., “Characterization and modeling of native MOSFETs down to 4.2 K,” IEEE Trans. Electron Devices, vol. 68, no. 9, pp. 4267–4273, Sep. 2021, doi: 10.1109/TED.2021.3099775.
[10]
Y. Liu, L. Lang, Y. Chang, Y. Shan, X. Chen and Y. Dong, “Cryogenic Characteristics of Multinanoscales Field-Effect Transistors,” IEEE Trans. Electron Devices, vol. 68, no. 2, pp. 456-463, 2021, doi: 10.1109/TED.2020.3041438.
[11]
H. Ying, B. R. Wier, J. Dark, N. E. Lourenco, L. Ge, A. P. Omprakash, M. Mourigal, D. Davidovic, and J. D. Cressler, “Operation of SiGe HBTs down to 70 mK,” IEEE Electron Device Lett., vol. 38, no. 1, pp. 12–15, Jan. 2017, doi: 10.1109/LED.2016.2633465.
[12]
D. Davidovic, H. Ying, J. Dark, B. R. Wier, L. Ge, N. E. Lourenco, A. P. Omprakash, M. Mourigal, and J. D. Cressler, “Tunneling, current gain, and transconductance in silicon-germanium heterojunction bipolar transistors operating at millikelvin temperatures,”Phys. Rev. Appl., vol. 8, no. 2, Aug. 2017, Art. no. 024015,
doi: 10.1103/physrevapplied.8.024015.
[13]
M. J. Curry, T. D. England, N. C. Bishop, G. Ten-Eyck, J. R. Wendt, T. Pluym, M. P. Lilly, S. M. Carr, and M. S. Carroll, “Cryogenic preamplification of a single-electron-transistor using a silicon-germanium heterojunction-bipolar-transistor,” Appl. Phys. Lett., vol. 106, no. 20, May 2015, Art. no. 203505, doi: 10.1063/1.4921308.
[14]
I. T. Vink, T. Nooitgedagt, R. N. Schouten, L. M. K. Vandersypen, and W. Wegscheider, “Cryogenic amplifier for fast real-time detection of single-electron tunneling,” Appl. Phys. Lett., vol. 91, no. 12, Sep. 2007, Art. no. 123512, doi: 10.1063/1.2783265.
[15]
S. Chen, C. Luo, Y. Zhang, J. Xu, Q. Hu, Z. Zhang, and G. Guo, “Current gain enhancement for silicon-on-insulator lateral bipolar junction transistors operating at liquid-helium temperature,” IEEE Electron Device Lett., vol. 41, no. 6, pp. 800–803, June 2020, doi: 10.1109/LED.2020.2985674.
[16]
J. D. Cressler, J. H. Comfort, E. F. Crabbe, G. L. Patton, J. M. C. Stork, J. Y.-C. Sun, and B. S. Meyerson, “On the profile design and optimization of epitaxial Si- and SiGe-base bipolar technology for 77 k applications. I. transistor DC design considerations,” IEEE Trans. Electron Devices, vol. 40, no. 3, pp. 525–541, Mar. 1993, doi: 10.1109/16.199358.
[17]
H. Homulle, L. Song, E. Charbon, and F. Sebastiano, “The cryogenic temperature behavior of bipolar, MOS, and DTMOS transistors in standard CMOS,”IEEE J. Electron Devices Soc., vol. 6, pp. 263–270, 2018, doi: 10.1109/JEDS.2018.2798281.
[18]
L. Song, H. Homulle, E. Charbon and F. Sebastiano, “Characterization of bipolar transistors for cryogenic temperature sensors in standard CMOS,” in 2016 IEEE SENSORS, 2016, pp. 1-3, doi: 10.1109/ICSENS.2016.7808759.
[19]
H. Homulle, “Cryogenic Electronics for the Read-Out of Quantum Processors,” PhD thesis, Technical Univ. Delf, 2019, doi:10.4233/uuid:e833f394-c8b1-46e2-86b8-da0c71559538.
[20]
Q. Hu, S. Chen, S.-L. Zhang, P. Solomon, and Z. Zhang, “Effects of substrate bias on low-frequency noise in lateral bipolar transistors fabricated on Silicon-on-Insulator substrate,” IEEE Electron Device Lett., vol. 41, no. 1, pp. 4–7, Jan. 2020, doi: 10.1109/LED.2019.2953362.
[21]
Q. Hu, X. Chen, H. Norstrom, S. Zeng, Y. Liu, F. Gustavsson, S.-L. Zhang, S. Chen, and Z. Zhang, “Current gain and low-frequency noise of symmetric lateral bipolar junction transistors on SOI,” inProc. 48th Eur. Solid-State Device Res. Conf. (ESSDERC), Sep. 2018, pp. 258–261, doi: 10.1109/ESSDERC.2018.8486918.
[22]
J.-B Yau, J. Cai, and T. H. Ning, “Substrate-Voltage Modulation of Currents in
Symmetric SOI Lateral Bipolar Transistors,” IEEE Trans. Electron Devices, vol. 63, no. 5, pp. 1835–1839, May 2016, doi: 10.1109/TED.2016.2543528.
[23]
R. S. Müller and T. I. Kamins, Device Electronics for Integrated Circuits, 3rd ed. New York, NY, USA: Wiley, 2003.
[24]
H. Rucker, J. Korn, and J. Schmidt, “Operation of sige HBTs at cryogenic temperatures,” inProc. IEEE Bipolar/BiCMOS Circuits Technol. Meeting (BCTM), Oct. 2017, pp. 17–20, doi: 10.1109/BCTM.2017.8112902.
[25]
F. Sischka, “GUMMEL-POON BIPOLAR MODEL,” Agilent Technologies, Munich, 1990). [Online]. Available: http://www.idea2ic.com/PlayWithSpice/pdf/G%20U%20M%20M%20E% 20L%20-%20P%20O%20O%20N.pdf
[26]
T. Yuan and T. H. Ning, Fundamentals of modern VLSI devices, Cambridge university press, 2021.
[27]
A. Beckers, F. Jazaeri, and C. Enz, “Cryogenic MOS transistor model,” IEEE Trans. Electron Devices, vol. 65, no. 9, pp. 3617-3625, Sept. 2018, doi: 10.1109/TED.2018.2854701.
[28]
“Spectre Circuit Simulator Reference, Product Version 19.1” Cadence Design Systems, Inc., Jan. 2020.
[29]
W. J. McCalla, “Fundamentals of computer-aided circuit simulation,” Springer Science $\&$ Business Media, 1987, doi:10.1007/978-1-4613-2011-1.
[30]
Y.-K Lin, J. P. Duarte, P. Kushwaha, H. Agarwal, H.-L Chang, A. Sachid, S. Salahuddin, and C. Hu, “Compact Modeling Source-to-Drain Tunneling in Sub-10-nm GAA FinFET With Industry Standard Model,” IEEE Trans. Electron Devices, vol. 64, no. 9, pp. 3576-3581, Sept. 2017, doi: 10.1109/TED.2017.2731162.
[31]
L. Zhang and M. Chan, “SPICE modeling of double-gate tunnel-FETs including channel transports,” IEEE Trans. Electron Devices, vol. 61, no. 2, pp. 300–307, Feb. 2014, doi:10.1109/TED.2013.2295237.
[32]
M. Schröter and X. Jin, “A Physics-Based Analytical Formulation for the Tunneling Current Through the Base of Bipolar Transistors Operating at Cryogenic Temperatures,” IEEE Trans. Electron Devices, vol. 70, no. 1, pp. 247-253, Jan. 2023, doi:10.1109/TED.2022.3223885.
[33]
A. J. Joseph, J. D. Cressler and D. M. Richey, “Operation of SiGe heterojunction bipolar transistors in the liquid-helium temperature regime,” IEEE Electron Device Lett., vol. 16, no. 6, pp. 268-270, June 1995, doi: 10.1109/55.790731.
[34]
Y. Hanbin, “Collector current transport mechanisms in SiGe HBTs operating at cryogenic temperatures,” PhD thesis, Georgia Institute of Technology, 2019, [Online]. Available: https://smartech.gatech.edu/handle/1853/61292.
[35]
M. Bucher, C. Lallement, C. Enz, F. Théodoloz, and F. Krummenacher, “The EPFL-EKV MOSFET model equations for simulation model version 2.6,” Dept. Electron. Lab, Swiss Federal Inst. Technol., Lausanne, Switzerland, Jun. 1997. [Online]. Available: https://www.epfl.ch/labs/iclab/wp-content/uploads/2019/02/ekv$\_$v262.pdf. |
Trapping of strangelets in the geomagnetic field
L. Paulucci
paulucci@fma.if.usp.br
Instituto de Física - Universidade de São Paulo
Rua do Matão, Travessa R, 187, 05508-090, Cidade Universitária
São Paulo SP, Brazil
J. E. Horvath
Instituto de Astronomia, Geofísica e Ciências
Atmosféricas - Universidade de São Paulo
Rua do Matão, 1226, 05508-900, Cidade Universitária
São Paulo SP, Brazil
G. A. Medina-Tanco
Instituto de Ciencias Nucleares, Universidad Nacional
Autónoma de México
A.P. 70-543, C.U. México D.F., México
(November 21, 2020)
Abstract
Strangelets coming from the interstellar medium (ISM) are an
interesting target to experiments searching for evidence of this
hypothetic state of hadronic matter. We entertain the possibility
of a trapped strangelet population, quite analogous to
ordinary nuclei and electron belts. For a population of
strangelets to be trapped by the geomagnetic field, these incoming
particles would have to fulfill certain conditions, namely having
magnetic rigidities above the geomagnetic cutoff and below a
certain threshold for adiabatic motion to hold. We show in this
work that, for fully ionized strangelets, there is a narrow window
for stable trapping. An estimate of the stationary population is
presented and the dominant loss mechanisms discussed. It is shown
that the population would be substantially enhanced with respect
to the ISM flux (up to two orders of magnitude) due to
quasi-stable trapping.
pacs: 94.30.Hn, 12.39.-x
I Introduction
In a celebrated paper Witten Witten (1984) elaborated on the
possibility Bodmer (1971); Chin and Kerman (1979); Terazawa (1979) that systems composed of
an deconfined Fermi gas of up, down and strange quarks could have
a lower energy per baryon than iron, thus being absolutely stable.
This hypothetical state (strange quark matter) could be created by
weak interactions introducing the massive $s$ quark, if the energy
“cost” of the mass is compensated by the availability of a new
Fermi sea associated to this extra flavor, thus lowering the Fermi
energy of the $u$ and $d$ quark seas.
Previous works have shown Farhi and Jaffe (1984) that this stability may
be realized for a wide range of parameters of strange quark matter
(SQM) in bulk on the basis of the MIT bag model. Calculations also
indicate that SQM can be absolutely stable within other models,
e.g. shell model Gilson and Jaffe (1993); Madsen (a), or not stable at all
Buballa . More recently, studies have indicated that a paired
version of SQM, the CFL (color-flavor locked) state seems to be
even more favorable energetically than the unpaired SQM, widening
the stability window Alford et al. (1999); Rapp et al. (2000); Rajagopal and Wilczeck ; Lugones and Horvath (2002).
For the description of finite size lumps of strange
matter, (termed strangelets) a few terms have to be added
to the bulk one in the free energy. A surface term suffices for $A\gg 10^{7}$, while other corrections are relevant for the lower
masses (see Madsen (2002) for a recent review). Large lumps will
have essentially the same number of quarks of bulk matter, with a
small depletion of the massive strange quark resulting in a net
positive charge. This is a feature also expected for small chunks
Gilson and Jaffe (1993); Madsen (2002), which thus resemble heavy nuclei.
In spite of theoretical controversies, it is generally agreed that
the ultimate SQM proof must be provided by experiments. The
experimental searches of strangelets started some 20 years ago and
have been reviewed recently in Klingenberg (2001); Finch . In addition to
direct production of strangelets in heavy ion collisions
Thomas and Jacobs ; Rusek et al. (1996); Van Buren (1999); Belz et al. (1996); Dittus et al. (1995); Appelquist et al. (1996); Ambrosini et al. (1996); Klingenberg (1999); Arsenescu et al. (2002),
cosmic rays may contain primaries in this state of matter, which could
eventually be detected directly or as a deposition in terrestrial
matter De Rujula and Glashow (1984); Bruegger et al. (1989); Isaac et al. ; Lu et al. .
Several cosmic ray events have been tentatively identified in the
past as primary strangelets (initially the Centauro events and the
HECRO-81 experiment Bjorken and McLerran (1979); Rybczynski
et al. (a); Saito et al. (1990)) for they present
features such as their high penetration in the atmosphere, low
charge-to-mass ratio and exotic secondaries Rybczynski
et al. (b). More
recently, at least one event recorded from the AMS-01 experiment
Choutko (2003), a mass spectrometer aboard the shuttle Discovery
during a 10-day flight in 1998, is being considered as possible
detection of a strangelets. While it is tempting to identify the
primary as a strangelet, the inevitable shell effects complicate
the analysis and preclude any firm conclusion as yet
Horvath and Paulucci (2006). It is not clear until today to what extent
the anomalous events can be originated by ordinary primaries or
rather forcefully require a truly exotic origin.
Considering the question of existence of strangelets among cosmic
ray primaries, a few injection (production) scenarios have been
considered. Witten originally suggested the merging of compact
stars as a likely site Witten (1984). In principle, injection spectra
and the total mass in the galaxy may be calculated knowing the
rate of the events and the total ejected mass in each of them.
These estimates are subject to some caveats, for example, while
the number of merging systems has been revised upwards Kalogera et al. (2004),
numerical work has shown that a substantial ejection of matter is
not guaranteed Kluzniak and Lee (2002) in a strange star-black hole system, and
the situation is unclear in the case of a fully relativistic SS-SS
system, which has only been partially addressed
Limousin et al. because the calculations had other goals. On
the other hand, strange matter formation in type II supernova
Benvenuto and Horvath (1989) has been preliminarily explored and in these events a
small fraction of strange matter may be ejected. A numerical
analysis has shown that the possible quark matter component of
cosmic rays primaries is compatible Medina-Tanco and Horvath (1996) with models in
which strangelets are ejected in either scenario.
While an uncertain flux from this “contamination” of the ISM is
expected Madsen (2005), we would like to discuss in this paper
another likely site to search for strangelets of cosmic origin.
Much in the same way heavy nuclei are present in the earth’s
magnetosphere bouncing between magnetic mirror points, strangelets
could also become trapped in specific regions of the magnetosphere
and their number density increased respect to the ISM flux,
provided some conditions for their capture by planetary magnetic
fields are met. This phenomenon is analogous to the Van Allen
belts, and has been first suggested in a former study Rosu .
A handful of experiments have probed the magnetosphere by
measuring the fluxes of the so-called “anomalous” cosmic ray
nuclei, and may already place interesting limits to strangelets as
well. Overall the existence and nature of exotic primaries is an
important issue. In addition to former and ongoing searches, there
will be a mass spectrometer placed at the International Space
Station, the AMS-02 experiment AMS ; Madsen (b), with one of its
goals to help the identification of this exotic component, of
crucial importance in testing the validity of the
Bodmer-Witten-Terazawa conjecture. We substantiate below the
strangelet belt idea, discuss the main features of this
population, and advocate for a search of this exotic component at
definite sites within existing uncertainties based on these
calculations.
II States of ionization and electronic recombination of strangelets in the ISM
As is well-known, unpaired (also referred as “normal” in this
work) SQM in bulk contains light $u$, $d$ and massive $s$ quarks
in $\beta$-equilibrium. Because of the depletion of the more
massive $s$ quark, a small fraction of electrons is also present
to maintain charge neutrality. On the other hand, SQM in a paired
CFL state is automatically neutral, since the equal number of
flavors is enforced by symmetry Rajagopal and Wilczek (2001). Actually, a
small positive charge is present because of the smaller abundance
of $s$ quarks near the surface in CFL strangelets Madsen (2002).
Therefore it is natural that CFL strangelets will be surrounded by
an electronic cloud in order to neutralize its total charge,
forming an exotic atom. The same happens for normal strange matter
if the strangelet radius is smaller than the electron Compton
wavelength, a condition satisfied whenever $A\ll 10^{7}$.
In the following and throughout the whole analysis presented here,
the strangelet rest mass will be assumed to be $\epsilon_{0}\,\,A\,\sim\,(930\,\times\,A)$ MeV, with $\epsilon_{0}$ the
asymptotic value of the energy per baryon of strange quark matter.
We will not consider the fact that the energy per baryon number
decreases with $A$ in sophisticated model calculations, given that
the uncertainties found in other parameter choices are expected to
be much larger than the error associated with this approximation.
Also the strange quark mass is considered to be $m_{s}\,=\,150\,MeV$ and the coupling gap of CFL strange quark matter, $\Delta\,=\,100\,MeV$ in this exploratory study. With these
assumptions, the net positive charge of strangelets is given
approximately by $Z=0.1\,A$ (low baryon number regime) in the
MIT bag model approach for normal strange matter and $Z=0.3\,A^{2/3}$ for the CFL model.
Strangelets from whatever astrophysical injection event would
travel through the interstellar medium and become ionized by
collisions. A simple analysis to evaluate the degree of ionization
of semi-relativistic strangelets surrounded by electronic clouds
due to these interactions was performed in a Bohr atom
approximation. Strangelets are partly neutralized by electrons
from the excitation of the vacuum if $Z\,\gg\,100$
Madsen and Larsen (2003), but for all cases of interest in this work the
baryon number range is such that we do not have to deal with this
effect.
We considered a two-body collision (incident electron - electron in
the strangelet cloud) instead of a multibody problem, which would be
much more difficult to handle. The stripping interactions are mainly
due to electrons with a Maxwellian speed distribution at a
temperature of $\sim\,100\,K$, an average condition of electrons
in the ISM.
The results are shown in figure 1 for strangelets with
total energy of $1\,GeV/A$. Considering the average density in the
interstellar medium to be $1\,particle/cm^{3}$, the mean free path
for an electronic collision which may or may not result in
ionization is of the order of $10^{15}\,cm$, which is very short
on astronomical standards.
The ionization degree became stable within a travelled distance of
a few $pc$ for $1\,GeV/A$ strangelets. For ultra-relativistic
strangelets (i.e., of the type of candidates that would produce a
Centauro event Lattes and Fujimoto (1980) $E/A\,\sim\,TeV$) the
calculations indicate always full ionization. Furthermore,
according to the model proposed by Werner and Salpeter
Werner and Salpeter (1969) for the radiation flux in the ISM, the influence of
the radiation field on ionization of strangelets will be
negligible unless the strangelet trajectory crosses a region
containing very energetic photons (i.e. the surroundings of a
Wolf-Rayet, O and B stars and/or regions of stellar formation).
We acknowledge that a Bohr atom treatment is a crude approach for
the electron distribution around the strangelet, since it does not
include quantum corrections as important as the spin-orbit
coupling and non-local effects, nor relativistic corrections for
many electrons bodies ($Z\,\geq\,40$). There is no general
expression for these corrections applicable in the case of atoms
with many electrons though; the existent models (e.g.,
Hartree-Fock calculations) are restricted to atoms with few
electrons, the same happening for experimental corrections. In
this way, the calculations presented here are rough estimates,
showing the general trend of the effects rather than providing
precise numerical values.
For low-energy particles the electronic capture can be as
important as the ionization process thus far discussed. An
approximate cross section for the capture of electrons of velocity
$v$ by a charged particle of atomic number $Z$ is given as
Massey and Burhop (1952) Bethe and Salpeter (1957)
$$\sigma_{c}=Z^{2}2^{2/3}\alpha^{4}\frac{h^{2}\nu^{2}}{m_{e}^{2}v^{2}c^{2}}\Big{%
(}\frac{m_{e}c^{2}}{h\nu}\Big{)}^{7/2}\times 6.65\times 10^{-25}\,cm^{2},$$
(1)
where $h\nu\approx E_{e}$ for $E_{e}>>I$, $I$ and $E_{e}$ being the
electron energies while bound to the nucleus and free in the ISM,
respectively, and $m_{e}c^{2}$ is the electron rest mass. This form of
the cross-section for radiative recombination is obtained relating
the capture of a bare nucleus of charge $Ze$ with the capture into
the corresponding state of a hydrogen atom, which is proportional
to the energy of the gamma emitted in the process and also to the
cross-section for the absorption of a quantum of frequency $\nu$
by a $H^{-}$ ion resulting in emission of an electron of velocity
$v$, and inversely proportional to the momentum of the electron
absorbed. In the case of a partially screened nucleus, the
cross-section is still given approximately by equation
(1), though a special calculation must be performed to
obtain the cross-section for capture into an orbital with quantum
number $n_{0}$, usually given in tables for ordinary nuclei.
The “atom” or “ion” formed by capturing an electron may also
lose this electron in further interactions. For light materials
the cross section for electron loss can be approximately expressed
for $v>v_{0}$ Bohr (1948) as
$$\sigma_{l}=8\pi a_{0}^{2}Z^{-2}\Big{(}\frac{v_{0}}{v}\Big{)}^{2},$$
(2)
where $a_{0}=\hbar^{2}/me^{2}=0.53\times 10^{-8}$ cm is the Bohr radius
and $v_{0}=e^{2}/\hbar$, whereas for intermediate $Z$ materials
$$\sigma_{l}=\pi a_{0}^{2}Z^{-1}\Big{(}\frac{v_{0}}{v}\Big{)},$$
(3)
because of the screening effect.
In summary, a comparison of eqs.(1), (2) and (3) shows that
electronic capture would only be important for high Z strangelets,
precisely where this simple picture can no longer be applied due
to vacuum excitation effects. That corresponds to a region
in baryon number which we believe to be of minimum relevance to the trapped population.
In summary, these results indicate that we can assume total
ionization as a good approximation to incoming ISM strangelets
that could form an ionization belt in the magnetosphere.
III Capture of strangelets in the geomagnetic field
The motion of ionized strangelets in the earth magnetosphere can be
studied by applying the Störmer theory in a dipolar magnetic
field. The movement analysis can be made in terms of the geomagnetic latitude
and the L parameter, where $L$ is the equatorial distance of a
field line to the axis of the dipole measured in units of the earth
radius.
The geomagnetic field is not a pure dipole field. Instead, most
magnetic models used for studying it include nearly 50 terms for
describing the potential field from which the magnetic field is
obtained in a sum of Legendre functions multiplied by oscillatory
coefficients in the azimuthal variable. Since the potential field
has a $r^{-(n+1)}$ dependence, the importance of high-order terms
decreases rapidly as one moves away from the earth surface. In
this way, the $n=1$ term, i. e., the dipole term, is the lowest
but dominant term, and most features of the trapped radiation
theory are analyzed based on a dipole field.
Charged particles with energy of order of $MeV$ in the inner part
of the magnetosphere ($L\ll 10$) rotate with a much higher
frequency than that of typical geomagnetic field variation (which
varies in time scales of, at most, few minutes). Under these
conditions, the magnetic moment is a conserved quantity (adiabatic
invariant). Therefore particles with high enough magnetic moments
become trapped in the dipolar field lines of the geomagnetic
field, with mirror points placed near the earth poles.
Particles with mirror points that allow penetration in the earth
atmosphere can be lost via collisions with atoms. All the
particles with mirror points placed inside the earth radius are
obviously lost, meaning that particles with
$|\alpha_{eq}|<\alpha_{E}$ or $|\pi-\alpha_{eq}|<\alpha_{E}$, where
$\alpha_{eq}$ is the equatorial pitch angle, are inside the earth
loss cone.
We will consider collisions mainly with the neutral nitrogen
molecule ($N_{2}$). The probability of interaction of trapped
particles which penetrate the atmosphere (suffering collisions
losses) can be taken as
$$P(s)=1-e^{-s/\lambda(s)}$$
(4)
at a certain point $s$, since each process is probabilistically
independent, being $\lambda$ the particle mean free path.
Generalizing the previous equation, it is necessary
to integrate over the particle path. Assuming that all the
strangelets which collide with particles in the atmosphere are
eventually removed from the trapped flux, we express the escape
probability as
$$P_{esc}=1-e^{-\int_{s}\sigma\Big{[}n(s^{\prime})+s^{\prime}\frac{dn}{ds^{%
\prime}}\Big{]}ds^{\prime}}$$
(5)
where $ds=LR_{E}\cos\lambda\sqrt{1+3\sin^{2}\lambda}d\lambda$ is the
arc along a field line, $\sigma$ is the particle cross-section
and $n(s)$ is the density of particles in
the atmosphere at a certain point $s$ of the strangelet’s path.
Since strangelets are hadrons we may take their relevant
interaction cross-section to be geometrical ($\sigma\propto\,A^{2/3}$).
The calculated loss cone for strangelets, assuming an exponential
profile of the atmospheric density is shown in figure
2 for different $L$ reflecting collisions with
atmospheric particles and the non-existence of a suitable mirror
point. It indicates, as expected, that the smaller the equatorial
pitch angle the easiest it is to remove a trapped particle.
In order for a particle to penetrate a certain region in the
magnetosphere its energy must be enough to overcome the local
geomagnetic cutoff rigidity. A solution was found by Störmer Stoermer (1955)
describing a special case of what he called the “forbidden cone”, which
gives the geomagnetic cutoff rigidity. In this way, the condition a particle
must fulfill to have access to a given region of the magnetosphere
can be written as Adams et al. (1981)
$$R_{particle}\,>\,\frac{59.6\,\cos^{4}\lambda}{L^{2}[1+(1-\cos\gamma\cos^{3}%
\lambda)^{1/2}]^{2}}\,\,\,\,GV$$
(6)
where $\lambda$ is the latitude and $\gamma$ the arrival direction of the
particle (east - west).
In the analysis of charged particles trapped in a magnetic field it is usually
considered that the motion of a given particle is a composition of three
different motions: the bouncing motion of a guiding center along the magnetic
field line; the rotational motion of the particle itself around that guiding
center; and the longitudinal drift of the guiding center. In this way, the
condition for a $triply-adiabatic$ motion is that the magnetic field intensity
must vary very slowly around a cyclotron orbit, imposing a maximum
energy allowed for stable trapping. The condition that
must be imposed for the cyclotron radius at the equator
is given by
$$R_{C}\,\Big{|}_{equator}=\,\frac{p_{\perp}}{qB}\,\ll\,\frac{B}{|\nabla_{\perp}%
B|}\Big{|}_{equator}$$
(7)
Figures 3 and 4 show those bounds for normal
and CFL strangelets, respectively, for $L\,=\,2$ in addition
to the minimum baryon number which is required for strangelet
stability Madsen (a). The existence of a minimum baryon
number is expected in all models of SQM because the energy needed
for producing the system increases as the baryon number decreases,
till it reaches a value above which the strange matter is
unstable. The value adopted has been $A_{min}\,=\,30$ (shown
with a vertical line) and may be trivially altered for any other
figure. Strangelets with very high baryon number, though allowed
for stable trapping, are not likely to be statistically
significant for detection in the magnetosphere due to a
substantial decrease of the interstellar flux expected as the
baryon number increases.
The upper bound (7) has been enforced in our calculations
to a 10% confidence level according to observations of anomalous
cosmic rays $L$-shell distributions Tylka (1993), and we
considered $E_{\perp}\sim E$, which means we are actually
underestimating the number of particles that could be
stably trapped in the geomagnetic field. Obviously, the
geomagnetic cutoff curve must be below the adiabaticity criteria
for stable trapping to occur. This is not the case for small
latitudes, but there is a narrow “window” in latitude starting
slightly above $30$ degrees at $L=2$ for strangelets from the
ISM flux to fulfill the conditions of capture and accumulate in
regions labelled by the $L$ parameter. However, the number of
accumulated particles is still interesting, as shown in the next
section.
When this calculation is repeated for the case of CFL strangelets,
the region allowed for stable trapping for CFL strangelets has a
different shape than that for normal strangelets. This feature is
due to the strong dependence of the charge upon $A$ of normal
strangelets ($Z\,\propto\,A$) resulting in constant values if
one considers momentum per baryon number, whereas for the CFL
strangelets charge is almost independent of $A$, leading to a
$\sim A^{-0.9}$ dependence in the momentum per baryon number
variable.
III.1 Trapped strangelet population
Even though strangelets can be captured and trapped in the earth’s
magnetic field, we must evaluate the possible maintenance of a
strangelet population to check whether there is an increase of the
flux. For this purpose, we must consider losses mechanisms.
In addition to the already analyzed losses by collisions with
neutral atmospheric particles, we have considered the inward
drift driven by asymmetric fluctuations of the geomagnetic field
as a dominant mechanism to diminish the strangelet population.
We will not consider in this work direct pitch angle diffusion.
Because of their large mass, strangelets are less likely to be
scattered appreciably in pitch angle by collisions. The net result
of multiple collisions with atmospheric particles would be a
reduction in the strangelets kinetic energy to thermal values and
minor changes in their pitch angle. Since we have already
considered that particles bouncing at a radial distance from the
surface of the earth below the atmosphere height scale (derived in
section II) would be eventually removed, we are in fact replacing
a diffusion equation in the $\cos(\alpha_{eq})$ variable for a
constant loss term (a sink function) directly related, though not
formally assigned, to pitch angle diffusion.
Radial diffusion must proceed by fluctuations in the third
invariant $\phi$, which is proportional to $L^{-1}$, due to
changes in the electric or magnetic fields that are more rapid
than the particle drift frequency. Because the gyration and bounce
periods are much shorter than the drift period, the first and
second adiabatic invariants are less likely to be affected by many of these
field perturbations.
Guided by the existing calculations and observations for anomalous
cosmic ray nuclei (hereafter ACR) trapping, we have considered third
invariant diffusion due asymmetric fluctuations in the geomagnetic
field, which is mainly driven by the solar wind pressure (sudden
compression and slow relaxation of the geomagnetic field).
The diffusion coefficient $D_{LL}$ is determined theoretically by
taking two consecutive steps Walt (1994). First, one has to
evaluate the radial displacement suffered by a particle under the
influence of the field disturbance, which is an idealized model of the
real disturbances occuring in the geomagnetic field.
The following procedure is taken in order to obtain
the diffusion coefficient as a function of the statistical features
of the disturbances alone. It consists of squaring this displacement and
taking the average over several
disturbances randomly occurring in time and over all possible particle’s
initial longitudes.
The diffusion coefficient due to magnetic field fluctuations for
equatorially trapped particles, with the assumption of efficient
phase mixing Walt (1994) can be expressed as
$$D_{LL}^{M}=\frac{\pi^{2}}{2}\Big{(}\frac{5}{7}\Big{)}^{2}\frac{R_{E}^{2}\,L^{1%
0}}{B_{0}^{2}}\nu_{drift}^{2}\,P_{A}(\nu_{drift})$$
(8)
where $P_{A}(\nu)$ is the power spectral density of the field
variation evaluated at the drift frequency. For off-equatorial
particles, the diffusion coefficient presents an exponential decay
with latitude.
Already in the case of nuclei, it is known that the complex
geometry and inhomogeneities in the geomagnetic field make
quantitative calculations ambiguous. The observed values of the
diffusion coefficient and their $L$ dependence will change with
global magnetic activity, and magnetic disturbances are known to
vary appreciably with time. We have assumed a $\nu^{-2}$
dependence of the power spectral density for simplicity
Walt (1994). The loss of more detailed information associated with
this approximation is that the diffusion coefficient becomes
independent of the energy of the particle entering the geomagnetic
field. In this case, the diffusion coefficient have a strong
dependence on the McIlwain parameter ($D_{LL}\propto L^{10}$) 111Radial diffusion caused by random variations in
the potential electric fields have a softer dependence on the
McIlwain parameter, being the resultant diffusion coefficient
$D_{LL}\propto L^{6}$ Falthammar (1968).. This indicates that its
influence is very important for particles trapped at higher
$L$-shells.
Typical values for changes in the trapped population
distribution ranges from a few hours at $L=6$ to hundreds of days
at $L=2$. Therefore if strangelets are captured by the geomagnetic
field their density must be higher for lower values of the $L$
parameter, which may result in a substantial increase of this
population compared to the ISM flux.
Some other losses mechanisms are of less importance in short time
scales, but have influence on long time scales, thus lowering the
residence time for trapped particles. This includes electrical
drift-resonant interactions between particles and fields,
especially in the pulsation frequency or VLF range
Baumjohann . Those phenomena are highly affected by the
solar wind activity.
The diffusion equation has been employed to study the trapped
strangelet flux
$$\frac{\partial f(\mu,J,L)}{\partial t}=\frac{\partial}{\partial L}\Big{[}\frac%
{D_{LL}}{L^{2}}\frac{\partial}{\partial L}(L^{2}f(\mu,J,L))\Big{]}$$
(9)
where $f$ is the distribution function, $D_{LL}$ is given by
equation (8) and $\mu$ and $J$ are the adiabatic
invariants magnetic moment and integral invariant, respectively.
The relation between the distribution function and the flux may
be given by $j(E,\alpha)=p^{2}L^{2}f(\mu,J,L)$. A
stationary population requires $\partial f/\partial t=0$, i.
e., the assumption that the source and loss terms are
instantaneously balanced is valid.
We assume a steady strangelet injection from the interstellar
medium at $L=6$ (the position of the maximum distribution function
is very insensitive to the chosen L-shell parameter for this
boundary condition) and derive the distribution function shape
between this maximum and $L\approx 1.05$ where it is null
(atmosphere particle interaction height), shown in Figure
5. We are not considering diffusion in pitch angle due
to interaction of particles with electromagnetic waves caused by
field variations, which alters the first adiabatic invariant.
The calculations were carried on with two values of the flux from
the ISM reaching the outer magnetosphere.
The first one, which will be called “standard”, is the one that
assumes the standard cosmic ray dependence on the strangelet flux,
$E^{-2.5}$. The total ISM strangelet flux that reaches the earth
as estimated by Madsen Madsen (2005) for a binary strange star
system coalescence scenario is given by
$$F\approx 2\times 10^{5}\,m^{-2}\,yr^{-1}\,sr^{-1}\,A^{-0.467}\,Z^{-1.2}\,max[R%
_{SM},R_{GC}]^{-1,2}\Lambda$$
(10)
where $R_{SM}$ and $R_{GC}$ are the solar modulation and geomagnetic
cutoff rigidities, respectively, and $\Lambda$ is an uncertain
parameter assumed to be of $O(1)$. In this way, the whole flux is fitted
with a $E^{-2.5}$ dependence with the constraints of minimum and maximum
energy respecting the values $R_{min}=5MVA/Z$ and $R_{max}=10^{6}GV$ Madsen (2005).
The second calculation, which will be called “improved”,
considers a more detailed characterization of the differential
flux, where for the region of interest in this work (rigidities of
few GV), the strangelet flux actually increases with a slope
of $R^{1.8}$. This flux was obtained from a fit to reference
Madsen (2005).
In either way, the flux entering the region of the magnetosphere
at $L_{max}$ has to fulfill the restrictions imposed for stable
trapping in the pitch angle and geomagnetic latitude of incidence
$$F_{in}=\int_{\lambda_{min}}^{\lambda_{max}}d\lambda\,P(\lambda)\,\int_{\alpha_%
{loss\,cone}}^{\pi/2}d\alpha_{eq}\,P(\alpha_{eq})\times F$$
(11)
The efficiency factors, $P(\lambda)$ and $P(\alpha_{eq})$ may be
easily identified: $P(\lambda)$ gives the fractional area of the spherical
section suitable for trapping discussed previously
$$P(\lambda)=\frac{2L^{2}\,(-cos\lambda)\int_{0}^{2\pi}d\phi}{2L^{2}\,\int_{0}^{%
\pi/2}cos\theta\int_{0}^{2\pi}d\phi}$$
where the factor $2$ comes from the symmetry in $\theta$ for both
hemispheres (north, south). $P(\alpha_{eq})$ limits the number of
particles entering the specific region of the magnetosphere with
an appropriate pitch angle to avoid the loss cone as already
discussed. We have also assumed an isotropic flux of particles,
since there is no theoretical prediction pointing to any
anisotropy in the arrival direction of strangelets, which means
that $j_{0}(cos\alpha_{eq})\,=\,constant$ is a reasonable
hypothesis:
$$P(\alpha_{eq})=4\frac{\alpha_{eq}}{\int_{0}^{\pi/2}\alpha_{eq}\,d\alpha_{eq}}$$
where the factor $4$ stands for the symmetry in the condition for a
given particle to belong to the loss cone:
$|\alpha_{eq}|<\alpha_{loss\,cone}$ and
$|\pi-\alpha_{eq}|<\alpha_{loss\,cone}$.
Solving the differential equation (9) and obtaining the corresponding
flux inward (in the -$\text{\^{e}}_{r}$ direction) for every $L$, it is possible to
determine the mean particle density at a given shell and, in this way, the
trapped strangelet flux.
The results are summarized in Tables 1 and 2 for $L=2$ (ACR belt
location) and $L=1.3$ (location of the maximum of the distribution
function) for the example of strangelets of $A=100$ and energy
corresponding to $R=1GV$.
The position of the peak of the distribution function in the
geomagnetic field (around $L=1.3$) is quite robust, it does not
appreciably change with the change in the $A/Z$ relation (CFL and
normal strangelets), nor with a change in the energy and baryon
number of the strangelets. This could be a consequence of the
assumption of the power spectral density as being proportional to
$\nu^{-2}$, what renders the diffusion coefficient independent of
the particle energy, therefore modifying the energy of the
particles does not affect their diffusion properties.
We observe that the trapped population is slightly more favored if
strange quark matter is in the CFL state, the difference between
the trapped fluxes for the two species increases with decreasing
energy exponent in the incident flux. It happens due to the
dependence on the baryonic number of the interstellar flux of
strangelets (eq. 10). Since the rigidity interval for
stable trapping is the same for both states for it only depends on
the geometrical characteristics of the geomagnetic field, the
difference on the number of particles trapped strongly depends on
the difference in the incoming flux. This dependence of the
integrated flux on the number of baryons that can be expressed as
$F_{ISM}\propto(0.125)^{-1.2}\,A^{-1.667}$ and $F_{ISM}\propto(0.3)^{-1.2}\,A^{-1.267}$ for normal and CFL strangelets,
respectively. In this way the flux of paired CFL strangelets is
lower than those without pairing, but only for low baryon number
($A<\sim 13$), that is, in a region where it is believed
strangelets are not stable at all. In the stability region the
flux for CFL strangelets is always higher than for normal
strangelets resulting in a higher trapped density. In this way,
the smaller difference seeing between strangelets with and without
pairing for the improved flux when comparing to that for the
standard flux is explained by the smaller difference in the
incoming flux due to the softer dependence on the atomic number
(the standard flux depends on $E^{-2.5}$, which for the same
rigidity depends on the particle’s atomic number; instead, the
improved flux depends on $R^{1.8}$ and the analysis was performed
in terms of same rigidity).
Additional considerations are relevant for the fate of a trapped
population of strangelets. It is well-known that the solar wind has
a strong influence on the ACR flux upon the earth. The most abundant
ACR heavy ion, oxygen, shows a strong intensity variation with the
solar cycle, having its interstellar flux of $8-27\,MeV/nucleon$
lowered up to two orders of magnitude during periods of solar
maximum activity Biswas (1996). During solar minimum, the trapped
flux at the earth magnetosphere is of the order of $\sim 5\times 10^{-4}\,particles\,cm^{-2}\,sr^{-1}\,s^{-1}\,(MeV/nucleon)^{-1}$, corresponding to an enhancement factor of $\sim 15$ Bobrovskaya et al. (2001), this experimental value being somewhat below
the theoretical expected one (higher than 25 Biswas (1996)). The
oxygen component corresponds to about $80\%$ of the trapped ACR,
while the $C/O$, $N/O$ and $Ne/O$ abundance ratio are $<0.005$,
$\sim 0.10-0.15$ and $\sim\,0.02-0.03$, respectively.
With the results obtained in this study, the trapped flux of
strangelets at $L<2$ would be of order $10^{-14}-10^{-15}\,particles\,cm^{-2}\,sr^{-1}\,s^{-1}\,(MeV/A)^{-1}$ at
rigidity $R\,=\,1\,GV$ for strangelets of baryon number
$A=100$. This represents an enhancement factor for trapped flux in
the regime of steady-state population comparing to the
interstellar flux at the same energy and $A$ of order $10$ and
$10^{2}$ for strangelets trapped at $L=2$ and $L=1.3$, respectively,
the values for CFL strangelets being of about twice the one for
normal strangelets ($q\sim 5.5$ and $11$, and $q\sim 162$ and
$314$ for CFL and normal strangelets at $L=2$ and $L=1.3$,
respectively). This results show that the strangelet flux could be
as high as a factor 10000 lower than that expected for carbon
during periods of maximum solar activity. Although we did not
consider the solar modulation in our calculations, it would act
significantly over those low energy strangelets Madsen (2005),
the region of interest in this study. In this manner, it could
have an important influence, similar to that detected for oxygen,
on the trapped density.
The advantage of a search for trapped strangelets in the geomagnetic
field performed during the solar maximum activity whether they are an
important component of the radiation belt or are to be measured
penetrating the atmosphere towards to surface of the earth
would be the reduced component of ACR, which could reduce dead time losses
in the detectors and possibly render a clearer identification of the primaries.
The proposed and widely accepted model for ACR trapping
Blake and Friesen (1977) assumes that the high mass-to-charge ratio of
singly-ionized ACRs enables them to penetrate deeply into the
magnetosphere. ACRs with trajectories near a low altitude mirror
point interact with particles in the upper atmosphere, loosing one
or all their remaining electrons. After stripping, the particle
gyroradius is reduced by a factor of $1/Z$, and the ion can become
stably trapped. As stated above, the results presented here were
obtained assuming fully-ionized strangelets, which have just the
“right” features to become trapped. However, some fraction of
the strangelets should reach earth’s atmosphere with an effective
charge slightly below their atomic number and suffer a process of
interaction similar to ACR’s, which is much less dependent on the
pitch angle and other variables. Finally, there is also the
possibility of quasi-stable trapping of ions with energies high
enough not to obey condition (7), but not too high as to
penetrate the magnetosphere without suffering any significant
depletion in their incident direction. These two additional
mechanisms could result in a further increase in the number
of trapped strangelets.
Conclusions
From the analysis presented here we conclude that non-relativistic
strangelets with $A\,<\,\sim\,10^{3}$ already ionized by
collisions with electrons in the ISM could be stably trapped by
the geomagnetic field. Assuming the existence of a strangelet
contamination in the ISM, its injection in the solar system and
given the geomagnetic geometry and the interaction of the magnetic
field with the solar wind, it looks very likely to have this
radiation belt surrounding the planet. If strangelets are to be a
component of the anomalous cosmic ray belt at $L\,\sim\,2$, we
have shown that, even considering the approximations taken during
the calculations presented here (which have the main consequence
of averaging the trapped population’s behavior), those particles
would be present with an enhancement factor comparing with the
interstellar flux of order $10^{1}$ and if we consider a new
particle belt (a strangelet belt) at $L\,\sim\,1.3$, the
enhancement factor could be as high as order $10^{2}$ in a
stationary population scenario 222It must be noted,
however, that for the ACR belt the theoretical values for the
enhancement factor are somewhat higher than the experimental ones,
a feature that would probably hold for the strangelet belt as
well.. These exotic baryons could in principle be detectable in
the earth magnetosphere depending on the chosen parameters for
each of the experiments (effective detection area, altitude and
type of orbiting, magnetic field for particle depletion and
others). In addition to the already mentioned capture of almost
fully ionized strangelets, additional trajectories leading to trapping (but not
obeying the adiabatic conditions) may exist, although they must be
calculated numerically, and could enhance even further the trapped
population, though most probably not affecting substantially the
results. Effects that could result in the reduction of the trapped
population are the diffusion driven by electric fields
fluctuations and phenomena directly related to enhanced solar
activity, which though less likely to affect the particles already
trapped at low $L$-shells, could have an influence on the particle
injection in the outer magnetosphere.
Overall, we believe our estimates to be on the conservative side
of the trapped flux, making the search of trapped strangelets a
feasible but difficult task. Needless to say, the detection of
those trapped particles having low $Z/A$ ratio would be extremely
important for determining the properties of cold, dense baryonic
matter.
Acknowledgements.
L.P. acknowledges the financial support received from
the Fundação de Amparo à Pesquisa do Estado de
São Paulo. J.E.H. wishes to acknowledge the CNPq
Agency (Brazil) for partial financial support.
References
Witten (1984)
E. Witten,
Phys. Rev. D 30,
272 (1984).
Bodmer (1971)
A. Bodmer,
Phys. Rev. D 4,
1601 (1971).
Chin and Kerman (1979)
S. A. Chin and
A. Kerman,
Phys. Rev. Lett. 43,
1292 (1979).
Terazawa (1979)
H. Terazawa,
Tokyo U. Report. pp. INS–336
(1979).
Farhi and Jaffe (1984)
E. Farhi and
L. Jaffe,
Phys. Rev. D 30,
2379 (1984).
Gilson and Jaffe (1993)
E. P. Gilson and
R. L. Jaffe,
Phys. Rev. Lett. 71,
332 (1993).
Madsen (a)
J. Madsen,
eprint astro-ph/9809032.
(8)
M. Buballa,
eprint hep-ph/0402234.
Lugones and Horvath (2002)
G. Lugones and
J. Horvath,
Phys. Rev. D 66,
074017 (2002).
Alford et al. (1999)
M. Alford,
K. Rajagopal,
and F. Wilczek,
Nucl. Phys. B 537,
433 (1999).
Rapp et al. (2000)
R. Rapp et al.,
Ann. Phys. (N. Y.) 280,
35 (2000).
(12)
K. Rajagopal and
F. Wilczeck,
for an overview of the CFL state, eprint hep-ph/0011333.
Madsen (2002)
J. Madsen,
J.Phys. G 28,
1737 (2002).
Klingenberg (2001)
R. Klingenberg,
J. Phys. G 27,
475 (2001).
(15)
E. Finch,
eprint nucl-ex/0605010.
(16)
J. Thomas and
P. Jacobs,
A guide to the high energy ion experiments,
UCRL-ID-119181.
Rusek et al. (1996)
A. Rusek et al.
(for the E886 collaboration),
Phys. Rev. C 54,
R15 (1996).
Van Buren (1999)
G. Van Buren
(for the E864 collaboration), J.
Phys. G 25, 411
(1999).
Belz et al. (1996)
J. Belz et al.
(for the BNL E888 collaboration),
Phys. Rev. Lett. 76,
3277 (1996).
Dittus et al. (1995)
F. Dittus et al.
(for the NA52 collaboration), in
International Conference on Strangeness in Hadronic
Matter, edited by
J. Rafelski
(American Institute of Physics, New York (AIP 340),
1995), p. 24.
Appelquist et al. (1996)
G. Appelquist
et al., Phys. Rev. Lett.
76, 3907 (1996).
Ambrosini et al. (1996)
G. Ambrosini
et al., Nucl. Phys.
A610, 306c
(1996).
Klingenberg (1999)
R. Klingenberg,
J. Phys. G 25,
R273 (1999).
Arsenescu et al. (2002)
R. Arsenescu
et al., New J. Phys.
4, 96 (2002).
De Rujula and Glashow (1984)
A. De Rujula and
S. L. Glashow,
Nature 312,
734 (1984).
Bruegger et al. (1989)
M. Bruegger
et al., Nature
337, 434 (1989).
(27)
M. C. P. Isaac
et al., eprint astro-ph/9806147.
(28)
Z.-T. Lu et al.,
eprint nucl-ex/0402015.
Bjorken and McLerran (1979)
J. D. Bjorken and
L. McLerran,
Phys. Rev. D 20,
2353 (1979).
Rybczynski
et al. (a)
M. Rybczynski,
Z. Wlodarczyk,
and G. Wilk,
eprint hep-ph/0109225.
Saito et al. (1990)
T. Saito et al.,
Phys. Rev. Lett. 65,
2094 (1990).
Rybczynski
et al. (b)
M. Rybczynski,
Z. Wlodarczyk,
and G. Wilk,
eprint hep-ph/0410065.
Choutko (2003)
V. Choutko
(for the AMS-01 Collaboration), in
Proc. 28th Internat. Cosmic Ray Conf., Tsukuba,
Japan (Universal Academic Press, Tokyo,
2003), p. 1765.
Horvath and Paulucci (2006)
J. E. Horvath and
L. Paulucci,
J. Phys. G 32,
B13 (2006).
Kalogera et al. (2004)
V. Kalogera
et al., Astrophys. J. Lett.
601, L179 (2004).
Kluzniak and Lee (2002)
W. Kluzniak and
W. Lee,
MNRAS 335, L29
(2002).
(37)
F. Limousin,
D. Gondek-Rosinska,
and
E. Gourgoulhon,
eprint gr-qc/0411127.
Benvenuto and Horvath (1989)
O. G. Benvenuto
and J. E.
Horvath, Phys. Rev. Lett.
63, 716 (1989).
Medina-Tanco and Horvath (1996)
G. A. Medina-Tanco
and J. E.
Horvath, Astrophys. J.
464, 364 (1996).
Madsen (2005)
J. Madsen,
Phys. Rev. D 71,
014026 (2005).
(41)
H. C. Rosu,
eprint hep-ph/9410028.
(42)
Home page of AMS experiment,
URL http://ams.cern.ch.
Madsen (b)
J. Madsen,
eprint hep-ph/0111417.
Rajagopal and Wilczek (2001)
K. Rajagopal and
F. Wilczek,
Phys. Rev. Lett. 86,
3492 (2001).
Madsen and Larsen (2003)
J. Madsen and
J. M. Larsen,
Phys. Rev. Lett. 90,
121102 (2003).
Lattes and Fujimoto (1980)
C. Lattes and
K. Fujimoto,
Phys. Repts. 65,
151 (1980).
Werner and Salpeter (1969)
M. W. Werner and
E. E. Salpeter,
MNRAS 145, 249
(1969).
Massey and Burhop (1952)
H. S. Massey and
E. H. S. Burhop,
Electronic and Ionic Impact Phenomena
(Oxford University Press, 1952).
Bethe and Salpeter (1957)
H. A. Bethe and
E. E. Salpeter,
Handbuch der Physik, vol. 35
(Springer–Verlag Berlin, 1957).
Bohr (1948)
N. Bohr, Kgl.
Dansk. Vid. Selsk. 18, 8
(1948).
Stoermer (1955)
C. Stoermer,
The polar aurora (Claredon Press,
Oxford, 1955).
Adams et al. (1981)
J. H. Adams,
R. Silberberg,
and C. H. Tsao,
Tech. Rep. NRL Memorandum Report 4506,
Naval Research Laboratory, Washington DC 20375-500,
USA (1981), See also, for example, D. F.
Smart and M. A. Shea, Adv. Space Res. 14(10), 787 (1994).
Tylka (1993)
A. J. Tylka, in
Proc. 23rd Internat. Cosmic Ray Conf., Calgary,
Canada (1993), vol. 3, p.
436.
Walt (1994)
M. Walt,
Introduction to geomagnetically trapped radiation
(Cambridge Press, 1994).
(55)
W. Baumjohann,
private communication.
Biswas (1996)
S. Biswas,
Space Science Rev. 75,
423 (1996).
Bobrovskaya et al. (2001)
V. V. Bobrovskaya
et al., Cosmic Research
39, 98 (2001).
Blake and Friesen (1977)
J. B. Blake and
L. M. Friesen, in
Proc. 15th Internat. Cosmic Ray Conf., Plovdiv,
Bulgaria (1977), vol. 2, p.
341.
Falthammar (1968)
C.-G. Falthammar, in
Earth’s Particles and Fields
(Ed.B.M. McCormac, Reinhold Book Co.,
1968), pp. 157–169. |
The phase diagram of the one-dimensional quantum sine-Gordon system ($\beta^{2}=4\pi)$ with a linear spatial modulation
Zhiguo Wang
Yumei Zhang
Department of Physics, Tongji University, Shanghai, 200092, China
(November 19, 2020)
Abstract
The one-dimensional quantum sine-Gordon system with a linear spatial
modulation is investigated in a special case, $\beta^{2}$ =4$\pi$. The
model is tranformed into a massive Thirring model and then is exactly
diagonalized, the energy spetrum of the model is obtained. Our result
clearly demonstrates that cancelling the cosine term without any considering
is unadvisable.
pacs: 05.70.Jk, 68.10.-m,87.22.Bt
The one-dimensional (1D) quantum sine-Gordon model${}^{\text{\@@cite[cite]{[\@@bibref{Number}{Sklyanin}{}{}]}}}$
may probably be the most useful quantum model since it can be used to
describe the most of the one- and two-dimensional (2D) models of either
fermi or bose system,${}^{\text{\@@cite[cite]{[\@@bibref{Number}{Minnhagen}{}{}]}},\text{\@@cite[cit%
e]{[\@@bibref{Number}{E.Fradkin}{}{}]},\@@cite[cite]{[\@@bibref{Number}{%
Gogolin}{}{}]}}}$ this fact attaches particular importance to the quantum
sine-Gordon model. Many works have been done about this model both in field
theory and in condensed matter physics.${}^{\text{\@@cite[cite]{[\@@bibref{Number}{Sklyanin}{}{}]}},\text{\@@cite[cite%
]{[\@@bibref{Number}{Coleman}{}{}]}},\text{\@@cite[cite]{[\@@bibref{Number}{%
Ingermanson}{}{}]}},\text{\@@cite[cite]{[\@@bibref{Number}{Mandelstam}{}{}]}},%
\text{\@@cite[cite]{[\@@bibref{Number}{Stevenson}{}{}]}}}$ It is exactly solvable by quantum inverse scattering method.${}^{\text{\@@cite[cite]{[\@@bibref{Number}{Sklyanin}{}{}]}}}$ By a variational method Coleman${}^{\text{\@@cite[cite]{[\@@bibref{Number}{Coleman}{}{}]}}}$ first discovered that the energy density of the system is
unbounded below when the coupling constant $\beta^{2}$ exceeds $8\pi$, so
there is a phase transition as the coupling constant varies. This
corresponds to the Kosterlitz-Thouless (K-T) phase transition by the
equivalence of the 2D Coulomb gas and sine-Gordon model. The soliton mode of
the sine-Gordon model corresponds to a one-fermion excitation in the fermi
picture, which was clarified soon later by Mandelstam by introducing a
Fermi-Bose relation.${}^{\text{\@@cite[cite]{[\@@bibref{Number}{Mandelstam}{}{}]}}}$
As there are other spatial variations in the cosine potential, the low
energy properties of the 1D quantum sine-Gordon system are more difficult to
be analysized. Here we shall discuss a simple case that there are a linear
spatial modulation in the cosine potential. The Hamlitonian reads as
$$H=\int\left\{\frac{1}{2}\left[\Pi^{2}(x)+\left(\frac{\partial\phi(x)}{\partial
x%
}\right)^{2}\right]-\frac{\alpha}{\beta^{2}}\cos\left(\beta\phi(x)+\lambda x%
\right)\right\}dx\text{ ,}$$
(1)
here $\phi(x)$ is a bose field operator, $\Pi(x)=-i\frac{\delta}{\delta\phi(x)}$ is its conjugate momentum, they satisfy the commutation relation
$$\left[\phi(x),\Pi(y)\right]=i\delta(x-y)\text{ . }$$
(2)
$\lambda$ is a spatial modulated parameter. In condensed matter physics $\lambda$ represents the fermi surface shifting from half filling in the
fermi picture.${}^{\text{\@@cite[cite]{[\@@bibref{Number}{Sun}{}{}]}}}$
In the presence of finite $\lambda$, most previous works suggested that the
above Hamlitonian (1) describes a massless free field and cancelled the
cosine potential directly. Has the cosine potential really no effect on the
system in this case?
Schulz discussed a one-dimensional quantum sine-Gordon system with an
additional gradient term.${}^{\text{\@@cite[cite]{[\@@bibref{Number}{Schulz}{}{}]}}}$ If we take a shift, $\phi+\frac{4m\pi x}{\beta a}\rightarrow\phi$, the above Hamiltonian (1) is
same as that of Schulz. But if one pays attention to the boundary condition
of bose operator $\phi$, he will find both model are different since the
boundary conditions of system will be altered under this shift, namely, $\int_{0}^{L}\nabla\phi(x)dx=0\rightarrow\int_{0}^{L}\nabla\phi(x)dx=\frac{%
\lambda}{\beta}L$, therefore the eigenstates will be also changed.
Schultz pointed out that a commensurate-incommensurate transition happens in
case of finite coefficient of the gradient term, this result implies that
directly cancelling the cosine potential is unsuited.
In order to find an unquestionable answer for this, we investigate this
Hamlitonian in a special case, $\beta^{2}$ =4$\pi$. First we transform the
model into the massive Thirring model by the operator identities between
fermions and bosons, and then exactly diagonalize the model using the
bogliubov transformation. When the cosine potential may be omitted is
obvious in our results.
For the case of a finite $\lambda$, the above system has rarely been
discussed before. After we use the bose-fermi relations${}^{\text{\@@cite[cite]{[\@@bibref{Number}{Takada}{}{}]}}}$
$$\frac{1}{2}\int:\Pi^{2}(x)+\left(\frac{\partial\phi(x)}{\partial x}\right)^{2}%
:dx=\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }$$
$$-i\int\left[\psi_{1}^{\dagger}(x)\frac{\partial}{\partial x}\psi_{1}(x)-\psi_{%
2}^{\dagger}(x)\frac{\partial}{\partial x}\psi_{2}(x)\right]dx\text{ ,}$$
(3)
$$\cos\left(2\sqrt{\pi}\phi(x)\right)=\pi\epsilon\left[\psi_{1}^{\dagger}(x)\psi%
_{2}(x)+\psi_{2}^{\dagger}(x)\psi_{1}(x)\right]\text{ ,}$$
(4)
the Bose Hamlitonian (1) in the case $\beta^{2}$ =4$\pi$ can be
transformed into a modified massive Thirring model
$$\displaystyle H$$
$$\displaystyle=$$
$$\displaystyle\int\left\{-i\left[\psi_{1}^{\dagger}(x)\frac{\partial}{\partial x%
}\psi_{1}(x)-\psi_{2}^{\dagger}(x)\frac{\partial}{\partial x}\psi_{2}(x)\right%
]\right.\text{ }$$
(5)
$$\displaystyle-\left.\frac{\alpha\epsilon}{4}\left[e^{i\lambda x}\psi_{1}^{%
\dagger}(x)\psi_{2}(x)+e^{-i\lambda x}\psi_{2}^{\dagger}(x)\psi_{1}(x)\right]%
\right\}dx\text{ .}$$
Here $\epsilon$ is an infinitesimal positive parameter, it is of the order
of the lattice constant in condensed matter physics. With following Fourier
transformations
$$\displaystyle c_{1k}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{\sqrt{L}}\int\psi_{1}(x)e^{i(k-\frac{\lambda}{2})x}dx%
\text{ ,}$$
$$\displaystyle\text{ }c_{2k}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{\sqrt{L}}\int\psi_{2}(x)e^{i(k+\frac{\lambda}{2})x}dx%
\text{ ,}$$
(6)
the Hamlitonian (5) is rewritten as
$$H=\sum_{k}\left[(k-\frac{\lambda}{2})c_{1k}^{\dagger}c_{1k}-(k+\frac{\lambda}{%
2})c_{2k}^{\dagger}c_{2k}\right]-\frac{\alpha\epsilon}{4}\sum_{k}\left[c_{1k}^%
{\dagger}c_{2k}+c_{2k}^{\dagger}c_{1k}\right]\text{
,}$$
(7)
where $\sum$ denotes the summation over momentum which is cut off at $\Lambda(\sim 1/\epsilon)$.
In order to diagonalize one particle terms we apply a Bogolubov
transformation to the Hamlitonian (7)
$$\displaystyle c_{1k}$$
$$\displaystyle=$$
$$\displaystyle u_{k}\beta_{k}+v_{k}\alpha_{k}^{\dagger}\text{ ,}$$
$$\displaystyle c_{2k}$$
$$\displaystyle=$$
$$\displaystyle u_{k}\alpha_{k}^{\dagger}-v_{k}\beta_{k}\text{ .}$$
(8)
The standard technique gives us a quasi-particle Hamlitonian
$$H=\sum_{k}\left[(\sqrt{k^{2}+\left(\frac{\alpha\epsilon}{4}\right)^{2}}-\frac{%
\lambda}{2})\beta_{k}^{\dagger}\beta_{k}+(\sqrt{k^{2}+\left(\frac{\alpha%
\epsilon}{4}\right)^{2}}+\frac{\lambda}{2})\alpha_{k}^{\dagger}\alpha_{k}%
\right]\text{ .}$$
(9)
The quasi-particle creation operator $\beta_{k}^{\dagger}$ and $\alpha_{k}^{\dagger}$ correspond to two different quasi-particles such as the
hole or particle. But now the excitations of them is not same. When $\alpha\epsilon>2\lambda$, both have excitation gap, while the excitation of $\beta_{k}^{\dagger}$ is gapless in the case $\alpha\epsilon<2\lambda$.
This shows that the original model can be treated as massless free system
with spatial modulation when the parameters satisfy
$$\alpha\epsilon<2\lambda\text{ .}$$
(10)
Otherwise, the original model can not be treated as a massless system. In this case, one should be always aware of that the micro-structure of the
ground state is essentially different with that of $\lambda=0$ case.
In summary, we have concluded that the cosine potential term can be
cancelled ,i.e., the model can be considered as a massless free system, when
the parameters satisfy equation (10). Although we only discussed in a
special case that $\beta^{2}=4\pi$, it is sufficient for showing that
cancelling the cosine potential term is unadvisable without considering the
varying of parameters.
References
[1]
E.K. Sklyanin, L.A. Takhtadzhyan and L.D. Faddeev,
Theor.Math.Phys., 40, 688(1980).
[2]
P.Minnhagen, Phys.Rev.Lett., 54, 2351(1985); Phys.Rev.,
B32, 3088(1985); Rev.Mod.Phys., 59, 1001(1987).
[3]
E.Fradkin, Field Theories of Condensed Matter Systems,
Addison-Wesley, 1991.
[4]
A.O. Gogolin, A.A. Nersesyan and A.M.Tsvelik, Bosonization
and Strongly Correlated Systems, Cambridge University Press, 1998.
[5]
S. Coleman, Phys.Rev. D11, 2088(1975).
[6]
R.Ingermanson, Nucl.Phys., B266, 620(1986).
[7]
S. Mandelstam, Phys.Rev., D11, 3026(1975).
[8]
P.M. Stevenson, Phys.Rev., D32, 1389(1985).
[9]
P. Sun and D. Schmeltzer, Phys.REv., B61, 349(2000).
[10]
H.J. Schulz, Phys. Rev., B22, 5274 (1980).
[11]
S. Takada and S. Misawa, Pro.Theor.Phys. 66, 101(1981). |
Supergravitons from one loop perturbative ${\cal N}=4$ SYM
Romuald A. Janik${}^{a}$ and Maciej Trzetrzelewski${}^{a,b}$
${}^{a}$ Institute of Physics,
Jagiellonian University,
Reymonta 4, 30-059 Kraków,
Poland
${}^{b}$ Department of Mathematics,
Royal Institute of Technology,
KTH 407.76, 100-44 Stockholm,
Sweden.
e-mail: ufrjanik@if.uj.edu.ple-mail: 33lewski@th.if.uj.edu.pl
Abstract
We determine the partition function of $\frac{1}{16}$BPS operators in ${\cal N}=4$ SYM
at weak coupling at the one-loop level in the planar limit. This
partition function is significantly different from the one computed at
zero coupling. We find that it coincides precisely with the partition
function of a gas of $\frac{1}{16}$BPS ‘supergravitons’ in $AdS_{5}\times S^{5}$.
1 Introduction
The AdS/CFT correspondence states an exact equivalence between ${\cal N}=4$
SYM gauge theory and type IIB superstrings in an $AdS_{5}\times S^{5}$
background [1]. It provides a fascinating new approach for
studying nonperturbative properties of gauge theory. On the other
hand, one can use the gauge theory knowledge to gain insight into the
behaviour of (super-)gravity at the quantum level
(see e.g. [2, 3, 4]). In general this is a
formidable problem but
progress can be made when studying configurations which preserve some
fraction of supersymmetry. A dictionary between $\frac{1}{2}$BPS
operators in gauge theory and dual geometries has been established in
[5]. $\frac{1}{4}$- and $\frac{1}{8}$BPS states have been discussed
from various points of view [6]. Of particular interest are the $\frac{1}{16}$BPS states [7] due to the existence $\frac{1}{16}$BPS black holes
[8]. At low energies, the gauge theory $\frac{1}{16}$BPS states should
correspond to a gas of
$\frac{1}{16}$BPS supergravitons, while at high energies these states should
account for the entropy of $\frac{1}{16}$BPS black holes.
In [7] $\frac{1}{16}$BPS states were counted on the gauge theory side at
zero coupling. It was found that the resulting partition function
overcounts both the $\frac{1}{16}$BPS supergraviton partition function (giving a
different energy scaling of entropy) and the $\frac{1}{16}$BPS black hole entropy
in the relevant parameter regimes. In that paper it was suggested that
once gauge theory interactions are turned on, many states which were
counted as $\frac{1}{16}$BPS at zero coupling would get anomalous dimensions, and
that the overcounting could be cured.
The aim of this paper is to perform the enumeration of $\frac{1}{16}$BPS operators in
perturbative ${\cal N}=4$ SYM to one-loop order. We do
the counting in the planar limit using the oscillator construction of
the one-loop dilatation operator of [9]. We find exact agreement with the $\frac{1}{16}$BPS supergraviton partition function.
The plan of the paper is as follows. In section 2 we review the
definition of $\frac{1}{16}$BPS states and fix notation. Then, in section 3, we
review the counting of these states in the free theory, and in section
4 we describe what has to be done to perform the calculation at one
loop. In section 5 we review the supergravity result for the $\frac{1}{16}$BPS supergraviton partition function. In section 6 we describe in some
details the construction of the one loop dilatation operator and, in
the following section, we determine the partition function of $\frac{1}{16}$BPS operators and perform some checks. In section 8 we compare the result
with the supergravity prediction and finaly, in section 9, we discuss the
possible extension to large but finite $N$. We close the paper with a
summary.
2 $\frac{1}{16}$BPS states
In this paper we consider $\frac{1}{16}$BPS states which by definition are
annihilated by the following two supercharges:
$$Q\equiv Q^{-\frac{1}{2},1},\quad\quad\quad\quad S\equiv S^{-\frac{1}{2},1},$$
(1)
where $Q^{\dagger}=S$ and $Q^{-\frac{1}{2},1}$, $S^{-\frac{1}{2},1}$ are as in
[7]. We would like to calculate the partition function over
these states. To do so it is convinient to introduce the anticommutator
$$\Delta\equiv 2\{S,Q\}.$$
(2)
The states annihilated by $S$ and $Q$
are exactly those annihilated by $\Delta$. Moreover these states are
in a 1 to 1 correspondence with the cohomology classes w.r.t. $Q$.
In general, states in ${\cal N}=4$ SYM can be labeled by the eigenvalue
of the dilatation operator $H$, two Lorentz spins $J_{1}$ and $J_{2}$,
and three $SU(4)_{R}$ charges $R_{1}$, $R_{2}$ and $R_{3}$ (we use the
notation of [7]). The anticommutator $\Delta$ can be evaluated
in terms of these quantum numbers. We have
$$\Delta=2\{S,Q\}=H-2J_{1}-\frac{3}{2}R_{1}-R_{2}-\frac{1}{2}R_{3}.$$
(3)
Hence we have to
calculate the partition function
$$Z_{\frac{1}{16}BPS}=\operatorname{tr}_{\Delta=0}x^{2H}z^{2J_{1}}y^{2J_{2}}v^{R%
_{2}}w^{R_{3}},$$
(4)
where we picked just
one possible choice of generating parameters. $R_{1}$ is of course
fixed by the condition $\Delta=0$. The above partition function
includes all single and multitrace operators. It counts all
operators annihilated by $Q$ and $S$. We thus do not restrict
ourselves to operators which are $\frac{1}{16}$BPS but not $\frac{1}{8}$BPS or
higher.
3 Gauge theory at zero coupling
In order to evaluate the partition function in gauge theory it is
convenient to use the oscillator representation, introduced in
[9], for all single trace
operators. In this picture, a single
trace operator $\operatorname{tr}O_{1}O_{2}O_{3}\ldots O_{L}$ is represented by $L$ sites
each occupied by the ‘elementary’ field $O_{i}$. Operators $O_{i}$ are in turn
represented by states in a Fock space generated by 4 bosonic
($a_{1}^{\dagger},a_{2}^{\dagger}$ and $b_{1}^{\dagger},b_{2}^{\dagger}$) creation
operators, and 4 fermionic ones ($c^{\dagger}_{1},c^{\dagger}_{2},c^{\dagger}_{3},c^{\dagger}_{4}$). The Fock space is narrowed by the
central charge constraint which relates the total number of
oscillators of various kinds on each site:
$$n_{a}-n_{b}+n_{c}=2.$$
(5)
For an explicit dictionary between operators and Fock space states see
[9].
The Lorentz spins $J_{1},J_{2}$ and the
$SU(4)$ charges are simply represented by the total number of various
oscillators111The sum $\sum_{i=1}^{3}q_{i}$ is defined as
$\frac{3}{2}R_{1}+R_{2}+\frac{1}{2}R_{3}$.:
$$\displaystyle R_{1}$$
$$\displaystyle=$$
$$\displaystyle{n_{c_{2}}}-{n_{c_{1}}},$$
$$\displaystyle R_{2}$$
$$\displaystyle=$$
$$\displaystyle{n_{c_{3}}}-{n_{c_{2}}},$$
$$\displaystyle R_{3}$$
$$\displaystyle=$$
$$\displaystyle{n_{c_{4}}}-{n_{c_{3}}},$$
$$\displaystyle\sum_{i=1}^{3}q_{i}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{2}({n_{c_{1}}}+{n_{c_{2}}}+{n_{c_{3}}}+{n_{c_{4}}})-2{n_%
{c_{1}}},$$
$$\displaystyle J_{1}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{2}({n_{a_{2}}}-{n_{a_{1}}}),$$
$$\displaystyle J_{2}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{2}({n_{b_{2}}}-{n_{b_{1}}}).$$
(6)
In the free theory, the
free dilatation operator $H_{0}$ also has a similar representation
$$H_{0}={n_{a_{1}}}+{n_{a_{2}}}+\frac{1}{2}({n_{c_{1}}}+{n_{c_{2}}}+{n_{c_{3}}}+%
{n_{c_{4}}}).$$
(7)
Consequently, in the free SYM theory, the condition $\Delta=0$ can
be evaluated to give
$$\Delta_{\lambda=0}=2{n_{a_{1}}}+2{n_{c_{1}}}=0.$$
(8)
Therefore, $\frac{1}{16}$BPS states are exactly the operators which do not have any
$a_{1}^{\dagger}$
or $c_{1}^{\dagger}$ operators in the oscillator representation. Since
all the spins and charges are expressed in terms of the total number
of oscillators of various kinds, it is convenient to keep track of
the number of oscillators of each type when counting $\frac{1}{16}$BPS operators.
We thus consider partition functions of the form
$$Z(a_{2},b_{1},b_{2},c_{2},c_{3},c_{4})=\sum_{\Delta=0}a_{2}^{n_{a_{2}}}b_{1}^{%
n_{b_{1}}}b_{2}^{n_{b_{2}}}c_{2}^{n_{c_{2}}}c_{3}^{n_{c_{3}}}c_{4}^{n_{c_{4}}}.$$
(9)
A simple counting over the Fock space states, taking into
consideration the central
charge constraint (5), gives for the ‘letter’ partition
function (partition function of operators at each site):
$$\displaystyle z_{B}$$
$$\displaystyle=$$
$$\displaystyle\frac{a_{2}^{2}+c_{2}c_{3}+c_{2}c_{4}+c_{3}c_{4}}{(1-b_{1}a_{2})(%
1-b_{2}a_{2})},$$
(10)
$$\displaystyle z_{F}$$
$$\displaystyle=$$
$$\displaystyle\frac{a_{2}(c_{2}+c_{3}+c_{4})+(b_{1}+b_{2}-a_{2}b_{1}b_{2})c_{2}%
c_{3}c_{4}}{(1-b_{1}a_{2})(1-b_{2}a_{2})},$$
(11)
where we made a separation into bosonic and fermionic states.
The partition function of single trace operators then follows from
$$Z_{s.t.}=-\sum_{n=1}^{\infty}\frac{\phi(n)}{n}\log\left(1-z_{B}(x^{n})-(-1)^{n%
+1}z_{F}(x^{n})\right),$$
(12)
where $x$ stands for generic arguments
(e.g. $x=(a_{2},b_{1},b_{2},c_{2},c_{3},c_{4})$ in our case).
Finally, the partition function of multitrace operators (at
$N_{c}=\infty$) is given by
$$Z=\exp\left(\sum_{n=1}^{\infty}\frac{1}{n}\left\{Z_{s.t.}^{B}(x^{n})+(-1)^{n+1%
}Z_{s.t.}^{F}(x^{n})\right\}\right).$$
(13)
The above formulas do not take into account finite $N$ effects which
appear, e.g. when certain long traces are equivalent to linear
combinations of shorter multitrace operators (due to the
Cayley-Hamilton theorem for finite matrices). For the specific case
of free SYM, the method of character expansions of
[10, 11] allows to perform an exact calculation at
finite $N$ starting directly from the letter partition function
(10). The resulting fixed $N$ partition function is
given by the formula
$$Z=\int DU\exp\left\{\sum_{n=1}^{\infty}\left(z_{B}(x^{n})+(-1)^{n+1}z_{F}(x^{n%
})\right)\frac{\operatorname{tr}U^{n}\operatorname{tr}U^{-n}}{n}\right\},$$
(14)
where the integral is over the
unitary group $U(N)$. For the case at hand this has been analyzed in
[7] for large $N$. For small values of parameters $x$ the
large $N$ limit of (14) does not depend on $N$
(reproducing effectively
(13)), while at a finite value of $x$ (strictly less than
1) the $\frac{1}{16}$BPS partition function exhibits a behavior $\log Z\sim N^{2}$.
4 Gauge theory at one loop
At one loop, $H=H_{0}+\lambda\delta H$ and the anomalous part is now a
nontrivial operator which acts on each two neighboring sites. The
complete one loop dilatation operator was constructed in
[9]. We discuss it in details in section 6 . The
condition $\Delta_{1-loop}=0$ now takes the form
$$\Delta_{1-loop}=\Delta_{\lambda=0}+\lambda\delta H=0.$$
(15)
Since at one loop the
eigenvalues of $\delta H$ are rational/radical expressions, for generic
transcendental $\lambda$ this condition picks out states which satisfy
both the free and one loop conditions separately, i.e. states
with ${n_{a_{1}}}={n_{c_{1}}}=0$ which do not get any anomalous dimensions $\delta H$
at one loop. We thus have to compute
$$Z=\sum_{\stackrel{{\scriptstyle{n_{a_{1}}}={n_{c_{1}}}=0}}{{\delta H=0}}}a_{2}%
^{n_{a_{2}}}b_{1}^{n_{b_{1}}}b_{2}^{n_{b_{2}}}c_{2}^{n_{c_{2}}}c_{3}^{n_{c_{3}%
}}c_{4}^{n_{c_{4}}}.$$
(16)
Note that now the
formula (12) no longer holds and we have to identify the number of
operators which do not get anomalous dimensions at one loop for each
$L$ independently.
We first determine the sum over single trace operators for fixed $L$ by
computing the above power series with some truncation on the number
of oscillators. This turns out to give enough information to guess
the analytical form of the generating function. Next, we test the
function on various configurations which were not used in the
process of obtaining the analytical form. The details of this
procedure are discussed in section 7.
Summing over $L$ gives the partition function of all single trace
operators. Then (13) may be used to get the partition
function of multitrace ones. Let us note that at one loop we do not
have a counterpart of the exact formula for finite $N$ valid for
zero coupling (14). We will, nevertheless, discuss
some aspects of the possible large $N$ behavior at the end of the
paper.
5 Supergraviton partition function
At strong coupling one can calculate the partition
function over $\frac{1}{16}$BPS states using the supergravity/superstring side of
the AdS/CFT correspondence. This has been considered in [7].
We now briefly review these results.
Since the $psu(2,2|4)$ supersymmetry algebra of the gauge theory is
also the symmetry group of superstrings in $AdS_{5}\times S^{5}$, we have
direct counterparts of $Q$ and $S$ operators and we can use them
to define the $\frac{1}{16}$BPS states.
In the low energy regime the partition function should be given by
supergravity fields which are annihilated by the $Q$ and $S$
operators. This has been done in [7] where the single particle
partition function
$$Z^{single}_{gravitons}=\sum_{\Delta=0}x^{2H}z^{2J_{1}}y^{2J_{2}}v^{R_{2}}w^{R_%
{3}},$$
(17)
was calculated with the result
$$\displaystyle Z^{single}_{gravitons}$$
$$\displaystyle=$$
$$\displaystyle\frac{bosons+fermions}{denominator},$$
(18)
$$\displaystyle denominator$$
$$\displaystyle=$$
$$\displaystyle(1-\frac{x^{2}}{w})(1-x^{2}v)(1-x^{2}\frac{w}{v})(1-x^{2}\frac{z}%
{y})(1-x^{2}zy),$$
(19)
$$\displaystyle bosons$$
$$\displaystyle=$$
$$\displaystyle vx^{2}+\frac{x^{2}}{w}+\frac{wx^{2}}{v}-\frac{x^{4}}{v}-\frac{vx%
^{4}}{w}-wx^{4}+2x^{6}+\frac{x^{6}z}{yv}$$
(20)
$$\displaystyle+\frac{vx^{6}z}{wy}+\frac{wx^{6}z}{y}-\frac{x^{8}z}{y}+\frac{x^{6%
}zy}{v}+\frac{vx^{6}zy}{w}+wx^{6}zy$$
$$\displaystyle-x^{8}zy+x^{4}z^{2}+x^{10}z^{2},$$
$$\displaystyle fermions$$
$$\displaystyle=$$
$$\displaystyle\frac{x^{3}}{y}+x^{3}y+\frac{x^{3}z}{v}+\frac{vx^{3}z}{w}+wx^{3}z%
-2x^{5}z+vx^{7}z$$
(21)
$$\displaystyle+\frac{x^{7}z}{w}+\frac{wx^{7}z}{v}+\frac{x^{7}z^{2}}{y}+x^{7}z^{%
2}y.$$
The full partition function is obtained by summation over the Fock
spaces of these particles using the formula
$$Z_{gravitons}=\exp\left(\sum_{n=1}^{\infty}\frac{1}{n}\left\{Z^{single,bos}_{%
gravitons}(x^{n},\ldots,w^{n})+(-1)^{n+1}Z^{single,fer}_{gravitons}(x^{n},%
\ldots,w^{n})\right\}\right).$$
(22)
We note that the above formula is identical in form to the one
obtained when passing from single- to multi-trace operators in gauge
theory (13).
It turns out that $Z_{gravitons}$ does not agree with the
result from free SYM theory [7]. In the case when $z=y=v=w=1$ even the
scaling of the entropy with energy is different.
Moreover, [7] obtained the partition function for $\frac{1}{8}$BPS
states by taking the limit $z\to 0$. The result again was in
disagreement with free SYM, but matched exactly the calculation made
using properties of the chiral ring of (interacting) SYM.
In section 8 we compare this supergraviton partition
function with perturbative computations at one loop in SYM.
When energies are large (compared with $N$) it is expected that the
partition function for $\frac{1}{16}$BPS states will have a
$$\log Z\sim N^{2},$$
(23)
behavior which should coincide with the one obtained from the $\frac{1}{16}$BPS black holes (see [7], section 5.3 for explicit formulas). In [7], a
similar qualitative behavior was obtained at zero coupling, although
the numerical details did not match. The motivation for this paper
was to investigate how much of this zero coupling result survives at
one loop.
6 The one loop dilatation operator
In this section we review the construction of the one loop
dilatation operator $\delta H$ in the oscillator picture [9].
The Fock space
Let us consider the space of operators which are traces of $L$
adjoint fields. We represent each of those fields as a state on
one of $L$ sites. Since the trace is cyclic invariant we restrict
ourselves to cyclic invariant states
of the Fock space at the end of
the calculation.
A generic state in Fock space is thus a linear combination of states
$$\mid s_{1}\rangle\otimes\ldots\otimes\mid s_{L}\rangle,$$
(24)
where on each site $i$ the
state $\mid s_{i}\rangle$ is obtained by acting with bosonic
$a_{1,i}^{\dagger},a_{2,i}^{\dagger},b_{1,i}^{\dagger},b_{2,i}^{\dagger}$ and fermionic $c_{1,i}^{\dagger},c_{2,i}^{\dagger},c_{3,i}^{\dagger},c_{4,i}^{\dagger}$ creation
operators on the Fock vacuum $\mid 0\rangle_{i}$. An arbitrary state
is labeled by the oscillator occupation numbers
$$\displaystyle\mid s_{i}\rangle$$
$$\displaystyle=$$
$$\displaystyle\mid n_{a_{1,i}},n_{a_{2,i}},n_{b_{1,i}},n_{b_{2,i}},n_{c_{1,i}},%
n_{c_{2,i}},n_{c_{3,i}},n_{c_{4,i}}\rangle$$
(25)
$$\displaystyle=$$
$$\displaystyle a_{1,i}^{\dagger\ n_{a_{1,i}}}a_{2,i}^{\dagger\ n_{a_{2,i}}}b_{1%
,i}^{\dagger\ n_{b_{1,i}}}b_{2,i}^{\dagger\ n_{b_{2,i}}}c_{1,i}^{\dagger\ n_{c%
_{1,i}}}c_{2,i}^{\dagger\ n_{c_{2,i}}}c_{3,i}^{\dagger\ n_{c_{3,i}}}c_{4,i}^{%
\dagger\ n_{c_{4,i}}}\mid 0\rangle_{i}.$$
As discussed in section 3, the occupation
numbers at each site are constrained by (5)
$$n_{a_{2,i}}+n_{a_{2,i}}-n_{b_{1,i}}-n_{b_{2,i}}+n_{c_{1,i}}+n_{c_{2,i}}+n_{c_{%
3,i}}+n_{c_{4,i}}=2.$$
(26)
The one loop dilatation
operator does not change the total number of oscillators of each
kind and only moves them from site to site. Therefore it acts within
the space with fixed total number of oscillators of any type for
fixed $L$. It follows that we can diagonalize $\delta H$ in subspaces
labeled by
$$[{n_{a_{1}}},{n_{a_{2}}},{n_{b_{1}}},{n_{b_{2}}},{n_{c_{1}}},{n_{c_{2}}},{n_{c%
_{3}}},{n_{c_{4}}};L].$$
(27)
The harmonic action
Let us now review the construction of $\delta H$ [9],
giving more details about the computer code implementation.
The action of the one-loop dilatation operator introduces an
interaction between only the neighboring sites (the last and the
first site are assumed to be neighbors). For this reason it is
enough to consider a pair of such sites
$$\mid v\rangle=\mid n_{a_{1}},n_{a_{2}},n_{b_{1}},n_{b_{2}},n_{c_{1}},n_{c_{2}}%
,n_{c_{3}},n_{c_{4}}\rangle\otimes\mid m_{a_{1}},m_{a_{2}},m_{b_{1}},m_{b_{2}}%
,m_{c_{1}},m_{c_{2}},m_{c_{3}},m_{c_{4}}\rangle,$$
(28)
where we dropped the index $i$ ( and $i+1$ ) for clarity. Our object
is now to calculate the hamiltonian matrix element between
(28) and an arbitrary other state
$$\mid v^{\prime}\rangle=\mid n^{\prime}_{a_{1}},n^{\prime}_{a_{2}},n^{\prime}_{%
b_{1}},n^{\prime}_{b_{2}},n^{\prime}_{c_{1}},n^{\prime}_{c_{2}},n^{\prime}_{c_%
{3}},n^{\prime}_{c_{4}}\rangle\otimes\mid m^{\prime}_{a_{1}},m^{\prime}_{a_{2}%
},m^{\prime}_{b_{1}},m^{\prime}_{b_{2}},m^{\prime}_{c_{1}},m^{\prime}_{c_{2}},%
m^{\prime}_{c_{3}},m^{\prime}_{c_{4}}\rangle.$$
(29)
If one is interested in calculating the element $\langle v^{\prime}\mid H\mid v\rangle$ then it turns out that the harmonic action [9]
can be described by the following set of rules
•
consider all the possibilities of oscillator hopping from site
$i\to i+1$ and from site $i+1\to i$ such that the state
(28) becomes (29)
•
to each such possibility associate a number
$$c_{n,n_{12},n_{21}}=(-1)^{1+n_{12}n_{21}}\frac{\Gamma(\frac{1}{2}n_{12}+\frac{%
1}{2}n_{21})\Gamma(1+\frac{1}{2}n-\frac{1}{2}n_{12}-\frac{1}{2}n_{21})}{\Gamma%
(1+\frac{1}{2}n)},$$
(30)
where $n_{12}$, $n_{21}$ are the numbers of oscillators hopping from
$i\to i+1$, $i+1\to i$ respectively, $n$ is the total number of
quanta at sites $i$ and $i+1$ in the beginning
•
include the $-1$ factors when the fermion oscillators are
hopping ”over” other fermions.
In particular, if a fermion is hopping form $i=1$ to $i=L$ or vice
versa then all fermions in between (for $1<i<L$) have to be
considered.
•
sum over all the possibilities and multiply the result by $\frac{\left\|\mid v^{\prime}\rangle\right\|}{\left\|\mid v\rangle\right\|}$
The harmonic action can be implemented in two independent ways. One is
to use the above rules as they are and compute the element $\langle v^{\prime}\mid H\mid v\rangle$ indirectly by evaluating
$$H\mid v\rangle=\sum_{v^{\prime}}H_{v,v^{\prime}}\mid v^{\prime}\rangle.$$
(31)
Second is to write down the formula for the matrix element $\langle v^{\prime}\mid H\mid v\rangle$ and compute it explicitly. It turns out to be
possible, we have
$$\langle v^{\prime}\mid H\mid v\rangle=\sqrt{\prod_{t\in T}\frac{{n^{\prime}}_{%
t}!}{n_{t}!}}\sum_{t^{\prime}\in T}\sum_{k_{t^{\prime}}=0}^{m_{t^{\prime}}}(-1%
)^{F}c_{x,y,z}\prod_{t^{\prime\prime}\in T}\binom{n_{t^{\prime\prime}}}{n_{t^{%
\prime\prime}}-n^{\prime}_{t^{\prime\prime}}+k_{t^{\prime\prime}}}\binom{m_{t^%
{\prime\prime}}}{k_{t^{\prime\prime}}},$$
$$T=\{a_{1},a_{2},b_{1},b_{2},c_{1},c_{2},c_{3},c_{4}\},$$
$$x=\sum_{t\in T}n_{t}+m_{t},\ \ \ \ y=\sum_{t\in T}\epsilon_{t}|n_{t}-n^{\prime%
}_{t}|+k_{t},\ \ \ \ z=\sum_{t\in T}\eta_{t}|n_{t}-n^{\prime}_{t}|+k_{t},$$
(32)
where $(-1)^{F}$ is the fermion number discussed in one of the rules and
where the parameters $\epsilon_{t}$, $\eta_{t}$ are defined in the
following way.
$\epsilon_{t}$ is equal $1$ if $n_{t}\geq n^{\prime}_{t}$ and $0$ otherwise,
$\eta_{t}=1-\epsilon_{t}$.
The above formula can be justified in the following way. Let us
consider $n_{a_{1}}$ bosons $a_{1}^{\dagger}$. We want them to hop form
$i\to i+1$ so that only $n^{\prime}_{a_{1}}$ of them are left ( clearly we
assume that $n_{a_{1}}\geq n^{\prime}_{a_{1}}$). Since they commute and are
indistinguishable the number of possibilities coincides with the
number of combinations $\binom{n_{a_{1}}}{n_{a_{1}}-n^{\prime}_{a_{1}}}$. Other
possibilities are when the number of such hops is
$n_{a_{1}}-n^{\prime}_{a_{1}}+k_{a_{1}}$ with $k_{a_{1}}>0$. Then, we have to hop back (from
$i+1\to i$) exactly $k_{a_{1}}$ oscillators $a_{1}^{\dagger}$. This can be done in
$\binom{m_{a_{1}}}{k_{a_{1}}}$ ways. Therefore, the net factor for given
$k_{a_{1}}$ is
$\binom{n_{a_{1}}}{n_{a_{1}}-n^{\prime}_{a_{1}}+k_{a_{1}}}\binom{m_{a_{1}}}{k_{%
a_{1}}}$.
To include all the possibilities we sum over all possible $k_{a_{1}}$’s,
i.e from $0$ to $m_{a_{1}}$.
For the other bosonic and fermionic operators $a_{2}^{\dagger},\ b_{1}^{\dagger},\ b_{2}^{\dagger},\ c_{1}^{\dagger},\ c_{2}^%
{\dagger},\ c_{3}^{\dagger},\ c_{4}^{\dagger}$ the analysis in analogous and gives
the corresponding factors as in (6). To include the $-1$
factors coming from hopping of fermions we weight the sum
(6) with the factor $(-1)^{F}$.
We have implemented the above construction of the one loop dilatation
operator independently in two different programs and verified that the
results agree. As a further check we reproduced various one loop
anomalous dimensions given in [9].
In order to complete the construction we project the Hilbert space
(24) to the
subspace of states which are invariant under cyclic permutations,
since only these states correspond to gauge theory single trace
operators.
We start with the hamiltonian matrix $H$ represented in the
non-cyclic invariant basis $B$ (24) constructed as
above. Then, we
construct a matrix
representation of an operator $T$ which translates the chain by one
site. The cyclic invariant states correspond to eigenvectors
$v_{1},\ldots,v_{n}$, $\ n\leq\#B$ of $T$ with an eigenvalue equal 1.
Now, we build the projection matrix
$P=[v_{1},\ldots,v_{n}]$ and perform the similarity transformation
$$H\to PHP^{T},$$
on $H$. The result is the hamiltonian matrix
represented in the cyclic invariant basis.
7 The one loop $\frac{1}{16}$BPS partition function
According to the general discussion in previous sections, the tree
level condition for the $\frac{1}{16}$BPS states contributing to the index is
$\Delta_{\lambda=0}=2n_{a_{1}}+2n_{c_{1}}=0$ hence from now on we take
$n_{a_{1,i}}=n_{c_{1,i}}=0$. Moreover, at one loop level the
condition $\Delta_{1-loop}=0$ is satisfied only for states which are
eigenstates corresponding to $0$ eigenvalue of the one loop dilatation
operator. Let
$D_{n_{a_{2}},n_{b_{1}},n_{b_{2}},n_{c_{2}},n_{c_{3}},n_{c_{4}},L}$ be the
number of such states in the sector with $n_{a_{2}}$, $n_{b_{1}}$,
$n_{b_{2}}$, $n_{c_{2}}$, $n_{c_{3}}$, $n_{c_{4}}$ number of quanta and $L$
sites respectively. The generating function we are looking for is
$$Z_{L}^{1/16th}(a_{2},b_{1},b_{2},c_{2},c_{3},c_{4})=\hskip-22.762205pt\sum_{%
\stackrel{{\scriptstyle n_{a_{2}},n_{b_{1}},n_{b_{2}}=0,\ldots,\infty}}{{n_{c_%
{2}},n_{c_{3}},n_{c_{4}}=0,\ldots,L}}}\hskip-22.762205ptD_{n_{a_{2}},n_{b_{1}}%
,n_{b_{2}},n_{c_{2}},n_{c_{3}},n_{c_{4}},L}a_{2}^{n_{a_{2}}}b_{1}^{n_{b_{1}}}b%
_{2}^{n_{b_{2}}}c_{2}^{n_{c_{2}}}c_{3}^{n_{c_{3}}}c_{4}^{n_{c_{4}}},$$
(the sum over fermionic variables runs from $0$ to $L$ due to the
Pauli exclusion principle). With use of computer code implementation of
the rules discussed in previous section, one can determine the
numbers $D_{n_{a_{2}},n_{b_{1}},n_{b_{2}},n_{c_{2}},n_{c_{3}},n_{c_{4}},L}$
exactly, but of course only for a finite number of configurations.
It is by no means obvious that such data can determine the whole
function
$Z_{L}^{1/16th}(a_{2},b_{1},b_{2},c_{2},c_{3},c_{4})$. Nevertheless, our analysis
shows that the Taylor expansion of the function
$Z_{L}^{1/16th}(a_{2},b_{1},b_{2},c_{2},c_{3},c_{4})$ coincides with the expansion of
certain rational function. The details of our computation are below.
For $L=2$ we analyzed the configuration with $0\leq n_{a_{2}},\ n_{b_{1}},\ n_{b_{2}}\leq 10$ and $0\leq n_{c_{2}},\ n_{c_{3}},\ n_{c_{4}}\leq 2$. There are $11^{3}3^{3}=35937$ such possibilities however
only $1494$ of them satisfy the central charge constraint.
For $L=3$ we took $0\leq n_{a_{2}},\ n_{b_{1}},\ n_{b_{2}}\leq 5$ and $0\leq n_{c_{2}},\ n_{c_{3}},\ n_{c_{4}}\leq 3$. There are $6^{3}4^{3}=13824$ such possibilities
among which only $849$ satisfy the central charge constraint.
For $L=4$ we analyzed the configuration with $0\leq n_{a_{2}},\ n_{b_{1}},\ n_{b_{2}}\leq 2$ and $0\leq n_{c_{2}},\ n_{c_{3}},\ n_{c_{4}}\leq 4$. There are $3^{3}5^{3}=3375$ such possibilities and $279$ which satisfy the central
charge constraint.
Let us now explain how the partition function was
reconstructed from the above data and consider in detail the case of
$L=2$. The computer analysis gives a polynomial
$$Z_{L=2}^{1/16th,cut}(a_{2},b_{1},b_{2},c_{2},c_{3},c_{4})=\hskip-22.762205pt%
\sum_{\stackrel{{\scriptstyle n_{a_{2}},n_{b_{1}},n_{b_{2}}=0,\ldots,10}}{{n_{%
c_{2}},n_{c_{3}},n_{c_{4}}=0,1,2}}}\hskip-22.762205ptD_{n_{a_{2}},n_{b_{1}},n_%
{b_{2}},n_{c_{2}},n_{c_{3}},n_{c_{4}},2}a_{2}^{n_{a_{2}}}b_{1}^{n_{b_{1}}}b_{2%
}^{n_{b_{2}}}c_{2}^{n_{c_{2}}}c_{3}^{n_{c_{3}}}c_{4}^{n_{c_{4}}},$$
which consists of $1494$ terms.
Our strategy to proceed is the following. If
the full partition function $Z_{L=2}^{1/16th}(a_{2},b_{1},b_{2},c_{2},c_{3},c_{4})$ is a
rational function then, in particular, so is $Z_{L=2}^{1/16th}(a_{2},1,1,1,1,1)$.
Therefore, the coefficients of $Z_{L=2}^{1/16th}(a_{2},1,1,1,1,1)$ should be
”easily” recognizable. Indeed, we have
$$\displaystyle Z_{L=2}^{1/16th,cut}(a_{2},1,1,1,1,1)=13+40a_{2}+72a_{2}^{2}+104%
a_{2}^{3}+136a_{2}^{4}+168a_{2}^{5}$$
$$\displaystyle\quad\quad\quad\quad+200a_{2}^{6}+232a_{2}^{7}+264a_{2}^{8}+296a_%
{2}^{9}+320a_{2}^{10},$$
(33)
which (except for the last term $320a_{2}^{10}$) is recognized as the
Taylor expansion of
$$\frac{13+14a_{2}+5a_{2}^{2}}{(1-a_{2})^{2}}.$$
Next, we turn on the variable $b_{1}$, i.e. we consider
$Z_{L=2}^{1/16th,cut}(a_{2},b_{1},1,1,1,1)$ and find the corresponding generating
function. Then, we proceed analogously with
$Z_{L=2}^{1/16th,cut}(a_{2},b_{1},b_{2},1,1,1)$ and find that
$$Z_{L=2}^{1/16th}(a_{2},b_{1},b_{2},1,1,1)=\frac{6+8a_{2}+3a_{2}^{2}+(3+3a_{2}+%
a_{2}^{2})(b_{1}+b_{2})+b_{1}b_{2}}{(1-b_{1}a_{2})(1-b_{2}a_{2})}.$$
The full $a_{2}$, $b_{1}$, $b_{2}$ dependence is now determined. In order
to find the $c_{2}$, $c_{3}$, $c_{4}$ dependence we do the following.
First, due to the Pauli exclusion principle the fermionic variables
cannot be in the denominator $(1-b_{1}a_{2})(1-b_{2}a_{2})$. Therefore they
enter only in the numerator in the form $c_{2}^{i}c_{3}^{j}c_{3}^{k}$, $i,j,k\leq L$. Second, the harmonic action is completely symmetric with
respect to fermionic oscillators. This implies that the numerator of
$Z_{L=2}^{1/16th}(a_{2},b_{1},b_{2},c_{2},c_{3},c_{4})$ is a completely symmetric function
with respect to $c_{2}$, $c_{3}$ and $c_{4}$. 222By the same
argument the full partition function has to be symmetric in bosonic
variables $b_{1}$, $b_{2}$ hence only the combinations $b_{1}+b_{2}$, $b_{1}b_{2}$ appear in the numerator. It is not obvious why there are no
other, higher order, combinations e.g. $b_{1}^{3}+b_{2}^{3}$. Clearly, this
must be a property of the harmonic action. . We therefore write
$Z_{L=2}^{1/16th}(a_{2},b_{1},b_{2},c_{2},c_{3},c_{4})$ as
$$\frac{1}{(1-b_{1}a_{2})(1-b_{2}a_{2})}\sum_{n=0}^{2}\sum_{m=0}^{2}\sum_{l_{1},%
l_{2},l_{3}=0}^{2}\tilde{D}_{n,m,l_{1},l_{2},l_{3}}A_{n}B_{m}\sigma_{l_{1},l_{%
2},l_{3}}(c_{2},c_{3},c_{4}),$$
(34)
where
$$A_{0}=1,\ A_{1}=a_{2},\ A_{2}=a_{2}^{2},$$
$$B_{0}=1,\ B_{1}=b_{1}+b_{2},\ B_{2}=b_{1}b_{2},$$
$\tilde{D}_{n,m,l_{1},l_{2},l_{3}}$ are some coefficients to be determined and
$\sigma_{l_{1},l_{2},l_{3}}(c_{2},c_{3},c_{4})$ are Schur polynomials
333Other bases of symmetric functions, e.g.
$(c_{2}+c_{3}+c_{4})^{i}(c_{2}c_{3}+c_{3}c_{4}+c_{4}c_{1})^{j}(c_{2}c_{3}c_{4})%
^{k}$ are possible.
However, our choice of Schur polynomials turns out to give simple
expression (36). defined as
$$\sigma_{n_{1},n_{2},n_{3}}(x_{1},x_{2},x_{3})=\left|\begin{array}[]{ccc}x_{1}^%
{n_{1}+2}&x_{2}^{n_{1}+2}&x_{3}^{n_{1}+2}\\
x_{1}^{n_{2}+1}&x_{2}^{n_{2}+1}&x_{3}^{n_{2}+1}\\
x_{1}^{n_{3}}&x_{2}^{n_{3}}&x_{3}^{n_{3}}\end{array}\right|/\left|\begin{array%
}[]{ccc}x_{1}^{2}&x_{2}^{2}&x_{3}^{2}\\
x_{1}&x_{2}&x_{3}\\
1&1&1\end{array}\right|.$$
(35)
The coefficients can be obtained by comparing the
Taylor expansion of (34) with
$Z_{L=2}^{1/16th,cut}(a_{2},b_{1},b_{2},c_{2},c_{3},c_{4})$.
The analysis for $L=3,4$ ( with the sum over $l_{1}$, $l_{2}$, $l_{3}$
in (34) from $0$ to $L$ ) is analogous. In this manner we
obtain a fairly simple rational generating functions for $L=2,3,4$.
Given those three functions, it was possible to guess the
partition function for arbitrary $L$.
The final result turns out to have a particularly simple expression in
terms of Schur polynomials namely
$$Z_{L}^{1/16th}(a_{2},b_{1},b_{2},c_{2},c_{3},c_{4})=\frac{P}{(1-a_{2}b_{1})(1-%
a_{2}b_{2})},$$
(36)
$$P=\sigma_{L,L,0}+a_{2}\sigma_{L,L-1,0}+a_{2}^{2}\sigma_{L-1,L-1,0}$$
$$+(b_{1}+b_{2})\left(\sigma_{L,L,1}+a_{2}\sigma_{L,L-1,1}+a_{2}^{2}\sigma_{L-1,%
L-1,1}\right)$$
$$+b_{1}b_{2}\left(\sigma_{L,L,2}+a_{2}\sigma_{L,L-1,2}+a_{2}^{2}\sigma_{L-1,L-1%
,2}\right),$$
where $\sigma_{n_{1},n_{2},n_{3}}=\sigma_{n_{1},n_{2},n_{3}}(c_{2},c_{3},c_{4})$ is the
Schur polynomial (35).
In order to test the above result further, we performed the analysis for
for $L=5$ with $0\leq n_{a_{2}},\ n_{b_{1}},\ n_{b_{2}}\leq 2$ and $0\leq n_{c_{2}},\ n_{c_{3}},\ n_{c_{4}}\leq 5$. There are $3^{3}6^{3}=5832$ such configurations and $414$ which satisfy the central
charge constraint. The corresponding generating function indeed
confirms (36).
In the remaining part of this section we perform other checks of (36).
The partition function for $\frac{1}{8}$BPS states
The $\frac{1}{8}$BPS states
are obtained by imposing the additional condition $J_{1}=0$ on the $\frac{1}{16}$BPS states [7]. This is equivalent to
setting the corresponding constraint on the numbers of quanta, namely
$$n_{a_{2}}=0.$$
(37)
The central charge condition in this case
$$2L+n_{b_{1}}+n_{b_{2}}=n_{c_{2}}+n_{c_{3}}+n_{c_{4}},$$
(38)
ensures that for given $L$ there is only a finite number of such
configurations.
It follows that the corresponding generating function is a
polynomial in the variables $b_{1}$, $b_{2}$, $c_{2}$, $c_{3}$, $c_{4}$. Since
$n_{c_{2}},n_{c_{3}},n_{c_{4}}\leq L$ the numbers $n_{b_{1}}$, $n_{b_{2}}$ are
also bounded, i.e. $n_{b_{1}},n_{b_{2}}\leq L$.
These simplifications allow us to perform explicit computations of
the generating function for these states for higher $L$ (we did it for
$L=6$ and $L=7$) and check these results with the general partition
function obtained in the previous section.
Indeed, we find that the resulting functions coincide with (36) after
setting $a_{2}=0$, i.e. they are
$$Z_{L}^{1/8th}(b_{1},b_{2},c_{2},c_{3},c_{4})=Z_{L}^{1/16th}(0,b_{1},b_{2},c_{2%
},c_{3},c_{4})$$
$$=\sigma_{L,L,0}(c_{2},c_{3},c_{4})+(b_{1}+b_{2})\sigma_{L,L,1}(c_{2},c_{3},c_{%
4})+b_{1}b_{2}\sigma_{L,L,2}(c_{2},c_{3},c_{4}).$$
Checks for operators with many derivatives
One puzzling feature of the generating function (36) is that
the denominators contain only two factors: $(1-a_{2}b_{1})(1-a_{2}b_{2})$. This suggests that the derivatives in $\frac{1}{16}$BPS states
are essentially commutative (we will discuss this point further in
section 9) which has a crucial impact on the singularity structure of
the $\frac{1}{16}$BPS partition function.
The test of $\frac{1}{8}$BPS states checks the numerator and does not
involve any derivative terms. In order to check for derivatives we
looked at the following configurations for $L=5$ and
$n_{c_{3}}=n_{c_{4}}=5$ : i) 7 and 10 $a_{2}b_{1}$ derivatives, i.e.
[0,7,7,0,0,5,5,5], [0,10,10,0,0,5,5,5]; ii) 6 derivatives of both
types, i.e. [0,6,6,0,0,5,5,5], [0,6,5,1,0,5,5,5], …[0,6,3,3,0,5,5,5], iii) states with derivatives and an additional
$c_{2}$ oscillator, i.e. [0,5,5,0,1,4,5,5], [0,5,4,1,1,4,5,5]. In all
cases, despite the large number of derivatives we found only a
single $\frac{1}{16}$BPS state in those sectors, which is consistent with (36).
Letter partition function
Finally, let us note that, although we guessed the partition function
for $\frac{1}{16}$BPS states starting from $L=2$ and proceeding to $L>2$,
substituting $L=1$ in (36), we recover the letter
partition function (10) which correspond to $\frac{1}{16}$BPS operators
in a $U(N)$ gauge theory. This is another consistency check of our
analytical formula (36).
8 Comparision with supergravity
The one loop $\frac{1}{16}$BPS partition function can be calculated from the length $L$
partition functions obtained in the previous section in two
steps. First, the single trace partition function is obtained through
$$Z_{s.t.}=\sum_{L=1}^{\infty}Z_{L}^{1/16th},$$
(39)
where we sum from $L=1$ since we are considering the partition
function in a $U(N)$ gauge theory444This will turn out to be
crucial for comparison with the supergraviton partition
functions.. Then, the full $\frac{1}{16}$BPS partition function is obtained by
passing to multitrace operators through
$$Z=\exp\left(\sum_{n=1}^{\infty}\frac{1}{n}\left\{Z^{B}_{s.t.}(x^{n},\ldots)+(-%
1)^{n+1}Z^{F}_{s.t.}(x^{n},\ldots)\right\}\right).$$
(40)
In this step we are using the fact that only the planar
one loop dilatation operator is considered. So we are in the strict $N\to\infty$
limit.
The first sum (39) can be carried out analytically, and
the result is
$$Z_{s.t.}=\frac{bosons+fermions}{denominator},$$
(41)
where
$$\displaystyle denominator$$
$$\displaystyle=$$
$$\displaystyle\!\!\!(1-a_{2}b_{1})(1-a_{2}b_{2})(1-c_{2}c_{3})(1-c_{2}c_{4})(1-%
c_{3}c_{4}),$$
$$\displaystyle bosons$$
$$\displaystyle=$$
$$\displaystyle\!\!\!a_{2}^{2}+c_{2}c_{3}+(c_{2}+c_{3})(1+(a_{2}(b_{1}+b_{2})-1)%
c_{2}c_{3})c_{4}$$
$$\displaystyle+c_{2}c_{3}(a_{2}b_{1}+a_{2}b_{2}-1+(1+b_{1}b_{2}+a_{2}^{2}b_{1}b%
_{2}-a_{2}(b_{1}+b_{2}))c_{2}c_{3})c_{4}^{2},$$
$$\displaystyle fermions$$
$$\displaystyle=$$
$$\displaystyle\!\!\!(b_{1}+b_{2})c_{2}c_{3}c_{4}+a_{2}^{2}(b_{1}+b_{2})c_{2}c_{%
3}c_{4}$$
$$\displaystyle+a_{2}(c_{3}+c_{4}+b_{1}b_{2}c_{2}^{2}c_{3}c_{4}(c_{3}+c_{4})+c_{%
2}(c_{3}c_{4}-1)(b_{1}b_{2}c_{3}c_{4}-1)).$$
Let us now compare this result with the single particle
supergraviton $\frac{1}{16}$BPS partition function (18). In order to
make the comparison possible we express the variables $x$, $v$, $w$, $z$
in terms of $a_{2}$, $c_{2}$, $c_{3}$, $c_{4}$ and $b_{1}$, $b_{2}$ in
terms of $y$. Using the definitions (3) and
(7) we find that $b_{1}=1/y$, $b_{2}=y$ and we obtain the
dictionary
$$\displaystyle x$$
$$\displaystyle=$$
$$\displaystyle c_{2}^{\frac{1}{3}}c_{3}^{\frac{1}{3}}c_{4}^{\frac{1}{3}},$$
(42)
$$\displaystyle v$$
$$\displaystyle=$$
$$\displaystyle c_{2}^{-\frac{2}{3}}c_{3}^{\frac{1}{3}}c_{4}^{\frac{1}{3}},$$
(43)
$$\displaystyle w$$
$$\displaystyle=$$
$$\displaystyle c_{2}^{-\frac{1}{3}}c_{3}^{-\frac{1}{3}}c_{4}^{\frac{2}{3}},$$
(44)
$$\displaystyle z$$
$$\displaystyle=$$
$$\displaystyle a_{2}c_{2}^{-\frac{2}{3}}c_{3}^{-\frac{2}{3}}c_{4}^{-\frac{2}{3}}.$$
(45)
Remarkably enough, with these substitutions the supergraviton $\frac{1}{16}$BPS partition function (18) coincides exactly with the
one loop single trace $\frac{1}{16}$BPS partition function (41). Thus, the full
$\frac{1}{16}$BPS partition functions coincide also, since the summation over
multitrace operators is mathematically equivalent to summation over
the supergraviton Fock spaces (c.f. (22) and
(40)).
As a byproduct we note that the resulting $\frac{1}{8}$BPS partition
function obtained from the one loop perturbative dilatation operator
exactly coincides with the gauge theory result obtained from the
chiral ring reasoning as in [7].
9 Discussion of large $N$ asymptotics
Ultimately we are interested in the behavior of the partition
function of $\frac{1}{16}$BPS states which scales like $\log Z\propto N^{2}$ and
which therefore can account for the entropy of $\frac{1}{16}$BPS charged black
holes in $AdS_{5}\times S^{5}$. In [7] a calculation in the free
theory showed that, for values of the chemical potentials below a
certain value, the partition function has a $N\to\infty$ limit,
while above that value one obtains $\log Z\propto N^{2}$ scaling.
This analysis follows from formula (14) which is
exact for any $N$.
At one loop, we do not have a similar exact formula since the $\frac{1}{16}$BPS states are very specific and form just a tiny fraction of all
operators made from the ‘letters’ (10). It is thus
interesting to understand whether staying within the $N=\infty$ phase
one can see the transition to the ‘black hole’ phase. Firstly, at
finite $N$, the number of states is
diminished due to trace identities following from the Cayley-Hamilton
theorem. Thus, there is a chance of observing $\log Z\propto N^{2}$ at a
certain
value of the parameters only when the corresponding $N=\infty$
partition function has a singularity there or is divergent.
Quite remarkably, our single trace partition function is finite for
all arguments less than 1. This is in stark contrast with the free
$\frac{1}{16}$BPS partition function which blows up much earlier (see section 5
in [7]). Let us note that there, this conclusion was reached
from the exact formula (14). However one can see
this behavior studying directly the single trace partition function
(12). We checked that calculating (12) as
a power series even for the simplest case of two
noncommutative letters, and studying its radius of convergence
recovers exactly the leading singularity (strictly below 1) which
coincides with the transition point obtained from
(14). This experiment gives us confidence that the
knowledge of $N=\infty$ partition function can be a reliable guide
to the singularity structure and hence to the position of the phase
transition.
In order to obtain some rough idea about the structure of the one loop
$\frac{1}{16}$BPS states we tried to see whether one can introduce some
‘effective letters’ which would then reproduce the one loop single
trace partition function (41). We define effective letter
functions $z_{B}^{eff.}(x)$ and $z_{F}^{eff.}(x)$ for bosons and
fermions respectively using the formula
$$Z_{s.t.}=-\sum_{n=1}^{\infty}\frac{\phi(n)}{n}\log\left(1-z_{B}^{eff.}(x^{n})-%
(-1)^{n+1}z_{F}^{eff.}(x^{n})\right).$$
(46)
where on the left hand side we put the generating function of single
trace $\frac{1}{16}$BPS operators obtained in the present paper.
Here, the most natural choice of
chemical potentials is $a_{2}=x^{3}$,
$c_{2}=c_{3}=c_{4}=x$ and $b_{1}=b_{2}=1$ (see
[7, 12]). Expanding the l.h.s and r.h.s of
(46)
it is possible to solve nonlinear equations for the coefficients of
the Taylor expansion of $z_{B}^{eff.}(x)$ and $z_{F}^{eff.}(x)$.
We have found unique solutions up to the 28th order of $x$. All the
coefficients are integers however they
become very large and negative which indicates that any
‘noncommutative’ letters drastically overcount the $\frac{1}{16}$BPS states.
The above experiment suggests that the building blocks of $\frac{1}{16}$BPS states
are predominantly commutative similarly to the building blocks of
$\frac{1}{8}$BPS states as discussed in section 6.1 of [7]. This is
further supported by the structure of the denominator in (41)
which essentially means that only the total number of blocks like
$a_{2}b_{1}$ etc. matters – and not their ordering.
Using the above guiding principle of commutativity we tried to apply
the ‘plethystic’ formalism of [13]. In this formalism, the
partition function at finite $N$ can be reconstructed from the
$N=\infty$ one in the following manner. Suppose that the single trace
bosonic and fermionic partition functions at $N=\infty$ are given by
$$Z_{s.t.}^{B}=\sum_{n=0}^{\infty}a_{n}x^{n},\quad\quad\quad\quad Z_{s.t.}^{F}=%
\sum_{n=0}^{\infty}b_{n}x^{n},$$
(47)
then the finite $N$ partition function $Z_{N}(x)$ is obtained from the
infinite product expansion:
$$\frac{\prod_{n=1}^{\infty}(1+gx^{n})^{b_{n}}}{\prod_{n=1}^{\infty}(1-gx^{n})^{%
a_{n}}}=\sum_{N}Z_{N}(x)g^{N},$$
(48)
which in fact exactly reproduces the finite $N$ structure of
$\frac{1}{8}$BPS states obtained from chiral ring arguments taking as
input only the $N=\infty$ result. However we
do not see any chance of a $\log Z\propto N^{2}$ behavior when we apply
this formalism to our partition function.
It has been suggested in the literature [14] that the dual states
contributing mainly to the black hole entropy would be of a
determinant type. For fixed $N$, states which are
determinants of some matrices can be expressed as combinations of
multitrace operators. So at least formally, these states are within
the space of states that we consider (which consists of all multitrace
operators).
Our conclusion is that in order to see the $\log Z\propto N^{2}$
scaling, one has to use the whole nonplanar one loop dilatation operator
the properties of which probably have a huge impact on the counting of
$\frac{1}{16}$BPS states with very many traces.
10 Summary
In this paper we determined the partition function of $\frac{1}{16}$BPS operators
in planar perturbative ${\cal N}=4$ SYM at one loop. We used the oscillator
representation of gauge theory operators and of the planar dilatation
operator.
In order to obtain the partition function we determined the number of
$\frac{1}{16}$BPS operators for a certain set of restrictions on the number of
oscillators and for operators of lengths less than 5. Then, we reconstructed a
generating function (assuming that it is a rational function) which
reproduced all these results. Subsequently we made numerous further
checks by evaluating the dilatation operator for higher length and
larger number of (some) oscillators and checking the result with the
proposed generating function.
The main result that we found is an exact agreement with the
partition function of $\frac{1}{16}$BPS supergravitons in $AdS_{5}\times S^{5}$. Consequently we also reproduce exactly, using the one loop
perturbative dilatation operator, the counting of
$\frac{1}{8}$BPS states which was previously done on the gauge theory
side using chiral ring reasonings [7].
Using the identification of single particle supergraviton states in
terms of short representations of $psu(2,2|4)$ (see [7]) and the
equality with the gauge theory partition function extracted in the
present work we may identify all $\frac{1}{16}$BPS states as descendents of $\operatorname{tr}Z^{L}$ operators. Thus these states will also persist to be $\frac{1}{16}$BPS at
higher loop orders. In the process of extracting the partition
function from the 1-loop hamiltonian data we did not use in any way
any information about the representation theory of $psu(2,2|4)$. The
fact that we recover all states in these multiplets is a further check
of this procedure.
However it is perhaps a bit surprising, in view of the applications to
black hole entropy, that we do not observe any other
new primary states (or their descendants). As a word of caution we
note that these might in principle appear for higher lengths and
oscillator occupancy numbers than we could check. However, given the
various checks and consistency with $\frac{1}{8}$BPS and extrapolation to
$L=1$ letters we do not think that this is very probable.
The huge reduction of the number of $\frac{1}{16}$BPS states with respect to the
free theory reinstated agreement with supergraviton partition function.
However, the transition to a phase with black hole like scaling which
was seen at zero coupling seems to disappear. The form of our
partition function suggests that the constituents generating the $\frac{1}{16}$BPS states behave much more like commutative objects than ‘noncommutative
letters’. We speculate that in order to see the black hole phase
explicitly from gauge theory, one has to use the complete nonplanar
dilatation operator.
Acknowledgments. RJ thanks Juan Maldacena for pointing
out this problem and Niels Obers, Javier Mas and Adam
Rej for interesting discussions. This
work has been supported in part by Polish Ministry of Science and
Information Technologies grant 1P03B04029 (2005-2008), RTN network
ENRAGE MRTN-CT-2004-005616, the Marie Curie ToK COCOS (contract
MTKD-CT-2004-517186) (RJ) and by the Marie Curie Research Training
Network ENIGMA (contract MRNT-CT-2004-5652) (MT). We would like to
thank the Isaac Newton Institute for hospitality when this work was being
finished.
References
[1]
J. M. Maldacena,
“The large N limit of superconformal field theories and supergravity,”
Adv. Theor. Math. Phys. 2 (1998) 231
[Int. J. Theor. Phys. 38 (1999) 1113], [hep-th/9711200];
S. S. Gubser, I. R. Klebanov and A. M. Polyakov,
“Gauge theory correlators from non-critical string theory,”
Phys. Lett. B 428 (1998) 105, [hep-th/9802109];
E. Witten,
“Anti-de Sitter space and holography,”
Adv. Theor. Math. Phys. 2 (1998) 253, [hep-th/9802150].
[2]
D. Berenstein,
“Large N BPS states and emergent quantum gravity,”
JHEP 0601 (2006) 125
[arXiv:hep-th/0507203].
[3]
G. Mandal,
“Fermions from half-BPS supergravity,”
JHEP 0508, 052 (2005)
[arXiv:hep-th/0502104].
[4]
L. Grant, L. Maoz, J. Marsano, K. Papadodimas and V. S. Rychkov,
“Minisuperspace quantization of ’bubbling AdS’ and free fermion droplets,”
JHEP 0508, 025 (2005)
[arXiv:hep-th/0505079].
[5]
H. Lin, O. Lunin and J. M. Maldacena,
“Bubbling AdS space and 1/2 BPS geometries,”
JHEP 0410, 025 (2004)
[arXiv:hep-th/0409174].
[6]
T. W. Brown, P. J. Heslop and S. Ramgoolam,
“Diagonal multi-matrix correlators and BPS operators in N=4 SYM,”
arXiv:0711.0176 [hep-th];
F. A. Dolan,
“Counting BPS operators in N=4 SYM,”
Nucl. Phys. B 790, 432 (2008)
[arXiv:0704.1038 [hep-th]];
M. Bianchi, F. A. Dolan, P. J. Heslop and H. Osborn,
“N = 4 superconformal characters and partition functions,”
Nucl. Phys. B 767, 163 (2007)
[arXiv:hep-th/0609179].
[7]
J. Kinney, J. M. Maldacena, S. Minwalla and S. Raju,
“An index for 4 dimensional super conformal theories,”
Commun. Math. Phys. 275, 209 (2007)
[arXiv:hep-th/0510251].
[8]
J. B. Gutowski and H. S. Reall,
“Supersymmetric AdS(5) black holes,”
JHEP 0402, 006 (2004)
[arXiv:hep-th/0401042];
J. B. Gutowski and H. S. Reall,
“General supersymmetric AdS(5) black holes,”
JHEP 0404, 048 (2004)
[arXiv:hep-th/0401129];
Z. W. Chong, M. Cvetic, H. Lu and C. N. Pope,
“General non-extremal rotating black holes in minimal five-dimensional
gauged supergravity,”
Phys. Rev. Lett. 95, 161301 (2005)
[arXiv:hep-th/0506029];
H. K. Kunduri, J. Lucietti and H. S. Reall,
“Supersymmetric multi-charge AdS(5) black holes,”
JHEP 0604 (2006) 036
[arXiv:hep-th/0601156].
[9]
N. Beisert,
“The complete one-loop dilatation operator of N = 4 super Yang-Mills
theory,”
Nucl. Phys. B 676, 3 (2004)
[arXiv:hep-th/0307015].
[10]
B. Sundborg,
“The Hagedorn transition, deconfinement and N = 4 SYM theory,”
Nucl. Phys. B 573, 349 (2000)
[arXiv:hep-th/9908001].
[11]
O. Aharony, J. Marsano, S. Minwalla, K. Papadodimas and M. Van Raamsdonk,
“The deconfinement and Hagedorn phase transitions in weakly coupled large N
gauge theories,”
Comptes Rendus Physique 5, 945 (2004).
[12]
P. J. Silva,
“Thermodynamics at the BPS bound for black holes in AdS,”
JHEP 0610, 022 (2006)
[arXiv:hep-th/0607056];
P. J. Silva,
“Phase transitions and statistical mechanics for BPS black holes in
AdS/CFT,”
JHEP 0703, 015 (2007)
[arXiv:hep-th/0610163].
[13]
S. Benvenuti, B. Feng, A. Hanany and Y. H. He,
“Counting BPS operators in gauge theories: Quivers, syzygies and
plethystics,”
arXiv:hep-th/0608050;
B. Feng, A. Hanany and Y. H. He,
“Counting gauge invariants: The plethystic program,”
JHEP 0703, 090 (2007)
[arXiv:hep-th/0701063].
[14]
M. Berkooz, D. Reichmann and J. Simon,
“A Fermi surface model for large supersymmetric AdS(5) black holes,”
JHEP 0701, 048 (2007)
[arXiv:hep-th/0604023]. |
The geometry of fractal percolation,
Michał Rams
Michał Rams, Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-956 Warsaw, Poland
rams@impan.gov.pl
and
Károly Simon
Károly Simon, Institute of Mathematics, Technical
University of Budapest, H-1529 B.O.box 91, Hungary
simonk@math.bme.hu
Abstract.
A well studied family of random fractals called fractal percolation is discussed. We focus on the projections of fractal percolation on the plane. Our goal is to present stronger versions of the classical Marstrand theorem, valid for almost every realization of fractal percolation. The extensions go in three directions:
•
the statements work for all directions, not almost all,
•
the statements are true for more general projections, for example radial projections onto a circle,
•
in the case $\dim_{H}>1$, each projection has not only positive Lebesgue measure but also has nonempty interior.
2000 Mathematics Subject Classification. Primary
28A80 Secondary 60J80, 60J85
Key words and phrases. Random fractals, Hausdorff dimension,
processes in random environment.
Rams was partially supported by the MNiSW grant N201 607640 (Poland).
The research of Simon was supported by OTKA Foundation
# K 104745
1. introduction
To model turbulence, Mandelbrot [13, 14] introduced a
statistically self-similar family of random Cantor sets.
Since that time this family has got at least three names in the literature: fractal percolation, Mandelbrot percolation and canonical curdling, among which we will use the first one.
In 1996 Lincoln Chayes [3] published an excellent survey giving an account about the most important results known in that time. His survey focused on the percolation related properties while we place emphasis on the geometric measure theoretical properties (projections and slices) of fractal percolation sets.
About the projections of a general Borel set the celebrated Marstrand Theorem gives the following information:
Theorem 1 ([15]).
Let $E\subset\mathbb{R}^{2}$ be a Borel set.
•
If $\dim_{\rm H}(E)<1$ then for Lebesgue almost all $\theta$ $\dim_{\rm H}(\mathrm{proj}_{\theta}(E))=\dim_{\rm H}(E)$.
•
If $\dim_{\rm H}(E)>1$ then for Lebesgue almost all $\theta$ we have $\mathcal{L}{\rm eb}(\mathrm{proj}_{\theta}(E))>0$.
where $\mathrm{proj}_{\theta}$ is the orthogonal projection in direction $\theta$.
In this paper we review some recent results which give more precise information in the special case of the projections of fractal percolation Cantor sets.
2. The construction and its immediate consequences
The construction consists of the infinite iteration of two steps. We start from the unit cube in $\mathbb{R}^{d}$.
•
All cubes we have after the $n$-th iteration of the process (they will be called level $n$ cubes) we subdivide into smaller cubes of equal size,
•
Among them some are retained and some are discarded. Retaining or discarding of different cubes are independent random events. The cubes that were retained are the level $n+1$ cubes.
Those points that have never been discarded form the fractal percolation set.
Please note that in literature the term fractal percolation is often used to denote object which we call homogeneous fractal percolation. That is, the fractal percolation for which all squares have equal probabilities of being retained.
2.1. An informal description of Fractal Percolation
We fix integer $M\geq 2$.
We partition the unit cube $Q\subset\mathbb{R}^{d}$ into $M^{d}$ congruent cubes of side length $M^{-1}$ and we assign a probability to each of the cubes in this partition (Figure 1 (a)).
We retain each of the cubes of this partition with the corresponding probability independently and discard it with one minus the corresponding probability.
The union of the retained squares is the first approximation of the random set to be constructed ( Figure 1 (b)). We obtain the second approximation by repeating this process independently of everything in each of the retained squares ( Figure 1 (c) and (d)). We continue this process at infinitum.
The object of our investigation is
the collection of those points which have not been discarded. It will be called fractal percolation set and denoted by ${E=(d,M,\mathbf{p})}$, where $\mathbf{p}$ is the chosen vector of the probabilities $\{p_{i}\}$.
In the special case when all $p_{i}$ are equal we obtain the homogeneous fractal percolation set which is denoted by $E^{h}=E^{h}(d,M,p)$.
2.2. Fractal percolation set in more details
For simplicity we give the construction on the plane but the definition works with obvious modifications in $\mathbb{R}^{d}$ for all $d\geq 1$. Besides the dimension of the ambient space the two other parameters of the construction are: the natural number $M\geq 2$ and
a vector of probabilities $\mathbf{p}\in[0,1]^{M^{2}}$ (note: not a probabilistic vector).
To shorten the notation we write $\mathcal{I}$ for the set of indices of $\mathbf{p}$:
$$\mathcal{I}:=\left\{1,\dots,M^{2}\right\}$$
The statistically self-similar random set which is the object of our study is defined as
(2.1)
$$E:=\bigcap_{n=1}^{\infty}E_{n},$$
where $E_{n}$ is the $n$-th approximation of $E$. The inductive definition of $E_{n}$ will occupy the rest of this subsection. Actually $E_{n}$ is the union of a random collection of level $n$ squares. First we define the level $n$ squares and then we introduce the random rule with which those level $n$ squares are selected whose union form $E_{n}$.
2.2.1. The process of subdivision
We divide the unit square $Q=\left[0,1\right]^{2}$ into $M^{2}$ congruent squares $Q_{1},\dots,Q_{M^{2}}$ of size $M^{-1}$ numbered according to lexicographical order (or any other order). These squares are the level one $M$-adic squares. Let
$$\mathcal{N}_{1}:=\left\{x_{i}\right\}_{i\in\mathcal{I}}$$
be the set of midpoints of the level one squares. For each midpoint $x_{i}$ we define the homothetic map $\varphi_{i}:Q\to Q_{i}$:
$$\varphi_{i}(y):=x_{i}+M^{-1}\cdot\left(y-\left(\frac{1}{2},\frac{1}{2}\right)%
\right).$$
For every $\mathbf{i}\in\mathcal{I}^{n}$, $\mathbf{i}=(i_{1},\dots,i_{n})$ we write
$$x_{\mathbf{i}}:=\varphi_{\mathbf{i}}\left(\frac{1}{2},\frac{1}{2}\right).$$
and we define the map
$$\varphi_{\mathbf{i}}(y):=x_{\mathbf{i}}+M^{-n}\cdot\left(y-\left(\frac{1}{2},%
\frac{1}{2}\right)\right).$$
To simplify the notation, we will not distinguish the set of the centers of level $n$ squares
$$\mathcal{N}_{n}:=\left\{\varphi_{\mathbf{i}}\left(\frac{1}{2},\frac{1}{2}%
\right):\mathbf{i}\in\mathcal{I}^{n}\right\}$$
and the family of level $n$-squares:
(2.2)
$$\left\{Q_{\mathbf{i}}:=\varphi_{\mathbf{i}}(Q):\mathbf{i}\in\mathcal{I}^{n}%
\right\}.$$
2.2.2. The process of retention
The square $Q=Q_{\emptyset}$ is retained.
For any $\mathbf{i}\in\mathcal{I}^{n}$ for which the square $Q_{\mathbf{i}}$ is retained and for each $j\in\mathcal{I}$, the square $Q_{\mathbf{i}j}$ is retained with probability $p_{j}$. The events ’$Q_{\mathbf{i}j}$ is retained’ and ’$Q_{\mathbf{i^{\prime}}j^{\prime}}$ is retained’ are independent whenever $\mathbf{i}\neq\mathbf{i^{\prime}}$ or $j\neq j^{\prime}$.
We define $E_{1}$ as the union of retained squares $Q_{i},i\in\mathcal{I}$. Similarly, $E_{n}$ is the union of retained squares $Q_{\mathbf{i}},\mathbf{i}\in\mathcal{I}^{n}$. We write
$$\mathcal{E}_{n}:=\left\{\mathbf{i}\in\mathcal{I}^{n}:Q_{\mathbf{i}}\mbox{ %
retained }\right\}.$$
2.3. The corresponding probability space and statistical self-similarity
The probability space corresponding to this random construction is best described by M.
Dekking [5]. For the convenience of the reader we repeat it here.
Let $\mathcal{T}$ be the $M^{d}$ array tree that is
$$\mathcal{T}:=\bigcup_{n=0}^{\infty}\mathcal{I}^{n},$$
where $\mathcal{I}^{0}:=\emptyset$ is the root of three . Let $\Omega:=\left\{0,1\right\}^{\mathcal{T}}$ that is $\Omega$ is the set of labeled trees where we label every node of $\mathcal{T}$ by $0$ or $1$. The probability measure $\mathbb{P}_{\mathbf{p}}$ on $\Omega$
is define in such a way that the family of labels $X_{\mathbf{i}}\in\left\{0,1\right\}$ of nodes $\mathbf{i}\in\mathcal{T}$ satisfy:
•
$\mathbb{P}_{\mathbf{p}}(X_{\emptyset}=1)=1$
•
$\mathbb{P}_{\mathbf{p}}(X_{i_{1},\dots,i_{n}})=p_{i_{n}}$
•
$\left\{X_{\mathbf{i}}\right\}_{\mathbf{i}\in\mathcal{T}}$ are independent.
Following [5] we define the survival set of level $n$ by
$$S_{n}:=\left\{\mathbf{i}\in\mathcal{I}^{n}:X_{i_{1}\dots,i_{k}}=1,\ \forall 1%
\leq k\leq n\right\}.$$
Then
$$E_{n}=\bigcup_{\mathbf{i}\in S_{n}}Q_{\mathbf{i}},\quad E=\bigcap_{n=1}^{%
\infty}E_{n}.$$
It follows from the construction that generalized fractal percolation set is statistically self-similar and the number of retained cubes form a branching process:
Lemma 2.
(a):
$\left\{\#\mathcal{E}_{n}\right\}$ is a branching process with average number of offsprings $\sum\limits_{i\in\mathcal{I}}p_{i}$. In particular if $p_{i}\equiv p$ then the
offspring distribution is $\texttt{Binomial}(M^{d},p)$.
(b):
For every $n\geq 1$ and $\mathbf{i}\in\mathcal{E}_{n}$ the rescaled copy $\varphi_{\mathbf{i}}^{-1}(E\cap Q_{\mathbf{i}})$ has the same distribution as $E$ itself.
(c):
The sets $\left\{E\cap Q_{\mathbf{i}}\right\}_{\mathbf{i}\in\mathcal{E}_{n}}$ are independent.
Using this it is not hard to prove that
(2.3)
$$E\neq\emptyset\mbox{ implies that }\dim_{\rm H}(E)=\dim_{\rm B}(E)=\frac{\log%
\sum\limits_{i\in\mathcal{I}}p_{i}}{\log M}\mbox{ a.s. }$$
This was proved by Kahane and Peyriere
[12], Hawkes [11], Falconer [8], Mauldin and Williams [16] independently. A canonical example of the inhomogeneous fractal percolation set is:
Example 3 (Random Sierpiński Carpet ).
Let $SC_{p}:=E(2,3,\mathbf{p})$, where using the notation of Figure 1 (c):
$$p_{5}=0\mbox{ and for }i\in\left\{1,\dots,9\right\}\setminus\left\{5\right\}:%
\ p_{i}=p.$$
3. Percolation and projection to coordinate axes
In this section we work on the plane so $Q=[0,1]^{2}$. The connectivity properties of $E^{h}(2,M,p)$ for an arbitrary $M\geq 2$ was first investigated by Chayes, Chayes and Durrett [2]. Dekking and Meester [6] gave a simpler proof and extended the scope of the theorem for some inhomogeneous fractal percolation sets like the random Sierpiński carpet $SC_{p}$. Here we summarize briefly some of the most interesting results of this area. For a much more detailed account see by L. Chayce [3].
We say that $E$ percolates if $E$ contains a connected set which intersects both the left and the right sides of $Q$. If $E$ percolates then $E$ has a large connected component.
3.1. The homogeneous case
The following very important result was proved by Chayce,Chayce, Durrett.
Theorem 4 ([2]).
Fix an arbitrary $M\geq 2$. Then there is a critical probability $\frac{1}{M}<p_{c}<1$ such that
(1)
If $p<p_{c}$ then $E^{h}(2,M,p)$ is a random dust that is totally disconnected almost surely.
(2)
If $p\geq p_{c}$ then $E^{h}(2,M,p)$ percolates with positive probability. This implies that $E^{h}(2,M,p)$ is not totally disconnected almost surely.
This shows a remarkable difference in between the fractal percolation and the usual percolation: in the latter case, the probability of percolation at critical parameter $p=p_{c}$ is 0.
3.2. The inhomogeneous case
Using some earlier works of Dekking and Grimmett [4], the results above were extended by Dekking and Meester [6]. They proved that by changing the components of $\mathbf{p}$ the inhomogeneous fractal percolation set $E(2,M,\mathbf{p})$ can go through the six stages below. Here the projection to the $x$-axis is denoted by $\mathrm{proj}_{x}$. That is $\mathrm{proj}_{x}(a,b)=a$.
The DM stages of $\mathbf{E(2,M,\mathbf{p})}$:
I:
$E=\emptyset$ almost surely.
II:
$\mathbb{P}\left(E\neq\emptyset\right)>0$ but $\dim_{\rm H}\left(\mathrm{proj}_{x}E\right)=\dim_{\rm H}\left(E\right)$ almost surely.
III:
$\dim_{\rm H}\left(\mathrm{proj}_{x}E\right)<\dim_{\rm H}\left(E\right)$ if $E\neq\emptyset$ but $\mathcal{L}{\rm eb}\left(\mathrm{proj}_{x}E\right)=0$ almost surely.
IV:
$0<\mathcal{L}{\rm eb}\left(\mathrm{proj}_{x}E\right)<1$ almost surely.
V:
$\mathcal{L}{\rm eb}\left(\mathrm{proj}_{x}E\right)=1$ holds with positive probability but $E$ does not percolate almost surely.
VI:
$E$ percolates with positive probability.
It was proved in [6] that the random Sierpiński Carpet $SC_{p}$ goes through all of these stages as we increase the value of $p$. The following theorem gives the precise answer when exactly a system appears in stages I,II,III.
Theorem 5 ([4], [8]).
Let $m_{r}$ be the sum of the probabilities in the $r$-th column, that is the expected number of squares in column $r$. Then
(1)
$E=\emptyset$ almost surely iff $\sum\limits_{i=1}^{M^{2}}p_{i}\leq 1$. Except when $\exists i$ such that $p_{i}=1$ and $p_{j}=0$ for all $i\neq j$. In this case $E$ is a singleton.
(2)
$\dim_{\rm H}(\mathrm{proj}_{x}(E))=\dim_{\rm H}(E)$ holds almost surely, iff $\sum\limits_{r=1}^{M}m_{r}\log m_{r}\leq 0$.
(3)
$\mathcal{L}{\rm eb}(\mathrm{proj}_{x}E)=0$ holds almost surely iff $\sum\limits_{r=1}^{M}\log m_{r}\leq 0$.
This result was strengthened by Falconer and Grimmett:
Theorem 6 ([9],[10]).
Assume that $m:=\min\left\{m_{r}\right\}>1$. Then $\mathrm{proj}_{x}(E)$ contains an interval almost surely, conditioned on non-extinction.
We will present the proof in the fifth section.
3.3. The DM stages for the homogeneous case
For the homogeneous case $m_{r}=M\cdot p$ Hence we obtain that almost surely:
•
If $0<p\leq\frac{1}{M^{2}}$ then $E=\emptyset$.
•
If $\frac{1}{M^{2}}<p\leq\frac{1}{M}$ then the system is in stage II.
•
If $\frac{1}{M}<p<p_{c}$ then the system is in stage V.
Stages III and IV do not appear in the homogeneous case.
4. The arithmetic sum/difference of two fractal percolations
There is a very nice and more detailed survey of this field due to M. Dekking [5].
In the previous section we studied the connectivity properties and the $90^{\circ}$ projections of random Cantor sets. In this section we consider sets which are products of inhomogeneous fractal percolation sets and we take their $45^{\circ}$, ($-45^{\circ}$) projections in order to study the arithmetic difference (arithmetic sum) respectively of independent copies of $E(1,M,\mathbf{p})$.
4.1. The arithmetic sum and its visualization
Let $A,B\subset\mathbb{R}$ be arbitrary. Then the arithmetic sum $A+B:=\left\{a+b:a\in A,b\in B\right\}$ is the $-45^{\circ}$-projection of $A\times B$ to the $x$-axis (this is the direction of the line $\ell_{a}$ on Figure 3). Similarly, we can visualize the arithmetic difference by taking the projection of the product set with the line of $+45^{\circ}$ angle.
The motivation for studying the arithmetic difference (or sum) of random Cantor sets comes from a conjecture of Palis which states that typically (in a natural sense which depends on the actual setup), the arithmetic difference of two dynamically defined Cantors is either small in the sense that it has Lebesgue measure zero or big in the sense that it contains some intervals, but at least typically, it does not occur that the arithmetic difference set is a set of positive Lebesgue measure with empty interior. This conjecture does not hold for the algebraic difference of inhomogeneous fractal percolation sets, but it holds in the homogeneous case. The way to prove this is via the $45^{\circ}$-projections of $E(1,M,\mathbf{p})\times E(1,M,\mathbf{p})$.
4.2. The product of two one dimensional fractal percolation versus a two dimensional fractal percolation.
We explain this relation in the case when $M=3$.
Assume that we are given the inhomogeneous fractal percolations $E(1,3,\mathbf{a})$, and $E(1,3,\mathbf{b})$, where $\mathbf{a}=(a_{1},a_{2},a_{3})$, $\mathbf{b}=(b_{1},b_{2},b_{3})$ are the vectors of probabilities. We define the vector $\mathbf{p}\in[0,1]^{9}$ as their product $\mathbf{p}=\mathbf{a}\bigotimes\mathbf{b}$ in the natural way which is suggested by looking at Figure 1 (a). That is:
$$p_{i}:=a_{u}\cdot b_{v}\mbox{ if }i-1=3*(v-1)+(u-1),\ 1\leq u,v\leq 3.$$
The reason that $E(2,3,\mathbf{p})$ and $E(1,3,\mathbf{a})\times E(1,3,\mathbf{b})$ are similar is explained in (a) and the essential difference between them is pointed out in (b) below:
(a):
Let $\mathbf{i}\in\left\{1,\dots,9\right\}^{n}$. Then the probability that $Q_{\mathbf{i}}$ is retained is the same during the construction of $E(2,3,\mathbf{p})$ and the construction of $E(1,3,\mathbf{a})\times E(1,3,\mathbf{b})$.
(b):
Let $K$ and $L$ be level $n$ squares for some $n$. Assume that both $K$ and $L$ are retained during the construction of $E(2,3,\mathbf{p})$ and $E(1,3,\mathbf{a})\times E(1,3,\mathbf{b})$. Then
•:
In the construction of $E=E(2,3,\mathbf{p})$ the sets $E\cap K$ and $E\cap L$ are independent.
•:
In the construction of $E(1,3,\mathbf{a})\times E(1,3,\mathbf{b})$ the sets $E\cap K$ and $E\cap L$ are independent iff $\mathrm{proj}_{x}K\neq\mathrm{proj}_{x}L$ and $\mathrm{proj}_{y}K\neq\mathrm{proj}_{y}L$ hold.
In dimension $d\geq 2$ the analogy is the same: the probability of the retention of a level $n$ cube is the same for the $d$-dimensional percolation and for the $d$-fold product of the corresponding one dimensional percolations. On the other hand, the future of what ever happens in two distinct retained level $n$ cubes is:
•
always independent in the $d$-dimensional percolation case,
•
independent for the $d$-fold product of the corresponding one dimensional fractal percolations iff the two cubes do not share any common projections to coordinate axes.b
4.3. The existence of an interval in the arithmetic difference set
Let $E_{1}:=E(1,M,\mathbf{p})$ and $E_{2}:=E(1,M,\mathbf{q})$. We define the cyclic cross correlation coefficients:
(4.1)
$$\gamma_{k}:=\sum\limits_{i=1}^{M}p_{i}q_{i-k(\mathrm{mod}\ M)}\mbox{ for }k=1,%
\dots,M.$$
Theorem 7 ([7]).
Assuming that $E_{1},E_{2}\not=\emptyset$, we have
(a):
If
$\forall i=1,\dots,M:\ \gamma_{i}>1$
then almost surely
$$E_{2}-E_{1}\mbox{ contains an
interval }.$$
(b):
If $\exists i\in\left\{1,\dots,M\right\}:\ \gamma_{i},\gamma_{i+1\bmod M}<1$ then almost surely
$$E_{2}-E_{1}\mbox{ does not contain any interval }.$$
In the homogeneous case and in the case when $M=3$ this gives complete characterization. Otherwise we can change to higher order Cantor sets (collapsing $n\geq 2$ steps of the construction into one) and we can apply the same theorem in that case. The fact that this can be done is not trivial because higher order fractal percolations are correlated. That is the way as the random set develops in one level $n$ square is dependent how it develops in some other squares. Nevertheless, M. Dekking and H. Don proved that this can be done by pointing out that the proof of the theorem above can be carried out for more general, correlated random sets than the inhomogeneous fractal percolations. This more general family includes the higher order fractal percolation sets.
4.4. The Lebesgue measure of the arithmetic difference set
Let $E_{2},E_{2}$ be two independent realizations of $E(1,M,\mathbf{p})$. Then
$$\gamma_{k}:=\sum\limits_{i=1}^{M}p_{i}p_{i-k(\mathrm{mod}\ M)}\mbox{ for }k=1,%
\dots,M.$$
Let $\Gamma:=\gamma_{1}\cdots\gamma_{M}$.
Theorem 8 ([17]).
If $\Gamma>1$ then
$\mathcal{L}{\rm eb}(E_{2}-E_{1})>0$.
Combined application of Theorems 7 and 8 yields that the Palis conjecture does not hold in the case when
for $M=3$ and $\mathbf{p}=(0.52,0.5,0.72)$.
Namely, in this case $\gamma_{1}=1.0388$ and $\gamma_{2}=\gamma_{3}=0.941$.
Let $E_{1},E_{2}$ be two independent copies of $E(1,3,\mathbf{p})$. Then by Theorem 7 there is no interval in $E_{1}-E_{2}$ (since there are two consecutive $\gamma$’s that are smaller than one) and by Theorem 8 we have $\mathcal{L}{\rm eb}\left(E_{1}-E_{2}\right)>0$ since $\gamma_{1}\cdot\gamma_{2}\cdot\gamma_{3}=1.0272>1$.
5. General projections: the opaque case
In this and in the following sections we study the projections of fractal percolation sets in general directions. In this section we consider the case $\dim_{\rm H}(E)>1$. Under some mild assumption, almost surely projections of $E$ have not only positive Lebesgue measure, as per Marstrand theorem, but also non-empty interior. Furthermore it holds for all and not only almost all directions. Moreover, this remains valid if we replace the orthogonal projection with a much more general family of projections.
One practical application
of our result is shown above (Figure 4).
One does not need to rotate such a set to use it as an umbrella.
We have already studied the horizontal and vertical projections. So we can restrict our attention to the directions $\alpha\in\mathcal{D}:=\left(0,90^{\circ}\right)$
A condition $A(\alpha)$, $\alpha\in\mathcal{D}$ on the vector of probabilities $\mathbf{p}$ will be defined below.
Theorem 9 ([19]).
Let $\alpha\in\mathcal{D}$. If $A(\alpha)$ holds and $E$ is nonempty then almost surely ${\rm proj}_{\alpha}(E)$ contains an interval.
Theorem 10 ([19]).
If $A(\alpha)$ holds for all $\alpha\in\mathcal{D}$ and $E\neq\emptyset$
then almost surely all projections ${\rm proj}_{\alpha}(E)$ contain an interval.
Remark 11.
The assertions of Theorems 9 and 10 remain valid if we replace $\mathrm{proj}_{\alpha}$ with more general families of projections, see [19, Section 6]. In particular, radial or co-radial projections (see Figure 5) are included.
Example 12.
If either
(1)
Homogeneous case: $p_{i}=p>M^{-1}$ for all $i$, or
(2)
Generalized random Sierpiński Carpet: $M=3$, $p_{5}=q$, $p_{i}=p$ for $i\neq 5$, and
$$p>\max\left(\frac{1}{3},\frac{1-q}{2}\right)$$
then Condition A($\alpha$) is satisfied for all $\alpha\in\mathcal{D}$.
Note that (1) is equivalent to $\dim_{\rm H}(E)>1$ almost surely.
5.1. Horizontal and vertical projections
Let us start by presenting the large deviation argument (by Falconer and Grimmett) working for horizontal and vertical projections. If
$\dim_{\rm H}\Lambda>1$ then from the dimension formula for some $n$ one can find a level $n$ column with (exponentially) many squares.
We prove inductively that in its every $N$-th level sub-column, $N>n$, we typically have exponentially many squares on each level (probability of existence of $N>n$ and an $N$-th level subcolumn which does not have exponentially many squares is super-exponentially small). When we move from level $n$ column to its level $n+1$ subcolumns, each square in the column gives birth to an expected number of $pM>1$ number of level $n+1$ squares in each of the subcolumns. By large deviation theorem there is only a superexponentially small probability that the number of level $n+1$
squares in a subcolumn is smaller than a fixed $\alpha\in(1,pM)$ multiple of the level $n$ squares in the column. By induction, if this exceptional situation does not happen (or happens only finitely many times), for each $N>n$ the number of squares of level $N$ in each subcolumn will be at least of order $\alpha^{N-n}$.
5.2. Condition A
Our goal in this subsection is to modify this argument to work in a more complicated situation of projections in general directions. Indeed, contrary to the horizontal/vertical projections case, here it is in general not true that if a line intersects a square of level $n$ then the expected number of squares of level $n+1$ it intersects is greater than 1. It is still true if the line intersects ’central’ part of the square, but not if it hits it close to the corners.
Nevertheless, we are able to find a modified version of the argument. We fix $\alpha\in\mathcal{D}$. We are going to consider $\Pi_{\alpha}$ instead of ${\rm proj}_{\alpha}$, i.e. we are projecting onto a diagonal $\Delta_{\alpha}$ of $Q$, see Figure 5. For any $\mathbf{i}\in\mathcal{I}^{n}$ the map $\Pi_{\alpha}\circ\varphi_{\mathbf{i}}:\Delta_{\alpha}\to\Delta_{\alpha}$ is a linear contraction of ratio $M^{-n}$. We will use its inverse: a map $\psi_{\alpha,\mathbf{i}}:\Pi_{\alpha}(Q_{\mathbf{i}})\to\Delta_{\alpha}$. It is a linear expanding map (of ratio $M^{n}$) and it is onto.
Consider the class of nonnegative real functions on $\Delta_{\alpha}$, vanishing on the endpoints. There is a natural random inverse Markov operator $G_{\alpha}$ defined as
$$G_{\alpha}f(x)=\sum_{i\in\mathcal{E}_{1};x\in\Pi_{\alpha}(Q_{i})}f\circ\psi_{%
\alpha,i}(x).$$
The corresponding operator on the $n$-th level is
$$G^{(n)}_{\alpha}f(x)=\sum\limits_{\mathbf{i}\in\mathcal{E}_{n};x\in\Pi_{\alpha%
}(Q_{\mathbf{i}}}f\circ\psi_{\alpha,\mathbf{i}}(x).$$
In particular for any $H\subset\Delta^{\alpha}$ we have
$$G_{\alpha}^{(n)}\mathbf{1}_{H}(x)=\#\left\{\mathbf{i}\in\mathcal{E}_{n}:x\in%
\Pi_{\alpha}\left(\varphi_{\mathbf{i}}(H)\right)\right\}.$$
Although $G^{(n)}_{\alpha}$ should not be thought of as the $n$-th iterate of $G_{\alpha}$, the expected value of $G^{(n)}_{\alpha}$
is the $n$-th iterate of the expected value of $G_{\alpha}$. Namely, let
$$F_{\alpha}=\mathbb{E}\left[G_{\alpha}\right]\mbox{ and }F_{\alpha}^{n}=\mathbb%
{E}\left[G_{\alpha}^{n}\right]$$
We then have the formulas
$$F_{\alpha}f(x)=\sum_{i\in\mathcal{I};x\in\Pi_{\alpha}(Q_{i})}p_{i}\cdot f\circ%
\psi_{\alpha,i}(x)$$
and
$$F^{n}_{\alpha}f(x)=\sum\limits_{\mathbf{i};x\in\Pi_{\alpha}(Q_{\mathbf{i}})}p_%
{\mathbf{i}}\cdot f\circ\psi_{\alpha,\mathbf{i}}(x),$$
where
$$p_{\mathbf{i}}=\prod_{k=1}^{n}p_{i_{k}}.$$
Definition 13.
We say the percolation model satisfies Condition A($\alpha$) if there exist closed intervals $I_{1}^{\alpha},I_{2}^{\alpha}\subset\Delta_{\alpha}$ and a positive integer $r_{\alpha}$ such that
i)
$I_{1}^{\alpha}\subset{\rm int}I_{2}^{\alpha},I_{2}^{\alpha}\subset{\rm int}%
\Delta_{\alpha}$,
ii)
$F_{\alpha}^{r_{\alpha}}\mathbf{1}_{I_{1}^{\alpha}}\geq 2\cdot\mathbf{1}_{I_{2}%
^{\alpha}}$.
It will be convenient to use additional notation. For $x\in\Delta_{\alpha}$, $\alpha\in\mathcal{D}$, and $I\subset\Delta_{\alpha}$ we denote
$$D_{n}(x,I,\alpha)=\{\mathbf{i}\in\mathcal{I}^{n};x\in\Pi_{\alpha}\circ\varphi_%
{\mathbf{i}}(I)\}.$$
That is, if we write $\ell^{\alpha}(x)$ for the line segment through $x\in\Delta_{\alpha}$ in direction $\alpha$, $D_{n}(x,I,\alpha)$ is the set of $\mathbf{i}$
for which $\ell^{\alpha}(x)$ intersects $\varphi_{\mathbf{i}}(I)$.
The point ii) of Definition 13 can then be written as
$$\forall_{x\in I_{2}^{\alpha}}\ \sum_{\mathbf{i}\in D_{r_{\alpha}}(x,I^{\alpha}%
_{1},\alpha)}p_{\mathbf{i}}\geq 2.$$
In other words, Condition $A(\alpha)$ is satisfied if for given $\alpha$ one can define ’small central’ and ’large central’ part of each square in such a way that for some $r\in\mathbb{N}$ if a line in direction $\alpha$ intersects the ’large central’ part of some $n$-th level square then the expected number of ’small central’ parts of its $n+r$-th level subsquares it intersects is uniformly greater than 1.
5.3. Consequences of Condition $A(\alpha)$
It is clear that if $A(\alpha)$ holds then one can apply the large deviation argument for projection in direction $\alpha$ - modulo a minor technical problem that the random variables in the large deviations theorem are not identically distributed.
A bit more complicated is the proof that almost surely all the projections contain intervals. It is based on the following robustness properties:
Proposition 14.
If condition A($\alpha$) holds for some $\alpha\in\mathcal{D}$ for some $I_{1}^{\alpha},I_{2}^{\alpha}$ and $r_{\alpha}$ then it will also hold in some neighbourhood $J\ni\alpha$. Moreover, for all $\theta\in J$ we can choose $I_{1}^{\theta}=I_{1}^{\prime},I_{2}^{\theta}=I_{2},r_{\theta}=r_{\alpha}$ not depending on $\theta$.
A natural corollary is that the whole range $\mathcal{D}$ can be presented as a countable union of closed intervals $J_{i}=[\alpha_{i}^{-},\alpha_{i}^{+}]$ such that for each $i$ Condition A($\alpha$) holds for all $\alpha\in J_{i}$ with the same $I_{1}^{i},I_{2}^{i},r_{i}$.
Proposition 15.
Let $I\subset B(I,\ell)\subset J\subset\Delta_{\alpha}$. If $\mathbf{i}\in D_{n}(x,I,\alpha)$ then $\mathbf{i}\in D_{n}(x,J,\beta)$ for all $\beta\in(\alpha-\ell M^{-n},\alpha+\ell M^{-n})$.
Hence, inside each $J_{i}$ one does not need to repeat the large deviation argument separately for each $\alpha$. At level $n$ it is enough to check it for approximately $M^{n}$ directions. As the number of directions one needs to check grows only exponentially fast with $n$, the proof goes through.
5.4. Checking Condition $A(\alpha)$
One last thing needed is an efficient way to check whether $A(\alpha)$ holds.
Definition 16.
We say that the fractal percolation model satisfies Condition B($\alpha$) if there exists a nonnegative continuous function $f:\Delta_{\alpha}\to\mathbb{R}$ such that $f$ is strictly positive except at the endpoints of $\Delta_{\alpha}$ and that
(5.1)
$$F_{\alpha}f\geq(1+\varepsilon)f$$
for some $\varepsilon>0$.
Proposition 17.
B($\alpha$) implies A($\alpha$).
In particular, for homogeneous case $p_{i}=p>M^{-1}$ for any $\alpha$ one can choose $f(x)$ as the length of the intersection of $Q$ with the line in direction $\alpha$ passing through $x$. It is easy to check that this function satisfies (5.1) for $\varepsilon=pM-1$.
5.5. Application: visibility
For a given set $E$, we define the visible subset (from direction $\alpha$) as the set of points $x\in E$ such that the half-line starting at $x$ and going in direction $\alpha$ does not meet any other point $y\in E$. Similarly, given $z\in\mathbb{R}^{2}$, the visible subset (from $z$) is the set of points $x\in E$ such that the interval $\overline{xz}$ does not meet any other point $y\in E$.
Let $E$ be a homogeneous fractal percolation with $p>M^{-1}$. By Theorem 9, $E$ is quite opaque: the orthogonal projection in any direction almost surely contain intervals. In particular, with large probability it contains large intervals. By stochastical self-similarity of $E$, the same is true for each $E\cap Q_{\mathbf{i}}$. Hence, not many points can be visible:
Theorem 18 ( [1]).
If $E$ is nonempty, almost surely the visible set from direction $\alpha$ has finite one-dimensional Hausdorff measure for each $\alpha$ and the visible set from point $z$ has Hausdorff dimension 1 for each $z\in\mathbb{R}^{2}$.
6. General projections: the transparent case
In this section we present results analogous to the second part of the Marstrand theorem. For homogeneous fractal percolation with Hausdorff dimension smaller than 1 almost surely $\dim_{\rm H}(\mathrm{proj}_{\alpha}(E))=\dim_{\rm H}E$ for all $\alpha$. Together with the results of the previous section, it implies
Theorem 19 ([18]).
In the homogeneous case, that is $E=E^{h}(2,M,p)$ for almost all realizations of $E$
(6.1)
$$\forall\alpha,\ \dim_{\rm H}(\mathrm{proj}_{\alpha}E)=\min\left\{1,\dim_{\rm H%
}(E)\right\}.$$
Principal Assumption for this Section: In this section we always work in the homogeneous case:
$$E=E^{h}(2,M,p),$$
where
(6.2)
$$M^{-2}<p\leq M^{-1}.$$
That is $p$ is chosen to ensure that $E\neq\emptyset$ with positive probability and
$\dim_{\rm H}(E)\leq 1$ almost surely conditioned on non-extinction.
To prove Theorem 19 one needs to analyze the structure of the slices of $E_{n}$:
Informal description of the structure of slices of $\mathbf{E_{n}}$ (which was defined as the $n$-th approximation of $E$):
Namely,
for almost all realizations of $E$ and for all straight lines $\ell$:
the number of level $n$ squares having nonempty intersection with $E$ is at most $\mathrm{const}\cdot n$.
On the other hand, almost surely for $n$ big enough, we can find some line of $45^{\circ}$ angle which intersects $\mathrm{const}\cdot n$ level $n$ squares.
Let $\mathcal{L}^{\varepsilon}$ be the set of lines on the plane whose angle is separated both from $0^{\circ}$ and $90^{\circ}$ at least by $\varepsilon$. Further for a line $\ell$ let $\mathcal{E}_{n}(\ell)$ be the set of retained level $n$ squares that intersect $\ell$. That is,
$$\mathcal{E}_{n}(\ell):=\left\{\mathbf{i}\in\mathcal{E}_{n}:Q_{\mathbf{i}}\cap%
\ell\neq\emptyset\right\}.$$
Theorem 20 ([18]).
For almost all realizations of $E$ we have
(6.3)
$$\forall\varepsilon\in\left(0,\frac{\pi}{2}\right),\ \exists N,\ \forall n\geq N%
,\ \forall\ell\in\mathfrak{L}^{\varepsilon};\quad\#\mathcal{E}_{n}(\ell)\leq%
\mathrm{const}\cdot n.$$
For simplicity, the proof in horizontal/vertical direction only (for general directions one needs to apply techniques presented in previous subsection). The proof is once again based on the large deviation argument, but working in the opposite direction. This time the expected number of squares in a subcolumn is smaller (by a constant bounded away from 1) than the number of squares in the column (and not greater, like in the opaque case). Hence, we can guarantee that if the column has sufficiently many squares for the large deviation theory to work, the number of squares in all subcolumns will shrink. This leads to an estimation on the possible rate of growth.
This estimation is sharp:
Proposition 21 ([18]).
There exists a constant $0<\lambda<1$ such that
for almost all realizations, conditioned on $E\neq\emptyset$, there exists an $N$ such that for all
$n>N$ there exists a line $\ell$ with
(6.4)
$$\#\mathcal{E}_{n}\left(\ell\right)>\lambda n.$$
Theorem 19 is an immediate consequence of Theorem 20.
7. The arithmetic sum of at least three fractal percolations
To study arithmetic sums of more than two fractal percolations we need to combine results of the previous three sections. Like in section 4, we look at the projection $(x_{1},\ldots,x_{d})\to\sum x_{i}$ from the cartesian product of fractal percolations to the real line. The proof is based on the large deviation argument presented in section 5. However, the main technical difficulty is the presence of dependencies. We will use the results from section 6 to bound their impact.
Let
$$E^{i}:=E^{h}\left(1,M,p_{i}\right),\ i=1,2,3,\quad p:=p_{1}\cdot p_{2}\cdot p_%
{3}\mbox{ and }E:=E^{h}\left(3,M,p\right).$$
Then
$$\dim_{\rm H}\left(E^{1}\times E^{2}\times E^{3}\right)=\dim_{\rm H}(E)=\frac{%
\log M^{3}\cdot p}{\log M}.$$
Moreover, the probability that a level $n$ cube $C$ is contained in any of the two random Cantor sets above is equal to $p^{n}$.
Let $S_{a}$ be the plane $\{\sum x_{i}=a\}$. We can write
$$E^{\mathrm{sum}}:=E^{1}+E^{2}+E^{3}=\left\{a:S_{a}\cap\left(E^{1}\times E^{2}%
\times E^{3}\right)\neq\emptyset\right\}.$$
That is we can consider $E^{\mathrm{sum}}$ as the projection of $E_{1}\times E_{2}\times E_{3}$ to the $x$-axis with planes orthogonal the vector $(1,1,1)$. So,
$E^{\mathrm{sum}}$ can contain an interval only if its dimension is greater than one, that is $p>M^{-2}$.
It is a sufficient condition as well:
Theorem 22 ([18]).
Let $d\geq 2$ and for $i=1,\dots,d$ let $E^{i}:=E^{h}(1,M,p_{i})$ satisfying
(7.1)
$$p:=\prod_{i=1}^{d}p_{i}>M^{-d+1}.$$
Then for every $\mathbf{b}=(b_{1},\dots,b_{d})\in\mathbb{R}^{d}$, $b_{i}\neq 0$ for all $i=1,\dots,d$
the sum $E^{\mathrm{sum}}_{\mathbf{b}}=\sum\limits_{i=1}^{d}b_{i}{E}^{i}$ contains an interval almost surely, conditioned on all ${E}^{i}$ being nonempty.
We explain the proof of this theorem in the special case when $d=3$ and $\mathbf{b}=(1,1,1)$. To verify that a certain $a\in E^{\mathrm{sum}}$ we need to prove that the $n$ approximation of the product intersects $S_{a}$, that is $(E^{1}\times E^{2}\times E^{3})_{n}\cap S_{a}\neq\emptyset$ for every $n$.
It follows from the dimension formula and (7.1)
that we have $M^{n(1+\tau)}$ retained level $n$ cubes for some $\tau>0$. By the pigeon hole principle for at least one $k=0,\dots,3M^{n}$ the plane $S_{kM^{-n}}$ intersects at least $M^{n\tau}$ retained level $n$ cubes. For such a $k$ we write $a=kM^{-n}$. So, $\#\left\{\mathcal{E}_{n}\cap S_{a}\right\}\geq M^{n\tau}$.
Fix an $0\leq m\leq M$. How many level $n+1$ retained cubes intersect
$S_{a+mM^{-(n+1)}}$? If the way $E^{1}\times E^{2}\times E^{3}$ develops in every level $n$ cube was independent then we could get that the answer by the large deviation argument: exponentially many except for an event with a super exponentially small probability.
We remind that the cubes are dependent if they have the same $x_{1},x_{2}$ or $x_{3}$ coordinate. Figure 7 shows the geometric position of (some of: we consider only the cubes with the same $x_{3}$ coordinate) cubes dependent on one chosen cube: $x_{1}+x_{2}+x_{3}={\rm const}$ and $x_{3}={\rm const}$ imply $x_{1}+x_{2}={\rm const}$. Potentially there could be exponentially many
such cubes. The key step of the proof is that
using a theorem analogous to Theorem 20 for $E^{1}\times E^{2}$ instead of
$E^{h}(2,M,p_{1}\cdot p_{2})$ one can check that on the red dashed line on Figure
7 there are only constant times $n$ retained squares, consequently
the $M^{n\tau}$ level $n$ cubes having non-empty intersection with $S_{a}$ (the blue
plane on Figure 7) can be divided into $\mathrm{const}\cdot n$ classes such that the coordinate axes projection of any two cubes in a class are different. The events inside each class are independent, hence we can use the large deviation theory separately for each class.
A technical comment: in order to be able to go with this procedure we may have to decrease $p_{1},p_{2},p_{3}$ in such a way that for the modified values we have
$$p_{1}\cdot p_{2}\cdot p_{3}>M^{-2}\mbox{ but }p_{i}\cdot p_{j}<M^{-1}\mbox{ %
for distinct }i,j\in\left\{1,2,3\right\}.$$
That is, $E^{1}\times E^{2}\times E^{3}$ is a big set in the sense that it has dimension greater than one but its all coordinate plane projections should be small sets having dimension smaller than one – only then the $n$-th approximates of the coordinate plane projections intersect every line in at most $\mathrm{const}\cdot n$ retained squares. However, the property of almost surely having intervals in the algebraic sum is monotonous with respect to $\{p_{i}\}$.
Hence among those level $n$ retained cubes that intersect the blue plane $S_{a}$ there cannot be more than $\mathrm{const}\cdot n$ on the red line (any coordinate plane parallel line) which imply that the number of cubes dependent on any one cube is polynomial ($\mathrm{const}\cdot n$). This bound on the dependency matrix lets us control the dependencies.
References
[1]
I. Arhosalo, E. Järvenpää, M. Järvenpää, M. Rams, and
P. Shmerkin.
Visible parts of fractal percolation.
Proceedings of the Edinburgh Mathematical Society (Series 2),
55(02):311–331, 2012.
[2]
J.T. Chayes, L. Chayes, and R. Durrett.
Connectivity properties of mandelbrot’s percolation process.
Probability theory and related fields, 77(3):307–324, 1988.
[3]
L. Chayes.
On the length of the shortest crossing in the super-critical phase of
mandelbrot’s percolation process.
Stochastic processes and their applications, 61(1):25–43,
1996.
[4]
F.M. Dekking and G.R. Grimmett.
Superbranching processes and projections of random cantor sets.
Probability theory and related fields, 78(3):335–355, 1988.
[5]
M. Dekking.
Random cantor sets and their projections.
Fractal Geometry and Stochastics IV, pages 269–284, 2009.
[6]
M. Dekking and R.W.J. Meester.
On the structure of mandelbrot’s percolation process and other random
cantor sets.
Journal of Statistical Physics, 58(5):1109–1126, 1990.
[7]
M. Dekking and K. Simon.
On the size of the algebraic difference of two random cantor sets.
Random Structures & Algorithms, 32(2):205–222, 2008.
[8]
K.J. Falconer.
Random fractals.
Math. Proc. Cambridge Philos. Soc, 100(3):559–582, 1986.
[9]
K.J. Falconer and G.R. Grimmett.
On the geometry of random cantor sets and fractal percolation.
Journal of Theoretical Probability, 5(3):465–485, 1992.
[10]
K.J. Falconer and G.R. Grimmett.
Correction: On the geometry of random cantor sets and fractal
percolation.
Journal of Theoretical Probability, 7(1):209–210, 1994.
[11]
J. Hawkes.
Trees generated by a simple branching process.
Journal of the London Mathematical Society, 2(2):373–384,
1981.
[12]
J.-P. Kahane and J. Peyriere.
Sur certaines martingales de benoit mandelbrot.
Advances in mathematics, 22(2):131–145, 1976.
[13]
B.B. Mandelbrot.
Intermittent turbulence in self-similar cascades- divergence of high
moments and dimension of the carrier.
Journal of Fluid Mechanics, 62(2):331–358, 1974.
[14]
B.B. Mandelbrot.
The fractal geometry of nature/revised and enlarged edition.
New York, WH Freeman and Co., 1983, 495 p., 1, 1983.
[15]
J. M. Marstrand.
Some fundamental geometrical properties of plane sets of fractional
dimensions.
Proceedings of the London Mathematical Society, 3(1):257–302,
1954.
[16]
R.D. Mauldin and S.C. Williams.
Random recursive constructions: asymptotic geometric and topological
properties.
Trans. Amer. Math. Soc, 295(1):325–346, 1986.
[17]
P. Mora, K. Simon, and B. Solomyak.
The lebesgue measure of the algebraic difference of two random cantor
sets.
Indagationes Mathematicae, 20(1):131–149, 2009.
[18]
M. Rams and K. Simon.
The dimension of projections of fractal percolations.
preprint.
[19]
M. Rams and K. Simon.
Projections of fractal percolations.
To appear in Ergodic Theory and Dynamical Systems. |
Open string amplitudes of closed topological vertex
Kanehisa Takasaki
Department of Mathematics, Kinki University
3-4-1 Kowakae, Higashi-Osaka, Osaka 577-8502, Japan
Toshio Nakatsu
Institute for Fundamental Sciences, Setsunan University
17-8 Ikeda Nakamachi, Neyagawa, Osaka 572-8508, Japan
E-mail: takasaki@math.h.kyoto-u.ac.jpE-mail: nakatsu@mpg.setsunan.ac.jp
()
Abstract
The closed topological vertex is the simplest “off-strip” case
of non-compact toric Calabi-Yau threefolds with acyclic web diagrams.
By the diagrammatic method of topological vertex,
open string amplitudes of topological string theory therein
can be obtained by gluing a single topological vertex
to an “on-strip” subdiagram of the tree-like web diagram.
If non-trivial partitions are assigned to just two parallel
external lines of the web diagram, the amplitudes can be calculated
with the aid of techniques borrowed from the melting crystal models.
These amplitudes are thereby expressed as matrix elements,
modified by simple prefactors, of an operator product on the Fock space
of 2D charged free fermions. This fermionic expression can be used
to derive $q$-difference equations for generating functions of
special subsets of the amplitudes. These $q$-difference equations
may be interpreted as the defining equation of a quantum mirror curve.
2010 Mathematics Subject Classification:
17B81, 33E20, 81T30
Key words: closed topological vertex, open string amplitude,
free fermion, quantum torus, shift symmetry,
q-difference equation, mirror curve
1 Introduction
The closed topological vertex [1]
(see Figure 1) is the simplest “off-strip” case
of non-compact toric Calabi-Yau threefolds with acyclic web diagrams.
The diagrammatic method of topological vertex [2]
provides us with a combinatorial construction of A-model
topological string theory in local toric geometry.
Just like the “on-strip” case [3],
the partition function of topological string theory
on the closed topological vertex can be calculated
in a closed form [4, 5], which
agrees with algebro-geometric results [1].
In this paper, we calculate open string amplitudes
in the case where non-trivial partitions are assigned
to just two parallel external lines of the web diagram,
and derive $q$-difference equations satisfied
by generating functions of special subsets of these amplitudes.
In the perspectives of mirror geometry
of topological string theory [6, 7],
the $q$-difference equations may be interpreted
as the defining equations of a “quantum mirror curve”.
This quantum mirror curve will be a new example of quantum curves
in the topological recursion program [8].
To calculate the open string amplitudes in question,
we use techniques that were developed
in our previous work on the melting crystal models
[9, 10, 11, 12].
A clue of these techniques is the notion of
‘‘shift symmetries’’ in a quantum torus algebra.
This algebra is realized by operators on the Fock space
of 2D charged free fermions111
The same fermionic realization of
the quantum torus algebra appears in the work
of Okounkov and Pandharipande [13]
on the Gromov-Witten invariants of $\mathbb{C}\mathbb{P}^{1}$..
The shift symmetries act on a set of basis elements $V^{(k)}_{m}$
of this algebra so as to shift the indices $k,m$
in a certain way. This enables us to relate
the commutative subalgebra spanned by $V^{(k)}_{0}$’s222
Its role as symmetries in the KP and 2D Toda hierarchies
was independently studied by Harnad and Orlov [14].
to the $U(1)$ current algebra spanned by $V^{(0)}_{m}$’s.
In our previous work, this algebraic machinery is used
to convert the partition functions of the melting crystal models
to tau functions of the 2D Toda hierarchy.
In this paper, we employ the same method to express
the open string amplitudes as matrix elements,
modified by simple prefactors, of an operator product
on the fermionic Fock space.
Our calculations starts from a cut-and-glue description
of the amplitudes [5]. Namely,
the web diagram is cut into two subdiagrams by removing
an internal line, and glued together along this line
after calculating the contributions of these two parts.
One of them is a single topological vertex,
and the other is an on-strip diagram for which
the well known result [3] can be used.
To glue these two parts again, we have to calculate a sum.
This is the place where the aforementioned techniques are used.
The result of these calculations is somewhat surprising:
The main part of the final expression of the open string amplitudes
looks like the open string amplitudes of a new on-strip diagram.
The final expression of the open string amplitudes
enables us to derive $q$-difference equations
for the generating functions of special subsets
of the amplitudes. The generating functions
are the Baker-Akhiezer functions
in the context of integrable hierarchies,
and play the role of “wave functions”
of a probe D-brane [6, 7].
Our result is an extension of known results
on the resolved conifold [15, 16, 17]
and more general on-strip geometry [18].
The structure of the $q$-difference equation is,
so to speak, a mixture of the $q$-difference equations
of the quantum dilogarithmic functions [19, 20]
and the basic hypergeometric equations that appear
in the resolved conifold and more general on-strip geometry.
Our result shows that quantum mirror curves beyond
on-strip geometry can have an intricate origin.
This paper is organized as follows. In Section 2,
the diagrammatic construction of the open string amplitudes
are reformulated in a partially summed form. Fermionic tools
for subsequent calculations are also reviewed here.
In Section 3, the techniques borrowed from the melting crystal models
are used to calculate the amplitudes in terms of fermions.
In Section 4, the fermionic expression of the amplitudes
is further converted to a final form. A technical clue therein
is the cyclic symmetry among “two-leg” topological vertices.
This well known symmetry is translated to a kind of
“operator-state correspondence” in the fermionic Fock space,
and used to rewrite the fermionic expression of the amplitudes.
In Section 5, the generating functions of special subsets
of the amplitudes are introduced, and shown
to satisfy $q$-difference equations. The structure
of the $q$-difference equations is examined
in the perspectives of mirror geometry.
In Section 6, these results are shown to be consistent
with a flop transition. Appendix A is a brief review
of the notion of on-strip amplitudes. Appendix B presents
another proof of the identities used in Section 4.
2 Construction of open string amplitudes
The setup for the open string amplitudes in question is shown
in Figure 2. $Q_{1},Q_{2},Q_{3}$ are Kähler parameters
on the internal lines. $\beta_{1}$ and $\beta_{2}$ are partitions
assigned to the two lower external lines.
The other external lines are given the trivial partition $\emptyset$.
Let $Z^{\mathrm{cv}}_{\beta_{1}\beta_{2}}$ denote the amplitude in this setup.
$Z^{\mathrm{cv}}_{\beta_{1}\beta_{2}}$ is a sum of weights over all possible values
of the partitions $\alpha_{1},\alpha_{2},\alpha_{3}$ on the internal lines.
The weight for a given configuration of $\alpha_{1},\alpha_{2},\alpha_{3}$
is a product of vertex weights and edge weights.
These weights depend on the parameter $q$ in the range $|q|<1$.
2.1 Vertex weights and gluing rules
The vertex weight at each vertex is
the topological vertex333
We follow a definition commonly used
in the recent literature [21, 22].
This definition differs from the earlier one
[2, 6] in that $q$ is replaced by $q^{-1}$
and an overall factor of the form
$q^{\kappa(\lambda)/2+\kappa(\mu)/2+\kappa(\nu)/2}$
is multiplied.
$$C_{\lambda\mu\nu}=q^{\kappa(\mu)/2}s_{\,{}^{\mathrm{t}}\!\,\nu}(q^{-\rho})\sum%
_{\eta\in\mathcal{P}}s_{\,{}^{\mathrm{t}}\!\,\lambda/\eta}(q^{-\nu-\rho})s_{%
\mu/\eta}(q^{-\,{}^{\mathrm{t}}\!\,\nu-\rho}),$$
(2.1)
where the sum with respect to $\eta$ ranges over the set $\mathcal{P}$
of all partitions. $\lambda=(\lambda_{i})_{i=1}^{\infty}$,
$\mu=(\mu_{i})_{i=1}^{\infty}$ and $\nu=(\nu_{i})_{i=1}^{\infty}$
are the partitions assigned to the three legs
of the vertex that are ordered anti-clockwise,
and $\,{}^{\mathrm{t}}\!\,\nu$ denotes the conjugate (or transposed) partition of $\nu$.
$\kappa(\mu)$ is the second Casimir invariant
$$\kappa(\mu)=\sum_{i=1}^{\infty}\mu_{i}(\mu_{i}-2i+1)=\sum_{i=1}^{\infty}\left(%
(\mu_{i}-i+1/2)^{2}-(-i+1/2)^{2}\right).$$
$s_{\,{}^{\mathrm{t}}\!\,\nu}(q^{-\rho})$, $s_{\,{}^{\mathrm{t}}\!\,\lambda/\eta}(q^{-\nu-\rho})$ and
$s_{\mu/\eta}(q^{-\,{}^{\mathrm{t}}\!\,\nu-\rho})$ are special values
of the infinite-variate Schur function $s_{\,{}^{\mathrm{t}}\!\,\nu}(\boldsymbol{x})$
and the skew Schur functions $s_{\,{}^{\mathrm{t}}\!\,\lambda/\eta}(\boldsymbol{x}),\,s_{\mu/\eta}(%
\boldsymbol{x})$,
$\boldsymbol{x}=(x_{1},x_{2},\ldots)$, at
$$q^{-\rho}=(q^{i-1/2})_{i=1}^{\infty},\quad q^{-\nu-\rho}=(q^{-\nu_{i}+i-1/2})_%
{i=1}^{\infty},\quad q^{-\,{}^{\mathrm{t}}\!\,\nu-\rho}=(q^{-\,{}^{\mathrm{t}}%
\!\,\nu_{i}+i-1/2})_{i=1}^{\infty}.$$
The vertex weight enjoy the cyclic symmetry
$$C_{\lambda\mu\nu}=C_{\mu\nu\lambda}=C_{\nu\lambda\mu}$$
(2.2)
that can be deduced from the crystal interpretation
of the topological vertex [23].
The vertex weights $C_{\lambda\mu\nu}$ and $C_{\lambda^{\prime}\mu^{\prime}\nu^{\prime}}$
at two vertices connecting an internal line are glued together
by the following rules:
(i)
The partitions on the internal line,
say $\lambda$ and $\lambda^{\prime}$, are matched as
$$\lambda^{\prime}=\,{}^{\mathrm{t}}\!\,\lambda.$$
(ii)
The product of the vertex weights
is multiplied by the edge weight
$$(-Q)^{|\lambda|}(-1)^{n|\lambda|}q^{-n\kappa(\lambda)/2},$$
where $Q$ is the Kähler parameter of the internal line,
and $n$ is an integer called “the framing number”.
The framing number is defined as
$$n=v^{\prime}\wedge v=w^{\prime}\wedge w,$$
(2.3)
where $v,w$ and $v^{\prime},w^{\prime}$ are vectors in the web diagram
that emanate from the two vertices (see Figure 3).
The wedge product means the determinant
of the $2\times 2$ matrix formed by the two vectors,
i.e.,
$$v^{\prime}\wedge v=v^{\prime}_{1}v_{2}-v^{\prime}_{2}v_{1}$$
for $v^{\prime}=(v^{\prime}_{1},v^{\prime}_{2})$ and $v=(v_{1},v_{2})$.
These vectors $v,w$ and $v^{\prime},w^{\prime}$ are chosen along
with the third vectors $u,u^{\prime}$, $u+u^{\prime}=0$,
in such a way that $u,v,w$ and $u^{\prime},v^{\prime},w^{\prime}$ are ordered
anti-clockwise and satisfy the zero-sum relations
$$u+v+w=0,\quad u^{\prime}+v^{\prime}+w^{\prime}=0.$$
These sets of vectors are uniquely determined
as far as the toric diagram is fully triangulated
(i.e., the area of each triangle is $1/2$).
2.2 Reformulation of amplitude
The amplitude $Z^{\mathrm{cv}}_{\beta_{1}\beta_{2}}$ is given by a sum
of the product of these weights
over $\alpha_{1},\alpha_{2},\alpha_{3}\in\mathcal{P}$.
Following Sułkowski’s formulation [5],
we decompose this sum to a partial with
sum respect to $\alpha_{1},\alpha_{2}$ at the first stage
and a sum with respect to $\alpha_{3}$ at the next stage.
The full amplitude can be thus reformulated as
$$Z^{\mathrm{cv}}_{\beta_{1}\beta_{2}}=\sum_{\alpha_{3}\in\mathcal{P}}Z_{\beta_{%
1}\beta_{2}|\alpha_{3}}(-Q_{3})^{|\alpha_{3}|}C_{\,{}^{\mathrm{t}}\!\,\alpha_{%
3}\emptyset\emptyset}.$$
(2.4)
$Z_{\beta_{1}\beta_{2}|\alpha_{3}}$ is the partial sum with respect
to $\alpha_{1},\alpha_{2}$ and represents the contribution
from the lower part of the web diagram. This part is glued
with the upper part via the internal line carrying $\alpha_{3}$.
$(-Q_{3})^{|\alpha_{3}|}$ is the edge weight of this internal line.
Note that the framing number (2.3)
in this case is equal to $0$.
$C_{\,{}^{\mathrm{t}}\!\,\alpha_{3}\emptyset\emptyset}$ is the contribution
from the upper part of the web diagram.
By the cyclic symmetry (2.2),
this vertex weight reduces to a special value
of the Schur function:
$$C_{\,{}^{\mathrm{t}}\!\,\alpha_{3}\emptyset\emptyset}=C_{\emptyset\emptyset\,{%
}^{\mathrm{t}}\!\,\alpha_{3}}=s_{\alpha_{3}}(q^{-\rho}).$$
(2.5)
The partial sum $Z_{\beta_{1}\beta_{2}|\alpha_{3}}$ itself
may be thought of as an open string amplitude
of the web diagram (called “double-$\mathbb{P}^{1}$”)
shown in Figure 4.
Since this is a diagram “on a strip”,
the associated open string amplitude can be calculated
by the well known result [3] (see Appendix A):
$$\displaystyle Z_{\beta_{1}\beta_{2}|\alpha_{3}}$$
$$\displaystyle=s_{\,{}^{\mathrm{t}}\!\,\beta_{1}}(q^{-\rho})s_{\,{}^{\mathrm{t}%
}\!\,\beta_{2}}(q^{-\rho})s_{\,{}^{\mathrm{t}}\!\,\alpha_{3}}(q^{-\rho})\prod_%
{i,j=1}^{\infty}(1-Q_{1}Q_{2}q^{-\beta_{1i}-\,{}^{\mathrm{t}}\!\,\beta_{2j}+i+%
j-1})^{-1}$$
$$\displaystyle\quad\mbox{}\times\prod_{i,j=1}^{\infty}(1-Q_{1}q^{-\beta_{1i}-%
\alpha_{3j}+i+j-1})\prod_{i,j=1}^{\infty}(1-Q_{2}q^{-\,{}^{\mathrm{t}}\!\,%
\alpha_{3i}-\,{}^{\mathrm{t}}\!\,\beta_{2j}+i+j-1}).$$
(2.6)
Plugging these building blocks into (2.4),
we obtain the following expression of $Z^{\mathrm{cv}}_{\beta_{1}\beta_{2}}$:
$$\displaystyle Z^{\mathrm{cv}}_{\beta_{1}\beta_{2}}$$
$$\displaystyle=s_{\,{}^{\mathrm{t}}\!\,\beta_{1}}(q^{-\rho})s_{\,{}^{\mathrm{t}%
}\!\,\beta_{2}}(q^{-\rho})\prod_{i,j=1}^{\infty}(1-Q_{1}Q_{2}q^{-\beta_{1i}-\,%
{}^{\mathrm{t}}\!\,\beta_{2j}+i+j-1})^{-1}$$
$$\displaystyle\quad\mbox{}\times\sum_{\alpha_{3}\in\mathcal{P}}s_{\,{}^{\mathrm%
{t}}\!\,\alpha_{3}}(q^{-\rho})s_{\alpha_{3}}(q^{-\rho})(-Q_{3})^{|\alpha_{3}|}%
\prod_{i,j=1}^{\infty}(1-Q_{1}q^{-\beta_{1i}-\alpha_{3j}+i+j-1})$$
$$\displaystyle\quad\quad\quad\mbox{}\times\prod_{i,j=1}^{\infty}(1-Q_{2}q^{-\,{%
}^{\mathrm{t}}\!\,\alpha_{3i}-\,{}^{\mathrm{t}}\!\,\beta_{2j}+i+j-1}).$$
(2.7)
Note here that the sum with respect to $\alpha_{3}$ resembles
the partition function of the modified melting model
[11, 12]: The main part of
the Boltzmann weight therein takes
the product form $s_{\,{}^{\mathrm{t}}\!\,\alpha_{3}}(q^{-\rho})s_{\alpha_{3}}(q^{-\rho})$,
and this weight is deformed by external potentials depending
on $\alpha_{3}$. To calculate this sum, we use the machinery
of 2D charged free fermions.
2.3 Fermionic Fock space and operators
The setup of the fermionic Fock space and operators
is the same as used for the melting crystal models
[9, 10, 11, 12].
Let $\psi_{n},\psi^{*}_{n}$, $n\in\mathbb{Z}$, denote the Fourier modes
of the 2D charged free fermion fields $\psi(z),\psi^{*}(z)$.
They satisfy the anti-commutation relations
$$\psi_{m}\psi^{*}_{n}+\psi^{*}_{n}\psi_{m}=\delta_{m+n,0},\quad\psi_{m}\psi_{n}%
+\psi_{n}\psi_{m}=0,\quad\psi^{*}_{m}\psi^{*}_{n}+\psi^{*}_{n}\psi^{*}_{m}=0.$$
The associated Fock space and its dual space
are decomposed to the charge-$s$ sectors for $s\in\mathbb{Z}$.
It is only the charge-$0$ sector that is relevant to
the calculation of (2.7). An orthonormal basis
of the charge-$0$ sector is given the ground states
$$\displaystyle\langle 0|$$
$$\displaystyle=\langle-\infty|\cdots\psi^{*}_{-i+1}\cdots\psi^{*}_{-1}\psi^{*}_%
{0},$$
$$\displaystyle|0\rangle$$
$$\displaystyle=\psi_{0}\psi_{1}\cdots\psi_{i-1}\cdots|-\infty\rangle$$
and the excited states
$$\displaystyle\langle\lambda|$$
$$\displaystyle=\langle-\infty|\cdots\psi^{*}_{\lambda_{i}-i+1}\cdots\psi^{*}_{%
\lambda_{2}-1}\psi^{*}_{\lambda_{1}},$$
$$\displaystyle|\lambda\rangle$$
$$\displaystyle=\psi_{-\lambda_{1}}\psi_{-\lambda_{2}+1}\cdots\psi_{-\lambda_{i}%
+i-1}\cdots|-\infty\rangle$$
labelled by partitions. The normal ordered product
${:}\psi_{m}\psi^{*}_{n}{:}$ is defined as
$${:}\psi_{m}\psi^{*}_{n}{:}=\psi_{m}\psi^{*}_{n}-\langle 0|\psi_{m}\psi^{*}_{n}%
|0\rangle.$$
The following operators on the Fock space are used
as fundamental tools in our calculations.
(i)
The zero-modes
$$L_{0}=\sum_{n\in\mathbb{Z}}n{:}\psi_{-n}\psi^{*}_{n}{:},\quad W_{0}=\sum_{n\in%
\mathbb{Z}}n^{2}{:}\psi_{-n}\psi^{*}_{n}{:}$$
of the Virasoro and $W_{3}$ algebras and the Fourier modes
$$J_{m}=\sum_{n\in\mathbb{Z}}{:}\psi_{-n}\psi^{*}_{n+m}{:},\quad m\in\mathbb{Z},$$
of the fermionic current ${:}\psi(z)\psi^{*}(z){:}$.
(ii)
The fermionic realization
$$K=\sum_{n\in\mathbb{Z}}(n-1/2)^{2}{:}\psi_{-n}\psi^{*}_{n}{:}=W_{0}-L_{0}+J_{0%
}/4$$
of the so called “cut-and-join operator” [24, 25].
(iii)
The basis elements
$$V^{(k)}_{m}=q^{-km/2}\sum_{n\in\mathbb{Z}}q^{kn}{:}\psi_{m-n}\psi^{*}_{n}{:},%
\quad k,m\in\mathbb{Z},$$
of a fermionic realization of the quantum torus algebra
[9, 13].
(iv)
The vertex operators [26, 27]
$$\Gamma_{\pm}(z)=\exp\left(\sum_{k=1}^{\infty}\frac{z^{k}}{k}J_{\pm k}\right),%
\quad\Gamma^{\prime}_{\pm}(z)=\exp\left(-\sum_{k=1}^{\infty}\frac{(-z)^{k}}{k}%
J_{\pm k}\right)$$
and the multi-variable extensions
$$\Gamma_{\pm}(\boldsymbol{x})=\prod_{i\geq 1}\Gamma_{\pm}(x_{i}),\quad\Gamma^{%
\prime}_{\pm}(\boldsymbol{x})=\prod_{i\geq 1}\Gamma^{\prime}_{\pm}(x_{i}).$$
The matrix elements of these operators are well known.
$J_{0},L_{0},W_{0},K$ are diagonal with respect to
the basis $\{|\lambda\rangle\}_{\lambda\in\mathcal{P}}$
in the charge-$0$ sector:
$$\displaystyle\langle\lambda|J_{0}|\mu\rangle=0,\quad\langle\lambda|L_{0}|\mu%
\rangle=\delta_{\lambda\mu}|\lambda|,$$
$$\displaystyle\langle\lambda|W_{0}|\mu\rangle=\delta_{\lambda\mu}\left(\kappa(%
\lambda)+|\lambda|\right),\quad\langle\lambda|K|\mu\rangle=\delta_{\lambda\mu}%
\kappa(\lambda).$$
(2.8)
The matrix elements of $\Gamma_{\pm}(\boldsymbol{x})$ and $\Gamma^{\prime}_{\pm}(\boldsymbol{x})$
are skew Schur functions [28, 29]:
$$\displaystyle\langle\lambda|\Gamma_{-}(\boldsymbol{x})|\mu\rangle=\langle\mu|%
\Gamma_{+}(\boldsymbol{x})|\lambda\rangle=s_{\lambda/\mu}(\boldsymbol{x}),$$
$$\displaystyle\langle\lambda|\Gamma^{\prime}_{-}(\boldsymbol{x})|\mu\rangle=%
\langle\mu|\Gamma^{\prime}_{+}(\boldsymbol{x})|\lambda\rangle=s_{\,{}^{\mathrm%
{t}}\!\,\lambda/\,{}^{\mathrm{t}}\!\,\mu}(\boldsymbol{x}).$$
(2.9)
3 Calculation of sum in (2.7)
Let us proceed to calculation of the sum in (2.7).
This comprises two steps. In the first step,
we express $s_{\,{}^{\mathrm{t}}\!\,\alpha_{3}}(q^{-\alpha})$ and
$s_{\alpha_{3}}(q^{-\rho})$ in a fermionic form,
and convert the c-number factors $\prod_{i,j=1}^{\infty}(1-Q_{1}\cdots)$
and $\prod_{i,j=1}^{\infty}(1-Q_{2}\cdots)$ to operators
inserted in the fermionic expression of the Schur functions.
The sum with respect to $\alpha_{3}$ thereby
turns into the vacuum expectation value
of an operator product on the Fock space.
In the second step, we use the “shift symmetries”
of the quantum torus algebra
[9, 10, 11, 12]
to rewrite the vacuum expectation value further.
These calculations are more or less parallel to the way
the partition functions of the various melting crystal models
are converted to tau functions of the 2D Toda hierarchy.
3.1 Step 1: Translation to fermionic language
The infinite products $\prod_{i,j}^{\infty}(1-Q_{1}\cdots)$
and $\prod_{i,j=1}^{\infty}(1-Q_{2}\cdots)$
can be re-expressed in an exponential form as
$$\prod_{i,j=1}^{\infty}(1-Q_{1}q^{-\beta_{1i}-\alpha_{3j}+i+j-1})=\exp\left(-%
\sum_{i,k=1}^{\infty}\frac{(Q_{1}q^{-\beta_{1i}+i})^{k}}{k}\sum_{j=1}^{\infty}%
q^{-k(\alpha_{3j}-j+1)}\right)$$
and
$$\prod_{i,j=1}^{\infty}(1-Q_{2}q^{-\,{}^{\mathrm{t}}\!\,\alpha_{3i}-\,{}^{%
\mathrm{t}}\!\,\beta_{2j}+i+j-1})=\exp\left(-\sum_{j,k=1}^{\infty}\frac{(Q_{2}%
q^{-\,{}^{\mathrm{t}}\!\,\beta_{2j}+j})^{k}}{k}\sum_{i=1}^{\infty}q^{-k(\,{}^{%
\mathrm{t}}\!\,\alpha_{3i}-i+1)}\right).$$
We convert these c-number factors to operators
inserted in the fermionic expression
$$s_{\,{}^{\mathrm{t}}\!\,\alpha_{3}}(q^{-\rho})=\langle 0|\Gamma^{\prime}_{+}(q%
^{-\rho})|\alpha_{3}\rangle,\quad s_{\alpha_{3}}(q^{-\rho})=\langle\alpha_{3}|%
\Gamma_{-}(q^{-\rho})|0\rangle$$
of the special values of the Schur functions.
To this end, let us note that
$\sum_{j=1}^{\infty}q^{-k(\alpha_{3j}-j+1)}$ and
$\sum_{j=1}^{\infty}q^{-k(\,{}^{\mathrm{t}}\!\,\alpha_{3i}-i+1)}$ are
related to eigenvalues of $V^{(\pm k)}_{0}$’s as shown below.
Lemma 1.
For any $k>0$ and any $\lambda\in\mathcal{P}$,
$$\displaystyle\left(V^{(-k)}_{0}+\frac{1}{1-q^{k}}\right)|\lambda\rangle$$
$$\displaystyle=\sum_{i=1}^{\infty}q^{-k(\lambda_{i}-i+1)}|\lambda\rangle,$$
(3.1)
$$\displaystyle\left(V^{(k)}_{0}-\frac{q^{k}}{1-q^{k}}\right)|\lambda\rangle$$
$$\displaystyle=-q^{k}\sum_{i=1}^{\infty}q^{-k(\,{}^{\mathrm{t}}\!\,\lambda_{i}-%
i+1)}|\lambda\rangle.$$
(3.2)
Remark 1.
(3.1) and (3.2)
imply the relations
$$\displaystyle\langle\lambda|\left(V^{(-k)}_{0}+\frac{1}{1-q^{k}}\right)$$
$$\displaystyle=\langle\lambda|\sum_{i=1}^{\infty}q^{-k(\lambda_{i}-i+1)},$$
$$\displaystyle\langle\lambda|\left(V^{(k)}_{0}-\frac{q^{k}}{1-q^{k}}\right)$$
$$\displaystyle=-\langle\lambda|q^{k}\sum_{i=1}^{\infty}q^{-k(\,{}^{\mathrm{t}}%
\!\,\lambda_{i}-i+1)}$$
in the dual Fock space as well.
Proof.
It is straightforward to derive (3.1):
$$\displaystyle V^{(-k)}_{0}|\lambda\rangle$$
$$\displaystyle=\sum_{j=1}^{\infty}(q^{-k(\lambda_{j}-j+1)}-q^{-k(-j+1)})|\lambda\rangle$$
$$\displaystyle=\left(\sum_{j=1}^{\infty}q^{-k(\lambda_{j}-j+1)}-\frac{1}{1-q^{k%
}}\right)|\lambda\rangle.$$
The subtraction term $q^{-k(-j+1)}$ in this calculation
originates in the normal ordering
$${:}\psi_{-n}\psi^{*}_{n}{:}=\begin{cases}\psi_{-n}\psi^{*}_{n}&\text{for $n>0$%
},\\
\psi_{-n}\psi^{*}_{n}-1&\text{for $n\leq 0$}.\end{cases}$$
It is not straightforward to derive (3.2).
Let $n$ be an integer greater than or equal
to the length of $\lambda$.
Accordingly, $\lambda_{i}=i$ for $i>n$.
Since the set of all integers $i\leq n$
can be divided into two disjoint sets as
$$\{i\mid i\leq n\}=\{\,{}^{\mathrm{t}}\!\,\lambda_{i}-i+1\mid i\geq 1\}\cup\{-%
\lambda_{i}+i\mid 1\leq i\leq n\},$$
one obtains the identity
$$\sum_{i=1}^{\infty}q^{-k(\,{}^{\mathrm{t}}\!\,\lambda_{i}-i+1)}+\sum_{i=1}^{n}%
q^{-k(-\lambda_{i}+i)}=\sum_{i=-\infty}^{n}q^{-ki}=\frac{q^{-kn}}{1-q^{k}},$$
which implies that
$$\displaystyle\sum_{i=1}^{\infty}q^{-k(\,{}^{\mathrm{t}}\!\,\lambda_{i}-i+1)}$$
$$\displaystyle=-\sum_{i=1}^{n}q^{-k(-\lambda_{i}+i)}+\frac{q^{-kn}}{1-q^{k}}$$
$$\displaystyle=-\sum_{i=1}^{n}(q^{-k(-\lambda_{i}+i)}-q^{-ki})+\frac{1}{1-q^{k}}$$
$$\displaystyle=-q^{-k}\sum_{i=1}^{n}(q^{k(\lambda_{i}-i+1)}-q^{k(-i+1)})+\frac{%
1}{1-q^{k}}.$$
Consequently,
$$\displaystyle V^{(k)}_{0}|\lambda\rangle$$
$$\displaystyle=\sum_{i=1}^{n}(q^{k(\lambda_{i}-i+1)}-q^{k(-i+1)})|\lambda\rangle$$
$$\displaystyle=\left(-q^{k}\sum_{i=1}^{\infty}q^{-k(\,{}^{\mathrm{t}}\!\,%
\lambda_{i}-i+1)}+\frac{q^{k}}{1-q^{k}}\right)|\lambda\rangle.$$
(3.2) can be thus derived.
∎
By (3.1) and (3.2),
the c-number factors $\prod_{i,j=1}^{\infty}(1-Q_{1}\cdots)$
and $\prod_{i,j=1}^{\infty}(1-Q_{2}\cdots)$ can be converted
to operators on the Fock space as
$$\displaystyle\prod_{i,j=1}^{\infty}(1-Q_{1}q^{-\beta_{1i}-\alpha_{3j}+i+j-1})%
\cdot s_{\,{}^{\mathrm{t}}\!\,\alpha_{3}}(q^{-\rho})$$
$$\displaystyle=\langle 0|\Gamma^{\prime}_{+}(q^{-\rho})\exp\left(-\sum_{i,k=1}^%
{\infty}\frac{(Q_{1}q^{-\beta_{1i}+i})^{k}}{k}\left(V^{(-k)}_{0}+\frac{1}{1-q^%
{k}}\right)\right)|\alpha_{3}\rangle$$
(3.3)
and
$$\displaystyle\prod_{i,j=1}^{\infty}(1-Q_{2}q^{-\,{}^{\mathrm{t}}\!\,\alpha_{3i%
}-\,{}^{\mathrm{t}}\!\,\beta_{2j}+i+j-1})\cdot s_{\alpha_{3}}(q^{-\rho})$$
$$\displaystyle=\langle\alpha_{3}|\exp\left(\sum_{j,k=1}^{\infty}\frac{(Q_{2}q^{%
-\,{}^{\mathrm{t}}\!\,\beta_{2j}+j-1})^{k}}{k}\left(V^{(k)}_{0}-\frac{q^{k}}{1%
-q^{k}}\right)\right)\Gamma_{-}(q^{-\rho})|0\rangle.$$
(3.4)
Moreover, the factor $(-Q_{3})^{|\alpha_{3}|}$ can be identified
with the diagonal matrix element of $(-Q_{3})^{L_{0}}$.
Having derived these building blocks, we can now use
the partition of unity
$$\sum_{\alpha_{3}\in\mathcal{P}}|\alpha_{3}\rangle\langle\alpha_{3}|=1$$
in the charge-$0$ sector to rewrite the sum
in (2.7) to the vacuum expectation value
of an operator product:
$$\displaystyle\sum_{\alpha_{3}\in\mathcal{P}}s_{\,{}^{\mathrm{t}}\!\,\alpha_{3}%
}(q^{-\rho})s_{\alpha_{3}}(q^{-\rho})(-Q_{3})^{|\alpha_{3}|}\prod_{i,j=1}^{%
\infty}(1-Q_{1}\cdots)\prod_{i,j=1}^{\infty}(1-Q_{2}\cdots)$$
$$\displaystyle=\langle 0|\Gamma^{\prime}_{+}(q^{-\rho})\exp\left(-\sum_{i,k=1}^%
{\infty}\frac{(Q_{1}q^{-\beta_{1i}+i})^{k}}{k}\left(V^{(-k)}_{0}+\frac{1}{1-q^%
{k}}\right)\right)(-Q_{3})^{L_{0}}$$
$$\displaystyle\quad\mbox{}\times\exp\left(\sum_{j,k=1}^{\infty}\frac{(Q_{2}q^{-%
\,{}^{\mathrm{t}}\!\,\beta_{2j}+j-1})^{k}}{k}\left(V^{(k)}_{0}-\frac{q^{k}}{1-%
q^{k}}\right)\right)\Gamma_{-}(q^{-\rho})|0\rangle.$$
(3.5)
3.2 Step 2: Use of shift symmetries
Let us recall the following consequence of
the shift symmetries of the quantum torus algebra
[9, 10, 11]. Note that the last one
(3.8) is modified from the previous formulation
in terms of $W_{0}$.
Lemma 2.
$$\displaystyle\Gamma^{\prime}_{-}(q^{-\rho})\Gamma^{\prime}_{+}(q^{-\rho})\left%
(V^{(-k)}_{0}+\frac{1}{1-q^{k}}\right)=V^{(-k)}_{k}\Gamma^{\prime}_{-}(q^{-%
\rho})\Gamma^{\prime}_{+}(q^{-\rho}),$$
(3.6)
$$\displaystyle\left(V^{(k)}_{0}-\frac{q^{k}}{1-q^{k}}\right)\Gamma_{-}(q^{-\rho%
})\Gamma_{+}(q^{-\rho})=\Gamma_{-}(q^{-\rho})\Gamma_{+}(q^{-\rho})(-1)^{k}V^{(%
k)}_{k},$$
(3.7)
$$\displaystyle V^{(-k)}_{k}=q^{-k/2}q^{K/2}J_{k}q^{-K/2},\quad V^{(k)}_{-k}=q^{%
k/2}q^{K/2}J_{-k}q^{-K/2}.$$
(3.8)
We use these operator identities to rewrite
(3.5) further. Let us first examine
the left side of $(-Q_{3})^{L_{3}}$ in (3.5).
Upon inserting $q^{-K/2}\Gamma^{\prime}_{-}(q^{-\rho})$
to the right of $\langle 0|$ as
$$\langle 0|\Gamma^{\prime}_{+}(q^{-\rho})=\langle 0|q^{-K/2}\Gamma^{\prime}_{-}%
(q^{-\rho})\Gamma^{\prime}_{+}(q^{-\rho}),$$
we can use (3.6) and (3.8) to rewrite
the left side of $(-Q_{3})^{L_{3}}$ as
$$\displaystyle\langle 0|\Gamma^{\prime}_{+}(q^{-\rho})\exp\left(-\sum_{i,k=1}^{%
\infty}\frac{(Q_{1}q^{-\beta_{1i}+i})^{k}}{k}\left(V^{(-k)}_{0}+\frac{1}{1-q^{%
k}}\right)\right)$$
$$\displaystyle=\langle 0|q^{-K/2}\exp\left(-\sum_{i,k=1}^{\infty}\frac{(Q_{1}q^%
{-\beta_{1i}+i})^{k}}{k}V^{(-k)}_{k}\right)\Gamma^{\prime}_{-}(q^{-\rho})%
\Gamma^{\prime}_{+}(q^{-\rho})$$
$$\displaystyle=\langle 0|\exp\left(-\sum_{i,k=1}^{\infty}\frac{(Q_{1}q^{-\beta_%
{1i}+i})^{k}}{k}q^{-k/2}J_{k}\right)q^{-K/2}\Gamma^{\prime}_{-}(q^{-\rho})%
\Gamma^{\prime}_{+}(q^{-\rho})$$
$$\displaystyle=\langle 0|\exp\left(-\sum_{i,k=1}^{\infty}\frac{(Q_{1}q^{-\beta_%
{1i}+i-1/2})^{k}}{k}J_{k}\right)q^{-K/2}\Gamma^{\prime}_{-}(q^{-\rho})\Gamma^{%
\prime}_{+}(q^{-\rho}).$$
The exponential operator in the last line is essentially
a vertex operator,
$$\exp\left(-\sum_{i,k=1}^{\infty}\frac{(Q_{1}q^{-\beta_{1i}+i-1/2})^{k}}{k}J_{k%
}\right)=(-Q_{1})^{-L_{0}}\Gamma^{\prime}_{+}(q^{-\beta_{1}-\rho})(-Q_{1})^{L_%
{0}},$$
hence
$$\displaystyle\langle 0|\Gamma^{\prime}_{+}(q^{-\rho})\exp\left(-\sum_{i,k=1}^{%
\infty}\frac{(Q_{1}q^{-\beta_{1i}+i})^{k}}{k}\left(V^{(-k)}_{0}+\frac{1}{1-q^{%
k}}\right)\right)$$
$$\displaystyle=\langle 0|\Gamma^{\prime}_{+}(q^{-\beta_{1}-\rho})(-Q_{1})^{L_{0%
}}q^{-K/2}\Gamma^{\prime}_{-}(q^{-\rho})\Gamma^{\prime}_{+}(q^{-\rho}).$$
(3.9)
In exactly the same manner, using (3.7) and (3.8),
we can rewrite the right side of $(-Q_{3})^{L_{0}}$ as
$$\displaystyle\exp\left(\sum_{j,k=1}^{\infty}\frac{(Q_{2}q^{-\,{}^{\mathrm{t}}%
\!\,\beta_{2j}+j-1})^{k}}{k}\left(V^{(k)}_{0}-\frac{q^{k}}{1-q^{k}}\right)%
\right)\Gamma_{-}(q^{-\rho})|0\rangle$$
$$\displaystyle=\Gamma_{-}(q^{-\rho})\Gamma_{+}(q^{-\rho})q^{K/2}(-Q_{2})^{L_{0}%
}\Gamma_{-}(q^{-\,{}^{\mathrm{t}}\!\,\beta_{2}-\rho})|0\rangle.$$
(3.10)
Plugging (3.9) and (3.10)
into (3.5) yields the following expression
of the sum in (2.7):
$$\displaystyle\sum_{\alpha_{3}\in\mathcal{P}}s_{\,{}^{\mathrm{t}}\!\,\alpha_{3}%
}(q^{-\rho})s_{\alpha_{3}}(q^{-\rho})(-Q_{3})^{|\alpha_{3}|}\prod_{i,j=1}^{%
\infty}(1-Q_{1}\cdots)\prod_{i,j=1}^{\infty}(1-Q_{2}\cdots)$$
$$\displaystyle=\langle 0|\Gamma^{\prime}_{+}(q^{-\beta_{1}-\rho})(-Q_{1})^{L_{0%
}}q^{-K/2}\Gamma^{\prime}_{-}(q^{-\rho})\Gamma^{\prime}_{+}(q^{-\rho})(-Q_{3})%
^{L_{0}}$$
$$\displaystyle\quad\mbox{}\times\Gamma_{-}(q^{-\rho})\Gamma_{+}(q^{-\rho})q^{K/%
2}(-Q_{2})^{L_{0}}\Gamma_{-}(q^{-\,{}^{\mathrm{t}}\!\,\beta_{2}-\rho})|0\rangle.$$
(3.11)
4 Final expression of open string amplitudes
We have thus derived the following intermediate expression
of $Z^{\mathrm{cv}}_{\beta_{1}\beta_{2}}$:
$$\displaystyle Z^{\mathrm{cv}}_{\beta_{1}\beta_{2}}$$
$$\displaystyle=s_{\,{}^{\mathrm{t}}\!\,\beta_{1}}(q^{-\rho})s_{\,{}^{\mathrm{t}%
}\!\,\beta_{2}}(q^{-\rho})\prod_{i,j=1}^{\infty}(1-Q_{1}Q_{2}q^{-\beta_{1i}-\,%
{}^{\mathrm{t}}\!\,\beta_{2j}+i+j-1})^{-1}$$
$$\displaystyle\quad\mbox{}\times\langle 0|\Gamma^{\prime}_{+}(q^{-\beta_{1}-%
\rho})(-Q_{1})^{L_{0}}q^{-K/2}\Gamma^{\prime}_{-}(q^{-\rho})\Gamma^{\prime}_{+%
}(q^{-\rho})(-Q_{3})^{L_{0}}$$
$$\displaystyle\quad\mbox{}\times\Gamma_{-}(q^{-\rho})\Gamma_{+}(q^{-\rho})q^{K/%
2}(-Q_{2})^{L_{0}}\Gamma_{-}(q^{-\,{}^{\mathrm{t}}\!\,\beta_{2}-\rho})|0\rangle.$$
(4.1)
As a final step, we use the following relations
in the fermionic Fock space that can be derived
from a special case of the cyclic symmetry (2.2).
This is a kind of operator-state correspondence
that maps vertex operators
of the form $\Gamma_{\pm}(q^{-\,{}^{\mathrm{t}}\!\,\lambda-\rho})$
and $\Gamma^{\prime}_{\pm}(q^{-\lambda-\rho})$
to the state vectors $\langle\,{}^{\mathrm{t}}\!\,\lambda|$
and $|\,{}^{\mathrm{t}}\!\,\lambda\rangle$ in the Fock space.
Lemma 3.
For any $\lambda\in\mathcal{P}$,
$$\displaystyle s_{\,{}^{\mathrm{t}}\!\,\lambda}(q^{-\rho})\Gamma^{\prime}_{-}(q%
^{-\lambda-\rho})|0\rangle$$
$$\displaystyle=q^{K/2}\Gamma_{-}(q^{-\rho})\Gamma_{+}(q^{-\rho})|\,{}^{\mathrm{%
t}}\!\,\lambda\rangle,$$
(4.2)
$$\displaystyle s_{\,{}^{\mathrm{t}}\!\,\lambda}(q^{-\rho})\Gamma_{-}(q^{-\,{}^{%
\mathrm{t}}\!\,\lambda-\rho})|0\rangle$$
$$\displaystyle=q^{\kappa(\lambda)/2}q^{-K/2}\Gamma^{\prime}_{-}(q^{-\rho})%
\Gamma^{\prime}_{+}(q^{-\rho})|\,{}^{\mathrm{t}}\!\,\lambda\rangle.$$
(4.3)
Remark 2.
There are a number of apparently different, but equivalent
forms of these relations. For example, one can use
the well known identity [28]
$$s_{\,{}^{\mathrm{t}}\!\,\lambda}(q^{-\rho})=q^{\kappa(\lambda)/2}s_{\lambda}(q%
^{-\rho})$$
(4.4)
to rewrite (4.3) as
$$s_{\lambda}(q^{-\rho})\Gamma_{-}(q^{-\,{}^{\mathrm{t}}\!\,\lambda-\rho})|0%
\rangle=q^{-K/2}\Gamma^{\prime}_{-}(q^{-\rho})\Gamma^{\prime}_{+}(q^{-\rho})|%
\,{}^{\mathrm{t}}\!\,\lambda\rangle.$$
(4.5)
(4.5), in turn, is equivalent to (4.2)
(with $\lambda$ being replaced by $\,{}^{\mathrm{t}}\!\,\lambda$)
as one can see from the identities
$$\displaystyle\langle\,{}^{\mathrm{t}}\!\,\mu|\Gamma_{-}(q^{-\,{}^{\mathrm{t}}%
\!\,\lambda-\rho})|0\rangle$$
$$\displaystyle=\langle\mu|\Gamma^{\prime}_{-}(q^{-\,{}^{\mathrm{t}}\!\,\lambda-%
\rho})|0\rangle,$$
$$\displaystyle\langle\,{}^{\mathrm{t}}\!\,\mu|\Gamma^{\prime}_{-}(q^{-\rho})%
\Gamma^{\prime}_{+}(q^{-\rho})|\,{}^{\mathrm{t}}\!\,\lambda\rangle$$
$$\displaystyle=\langle\mu|\Gamma_{-}(q^{-\rho})\Gamma_{+}(q^{-\rho})|\lambda\rangle$$
and the fact that $\langle\lambda|$ and $|\lambda\rangle$
are eigenvector of $K$ with eigenvalue $\kappa(\lambda)$.
(4.2), (4.3) and (4.5) imply the relations
$$\displaystyle s_{\,{}^{\mathrm{t}}\!\,\lambda}(q^{-\rho})\langle 0|\Gamma^{%
\prime}_{+}(q^{-\lambda-\rho})$$
$$\displaystyle=\langle\,{}^{\mathrm{t}}\!\,\lambda|\Gamma_{-}(q^{-\rho})\Gamma_%
{+}(q^{-\rho})q^{K/2}$$
$$\displaystyle s_{\,{}^{\mathrm{t}}\!\,\lambda}(q^{-\rho})\langle 0|\Gamma_{+}(%
q^{-\,{}^{\mathrm{t}}\!\,\lambda-\rho})$$
$$\displaystyle=q^{\kappa(\lambda)/2}\langle\,{}^{\mathrm{t}}\!\,\lambda|\Gamma^%
{\prime}_{-}(q^{-\rho})\Gamma^{\prime}_{+}(q^{-\rho})q^{-K/2},$$
$$\displaystyle s_{\lambda}(q^{-\rho})\langle 0|\Gamma_{+}(q^{-\,{}^{\mathrm{t}}%
\!\,\lambda-\rho})$$
$$\displaystyle=\langle\,{}^{\mathrm{t}}\!\,\lambda|\Gamma^{\prime}_{-}(q^{-\rho%
})\Gamma^{\prime}_{+}(q^{-\rho})q^{-K/2}$$
in the dual Fock space as well.
Proof.
The topological vertex has the fermionic expression
$$\displaystyle C_{\lambda\mu\nu}$$
$$\displaystyle=q^{\kappa(\mu)/2}s_{\,{}^{\mathrm{t}}\!\,\nu}(q^{-\rho})\langle%
\,{}^{\mathrm{t}}\!\,\lambda|\Gamma_{-}(q^{-\nu-\rho})\Gamma_{+}(q^{-\,{}^{%
\mathrm{t}}\!\,\nu-\rho})|\mu\rangle$$
$$\displaystyle=q^{\kappa(\mu)/2}s_{\,{}^{\mathrm{t}}\!\,\nu}(q^{-\rho})\langle%
\lambda|\Gamma^{\prime}_{-}(q^{-\nu-\rho})\Gamma^{\prime}_{+}(q^{-\,{}^{%
\mathrm{t}}\!\,\nu-\rho})|\,{}^{\mathrm{t}}\!\,\mu\rangle.$$
(4.6)
The “two-leg” case $C_{\mu\emptyset\lambda}=C_{\lambda\mu\emptyset}$
of the cyclic symmetry (2.2)444Zhou [30]
gave a direct proof of the two-leg cyclic symmetry
without relying on the crystal interpretation [23].
We present another direct proof in Append B that employs
the same techniques as used in Section 3.
thereby turns into the relation
$$s_{\,{}^{\mathrm{t}}\!\,\lambda}(q^{-\rho})\langle 0|\Gamma^{\prime}_{+}(q^{-%
\lambda-\rho})|\mu\rangle=\langle\,{}^{\mathrm{t}}\!\,\lambda|\Gamma_{-}(q^{-%
\rho})\Gamma_{+}(q^{-\rho})q^{K/2}|\mu\rangle.$$
(4.7)
among matrix elements of operators on the Fock space.
Since this identity holds for any $\mu$,
one obtains (4.2) in the dual form.
Similarly, the symmetry relation $C_{\emptyset\mu\lambda}=C_{\mu\lambda\emptyset}$ yields the identity
$$s_{\,{}^{\mathrm{t}}\!\,\lambda}(q^{-\rho})\langle\mu|q^{K/2}\Gamma_{-}(q^{-\,%
{}^{\mathrm{t}}\!\,\lambda-\rho})|0\rangle=\langle\mu|\Gamma^{\prime}_{-}(q^{-%
\rho})\Gamma^{\prime}_{+}(q^{-\rho})|\,{}^{\mathrm{t}}\!\,\lambda\rangle q^{%
\kappa(\lambda)/2},$$
(4.8)
and this implies (4.3).
∎
We can use the specialization
$$\displaystyle s_{\,{}^{\mathrm{t}}\!\,\beta_{1}}(q^{-\rho})\langle 0|\Gamma^{%
\prime}_{+}(q^{-\beta_{1}-\rho})$$
$$\displaystyle=\langle\,{}^{\mathrm{t}}\!\,\beta_{1}|\Gamma_{-}(q^{-\rho})%
\Gamma_{+}(q^{-\rho})q^{K/2},$$
$$\displaystyle s_{\,{}^{\mathrm{t}}\!\,\beta_{2}}(q^{-\rho})\Gamma_{-}(q^{-\,{}%
^{\mathrm{t}}\!\,\beta_{2}-\rho})|0\rangle$$
$$\displaystyle=q^{\kappa(\beta_{2})/2}q^{-K/2}\Gamma^{\prime}_{-}(q^{-\rho})%
\Gamma^{\prime}_{+}(q^{-\rho})|\,{}^{\mathrm{t}}\!\,\beta_{2}\rangle$$
of (4.2) and (4.3)
to $\lambda=\beta_{1}$ and $\lambda=\beta_{2}$
to rewrite (4.1) as
$$\displaystyle Z^{\mathrm{cv}}_{\beta_{1}\beta_{2}}$$
$$\displaystyle=q^{\kappa(\beta_{2})/2}\prod_{i,j=1}^{\infty}(1-Q_{1}Q_{2}q^{-%
\beta_{1i}-\,{}^{\mathrm{t}}\!\,\beta_{2j}+i+j-1})^{-1}$$
$$\displaystyle\quad\mbox{}\times\langle\,{}^{\mathrm{t}}\!\,\beta_{1}|\Gamma_{-%
}(q^{-\rho})\Gamma_{+}(q^{-\rho})q^{K/2}(-Q_{1})^{L_{0}}q^{-K/2}\Gamma^{\prime%
}_{-}(q^{-\rho})\Gamma^{\prime}_{+}(q^{-\rho})(-Q_{3})^{L_{0}}$$
$$\displaystyle\quad\mbox{}\times\Gamma_{-}(q^{-\rho})\Gamma_{+}(q^{-\rho})q^{K/%
2}(-Q_{2})^{L_{0}}q^{-K/2}\Gamma^{\prime}_{-}(q^{-\rho})\Gamma^{\prime}_{+}(q^%
{-\rho})|\,{}^{\mathrm{t}}\!\,\beta_{2}\rangle.$$
Since $q^{K/2}$’s and $q^{-K/2}$’s in this expression cancel out as
$$q^{K/2}(-Q_{1})^{L_{0}}q^{-K/2}=(-Q_{1})^{L_{0}},\quad q^{K/2}(-Q_{2})^{L_{0}}%
q^{-K/2}=(-Q_{2})^{L_{0}},$$
(4.9)
we arrive at the following final expression of $Z^{\mathrm{cv}}_{\beta_{1}\beta_{2}}$.
Theorem 1.
The open string amplitude $Z^{\mathrm{cv}}_{\beta_{1}\beta_{2}}$ can be expressed as
$$\displaystyle Z^{\mathrm{cv}}_{\beta_{1}\beta_{2}}$$
$$\displaystyle=q^{\kappa(\beta_{2})/2}\prod_{i,j=1}^{\infty}(1-Q_{1}Q_{2}q^{-%
\beta_{1i}-\,{}^{\mathrm{t}}\!\,\beta_{2j}+i+j-1})^{-1}$$
$$\displaystyle\quad\mbox{}\times\langle\,{}^{\mathrm{t}}\!\,\beta_{1}|\Gamma_{-%
}(q^{-\rho})\Gamma_{+}(q^{-\rho})(-Q_{1})^{L_{0}}\Gamma^{\prime}_{-}(q^{-\rho}%
)\Gamma^{\prime}_{+}(q^{-\rho})(-Q_{3})^{L_{0}}$$
$$\displaystyle\quad\mbox{}\times\Gamma_{-}(q^{-\rho})\Gamma_{+}(q^{-\rho})(-Q_{%
2})^{L_{0}}\Gamma^{\prime}_{-}(q^{-\rho})\Gamma^{\prime}_{+}(q^{-\rho})|\,{}^{%
\mathrm{t}}\!\,\beta_{2}\rangle.$$
(4.10)
It is remarkable that the main part
$\langle\,{}^{\mathrm{t}}\!\,\beta_{1}|\cdots|\,{}^{\mathrm{t}}\!\,\beta_{2}\rangle$
of this expression coincides with the open string amplitude
of the on-strip web diagram shown in Figure 5.
Thus, speaking schematically, gluing the one-leg vertex
(2.5) to the on-strip web diagram of Figure 4
generates another on-strip web diagram and its correction
$q^{\kappa(\beta_{2})/2}\prod_{i,j=1}^{\infty}(1-Q_{1}Q_{2}\cdots)$.
This structure of (4.10) is a key
to derive $q$-difference equations for generating functions.
5 $q$-difference equations for generating functions
The foregoing expression (4.10)
of the open string amplitudes can be used
to derive $q$-difference equation for the generating functions
$$\displaystyle\Psi(x)$$
$$\displaystyle=\frac{1}{Z^{\mathrm{cv}}_{\emptyset\emptyset}}\sum_{k=0}^{\infty%
}Z^{\mathrm{cv}}_{(1^{k})\emptyset}x^{k},$$
(5.1)
$$\displaystyle\tilde{\Psi}(x)$$
$$\displaystyle=\frac{1}{Z^{\mathrm{cv}}_{\emptyset\emptyset}}\sum_{k=0}^{\infty%
}Z^{\mathrm{cv}}_{(k)\emptyset}x^{k}$$
(5.2)
of special subsets of the normalized amplitudes
$Z^{\mathrm{cv}}_{\beta_{1}\beta_{2}}/Z^{\mathrm{cv}}_{\emptyset\emptyset}$.
Note that $(1^{k})$ ($k$-copies of $1$) and $(k)$
represent Young diagrams with a single column or row.
These generating functions are
the Baker-Akhiezer functions555
Speaking more precisely, it is rather $\Psi(-x)$
and $\tilde{\Psi}(x)$ that literally correspond
to the dual pair of Baker-Akhiezer functions.
Because of this, the $q$-difference equations
for $\Psi(x)$ and $\tilde{\Psi}(x)$ presented below
are not fully symmetric. This is also the case
for another pair $\Phi(x)$ and $\tilde{\Phi}(x)$
of generating functions introduced below.
of an integrable hierarchy, and $x$ amounts
to the spectral variable therein [18].
One can derive $q$-difference equations for
generating functions
of $Z^{\mathrm{cv}}_{\emptyset(1^{k})}/Z^{\mathrm{cv}}_{\emptyset\emptyset}$
and $Z^{\mathrm{cv}}_{\emptyset(k)}/Z^{\mathrm{cv}}_{\emptyset\emptyset}$ as well,
though they become slightly more complicated
because of the presence of the factor $q^{\kappa(\beta_{2})/2}$.
5.1 Derivation of $q$-difference equation
A key towards the derivation of a $q$-difference equation
is to compare $\Psi(x)$ and $\tilde{\Psi}(x)$ with
another pair of generating functions
$$\displaystyle\Phi(x)$$
$$\displaystyle=\frac{1}{Y_{\emptyset\emptyset}}\sum_{k=0}^{\infty}Y_{(1^{k})%
\emptyset}x^{k},$$
(5.3)
$$\displaystyle\tilde{\Phi}(x)$$
$$\displaystyle=\frac{1}{Y_{\emptyset\emptyset}}\sum_{k=0}^{\infty}Y_{(k)%
\emptyset}x^{k}$$
(5.4)
obtained from the the main part
$$\displaystyle Y_{\beta_{1}\beta_{2}}$$
$$\displaystyle=\langle\,{}^{\mathrm{t}}\!\,\beta_{1}|\Gamma_{-}(q^{-\rho})%
\Gamma_{+}(q^{-\rho})(-Q_{1})^{L_{0}}\Gamma^{\prime}_{-}(q^{-\rho})\Gamma^{%
\prime}_{+}(q^{-\rho})(-Q_{3})^{L_{0}}$$
$$\displaystyle\quad\mbox{}\times\Gamma_{-}(q^{-\rho})\Gamma_{+}(q^{-\rho})(-Q_{%
2})^{L_{0}}\Gamma^{\prime}_{-}(q^{-\rho})\Gamma^{\prime}_{+}(q^{-\rho})|\,{}^{%
\mathrm{t}}\!\,\beta_{2}\rangle$$
(5.5)
of the fermionic expression (4.10) of $Z^{\mathrm{cv}}_{\beta_{1}\beta_{2}}$.
Let us first note the following relation between
the coefficients of $\Psi(x)$ and $\Phi(x)$.
Lemma 4.
The coefficients of the expansion
$$\Psi(x)=\sum_{k=0}^{\infty}a_{k}x^{k},\quad\Phi(x)=\sum_{k=0}^{\infty}b_{k}x^{%
k},\quad a_{0}=b_{0}=1,$$
are related as
$$a_{k}=b_{k}\prod_{i=1}^{k}(1-Q_{1}Q_{2}q^{i-1})^{-1}\quad\text{for $k\geq 1$}.$$
(5.6)
Proof.
$a_{k}/b_{k}$ is given by the ratio of the values
of the prefactor in (4.10)
for $\beta_{1}=(1^{k}),\,\beta_{2}=\emptyset$
and $\beta_{1}=\beta_{2}=\emptyset$:
$$\displaystyle\frac{a_{k}}{b_{k}}$$
$$\displaystyle=\prod_{i,j=1}^{\infty}(1-Q_{1}Q_{2}q^{-\beta_{1i}-\,{}^{\mathrm{%
t}}\!\,\beta_{2j}+i+j-1})^{-1}/\prod_{i,j=1}^{\infty}(1-Q_{1}Q_{2}q^{i+j-1})^{%
-1}$$
$$\displaystyle=\prod_{i=1}^{k}\prod_{j=1}^{\infty}(1-Q_{1}Q_{2}q^{i+j-2})^{-1}/%
\prod_{i=1}^{k}\prod_{j=1}^{\infty}(1-Q_{1}Q_{2}q^{i+j-1})^{-1}$$
$$\displaystyle=\prod_{i=1}^{k}(1-Q_{1}Q_{2}q^{i-1})^{-1}.$$
∎
The next step is to derive a $q$-difference equation for $\Phi(x)$.
Lemma 5.
$\Phi(x)$ can be expressed in the infinite-product form
$$\Phi(x)=\prod_{i=1}^{\infty}\frac{(1-Q_{1}q^{i-1/2}x)(1-Q_{1}Q_{2}Q_{3}q^{i-1/%
2}x)}{(1-q^{i-1/2}x)(1-Q_{1}Q_{3}q^{i-1/2}x)},$$
(5.7)
and satisfies the $q$-difference equation
$$\Phi(qx)=\frac{(1-q^{1/2}x)(1-Q_{1}Q_{3}q^{1/2}x)}{(1-Q_{1}q^{1/2}x)(1-Q_{1}Q_%
{2}Q_{3}q^{1/2}x)}\Phi(x).$$
(5.8)
Remark 3.
The infinite product $\prod_{i=1}^{\infty}(1-q^{i-1/2}x)^{-1}$
is an expression of the quantum dilogarithmic function [19, 20].
Thus $\Phi(x)$ is a multiplicative combination
of four quantum dilogarithmic functions.
Proof.
$Y_{(1^{k})\emptyset}$ can be expressed as
$$\displaystyle Y_{(1^{k})\emptyset}$$
$$\displaystyle=\langle(k)|\Gamma_{-}(q^{-\rho})\Gamma_{+}(q^{-\rho})(-Q_{1})^{L%
_{0}}\Gamma^{\prime}_{-}(q^{-\rho})\Gamma^{\prime}_{+}(q^{-\rho})(-Q_{3})^{L_{%
0}}$$
$$\displaystyle\quad\mbox{}\times\Gamma_{-}(q^{-\rho})\Gamma_{+}(q^{-\rho})(-Q_{%
2})^{L_{0}}\Gamma^{\prime}_{-}(q^{-\rho})\Gamma^{\prime}_{+}(q^{-\rho})|0\rangle.$$
By the fundamental properties
$$\sum_{k=0}^{\infty}x^{k}\langle(k)|=\langle 0|\Gamma_{+}(x),\quad\sum_{k=0}^{%
\infty}x^{k}\langle(1^{k})|=\langle 0|\Gamma^{\prime}_{+}(x),$$
(5.9)
of the single-variate vertex operators [26, 27],
the generating function of $Y_{(1^{k})\emptyset}$’s
can be expressed as
$$\displaystyle\sum_{k=0}^{\infty}Y_{(1^{k})\emptyset}x^{k}$$
$$\displaystyle=\langle 0|\Gamma_{+}(x)\Gamma_{-}(q^{-\rho})\Gamma_{+}(q^{-\rho}%
)(-Q_{1})^{L_{0}}\Gamma^{\prime}_{-}(q^{-\rho})\Gamma^{\prime}_{+}(q^{-\rho})(%
-Q_{3})^{L_{0}}$$
$$\displaystyle\quad\mbox{}\times\Gamma_{-}(q^{-\rho})\Gamma_{+}(q^{-\rho})(-Q_{%
2})^{L_{0}}\Gamma^{\prime}_{-}(q^{-\rho})\Gamma^{\prime}_{+}(q^{-\rho})|0\rangle.$$
One can now use the commutation relations [26, 27]
$$\displaystyle\Gamma_{+}(x)\Gamma_{-}(y)$$
$$\displaystyle=(1-xy)^{-1}\Gamma_{-}(y)\Gamma_{+}(x),$$
$$\displaystyle\Gamma^{\prime}_{+}(x)\Gamma^{\prime}_{-}(y)$$
$$\displaystyle=(1-xy)^{-1}\Gamma^{\prime}_{-}(y)\Gamma^{\prime}_{+}(x),$$
$$\displaystyle\Gamma_{+}(x)\Gamma^{\prime}_{-}(y)$$
$$\displaystyle=(1+xy)\Gamma^{\prime}_{-}(y)\Gamma_{+}(x),$$
$$\displaystyle\Gamma^{\prime}_{+}(x)\Gamma_{-}(y)$$
$$\displaystyle=(1+xy)\Gamma_{-}(y)\Gamma^{\prime}_{+}(x)$$
(5.10)
of the single-variate vertex operators
to move $\Gamma_{+}(x)$ to the right
until it hits $|0\rangle$ and disappears.
This yields the infinite-product expression
$$\sum_{k=0}^{\infty}Y_{(1^{k})\emptyset}x^{k}=\prod_{i=1}^{\infty}\frac{(1-Q_{1%
}q^{i-1/2}x)(1-Q_{1}Q_{2}Q_{3}q^{i-1/2}x)}{(1-q^{i-1/2}x)(1-Q_{1}Q_{3}q^{i-1/2%
}x)}Y_{\emptyset\emptyset}$$
of the unnormalized generating function,
hence the expression (5.7) of $\Phi(x)$.
The $q$-difference equation (5.8)
is an immediate consequence of (5.7).
∎
To derive a $q$-difference equation for $\Psi(x)$,
let us rewrite (5.8) as
$$\displaystyle(1-Q_{1}(1+Q_{2}Q_{3})q^{1/2}x+Q_{1}^{2}Q_{2}Q_{3}qx^{2})\Phi(qx)$$
$$\displaystyle=(1-(1+Q_{1}Q_{3})q^{1/2}x+Q_{1}Q_{3}qx^{2})\Phi(x)$$
and extract the coefficients of $x^{k}$.
This yields the recursion relations
$$\displaystyle q^{k}b_{k}-Q_{1}(1+Q_{2}Q_{3})q^{1/2}q^{k-1}b_{k-1}+Q_{1}^{2}Q_{%
2}Q_{3}qq^{k-2}b_{k-2}$$
$$\displaystyle=b_{k}-(1+Q_{1}Q_{3})q^{1/2}b_{k-1}+Q_{1}Q_{3}qb_{k-2}$$
(5.11)
for $b_{k}$’s. Note that these relations hold for all $k\in\mathbb{Z}$
if $b_{k}$’s for $k<0$ are understood to be $0$.
By (5.6), these recursion relations turn
into the recursion relations
$$\displaystyle(1-Q_{1}Q_{2}q^{k-2})(1-Q_{1}Q_{2}q^{k-1})q^{k}a_{k}$$
$$\displaystyle\mbox{}-Q_{1}(1+Q_{2}Q_{3})q^{1/2}(1-Q_{1}Q_{2}q^{k-2})q^{k-1}a_{%
k-1}+Q_{1}^{2}Q_{2}Q_{3}qq^{k-2}a_{k-2}$$
$$\displaystyle=(1-Q_{1}Q_{2}q^{k-2})(1-Q_{1}Q_{2}q^{k-1})a_{k}$$
$$\displaystyle\quad\mbox{}-(1+Q_{1}Q_{3})q^{1/2}(1-Q_{1}Q_{2}q^{k-2})a_{k-1}+Q_%
{1}Q_{3}qa_{k-2}$$
(5.12)
for $a_{k}$’s. Multiplying these equations by $x^{k}$
and taking the sum over $k=0,1,\ldots$,
we can derive a $q$-difference equation for $\Phi(x)$.
To state this result in a compact form, let us use
the shift operator $q^{x\partial_{x}}$, $\partial_{x}=\partial/\partial x$,
that acts on a function $f(x)$ of $x$ as
$$q^{x\partial_{x}}f(x)=f(qx).$$
Theorem 2.
$\Psi(x)$ satisfies the $q$-difference equation
$$\displaystyle(1-Q_{1}Q_{2}q^{-2}q^{x\partial_{x}})(1-Q_{1}Q_{2}q^{-1}q^{x%
\partial_{x}})\Psi(qx)$$
$$\displaystyle\mbox{}-Q_{1}(1+Q_{2}Q_{3})q^{1/2}x(1-Q_{1}Q_{2}q^{-1}q^{x%
\partial_{x}})\Psi(qx)+Q_{1}^{2}Q_{2}Q_{3}qx^{2}\Psi(qx)$$
$$\displaystyle=(1-Q_{1}Q_{2}q^{-2}q^{x\partial_{x}})(1-Q_{1}Q_{2}q^{-1}q^{x%
\partial_{x}})\Psi(x)$$
$$\displaystyle\quad\mbox{}-(1+Q_{1}Q_{3})q^{1/2}x(1-Q_{1}Q_{2}q^{-1}q^{x%
\partial_{x}})\Psi(x)+Q_{1}Q_{3}qx^{2}\Psi(x).$$
(5.13)
A $q$-difference equation for $\tilde{\Psi}(x)$ can be derived
in the same way from the $q$-difference equation
$$\tilde{\Phi}(qx)=\frac{(1+Q_{1}q^{1/2}x)(1+Q_{1}Q_{2}Q_{3}q^{1/2}x)}{(1+q^{1/2%
}x)(1+Q_{1}Q_{3}q^{1/2}x)}\tilde{\Phi}(x)$$
(5.14)
for $\tilde{\Phi}(x)$ and the relation
$$\tilde{a}_{k}=\tilde{b}_{k}\prod_{i=1}^{k}(1-Q_{1}Q_{2}q^{1-i})^{-1}\quad\text%
{for $k\geq 1$}$$
(5.15)
between the coefficients of the expansion
$$\tilde{\Psi}(x)=\sum_{k=0}^{\infty}\tilde{a}_{k}x^{k},\quad\tilde{\Phi}(x)=%
\sum_{k=0}^{\infty}\tilde{b}_{k}x^{k},\quad\tilde{a}_{0}=\tilde{b}_{0}=1,$$
of $\tilde{\Psi}(x)$ and $\tilde{\Phi}(x)$.
We omit the detail of calculations and show the result:
Theorem 3.
$\tilde{\Psi}(x)$ satisfies the $q$-difference equation
$$\displaystyle(1-Q_{1}Q_{2}q^{2}q^{-x\partial_{x}})(1-Q_{1}Q_{2}qq^{-x\partial_%
{x}})\tilde{\Psi}(q^{-1}x)$$
$$\displaystyle\mbox{}+Q_{1}(1+Q_{2}Q_{3})q^{-1/2}x(1-Q_{1}Q_{2}qq^{-x\partial_{%
x}})\tilde{\Psi}(q^{-1}x)+Q_{1}^{2}Q_{2}Q_{3}q^{-1}x^{2}\tilde{\Psi}(q^{-1}x)$$
$$\displaystyle=(1-Q_{1}Q_{2}q^{2}q^{-x\partial_{x}})(1-Q_{1}Q_{2}qq^{-x\partial%
_{x}})\tilde{\Psi}(x)$$
$$\displaystyle\quad\mbox{}+(1+Q_{1}Q_{3})q^{-1/2}x(1-Q_{1}Q_{2}qq^{-x\partial_{%
x}})\tilde{\Psi}(x)+Q_{1}Q_{3}q^{-1}x^{2}\tilde{\Psi}(x).$$
(5.16)
Remark 4.
Both sides of (5.13) and (5.16)
can be rewritten as
$$\displaystyle(1-Q_{1}Q_{2}q^{-2}q^{x\partial_{x}}-Q_{1}q^{1/2}x)(1-Q_{1}Q_{2}q%
^{-1}q^{x\partial_{x}}-Q_{1}Q_{2}Q_{3}q^{1/2}x)\Psi(qx)$$
$$\displaystyle=(1-Q_{1}Q_{2}q^{-2}q^{x\partial_{x}}-q^{1/2}x)(1-Q_{1}Q_{2}q^{-1%
}q^{x\partial_{x}}-Q_{1}Q_{3}q^{1/2}x)\Psi(x)$$
(5.17)
and
$$\displaystyle(1-Q_{1}Q_{2}q^{2}q^{-x\partial_{x}}+Q_{1}q^{-1/2}x)(1-Q_{1}Q_{2}%
qq^{-x\partial_{x}}+Q_{1}Q_{2}Q_{3}q^{-1/2}x)\tilde{\Psi}(q^{-1}x)$$
$$\displaystyle=(1-Q_{1}Q_{2}q^{2}q^{-x\partial_{x}}+q^{-1/2}x)(1-Q_{1}Q_{2}qq^{%
-x\partial_{x}}+Q_{1}Q_{3}q^{-1/2}x)\tilde{\Psi}(x).$$
(5.18)
This expression corresponds to writing $q$-difference equations
for $\Phi(x)$ and $\tilde{\Phi}(x)$ as
$$(1-Q_{1}q^{1/2}x)(1-Q_{1}Q_{2}Q_{3}q^{1/2}x)\Phi(qx)=(1-q^{1/2}x)(1-Q_{1}Q_{3}%
q^{1/2}x)\Phi(x)$$
(5.19)
and
$$(1+Q_{1}q^{-1/2}x)(1+Q_{1}Q_{2}Q_{3}q^{-1/2}x)\tilde{\Phi}(q^{-1}x)=(1+q^{-1/2%
}x)(1+Q_{1}Q_{3}q^{-1/2}x)\tilde{\Phi}(x).$$
(5.20)
These $q$-difference equations are transformed
to the foregoing ones for $\Psi(x)$ and $\tilde{\Psi}(x)$
by the transformation (5.6) and (5.15)
of the coefficients.
5.2 Structure of $q$-difference operators
Let us rewrite (5.13) as
$$H(x,q^{x\partial_{x}})\Psi(x)=0$$
(5.21)
and examine the structure of the $q$-difference operator
$H$. This operator reads
$$\displaystyle H(x,q^{x\partial_{x}})$$
$$\displaystyle=(1-Q_{1}Q_{2}q^{-2}q^{x\partial_{x}})(1-Q_{1}Q_{2}q^{-1}q^{x%
\partial_{x}})$$
$$\displaystyle\quad\mbox{}-(1+Q_{1}Q_{3})q^{1/2}x(1-Q_{1}Q_{2}q^{-1}q^{x%
\partial_{x}})+Q_{1}Q_{3}qx^{2}$$
$$\displaystyle\quad\mbox{}-(1-Q_{1}Q_{2}q^{-2}q^{x\partial_{x}})(1-Q_{1}Q_{2}q^%
{-1}q^{x\partial_{x}})q^{x\partial_{x}}$$
$$\displaystyle\quad\mbox{}+Q_{1}(1+Q_{2}Q_{3})q^{1/2}x(1-Q_{1}Q_{2}q^{-1}q^{x%
\partial_{x}})q^{x\partial_{x}}-Q_{1}^{2}Q_{2}Q_{3}qx^{2}q^{x\partial_{x}}.$$
(5.22)
Remarkably, $H(x,q^{x\partial_{x}})$ can be factorized as
$$H(x,q^{x\partial_{x}})=(1-Q_{1}Q_{2}q^{-2}q^{x\partial_{x}})K(x,q^{x\partial_{%
x}}),$$
(5.23)
where
$$\displaystyle K(x,q^{x\partial_{x}})$$
$$\displaystyle=(1-Q_{1}Q_{2}q^{-1}q^{x\partial_{x}})(1-q^{x\partial_{x}})-(1+Q_%
{1}Q_{3})q^{1/2}x$$
$$\displaystyle\quad\mbox{}+Q_{1}(1+Q_{2}Q_{3})q^{1/2}xq^{x\partial_{x}}+Q_{1}Q_%
{3}qx^{2}.$$
(5.24)
This is also the case for the $q$-difference equation
(5.16) for $\tilde{\Psi}(x)$.
The $q$-difference operator $\tilde{H}(x,q^{x\partial_{x}})$
in the expression
$$\tilde{H}(x,q^{x\partial_{x}})\tilde{\Psi}(x)=0$$
(5.25)
of (5.16) reads
$$\displaystyle\tilde{H}(x,q^{x\partial_{x}})$$
$$\displaystyle=(1-Q_{1}Q_{2}q^{2}q^{-x\partial_{x}})(1-Q_{1}Q_{2}qq^{-x\partial%
_{x}})$$
$$\displaystyle\quad\mbox{}+(1+Q_{1}Q_{3})q^{1/2}x(1-Q_{1}Q_{2}qq^{-x\partial_{x%
}})+Q_{1}Q_{3}qx^{2}$$
$$\displaystyle\quad\mbox{}-(1-Q_{1}Q_{2}q^{2}q^{-x\partial_{x}})(1-Q_{1}Q_{2}qq%
^{-x\partial_{x}})q^{-x\partial_{x}}$$
$$\displaystyle\quad\mbox{}-Q_{1}(1+Q_{2}Q_{3})q^{1/2}x(1-Q_{1}Q_{2}qq^{-x%
\partial_{x}})q^{-x\partial_{x}}-Q_{1}^{2}Q_{2}Q_{3}qx^{2}q^{-x\partial_{x}}.$$
(5.26)
This operator can be factorized as
$$\tilde{H}(x,q^{x\partial_{x}})=(1-Q_{1}Q_{2}q^{2}q^{-x\partial_{x}})\tilde{K}(%
x,q^{x\partial_{x}}),$$
(5.27)
where
$$\displaystyle\tilde{K}(x,q^{x\partial_{x}})$$
$$\displaystyle=(1-Q_{1}Q_{2}qq^{-x\partial_{x}})(1-q^{-x\partial_{x}})+(1+Q_{1}%
Q_{3})q^{1/2}x$$
$$\displaystyle\quad\mbox{}-Q_{1}(1+Q_{2}Q_{3})q^{1/2}xq^{-x\partial_{x}}+Q_{1}Q%
_{3}qx^{2}.$$
(5.28)
Let us note here that the action of
$1-Q_{1}Q_{2}q^{-2}q^{x\partial_{x}}$ and $1-Q_{1}Q_{2}q^{2}q^{-x\partial_{x}}$
on the space of power series of $x$ is invertible
as far as $Q_{1}$ and $Q_{2}$ take generic values,
i.e., apart from the exceptional cases
where $Q_{1}Q_{2}=q^{n}$, $n\in\mathbb{Z}$.
Therefore these factors can be removed
from the $q$-difference equations (5.21)
and (5.25). Actually,
this genericity is implicitly assumed in the transformations
(5.6) and (5.15)
of these generating functions. Thus we find
the following refinement of Theorems 2 and 3.
Theorem 4.
For generic values of $Q_{1}$ and $Q_{2}$,
the $q$-difference equations (5.13)
and (5.16) can be reduced to
$$K(x,q^{x\partial_{x}})\Psi(x)=0,\quad\tilde{K}(x,q^{x\partial_{x}})\tilde{\Psi%
}(x)=0.$$
(5.29)
This result fits well into the perspectives
of mirror geometry of topological string theory
on non-compact toric Calabi-Yau threefolds [6, 7].
$\Psi(x)$ and $\tilde{\Psi}(x)$ may be thought of
as wave functions of a probe D-brane.
In this interpretation, a $q$-difference equation
satisfied by these functions defines
a quantum mirror curve.
The $q$-difference equations (5.29)
indeed have such a characteristic.
In the classical limit as $q\to 1$,
the non-commutative polynomials $K(x,q^{x\partial_{x}})$
and $\tilde{K}(x,q^{-x\partial_{x}})$ turn into the ordinary polynomials
$$\displaystyle K_{\mathrm{cl}}(x,y)$$
$$\displaystyle=(1-Q_{1}Q_{2}y)(1-y)-(1+Q_{1}Q_{3})x$$
$$\displaystyle\quad\mbox{}+Q_{1}(1+Q_{2}Q_{3})xy+Q_{1}Q_{3}x^{2}$$
(5.30)
in $(x,y)$ and
$$\displaystyle\tilde{K}_{\mathrm{cl}}(x,y)$$
$$\displaystyle=(1-Q_{1}Q_{2}y^{-1})(1-y^{-1})+(1+Q_{1}Q_{3})x$$
$$\displaystyle\quad\mbox{}-Q_{1}(1+Q_{2}Q_{3})xy^{-1}+Q_{1}Q_{3}x^{2}$$
(5.31)
in $(x,y^{-1})$. As expected from the perspectives
of mirror geometry, the Newton polygons of these polynomials
have the same shape as the toric diagram in Figure 1.
6 Flop transition
Let us examine the flop transition from Figure 1
to Figure 6. After this move, the previous setup
for defining the amplitude $Z^{\mathrm{cv}}_{\beta_{1}\beta_{2}}$
turns into the setup shown in Figure 7.
Note that the Kähler parameters after the flop transition
are denoted by $P_{1},P_{2},P_{3}$; they are expected
to be related to the Kähler parameters $Q_{1},Q_{2},Q_{3}$
before the transition by birational transformations.
Our method for calculating $Z^{\mathrm{cv}}_{\beta_{1}\beta_{2}}$
can be extended to the amplitude $\hat{Z}^{\mathrm{cv}}_{\beta_{1}\beta_{2}}$
of Figure 7 as follows.
The sum over $\alpha_{1},\alpha_{2},\alpha_{3}\in\mathcal{P}$ can be
decomposed to a partial sum $\hat{Z}^{\alpha_{3}}_{\beta_{1}\beta_{2}}$
with respect to $\alpha_{1},\alpha_{2}$ at the first stage
and a sum with respect to $\alpha_{3}$ at the next stage as
$$\hat{Z}^{\mathrm{cv}}_{\beta_{1}\beta_{2}}=\sum_{\alpha_{3}\in\mathcal{P}}\hat%
{Z}_{\beta_{1}\beta_{2}|\alpha_{3}}(-P_{3})^{|\alpha_{3}|}(-1)^{|\alpha_{3}|}q%
^{-\kappa(\alpha_{3})/2}C_{\,{}^{\mathrm{t}}\!\,\alpha_{3}\emptyset\emptyset}.$$
(6.1)
The extra factor $(-1)^{|\alpha_{3}|}q^{-\kappa(\alpha_{3})/2}$ is inserted
by the gluing rule. The framing number (2.3)
along the internal line carrying $\alpha_{3}$ is equal to $1$.
The partial sum $\hat{Z}_{\beta_{1}\beta_{2}|\alpha_{3}}$
is an open string amplitude of the double-$\mathbb{P}^{1}$ diagram
shown in Figure 8. Since this is an on-strip diagram,
the amplitude can be calculated explicitly as
$$\displaystyle\hat{Z}_{\beta_{1}\beta_{2}|\alpha_{3}}$$
$$\displaystyle=s_{\,{}^{\mathrm{t}}\!\,\beta_{1}}(q^{-\rho})s_{\,{}^{\mathrm{t}%
}\!\,\beta_{2}}(q^{-\rho})s_{\,{}^{\mathrm{t}}\!\,\alpha_{3}}(q^{-\rho})\prod_%
{i,j=1}^{\infty}(1-P_{2}q^{-\beta_{1i}-\,{}^{\mathrm{t}}\!\,\beta_{2j}+i+j-1})%
^{-1}$$
$$\displaystyle\quad\mbox{}\times\prod_{i,j=1}^{\infty}(1-P_{1}q^{-\,{}^{\mathrm%
{t}}\!\,\alpha_{3i}-\,{}^{\mathrm{t}}\!\,\beta_{1j}+i+j-1})\prod_{i,j=1}^{%
\infty}(1-P_{1}P_{2}q^{-\,{}^{\mathrm{t}}\!\,\alpha_{3i}-\,{}^{\mathrm{t}}\!\,%
\beta_{2j}+i+j-1}).$$
(6.2)
This amplitude is related to its counterpart $Z_{\beta_{1}\beta_{2}|\alpha_{3}}$
by the same flop operation as the move
from Figure 1 to Figure 6.
One see form (2.6) and (6.2)
that $\hat{Z}_{\beta_{1}\beta_{2}|\alpha_{3}}$ is almost identical
to $Z_{\beta_{1}\beta_{2}|\alpha_{3}}$ if the Kähler parameters
are related as
$$P_{1}P_{2}=Q_{2},\quad P_{2}=Q_{1}Q_{2}.$$
(6.3)
The only discrepancy lies in the infinite products
$\prod_{i,j=1}^{\infty}(1-Q_{1}q^{\cdots})$ in (2.6)
and $\prod_{i,j=1}^{\infty}(1-P_{1}q^{\cdots})$ in (6.2).
Substituting (6.2) and (2.5)
in (6.1), we obtain the following expression
of $\hat{Z}^{\mathrm{cv}}_{\beta_{1}\beta_{2}}$:
$$\displaystyle\hat{Z}^{\mathrm{cv}}_{\beta_{1}\beta_{2}}$$
$$\displaystyle=s_{\,{}^{\mathrm{t}}\!\,\beta_{1}}(q^{-\rho})s_{\,{}^{\mathrm{t}%
}\!\,\beta_{2}}(q^{-\rho})\prod_{i,j=1}^{\infty}(1-P_{2}q^{-\beta_{1i}-\,{}^{%
\mathrm{t}}\!\,\beta_{2j}+i+j-1})^{-1}$$
$$\displaystyle\quad\mbox{}\times\sum_{\alpha_{3}\in\mathcal{P}}s_{\alpha_{3}}(q%
^{-\rho})^{2}P_{3}^{|\alpha_{3}|}\prod_{i,j=1}^{\infty}(1-P_{1}q^{-\,{}^{%
\mathrm{t}}\!\,\alpha_{3i}-\,{}^{\mathrm{t}}\!\,\beta_{1j}+i+j-1})$$
$$\displaystyle\quad\quad\quad\mbox{}\times\prod_{i,j=1}^{\infty}(1-P_{1}P_{2}q^%
{-\,{}^{\mathrm{t}}\!\,\alpha_{3i}-\,{}^{\mathrm{t}}\!\,\beta_{2j}+i+j-1}).$$
(6.4)
Note that we have used the identity (4.4) as well
to rewrite the first part of the summand as
$$s_{\,{}^{\mathrm{t}}\!\,\alpha_{3}}(q^{-\rho})s_{\alpha_{3}}(q^{-\rho})q^{-%
\kappa(\alpha_{3})/2}=s_{\alpha_{3}}(q^{-\rho})^{2}.$$
Thus, in contrast with (2.7),
the sum in this case resembles the partition function
of the ordinary melting crystal model [9, 10]
for which the main part of the Boltzmann weight
is $s_{\alpha_{3}}(q^{-\rho})^{2}$
rather than $s_{\,{}^{\mathrm{t}}\!\,\alpha_{3}}(q^{-\rho})s_{\alpha_{3}}(q^{-\rho})$.
The sum in (6.4) can be calculated
in more or less the same way as the case of (2.7).
Let us show the final result only.
Theorem 5.
The open string amplitude $\hat{Z}^{\mathrm{cv}}_{\beta_{1}\beta_{2}}$
can be expressed as
$$\displaystyle\hat{Z}^{\mathrm{cv}}_{\beta_{1}\beta_{2}}$$
$$\displaystyle=q^{\kappa(\beta_{1})/2+\kappa(\beta_{2})/2}\prod_{i,j=1}^{\infty%
}(1-P_{2}q^{-\beta_{1i}-\,{}^{\mathrm{t}}\!\,\beta_{2j}+i+j-1})^{-1}$$
$$\displaystyle\quad\mbox{}\times\langle\,{}^{\mathrm{t}}\!\,\beta_{1}|\Gamma^{%
\prime}_{-}(q^{-\rho})\Gamma^{\prime}_{+}(q^{-\rho})(-P_{1})^{L_{0}}\Gamma_{-}%
(q^{-\rho})\Gamma_{+}(q^{-\rho})P_{3}^{L_{0}}$$
$$\displaystyle\quad\mbox{}\times\Gamma_{-}(q^{-\rho})\Gamma_{+}(q^{-\rho})(-P_{%
1}P_{2})^{L_{0}}\Gamma^{\prime}_{-}(q^{-\rho})\Gamma^{\prime}_{+}(q^{-\rho})|%
\,{}^{\mathrm{t}}\!\,\beta_{2}\rangle.$$
(6.5)
The main part $\langle\,{}^{\mathrm{t}}\!\,\beta_{1}|\cdots|\,{}^{\mathrm{t}}\!\,\beta_{2}\rangle$
of this expression is essentially the open string amplitude
of the web diagram shown in Figure 9.
This web diagram can be derived the web diagram
of Figure 5 by the same flop operation
as the move from Figure 1 to Figure 6.
To see how this part is related to the main part
of (4.10), let us use the commutation relations
(5.10) to exchange the order of
the first four vertex operators therein as
$$\displaystyle\langle\,{}^{\mathrm{t}}\!\,\beta_{1}|\Gamma^{\prime}_{-}(q^{-%
\rho})\Gamma^{\prime}_{+}(q^{-\rho})(-P_{1})^{L_{0}}\Gamma_{-}(q^{-\rho})%
\Gamma_{+}(q^{-\rho})P_{3}^{L_{0}}$$
$$\displaystyle=\langle\,{}^{\mathrm{t}}\!\,\beta_{1}|(-P_{1})^{L_{0}}\Gamma^{%
\prime}_{-}(-P_{1}^{-1}q^{-\rho})\Gamma^{\prime}_{+}(-P_{1}q^{-\rho})\Gamma_{-%
}(q^{-\rho})\Gamma_{+}(q^{-\rho})P_{3}^{L_{0}}$$
$$\displaystyle=(-P_{1})^{|\beta_{1}|}\prod_{i,j=1}^{\infty}(1-P_{1}q^{i+j-1})(1%
-P_{1}^{-1}q^{i+j-1})^{-1}$$
$$\displaystyle\quad\mbox{}\times\langle\,{}^{\mathrm{t}}\!\,\beta_{1}|\Gamma_{-%
}(q^{-\rho})\Gamma_{+}(q^{-\rho})\Gamma^{\prime}_{-}(-P_{1}^{-1}q^{-\rho})%
\Gamma^{\prime}_{+}(-P_{1}q^{-\rho})P_{3}^{L_{0}}$$
$$\displaystyle=(-P_{1})^{|\beta_{1}|}\prod_{i,j=1}^{\infty}(1-P_{1}q^{i+j-1})(1%
-P_{1}^{-1}q^{i+j-1})^{-1}$$
$$\displaystyle\quad\mbox{}\times\langle\,{}^{\mathrm{t}}\!\,\beta_{1}|\Gamma_{-%
}(q^{-\rho})\Gamma_{+}(q^{-\rho})(-P_{1}^{-1})^{L_{0}}\Gamma^{\prime}_{-}(q^{-%
\rho})\Gamma^{\prime}_{+}(q^{-\rho})(-P_{1}P_{3})^{L_{0}}.$$
This shows that if the two sets of Kähler parameters
are matched as
$$P_{1}^{-1}=Q_{1},\quad P_{1}P_{3}=Q_{3},\quad P_{1}P_{2}=Q_{2},$$
(6.6)
$\hat{Z}^{\mathrm{cv}}_{\beta_{1}\beta_{2}}$ and $Z^{\mathrm{cv}}_{\beta_{1}\beta_{2}}$
are related as
$$\hat{Z}^{\mathrm{cv}}_{\beta_{1}\beta_{2}}=q^{\kappa(\beta_{1})/2}(-P_{1})^{|%
\beta_{1}|}\prod_{i,j=1}^{\infty}(1-P_{1}q^{i+j-1})(1-P_{1}^{-1}q^{i+j-1})^{-1%
}\cdot Z^{\mathrm{cv}}_{\beta_{1}\beta_{2}}.$$
(6.7)
Note that (6.6) is consistent with (6.3).
These matching rules of parameters agree with the known result
for the partition functions [3, 5].
7 Conclusion
Let us summarize what we have done in this paper.
Calculation of open string amplitudes
We reformulated the open string amplitude $Z^{\mathrm{cv}}_{\beta_{1}\beta_{2}}$
of Figure 2 in the partially summed form (2.4),
and derived the reduced expression (2.7).
The main part of (2.7) turns out
to be similar to the partition function
of the modified melting crystal model. Firstly,
the main part $s_{\,{}^{\mathrm{t}}\!\,\alpha_{3}}(q^{-\rho})s_{\alpha_{3}}(q^{-\rho})$
of the summand is exactly the same. Secondly,
the other part can be described by matrix elements
of the diagonal operators $V^{(\pm)}_{0}$
in the quantum torus algebra.
This is also a characteristic of the external potentials
in the melting crystal models. We could thereby apply
the method for the melting crystal models to derive
the fermionic expression (4.1)
of $Z^{\mathrm{cv}}_{\beta_{1}\beta_{2}}$. This expression was
further converted to the final expression
(4.10) of $Z^{\mathrm{cv}}_{\beta_{1}\beta_{2}}$,
which is a product of a simple prefactor and
the open string amplitude $Y_{\beta_{1}\beta_{2}}$
of a new on-strip diagram.
Derivation of $q$-difference equations
We derived $q$-difference equations for
the generating functions $\Psi(x),\,\tilde{\Psi}(x)$
of the normalized amplitudes
$Z^{\mathrm{cv}}_{\beta_{1}\beta_{2}}/Z^{\mathrm{cv}}_{\emptyset\emptyset}$
specialized to $\beta_{1}=(1^{k}),\,(k)$, $k=0,1,2,\ldots$,
and $\beta_{2}=\emptyset$. The derivation makes
full use of the factorized form of (4.10).
Namely, we first derived the $q$-difference equations
(5.8) and (5.14)
for the generating functions $\Phi(x),\,\tilde{\Phi}(x)$
obtained from $Y_{\beta_{1}\beta_{2}}/Y_{\emptyset\emptyset}$.
These equations are transformed
to the $q$-difference equations (5.13)
and (5.16) for $\Psi(x),\,\tilde{\Psi}(x)$.
This is the place where the prefactor of $Y_{\beta_{1}\beta_{2}}$
in (4.10) plays a role.
We examined the structure of these $q$-difference equations
and found that they can be reduced to the simpler equations
(5.29). It is these reduced equations
that should be interpreted as the defining equation
of a quantum mirror curve.
Flop transition
We considered the flop transition from Figure 1
to Figure 6. The open string amplitude
$\hat{Z}^{\mathrm{cv}}_{\beta_{1}\beta_{2}}$ after the transition
can be calculated in much the same way
as in the case of $Z^{\mathrm{cv}}_{\beta_{1}\beta_{2}}$.
We confirmed that $\hat{Z}^{\mathrm{cv}}_{\beta_{1}\beta_{2}}$
can be matched to the amplitude $Z^{\mathrm{cv}}_{\beta_{1}\beta_{2}}$
by the birational transformations (6.6)
of the Kähler parameters.
On the other hand, we have been unable to derive
$q$-difference equations in other configurations
of partitions on the external lines of the web diagram.
A main reason lies in the emergence of operators $q^{\pm K/2}$
that do not cancel out as shown in (4.9).
Since such factors remain in the operator product
of a fermionic expression of the amplitude,
the method of Section 5 does not work.
We have encountered the same difficulty in an attempt
to extend the result of this paper to more general
tree-like diagrams studied by Karp, Liu and Mariño [4].
We believe that this difficulty is of quite technical nature
and can be overcome. A promising idea will be to use
various identities satisfied by dilogarithmic functions of
non-commutative variables [19, 20].
Acknowledgements
The authors are grateful to Motohico Mulase
for valuable comments.
This work is partly supported by JSPS Kakenhi Grant
No. 24540223, No. 25400111 and No. 15K04912.
Appendix A Amplitudes of on-strip geometry
The toric diagram of on-strip geometry is a triangulation
of the strip of height $1$ to triangles of area $1/2$
(see Figure 10).
The associated web diagram is a connected acyclic graph.
If the toric graph comprises $N$ triangles,
the web diagram has $N$ vertices, $N-1$ internal lines
and $N+2$ external lines. The $N$ external lines
other than the leftmost and rightmost ones are vertical.
For brevity, the external lines are also referred to
as “legs”.
We assign the Kähler parameters $Q_{1},\ldots,Q_{N-1}$
to the internal lines, the partitions $\beta_{1},\ldots,\beta_{N}$
to the vertical external lines, and the partitions $\alpha_{0},\alpha_{N}$
to the leftmost and rightmost external lines.
Let $Z^{\alpha_{0}\alpha_{N}}_{\beta_{1}\cdots\beta_{N}}$ denote
the open string amplitude in this setup.
This amplitude is defined as a sum of the product
of vertex and edge weights with respect to
the partitions $\alpha_{1},\ldots,\alpha_{N-1}$ on the internal lines.
In the case of $\alpha_{0}=\alpha_{N}=\emptyset$,
Iqbal and Kashani-Poor [3] calculated this sum
in a closed form by skillful use of the Cauchy identities
for skew Schur functions. Their result can be reformulated,
without restriction to $\alpha_{0}=\alpha_{N}=\emptyset$,
in the language of fermions [31, 32, 33].
Following the notations of Nagao [32]
and Sułkowski [33], let us define
the sign (or type) $\sigma_{n}=\pm 1$ of the $n$-th vertex as:
(i)
$\sigma_{n}=+1$ if the vertical leg points up,
(ii)
$\sigma_{n}=-1$ if the vertical leg points down.
For example, in the case of the web diagram of Figure 10,
$$\sigma_{1}=-1,\quad\sigma_{2}=+1,\quad\sigma_{3}=+1,\quad\sigma_{4}=-1,\quad%
\sigma_{5}=-1.$$
These data are used to show the types of vertex operators as
$$\Gamma^{\sigma}_{\pm}(\boldsymbol{x})=\begin{cases}\Gamma_{\pm}(\boldsymbol{x}%
)&\text{if $\sigma=+1$},\\
\Gamma^{\prime}_{\pm}(\boldsymbol{x})&\text{if $\sigma=-1$}.\end{cases}$$
Let us further introduce the auxiliary notations
$$\beta^{(n)}=\begin{cases}\beta_{n}&\text{if $\sigma_{n}=+1$},\\
\,{}^{\mathrm{t}}\!\,\beta_{n}&\text{if $\sigma_{n}=-1$},\end{cases}\qquad Q_{%
mn}=Q_{m}Q_{m+1}\cdots Q_{n-1}.$$
With these notations, the fermionic expression
of $Z^{\alpha_{0}\alpha_{N}}_{\beta_{1}\cdots\beta_{N}}$ read
$$\displaystyle Z^{\alpha_{0}\alpha_{N}}_{\beta_{1}\cdots\beta_{N}}$$
$$\displaystyle=q^{(1-\sigma_{1})\kappa(\alpha_{0})/4}q^{(1+\sigma_{N})\kappa(%
\alpha_{N})/4}s_{\,{}^{\mathrm{t}}\!\,\beta_{1}}(q^{-\rho})\cdots s_{\,{}^{%
\mathrm{t}}\!\,\beta_{N}}(q^{-\rho})$$
$$\displaystyle\quad\mbox{}\times\langle\,{}^{\mathrm{t}}\!\,\alpha_{0}|\Gamma^{%
\sigma_{1}}_{-}(q^{-\beta^{(1)}-\rho})\Gamma^{\sigma_{1}}_{+}(q^{-\,{}^{%
\mathrm{t}}\!\,\beta^{(1)}-\rho})(\sigma_{1}Q_{1}\sigma_{2})^{L_{0}}\cdots$$
$$\displaystyle\quad\mbox{}\times\Gamma^{\sigma_{N-1}}_{-}(q^{-\beta^{(N-1)}-%
\rho})\Gamma^{\sigma_{N-1}}_{+}(q^{-\,{}^{\mathrm{t}}\!\,\beta^{(N-1)}-\rho})(%
\sigma_{N-1}Q_{N-1}\sigma_{N})^{L_{0}}$$
$$\displaystyle\quad\mbox{}\times\Gamma^{\sigma_{N}}_{-}(q^{-\beta^{(N)}-\rho})%
\Gamma^{\sigma_{N}}_{+}(q^{-\,{}^{\mathrm{t}}\!\,\beta^{(N)}-\rho})|\alpha_{N}\rangle.$$
(A.1)
In the case where $N=1$, this formula reduces
to the fermionic expression (4.6)
of the vertex weight itself. Starting from (4.6),
one can prove this formula by induction.
If $\alpha_{0}=\alpha_{N}=\emptyset$,
one can use the commutation relations (5.10)
to move $\Gamma^{\sigma}_{-}$’s to the left and
$\Gamma^{\sigma}_{+}$’s to the right until
they hit $\langle 0|$ and $|0\rangle$ and disappear.
This yields the explicit formula
$$\displaystyle Z^{\emptyset\emptyset}_{\beta_{1}\cdots\beta_{N}}$$
$$\displaystyle=s_{\,{}^{\mathrm{t}}\!\,\beta_{1}}(q^{-\rho})\cdots s_{\,{}^{%
\mathrm{t}}\!\,\beta_{N}}(q^{-\rho})$$
$$\displaystyle\mbox{}\times\prod_{1\leq<m<\leq<n}\prod_{i,j=1}^{\infty}(1-Q_{mn%
}q^{-\,{}^{\mathrm{t}}\!\,\beta^{(m)}_{i}-\beta^{(n)}_{j}+i+j-1})^{-\sigma_{m}%
\sigma_{n}}$$
(A.2)
of Iqbal and Kashani-Poor [3].
Appendix B Direct proof of two-leg cyclic symmetry
As another application of the techniques used
in Section 3, we present a direct proof of the identities
(4.7) and (4.8) that amounts
to the cyclic symmetry of two-leg vertices.
Actually, these two identities are equivalent,
and can be reduced to the following one:
$$s_{\lambda}(q^{-\rho})s_{\mu}(q^{-\lambda-\rho})=\langle\mu|q^{-K/2}\Gamma^{%
\prime}_{-}(q^{-\rho})\Gamma^{\prime}_{+}(q^{-\rho})q^{-K/2}|\lambda\rangle.$$
(B.1)
It is this identity that we prove here.
Note that this identity implies the non-trivial relation
$$s_{\lambda}(q^{-\rho})s_{\mu}(q^{-\lambda-\rho})=s_{\mu}(q^{-\rho})s_{\lambda}%
(q^{-\mu-\rho}),$$
(B.2)
from which the equivalence of (4.7)
and (4.8) follows.
We prove (B.1) by generating functions.
Namely, we construct generating functions of both sides
by the Schur functions $s_{\mu}(\boldsymbol{x})$, $\boldsymbol{x}=(x_{1},x_{2},\ldots)$,
and confirm that these generating functions are identical.
It is easy to calculate the generating function
of the left side of (B.1).
By the Cauchy identity
$$\sum_{\mu\in\mathcal{P}}s_{\mu}(\boldsymbol{x})s_{\mu}(\boldsymbol{y})=\prod_{%
i,j=1}^{\infty}(1-x_{i}y_{j})^{-1},\quad\boldsymbol{y}=(y_{1},y_{2},\ldots),$$
(B.3)
of the Schur functions [28],
the generating function of the left side
of (B.1) can be expressed as
$$\sum_{\mu\in\mathcal{P}}s_{\mu}(\boldsymbol{x})s_{\lambda}(q^{-\rho})s_{\mu}(q%
^{-\lambda-\rho})=s_{\lambda}(q^{-\rho})\prod_{i,j=1}^{\infty}(1-x_{i}q^{-%
\lambda_{j}+j-1/2})^{-1}.$$
(B.4)
On the other hand, constructing the generating function
of the left side of (B.1) amounts to
inserting $\Gamma_{+}(\boldsymbol{x})$ to the right of $\langle 0|$ as
$$\displaystyle\sum_{\mu\in\mathcal{P}}s_{\mu}(\boldsymbol{x})\langle\mu|q^{-K/2%
}\Gamma^{\prime}_{-}(q^{-\rho})\Gamma^{\prime}_{+}(q^{-\rho})q^{-K/2}|\lambda\rangle$$
$$\displaystyle=\langle 0|\Gamma_{+}(\boldsymbol{x})q^{-K/2}\Gamma^{\prime}_{-}(%
q^{-\rho})\Gamma^{\prime}_{+}(q^{-\rho})q^{-K/2}|\lambda\rangle$$
$$\displaystyle=\langle 0|\exp\left(\sum_{i,k=1}^{\infty}\frac{x_{i}^{k}}{k}J_{k%
}\right)q^{-K/2}\Gamma^{\prime}_{-}(q^{-\rho})\Gamma^{\prime}_{+}(q^{-\rho})q^%
{-K/2}|\lambda\rangle.$$
The subsequent calculations are very similar to Section 3.
One can use (3.6) and (3.8) to rewrite
the last quantity as
$$\displaystyle\langle 0|\exp\left(\sum_{i,k=1}^{\infty}\frac{x_{i}^{k}}{k}J_{k}%
\right)q^{-K/2}\Gamma^{\prime}_{-}(q^{-\rho})\Gamma^{\prime}_{+}(q^{-\rho})q^{%
-K/2}|\lambda\rangle$$
$$\displaystyle=\langle 0|q^{-K/2}\Gamma^{\prime}_{-}(q^{-\rho})\Gamma^{\prime}_%
{+}(q^{-\rho})\exp\left(\sum_{i,k=1}^{\infty}\frac{x_{i}^{k}q^{k/2}}{k}\left(V%
^{(-k)}_{0}+\frac{1}{1-q^{k}}\right)\right)q^{-K/2}|\lambda\rangle$$
$$\displaystyle=\langle 0|\Gamma^{\prime}_{+}(q^{-\rho})q^{-K/2}\exp\left(\sum_{%
i,k=1}^{\infty}\frac{x_{i}^{k}q^{k/2}}{k}\left(V^{(-k)}_{0}+\frac{1}{1-q^{k}}%
\right)\right)|\lambda\rangle.$$
Note that the order of $\exp(\cdots)$ and $q^{K/2}$
has been exchanged because $V^{(-k)}_{0}$ commutes with $q^{K/2}$.
By (3.1), the action of $\exp(\cdots)$
on $|\lambda\rangle$ can be expressed as
$$\displaystyle\exp\left(\sum_{i,k=1}^{\infty}\frac{x_{i}^{k}q^{k/2}}{k}\left(V^%
{(-k)}_{0}+\frac{1}{1-q^{k}}\right)\right)|\lambda\rangle$$
$$\displaystyle=\exp\left(\sum_{i,k=1}^{\infty}\frac{x_{i}^{k}q^{k/2}}{k}\sum_{j%
=1}^{\infty}q^{-k(\lambda_{j}-j+1)}\right)|\lambda\rangle$$
$$\displaystyle=\prod_{i,j=1}^{\infty}\exp\left(\sum_{k=1}^{\infty}\frac{(x_{i}q%
^{-\lambda_{j}+j-1/2})^{k}}{k}\right)|\lambda\rangle$$
$$\displaystyle=\prod_{i,j=1}^{\infty}(1-x_{i}q^{-\lambda_{j}+j-1/2})^{-1}|%
\lambda\rangle.$$
Thus the generating function of the right side of (B.1)
turns out to take such a form as
$$\displaystyle\sum_{\mu\in\mathcal{P}}s_{\mu}(\boldsymbol{x})\langle\mu|q^{-K/2%
}\Gamma^{\prime}_{-}(q^{-\rho})\Gamma^{\prime}_{+}(q^{-\rho})q^{-K/2}|\lambda\rangle$$
$$\displaystyle=\langle 0|\Gamma^{\prime}_{+}(q^{-\rho})q^{-K/2}|\lambda\rangle%
\prod_{i,j=1}^{\infty}(1-x_{i}q^{-\lambda_{j}+j-1/2})^{-1}$$
$$\displaystyle=q^{-\kappa(\lambda)/2}s_{\,{}^{\mathrm{t}}\!\,\lambda}(q^{-\rho}%
)\prod_{i,j=1}^{\infty}(1-x_{i}q^{-\lambda_{j}+j-1/2})^{-1}.$$
By (4.4), this coincides with (B.4).
References
[1]
J. Bryan and D. Karp,
The closed topological vertex via the Cremona transform,
J. Algebraic Geom. 14 (2005), 529–542.
arXiv:math/0311208 [math.AG].
[2]
M. Aganagic, A. Klemm, M. Mariño and C. Vafa,
The topological vertex,
Commun. Math. Phys. 254 (2005), 425–478.
arXiv:hep-th/0305132.
[3]
A. Iqbal and A.-K. Kashani-Poor,
The vertex on a strip,
Adv. Theor. Math. Phys. 10 (2006), 317–343.
arXiv:hep-th/0410174.
[4]
D. Karp, C. M. Liu and M. Mariño,
The local Gromov-Witten invariants of configurations
of rational curves,
Geometry and Topology 10 (2006), 115–168.
arXiv:math/0506488 [math.AG].
[5]
P. Sułkowski,
Crystal model for the closed topological vertex geometry,
JHEP 0612 (2006), 030.
arXiv:hep-th/0606055.
[6]
M. Aganagic, R. Dijkgraaf, A. Klemm, M. Mariño and C. Vafa,
Topological strings and integrable hierarchies,
Comm. Math. Phys. 261 (2006), 451–516.
arXiv:hep-th/0312085.
[7]
R. Dijkgraaf and C. Vafa,
Two dimensional Kodaira-Spencer theory and
three dimensional Chern-Simons gravity,
arXiv:0711.1932 [hep-th].
[8]
S. Gukov and P. Sułkowski,
A-polynomial, B-model, and quantization,
JHEP 1202 (2012), 070.
arXiv:1108.0002 [hep-th].
[9]
T. Nakatsu and K. Takasaki,
Melting crystal, quantum torus and Toda hierarchy,
Commun. Math. Phys. 285 (2009), 445–468.
arXiv:0710.5339 [hep-th].
[10]
T. Nakatsu and K. Takasaki,
Integrable structure of melting crystal model with external potentials,
M.-H. Saito, S. Hosono and K. Yoshioka (eds.),
New Developments in Algebraic Geometry,
Integrable Systems and Mirror Symmetry,
Adv. Stud. Pure Math. vol. 59,
Mathematical Society of Japan, Tokyo, 2010, pp. 201–223.
arXiv:0807.4970 [math-ph].
[11]
K. Takasaki,
Modified melting crystal model and Ablowitz-Ladik hierarchy,
J. Phys. A: Math. Theor. 46 (2013), 245202.
arXiv:1302.6129 [math-ph].
[12]
K. Takasaki,
Orbifold melting crystal models and reductions of Toda hierarchy,
J. Phys. A: Math. Theor. 48 (2015), 215201.
arXiv:1410.5060 [math-ph].
[13]
A. Okounkov and R. Pandharipande,
Gromov-Witten theory, Hurwitz theory, and completed cycles,
Annals of Math. 163 (2006), 517–560.
arXiv:math.AG/0204305.
[14]
J. Harnad and A. Y. Orlov,
Convolution symmetries of integrable hierarchies, matrix models
and $\tau$ functions,
P. Deift and P. Forrester (eds.),
Random Matrix Theory, Interacting Particle Systems,
and Integrable Systems, MSRI Publications, vol. 65,
Cambridge University Press, 2014, pp. 247–275.
arXiv:0901.0323 [math-ph].
[15]
A. K. Kashani-Poor,
The wave function behavior of the open topological string
partition function on the conifold,
JHEP 04 (2007), 004.
arXiv:hep-th/0606112
[16]
S. Hyun and S.-H. Yi,
Non-compact topological branes on conifold,
JHEP 0611 (2006), 075.
arXiv:hep-th/0609037.
[17]
J. Zhou,
Quantum mirror curves for $\mathbb{C}^{3}$
and the resolved conifold,
arXiv:1207.0598 [math.AG].
[18]
K. Takasaki,
Remarks on partition functions of topological string theory
on generalized conifolds,
arXiv:1301.4548 [math-ph].
[19]
L. Faddeev and A. Yu. Volkov,
Abelian current algebra and the Virasoro algebra on the lattice,
Phys. Lett. B315 (1993), 311–318,
arXiv:hep-th/9307048.
[20]
L. D. Faddeev and R. M. Kashaev,
Quantum dilogarithm,
Mod. Phys. Lett. A9 (1994), 427–434,
arXiv:hep-th/9310070.
[21]
A. Iqbal, C. Kozçaz and C. Vafa,
The refined topological vertex,
JHEP 0910 (2009), 069.
arXiv:hep-th/0701156.
[22]
M. Taki,
Refined topological vertex and instanton counting,
JHEP 803 (2008), 048.
arXiv:0710.1776 [hep-th].
[23]
A. Okounkov, N. Reshetikhin and C. Vafa,
Quantum Calabi-Yau and classical crystals,
P. Etingof, V. Retakh and I.M. Singer (eds.),
The unity of mathematics,
Progr. Math. vol. 244, Birkhäuser, 2006, pp. 597–618.
arXiv:hep-th/0309208.
[24]
I. P. Goulden and D. M. Jackson,
Transitive factorisations into transpositions
and holomorphic mappings on the sphere,
Proc. Amer. Math. Soc. 125 (1997), 51–60.
[25]
M. E. Kazarian and S. K. Lando,
An algebro-geometric proof of Witten’s conjecture,
J. Amer. Math. Soc. 20 (2007), 1079–1089.
arXiv:math/0601760.
[26]
A. Okounkov and N. Reshetikhin,
Correlation function of Schur process with application
to local geometry of a random 3-dimensional young diagram,
J. Amer. Math. Soc. 16, (2003), 581–603.
arXiv:math/0107056.
[27]
J. Bryan and B. Young,
Generating functions for coloured 3D Young diagrams
and the Donaldson-Thomas invariants of orbifolds,
Duke Math. J. 152 (2010), 115–153.
arXiv:0802.3948 [math.CO].
[28]
I. G. Macdonald,
Symmetric functions and Hall polynomials,
Oxford University Press, 1995.
[29]
T. Miwa, M. Jimbo and E. Date,
Solitons: Differential equations, symmetries,
and infinite-dimensional algebras,
Cambridge University Press, 2000.
[30]
J. Zhou,
A conjecture on Hodge integrals,
arXiv:math.AG/0310282.
[31]
T. Eguchi and H. Kanno,
Geometric transitions, Chern-Simons gauge theory
and Veneziano type amplitudes,
Phys. Lett. B585 (2004), 163–172.
arXiv:hep-th/0312223.
[32]
K. Nagao,
Non-commutative Donaldson-Thomas theory
and vertex operators,
Geometry and Topology 15 (2011) 1509–1543.
arXiv:0910.5477 [math.AG].
[33]
P. Sułkowski,
Wall-crossing, free fermions and crystal melting,
Comm. Math. Phys. 301 (2011), 517–562.
arXiv:0910.5485 [hep-th]. |
The functor of toric varieties associated with Weyl chambers
and Losev-Manin moduli spaces
Victor Batyrev
Mathematisches Institut, Universität Tübingen,
Auf der Morgenstelle 10, 72076 Tübingen, Germany
batyrev@everest.mathematik.uni-tuebingen.de
and
Mark Blume
Mathematisches Institut, Universität Münster,
Einsteinstrasse 62, 48149 Münster, Germany
mark.blume@uni-muenster.de
Abstract.
A root system $R$ of rank $n$ defines an $n$-dimensional
smooth projective toric variety $X(R)$ associated with
its fan of Weyl chambers. We give a simple description of
the functor of $X(R)$ in terms of the root system $R$
and apply this result in the case of root systems of type
$A$ to give a new proof of the fact that the toric variety
$X(A_{n})$ is the fine moduli space $\overline{L}_{n+1}$ of
stable $(n+1)$-pointed chains of projective lines
investigated by Losev and Manin.
The second author was supported by DFG-Schwerpunkt 1388 Darstellungstheorie.
Introduction
Let $R\subset E$ be a root system of rank $n$ in an $n$-dimensional
Euclidean space $E$ and let $M(R)\subset E$ be its root lattice.
The toric variety $X(R)$ corresponding to the root system $R$ is the
smooth projective toric variety associated with the fan of Weyl
chambers $\Sigma(R)$ in the dual space $E^{*}$ with respect to the
lattice $N(R)\subset E^{*}$ dual to the root lattice $M(R)$.
It was shown by Klyachko in [Kl85] (see also [Kl95]) that if $G$
is a semisimple algebraic group corresponding to $R$ and $B$ is
a Borel subgroup in $G$ then the toric variety $X(R)$ can be
characterised as the closure of a general orbit of a maximal torus
$T\subset G$ acting on the flag variety $G/B$.
The natural representation of the Weyl group $W(R)$ on the cohomology
of $X(R)$ has been studied by Procesi [Pr90],
Dolgachev-Lunts [DL94], and Stembridge [St94].
The present paper is inspired by a paper of Losev and Manin [LM00],
in which fine moduli spaces $\overline{L}_{n}$ of stable $n$-pointed
chains of projective lines were constructed and it was observed that
the Losev-Manin moduli space $\overline{L}_{n}$ is the toric variety
associated with the polytope called the $(n-1)$-dimensional
permutohedron studied by Kapranov [Ka93, (4.3)].
These toric varieties form the $A_{n}$-family of the toric varieties
associated with root systems: the Losev-Manin moduli space
$\overline{L}_{n+1}$ coincides with the toric variety $X(A_{n})$.
Moreover, in [LM00] is was shown that the homology groups of
$\overline{L}_{n+1}$ $(n\geq 0)$ together with the natural action of
the Weyl group $W(A_{n})\cong S_{n+1}$ are closely related to the so called
commutativity equations (see also the recent paper of Shadrin and
Zvonkine [SZ09]).
We remark that these varieties are special examples of
toric varieties obtained as equivariant blowups of $\mathds{P}^{n}$
considered recently by Bloch and Kreimer [BK08, §3].
The Losev-Manin moduli space $\overline{L}_{n}$ is an equivariant
compactification of a maximal torus
$T\cong(\mathds{C}^{*})^{n}/\mathds{C}^{*}\subset\operatorname{PGL}(n,\mathds{C})$, where the torus $T$ can
be identified as the moduli space of $n$ points in
$\mathds{P}^{1}\setminus\{0,\infty\}$ up to automorphisms of $\mathds{P}^{1}$ fixing
$0$ and $\infty$. The boundary components of $\overline{L}_{n}$
parametrise certain types of $n$-pointed reducible rational curves.
There are some similarities and relations between the Losev-Manin
moduli spaces and the well-known Grothendieck-Knudsen moduli spaces.
The Losev-Manin moduli spaces $\overline{L}_{n}$ parametrise
isomorphism classes of chains of projective lines with two
poles and $n$ marked points that may coincide, whereas the
Grothendieck-Knudsen moduli spaces $\overline{M}_{0,n+2}$
parametrise isomorphism classes of trees of projective lines
with $n+2$ marked points that may not coincide.
They are related by surjective birational morphisms
$\overline{M}_{0,n+2}\to\overline{L}_{n}$ dependent on the choice
of two different elements $i,j\in\{1,\ldots,n+2\}$.
Both form a particular case of moduli spaces of weighted
pointed stable curves as introduced by Hassett [Ha03].
Our main objective was to generalise the result of Losev and Manin
to other root systems $R$. This problem was mentioned by Losev and
Manin in the introduction of their paper [LM00].
In [BB11] we present results in this direction for $R$
a classical root system.
To investigate interpretations of the toric varieties $X(R)$
associated with root systems as moduli spaces, it is natural first
to investigate their functors of points. The functor of toric varieties
in general was described by Mumford in [AMRT, Ch. I]; a different
description was proposed by Cox [Co95] for smooth toric varieties.
In the present paper we propose another description of the functor
of the toric varieties $X(R)$ for root systems $R$, which is based on
projection maps $X(R)\to\mathds{P}^{1}$ and done with a view toward interpretations
of these varieties as moduli spaces of pointed trees of projective lines.
Outline of the paper.
In the first section of this paper, we derive some general results
about the toric varieties $X(R)$ associated with arbitrary root
systems $R$. Important are functorial properties with respect to
maps of root systems. For example, any pair of opposite roots
$\{\pm\alpha\}\subset R$, i.e. a root subsystem of type $A_{1}$,
gives rise to a projection $X(R)\to X(A_{1})\cong\mathds{P}^{1}$. Morphisms
constructed this way appear in many variants in the following.
As a main result, we give a description of the functor of the
toric varieties $X(R)$ in terms of the root system $R$. We use
the property of the spaces $X(R)$ that morphisms $Y\to X(R)$ are
uniquely determined by their compositions with all the
projection maps $X(R)\to\mathds{P}^{1}$ given by the root subsystems
$\{\pm\alpha\}\subset R$ of type $A_{1}$. Further, the relations
between these morphisms are given by the root subsystems of type
$A_{2}$ in $R$.
For the rest of the present paper, we are concerned with the toric
varieties $X(A_{n})$ associated with root systems of type $A$ and
their interpretation as Losev-Manin moduli spaces $\overline{L}_{n+1}$.
We consider the toric varieties $X(A_{n})$ in Section 2.
We review some results concerning the (co)homology of $X(A_{n})$,
we give a basis for the homology and, in a simple way, derive the
relations between torus invariant cycles used in [LM00, Section 3].
Further, we comment on primitive collections and relations of the
toric variety $X(A_{n})$ and apply this to show that the
anticanonical class of $X(A_{n})$ is a semiample divisor.
This implies that $X(A_{n})$ is an almost Fano variety.
The anticanonical divisor defines a birational toric morphism
to the Gorenstein toric Fano variety $\mathds{P}_{\Delta(A_{n})}$
corresponding to the reflexive polytope
$\Delta(A_{n})=(\textrm{convex hull of all roots of $A_{n}$})$.
In Section 3, we give a new proof of the fact that
the toric varieties $X(A_{n})$ are the fine moduli spaces
$\overline{L}_{n+1}$ of $(n+1)$-pointed chains of projective
lines introduced by Losev and Manin.
We use the functorial properties of toric varieties associated
with root systems developed in Subsection 1.2
to construct the universal curve $X(A_{n+1})\to X(A_{n})$ in
Subsection 3.2.
Our result about the functor of $X(R)$ (Subsection 1.3)
is used in the case of root systems of type $A$ in Subsection
3.3 to show that the functor of $X(A_{n})$ is
isomorphic to the moduli functor of $(n+1)$-pointed chains of
projective lines.
This provides an alternative proof of the fact that this
moduli problem admits a fine moduli space $\overline{L}_{n+1}$
and furthermore shows that it coincides with the toric variety
$X(A_{n})$.
We will see that the data describing morphisms $Y\to X(A_{n})$
correspond in a natural way to parameters in equations describing
stable $(n+1)$-pointed chains of projective lines over $Y$ embedded
in $(\mathds{P}^{1}_{Y})^{n+1}$.
1. Toric varieties associated with root systems and their functor
1.1. The toric variety $X(R)$
Let $R$ be a (reduced and crystallographic) root system in a
Euclidean space $E$. With $R$ we associate a toric variety $X(R)$
([Pr90], [DL94]).
Let $M(R)$ be the root lattice of $R$, i.e. the lattice in
$E$ generated by the roots of $R$, and let $N(R)$ be the lattice
dual to $M(R)$.
For any set of simple roots $S$, we have a cone $\sigma_{S}:=S^{\vee}=\{v\in N(R)_{\mathds{Q}};\langle u,v\rangle\geq 0\;%
\textit{for all}\;u\in S\}$
in the vector space $N(R)_{\mathds{Q}}$, the (closed) Weyl chamber
corresponding to $S$.
Definition 1.1.
We define $\Sigma(R)$ to be the fan in the lattice
$N(R)$ that consists of the Weyl chambers of the root system $R$
and all their faces. Let $X(R)$ be the toric variety associated
with the fan $\Sigma(R)$.
For $v\in N(R)_{\mathds{Q}}$ let $\sigma_{v}\in\Sigma(R)$ be the cone minimal in
$\Sigma(R)$ containing $v$. Equivalently, $\sigma_{v}$ is the cone
dual to the roots in $v^{\vee}=\{u\in M(R)_{\mathds{Q}};\langle u,v\rangle\geq 0\}$, i.e. $\sigma_{v}=(v^{\vee}\cap R)^{\vee}$.
In particular, for a general choice of $v$ this is the cone $\sigma_{S}$
dual to the set of simple roots $S\subset R$ of the set of positive
roots $v^{\vee}\cap R$ defined by $v$, i.e. the Weyl chamber for $S$.
The Weyl chambers cover $N(R)_{\mathds{Q}}$, so the fan $\Sigma(R)$ is complete.
Note that each set of simple roots forms a basis of the root lattice
$M(R)$ and $\sigma_{S}^{\vee}\cap M(R)=\langle S\rangle$ is the submonoid
of $M(R)$ generated by $S$. $X(R)$ is covered by the open subvarieties
$U_{S}:=\operatorname{Spec}\mathds{Z}[\sigma_{S}^{\vee}\cap M(R)]=\operatorname%
{Spec}\mathds{Z}[\langle S\rangle]\cong\mathds{A}^{\dim M(R)}$ for all the different sets of simple roots $S$.
The toric variety $X(R)$ is smooth and projective. It carries in
a natural way the action of the Weyl group $W(R)$ of the root
system $R$. The Weyl group permutes the sets of simple roots
and this way it acts simply transitive on the set of Weyl chambers.
The corresponding action on $X(R)$ permutes the open sets $U_{S}$, it
is a simply transitive action on the set of torus fixed points of
$X(R)$.
The root lattice $M(R)$ of $R$ is the lattice of characters of the
dense torus $T(R)$ in $X(R)$. This way, any element $u\in M(R)$
determines a character $x^{u}$ of $T(R)$, i.e. a rational function
on $X(R)$.
Example 1.2.
The toric variety $X(A_{1})$ is isomorphic to $\mathds{P}^{1}$.
Remark 1.3.
For two root systems $R_{1},R_{2}$ there is an isomorphism of fans
$\Sigma(R_{1}\times R_{2})\cong\Sigma(R_{1})\times\Sigma(R_{2})$ and thus
an isomorphism of toric varieties $X(R_{1}\times R_{2})\cong X(R_{1})\times X(R_{2})$.
1.2. Morphisms for maps of root systems and closures of
torus orbits
First, we show that maps between root systems coming from linear
maps of the ambient vector spaces induce toric morphisms
of the associated toric varieties.
Proposition 1.4.
Let $R,R^{\prime}$ be root systems in Euclidean spaces $E,E^{\prime}$.
Then a map of vector spaces $\mu\colon E^{\prime}\to E$ such that
$\mu(R^{\prime})\subset\{a\alpha;\>\alpha\in R,a\in\mathds{Z}\}$
induces a toric morphism of the associated toric varieties
$X(\mu)\colon X(R)\to X(R^{\prime})$.
Proof.
The map of vector spaces $\mu\colon E^{\prime}\to E$ induces a map of the
root lattices $\mu\colon M(R^{\prime})\to M(R)$ because $\mu(R^{\prime})\subset\{a\alpha;\alpha\in R,a\in\mathds{Z}\}$.
Let $\nu\colon N(R)\to N(R^{\prime})$ be the dual map of the dual lattices.
We have to show that each cone of $\Sigma(R)$ is mapped by
$\nu:N(R)_{\mathds{Q}}\to N(R^{\prime})_{\mathds{Q}}$ into a cone of $\Sigma(R^{\prime})$.
Let $v\in N(R)_{\mathds{Q}}$, we show that $\nu(\sigma_{v})\subseteq\sigma_{\nu(v)}$
(where as above $\sigma_{v}=(v^{\vee}\cap R)^{\vee}$ is the cone minimal
in $\Sigma(R)$ containing $v$; in the same way the cone $\sigma_{\nu(v)}$
of $\Sigma(R^{\prime})$ is defined). It suffices to show that
$\mu(\nu(v)^{\vee}\cap R^{\prime})\subseteq\langle v^{\vee}\cap R\rangle$.
This is true, since $\mu(R^{\prime})\subset\{a\alpha;\alpha\in R,a\in\mathds{Z}\}$
by assumption and $\mu(\nu(v)^{\vee})\subseteq v^{\vee}$ because
$\langle u^{\prime},\nu(v)\rangle=\langle\mu(u^{\prime}),v\rangle$ for any
$u^{\prime}\in M(R^{\prime})_{\mathds{Q}}$.
∎
We have two special cases:
(1) Root subsystems induce proper surjective morphisms.
Let $R\subset E$ be a root system and $R^{\prime}\subset E^{\prime}$ a root
system in a subspace $E^{\prime}\subseteq E$ such that $R^{\prime}\subseteq R$.
Then $\mu\colon M(R^{\prime})\to M(R)$ is injective, its dual
$\nu\colon N(R)\to N(R^{\prime})$ is surjective and we have a proper
surjective morphism $X(R)\to X(R^{\prime})$ which locally is given by
inclusions of coordinate rings $\mathds{Z}[\sigma_{\nu(v)}^{\vee}\cap M(R^{\prime})]\to\mathds{Z}[\sigma_{v}^{%
\vee}\cap M(R)]$.
(2) Projections of root systems induce closed embeddings.
Let $R\subset E$, $R^{\prime}\subset E^{\prime}$ be root systems and $\mu\colon E^{\prime}\to E$
a homomorphism of vector spaces such that $R\subseteq\mu(R^{\prime})\subset\{a\alpha;\alpha\in R,a\in\mathds{Z}\}$.
Then $\mu\colon M(R^{\prime})\to M(R)$ is surjective and for $v\in N(R)_{\mathds{Q}}$
induces a surjection $\langle\nu(v)^{\vee}\cap R^{\prime}\rangle\to\langle v^{\vee}\cap R\rangle$, the map $\nu\colon N(R)\to N(R^{\prime})$
is injective and $\nu^{-1}(\sigma_{\nu(v)})=\sigma_{v}$ for $v\in N(R)_{\mathds{Q}}$.
We have a closed embedding $X(R)\to X(R^{\prime})$ which locally is given by
surjective maps of coordinate rings
$\mathds{Z}[\sigma_{\nu(v)}^{\vee}\cap M(R^{\prime})]\to\mathds{Z}[\sigma_{v}^{%
\vee}\cap M(R)]$.
Example 1.5.
The first case in particular occurs if the root system $R^{\prime}$ is of
the form $R^{\prime}=R\cap E^{\prime}$, i.e. cut out by a subspace $E^{\prime}\subseteq E$.
Then the morphism $X(R)\to X(R^{\prime})$ is locally given by inclusions of
coordinate rings $\mathds{Z}[\sigma^{\vee}\!\cap M(R^{\prime})]\to\mathds{Z}[\sigma^{\vee}\!\cap
M%
(R)]$ for $\sigma\in\Sigma(R)$.
For example consider root subsystems $\{\pm\alpha\}\subseteq R$
consisting of two opposite roots, i.e. isomorphic to $A_{1}$.
Each of these gives rise to a projection $\varphi_{\{\pm\alpha\}}\colon X(R)\to X(A_{1})\cong\mathds{P}^{1}$.
Example 1.6.
For any root system $R$ the projections $X(R)\to\mathds{P}^{1}$ for all
root subsystems $A_{1}\cong R^{\prime}=\{\pm\alpha\}\subseteq R$ form
a morphism $X(R)\to\prod_{A_{1}\cong R^{\prime}\subseteq R}\mathds{P}^{1}$.
This morphism is an instance of the second case: it
corresponds to the projection of root systems
$\prod_{A_{1}\cong R^{\prime}\subseteq R}R^{\prime}\to R$.
(A variant of this closed embedding has been considered
in [BJ07].)
In the second case we can describe the equations for $X(R)$
in $X(R^{\prime})$.
Proposition 1.7.
Let $R\subset E$, $R^{\prime}\subset E^{\prime}$ be root systems and $\mu\colon E^{\prime}\to E$
a homomorphism of vector spaces such that $R\subseteq\mu(R^{\prime})\subset\{a\alpha;\alpha\in R,a\in\mathds{Z}\}$.
Then the image of the closed embedding $X(\mu)\colon X(R)\to X(R^{\prime})$
is determined by the equations $x^{u^{\prime}}=1$ for $u^{\prime}\in\ker(\mu)\cap M(R^{\prime})$.
Locally, the subvariety $X(R)\cap U_{S^{\prime}}\subseteq U_{S^{\prime}}$ for any set of
simple roots $S^{\prime}$ of $R^{\prime}$ is given by the equations $\prod_{i}x^{\alpha_{i}}=\prod_{j}x^{\beta_{j}}$ for collections of simple roots
$\alpha_{i},\beta_{j}\in S^{\prime}$ such that
$\sum_{i}\alpha_{i}-\sum_{j}\beta_{j}\in\ker(\mu)$.
Proof.
Let $v\in N(R)_{\mathds{Q}}$ be an element in the interior of some Weyl chamber,
let $S$ be the set of simple roots of $R$ with respect to $v$ and
let $S^{\prime}$ be the set of simple roots of $R^{\prime}$ with respect to $\nu(v)$.
Then $X(\mu)^{-1}(U_{S^{\prime}})=U_{S}$ and the inclusion $U_{S}\to U_{S^{\prime}}$
corresponds to the surjective map of coordinate rings
$\mathds{Z}[\langle S^{\prime}\rangle]\to\mathds{Z}[\langle S\rangle]$ given by the surjection
$\langle S^{\prime}\rangle\to\langle S\rangle$ determined by $\mu$.
We have $\langle S\rangle\cong\langle S^{\prime}\rangle/\!\!\sim$, where $\sim$ is
the equivalence relation $s_{1}\sim s_{2}\Leftrightarrow s_{1}-s_{2}\in\ker(\mu)$.
Thus $\mathds{Z}[\langle S\rangle]\cong\mathds{Z}[\langle S^{\prime}\rangle]/I$, where $I$ is the
ideal generated by $x^{s_{1}}-x^{s_{2}}$ for $s_{1},s_{2}\in\langle S^{\prime}\rangle$
such that $s_{1}-s_{2}\in\ker(\mu)$.
We can write $s_{1}$ (resp. $s_{2}$) as sums $s_{1}=\sum_{i}\alpha_{i}$
(resp. $s_{2}=\sum_{j}\beta_{j}$) of simple roots $\alpha_{i},\beta_{j}\in S^{\prime}$.
∎
Example 1.8.
In the case of the embedding $X(R)\to\prod_{A_{1}\cong R^{\prime}\subseteq R}\mathds{P}^{1}$,
the equations come from the linear relations between the roots of the
root system $R$. This will be discussed in detail in Subsection
1.3.
Next, we consider closures of torus orbits in $X(R)$. In general,
such orbit closures are again toric varieties and are in bijection
with the cones of the fan (see e.g. [Ful, 3.1]). We will
see that in the case of toric varieties associated with root systems
$R$ the orbit closures are again toric varieties associated with
certain root subsystems of $R$.
Proposition 1.9.
The closure of the torus orbit $Z\subseteq X(R)$ corresponding to
a cone $\tau\in\Sigma(R)$ is isomorphic to the toric variety $X(R^{\prime})$
associated with the root subsystem $R^{\prime}=R\cap E^{\prime}\subset E^{\prime}$ of $R$
cut out by the subspace $E^{\prime}=\tau^{\bot}\subseteq E$ with root lattice
$M(R^{\prime})=M(R)\cap E^{\prime}$.
Let $S$ be a set of simple roots of $R$ such that $\tau$ is contained
in the Weyl chamber $\sigma_{S}$ and put $S^{\prime}=S\cap E^{\prime}$. Then
$S^{\prime}$ is a set of simple roots of the root system $R^{\prime}$.
The orbit closure $Z$ is covered by the open sets $Z\cap U_{S}$ for
such sets of simple roots $S$, the closed subvariety
$Z\cap U_{S}\subseteq U_{S}$ is given by the equations $x^{\alpha}=0$ for
$\alpha\in S\setminus S^{\prime}$.
Proof.
Let $S$ be a set of simple roots of $R$ such that $\tau$ is a face
of $\sigma_{S}$. Then $E^{\prime}=\tau^{\bot}$ cuts out a face of
$\langle S\rangle_{\mathds{Q}_{\geq 0}}$ generated by $S^{\prime}=S\cap E^{\prime}$, the set
$S^{\prime}$ is a set of simple roots of the root system $R^{\prime}=R\cap E^{\prime}$
and $M(R^{\prime})=M(R)\cap E^{\prime}$.
By the general theory of toric varieties (see e.g. [Ful, 3.1])
the orbit closure corresponding to the cone $\tau$ is a toric variety
covered by the affine charts $Z\cap\operatorname{Spec}\mathds{Z}[\sigma_{S}^{\vee}\cap M(R)]=\operatorname{%
Spec}\mathds{Z}[\sigma_{S}^{\vee}\cap M(R)\cap\tau^{\bot}]$ for the maximal cones
$\sigma_{S}\in\Sigma(R)$ such that $\tau$ is a face of $\sigma_{S}$.
In the present case $\sigma_{S}^{\vee}\cap M(R)\cap\tau^{\bot}=\langle S\rangle\cap\tau^{\bot}=%
\langle S^{\prime}\rangle$, and so $Z$ is
covered by the open sets $Z\cap U_{S}=\operatorname{Spec}\mathds{Z}[\langle S^{\prime}\rangle]$ and
isomorphic to the toric variety $X(R^{\prime})$ associated with the root
system $R^{\prime}$.
The inclusion $Z\cap U_{S}=\operatorname{Spec}\mathds{Z}[\langle S^{\prime}\rangle]\subseteq%
\operatorname{Spec}\mathds{Z}[\langle S\rangle]=U_{S}$ is given by the homomorphism
$\mathds{Z}[\langle S\rangle]\to\mathds{Z}[\langle S^{\prime}\rangle]$, $x^{u}\mapsto x^{u}$
if $u\in\langle S^{\prime}\rangle$ and $x^{u}\mapsto 0$ otherwise. Thus the
closed subvariety $Z\cap U_{S}\subseteq U_{S}$ is determined by the
equations $x^{\alpha}=0$ for $\alpha\in S\setminus S^{\prime}$.
∎
Concerning the situation of the proposition we have two further remarks.
Remark 1.10.
The Dynkin diagram of $R^{\prime}$ is the subdiagram of the Dynkin diagram of
$R$ formed with respect to the set of simple roots $S$, that arises
after leaving out the vertices (and adjacent edges) corresponding
to the roots $\alpha\in S\setminus S^{\prime}$.
Usually, the root system $R^{\prime}$ will be reducible and decompose as
$R^{\prime}\cong\prod_{i}R_{i}$ into a number of irreducible root systems $R_{i}$
corresponding to the connected components of the Dynkin diagram of $R^{\prime}$.
Remark 1.11.
Since the fan $\Sigma(R)$ is symmetric under reflection in the
origin, also $-\tau$ is a cone of $\Sigma(R)$ and, apart from the
inclusion $i^{+}\colon X(R^{\prime})\to X(R)$ of $X(R^{\prime})$ as orbit closure
corresponding to $\tau$, there is another inclusion $i^{-}\colon X(R^{\prime})\to X(R)$ that embeds $X(R^{\prime})$ as orbit closure corresponding to $-\tau$.
Any such root subsystem $R^{\prime}\subset R$ comes with a proper surjective
morphism $X(R)\to X(R^{\prime})$, the two inclusions then are sections with
respect to this morphism.
Consider the particular case of a one-dimensional cone
$\tau=\langle v\rangle_{\mathds{Q}_{\geq 0}}$: for $\tau,-\tau$ we have the
two torus invariant divisors isomorphic to $X(R^{\prime})$, $R^{\prime}=R\cap v^{\bot}$,
given as the images $i^{\pm}(X(R^{\prime}))\subseteq X(R)$ and defined by the equations
$x^{\alpha}=0$ for $\alpha\in R$ such that $\langle\alpha,\pm v\rangle>0$.
1.3. The functor of $X(R)$
We will give a description of the functor of the toric variety
$X(R)$ in terms of the root system $R$.
This is done via the proper surjective toric morphisms $X(R)\to\mathds{P}^{1}$
for root subsystems isomorphic to $A_{1}$ forming the closed embedding
$X(R)\to\prod_{A_{1}\cong R^{\prime}\subseteq R}\mathds{P}^{1}$ and by the use
of the functor of $\mathds{P}^{1}$.
Remark 1.12.
We recall the well known description of the functor
of $\mathds{P}^{1}$.
For any scheme $Y$ a morphism $Y\to\mathds{P}^{1}$ is uniquely determined
(with respect to chosen coordinates on $\mathds{P}^{1}$) by the data consisting
of a line bundle $\mathscr{L}$ on $Y$ together with two sections
$t_{-},t_{+}$ that generate $\mathscr{L}$ up to isomorphisms of line bundles
with two sections (or equivalently by a line bundle $\mathscr{L}$
together with a surjective homomorphism $\mathcal{O}_{Y}^{\oplus 2}\to\mathscr{L}$
up to isomorphism). The functor that associates to a scheme $Y$
such data and to a morphism $Y^{\prime}\to Y$ the map given by pull-back of
line bundles and sections then is isomorphic to the functor
$\operatorname{Mor}(\>\cdot\>,\mathds{P}^{1})$. An isomorphism is determined by the universal
data consisting of the twisting sheaf together with homogeneous
coordinates $(\mathcal{O}_{\mathds{P}^{1}}(1),x_{0},x_{1})$ on $\mathds{P}^{1}$ corresponding
to ${\rm id}_{\mathds{P}^{1}}\in\operatorname{Mor}(\mathds{P}^{1},\mathds{P}^{1})$.
Another equivalent formulation is as follows: take as data
open sets $U_{-},U_{+}\subseteq Y$ and regular functions $f_{-}\in\mathcal{O}_{Y}(U_{-})$,
$f_{+}\in\mathcal{O}_{Y}(U_{+})$ such that $Y=U_{-}\cup U_{+}$, $f_{-}f_{+}=1$ on
$U_{-}\cap U_{+}$ and $\{f_{-}\neq 0\}=U_{-}\cap U_{+}=\{f_{+}\neq 0\}$.
Such data $(U_{-},U_{+},f_{-},f_{+})$ corresponds to a line bundle with
two generating sections $(\mathscr{L},t_{-},t_{+})$ uniquely determined
up to isomorphism by $U_{-}=\{t_{+}\neq 0\}$, $U_{+}=\{t_{-}\neq 0\}$ and
$f_{-}=t_{-}/t_{+}$ on $U_{-}$, $f_{+}=t_{+}/t_{-}$ on $U_{+}$.
By Subsection 1.2, root subsystems of a
root system $R$ isomorphic to $A_{1}$ define toric morphisms
$X(R)\to\mathds{P}^{1}$. These morphisms can also be described in terms of
the preceding remark.
Example 1.13.
Any root subsystem of $R$ isomorphic to $A_{1}$, i.e. an unordered pair
of opposite roots $\{\pm\alpha\}$ in $R$, defines a morphism $X(R)\to\mathds{P}^{1}$.
The data $\{(U_{\alpha},f_{\alpha}),(U_{-\alpha},f_{-\alpha})\}$ are defined
in terms of the rational functions $x^{\alpha},x^{-\alpha}$ associated
with the roots $\alpha,-\alpha$: let $U_{\alpha}$ be the open subset of
$X(R)$ where $x^{\alpha}$ is regular, $f_{\alpha}:=x^{\alpha}|_{U_{\alpha}}$
and the same for $-\alpha$.
The rational functions $x^{\alpha},x^{-\alpha}$ have no common
zeros or poles because any half-space in $M(R)$ contains at least
one of the roots $\alpha,-\alpha$.
We will denote these morphisms by
$$\varphi_{\{\pm\alpha\}}\colon\>X(R)\>\to\>\mathds{P}^{1}_{\{\pm\alpha\}}$$
We consider $\mathds{P}^{1}_{\{\pm\alpha\}}$ as a copy of $\mathds{P}^{1}$ with chosen
homogeneous coordinates $z_{\alpha},z_{-\alpha}$.
The toric morphism $\varphi_{\{\pm\alpha\}}$ corresponds to a map of
lattices $\mathds{Z}u_{\alpha}\to M(R)$, $u_{\alpha}\mapsto\alpha$ and hence
to a homomorphism of algebras $\mathds{Z}[y_{\alpha}^{\pm}]\to\mathds{Z}[M(R)]$,
$y_{\alpha}\mapsto x^{\alpha}$, where $y_{\alpha}=z_{\alpha}/z_{-\alpha}$,
i.e. the pull-back of the rational function
$y_{\alpha}=z_{\alpha}/z_{-\alpha}$ on $\mathds{P}^{1}$ via $\varphi_{\{\pm\alpha\}}$
is the rational function $x^{\alpha}$ on $X(R)$.
Also by the preceding subsection, the collection of these morphisms
defines a closed embedding
$$\textstyle\varphi\colon\;X(R)\;\to\;\prod_{A_{1}\cong R^{\prime}\subseteq R}%
\mathds{P}^{1}_{R^{\prime}}\,.$$
We choose a set of positive roots $R^{+}$ of $R$. This toric morphism
corresponds to a surjective map of lattices
$\mu\colon\bigoplus_{\alpha\in R^{+}}\mathds{Z}u_{\alpha}\to M(R)$,
$u_{\alpha}\mapsto\alpha$ or of algebras
$\bigotimes_{\alpha\in R^{+}}\mathds{Z}[y_{\alpha}^{\pm}]\to\mathds{Z}[M(R)]$,
$y_{\alpha}\mapsto x^{\alpha}$.
The equations describing $X(R)$ in $\prod_{A_{1}\cong R^{\prime}\subseteq R}\mathds{P}^{1}_{R^{\prime}}$ come from elements in the kernel of $\mu$, and such an element
$u=\sum_{i}l_{i}u_{\alpha_{i}}\in\ker(\mu)$ corresponds to a linear relation
$\sum_{i}l_{i}\alpha_{i}=0$ among the positive roots of the root system $R$.
For any such element we have an equation $\prod_{i}y_{\alpha_{i}}^{l_{i}}=1$
or equivalently a homogeneous equation
$\prod_{i}z_{\alpha_{i}}^{l_{i}}=\prod_{i}z_{-\alpha_{i}}^{l_{i}}$.
Example 1.14.
Consider the toric variety $X(A_{2})$ associated with the root
system $A_{2}=\{\pm\alpha,\pm\beta,\pm(\gamma=\alpha+\beta)\}$
and its embedding
$X(A_{2})\to\mathds{P}^{1}_{\{\pm\alpha\}}\times\mathds{P}^{1}_{\{\pm\beta\}}%
\times\mathds{P}^{1}_{\{\pm\gamma\}}$.
There is a one-dimensional space of linear relations generated
by the relation $\alpha+\beta=\gamma$, so $X(A_{2})\subset\mathds{P}^{1}_{\{\pm\alpha\}}\times\mathds{P}^{1}_{\{\pm\beta%
\}}\times\mathds{P}^{1}_{\{\pm\gamma\}}$ is determined by the homogeneous equation
$z_{\alpha}z_{\beta}z_{-\gamma}=z_{-\alpha}z_{-\beta}z_{\gamma}$.
In general, for any root subsystem in $R$ isomorphic to $A_{2}$,
there is a linear relation of the form $\alpha+\beta=\gamma$.
We show that these generate the space of all linear relations.
Proposition 1.15.
Let $R$ be a root system. Then the space of linear relations
between the positive roots of $R$ is generated by the relations
$\alpha+\beta=\gamma$ for root subsystems
$\{\pm\alpha,\pm\beta,\pm(\gamma=\alpha+\beta)\}$
of $R$ isomorphic to $A_{2}$.
Proof.
We show that the kernel of the map of lattices
$\mu\colon\bigoplus_{\alpha\in R^{+}}\mathds{Z}u_{\alpha}\to M(R)$
is generated by elements of the form $u_{\alpha}+u_{\beta}-u_{\gamma}$.
Since the simple roots $\alpha_{i}$ form a basis of the lattice $M(R)$,
the lattice $\ker(\mu)$ is generated by elements of the form
$u_{\beta}-\sum_{i}l_{i}u_{\alpha_{i}}$, where $\beta$ is a positive
root and thus a linear combination of simple roots
$\beta=\sum_{i}l_{i}\alpha_{i}$ with $l_{i}\in\mathds{Z}_{\geq 0}$.
The statement now follows from the fact that starting with the set
of simple roots $S_{0}:=S$ one obtains all positive roots by successively
adding roots that are sums oftwo roots already obtained, i.e. $S_{i+1}=S_{i}\cup\{\gamma\in R;\>\textit{$\gamma=\alpha+\beta$
for some $\alpha,\beta\in S_{i}$}\}$(see [Bou, Ch.4, §1.6, Prop.19] or [FH, §21.3]).
∎
Corollary 1.16.
The image of the closed embedding $\varphi\colon\;X(R)\;\to\;\prod_{A_{1}\cong R^{\prime}\subseteq R}\mathds{P}^{%
1}_{R^{\prime}}$
is determined by the homogeneous equations
$z_{\alpha}z_{\beta}z_{-\gamma}=z_{-\alpha}z_{-\beta}z_{\gamma}$
for root subsystems
$\{\pm\alpha,\pm\beta,\pm(\gamma=\alpha+\beta)\}$ of $R$
isomorphic to $A_{2}$.
With a view to the closed embedding $\varphi\colon\;X(R)\;\to\;\prod_{A_{1}\cong R^{\prime}\subseteq R}\mathds{P}^{%
1}_{R^{\prime}}$, we formulate a description
of the functor of $X(R)$ by characterising a morphism $Y\to X(R)$
in terms of the family of morphisms $Y\to\mathds{P}^{1}_{\{\pm\alpha\}}$
for all root subsystems of $R$ isomorphic to $A_{1}$ that satisfy
compatibility conditions coming from the root subsystems of $R$
isomorphic to $A_{2}$.
Definition 1.17.
Let $R$ be a root system.
We define a contravariant functor$F_{R}\colon(\textrm{schemes})\to(\textrm{sets})$ that associates to a scheme $Y$ the following
data, called $R$-data:
a family $(U_{\alpha},f_{\alpha})_{\alpha\in R}$ consisting of open
sets $U_{\alpha}\subseteq Y$ and regular functions
$f_{\alpha}\in\mathcal{O}_{Y}(U_{\alpha})$ that satisfy the conditions,
(i)
$\textit{for all}\;\alpha\in R\colon\;Y=U_{\alpha}\cup U_{-\alpha}$,
$\{f_{\alpha}\neq 0\}=U_{\alpha}\cap U_{-\alpha}\;\textit{and}\;f_{\alpha}f_{-%
\alpha}=1\;\textit{on}\;U_{\alpha}\cap U_{-\alpha}$ ,
(ii)
$\textit{for all}\;\alpha,\beta,\gamma\in R\colon\;\textit{if}\;\gamma=\alpha+%
\beta,\;\textit{then}\;U_{\alpha}\cap U_{\beta}\subseteq U_{\gamma}\;\textit{%
and}\;f_{\alpha}f_{\beta}=f_{\gamma}\;\textit{on}\;U_{\alpha}\cap U_{\beta}$ ,
or, equivalently, a family $(\mathscr{L}_{\{\pm\alpha\}},\{t_{\alpha},t_{-\alpha}\})_{\{\pm\alpha\}%
\subseteq R}$ of line bundles with two
generating sections that satisfy
(ii)’
$\textit{for all}\;\alpha,\beta,\gamma\in R:\;\textit{if}\;\,\gamma=\alpha+%
\beta,\;\textit{then}\;\,t_{\alpha}t_{\beta}t_{-\gamma}=t_{-\alpha}t_{-\beta}t%
_{\gamma}$ ,
up to isomorphism of line bundles with a pair of sections.
To a morphism $h\colon Y^{\prime}\to Y$ we associate the map
$F_{R}(h)\colon F_{R}(Y)\to F_{R}(Y^{\prime})$ given by pull-back of open
sets and functions or line bundles with sections.
Example 1.18.
$R$-data over a field $K$ can be written as a collection
$((t_{\alpha}\!:\!t_{-\alpha}))_{\{\pm\alpha\}\subseteq R}$
of ratios of elements of $K$ (such that for any $\alpha$ not both
$t_{\alpha},t_{-\alpha}$ are zero) that satisfy the equations
$t_{\alpha}t_{\beta}t_{-\gamma}=t_{-\alpha}t_{-\beta}t_{\gamma}$
for $\gamma=\alpha+\beta$.
Remark 1.19.
On $X(R)$ we have the following $R$-data, called the universal
$R$-data, coming from the morphisms $\varphi_{\{\pm\alpha\}}\colon X(R)\to\mathds{P}^{1}_{\{\pm\alpha\}}$, i.e. for $\alpha\in R$
consider the rational function $x^{\alpha}$, define $U_{\alpha}$
as the open set where $x^{\alpha}$ is regular and put
$f_{\alpha}:=x^{\alpha}|_{U_{\alpha}}$.
Theorem 1.20.
The toric variety $X(R)$ associated with the root system $R$
together with the universal $R$-data represents the functor $F_{R}$.
Proof.
We show that there is an isomorphism of functors
$\operatorname{Mor}(\>\cdot\>,X(R))\cong F_{R}$ such that the identity in
$\operatorname{Mor}(X(R),X(R))$ corresponds to the universal $R$-data on $X(R)$
denoted by $D_{0}\in F_{R}(X(R))$.
By $\Phi(Y)\colon\operatorname{Mor}(Y,X(R))\to F_{R}(Y)$, $h\mapsto h^{*}(D_{0})$,
we have defined a morphism of functors $\Phi\colon\operatorname{Mor}(\>\cdot\>,X(R))\to F_{R}$.
On the other hand, for $R$-data $D$ on $Y$ we have a morphism
$\varphi_{D}\colon Y\to X(R)\subseteq\prod_{\{\pm\alpha\}}\mathds{P}^{1}_{\{\pm%
\alpha\}}$,
where $X(R)$ is considered as a closed subvariety of
$\prod_{\{\pm\alpha\}}\mathds{P}^{1}_{\{\pm\alpha\}}$ via the embedding
$\varphi\colon\>X(R)\>\to\>\prod_{\{\pm\alpha\}}\mathds{P}^{1}_{\{\pm\alpha\}}$.
In particular it is $\varphi_{D_{0}}={\rm id}_{X(R)}$.
The maps $\Psi(Y)\colon F_{R}(Y)\to\operatorname{Mor}(Y,X(R))$, $D\mapsto\varphi_{D}$
form a morphism of functors $\Psi\colon F_{R}\to\operatorname{Mor}(\>\cdot\>,$ $X(R))$,
because for any morphism of schemes $h\colon Y^{\prime}\to Y$ it is
$\varphi_{h^{*}(D)}=\varphi_{D}\circ h$. This is true since the maps
$\varphi_{D,\{\pm\alpha\}}\colon Y\to\mathds{P}^{1}_{\{\pm\alpha\}}$ associated
with the part of $R$-data for pairs of roots $\{\pm\alpha\}$
satisfy $\varphi_{h^{*}(D),\{\pm\alpha\}}=\varphi_{D,\{\pm\alpha\}}\circ h$.
Thus, we have two morphisms of functors $\Phi$ and $\Psi$; these
are inverse to each other.
∎
Remark 1.21.
The universal $R$-data on $X(R)$ gives rise to $R$-data
$((t_{\alpha}:t_{-\alpha}))_{\{\pm\alpha\}\subseteq R}$
over points of $X(R)$ having the following properties:
$\bullet$ Over the affine chart $U_{S}$ for a set of simple roots
$S$ of $R$, we have $(t_{\alpha}:t_{-\alpha})\neq(1:0)$ if
$\alpha\in\langle S\rangle$, and over the torus fixed point of $U_{S}$,
we have $(t_{\alpha}:t_{-\alpha})=(0:1)$ for $\alpha\in\langle S\rangle$.
$\bullet$ Over the torus invariant divisor corresponding to a one-dimensional
cone generated by $v$, we have $(t_{\alpha}:t_{-\alpha})=(0:1)$ if
$\langle\alpha,v\rangle>0$ (cf. Remark 1.11).
2. Toric varieties associated with root systems of type $A$
2.1. Toric varieties $X(A_{n})$
Consider an $(n+1)$-dimensional Euclidean vector space with basis
$u_{1},\ldots,u_{n+1}$. The root system $A_{n}$ in the $n$-dimensional
subspace $E=\{\sum_{i}a_{i}u_{i};\linebreak\sum_{i}a_{i}=0\}$ consists of the
$n(n+1)$ roots
$$u_{i}-u_{j}\quad\textit{for}\;\;i,j\in\{1,\ldots,n+1\},\;i\neq j.$$
The lattice $N(A_{n})\cong\mathds{Z}^{n}$ dual to the root lattice $M(A_{n})\cong\mathds{Z}^{n}$
has a generating system $v_{1},\ldots,v_{n+1}$ with one relation
$\sum_{i}v_{i}=0$, it is a quotient of the lattice dual to
$\bigoplus_{i=1}^{n+1}\mathds{Z}u_{i}$ with basis $v_{1},\ldots,v_{n+1}$
dual to $u_{1},\ldots,u_{n+1}$.
The sets of simple roots of the root system $A_{n}$ are of the form
$$S=\{u_{i_{1}}-u_{i_{2}},u_{i_{2}}-u_{i_{3}},\ldots,u_{i_{n}}-u_{i_{n+1}}\}$$
for some ordering $i_{1},\ldots,i_{n+1}$ of the set $\{1,\ldots,{n+1}\}$.
The maximal cone $\sigma_{S}=S^{\vee}$ of $\Sigma(A_{n})$, i.e. the Weyl
chamber corresponding to $S$, consists of those elements
$\sum_{i}a_{i}v_{i}\in N(A_{n})$ that satisfy
$a_{i_{1}}\geq a_{i_{2}},\ldots,a_{i_{n}}\geq a_{i_{n+1}}$ or
equivalently of non-negative linear combinations of
$$v_{i_{1}},\;v_{i_{1}}+v_{i_{2}},\;\ldots\,,\;v_{i_{1}}+\cdots+v_{i_{n}}.$$
So, the fan $\Sigma(A_{n})$ can be described as follows (and coincides
with that of [LM00, (2.5), (2.6)]):
there are $2^{n+1}-2$ one-dimensional cones of the fan $\Sigma(A_{n})$,
and these are generated by the elements $v_{A}=\sum_{i\in A}v_{i}\in N(A_{n})$
for $A\in\mathcal{A}$ where
$$\mathcal{A}=\{A\>|\>\emptyset\neq A\subsetneq\{1,\ldots,n+1\}\}.$$
A family $(v_{A^{(i)}})_{i=1,\ldots,k}$ corresponding to a collection
of pairwise different sets $A^{(1)},\ldots,A^{(k)}$ $\in\mathcal{A}$
generates a $k$-dimensional cone of $\Sigma(A_{n})$ whenever these sets
can be ordered such that $A^{(i_{1})}\subset A^{(i_{2})}\subset\cdots\subset A^{(i_{k})}$.
The fan $\Sigma(A_{n})$ defines an $n$-dimensional smooth projective
toric variety $X(A_{n})$. It is covered by the $(n+1)!$ open subvarieties
$U_{S}=\operatorname{Spec}\mathds{Z}[\sigma_{S}^{\vee}\cap M(A_{n})]$ for the sets of simple roots
$S$ corresponding to strict orderings of the set $\{1,\ldots,n+1\}$.
If $S=\{u_{i_{1}}-u_{i_{2}},u_{i_{2}}-u_{i_{3}},\ldots,u_{i_{n}}-u_{i_{n+1}}\}$,
then $\mathds{Z}[\sigma_{S}^{\vee}\cap M(A_{n})]=\mathds{Z}[x_{i_{1}}/x_{i_{2}},%
\ldots,x_{i_{n}}/x_{i_{n+1}}]$.
The Weyl group of the root system $A_{n}$ is the symmetric group $S_{n+1}$;
it acts on $\Sigma(A_{n})$ and on $X(A_{n})$ by permuting the Weyl chambers
and the open sets $U_{S}$.
By results of the last subsection (Proposition 1.9
and Remark 1.10), the closures of torus orbits in
$X(A_{n})$ are isomorphic to products $\prod_{i}X(A_{n_{i}})$. In particular,
the torus invariant divisor corresponding to the cone generated by
$v_{i_{1}}+\cdots+v_{i_{k}}$ is of the form $X(A_{n-k})\times X(A_{k-1})$.
Example 2.1.
The fans $\Sigma(A_{1})$ and $\Sigma(A_{2})$ are as Figure 1.
There are other descriptions of the toric variety $X(A_{n})$:
$X(A_{n})$ as blow-up of $\mathds{P}^{n}$.
$X(A_{n})$ can be constructed by a sequence of toric blow-ups starting
with $\mathds{P}^{n}$ by first blowing up the $n+1$ torus fixed points, then
blowing up the strict transform of the lines joining two of these
points, then the strict transform of the planes through any three
of these points and so on (see [Pr90, Ch.3], [Ka93, (4.3.13)],
[DL94, (5.1)]).
$X(A_{n})$ as toric variety associated with the
lattice polytope called permutohedron.
The $n$-dimensional permutohedron is defined as the convex
hull in $\mathds{Q}^{n+1}$ of the $S_{n+1}$-orbit of the point
$(1,2,\ldots,n+1)$ where the symmetric group $S_{n+1}$
acts by permuting coordinates.
One can show that the fan for this polytope is the fan
$\Sigma(A_{n})$. The toric variety $X(A_{n})$ also appears as
“permutohedral space $\Pi^{n}\,$”
in [Ka93, (4.3.10) through (4.3.13)].
2.2. (Co)homology of $X(A_{n})$
We know that the integral cohomology is torsion-free and confined
to the even degrees. Standard methods from toric geometry
(see e.g. [Dan, (10.8)]) furthermore imply the
description of the cohomology ring of the toric variety $X(A_{n})$
over the complex numbers (see also [LM00, (2.7)]) as
$$H^{*}(X(A_{n}),\mathds{Z})\;\cong\;\mathds{Z}[\,l_{A};A\in\mathcal{A}\,]/(R_{1%
}+R_{2}),$$
where $R_{1}$ is the ideal generated by the elements
$r_{i,j}=\sum_{i\in A}l_{A}-\sum_{j\in A}l_{A}$ for $i,j\in\{1,\ldots,n+1\}$,
$i\neq j$, and $R_{2}$ the ideal generated by the elements
$r_{A,A^{\prime}}=l_{A}l_{A^{\prime}}$ for $A,A^{\prime}\in\mathcal{A}$ such that
$A\not\subseteq A^{\prime}$, $A^{\prime}\not\subseteq A$ (these correspond to
the primitive collections of the fan $\Sigma(A_{n})$, see next
subsection).
The Betti numbers and Poincaré polynomials of the varieties $X(A_{n})$
over the complex numbers are calculated in [LM00, (2.3)]; see
also [St94, Section 6], [DL94, Section 4] for a description
in terms of the Eulerian numbers and see [Kl85], [Kl95],
[St94] for the general case of toric varieties associated
with root systems. In particular, we know that the rank of
$H^{*}(X(A_{n}),\mathds{Z})$ is $(n+1)!$, i.e. the number of maximal cones
of $\Sigma(A_{n})$.
The $\mathds{Z}$-module $\mathds{Z}[\,l_{A}:A\in\mathcal{A}\,]/(R_{1}+R_{2})$ is generated
by the classes of square-free monomials (see [Dan, (10.7.1)]).
We can restrict to monomials each of which has only factors
corresponding to one-dimensional faces of one maximal cone.
Such a monomial $\prod_{i=1}^{m}l_{A^{(i)}}$ corresponds to an
$m$-dimensional face of the respective maximal cone and, on the
other hand, to a collection $A^{(1)}\subsetneq\cdots\subsetneq A^{(m)}$
of elements of $\mathcal{A}$.
We denote by $G$ the $\mathds{Z}$-submodule of $\mathds{Z}[\,l_{A};A\in\mathcal{A}\,]$
generated by these monomials (called “good monomials” in
[LM00, (2.8)]). We have the canonical isomorphism of $\mathds{Z}$-modules
$G/U\cong\mathds{Z}[\,l_{A};A\in\mathcal{A}\,]/(R_{1}+R_{2})$ where
$U=(R_{1}+R_{2})\cap G$.
The module $G/U$ can be identified with the homology module
$H_{*}(X(A_{n}),\mathds{Z})$ (cf. [LM00, (2.9.2)]).
In [LM00, (2.8.2)] the following generators of the module of
relations $U$ are given. For a collection $A^{(1)}\subsetneq\cdots\subsetneq A^{(m)}$ of elements of $\mathcal{A}$ and
$k\in\{1,\ldots,m+1\}$, $i,j\in A^{(k)}\setminus A^{(k-1)}$ (put
$A^{(0)}=\emptyset$, $A^{(m+1)}=\{1,\ldots,n+1\}$), $i\neq j$, let
$$\textstyle r_{i,j}((A^{(h)})_{h},k)=\Big{(}\sum_{\tiny\hskip-2.845276pt\begin{%
array}[]{l}i\!\in\!A\\
j\!\not\in\!A\\
\end{array}\hskip-3.414331pt}l_{A}-\sum_{\tiny\hskip-2.845276pt\begin{array}[]%
{l}j\!\in\!A\\
i\!\not\in\!A\\
\end{array}\hskip-3.414331pt}l_{A}\Big{)}\prod_{h=1}^{m}l_{A^{(h)}}$$
where the sums run over sets $A\in\mathcal{A}$ such that
$A^{(k-1)}\subsetneq A\subsetneq A^{(k)}$.
The maximal cones of the fan $\Sigma(A_{n})$ correspond to collections
$A^{(1)}\subsetneq\cdots\subsetneq A^{(n)}$ of elements of
$\mathcal{A}$ and these correspond to permutations $\sigma\in S_{n+1}$
via $\{\sigma(1),\ldots,\sigma(k)\}=A^{(k)}$ for $k=1,\ldots,n$.
The descent set of a permutation $\sigma\in S_{n+1}$ is the set
$$\operatorname{Desc}(\sigma)=\{k\in\{1,\ldots,n\};\>\sigma(k)>\sigma(k+1)\}\,.$$
For any $\sigma\in S_{n+1}$ we define a monomial in $G$ by
$$\textstyle l^{\sigma}=\prod_{k\not\in\operatorname{Desc}(\sigma)}l_{\{\sigma(1%
),\ldots,\sigma(k)\}}\,.$$
Proposition 2.2.
The classes of the monomials $l^{\sigma}$ for $\sigma\in S_{n+1}$ form
a basis of the homology module $G/U=H_{*}(X(A_{n}),\mathds{Z})$.
The module of relations $U$ is generated by the elements
$r_{i,j}((A^{(h)})_{h},k)$ for collections $A^{(1)}\subsetneq\ldots\subsetneq A^{(m)}$ of elements of $\mathcal{A}$ and
$k\in\{1,\ldots,m+1\}$, $i,j\in A^{(k)}\setminus A^{(k-1)}$
(put $A^{(0)}=\emptyset$, $A^{(m+1)}=\{1,\ldots,n+1\}$),
$i\neq j$.
Proof.
We have $(n+1)!$ distinct monomials $l^{\sigma}$, and this number coincides
with the rank of $G/U$. Thus it remains to show that every monomial in
$G$ via the relations $r_{i,j}((A^{(h)})_{h},k)$ is equivalent to a
linear combination of the monomials $l^{\sigma}$.
For a monomial $\prod_{k=1}^{m}l_{A^{(k)}}$ corresponding to a collection
$A^{(1)}\subsetneq\cdots\subsetneq A^{(m)}$ we define the number
$d(\prod_{k=1}^{m}l_{A^{(k)}}):=|\{k\in\{1,\ldots,m\};\>\min P_{k}>\max P_{k+1}%
\}|\in\mathds{Z}_{\geq 0}$ in terms of the associated
partition $P_{1}=A^{(1)}$, $P_{2}=A^{(2)}\setminus A^{(1)}$, $\ldots\;$,
$P_{m}=A^{(m)}\setminus A^{(m-1)}$, $P_{m+1}=\{1,\ldots,n+1\}\setminus A^{(m)}$ of the set $\{1,\ldots,n+1\}$. The monomials $y\in G$ satisfying
$d(y)=0$ are exactly the monomials of the form $l^{\sigma}$.
On the other hand we have an ordering $\prec$ of the monomials of $G$:
take the partition $(P_{i})_{i=1,\ldots,m+1}$ associated with a monomial
and consider the sequence (corresponding to a permutation of the set
$\{1,\ldots,n+1\}$) that arises by taking first the elements of $P_{m+1}$
then those of $P_{m}$ and so on and by ordering the elements of each
$P_{i}$ according to their size, and on these sequences we take the
lexicographic order.
We show that every monomial in $G$ modulo $U$ is equivalent to a
linear combination of the monomials $l^{\sigma}$, $\sigma\in S_{n+1}$,
by showing that every monomial $y\in G$ with $d(y)>0$ modulo a
relation is equivalent to a linear combination of monomials $y^{\prime}$
with $y\prec y^{\prime}$.
In fact, let $A^{(1)}\subsetneq\cdots\subsetneq A^{(m)}$ be a
collection of elements of $\mathcal{A}$ with associated partition
$(P_{k})_{k=1,\ldots,m+1}$ such that the corresponding monomial
$y=\prod_{k=1}^{m}l_{A^{(k)}}$ satisfies $d(y)>0$.
Take $k\in\{1,\ldots,m\}$ such that $i:=\min P_{k}>\max P_{k+1}=:j$,
then
$$\textstyle r_{i,j}((A^{(h)})_{h\neq k},k)=\Big{(}\sum_{\tiny\hskip-2.845276pt%
\begin{array}[]{l}i\!\in\!A\\
j\!\not\in\!A\\
\end{array}\hskip-3.414331pt}l_{A}-\sum_{\tiny\hskip-2.845276pt\begin{array}[]%
{l}j\!\in\!A\\
i\!\not\in\!A\\
\end{array}\hskip-3.414331pt}l_{A}\Big{)}\prod_{h\neq k}l_{A^{(h)}}\,,$$
where the sums run over sets $A\in\mathcal{A}$ such that $A^{(k-1)}\subsetneq A\subsetneq A^{(k+1)}$, is a relation that contains $y$
as the monomial minimal with respect to $\prec$.
∎
Remark 2.3.
The above set of generators for the module of relations is used in
[LM00, Section 3] in the context of solutions to the
commutativity equations; the respective statement [LM00, (2.9)]
is proven in a different way in that paper.
Our statement involves a basis of homology which, in the general case
of toric varieties associated with root systems, is given in
[Kl85], [Kl95] (see [BB11, Rem. 5.4] for more
explanations).
2.3. Primitive collections and the morphism $X(A_{n})\to\mathds{P}_{\Delta(A_{n})}$
It was observed in [Ba91] that any $n$-dimensional smooth
projective toric variety $X$ corresponding to a fan $\Sigma$
can be described by the set of primitive collections among the
generators of the one-dimensional cones of $\Sigma$ together with
the corresponding primitive relations.
A primitive collection of the fan $\Sigma$ is a set of generators of
one-dimensional cones that does not generate a cone of $\Sigma$, but all
of its proper subsets generate a cone of $\Sigma$.
Theorem 2.4.
([Ba91, Thm. 2.15], see also
[CR08, Thm. 1.4, Prop. 1.10]).
The Mori cone $\operatorname{NE}(X)\subset A_{1}(X)\otimes_{\mathds{Z}}\mathds{R}$ of effective $1$-cycles
is generated by the primitive relations.
Equivalently, the nef-cone $\operatorname{Nef}(X)\subset\operatorname{Pic}(X)\otimes_{\mathds{Z}}\mathds{R}$ (which
coincides with the closure of the ample cone) is given by line bundles
corresponding to piecewise linear functions $\varphi$ satisfying
$\varphi(w_{1})+\cdots+\varphi(w_{k})\geq\varphi(w_{1}+\cdots+w_{k})$ for all primitive
collections $\{w_{1},\ldots,w_{k}\}$ of the fan $\Sigma$.
In the case $\Sigma=\Sigma(A_{n})$, the primitive collections
consist of two elements $v_{A},v_{A^{\prime}}$ (again put
$v_{A}=\sum_{i\in A}v_{i}$) corresponding to non comparable elements
of $\mathcal{A}$, i.e. elements $A,A^{\prime}\in\mathcal{A}$ such that
$A\not\subseteq A^{\prime}$, $A^{\prime}\not\subseteq A$. The corresponding
primitive relation has one of the following four forms
(put $I=\{1,\ldots,n+1\}$):
(1) $v_{A}+v_{A^{\prime}}=0$, if $A\cup A^{\prime}=I$ and $A\cap A^{\prime}=\emptyset$;
(2) $v_{A}+v_{A^{\prime}}=v_{A\cup A^{\prime}}$, if $A\cup A^{\prime}\neq I$ and
$A\cap A^{\prime}=\emptyset$;
(3) $v_{A}+v_{A^{\prime}}=v_{A\cap A^{\prime}}$, if $A\cup A^{\prime}=I$ and
$A\cap A^{\prime}\neq\emptyset$;
(4) $v_{A}+v_{A^{\prime}}=v_{A\cap A^{\prime}}+v_{A\cup A^{\prime}}$, if
$A\cup A^{\prime}\neq I$ and $A\cap A^{\prime}\neq\emptyset$.
Corollary 2.5.
Let $D=\sum_{A\in\mathcal{A}}a_{A}D_{A}$ be a torus-invariant divisor
in $X(A_{n})$, where $D_{A}$ is the torus-invariant prime divisor
corresponding to $v_{A}$. We put $a_{I}=a_{\emptyset}=0$.
Then $D$ is ample (resp. semiample, or equivalently nef)
if and only if
$$a_{A}+a_{A^{\prime}}>a_{A\cap A^{\prime}}+a_{A\cup A^{\prime}}$$
(resp. $\geq$) for all primitive collections $\{v_{A},v_{A^{\prime}}\}$.
Corollary 2.6.
The anticanonical class of $X(A_{n})$ is semiample, or equivalently nef;
$X(A_{n})$ is an almost Fano variety.
Proof.
By corollary 2.5 $-K_{X(A_{n})}$ is semiample
(and ample if there is no primitive relation of type (4), i.e.
if $n\leq 2$). That $X(A_{n})$ is an almost Fano variety means that
$-K_{X(A_{n})}$ is semiample and big, where the second property follows
from the fact that the corresponding polytope is full-dimensional.
∎
Being semiample and big, the anticanonical class defines a toric
morphism $X(A_{n})\to\mathds{P}(H^{0}(\mathcal{O}_{X(A_{n})}(-K_{X(A_{n})})))$, which is birational
onto its image, but not necessarily an embedding.
Proposition 2.7.
The anticanonical class defines a birational toric morphism
$X(A_{n})\to\mathds{P}_{\Delta(A_{n})}$, where $\mathds{P}_{\Delta(A_{n})}$ is
the Gorenstein toric Fano variety associated with the reflexive polytope
$$\Delta(A_{n})\;=\;\{m\in M_{\mathds{Q}};\>\left<m,v_{A}\right>\geq-1\;\textit{%
for}\;A\in\mathcal{A}\}\;=\;\operatorname{conv}\{\alpha_{ij};\>i,j\in I,i\neq j\}$$
and $\alpha_{ij}=u_{i}-u_{j}$ are the roots of the root system $A_{n}$.
Proof.
In general, the image of the morphism given by a semiample divisor
$\sum_{i}a_{i}D_{i}$ (sum over the torus invariant prime divisors $D_{i}$)
is the polarised toric variety corresponding to the polytope
$\{m\in M_{\mathds{Q}};\>\left<m,v_{i}\right>\geq-a_{i}\;\textit{for all}\;i\}$
($v_{i}\in N$ being the lattice generators of the one dimensional
cones corresponding to $D_{i}$).
In the present case, we have the anticanonical divisor
$-K_{X(A_{n})}=\sum_{A\in\mathcal{A}}D_{A}$ on $X(A_{n})$ giving rise
to the polytope $\Delta(A_{n})=\{m\in M_{\mathds{Q}};\>\left<m,v_{A}\right>\geq-1\;\textit{for}%
\;A\in\mathcal{A}\}$.
This polytope $\Delta(A_{n})$ coincides with the convex hull of
the roots of $A_{n}$. Clearly, every root $\alpha_{ij}$ satisfies
$\left<m,v_{A}\right>\geq-1$ for all $A\in\mathcal{A}$.
On the other hand, a given element
$m=(m_{1},\ldots,m_{n+1})\in\Delta(A_{n})$ we can write as
$m=\sum_{i\in B,j\in C}a_{ij}\alpha_{ij}$ with $a_{ij}\geq 0$
and $B=\{i;\>m_{i}>0\}$, $C=\{i;\>m_{i}<0\}$, and the condition
$\left<m,v_{C}\right>\geq-1$ gives $\sum a_{ij}\leq 1$, so
$m\in\operatorname{conv}\{\alpha_{ij}\>|\>i\neq j\}$.
The polytope $\Delta(A_{n})$ is reflexive ($0$ being its only inner lattice
point, the other lattice points in $\Delta(A_{n})$ are the roots),
equivalently $\mathds{P}_{\Delta(A_{n})}$ is a Gorenstein toric Fano variety.
∎
Remark 2.8.
The fan $\Sigma_{\Delta(A_{n})}$ determined by $\Delta(A_{n})$
consists of the cones
$$\sigma_{B_{1},B_{2}}=\left<v_{A};\>B_{1}\subseteq A\subseteq B_{2}\right>$$
for $B_{1},B_{2}\in\mathcal{A}$, $B_{1}\subseteq B_{2}$.
The cone $\sigma_{B_{1},B_{2}}$ has dimension $|B_{2}\setminus B_{1}|+1$ and
is generated by $2^{|B_{2}\setminus B_{1}|}$ elements $v_{A}$.
We see that $\mathds{P}_{\Delta(A_{n})}$ is singular for $n\geq 3$, the
singular locus consisting of the torus orbits of codimension
at least $3$.
The morphism $X(A_{n})\to\mathds{P}_{\Delta(A_{n})}$ is a crepant resolution
(an MPCP-desingularisation in the sense of [Ba94]). It is given
by subdividing each $d$-dimensional cone $\sigma_{B_{1},B_{2}}$
into $(d-1)!$ $d$-dimensional cones generated by
$v_{B_{1}},v_{B_{1}\cup\{a_{1}\}},\;\ldots\;,v_{B_{1}\cup\{a_{1},\ldots,a_{d-1}%
\}}\!=\!v_{B_{2}}$ corresponding to permutations of the set $B_{2}\setminus B_{1}$.
A set $\{v_{A^{(1)}},\ldots,v_{A^{(k)}}\}$ is contained in a single
cone of $\Sigma_{\Delta(A_{n})}$, if $\bigcap_{i}A^{(i)}\neq\emptyset$
and $\bigcup_{i}A^{(i)}\neq I$. In particular, the primitive collections
of types (1), (2) and (3) survive.
3. Losev-Manin moduli spaces
3.1. The moduli functor of $A_{n}$-curves
We begin with a presentation of the Losev-Manin moduli spaces
$\overline{L}_{n}$ introduced in [LM00] and the corresponding moduli
functor of stable $n$-pointed chains of projective lines (here called
$A_{n-1}$-curves).
Consider a complex projective line $\mathds{P}^{1}_{\mathds{C}}$ with two distinct
closed points $s_{-},s_{+}\in\mathds{P}^{1}_{\mathds{C}}$ called poles. The moduli space
of $n$ distinguishable points in $\mathds{P}^{1}_{\mathds{C}}\setminus\{s_{-},s_{+}\}$ is
a torus $(\mathds{C}^{*})^{n}/\mathds{C}^{*}\cong(\mathds{C}^{*})^{n-1}$, here we divide out
the automorphism group $\mathds{C}^{*}$ of the projective line with
two poles $(\mathds{P}^{1}_{\mathds{C}},s_{-},s_{+})$.
This space has a natural compactification $\overline{L}_{n}$ such
that its boundary parametrises isomorphism classes of certain
types of reducible $n$-pointed curves.
Definition 3.1.
A chain of projective lines of length $m$ over an algebraically
closed field $K$ is a projective curve $C=C_{1}\cup\cdots\cup C_{m}$ over
$K$ such that each irreducible component $C_{j}$ of $C$ is a projective
line with poles $p^{-}_{j},p^{+}_{j}$ and these components intersect as follows:
different components $C_{i}$ and $C_{j}$ intersect only if $|i-j|=1$
and in this case $C_{j},C_{j+1}$ intersect transversally at the
single point $p^{+}_{j}=p^{-}_{j+1}$.
For $p^{-}_{1}\in C_{1}$ (resp. $p^{+}_{m}\in C_{m}$) we write $s_{-}$ (resp. $s_{+}$).
Two chains of projective lines $(C,s_{-},s_{+})$ and $(C^{\prime},s_{-}^{\prime},s_{+}^{\prime})$
are called isomorphic if there is an isomorphism $\varphi\colon C\to C^{\prime}$
such that $\varphi(s_{-})=s_{-}^{\prime}$ and $\varphi(s_{+})=s_{+}^{\prime}$.
Definition 3.2.
An $n$-pointed chain of projective lines
$(C,s_{-},s_{+},s_{1},\ldots,s_{n})$ is a chain of projective
lines together with closed points $s_{i}\in C$ different
from $s_{-},s_{+}$ and the intersection points of components (see Figure 2).
Two $n$-pointed chains of projective lines
$(C,s_{-},s_{+},s_{1},\ldots,s_{n})$ and $(C^{\prime},s_{-}^{\prime},s_{+}^{\prime},s^{\prime}_{1},\ldots,s^{\prime}_{n})$
are called isomorphic if there is an isomorphism $\varphi\colon(C,s_{-},s_{+})\to(C^{\prime},s_{-}^{\prime},s_{+}^{\prime})$ of the underlying chains of projective
lines such that $\varphi(s_{j})=s^{\prime}_{j}$ for all $j\in\{1,\ldots,n\}$.
An $n$-pointed chain of projective lines $(C,s_{-},s_{+},s_{1},\ldots,s_{n})$
is called stable if each component of $C$ contains at least one
of the points $s_{j}$.
An $A_{n}$-curve over an algebraically closed field $K$ is a stable
$(n+1)$-pointed chain of projective lines over $K$.
Definition 3.3.
An $A_{n}$-curve of length $m$ decomposes into irreducible components
$C=C_{1}\cup\cdots\cup C_{m}$ with $s_{-}\in C_{1}$, $s_{+}\in C_{m}$. We will call
the resulting decomposition
$$\{1,\ldots,n+1\}=P_{1}\sqcup\cdots\sqcup P_{m}$$
such that $i\in P_{k}$ if and only if $s_{i}\in C_{k}$ the
combinatorial type of the $A_{n}$-curve $C$.
We will also write this in the form $s_{i_{1}}\ldots s_{i_{l}}|s_{i_{l+1}}\ldots|\ldots$ with the sections for the different
sets $P_{k}$ separated by the symbol “$\,|\,$” starting on the left with
$P_{1}$.
Definition 3.4.
Let $Y$ be a scheme. An $A_{n}$-curve over $Y$ is a collection
$(\pi\colon C\to Y,s_{-},s_{+},s_{1},\ldots,s_{n+1})$,
where $C$ is a scheme, $\pi$ is a proper flat morphism of schemes
and $s_{-},s_{+},s_{1},$ $\ldots,s_{n+1}\colon Y\to C$ are
sections such that for any geometric point $y$ of $Y$ the
collection $(C_{y},(s_{-})_{y},(s_{+})_{y},(s_{1})_{y},$ $\ldots,(s_{n+1})_{y})$
is an $A_{n}$-curve over $y$.
An isomorphism of $A_{n}$-curves over $Y$ is an isomorphism of
$Y$-schemes that is compatible with the sections.
We define the moduli functor of $A_{n}$-curves as the
contravariant functor
$$\begin{array}[]{crcl}\overline{L}_{n+1}:&(\textrm{schemes})&\to&(\textrm{sets}%
)\\
&Y&\mapsto&\left\{\textrm{$A_{n}$-curves over $Y$}\right\}\,/\sim\end{array}$$
that associates to a scheme $Y$ the set of isomorphism classes
of $A_{n}$-curves over $Y$ and to a morphism of schemes the map
obtained by pulling back $A_{n}$-curves.
It is shown in [LM00, (2.2)] that the functor
$\overline{L}_{n+1}$ is represented by a smooth projective variety,
denoted by the same symbol $\overline{L}_{n+1}$, that
is, we have a fine moduli space of $A_{n}$-curves.
Further it is argued in [LM00, (2.6.3)] that
$\overline{L}_{n+1}$ is isomorphic to the toric variety
$X(A_{n})$ associated with the root system $A_{n}$.
In [LM00, (2.1)] an inductive construction of $\overline{L}_{n+1}$
together with the universal curve $C_{n+1}\to\overline{L}_{n+1}$
is given using arguments of [Kn83].
As in the case of the similar moduli spaces $\overline{M}_{0,n}$,
the universal curve $C_{n+1}$ over $\overline{L}_{n+1}$ is isomorphic
to the next moduli space $\overline{L}_{n+2}$.
Example 3.5.
(1) $C_{1}\to\overline{L}_{1}$ is isomorphic to $\mathds{P}^{1}\to pt$ with
$0,\infty$ and $1$-section. This reflects the fact that
any $A_{0}$-curve over a scheme $Y$ is isomorphic to the trivial
projective bundle $\mathds{P}^{1}_{Y}$ with $0,\infty$ and $1$-section.
(2) $\overline{L}_{2}$ is isomorphic to $\mathds{P}^{1}$. The fibres of the
universal curve $C_{2}$ over $\mathds{P}^{1}\setminus\{(0:1),(1:0)\}$ are
pointed $(\mathds{P}^{1},s_{-},s_{+})$, and over $(0:1)$ and $(1:0)$ the fibres
are pointed chains consisting of two components (see Figure 3).
3.2. The universal curve
We construct the universal $A_{n}$-curve $X(A_{n+1})\to X(A_{n})$
with its sections, which later will be seen to coincide with the
universal curve over the Losev-Manin moduli space $\overline{L}_{n+1}$,
in terms of the functorial properties of toric varieties associated
with root systems developed in subsection 1.2.
Construction 3.6 (The universal $A_{n}$-curve).
Consider the root subsystem $A_{n}\subset A_{n+1}$
consisting of the roots $\beta_{ij}=u_{i}-u_{j}$ for
$i,j\in\{1,\ldots,n+1\}$. The inclusion of root systems
$A_{n}\subset A_{n+1}$ determines a proper surjective morphism
$X(A_{n+1})\to X(A_{n})$.
There are the $n+1$ additional pairs of opposite roots $\pm\alpha_{i}$,
$\alpha_{i}=u_{i}-u_{n+2}$ for $i\in\{1,\ldots,n+1\}$.
For each of the root subsystems $\{\pm\alpha_{i}\}\cong A_{1}$ in $A_{n+1}$
not contained in $A_{n}$ we have a section $s_{i}\colon X(A_{n})\to X(A_{n+1})$
corresponding to the projection of the root system $A_{n+1}$
onto the root subsystem $A_{n}$ with kernel generated by $\alpha_{i}$.
The image of $s_{i}$ can be described by the equation $x^{\alpha_{i}}=1$.
On the other hand, the inclusion $\{\pm\alpha_{i}\}\subset A_{n+1}$
defines a projection $X(A_{n+1})\to X(A_{1})\cong\mathds{P}^{1}_{\{\pm\alpha_{i}\}}$.
If we choose coordinates of $\mathds{P}^{1}_{\{\pm\alpha_{i}\}}$ as in Subsection
1.3, then the image of the section $s_{i}$ is given by
the preimage of the point $(1:1)\in\mathds{P}^{1}_{\{\pm\alpha_{i}\}}$.
Further, we have two sections $s_{\pm}\colon X(A_{n})\to X(A_{n+1})$
which are inclusions of $X(A_{n})$ into $X(A_{n+1})$ as torus
invariant divisors (cf. Remark 1.11).
The image of $s_{-}$ (resp. $s_{+}$) corresponds to the ray of the fan
$\Sigma(A_{n+1})$ generated by $-v_{n+2}$ (resp. $v_{n+2}$) and is
given by the equations $x^{\alpha_{i}}=0$ (resp. $x^{-\alpha_{i}}=0$).
Proposition 3.7.
The collection
$(X(A_{n+1})\!\to\!X(A_{n}),s_{-},s_{+},s_{1},\ldots,s_{n+1})$
is an $A_{n}$-curve over $X(A_{n})$.
Proof.
The morphism $X(A_{n+1})\to X(A_{n})$ is proper.
To show flatness, consider the covering by affine toric charts:
for any set of simple roots $S$ of the root system $A_{n}$
there are $n+2$ affine spaces $U_{S_{j}}\cong\mathds{A}^{n+1}$ lying over
$U_{S}\cong\mathds{A}^{n}$ corresponding to $n+2$ sets of simple roots
$S_{j}$ of the root system $A_{n+1}$ such that the submonoid
$\langle S_{j}\rangle$ of the root lattice $M(A_{n+1})$ generated
by $S_{j}$ contains $S$. For example, if
$S=\{u_{1}-u_{2},\ldots,u_{n}-u_{n+1}\}$ we have
$S_{j}=\{\ldots,u_{j-1}-u_{j},u_{j}-u_{n+2},u_{n+2}-u_{j+1},u_{j+1}-u_{j+2},\ldots\}$ for $j=0,\ldots,n+1$
(in particular $S_{0}=\{u_{n+2}-u_{1},u_{1}-u_{2},\ldots\}$ and
$S_{n+1}=\{\ldots,u_{n}-u_{n+1},u_{n+1}-u_{n+2}\}$).
The maps $U_{S_{j}}\to U_{S}$ are flat morphisms $\mathds{A}^{n+1}\to\mathds{A}^{n}$.
Indeed, they are given by the identity on a polynomial ring in $n-1$
variables tensored with a map of the form
$\mathds{Z}[z]\to\mathds{Z}[x,y]$, $z\mapsto xy$ or $z\mapsto y$.
By what is shown below, the fibres are $A_{n}$-curves:
Remark 3.10 describes the universal
$A_{n}$-curve in terms of equations given by $A_{n}$-data, and
Proposition 3.12 shows that such equations
define an $A_{n}$-curve once it is known that any $A_{n}$-data over
a field arises as in Proposition 3.12 from an
$A_{n}$-curve. This is shown in Lemma 3.13.
∎
Definition 3.8.
We call the object $(X(A_{n+1})\!\to\!X(A_{n}),s_{-},s_{+},s_{1},\ldots,s_{n+1})$
of Construction 3.6 the universal $A_{n}$-curve over
$X(A_{n})$.
Example 3.9.
The universal curve $C_{2}$ over $\overline{L}_{2}$ is pictured as
Figure 3 and later seen to coincide with $X(A_{2})$ over $X(A_{1})$.
Here, in Figure 4, we draw the corresponding inclusion of root systems
(left) and the map of fans $\Sigma(A_{2})\to\Sigma(A_{1})$ (right).
We apply the embedding into a product of projective lines
$$\textstyle X(R)\to\prod_{A_{1}\cong R^{\prime}\subseteq R}\mathds{P}^{1}_{R^{%
\prime}}=:P(R)$$
considered in Subsections 1.2 and
1.3 to the situation of Construction
3.6.
Remark 3.10.
Consider $X(A_{n+1})$ (resp. $X(A_{n})$) as embedded into
the product of projective lines $P(A_{n+1})$ (resp. $P(A_{n})$).
Then the morphism $X(A_{n+1})\to X(A_{n})$ is induced by the projection
onto the subproduct $P(A_{n+1})\to P(A_{n})$.
The subvarieties $X(A_{n+1})\subseteq P(A_{n+1})$ (resp.
$X(A_{n})\subseteq P(A_{n})$) are determined by the homogeneous equations
$z_{\alpha}z_{\beta}z_{-\gamma}=z_{-\alpha}z_{-\beta}z_{\gamma}$
for roots $\alpha,\beta,\gamma$ such that $\alpha+\beta=\gamma$,
i.e. root subsystems of type $A_{2}$ in $A_{n+1}$ (resp. $A_{n}$).
If we first consider the product $P(A_{n+1})$ and the equations coming
from root subsystems of type $A_{2}$ in $A_{n}$, this gives
$$\textstyle P(A_{n+1}/A_{n})_{X(A_{n})}=\big{(}\prod_{A_{1}\cong R\subseteq A_{%
n+1}\setminus A_{n}}\mathds{P}^{1}_{R}\big{)}_{X(A_{n})}=\big{(}\prod_{i=1}^{n%
+1}\mathds{P}^{1}_{\{\pm\alpha_{i}\}}\big{)}_{X(A_{n})}\,.$$
Therein, $X(A_{n+1})$ is the closed subscheme given by
the homogeneous equations
(1)
$$t_{\beta_{ij}}z_{\alpha_{j}}z_{-\alpha_{i}}=t_{-\beta_{ij}}z_{-\alpha_{j}}z_{%
\alpha_{i}},\quad i,j\in\{1,\ldots,n+1\},i\neq j,\;\beta_{ij}=\alpha_{i}-%
\alpha_{j}$$
where $t_{\pm\beta_{ij}}$ are the homogeneous coordinates of
$\mathds{P}^{1}_{\{\pm\beta_{ij}\}}$ (consider $X(A_{n})$ as embedded into
$\prod_{R}\mathds{P}^{1}_{R}$) or equivalently the two generating
sections of the line bundle $\mathscr{L}_{\{\pm\beta_{ij}\}}$ being
part of the universal $A_{n}$-data on $X(A_{n})$.
The sections $s_{i}\colon X(A_{n})\to X(A_{n+1})$ for $i\in\{1,\ldots,n+1\}$
are given by the additional equations $z_{\alpha_{i}}=z_{-\alpha_{i}}$.
The sections $s_{-}$ (resp. $s_{+}$) are given by $z_{\alpha_{1}}=\;\cdots\;=z_{\alpha_{n+1}}=0$ (resp. $z_{-\alpha_{1}}=\;\cdots\;=z_{-\alpha_{n+1}}=0$).
Example 3.11.
The universal $A_{1}$-curve $X(A_{2})\subset(\mathds{P}^{1}_{\{\pm\alpha_{1}\}}\times\mathds{P}^{1}_{\{\pm%
\alpha_{2}\}})_{X(A_{1})}$
over $X(A_{1})$ is given by the homogeneous equation
$$t_{\beta_{12}}z_{\alpha_{2}}z_{-\alpha_{1}}=t_{-\beta_{12}}z_{-\alpha_{2}}z_{%
\alpha_{1}}$$
where $\beta_{12}=u_{1}-u_{2}$ and
$(\mathscr{L}_{\{\pm\beta_{12}\}},\{t_{\beta_{12}},t_{-\beta_{12}}\})$
is the universal $A_{1}$-data on $X(A_{1})\cong\mathds{P}^{1}$.
We picture in Figure 5 the $A_{1}$-curves in $\mathds{P}^{1}\times\mathds{P}^{1}$ defined
by this equation over the points given by
$(t_{\beta_{12}}:t_{-\beta_{12}})=(0:1),(a:b),(1:0)$. In the
second case, we write the marked points in terms of the homogeneous
coordinates $(z_{-\alpha_{2}}:z_{\alpha_{2}})$.
By Remark 3.10 the universal $A_{n}$-curve
over $X(A_{n})$ can be embedded into a product $P(A_{n+1}/A_{n})_{X(A_{n})}\cong(\mathds{P}^{1})^{n+1}_{X(A_{n})}$ and the embedded curve is given by
equations (1) determined by the universal
$A_{n}$-data.
We show that any $A_{n}$-curve $C$ over a field can be embedded into
a product $(\mathds{P}^{1})^{n+1}$ and extract $A_{n}$-data such that $C$
is described by the same equations as the universal curve.
Proposition 3.12.
Let $(C,s_{-},s_{+},s_{1},\ldots,s_{n+1})$ be an $A_{n}$-curve
over a field. For $i\in\{1,\ldots,n+1\}$ let $z_{\alpha_{i}},z_{-\alpha_{i}}$
be a basis of $H^{0}(C,\mathcal{O}_{C}(s_{i}))$ such that $z_{\alpha_{i}}(s_{-})=0$,
$z_{-\alpha_{i}}(s_{+})=0$, $z_{\alpha_{i}}(s_{i})=z_{-\alpha_{i}}(s_{i})\neq 0$.
We will write $\mathds{P}_{\{\pm\alpha_{i}\}}$ for $\mathds{P}(H^{0}(C,\mathcal{O}_{C}(s_{i})))$.
Then by
$$(t_{\beta_{ij}}:t_{-\beta_{ij}})=(z_{-\alpha_{j}}(s_{i}):z_{\alpha_{j}}(s_{i}))$$
for $\beta_{ij}=\alpha_{i}-\alpha_{j}$, $i\neq j$, we can define $A_{n}$-data
$(t_{\beta_{ij}}:t_{-\beta_{ij}})_{\{\pm\beta_{ij}\}\subseteq A_{n}}$,
and the morphism
$$\textstyle C\;\to\;\prod_{i=1}^{n+1}\mathds{P}^{1}_{\{\pm\alpha_{i}\}}=P(A_{n+%
1}/A_{n})$$
is an isomorphism onto the closed subvariety $C^{\prime}\subseteq P(A_{n+1}/A_{n})$
determined by the homogeneous equations
(2)
$$t_{\beta_{ij}}z_{\alpha_{j}}z_{-\alpha_{i}}=t_{-\beta_{ij}}z_{-\alpha_{j}}z_{%
\alpha_{i}},\quad i,j\in\{1,\ldots,n+1\},i\neq j,\;\beta_{ij}=\alpha_{i}-%
\alpha_{j}\,.$$
Furthermore, $C^{\prime}$ together with the marked points $s_{i}^{\prime}$
defined by the additional equations $z_{\alpha_{i}}=z_{-\alpha_{i}}$
and $s_{-}^{\prime}$ (resp. $s_{+}^{\prime}$) defined by $z_{\alpha_{i}}=0$ for
$i=1,\ldots,n+1$ (resp. $z_{-\alpha_{i}}=0$ for $i=1,\ldots,n+1$)
is an $A_{n}$-curve and we have an isomorphism of $A_{n}$-curves
$(C,s_{-},s_{+},s_{1},\ldots,$ $s_{n+1})\to(C^{\prime},s_{-}^{\prime},s_{+}^{\prime},s_{1}^{\prime},\ldots,s_{%
n+1}^{\prime})$.
Proof.
The data $(t_{\beta_{ij}}:t_{-\beta_{ij}})$ is defined as the position
of a marked point $s_{i}$ relative to another marked point $s_{j}$
of $C$. Note that the two ways to define $(t_{\beta_{ij}}:t_{-\beta_{ij}})$
are equivalent because $(z_{\alpha_{i}}(s_{j}):z_{-\alpha_{i}}(s_{j}))=(z_{-\alpha_{j}}(s_{i}):z_{%
\alpha_{j}}(s_{i}))$.
The data $(t_{\beta_{ij}}:t_{-\beta_{ij}})$ relates the bases
$z_{\alpha_{i}},z_{-\alpha_{i}}$ and $z_{\alpha_{j}},z_{-\alpha_{j}}$
via the equation
$t_{\beta_{ij}}z_{\alpha_{j}}z_{-\alpha_{i}}$
$=t_{-\beta_{ij}}z_{-\alpha_{j}}z_{\alpha_{i}}$.
If the corresponding marked points are not contained in the
same component of $C$, both sides of the equation are zero.
Otherwise, both are homogeneous coordinates of the same component
of $C$ and we have the formula
$(t_{\beta_{ij}}z_{-\alpha_{i}}:t_{-\beta_{ij}}z_{\alpha_{i}})=(z_{-\alpha_{j}}%
:z_{\alpha_{j}})$ which can be checked at the two poles
and the marked point $s_{i}$.
It follows that the morphism $C\to\mathds{P}(A_{n+1}/A_{n})$ maps $C$ to the
subscheme $C^{\prime}$ defined by the equations in (2).
The collection
$(t_{\beta_{ij}}:t_{-\beta_{ij}})_{\{\pm\beta_{ij}\}\subseteq A_{n}}$
is $A_{n}$-data: we have to show that the equations
$t_{\beta_{ij}}t_{\beta_{jk}}t_{-\beta_{ik}}=t_{-\beta_{ij}}t_{-\beta_{jk}}t_{%
\beta_{ik}}$
are satisfied.
If $s_{i},s_{j},s_{k}$ are not contained in the same component, then
both sides are zero. If $s_{i},s_{j},s_{k}$ are in the
same component, then $(t_{\beta_{ik}}:t_{-\beta_{ik}})=(z_{-\alpha_{k}}(s_{i}):z_{\alpha_{k}}(s_{i})%
)=(t_{\beta_{jk}}z_{-\alpha_{j}}(s_{i}):t_{-\beta_{jk}}z_{\alpha_{j}}(s_{i}))=%
(t_{\beta_{jk}}t_{\beta_{ij}}:t_{-\beta_{jk}}t_{-\beta_{ij}})$
making use of $(z_{-\alpha_{k}}:z_{\alpha_{k}})=(t_{\beta_{jk}}z_{-\alpha_{j}}:t_{-\beta_{jk}%
}z_{\alpha_{j}})$.
We show that $C\to C^{\prime}\subseteq P(A_{n+1}/A_{n})$ is an isomorphism.
The curve $C$ decomposes into irreducible components
$C=C_{1}\cup\ldots\cup C_{m}$. Let $\{1,\ldots,n+1\}=P_{1}\sqcup\ldots\sqcup P_{m}$ be the combinatorial type of $C$ (see Definition
3.3).
For $k\in\{1,\ldots,m\}$ the morphism $C_{k}\to\prod_{i\in P_{k}}\mathds{P}^{1}_{\{\pm\alpha_{i}\}}=:P(P_{k})$ is an isomorphism onto
$C^{\prime\prime}_{k}\subseteq P(P_{k})$ given by the equations in (2)
involving only coordinates $z_{\pm\alpha_{i}}$ for $i\in P_{k}$.
The equations in (2) involving coordinates
for roots $\alpha_{i},\alpha_{j}$ for $i\in P_{k}$, $j\in P_{k^{\prime}}$,
$k\neq k^{\prime}$, are of the form $z_{\alpha_{j}}z_{-\alpha_{i}}=0$
if $k<k^{\prime}$ and $z_{-\alpha_{j}}z_{\alpha_{i}}=0$ if $k^{\prime}<k$.
So, the equations in (2) containing coordinates
for some $i\in P_{k}$ define a subvariety of
$P(A_{n+1}/A_{n})=\prod_{i\in P_{j},j<k}\mathds{P}^{1}_{\{\pm\alpha_{i}\}}%
\times P(P_{k})\times\prod_{i\in P_{j},k<j}\mathds{P}^{1}_{\{\pm\alpha_{i}\}}$
consisting of the irreducible components
$C^{\prime}_{k}=\prod_{i\in P_{k^{\prime}},k^{\prime}<k}\{z_{-\alpha_{i}}\!=\!0%
\}\times C^{\prime\prime}_{k}\times\prod_{i\in P_{k^{\prime}},k<k^{\prime}}\{z%
_{\alpha_{i}}\!=\!0\}$,
$\prod_{i\in P_{j},j<k}\mathds{P}^{1}_{\{\pm\alpha_{s}\}}\times\prod_{i\in P_{j%
},k\leq j}\{z_{\alpha_{i}}=0\}$,
$\prod_{i\in P_{j},j\leq k}\{z_{-\alpha_{i}}=0\}\times\prod_{i\in P_{j},k<j}%
\mathds{P}^{1}_{\{\pm\alpha_{s}\}}$.
For each $k$ we have an isomorphism $C_{k}\to C^{\prime}_{k}$, and these
form the isomorphism $C\to C^{\prime}$.
The sections of $C^{\prime}$ were defined such that the isomorphism
$C\to C^{\prime}$ is an isomorphism of $A_{n}$-curves.
∎
Lemma 3.13.
Any $A_{n}$-data over a field arises as $A_{n}$-data extracted from
an $A_{n}$-curve by the method of Proposition 3.12.
Proof.
Let
$(t_{\beta_{ij}}:t_{-\beta_{ij}})_{\{\pm\beta_{ij}\}\subseteq A_{n}}$
be $A_{n}$-data over a field.
We define an ordering $\prec$ on the set $\{1,\ldots,n+1\}$.
For $i\neq j$ define $i\prec j$ (resp. $i\preceq j$) if
$(t_{\beta_{ij}}:t_{-\beta_{ij}})=(1:0)$ (resp.
$(t_{\beta_{ij}}:t_{-\beta_{ij}})\neq(0:1)$) where
$\beta_{ij}=\alpha_{i}-\alpha_{j}$.
This ordering defines a decomposition
$\{1,\ldots,n+1\}=P_{1}\sqcup\cdots\sqcup P_{m}$
into nonempty equivalence classes such that $i\prec j$ if and only if
$i\in P_{k},\,j\in P_{k^{\prime}}$ for $k<k^{\prime}$.
We construct an $A_{n}$-curve such that the $A_{n}$-data extracted from
it by the method of Proposition 3.12 is the
given $A_{n}$-data.
Take a chain of projective lines $(C,s_{-},s_{+})$ of length $m$.
For each component $C_{k}$ we can choose for each $i\in P_{k}$ a point
$s_{i}\in C_{k}$ different from the poles $p_{k}^{\pm}$, such that their relative
positions are given by the data $(t_{\beta_{ij}}:t_{-\beta_{ij}})$,
i.e. $s_{i}=(t_{\beta_{ij}}:t_{-\beta_{ij}})$ with respect to coordinates
of $C_{k}$ such that $p_{k}^{-}=(1:0)$, $p_{k}^{+}=(0:1)$, $s_{j}=(1:1)$.
This is possible, the compatibility is assured by the conditions
of $A_{n}$-data.
∎
Considering the universal $A_{n}$-curve, the combinatorial types of
the geometric fibres determine a stratification of $X(A_{n})$ which
coincides with the stratification of this toric variety into torus
orbits.
Proposition 3.14.
Over the torus orbit in $X(A_{n})$ corresponding to the one-dimensional
cone generated by $v_{i_{1}}+\cdots+v_{i_{k}}$ we have the combinatorial type
$$s_{i_{n+1}}\ldots s_{i_{k+1}}|s_{i_{k}}\ldots s_{i_{1}}\,.$$
Proof.
The universal $A_{n}$-data over each point of the closure of the orbit
corresponding to a generator of a one-dimensional cone generated by
$v$ has the property $(t_{\beta}:t_{-\beta})=(0:1)$ if
$\langle\beta,v\rangle>0$ (see Remark 1.21).
For $v=v_{i_{1}}+\cdots+v_{i_{k}}$ this in particular implies
$(t_{\beta_{ij}}:t_{-\beta_{ij}})=(0:1)$ if $i\in\{i_{1},\ldots,i_{k}\}$
and $j\in\{i_{k+1},\ldots,i_{n+1}\}$.
We obtain for points in this torus orbit the above combinatorial type.
∎
The proposition describes the combinatorial types over the torus
orbits in $X(A_{n})$ of codimension one. The combinatorial type over a
lower-dimensional torus orbit is given by the partition that arises
as common refinement of the partitions for the torus orbits of
codimension one that contain the respective orbit in their closure.
3.3. Isomorphism between the functor of $X(A_{n})$
and the moduli functor
We show that the moduli functor of $A_{n}$-curves is isomorphic to
the functor of $X(A_{n})$ of Subsection 1.3.
This implies that the toric variety $X(A_{n})$ is a fine
moduli space of $A_{n}$-curves and hence coincides with
the Losev-Manin moduli space $\overline{L}_{n+1}$.
To relate $A_{n}$-curves to $A_{n}$-data, we consider an embedding
of arbitrary $A_{n}$-curves over a scheme $Y$ into a product
$(\mathds{P}^{1})_{Y}^{n+1}$ that generalises the embedding in Proposition
3.12 to the relative situation.
The main tools are the following contraction morphisms.
The technical details are standard, compare to similar
constructions in [Kn83], [GHP88].
Construction 3.15.
Let $(\pi\colon C\to Y,s_{-},s_{+},s_{1},\ldots,s_{n+1})$
be an $A_{n}$-curve over $Y$. Each subset of sections
$\{s_{i_{1}},\ldots,s_{i_{k}}\}\subseteq\{s_{1},\ldots,s_{n}\}$ defines
a line bundle $\mathcal{O}_{C}(s_{i_{1}}+\cdots+s_{i_{k}})$ on $C$ and a locally
free sheaf $\pi_{*}\mathcal{O}_{C}(s_{i_{1}}+\cdots+s_{i_{k}})$ of rank $k+1$ on $Y$.
The natural surjective homomorphism $\pi^{*}\pi_{*}\mathcal{O}_{C}(s_{i_{1}}+\cdots+s_{i_{k}})\to\mathcal{O}_{C}(s_%
{i_{1}}+\cdots+s_{i_{k}})$ induces a morphism
$p_{\{i_{1},\ldots,i_{k}\}}\colon C\to C_{\{i_{1},\ldots,i_{k}\}}\subseteq%
\mathds{P}_{Y}(\pi_{*}\mathscr{\mathcal{O}}_{C}(s_{i_{1}}+\cdots+s_{i_{k}}))$.
This construction commutes with base change. In particular,
for each fibre $C_{y}$, $y\in Y$, the morphism
$(p_{\{i_{1},\ldots,i_{k}\}})_{y}\colon C_{y}\to\mathds{P}_{Y}(\pi_{*}\mathscr{%
\mathcal{O}}_{C}(s_{i_{1}}+\cdots+s_{i_{k}}))_{y}$ arises by
applying the above construction to the line bundle
$\mathcal{O}_{C_{y}}(s_{i_{1}}(y)+\cdots+s_{i_{k}}(y))$ on $C_{y}$; it is an
isomorphism on the components containing at least one of the
marked points $s_{i_{j}}(y)$ and contracts all other components.
The image of $p_{\{i_{1},\ldots,i_{k}\}}$ with the sections
$p_{\{i_{1},\ldots,i_{k}\}}\circ s_{\pm}$, $p_{\{i_{1},\ldots,i_{k}\}}\circ s_{i_{j}}$ is an $A_{k-1}$-curve.
These morphisms are functorial with respect to multiple
inclusions of sets of sections.
We will make use of the following particular cases:
(1) Contraction with respect to one section onto a $\mathds{P}^{1}$-bundle.
For each section $s_{i}$ of $C$ there is a contraction morphism
$p_{i}\colon C\to C_{i}=\mathds{P}_{Y}(\pi_{*}\mathscr{\mathcal{O}}_{C}(s_{i}))$.
Since the $\mathds{P}^{1}$-bundle $\mathds{P}_{Y}(\pi_{*}\mathscr{\mathcal{O}}_{C}(s_{i}))$ has
three disjoint sections $p_{i}\circ s_{-},p_{i}\circ s_{+},p_{i}\circ s_{i}\colon Y\to\mathds{P}_{Y}(%
\pi_{*}\mathscr{\mathcal{O}}_{C}(s_{i}))$, there is an isomorphism
$\mathds{P}_{Y}(\pi_{*}\mathscr{\mathcal{O}}_{C}(s_{i}))\cong\mathds{P}^{1}_{Y}$ that identifies
$p_{i}\circ s_{-},p_{i}\circ s_{+},p_{i}\circ s_{i}$ with the
$(1:0),(0:1),(1:1)$-section of $\mathds{P}^{1}_{Y}$ (see Figure 6).
(2) Contraction with respect to two sections onto an $A_{1}$-curve.
Let $s_{i_{1}},s_{i_{2}}$ be two of the sections of $C$. Then there is
a contraction morphism $p_{\{i_{1},i_{2}\}}\colon C\to C_{\{i_{1},i_{2}\}}$
and $C_{\{i_{1},i_{2}\}}$ is an $A_{1}$-curve over $Y$.
The curve $C_{\{i_{1},i_{2}\}}$ contains the information about the
relative positions of the sections $s_{i_{1}},s_{i_{2}}$ in $C$.
This data relates the two contraction morphisms with respect to
the sections $s_{i_{1}}$ and $s_{i_{2}}$.
Construction 3.16.
Let $(C\to Y,s_{-},s_{+},s_{1},s_{2})$ be an $A_{1}$-curve. Then we
have the morphism $p_{2}\colon C\to\mathds{P}^{1}_{Y}$ for the section
$s_{2}$ such that, with respect to homogeneous coordinates
$z_{0}^{(2)},z_{1}^{(2)}$, the sections $s_{-},s_{+},s_{2}$ become the
$(1:0),(0:1),(1:1)$-section of $\mathds{P}^{1}_{Y}$. The section $s_{1}$
determines $(t_{1,2}:t_{2,1}):=p_{2}\circ s_{1}$. This can be
rewritten as a line bundle with two generating sections
$(\mathscr{L}_{\{1,2\}},\{t_{1,2},t_{2,1}\})$ up to isomorphism.
Equivalently, we can consider the morphism $p_{1}\colon C\to\mathds{P}^{1}_{Y}$
for the section $s_{1}$. Then $s_{-},s_{+},s_{1}$ become the
$(1:0),(0:1),(1:1)$-section with respect to homogeneous coordinates
$z_{0}^{(1)},z_{1}^{(1)}$ and $p_{1}\circ s_{2}=(t_{2,1}:t_{1,2})$.
Lemma 3.17.
The morphism $p_{1}\times p_{2}\colon C\to\mathds{P}^{1}_{Y}\times\mathds{P}^{1}_{Y}$ maps
$C$ to the closed subvariety given by the homogeneous equation
$$t_{1,2}z_{1}^{(2)}z_{0}^{(1)}=t_{2,1}z_{0}^{(2)}z_{1}^{(1)}\,.$$
Proof.
It suffices to show the lemma for the strata of $Y$ corresponding
to the three possible combinatorial types of $A_{1}$-curves.
On the strata corresponding to reducible curves the above
equation becomes $z_{1}^{(2)}z_{0}^{(1)}=0$ (resp. $z_{0}^{(2)}z_{1}^{(1)}=0$). These equations are satisfied,
e.g. in the first case $z_{1}^{(2)}=0$ on the component
containing $s_{-},s_{1}$ and $z_{0}^{(1)}=0$ on the component
containing $s_{+},s_{2}$.
On the remaining stratum $Y^{\prime}\subseteq Y$ both $(z_{1}^{(1)}:z_{0}^{(1)})$
and $(z_{1}^{(1)}:z_{0}^{(1)})$ are homogeneous coordinates of $\mathds{P}^{1}_{Y^{\prime}}$
and related by $(t_{1,2}z_{1}^{(2)}:t_{2,1}z_{0}^{(2)})=(z_{1}^{(1)}:z_{0}^{(1)})$.
∎
(3) Contraction with respect to three sections onto an $A_{2}$-curve.
Let $s_{i_{1}},s_{i_{2}},s_{i_{3}}$ be three of the sections of $C$. Then
there is a contraction morphism $p_{\{i_{1},i_{2},i_{3}\}}\colon C\to C_{\{i_{1},i_{2},i_{3}\}}$ and $C_{\{i_{1},i_{2},i_{3}\}}$ is an $A_{2}$-curve
over $Y$. This curve contains the information about the relative
positions of the pairs of two sections in a set of three sections
$s_{i_{1}},s_{i_{2}},s_{i_{3}}$ of $C$. These data are related by one equation:
Lemma 3.18.
Let $(C,s_{-},s_{+},s_{1},s_{2},s_{3})$ be an $A_{2}$-curve over $Y$.
Then the collection of $A_{1}$-data
$\{(\mathscr{L}_{\{1,2\}},\{t_{1,2},t_{2,1}\}),(\mathscr{L}_{\{2,3\}},\{t_{2,3}%
,t_{3,2}\}),(\mathscr{L}_{\{3,1\}},\{t_{3,1},t_{1,3}\})\}$
extracted by the method of Construction 3.16
from the $A_{1}$-curves $C_{\{i_{1},i_{2}\}},C_{\{i_{2},i_{3}\}},C_{\{i_{3},i_{1}\}}$
is $A_{2}$-data, i.e. the sections satisfy the equation
$$t_{1,2}t_{2,3}t_{3,1}=t_{2,1}t_{3,2}t_{1,3}$$
in $\mathscr{L}_{\{1,2\}}\otimes\mathscr{L}_{\{2,3\}}\otimes\mathscr{L}_{\{3,1\}}$.
Proof.
Again, this equation is satisfied on a closed subset and it suffices
to consider the situation for the strata corresponding to the
different combinatorial types.
If the sections are not contained in the same component, then
both sides vanish.
Over the remaining stratum $Y^{\prime}\subseteq Y$ we have a bundle $\mathds{P}^{1}_{Y^{\prime}}$.
Using the formula
$(t_{i,j}z_{1}^{(j)}:t_{j,i}z_{0}^{(j)})=(z_{1}^{(i)}:z_{0}^{(i)})$
for the homogeneous coordinates we obtain
$(z_{1}^{(1)}:z_{0}^{(1)})=(t_{1,2}z_{1}^{(2)}:t_{2,1}z_{0}^{(2)})=(t_{1,2}t_{2%
,3}z_{1}^{(3)}:t_{2,1}t_{3,2}z_{0}^{(3)})=(t_{1,2}t_{2,3}t_{3,1}z_{1}^{(1)}:t_%
{2,1}t_{3,2}t_{1,3}z_{0}^{(1)})$,
so $(t_{1,2}t_{2,3}t_{3,1}:t_{2,1}t_{3,2}t_{1,3})=(1:1)$.
∎
For an $A_{n}$-curve $(C\to Y,s_{-},s_{+},s_{1},\ldots,s_{n})$ we have
the contraction morphisms $p_{i}\colon C\to\mathds{P}_{Y}(\pi_{*}\mathscr{\mathcal{O}}_{C}(s_{i}))\cong(%
\mathds{P}^{1}_{\{\pm\alpha_{i}\}})_{Y}$ where on $(\mathds{P}^{1}_{\{\pm\alpha_{i}\}})_{Y}$
we have homogeneous coordinates $z_{-\alpha_{i}},z_{\alpha_{i}}$
such that, in these coordinates, $s_{-},s_{+},s_{i}$ become the
$(1:0),(0:1),(1:1)$-section of $\mathds{P}^{1}_{Y}$. We have the roots
$\alpha_{i}=u_{i}-u_{n+2}$ and $\beta_{ij}=\alpha_{i}-\alpha_{j}=u_{i}-u_{j}$.
Theorem 3.19.
There is an isomorphism between the functor $F_{A_{n}}$ and the
moduli functor of $A_{n}$-curves $\overline{L}_{n+1}$
such that the universal $A_{n}$-data on $X(A_{n})$ is mapped to
the universal $A_{n}$-curve over $X(A_{n})$.
Proof.
Let $Y$ be a scheme. For $A_{n}$-data on $Y$ we construct an
$A_{n}$-curve $C$ over $Y$ via equations in $P(A_{n+1}/A_{n})_{Y}=\prod_{i=1}^{n+1}(\mathds{P}^{1}_{\{\pm\alpha_{i}\}})_{Y}$ as in Remark
3.10 with the given $A_{n}$-data on $Y$
replacing the universal $A_{n}$-data on $X(A_{n})$.
This is an $A_{n}$-curve since all $A_{n}$-data are pull-back of the
universal $A_{n}$-data on $X(A_{n})$ and so the constructed curve
is a pull-back of the universal $A_{n}$-curve over $X(A_{n})$.
In the other direction, given an $A_{n}$-curve on $Y$ we
extract $A_{n}$-data. For each pair of distinct sections
$s_{i},s_{j}$ we have a contraction morphism
$C\to C_{\{i,j\}}$ onto an $A_{1}$-curve over $Y$.
From $(C_{\{i,j\}},s_{i},s_{j})$ we extract $A_{1}$-data
$(\mathscr{L}_{\{\pm\beta_{ij}\}},t_{\beta_{ij}},t_{-\beta_{ij}}):=(\mathscr{L}%
_{\{i,j\}},t_{i,j},t_{j,i})$
as in Construction 3.16.
That the collection of all these data forms $A_{n}$-data
follows from Lemma 3.18.
Both constructions commute with base-change and thus define
morphisms of functors $F_{A_{n}}\to\overline{L}_{n+1}$
and $\overline{L}_{n+1}\to F_{A_{n}}$.
We show that they are inverse to each other.
Starting with an $A_{n}$-curve $C$ over $Y$, we extract data
$(\mathscr{L}_{\{\pm\beta_{ij}\}},\{t_{-\beta_{ij}},t_{\beta_{ij}}\})_{\{\pm%
\beta_{ij}\}\subseteq A_{n}}$ and from these
$A_{n}$-data we construct an $A_{n}$-curve $C^{\prime}\subseteq P(A_{n+1}/A_{n})_{Y}$
as in Remark 3.10.
We show that the product of the contraction morphisms $p_{i}$
defines an isomorphism of $A_{n}$-curves $C\to C^{\prime}\subseteq P(A_{n+1}/A_{n})_{Y}$.
The morphism $\prod_{i}p_{i}\colon C\to P(A_{n+1}/A_{n})_{Y}$ factors through
the inclusion $C^{\prime}\subseteq P(A_{n+1}/A_{n})_{Y}$ by Lemma 3.17.
The morphism $C\to C^{\prime}$ is an isomorphism, because it is an isomorphism
on the fibres by Proposition 3.12 and the curves
are flat over $Y$.
The sections and the involution of $C^{\prime}$ were defined such that
$C\to C^{\prime}$ is an isomorphism of $A_{n}$-curves.
Starting with $A_{n}$-data we construct an $A_{n}$-curve and extract
$A_{n}$-data from it. It is easily verified that this $A_{n}$-data
coincides with the original $A_{n}$-data, since the contraction
morphism with respect to a section of an embedded $A_{n}$-curve
$C^{\prime}\subseteq P(A_{n+1}/A_{n})_{Y}$ is induced by projection
onto the corresponding factor.
∎
References
[1]
[AMRT]
A. Ash, D. Mumford, M. Rapoport and
Y. Tai, Smooth compactification of locally symmetric
varieties, Math. Sci. Press, Brookline, Mass., 1975.
[Ba91]
V. Batyrev, On the classification
of smooth projective toric varieties, Tohoku Math. J. 43
(1991), 569–585.
[Ba94]
V. Batyrev, Dual polyhedra and
mirror symmetry for Calabi-Yau hypersurfaces in toric varieties,
J. Algebraic Geom. 3 (1994), 493–535, arXiv:alg-geom/9310003.
[BB11]
V. Batyrev and M. Blume,
On generalisations of Losev-Manin moduli spaces for
classical root systems, Pure and Applied Mathematics Quarterly 7 (2011)
(Special Issue: In memory of Eckart Viehweg), 1053–1084,
arXiv:0912.2898.
[BK08]
S. Bloch and D. Kreimer, Mixed
Hodge structures and renormalization in physics,
Commun. Number Theory Phys. 2 (2008), 637–718.
[Bou]
N. Bourbaki, Groupes et algèbres
de Lie (Ch. 4-6), Hermann, Paris, 1968.
[BJ07]
M. Brion and R. Joshua, Equivariant
Chow ring and Chern classes of wonderful symmetric varieties
of minimal rank, Transform. Groups 13 (2008), 471–493.
[Co95]
D. Cox, The functor of a smooth
toric variety, Tohoku Math. J. 47 (1995), 251–262.
[CR08]
D. Cox and C. v. Renesse, Primitive
collections and toric varieties, Tohoku Math. J. 61 (2009), 309–332.
[Dan]
V. Danilov, The Geometry of toric
varieties, Russian Math. Surveys 33 (1978), 97–154.
[DL94]
I. Dolgachev and V. Lunts, A character
formula for the representation of a Weyl group in the cohomology
of the associated toric variety, J. Algebra 168 (1994), 741–772.
[FH]
W. Fulton and J. Harris, Representation theory,
Springer-Verlag, New York, Grad. Texts in Math. 129, 1991.
[Ful]
W. Fulton, Introduction to toric varieties,
Ann. of Math. Stud. 131, Princeton University Press, Princeton, NJ, 1993.
[GHP88]
L. Gerritzen, F. Herrlich and M. van der Put,
Stable $n$-pointed trees of projective lines,
Nederl. Akad. Wetensch., Indag. Math. 50 (1988), 131–163.
[Ha03]
B. Hassett, Moduli spaces of weighted
pointed stable curves, Adv. Math. 173 (2003), 316–352.
[Ka93]
M. Kapranov, Chow quotients of
Grassmannians I, Adv. Soviet Math. 16 (1993), 29–110.
[Kl85]
A. Klyachko, Orbits of a maximal torus
on a flag space, Functional Anal. Appl. 19 (1985), 65–66.
[Kl95]
A. Klyachko, Toric varieties and flag
spaces, Trudy Mat. Inst. Steklov. 208 (1995), Teor. Chisel,
Algebra i Algebr. Geom., 139–162.
[Kn83]
F. Knudsen, The projectivity of the
moduli space of stable curves II: The stacks $M_{g,n}$,
Math. Scand. 52 (1983), 161–199.
[LM00]
A. Losev and Yu. Manin,
New moduli spaces of pointed curves and pencils of
flat connections, Michigan Math. J. 48 (2000), 443–472.
[Pr90]
C. Procesi, The toric variety associated
to Weyl chambers, Mots, 153–161, Lang. Raison. Calc., Hermès,
Paris, 1990.
[SZ09]
S. Shadrin and D. Zvonkine, A group
action on Losev-Manin cohomological field theories, arXiv:0909.0800.
[St94]
J. Stembridge, Some permutation
representations of Weyl groups associated with the cohomology of
toric varieties, Adv. Math. 106 (1994), 244–301. |
Finite Chevalley groups and loop groups
Masaki Kameko
Department of Mathematics
Faculty of Contemporary Society
Toyama University of International Studies
65-1 Higashikuromaki
Toyama, 930-1292
Japan
kameko@tuins.ac.jp
Abstract.
Let $p$, $\ell$ be distinct primes and let $q$ be a power of $p$. Let $G$ be a connected compact Lie group.
We show that there exists an integer $b$ such that the mod $\ell$ cohomology of the classifying space of a finite Chevalley group $G(\mathbb{F}_{q})$ is isomorphic to the mod $\ell$ cohomology of the classifying space of the loop group $\mathcal{L}G$ for $q=p^{ab}$, $a\geq 1$.
1. Introduction
Let $p$, $\ell$ be distinct primes and let $q$ be a power of $p$. We denote by
$\mathbb{F}_{q}$ the finite field with $q$-elements.
Let $G$ be a connected compact Lie group.
There exists a reductive complex linear algebraic group $G(\mathbb{C})$ associated with $G$, called the complexification of $G$.
One may consider $G(\mathbb{C})$ as $\mathbb{C}$-rational points of a group scheme over $\mathbb{C}$ obtained by the base-change of a reductive integral affine group scheme $G_{\mathbb{Z}}$, so-called Chevalley group scheme, with the complex analytic topology.
For a field $k$, taking the $k$-rational points of the group scheme
$$G_{k}=G_{\mathbb{Z}}\times_{\mathrm{Spec}(\mathbb{Z})}\mathrm{Spec}(k)$$
over $k$, we have the (possibly infinite) Chevalley group
$$G(k)=\mathrm{Hom}_{\mathrm{Sch}/k}(\mathrm{Spec}(k),G_{k}),$$
where $\mathrm{Sch}/k$ is the category of schemes over $k$. We consider the Chevalley group $G(k)$ as a discrete group unless otherwise is clear from the context.
Denote by $\overline{\mathbb{F}}_{p}$ the algebraic closure of the finite field $\mathbb{F}_{q}$.
We may consider the finite Chevalley group $G(\mathbb{F}_{q})$ as the fixed point set $G(\overline{\mathbb{F}}_{p})^{\phi^{q}}$ where
$$\phi^{q}:G(\overline{\mathbb{F}}_{p})\to G(\overline{\mathbb{F}}_{p})$$
is the Frobenius map induced by the Frobenius homomorphism $\phi^{q}:\overline{\mathbb{F}}_{p}\to\overline{\mathbb{F}}_{p}$ sending $x$ to $x^{q}$.
In [9], Quillen computed the mod $\ell$ cohomology of a finite general linear group $GL_{n}(\mathbb{F}_{q})$. The finite general linear group $GL_{n}(\mathbb{F}_{q})$ is a finite Chevalley group associated with the unitary group $U(n)$.
We recall Quillen’s computation from the viewpoint of the the following Theorem 1.1 due to Friedlander [2, Theorem 12.2], Friedlander-Mislin [3, Theorem 1.4].
Throughout the rest of this paper, we fix a connected compact Lie group $G$ and associated reductive integral affine group scheme $G_{\mathbb{Z}}$.
Let $BG^{\wedge}$ be the Bousfield-Kan $\mathbb{Z}/\ell$-completion of the classifying space $BG$ of the connected compact Lie group $G$.
We write $H^{*}(X)$, $\tilde{H}^{*}(X)$ for the mod $\ell$ cohomology, reduced mod $\ell$ cohomology of a space $X$, respectively. We also write $H_{*}(X)$, $\tilde{H}_{*}(X)$ for the mod $\ell$ homology, reduced mod $\ell$ homology of $X$, respectively.
We denote by $\mathrm{fib}(\alpha)$, $\pi_{0}:P_{\alpha}\to X$ the homotopy fibre, mapping track of a map $\alpha:A\to X$.
That is,
$$P_{\alpha}=\{(a,\lambda)\in A\times X^{I}\;|\;\alpha(a)=\lambda(1)\},$$
$\pi_{0}((a,\lambda))=\lambda(0)$
and
$\mathrm{fib}(\alpha)=\pi_{0}^{-1}(*),$
where $I=[0,1]$ is the unit interval, $X^{I}$ is the set of continuous maps from $I$ to $X$, $*$ is the base-point of $X$.
Theorem 1.1 (Friedlander-Mislin).
There exist maps
$$D:BG(\overline{\mathbb{F}}_{p})\to BG^{\wedge}$$
and
$$\phi^{q}:BG^{\wedge}\to BG^{\wedge}$$
satisfying the following three conditions:
(1)
The induced homomorphism
$D^{*}:H^{*}(BG^{\wedge})\to H^{*}(BG(\overline{\mathbb{F}}_{p}))$
is an isomorphism.
(2)
$\phi^{q}\circ D=D\circ\phi^{q}$ where
$\phi^{q}:BG(\overline{\mathbb{F}}_{p})\to BG(\overline{\mathbb{F}}_{p})$
is the Frobenius map induced by the Frobenius homomorphism $\phi^{q}:\overline{\mathbb{F}}_{p}\to\overline{\mathbb{F}}_{p}$.
(3) There exists a map
$\mathrm{fib}(D_{q})\to\mathrm{fib}(\Delta)$
induces an isomorphism
$H^{*}(\mathrm{fib}(\Delta))\to H^{*}(\mathrm{fib}(D_{q})),$
where the above map is obtained from the following homotopy commutative diagram by choosing a suitable homotopy:
$$\diagram\node{BG(\mathbb{F}_{q})}\arrow{e}\arrow{s,l}{D_{q}}\node{BG^{\wedge}}%
\arrow{s,r}{\Delta}\\
\node{BG^{\wedge}}\arrow{e,t}{(1\times\phi^{q})\circ\Delta}\node{BG^{\wedge}%
\times BG^{\wedge},}$$
where $D_{q}=D\circ i_{q}$, $i_{q}=BG(\mathbb{F}_{q})\to BG(\overline{\mathbb{F}}_{p})$ is the map induced by the inclusion of $\mathbb{F}_{q}$ into $\overline{\mathbb{F}}_{p}$ and
$\Delta:BG^{\wedge}\to BG^{\wedge}\times BG^{\wedge}$ is the diagonal map.
Remark 1.1.
In [2, Proposition 8.8], Friedlander constructed a chain of maps between simplicial sets
$\displaystyle\mathop{\mathrm{holim}}_{\leftarrow}\thinspace(\mathbb{Z}/\ell)_{%
\infty}(BG_{\overline{\mathbb{F}}_{p}})_{\mathrm{et}}$ and $(\mathbb{Z}/\ell)_{\infty}\mathrm{Sing}_{\bullet}(BG(\mathbb{C}))$,
where $G(\mathbb{C})$ is given the complex analytic topology.
He showed that these maps are weak homotopy equivalences. We take $BG^{\wedge}$ to be the geometric realization of the simplicial set $\displaystyle\mathop{\mathrm{holim}}_{\leftarrow}\thinspace(\mathbb{Z}/\ell)_{%
\infty}(BG_{\overline{\mathbb{F}}_{p}})_{\mathrm{et}}$, so that the Forbenius map $\phi^{q}:BG^{\wedge}\to BG^{\wedge}$ is induced by the map defined on $G_{\overline{\mathbb{F}}_{p}}$. Therefore, the map $\phi^{q}$ is an automorphism and there holds
$$\underbrace{\phi^{q}\circ\dots\circ\phi^{q}}_{e-\text{times}}=\phi^{q^{e}}.$$
We emphasize here that the equality holds in the category of sets and maps, not in the homotopy category.
Remark 1.2.
For a discrete group $H$, we may identify the classifying space $BH$ with the geometric realization of $\displaystyle\mathop{\mathrm{holim}}_{\leftarrow}\thinspace(BH_{k})_{\mathrm{%
et}}$, where $k$ is an algebraically closed field and $H_{k}$ is a group scheme $H\otimes\mathrm{Spec}(k)$.
The map $D$ in Theorem 1.1 is induced by the obvious homomorphism of group schemes
$$G(k)_{k}=\mathrm{Hom}_{Sch/k}(\mathrm{Spec}(k),G_{k})\otimes\mathrm{Spec}(k)%
\to G_{k}.$$
See Friedlander-Mislin [3, Section 2] for detail.
Thus, we have the equality $\phi^{q}\circ D=D\circ\phi^{q}$ in Theorem 1.1.
Now, we recall Quillen’s computation of the mod $\ell$ cohomology of $GL_{n}(\mathbb{F}_{q})$.
The first part of Quillen’s computation is the homotopy theoretical interpretation of the problem.
For a map $f:\thinspace X\to X$, let us define a space $\mathcal{L}_{f}X$ by
$$\mathcal{L}_{f}X=\{\lambda\in X^{I}\;|\;\lambda(1)=f(\lambda(0))\}.$$
We call this space the twisted loop space of $f$ following the terminology of [5]. Let $\pi_{0}:\mathcal{L}_{f}X\to X$ be the evaluation map at $0$.
Let $\pi_{0}:P_{\Delta}\to X\times X$ be the mapping track of the diagonal map $\Delta:X\to X\times X$.
Associated with the diagram in Theorem 1.1 (3), we have the following fibre square:
$$\diagram\node{\mathcal{L}_{f}X}\arrow{e,t}{g}\arrow{s,r}{\pi_{0}}\node{P_{%
\Delta}}\arrow{s,r}{\pi_{0}}\\
\node{X}\arrow{e,t}{(1\times f)\circ\Delta}\node{X\times X,}$$
where $g$ is given by
$$g(\lambda)=(\lambda(0),\lambda^{\prime})\in X\times(X\times X)^{I},$$
and $\displaystyle\lambda^{\prime}(t)=(\lambda(\frac{t}{2}),\lambda(1-\frac{t}{2}))$.
Theorem 1.1 (3) implies that $H^{*}(\mathcal{L}_{f}X)$ is isomorphic to $H^{*}(BG(\mathbb{F}_{q}))$ for $X=BG^{\wedge}$, $f=\phi^{q}$.
Thus, the computation of the mod $\ell$ cohomology of a finite Chevalley group is nothing but the computation of the mod $\ell$ cohomology of the twisted loop space $\mathcal{L}_{f}X$.
The second part of Quillen’s computation is the computation using the Eilenberg-Moore spectral sequence.
For a twisted loop space, there exists the Eilenberg-Moore spectral sequence converging to the associated graded algebra of the mod $\ell$ cohomology of the twisted loop space $\mathcal{L}_{f}X.$
Let us write $A$ for $H^{*}(X)$.
The $E_{2}$-term of the Eilenberg-Moore spectral sequence is given by
$\mathrm{Tor}_{A\otimes A}(A,A).$
If the induced homomorphism
${f}^{*}:A\to A$
is the identity homomorphism and if $A$ is a polynomial algebra, then the above $E_{2}$-term is a polynomial tensor exterior algebra $A\otimes V$ where $V=\mathrm{Tor}_{A}(\mathbb{Z}/\ell,\mathbb{Z}/\ell)$,
and since as an algebra over $\mathbb{Z}/\ell$, it is generated by $\mathrm{Tor}^{0}_{A\otimes A}(A,A)$
and $\mathrm{Tor}^{-1}_{A\otimes A}(A,A)$, the spectral sequences collapses at the $E_{2}$-level.
On the other hand, it is well-known that there exists a homotopy equivalence between the classifying space of the loop group $\mathcal{L}G$ and the free loop space $\mathcal{L}BG$, where
$$\mathcal{L}X=\{\lambda\in X^{I}\;|\;\lambda(1)=\lambda(0)\}.$$
See Proposition 2.4 in [1].
For the free loop space $\mathcal{L}BG$,
we have the following fibre square:
$$\diagram\node{\mathcal{L}X}\arrow{e}\arrow{s,l}{\pi_{0}}\node{P_{\Delta}}%
\arrow{s,r}{\pi_{0}}\\
\node{X}\arrow{e,t}{\Delta}\node{X\times X,}$$
where $\pi_{0}$ is the evaluation map at $0$, so that $\pi_{0}(\lambda)=\lambda(0)$, and $\pi_{0}:P_{\Delta}\to X\times X$ is the mapping track of the diagonal map $\Delta:X\to X\times X$.
As in the case of finite Chevalley groups, there exists the Eilenberg-Moore spectral sequence
$$\mathrm{Tor}_{A\otimes A}(A,A)\Rightarrow\mathrm{gr}\thinspace H^{*}(\mathcal{%
L}BG).$$
Thus, it is easy to see that if $A=H^{*}(BG)$ is a polynomial algebra, the $E_{2}$-term of the spectral sequence is equal to the polynomial tensor exterior algebra $A\otimes V=A\otimes\mathrm{Tor}_{A}(\mathbb{Z}/\ell,\mathbb{Z}/\ell)$, the spectral sequence collapses at the $E_{2}$-level as in the case of finite Chevalley groups.
Therefore, if $H^{*}(BG)$ is a polynomial algebra,
the mod $\ell$ cohomology of the free loop space of the classifying space $BG$ is isomorphic to the mod $\ell$ cohomology of the finite Chevalley group $G(\mathbb{F}_{q})$ as a graded $\mathbb{Z}/\ell$-module.
Even if $H^{*}(BG)$ is not a polynomial algebra over $\mathbb{Z}/\ell$, if the induced homomorphism ${\phi^{q}}^{*}:A\to A$ is the identity homomorphism, $E_{2}$-terms of the above Eilenberg-Moore spectral sequences are the same.
Observing this phenomenon, Tezuka in [10] asked the following:
Conjecture 1.1.
If $\ell|q-1$ (resp. $4|q-1$) when $\ell$ is odd (resp. even),
there exists a ring isomorphism
between $H^{*}(BG(\mathbb{F}_{q}))$ and $H^{*}(\mathcal{L}BG)$.
In conjunction with this conjecture, in this paper, we prove the following result:
Theorem 1.2.
There exists an integer $b$ such that, for $q=p^{ab}$ where $a$ is an arbitrary positive integer, there exists an
isomorphism of graded $\mathbb{Z}/\ell$-modules
$$H^{*}(BG(\mathbb{F}_{q}))=H^{*}(\mathcal{L}BG).$$
Remark 1.3.
Although we give an example of the integer $b$ in Theorem 1.2 in Section 2 as a function of the graded $\mathbb{Z}/\ell$-module
$H^{*}(G)$, it is not at all the best possible.
Remark 1.4.
If $H^{*}(BG)$ is not a polynomial algebra, it is not easy to compute the mod $\ell$ cohomology of $BG(\mathbb{F}_{q})$ and $\mathcal{L}BG$. The only computational results in the literature are the computation of the mod 2 cohomology of $B\mathrm{Spin}_{10}(\mathbb{F}_{q})$ and $\mathcal{L}B\mathrm{Spin}(10)$ in [6] and [7] for $\ell=2$ and
the mod $3$ cohomology of $\mathcal{L}BPU(3)$ for $\ell=3$ in [7].
When we want to show that the cohomology of a space $X$ is isomorphic to the cohomology of another space $Y$, we usually try to construct a chain of maps
$$X=X_{0}\stackrel{{\scriptstyle f_{0}}}{{\longleftarrow}}X_{1}\stackrel{{%
\scriptstyle f_{1}}}{{\longrightarrow}}X_{2}\longleftarrow\dots\longleftarrow X%
_{n}\stackrel{{\scriptstyle f_{n}}}{{\longrightarrow}}X_{n+1}=Y$$
such that maps $f_{k}$’s induce isomorphisms in mod $\ell$ cohomology.
For example, Theorem 1.1 is proved by this method.
However, when we try to prove Theorem 1.2 or Conjecture 1.1, we can not construct such a chain of maps.
Consider the case $G=S^{1}$.
Then, we have $G(\mathbb{F}_{q})=\mathbb{Z}/(q-1)$, $G(\mathbb{C})=\mathbb{C}\backslash\{0\}$.
So, we have $H^{*}(BG(\mathbb{F}_{q});\mathbb{Q})=\mathbb{Q}$, $H^{*}(\mathcal{L}BG;\mathbb{Q})=\mathbb{Q}[y]\otimes\Lambda(x)$.
If there exists a chain of maps such as above, then they also induce isomorphisms of Bockstein spectral sequences. This contradicts the above observation on the rational (and integral) cohomology
of $BG(\mathbb{F}_{q})$ and $\mathcal{L}BG$.
Thus, in the proof of Theorem 1.2, we construct maps which induce monomorphisms among Leray-Serre spectral sequences associated with fibrations
$\pi_{0}:\mathcal{L}_{f}X\to X$, $\pi_{0}:\mathcal{L}X\to X$, $\pi_{0}:\mathcal{L}_{f}X\times_{X}P_{\alpha}\to X$, where $X=BG^{\wedge}$, $f=\phi^{q}$ and $\alpha:A\to X$ is a certain map we define in Section 3.
By
comparing the image of Leray-Serre spectral sequences, we construct an isomorphism between
Leray-Serre spectral sequences associated with fibrations $\pi_{0}:\mathcal{L}_{f}X\to X$ and $\pi_{0}:\mathcal{L}X\to X$. This isomorphism could not be realized by a chain of maps.
We announced and outlined the proof of Theorem 1.2 in [4]. By choosing $\phi^{q}$ and $D$ as in Remarks 1.1, 1.2, in this paper, we can give a simpler proof for Theorem 1.2.
In Section 2, we define the integer $b$ as a function of a graded $\mathbb{Z}/\ell$-module $H^{*}(G)$.
In Section 3, we give a proof of Theorem 1.2 assuming Lemma 3.2.
In Section 4, we prove Lemma 3.2.
Since there exists no map realizing the
isomorphism between $H^{*}(BG(\mathbb{F}_{q}))$ and $H^{*}(\mathcal{L}BG)$, it is difficult to believe the existence of such isomorphism for arbitrary connected compact Lie group $G$.
It is my pleasure to thank M. Tezuka for informing me of Conjecture 1.1.
The author is partially supported by Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (C) 19540105.
2. The integer $b$
In this section, we define the integer $b$ in Theorem 1.2.
We define the integer $b$ as
$$b={e_{1}}^{\dim G}{e_{2}}$$
and we define $e_{1}$, $e_{2}$ as follows:
For the sake of notational simplicity, let $V=H^{*}(G)$. We have isomorphisms
$$V=H^{*}_{\mathrm{et}}(G_{\overline{\mathbb{F}}_{p}},\mathbb{Z}/\ell)=H^{*}(%
\mathrm{fib}(D_{q}))=H^{*}(\mathrm{fib}(\Delta))=H^{*}(\Omega BG^{\wedge}).$$
Denote by $GL(V)$ be the group of automorphisms of $V$ and we also denote by $|GL(V)|$ the order of the finite group $GL(V)$.
Let
$$\displaystyle e_{1}$$
$$\displaystyle=(\ell|GL(V)|)^{2\dim G}\quad\text{and}$$
$$\displaystyle e_{2}$$
$$\displaystyle=|GL(V\otimes V)|.$$
Before we proceed to lemmas, we set up some notations. Let us consider a commutative diagram.
$$\diagram\node{A}\arrow{e,t}{g}\arrow{s,l}{\alpha}\node{B}\arrow{s,r}{\beta}\\
\node{X}\arrow{e,t}{f}\node{Y.}$$
We write $f:X^{I}\to Y^{I}$ for the map induced by $f:X\to Y$, so that
$$f(\lambda)(t)=f(\lambda(t)).$$
We also write $g\times f$ for the map $A\times X^{I}\to B\times Y^{I}$ induced by $f$, $g$ and the restriction of $g\times f:\thinspace A\times X^{I}\to B\times Y^{I}$ to the mapping tracks $P_{\alpha}\to P_{\beta}$ and the homotopy fibres $\mathrm{fib}(\alpha)\to\mathrm{fib}(\beta)$.
Let $q^{\prime}=q^{e}$ and $q$ is a power of $p$.
The inclusion of $\mathbb{F}_{q}$ into $\mathbb{F}_{q^{\prime}}$ induces maps
$$i:\thinspace BG(\mathbb{F}_{q})\to BG(\mathbb{F}_{q^{\prime}})$$
and
$$i\times 1:\thinspace\mathrm{fib}(D_{q})\longrightarrow\mathrm{fib}(D_{q^{%
\prime}}).$$
As for $e_{1}$, we have the following lemma. A variant of this lemma is used in the proof of Theorem 1.4 in [3].
Lemma 2.1.
Suppose that $e$ is divisible by $e_{1}$.
Then, the induced homomorphism
$$(i\times 1)^{*}:\tilde{H}^{*}(\mathrm{fib}(D_{q^{\prime}}))\to\tilde{H}^{*}(%
\mathrm{fib}(D_{q}))$$
is zero.
Proof.
In general,
we have
$$\Delta_{*}({(i\times 1)}_{*}(x))=1\otimes{(i\times 1)}_{*}(x)+\sum{(i\times 1)%
}_{*}(y^{\prime})\otimes{(i\times 1)}_{*}(y^{\prime\prime})+{(i\times 1)}_{*}(%
x)\otimes 1,$$
where $\deg y^{\prime}<\deg x$ or $\deg y^{\prime\prime}<\deg x$.
Hence, if ${(i\times 1)}_{*}(y)=0$ for $\deg y<\deg x$, then we have
$$\Delta_{*}({(i\times 1)}_{*}(x))=1\otimes{(i\times 1)}_{*}(x)+{(i\times 1)}_{*%
}(x)\otimes 1.$$
So, if $(i\times 1)_{*}(y)=0$ for $\deg y<\deg x$, ${(i\times 1)}_{*}(x)$ is primitive.
The Frobenius map $\phi^{q}$ is an element of the Galois group
$\mathrm{Gal}(\overline{\mathbb{F}}_{p}/\mathbb{F}_{p})$, the induced homomorphism
${\phi^{q}}^{*}:H_{\mathrm{et}}^{*}(G_{\overline{\mathbb{F}}_{p}},\mathbb{Z}/%
\ell)\to H^{*}_{\mathrm{et}}(G_{\overline{\mathbb{F}}_{p}},\mathbb{Z}/\ell)$
is an automorphism in $GL(V)$.
Suppose that $q^{\prime}=q^{e}$ and $e$ is divisible by $\ell\cdot m$ where $m$ is the order of
$${\phi^{q}}^{*}:V\to V$$
as an element in $GL(V)$. Consider the induced homomorphism
$${(i\times 1)}_{*}:\thinspace\tilde{H}_{*}(\mathrm{fib}(D_{q}))\to\tilde{H}_{*}%
(\mathrm{fib}(D_{q^{\prime}})).$$
The isomorphism between $H_{\mathrm{et}}^{*}(G_{\overline{\mathbb{F}}_{p}},\mathbb{Z}/\ell)$ and $H^{*}(\mathrm{fib}(D_{q}))$ is given by
the Lang map $(1/\phi^{q}):G({\overline{\mathbb{F}}_{p}})/G(\mathbb{F}_{q})\to G({\overline{%
\mathbb{F}}_{p}})$ defined by $(1/\phi^{q})(g)=g\cdot(\phi^{q}(g))^{-1}$.
Thus, the map ${(i\times 1)}_{*}$ corresponds to a homomorphism $\theta^{q^{\prime}/q}:G(\overline{\mathbb{F}}_{p})\to G(\overline{\mathbb{F}}_%
{p})$
given by the diagram
$$\diagram\node{G({\overline{\mathbb{F}}_{p}})/G(\mathbb{F}_{q})}\arrow{e,t}{\pi%
}\arrow{s,l}{1/\phi^{q}}\node{G({\overline{\mathbb{F}}_{p}})/G(\mathbb{F}_{q^{%
\prime}})}\arrow{s,r}{1/\phi^{q^{\prime}}}\\
\node{G({\overline{\mathbb{F}}_{p}})}\arrow{e,t}{\theta^{q^{\prime}/q}}\node{G%
({\overline{\mathbb{F}}_{p}}),}$$
where $\pi$ is the obvious projection.
In other words, $\theta^{q^{\prime}/q}$ is given by
$$\theta^{q^{\prime}/q}(g)=g\cdot\phi^{q}(g)\cdot\dots\cdot(\phi^{q})^{e-1}(g).$$
Thus, ${(i\times 1)}_{*}(x)$ is given by
$${(i\times 1)}_{*}(x)=(\mu\circ(1\times\phi^{q}\times\dots\times\phi^{q^{e-1}})%
\circ\Delta)_{*}(x)$$
If $x$ is primitive, we have
$$\displaystyle{(i\times 1)}_{*}(x)$$
$$\displaystyle=\sum_{t=0}^{e-1}(\phi^{q})_{*}^{t}(x)$$
$$\displaystyle=(e/m)\cdot\left(\sum_{t=0}^{m-1}(\phi^{q})_{*}^{t}(x)\right)$$
$$\displaystyle=0.$$
Thus, if $e$ is divisible by $(\ell\cdot m)^{2}$ and if ${(i\times 1)}_{*}(y)=0$ for $\deg y<j$, then ${(i\times 1)}_{*}(x)=0$ for $\deg x\leq j$.
Therefore, if $G$ is connected and if $e$ is divisible by $(\ell\cdot m)^{2k}$, the induced homomorphism
$$\tilde{H}^{*}(\mathrm{fib}(D_{q^{\prime}}))\to\tilde{H}^{*}(\mathrm{fib}(D_{q}))$$
is zero up to degree $k$. Let $k=\dim G$. Since, by definition, $H^{j}(\mathrm{fib}(D_{q}))=\{0\}$ for $j>k$, we have Lemma 2.1.
∎
As for the integer $e_{2}$, we prove the following:
Lemma 2.2.
Suppose that $e$ is divisible by $e_{2}$.
Then, the induced homomorphism
$$(1\times(1\times\phi^{q^{\prime}}))^{*}:H^{*}(\mathrm{fib}(\Delta\circ D_{q}))%
\to H^{*}(\mathrm{fib}(\Delta\circ D_{q}))$$
is the identity homomorphism.
Proof.
Let $p_{i}:BG^{\wedge}\times BG^{\wedge}\to BG^{\wedge}$ be the projections onto the first and second factors for $i=1,2$.
Consider the diagram
$$\diagram\node{\Omega BG^{\wedge}}\arrow{e,t}{=}\arrow{s}\node{\Omega BG^{%
\wedge}}\arrow{e,t}{\Omega\Delta}\arrow{s}\node{\Omega(BG^{\wedge}\times BG^{%
\wedge})}\arrow{s}\\
\node{\mathrm{fib}(\Delta\circ D_{q})}\arrow{e,t}{1\times{p_{1}}}\arrow{s,l}{1%
\times{p_{2}}}\node{\mathrm{fib}(D_{q})}\arrow{e,t}{D_{q}\times\Delta}\arrow{s%
,l}{p_{1}}\node{\mathrm{fib}(\Delta)}\arrow{s,r}{p_{1}}\\
\node{\mathrm{fib}(D_{q})}\arrow{e,t}{p_{1}}\node{BG(\mathbb{F}_{q})}\arrow{e,%
t}{D_{q}}\node{BG^{\wedge}}$$
The induced homomorphism
$$(\Omega\Delta)_{*}:\thinspace H_{*}(\Omega BG^{\wedge}){\longrightarrow}H_{*}(%
\Omega(BG^{\wedge}\times BG^{\wedge}))$$
is a monomorphism and $\pi_{1}(BG^{\wedge})=\{0\}$. So, $\pi_{1}(BG(\mathbb{F}_{q}))$ acts trivially on
$H_{*}(\Omega BG^{\wedge})$. Therefore, $\pi_{1}(\mathrm{fib}(D_{q}))$ also acts trivially on $H_{*}(\Omega BG^{\wedge})$. Thus, the local coefficient of the induced fibre sequence
$$\Omega BG^{\wedge}\to\mathrm{fib}(\Delta\circ D_{q})\stackrel{{\scriptstyle 1%
\times{p_{2}}}}{{\longrightarrow}}\mathrm{fib}(D_{q})$$
is trivial.
Hence, the $E_{2}$-term of the Leray-Serre spectral sequence for the cohomology of $\mathrm{fib}(\Delta\circ D_{q})$ is
given by
$$V\otimes V=H^{*}(\mathrm{fib}(D_{q}))\otimes H^{*}(\Omega BG^{\wedge}).$$
Therefore, we have that
$$\dim_{\mathbb{Z}/\ell}H^{*}(\mathrm{fib}(\Delta\circ D_{q}))\leq\dim_{\mathbb{%
Z}/\ell}(V\otimes V).$$
Since the map $\phi^{q}:BG^{\wedge}\to BG^{\wedge}$ is an automorphism,
the induced map
$$1\times(1\times\phi^{q}):\mathrm{fib}(\Delta\circ D_{q})\to\mathrm{fib}(\Delta%
\circ D_{q})$$
is also an automorphism. Since
$$\dim_{\mathbb{Z}/\ell}H^{*}(\mathrm{fib}(\Delta\circ D_{q}))\leq\dim_{\mathbb{%
Z}/\ell}(V\otimes V),$$
$e_{2}$ is divisible by the order of
$$(1\times(1\times\phi^{q}))^{*}:H^{*}(\mathrm{fib}(\Delta\circ D_{q}))\to H^{*}%
(\mathrm{fib}(\Delta\circ D_{q})).$$
Hence, if $e$ is divisible by $e_{2}$, we have
$$(1\times(1\times\phi^{q^{\prime}}))^{*}=((1\times(1\times\phi^{q}))^{*})^{e}=1.\qed$$
3. Proof of Theorem 1.2
Let $X$ be a space and let $f:X\to X$ be a self-map of $X$ with a non-empty fixed point set.
Let $\alpha:A\to X$ be a map such that
$$f\circ\alpha=\alpha.$$
We choose a base-point $*$ in $A$, $X$, so that both $f$, $\alpha$ are base-point preserving.
Firstly, we define a map
$$\varphi:\mathcal{L}_{f}X\times_{X}\mathcal{L}_{f}X\to\mathcal{L}X,$$
where
$$\mathcal{L}_{f}X\times_{X}\mathcal{L}_{f}X=\{(\lambda_{1},\lambda_{2})\in%
\mathcal{L}_{f}X\times\mathcal{L}_{f}X\;|\;\lambda_{1}(0)=\lambda_{2}(0)\}.$$
The map $\varphi$ is defined by
$$\varphi(\lambda_{1},\lambda_{2})(t)=\begin{cases}\lambda_{1}(2t)&\text{for $%
\displaystyle 0\leq t\leq\frac{1}{2}$,}\\
\lambda_{2}(2-2t)&\text{for $\displaystyle\frac{1}{2}\leq t\leq 1$.}\end{cases}$$
Since $\lambda_{1}(1)=f(\lambda_{1}(0))$, $\lambda_{2}(1)=f(\lambda_{2}(0))$ and $\lambda_{1}(0)=\lambda_{2}(0)$, this map is well-defined.
Next, we define a map from $P_{\alpha}$ to $\mathcal{L}_{f}X$, say $\psi:P_{\alpha}\to\mathcal{L}_{f}X$, by
$$\psi((a,\lambda))(t)=\begin{cases}\lambda(2t)&\text{for $\displaystyle 0\leq t%
\leq\frac{1}{2}$,}\\
f(\lambda(2-2t))&\text{for $\displaystyle\frac{1}{2}\leq t\leq 1$.}\end{cases}$$
Since $\lambda(1)=f(\lambda(1))$, this map is also well-defined.
Now, we consider the following diagram:
$$\diagram\node{\mathcal{L}_{f}X}\node{\mathcal{L}_{f}X\times_{X}\mathcal{L}_{f}%
X}\arrow{w,t}{p_{1}}\arrow{e,t}{\varphi}\node{\mathcal{L}X}\\
\node[2]{\mathcal{L}_{f}X\times_{X}P_{\alpha}.}\arrow{n,r}{1\times\psi}$$
where
$$\mathcal{L}_{f}X\times_{X}P_{\alpha}=\{(\lambda_{1},(a,\lambda_{2}))\in%
\mathcal{L}_{f}X\times P_{\alpha}\;|\;\lambda_{1}(0)=\lambda_{2}(0),\;\alpha(a%
)=\lambda_{2}(1)\},$$
$p_{1}$ is the projection onto the first factor and $\pi_{0}\circ p_{1}=\pi_{0}$, $\pi_{0}\circ\varphi=\pi_{0}$, $\pi_{0}\circ(1\times\psi)=\pi_{0}$.
Let us denote by $E_{r}(Y)$ the Leray-Serre spectral sequence associated with a fibration $\xi:Y\to X$. Then we have the following diagram of spectral sequences:
$$\diagram\node{E_{r}(\mathcal{L}_{f}X)}\arrow{e,t}{p_{1}^{*}}\node{E_{r}(%
\mathcal{L}_{f}X\times_{X}\mathcal{L}_{f}X)}\arrow{s,r}{1\times\psi^{*}}\node{%
E_{r}(\mathcal{L}X)}\arrow{w,t}{\varphi^{*}}\\
\node[2]{E_{r}(\mathcal{L}_{f}X\times_{X}P_{\alpha}).}$$
By abuse of notation, we denote by $\psi:\thinspace\mathrm{fib}(\alpha)\to\Omega X$ the restriction of $\psi:\thinspace P_{\alpha}\to\mathcal{L}_{f}X$ to fibres.
Let us consider a sufficient condition for the induced homomorphism
$$\psi^{*}:\tilde{H}^{*}(\Omega X)\to\tilde{H}^{*}(\mathrm{fib}(\alpha))$$
to be zero.
Again, by abuse of notation, we denote by
$\varphi:\Omega X\times\Omega X\to\Omega X$
the restriction of $\varphi:\mathcal{L}_{f}X\times_{X}\mathcal{L}_{f}X\to\mathcal{L}X$ to fibres.
Lemma 3.1.
If the induced homomorphism
$$(1\times(1\times f))^{*}:H^{*}(\mathrm{fib}(\Delta\circ\alpha))\to H^{*}(%
\mathrm{fib}(\Delta\circ\alpha))$$
is the identity homomorphism, then the induced homomorphism
$$\psi^{*}:\tilde{H}^{*}(\Omega X)\to\tilde{H}^{*}(\mathrm{fib}(\alpha))$$
is zero.
Proof.
The map $\psi:\mathrm{fib}(\alpha)\to\Omega X$ factors through
$$\mathrm{fib}(\alpha)\stackrel{{\scriptstyle 1\times\Delta}}{{\longrightarrow}}%
\mathrm{fib}(\Delta\circ\alpha)\stackrel{{\scriptstyle 1\times(1\times f)}}{{%
\longrightarrow}}\mathrm{fib}(\Delta\circ\alpha)\stackrel{{\scriptstyle\varphi%
}}{{\longrightarrow}}\Omega X.$$
It is clear that the composition $\varphi\circ\Delta$ is null homotopic since an obvious null homotopy $h_{s}$ is given by
$$h_{s}((a,\lambda))(t)=\begin{cases}\lambda(2st)&\text{for $\displaystyle 0\leq
t%
\leq\frac{1}{2}$,}\\
\lambda(2s-2st)&\text{for $\displaystyle\frac{1}{2}\leq t\leq 1$.}\end{cases}$$
Therefore, we have
$$\displaystyle(1\times\Delta)^{*}((1\times(1\times f))^{*}(\varphi^{*}(x)))$$
$$\displaystyle=(1\times\Delta)^{*}(\varphi^{*}(x))$$
$$\displaystyle=0.$$
for $x\in\tilde{H}^{*}(\Omega X)$.
∎
We also need the following lemmas in the proof of Theorem 1.2.
Lemma 3.2.
Suppose that $X$ is simply connected, that $H^{i}(\mathrm{fib}(\alpha))=0$ for $i>k$ and
that there exists a sequence of maps
$$\diagram\node{A_{0}}\arrow{e,t}{i_{0}}\node{A_{1}}\arrow{e,t}{i_{1}}\node{A_{2%
}}\arrow{e}\node{\dots}\arrow{e}\node{A_{k}}\arrow{e,t}{i_{k}}\node{A}\arrow{e%
,t}{\alpha}\node{X}$$
such that the induced homomorphism $\tilde{H}^{*}(\mathrm{fib}(\alpha_{j}))\to\tilde{H}^{*}(\mathrm{fib}(\alpha_{j%
-1}))$
is zero for $j=1,2,\dots,k$.
Then the projection on the first factor $p_{1}:Y\times_{X}P_{\alpha}\to Y$
induces a monomorphism $p_{1}^{*}:\thinspace E_{r}(Y)\to E_{r}(Y\times_{X}P_{\alpha})$ of Leray-Serre spectral sequences
for arbitrary fibration $\xi:Y\to X$.
We need the following lemma to compare the spectral sequences.
Lemma 3.3.
Let
$$E_{r}^{\prime}\stackrel{{\scriptstyle\rho_{r}^{\prime}}}{{\longrightarrow}}E_{%
r}\stackrel{{\scriptstyle\rho^{\prime\prime}_{r}}}{{\longleftarrow}}E_{r}^{%
\prime\prime}$$
be homomorphisms of spectral sequences.
Suppose that
(1)
$\mathrm{Im}\,\rho_{2}^{\prime}=\mathrm{Im}\,\rho_{2}^{\prime\prime}$,
(2)
$\rho_{r}^{\prime}$ is a monomorphism for $r\geq 2$.
Then, there exists a unique homomorphism of spectral sequences
$$\{\tau_{r}:E_{r}^{\prime\prime}\to E_{r}^{\prime}\;|\;r\geq 2\}$$
such that $\rho^{\prime}_{r}\circ\tau_{r}=\rho_{r}^{\prime\prime}$ for $r\geq 2$. In particular, if $\rho^{\prime\prime}_{2}$ is also a monomorphism, then $\rho^{\prime\prime}_{r}$ is a monomorphism and $\tau_{r}$ is an isomorphism for $r\geq 2$.
Proof.
We define $\tau_{2}(x^{\prime\prime})$ by
$$\rho^{\prime}_{2}(\tau_{2}(x^{\prime\prime}))=\rho_{2}^{\prime\prime}(x^{%
\prime\prime}).$$
Since $\mathrm{Im}\thinspace\rho_{2}^{\prime}=\mathrm{Im}\thinspace\rho_{2}^{\prime\prime}$ and $\rho_{2}^{\prime}$ is a monomorphism,
it is well-defined and we have
$$\rho_{2}^{\prime}\circ\tau_{2}=\rho_{2}^{\prime\prime}$$
at the $E_{2}$-level.
Suppose that we have
$$\rho_{r}^{\prime}\circ\tau_{r}=\rho_{r}^{\prime\prime}.$$
Then, we want to show that
$$d_{r}^{\prime}(\tau_{r}(x^{\prime\prime}))=\tau_{r}(d_{r}^{\prime\prime}(x^{%
\prime\prime})).$$
Since $\rho_{r}^{\prime}$ is a monomorphism, it suffices to show that
$$\rho^{\prime}_{r}(d_{r}^{\prime}(\tau_{r}(x^{\prime\prime})))=\rho^{\prime}_{r%
}(\tau_{r}(d_{r}^{\prime\prime}(x^{\prime\prime}))).$$
It is easily verified as follows:
$$\displaystyle\rho^{\prime}_{r}(d_{r}^{\prime}(\tau_{r}(x^{\prime\prime})))$$
$$\displaystyle=d_{r}(\rho^{\prime}_{r}(\tau_{r}(x^{\prime\prime})))$$
$$\displaystyle=d_{r}(\rho^{\prime\prime}_{r}(x^{\prime\prime}))$$
$$\displaystyle=\rho^{\prime\prime}_{r}(d_{r}^{\prime\prime}(x^{\prime\prime}))$$
$$\displaystyle=\rho_{r}^{\prime}(\tau_{r}(d_{r}^{\prime\prime}(x^{\prime\prime}%
))).$$
Then, $\tau_{r}$ induces a homomorphism
$$\tau_{r+1}:\thinspace E^{\prime\prime}_{r+1}\to E^{\prime}_{r+1}$$
such that
$$\rho^{\prime}_{r+1}\circ\tau_{r+1}=\rho^{\prime\prime}_{r+1}.$$
Continue this process, we have a homomorphism of spectral sequence
$$\tau_{r}:\thinspace E_{r}^{\prime\prime}\to E_{r}^{\prime}$$
for $r\geq 2$. It is clear that if $\rho_{2}^{\prime\prime}$ is a monomorphism, then $\tau_{2}$ is an isomorphism.
It is also clear from the construction that $\rho_{r}^{\prime\prime}$ is a monomorphism for $r\geq 2$ and $\tau_{r}$ is an isomorphism for $r\geq 2$.
∎
Now, we prove Theorem 1.2 assuming Lemma 3.2.
Proof of Theorem 1.2.
Let $k=\dim G$.
Let $q_{j}=p^{e_{1}^{j}}$ for $j=0,\dots,k$.
Let $q=p^{ab}={q_{k}}^{ae_{2}}$ ($a\geq 1$).
Let $X=BG^{\wedge}$, $A=BG(\mathbb{F}_{q_{k}})$, $\alpha=D_{q_{k}}$, $f=\phi^{q}$,
$A_{j}=BG(\mathbb{F}_{q_{j}})$ and $\alpha_{i}=D_{q_{j}}$ for $j=0,1,\dots,k$.
In order to prove Theorem 1.2, we consider the Leray-Serre spectral sequence $E_{r}(\mathcal{L}_{f}X)$, $E_{r}(\mathcal{L}X)$ and establish an isomorphism of spectral sequences $\tau:\thinspace E_{r}(\mathcal{L}X)\to E_{r}(\mathcal{L}_{f}X)$.
By Lemma 2.1, we have that the induced homomorphism
$$\tilde{H}^{*}(\mathrm{fib}(\alpha_{j}))\to\tilde{H}^{*}(\mathrm{fib}(\alpha_{j%
-1}))$$
is zero for $j=1,\dots,k$.
By Lemma 3.2, we have
a monomorphism
$$(1\times\psi)^{*}\circ p_{1}^{*}:E_{r}(\mathcal{L}_{f}X)\longrightarrow E_{r}(%
\mathcal{L}_{f}X\times_{X}P_{\alpha}).$$
The fibres of fibrations $\pi_{0}:\mathcal{L}_{f}X\to X$, $\pi_{0}:\mathcal{L}X\to X$, $\pi_{0}:\mathcal{L}_{f}X\times_{X}P_{\alpha}\to X$ are $\Omega X$, $\Omega X$, $\Omega X\times\mathrm{fib}(\alpha)$, respectively.
Identifying the $E_{2}$-terms $E_{2}(\mathcal{L}_{f}X)$, $E_{2}(\mathcal{L}X)$, $E_{2}(\mathcal{L}_{f}X\times_{X}P_{\alpha})$ of Leray-Serre spectral sequences with $H^{*}(X)\otimes H^{*}(\Omega X)$, $H^{*}(X)\otimes H^{*}(\Omega X)$, $H^{*}(X)\otimes H^{*}(\Omega X)\otimes H^{*}(\mathrm{fib}(\alpha))$, respectively, we have
$$\displaystyle(1\times\psi)^{*}(p_{1}^{*}(x\otimes y))$$
$$\displaystyle=x\otimes y\otimes 1\quad\text{and}$$
$$\displaystyle(1\times\psi)^{*}(\varphi^{*}(x\otimes y))$$
$$\displaystyle=\sum x\otimes y^{\prime}\otimes\psi^{*}(\chi(y^{\prime\prime})),$$
where $\varphi^{*}(y)=\sum y^{\prime}\otimes\chi(y^{\prime\prime})$ and $\chi:H^{*}(\Omega X)\to H^{*}(\Omega X)$ is the canonical anti-automorphism of connected Hopf algebra over $\mathbb{Z}/\ell$.
By the definition of $f$, we have
$f=(\phi^{q_{k}})^{ae_{2}}$ and $ae_{2}$ is divisible by $e_{2}$. So, by Lemma 2.2,
the induced homomorphism
$(1\times(1\times f))^{*}$ is the identity homomorphism.
By Lemma 3.1,
we obtain
$$\mathrm{Im}\,(1\times\psi)^{*}\circ p_{1}^{*}=\mathrm{Im}\,(1\times\psi)^{*}%
\circ\varphi^{*}$$
in the $E_{2}$-term $E_{2}(\mathcal{L}_{f}X\times_{X}P_{\alpha})=H^{*}(X)\otimes H^{*}(\Omega X)%
\otimes H^{*}(\mathrm{fib}(\alpha))$.
Therefore, using Lemma 3.3, we obtain an isomorphism between Leray-Serre spectral sequences $E_{r}(\mathcal{L}_{f}X)$ and $E_{r}(\mathcal{L}X)$.
∎
4. Proof of Lemma 3.2
In order to prove Lemma 3.2,
we need to recall the internal structure of Leray-Serre spectral sequence.
Let $\eta:\thinspace Z\to Y$, $\xi:\thinspace Y\to X$ be fibrations.
Without loss of generality, we may assume that $X$ is a $CW$ complex.
Suppose $X$ is a CW complex and denote its $n$-skelton by $X^{(n)}$. We denote $\xi^{-1}(X^{(n)})\subset Y$,
$\eta^{-1}(\xi^{-1}(X^{(n)}))$ by $F_{n}Y$, $F_{n}Z$, respectively and let $F_{n}Y=F_{n}Z=\emptyset$ for $n<0$.
For the sake of notational simplicity, we let
$$\displaystyle M_{m,n}(Y)$$
$$\displaystyle=H^{*}(F_{m}Y,F_{n}Y),$$
$$\displaystyle M_{m,n}(Z)$$
$$\displaystyle=H^{*}(F_{m}Z,F_{n}Z),$$
respectively.
We denote by $E_{r}(Y)$, $E_{r}(Z)$ the Leray-Serre spectral sequences associated with fibrations $\xi$, $\xi\circ\eta$, respectively.
Lemma 4.1.
Suppose that for $m\geq n\geq 0$, the induced homomorphism
$$\eta^{*}:\thinspace M_{m,n}(Y)\longrightarrow M_{m,n}(Z)$$
is a monomorphism. Then, the induced homomorphism
$$\eta^{*}:\thinspace E_{r}(Y)\to E_{r}(Z)$$
is also a monomorphism for $r\geq 2$.
Proof.
Let us consider the following diagram:
$$\diagram\node{H^{*}(Y,F_{s-r}Y)}\node[2]{M_{s+r,s}(Y)}\\
\node{H^{*}(Y,F_{s-1}Y)}\arrow{n,l}{j_{r}}\arrow{e,t}{i_{1}}\node{M_{s,s-1}(Y)%
}\arrow{ne,t}{\delta_{r}^{\prime}}\arrow{e,t}{\delta_{1}}\node{H^{*}(Y,F_{s}Y)%
}\arrow{n,r}{i_{r}}\\
\node{M_{s-1,s-r}(Y)}\arrow{n,l}{\delta_{r}}\arrow{ne,b}{\delta_{r}^{\prime%
\prime}}\node[2]{H^{*}(Y,F_{s+r}Y).}\arrow{n,r}{j_{r}}$$
Let
$Z_{r}(Y)=\mathrm{Ker}\thinspace\delta_{r}^{\prime}$ and
$B_{r}(Y)=\mathrm{Im}\thinspace\delta_{r}^{\prime\prime}.$
Then, there holds $E_{r}(Y)=Z_{r}(Y)/B_{r}(Y)$
for $r\geq 2$.
See standard text books, for instance, McCleary’s book [8], for detail.
We consider the same diagram and $Z_{r}(Z)$, $B_{r}(Z)$ for $Z$ in the same manner. Then, we have $E_{r}(Z)=Z_{r}(Z)/B_{r}(Z)$.
Thus,
in order to prove the injectivity of the induced homomorphism $\eta^{*}:E_{r}(Y)\to E_{r}(Z)$,
it suffices to show that
$${\eta}^{*}(Z_{r}(Y))\cap B_{r}(Z)={\eta}^{*}(B_{r}(Y)),$$
that is, if ${\eta}^{*}(x)\in\mathrm{Im}\thinspace\delta_{r}^{\prime\prime}$ in
$M_{s,s-1}(Z)$, then
$x\in\mathrm{Im}\thinspace\delta_{r}^{\prime\prime}$ in $M_{s,s-1}(Y)$.
We consider the following diagram:
$$\diagram\node{M_{s-1,s-r}(Z)}\arrow{e,t}{\delta_{r}^{\prime\prime}}\node{M_{s,%
s-1}(Z)}\arrow{e,t}{j_{r}^{\prime\prime}}\node{M_{s,s-r}(Z)}\\
\node{M_{s-1,s-r}(Y)}\arrow{n,r}{\eta^{*}}\arrow{e,t}{\delta_{r}^{\prime\prime%
}}\node{M_{s,s-1}(Y)}\arrow{n,r}{\eta^{*}}\arrow{e,t}{j_{r}^{\prime\prime}}%
\node{M_{s,s-r}(Y),}\arrow{n,r}{\eta^{*}}$$
where horizontal sequences are cohomology long exact sequences associated with triples $(F_{s}Z,F_{s-1}Z,F_{s-r}Z)$, $(F_{s}Y,F_{s-1}Y,F_{s-r}Y)$.
Suppose $\eta^{*}(x)\in\mathrm{Im}\thinspace\delta_{r}^{\prime\prime}$, then $j^{\prime\prime}_{r}(\eta^{*}(x))=0$. So, we have that $\eta^{*}(j_{r}^{\prime\prime}(x))=0$. Since $\eta^{*}$ is a monomorphism, we have $j^{\prime\prime}_{r}(x)=0$. Hence, we have $x\in\mathrm{Im}\thinspace\delta_{r}^{\prime\prime}$. This completes the proof.
∎
Now, we complete the proof of Lemma 3.2.
Recall that $P_{\alpha_{j}}\to X$ is the fibration with the fibre $\mathrm{fib}(\alpha_{j})$.
Let $Y_{j}=Y\times_{X}P_{\alpha_{j}}$ for $j=0,\dots,k$. There is a sequence of fibrations and fibre maps over $X$,
$$\diagram\node{Y_{0}}\arrow{e,t}{i^{\prime}_{0}}\node{Y_{1}}\arrow{e,t}{i^{%
\prime}_{1}}\node{\dots}\arrow{e,t}{i^{\prime}_{k-1}}\node{Y_{k}}\arrow{e,t}{i%
^{\prime}_{k}}\node{Y\times_{X}P_{\alpha}}\arrow{e,t}{p_{1}}\node{Y,}$$
where $i^{\prime}_{j}=1\times i_{j}\times 1$ for $j=0,\dots,k$.
We denote the projection onto the first factor from $Y_{j}$ to $Y$ by $\eta_{j}$.
The following diagram is a fibre square and the fibre of $\eta_{j}$ is also $\mathrm{fib}(\alpha_{j})$.
$$\diagram\node{Y_{j}}\arrow{s,l}{\eta_{j}}\arrow{e,t}{p_{2}}\node{P_{\alpha_{j}%
}}\arrow{s,r}{\pi_{0}}\\
\node{Y}\arrow{e,t}{\xi}\node{X.}$$
Proposition 4.1.
Let $Y^{\prime\prime}\subset Y^{\prime}\subset Y$ be subspaces of $Y$ and let $Y^{\prime\prime}_{j}=\eta_{j}^{-1}(Y^{\prime\prime})$, $Y^{\prime}_{j}=\eta_{j}^{-1}(Y^{\prime})$ be subspaces of $Y_{j}$.
Suppose that $X$ is simply connected and that $H^{i}(\mathrm{fib}(\alpha))=\{0\}$ for $i>k$.
The induced homomrphism
$$H^{*}(Y^{\prime},Y^{\prime\prime})\to H^{*}(Y^{\prime}\times_{X}P_{\alpha},Y^{%
\prime\prime}\times_{X}P_{\alpha})$$
is a monomorphism.
Proof.
We have relative fibrations
$$\mathrm{fib}(\alpha)\to(Y^{\prime}\times_{X}P_{\alpha},Y^{\prime\prime}\times_%
{X}P_{\alpha})\to(Y^{\prime},Y^{\prime\prime})\quad\text{and}\quad\mathrm{fib}%
(\alpha_{j})\to(Y^{\prime}_{j},Y^{\prime\prime}_{j})\to(Y^{\prime},Y^{\prime%
\prime})$$
for $j=0,\dots,k$.
There exist associated Leray-Serre spectral sequences
$$\quad E_{r}(Y^{\prime}\times_{X}P_{\alpha},Y^{\prime\prime}\times_{X}P_{\alpha%
})\quad\text{and}\quad E_{r}(Y_{j}^{\prime},Y^{\prime\prime}_{j}),$$
converging to
$$\mathrm{gr}\thinspace H^{*}(Y^{\prime}\times_{X}P_{\alpha},Y^{\prime\prime}%
\times_{X}P_{\alpha})\quad\text{and}\quad\mathrm{gr}\thinspace H^{*}(Y_{j}^{%
\prime},Y_{j}^{\prime\prime}),$$
for $j=0,\dots,k$, respectively.
The fundamental group $\pi_{1}(Y)$ acts on $H_{*}(\mathrm{fib}(\alpha_{j}))$. This action factors through $\pi_{1}(X)=\{0\}$. Therefore, the action of $\pi_{1}(Y)$ on $H_{*}(\mathrm{fib}(\alpha_{j}))$ is trivial. Hence, its action on $H^{*}(\mathrm{fib}(\alpha_{j}))$ is also trivial. Thus, the $E_{2}$-term of the Leray-Serre spectral sequence associated with
the relative fibration
$$\mathrm{fib}(\alpha_{j})\to(Y^{\prime}_{j},Y^{\prime\prime}_{j})\to(Y^{\prime}%
,Y^{\prime\prime}),$$
is
$$E_{2}(Y_{j}^{\prime},Y_{j}^{\prime\prime})=H^{*}(Y^{\prime},Y^{\prime\prime})%
\otimes H^{*}(\mathrm{fib}(\alpha_{j})).$$
Suppose that $d_{r_{k}}(y)=z$ in $E_{r_{k}}^{s,0}(Y^{\prime}\times_{X}P_{\alpha},Y^{\prime\prime}\times_{X}P_{%
\alpha})$ for some $y\in E_{r_{k}}^{s-r_{k},r_{k}-1}(Y^{\prime}\times_{X}P_{\alpha},Y^{\prime%
\prime}\times_{X}P_{\alpha})$.
Let $z_{k}={i^{\prime}_{k}}^{*}(z)$ and $y_{k}={i^{\prime}_{k}}^{*}(y)$.
Since $r_{k}-1>0$, we have ${i^{\prime}_{k-1}}^{*}(y_{k})=0$. Therefore, ${i^{\prime}_{k-1}}^{*}(z_{k})$ in $E_{r_{k}}^{s,0}(Y^{\prime}_{k-1},Y^{\prime\prime}_{k-1})$ is also zero.
So, for some $r_{k-1}<r_{k}$, ${i^{\prime}_{k-1}}^{*}(z_{k})$ in $E_{r_{k-1}}^{s,0}(Y^{\prime}_{k-1},Y^{\prime\prime}_{k-1})$ must be hit, that is, there exists $y_{k-1}$ in $E_{r_{k-1}}^{s-r_{k-1},r_{k-1}-1}(Y_{k-1}^{\prime},Y_{k-1}^{\prime\prime})$ such that $d_{r_{k-1}}(y_{k-1})=i_{k-1}^{*}(z_{k})$ in $E_{r_{k-1}}^{s,0}(Y^{\prime}_{k-1},Y^{\prime\prime}_{k-1})$.
Continuing this precess, we have a sequence of integers
$$2\leq r_{0}<r_{1}<\dots<r_{k}.$$
Hence, we have $r_{k}\geq k+2$. However, $d_{r}=0$ for $r\geq k+2$ in
$$E_{r}(Y^{\prime}\times_{X}P_{\alpha},Y^{\prime\prime}\times_{X}P_{\alpha}).$$
It is a contradiction. So, each element in $H^{s}(Y^{\prime},Y^{\prime\prime})=E_{r}^{s,0}(Y^{\prime}\times_{X}P_{\alpha},%
Y^{\prime\prime}\times_{X}P_{\alpha})$ is not hit in $E_{r}(Y^{\prime}\times_{X}P_{\alpha},Y^{\prime\prime}\times_{X}P_{\alpha})$ for $r\geq 2$. In other words, it is a permanent cocycle.
Therefore, the induced homomorphism
$$H^{*}(Y^{\prime},Y^{\prime\prime})\to H^{*}(Y^{\prime}\times_{X}P_{\alpha},Y^{%
\prime\prime}\times_{X}P_{\alpha})$$
is a monomorphism.
∎
By Proposition 4.1, we have that the induced homomorphism
$$\eta^{*}:M_{m,n}(Y)\to M_{m,n}(Z)$$
is a monomorphism
for $Z=Y\times_{X}P_{\alpha}$, $\eta=p_{1}:Y\times_{X}P_{\alpha}\to Y$.
Thus, Lemma 4.1 completes the proof of Lemma 3.2.
References
[1]
M. F. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London Ser. A 308 (1983), no. 1505, 523–615.
[2]
E. M. Friedlander, Étale homotopy of simplicial schemes, Ann. of Math. Stud., 104, Princeton Univ. Press, Princeton, N.J., 1982.
[3]
E. M. Friedlander and G. Mislin, Cohomology of classifying spaces of complex Lie groups and related discrete groups, Comment. Math. Helv. 59 (1984), no. 3, 347–361.
[4]
M. Kameko, On the cohomology of finite Chevalley groups and free loop spaces, Sūrikaisekikenkyūsho Kōkyūroku No. 1581 (2008), 45–54.
[5]
D. Kishimoto, Cohomology of twisted loop spaces, preprint.
[6]
S. N. Kleinerman, The cohomology of Chevalley groups of exceptional Lie type, Mem. Amer. Math. Soc. 39 (1982), no. 268, viii+82 pp.
[7]
K. Kuribayashi, M. Mimura and T. Nishimoto, Twisted tensor products related to the cohomology of the classifying spaces of loop groups, Mem. Amer. Math. Soc. 180 (2006), no. 849, vi+85 pp.
[8]
J. McCleary, A user’s guide to spectral sequences, Second edition, Cambridge Univ. Press, Cambridge, 2001.
[9]
D. Quillen, On the cohomology and $K$-theory of the general linear groups over a finite field, Ann. of Math. (2) 96 (1972), 552–586.
[10]
M. Tezuka, On the cohomology of finite Chevalley groups and free loop spaces of classifying spaces, Sūrikaisekikenkyūsho Kōkyūroku No. 1057 (1998), 54–55. |
Abstract
The difference from 4 to 6 $\sigma$ in the Hubble constant ($H_{0}$) between the values observed with the local (Cepheids and Supernovae Ia, SNe Ia) and the high-z probes (Cosmic Microwave Background obtained by the Planck data) still challenges the astrophysics and cosmology community.
Previous analysis has shown that there is an evolution in the Hubble constant that scales as $f(z)=\mathcal{H}_{0}/(1+z)^{\eta}$, where $\mathcal{H}_{0}$ is $H_{0}(z=0)$ and $\eta$ is the evolutionary parameter.
Here, we investigate if this evolution still holds by using the SNe Ia gathered in the Pantheon sample and the Baryon Acoustic Oscillations. We assume $H_{0}=70~{}{\,\textrm{km s}^{-1}\,\textrm{Mpc}^{-1}}$ as the local value and divide the Pantheon into three bins ordered in increasing values of redshift.
Similar to our previous analysis but varying two cosmological parameters contemporaneously ($H_{0}$, ${\Omega_{0m}}$ in the ${\Lambda}$CDM model and ${H_{0}}$, ${w_{a}}$ in the ${w_{0}w_{a}}$CDM model), for each bin we implement a Markov-Chain Monte Carlo analysis (MCMC) obtaining the value of $H_{0}$ assuming Gaussian priors to restrict the parameters spaces to values we expect from our prior knowledge of the current cosmological models and to avoid phantom Dark Energy models with ${w<-1}$.
Subsequently, the values of $H_{0}$ are fitted with the model $f(z)$.
Our results show that a decreasing trend with $\eta\sim 10^{-2}$ is still visible in this sample. The $\eta$ coefficient reaches zero in 2.0 $\sigma$ for the $\Lambda$CDM model up to 5.8 $\sigma$ for $w_{0}w_{a}$CDM model.
This trend, if not due to statistical fluctuations, could be explained through a hidden astrophysical bias, such as the effect of stretch evolution, or it requires new theoretical models, a possible proposition is the modified gravity theories, $f(R)$.
This analysis is meant to further cast light on the evolution of ${H_{0}}$ and it does not specifically focus on constraining the other parameters. This work is also a preparatory to understand how the combined probes still show an evolution of the $H_{0}$ by redshift and what is the current status of simulations on GRB cosmology to obtain the uncertainties on the $\Omega_{0m}$ comparable with the ones achieved through SNe Ia.
keywords: supernovae; Ia; cosmology; Hubble; tension; $\Lambda$CDM; evolution; modified; gravity; theories
\pubvolume
10
\issuenum1
\articlenumber24
\externaleditorAcademic Editor: Elena Moretti and Francesco Longo \datereceived5 November 2021
\dateaccepted25 January 2022
\datepublished29 January 2022
\hreflinkhttps://doi.org/10.3390/galaxies10010024 \TitleOn the Evolution of the Hubble Constant with the SNe Ia Pantheon Sample and Baryon Acoustic Oscillations: A Feasibility Study for GRB-Cosmology in 2030
\TitleCitationOn the Evolution of the Hubble Constant with the SNe Ia Pantheon Sample and Baryon Acoustic Oscillations: A Feasibility Study for GRB-Cosmology in 2030
\AuthorMaria Giovanna Dainotti ${}^{1,2,3,}$*\orcidA, Biagio De Simone ${}^{4,5}$\orcidB, Tiziano Schiavone ${}^{6,7}$\orcidC, Giovanni Montani ${}^{8,9}$\orcidD, Enrico Rinaldi ${}^{10,11,12}$\orcidE, Gaetano Lambiase ${}^{4,5}$\orcidF, Malgorzata Bogdan ${}^{13,14}$\orcidG and Sahil Ugale ${}^{15}$\orcidH
\AuthorNamesMaria Giovanna Dainotti, Biagio De Simone, Tiziano Schiavone, Giovanni Montani, Enrico Rinaldi, Gaetano Lambiase, Malgorzata Bogdan, Sahil Ugale
\AuthorCitationDainotti, M.G.; De Simone, B.; Schiavone, T.; Montani, G.;Rinaldi, E.; Lambiase, G.; Bogdan, M.; Ugale, S.
\corresCorrespondence: maria.dainotti@nao.ac.jp
1 Introduction
The $\Lambda$CDM model is one of the most accredited models, which implies an accelerated expansion phase Riess et al. (1998); Perlmutter et al. (1999).
Although it represents the favored paradigm, it is affected by great challenges: the fine-tuning, the coincidence Weinberg (1989); Peebles and Ratra (2003), and the Dark Energy nature’s problems.
More importantly, the ${H_{0}}$ tension represents a big challenge for modern cosmology.
Indeed, the $4.4$ up to 6.2 $\sigma$ discrepancy, depending on the sample used Riess et al. (2019); Camarena and Marra (2020); Wong et al. (2020), between the local value of $H_{0}$ obtained with Cepheids observations and SNe Ia, $H_{0}=74.03\pm 1.42{\,\textrm{km s}^{-1}\,\textrm{Mpc}^{-1}}$ Gómez-Valent and Amendola (2018); Reid et al. (2019), and the Planck data of Cosmic Microwave background radiation (CMB), $H_{0}=67.4\pm 0.5{\,\textrm{km s}^{-1}\,\textrm{Mpc}^{-1}}$ from the Planck Collaboration Aghanim et al. (2020) requires further investigation.
From now on, $H_{0}$ will be in the units ${\,\textrm{km s}^{-1}\,\textrm{Mpc}^{-1}}$.
We stress that other probes report values of $H_{0}\approx 72\pm 2$, similar to the value obtained with the SNe Ia.
Surely, to solve the Hubble tension it is necessary to use probes that are standard candles.
SNe Ia, considered one of the best standard candles, are observed only up to a low redshift range: the farthest so far discovered is at $z=2.26$ Rodney et al. (2016).
It is important for studying the evolution of the cosmological parameters to investigate probes at high redshift. One of the best candidates in this regard is represented by the Gamma-ray Bursts (GRBs).
GRBs are observed up to cosmological redshifts (the actual record is of $z=9.4$ (Cucchiara et al., 2011)) and surpassed even the quasars (the most distant being at $z=7.64$ Wang et al. (2021)). Due to their detectability at high redshift, GRBs allow extending the current Hubble diagram to new redshift ranges (Cardone et al., 2009, 2010, 2011; Dainotti et al., 2013; Postnikov et al., 2014).
Indeed, it is important to stress that once we have established if the Hubble constant undergoes redshift evolution, the Pantheon sample can safely be combined with other probes. Surely the advantage of the use of the SNe Ia is that their emission mechanism is pretty clear, namely they originate from the thermonuclear explosion of carbon–oxygen white dwarfs (C/O WDs).
For GRBs, more investigation about their progenitor mechanism is needed. We here stress that this work can be also preparatory to the work of future application of GRBs as cosmological tools together with SNe Ia and Baryon Acoustic Oscillations (BAOs) through well-established correlations among the prompt variables, such as: the Amati relation Amati, L. et al. (2002), which connects the peak in the $\nu F_{\nu}$ spectrum to the isotropic energy in the prompt emission ($E_{iso}$), the Yonetoku relation Yonetoku et al. (2004); Ito et al. (2019) between $E_{peak}$ and the peak luminosity of the prompt emission, $L_{peak}$, the Liang and Zhang relation Liang and Zhang (2005) between $E_{iso}$, the rest-frame break time of the GRB $t^{\prime}_{b}$ and the peak energy spectrum in the rest frame $E^{\prime}_{p}$, the Ghirlanda relation ($E_{peak}-E_{jet}=E_{iso}\times(1-cos\theta)$) Ghirlanda, G. et al. (2010), and the prompt-afterglow relations for the GRBs with the plateau emission investigated in (Dainotti et al., 2008, 2010, 2011, 2013, 2015a, 2015b, 2016; Dainotti, M. G. et al., 2017; Dainotti et al., 2017; Dainotti and Amati, 2018; Dainotti et al., 2018, 2020a, 2021a, 2021b; Vecchio et al., 2016), which have as common emission mechanism most likely the magnetar model, where a neutron star with an intense magnetic field undergoes a fast-spinning down (Duncan, 2001; Dall’Osso et al., 2012; Rowlinson et al., 2014; Rea et al., 2015; Stratta et al., 2018).
A feasibility study shows that GRBs can give relevant constraints on the cosmological parameters Dainotti et al. (2013); Amati et al. (2019). We here give a list of examples of other probes used for measuring the Hubble constant tension.
One of them is the use of data from time-delay measurements and strong lens systems Liao et al. (2019); Keeley et al. (2020).
On the contrary, additional probes carry similar values of $H_{0}$ to the ones of Planck, based on the Cosmic Chronometers (CC) ($H_{0}=67.06\pm 1.68$) in Gómez-Valent and Amendola (2018).
Besides, there is a series of independent probes, such as quasars Risaliti and Lusso (2019), the Tip of the Red-Giant Branch (TRGB) calibration through SNe Ia Freedman et al. (2019), and also GRBs
Cardone et al. (2009, 2010); Dainotti et al. (2013); Postnikov et al. (2014); Dutta et al. (2019); Yang et al. (2020), which bring estimates of $H_{0}$ ranging between the values obtained with local measurements (SNe Ia and Cepheids) and Baryon Acoustic Oscillations (BAO)+CMB.
Dainotti et al. (2021) discuss possible reasons behind the $H_{0}$ tension in the Pantheon sample: selection biases of parameters of the SNe Ia, unknown systematics, internal inconsistencies in the Planck data, or alternative theoretical interpretations compared to the standard cosmological model. Furthermore, the use of type 1 Active Galactic Nuclei (AGN) represents another promising cosmological probe given the peculiarity of their spectral emission Ingram et al. (2021).
To date, a wide range of different solutions to the Hubble constant tension has been provided by several groups (Agrawal et al., 2019; Arjona et al., 2021; Beltran2021, ; Banihashemi et al., 2021; Ballardini et al., 2021; Corona et al., 2021; Cyr-Racine et al., 2021; Valentino et al., 2021; Valentino and Melchiorri, 2021; Drees and Zhao, 2021; fernandez2021, ; firouzjahi2022cosmological, ; Ghose2021, ; Gu et al., 2021; Hart2021, ; Khalifeh and Jimenez, 2021; Li et al., 2021; Lulli et al., 2021; Mawas et al., 2021; Mehdi2021, ; Moreno-Pulido and Peracaula, 2021; Naidoo et al., 2021; Niedermann and Sloth, 2021; Nilsson and Park, 2021; paul2021, ; Ray et al., 2021; Schöneberg et al., 2021; Trott and Huterer, 2021; Ye et al., 2021; Zhou et al., 2021; Zhu et al., 2021; Alestas et al., 2022; Cea, 2022; Gurzadyan et al., 2022). Concerning the observational solutions, we here detail a series of proposals (das2021selfinteracting, ; 2016PhRvD..94j3523K, ; Yang et al., 2018; Di Valentino et al., 2018b, 2019; Pan et al., 2020; Di Valentino et al., 2021a, b, c, d; Di Valentino et al., 2018, 2021; Anchordoqui et al., 2021; Di Valentino et al., 2021, 2020a, 2020b; Allali et al., 2021; Anderson, 2021; Asghari and Sheykhi, 2021; Borghi2021, ; Brownsberger et al., 2021; Cyr-Racine, 2021; Farrugia2021, ; Greene2021, ; Khosravi and Farhang, 2021; Liu2022, ; Lu2021, ; Mantz et al., 2021; Mortsell et al., 2021a, b; Theodoropoulos and
Perivolaropoulos, 2021; Chang2022, ; Gómez-Valent, 2022; Pol et al., 2022; Wong et al., 2022; Rashkovetskyi2021, ; Romaniello et al., 2021; Gutiérrez-Luna et al., 2021; Luu, 2021; Alestas et al., 2021; Sakr and Sapone, 2021; Wang et al., 2021; Safari et al., 2022; Roth et al., 2021).
In Asencio et al. (2021), the simulations of data taken from the anomalously fast-colliding El Gordo galaxy clusters allow discussing the probability of observing such a scenario in a $\Lambda$CDM framework.
Ref. Javanmardi et al. (2021) perform a re-calibration of Cepheids in NGC 5584, thus obtaining a relation between the periods of Cepheids and their amplitude ratios (tighter than the one obtained in SH0ES Reid et al. (2019)) which could be useful to better estimate the value of $H_{0}$.
In Zhao and Xia (2021), the UV and X-ray data coming from quasars are used to constrain $H_{0}$ in the Finslerian cosmology. Ref. Roth et al. (2021) demonstrate that the Planetary Nebula Luminosity Function (PNLF) can be extended beyond the Cepheid distances, thus promoting it to be an additive probe for constraining $H_{0}$.
In Thakur et al. (2021), the analysis of Pan-STARRS telescope SNe Ia data provides a value of $H_{0}$ which lies between the SH0ES and Planck values.
Ref. Sharov and Sinyakov (2020) investigated how the $H_{0}$ measurements can depend on the choice of different probes (SNe, BAO, Cepheid, CC, etc.), showing also that through the set of filters on cosmological models, such as fiducial values for cosmological parameters ($w=-1$, with $w$ parameter for the equation of state, or $\Omega_{k}=0$, namely the curvature parameter set to zero), the tension can be alleviated.
Ref. Vagnozzi et al. (2021) extended a Hubble diagram up to redshift $z\sim 8$ combining galaxies and high-redshift quasars to test the late-time cosmic expansion history, giving a constraint on the upper-value of $H_{0}$ which is only marginally consistent with the results obtained by the Cepheids.
Ref. Staicova (2021) further tests the $w$CDM (with varying parameters of the equation of state), and oCDM models (with varying curvature) through the merging of BAOs, SNe Ia, CC, GRBs, and quasars data, after the analysis of the standard $\Lambda$CDM model.
In Krishnan et al. (2021), the combination of strongly lensed quasars and SNe Ia led the authors to conclude that the solution to the tension should be found outside of the Friedmann–Lemaitre–Robertson–Walker metric.
Ref. Li and Shapiro (2021); Chang2022 detect in the Stochastic Gravitational Wave Background a new method to alleviate the tension, while Mozzon et al. (2021); Abbott et al. (2021) focuses on the gravitational-wave signals from compact star mergers as probes that can give constraints on the $H_{0}$ value.
Ref. Mehrabi et al. (2021) combine the SNe Ia and the VLT-KMOS HII galaxies data to put new constraints on the cosmokinetic parameters.
The proposed solutions deal also with models that are alternative to the standard $\Lambda$CDM or, in other cases, that can extend it.
Ref. Li et al. (2015) constrain the Brans–Dicke (BD) theory through CMB and BAOs.
The TRGB method, combined with SNe Ia, gives a value of $H_{0}$ compatible with the one from CMB Freedman (2021).
Ref. Wu et al. (2021) obtain an $8\%$-precise value of $H_{0}$ through the Fast Radio Bursts (FRB).
In Perivolaropoulos and Skara (2021), the Cepheids calibration parameters are allowed to vary, thus leading to an estimated value of $H_{0}$ which is compatible with the CMB one.
The possibility that the Solar system’s proper motion may induce a bias in the measurement of $H_{0}$ has been subject to study in (Horstmann et al., 2021), finding out that there is no degeneracy between the cosmological parameters and the parameters of the Solar system motion.
Ref. Ferree and Bunn (2021) measure $H_{0}$ through the galaxies parallax having as reference the CMB rest-frame, being this parallax caused by the peculiar motions.
Ref. Luongo et al. (2021) verified through the measurements on GRBs and quasars that the Hubble constant has a bigger value in the sky directions aligned with the CMB dipole polarization, suggesting that a detachment from the FLRW should be considered.
Ref. de Souza et al. (2021); Fang and Yang (2021); Palmese et al. (2021); Yang et al. (2021) investigate how the dark sirens producing gravitational waves could help to probe $H_{0}$. Despite being a promising method, the incompleteness of galaxy catalogs may hinder the outcome of this method, thus (Gray et al., 2021) proposes a pixelated-sky approach to overcome the issue of event redshifts which are missing but may be retrieved through the galaxies present on the line of sight.
A review of the most promising emerging probes to measure the Hubble constant can be found in (Moresco et al., 2022).
Recent results on the measurements of the Hubble parameter and constant through the Third LIGO-Virgo-KAGRA Gravitational-Wave Transient Catalog (GWTC-3) can be found in (Collaboration et al., ).
An evolving trend for $H_{0}$ may be naturally predicted in Teleparallelism Nunes et al. (2017); Nunes (2018); Benetti et al. (2020); Räsänen (2009, 2010), as well as in modified gravity theories Sotiriou (2006); Nojiri and Odintsov (2007); Sotiriou and Faraoni (2010); Koksbang (2019b, c, 2020).
Refs. Odderskov et al. (2016); Nájera and Fajardo (2021, 2021a) study the $f(Q,T)$ models in Teleparallel Gravity through CC and SNe Ia, thus obtaining a value of $H_{0}$ compatible in 1 $\sigma$ with the SH0ES result. The linear theory of perturbation for the $f(Q,T)$ theory is investigated in (Nájera and Fajardo, 2021b), allowing the future tests of this model through CMB data.
In Linares Cedeño and Nucamendi (2021), the Unimodular Gravity model is constrained with Planck 2018 Aghanim et al. (2020), SH0ES, SNe Ia, and H0LiCOW collaboration Wong et al. (2020).
Furthermore, the Axi–Higgs model is tested with CMB, BAO, Weak Lensing data (WL), and SNe Fung et al. (2021): in another paper, it is shown how this model relaxes the Hubble tension Fung et al. (2021).
Ref. Shokri et al. (2021) describe the modified inflationary models considering constant-roll inflation.
Ref. Castellano et al. (2021) give boundaries on the Hubble constant value with the gravitino mass conjecture.
Ref. Tomita (2017a, b, 2018, 2019, 2020) show the role of cosmological second-order perturbations of the flat $\Lambda$CDM model in the $H_{0}$ tension.
Ref. Belgacem and Prokopec (2021) discuss how Dark Energy may be generated by quantum fluctuations of an inflating field and how the Hubble tension may be reduced by the spatial correlations induced by this effect.
The Dark Energy itself may be subject to evolution, as pointed out in Bernardo et al. (2022).
Ref. Ambjorn and Watabiki (2021) show how a modification of the Friedmann equation may naturally explain the inconsistency between the local and the cosmological measurements of the Hubble constant.
Ref. Di Bari et al. (2021) explain how the search for low-frequency gravitational waves (GWs) justifies the Hubble tension’s solution through the assumption of neutrino-dark sector interactions.
Ref. Duan et al. (2021) show how the $R_{K}^{(*)}$ anomalies (namely, the discrepancy between the theoretical ratio of the fractions $B\rightarrow K^{*}\mu^{+}\mu^{-}/B\rightarrow K^{*}e^{+}e^{-}$ for the dilepton invariant mass bins from the Standard Model and the observed one, see (Ghosh, 2017)) and the $H_{0}$ tension can be solved by Dirac neutrinos in a two-Higgs-doublet theory.
The introduction of models where the cosmological axio-dilation is present may lead to a solution of the Hubble tension Burgess et al. (2021).
Ref. Jiang and Piao (2021); Karwal et al. (2021); Nojiri et al. (2021); Tian and Zhu (2021) discuss how the Early Dark Energy models (EDE) can be used to alleviate the $H_{0}$ tension.
Ref. Linares Cedeño et al. (2021) analyze how the phantom Dark Energy models can give a limited reduction of the $H_{0}$ tension, while Hernández-Almada et al. (2021) explore how the Kaniadakis holographic Dark Energy model alleviates the $H_{0}$ tension.
In Hernández-Almada et al. (2021), the Viscous Generalized Chaplygin Gas (VGCG) model is used to diminish the Hubble tension. The holographic Dark Energy models are pointed as a possible solution through the study of unparticle cosmology Abchouyeh and van Putten (2021).
Ref. Wang (2018) test seven cosmological models through the constraints of SNe Ia, BAO, CMB, Planck lensing, and Cosmic Chronometers with the outcome that in the $\Lambda$CDM scenario a flat universe is favored.
Ref. Ye et al. (2021) discuss how the new physical scenarios before the recombination epoch imply the shift of cosmological parameters and how these shifts are related to the discrepancy between the local and non-local values of $H_{0}$.
Ref. Nguyen (2020) proposes that the $H_{0}$ tension may be solved if the speed of light is treated as a function of the scale factor (as in Barrow (1999)), and applies this scenario to SNe Ia data.
Ref. Artymowski et al. (2021); Yang et al. (2021) discuss the implementation of the alternative Phenomenologically Emergent Dark Energy model (PEDE), which can be also extended to a Generalized Emergent Dark Energy model (GEDE) with the addition of an extra free parameter.
This shows the possibility of obtaining the PEDE or the $\Lambda$CDM cosmology as sub-cases of the GEDE scenario.
Ref. Adil et al. (2021) consider a scenario of modified gravity predicting the increase of the expansion rate in the late-universe, thus proving that in this scenario the Hubble tension reduces significantly.
Ref. Vagnozzi (2021) study the $\Lambda$CDM model constrained, at the early-time universe, by the presence of the early Integrated Sachs–Wolfe (eISW) effect, proving that the early-time models aimed at attenuating the Hubble tension should be able to reproduce the same eISW effect just like the $\Lambda$CDM does.
The observations of a locally higher value for $H_{0}$ led to the discussion of local measurements, constraints, and modeling.
In this regards, the assumption of a local void Alestas et al. (2020); Kazantzidis et al. (2021); Martín and Rubio (2021) may produce locally an increased value for $H_{0}$.
The Universe appears locally inhomogeneous below a scale of roughly 100 Mpc.
The question some cosmologists are attempting to solve is whether local inhomogeneities have impact on cosmological measurements and the Hubble diagram. Many observables are related to photons paths, which may be directly affected by the
matter distribution. Many theoretical attempts were made during the last few decades to develop the necessary average prescription to evaluate the photon propagation on the observer’s past light cone based on covariant and gauge-invariant observables (Buchert, 2000; Gasperini et al., 2009, 2011; Fanizza et al., 2020). Local inhomogeneities and cosmic structure cause scattering and bias effects in the Hubble diagram, which are due to peculiar
velocities, selection effects, and gravitational lensing, but also
to non-linear relativistic corrections (Ben-Dayan et al., 2013; Fleury et al., 2017; Fanizza et al., 2020).
This question was addressed in (Adamek et al., 2019) utilising the N-body simulation of cosmic structure formation through the numerical
code gevolution. This non-perturbative approach pointed out
discrepancies in the luminosity distance between a homogeneous and
inhomogeneous scenario, showing, in particular, the presence of non-Gaussian effects at higher redshifts.
These studies related to distance indicators will become even more
significant considering the large number of the forthcoming surveys
designed to the observations on the Large Scale Structure of the Universe
in the next decade (for instance, the Euclid survey (Amendola et al., 2018; Fanizza, 2021) and the Vera C. Rubin Observatory’s LSST (Andreoni et al., 2021)). The effect of local structures in an inhomogeneous universe should be considered in the locally measured value of $H_{0}$ Ben-Dayan et al. (2014); Fanizza et al. (2021).
The local under-density interpretation was also studied in Milgromian dynamics Haslbauer et al. (2020); Asencio et al. (2021); Perivolaropoulos (2014), but in Castello et al. (2021) it is shown how this interpretation does not solve the tension.
Ref. Haslbauer et al. (2020) study the KBC local void which is in contrast with the $\Lambda$CDM, thus proposing the Milgromian dynamics as an alternative to standard cosmology. Milgromian dynamics are studied also in Banik and Zhao (2021) where, through the galactic structures and clusters, it is shown how this model can be consistent at different scales and alleviate the Hubble tension.
Ref. Alestas and Perivolaropoulos (2021) describe the late time approaches and their effect on the Hubble parameter. The bulk viscosity of the universe is also considered the link between the early and late universe values of $H_{0}$ Normann and Brevik (2021).
Ref. Bernal et al. (2021) explain how the local measurements over-constrain the cosmological models and propose the graphical analysis of the impact that these constraints have on the $H_{0}$ estimation through ad hoc triangular plots.
Ref. Thiele et al. (2021); Perivolaropoulos (2014); Grande and Perivolaropoulos (2011) describes the effects of inhomogeneities at small scales in the baryon density.
Ref. Dinda (2021) find out that the late time modifications can solve the tension between the $H_{0}$ SH0ES and CMB values through a parametrization of the comoving distance.
Ref. Marra and Perivolaropoulos (2021) propose to alleviate the Hubble tension considering an abrupt modification of the effective gravitational constant at redshift $z\approx 0.01$.
Other proposals are focused on the existence of different approaches.
Ref. Krishnan et al. (2020, 2021) show how $H_{0}$ evolves with redshift at local scales.
Ref. Krishnan et al. (2021b) discuss how the breakdown of Friedmann–Lemaitre–Robertson–Walker (FLRW Robertson (1935)) may be a plausible assumption to alleviate the Hubble tension. Ref. Gerardi et al. (2021) investigate the binary neutron stars mergers and, with the analysis of simulated catalogs, show their potential to help to alleviate the $H_{0}$ tension.
Ref. Escamilla-Rivera et al. (2021) explain how Gaussian process (GP) and locally weighted scatter plot smoothing are used in conjunction with simulation and extrapolation (LOESS-Simex) methods can reproduce different sets of data with a high level of precision, thus giving new perspectives on the Hubble tension through the simulation of Cosmic Chronometers, SNe Ia, and BAOs data sets.
Ref. Sun et al. (2021) focus on the GP and state the necessity of lower and upper bounds on the hyperparameters to obtain a reliable estimation of $H_{0}$.
On the other hand, Ref. Renzi and Silvestri (2020) suggested a novel approach to measure $H_{0}$ based on the distance duality relation, namely a method that connects the luminosity distance of a source to its angular diameter.
In this case, data do not require a calibration phase and the relative constraints are not dependent on the underlying cosmological model.
Ref. Gurzadyan and Stepanian (2021) showed how the tension can be solved with a modified weak-field General Relativity theory, thus defining a local $H_{0}$ and a global $H_{0}$ value.
Ref. Geng et al. (2021) investigated how a specific Dark Energy model in the generalized Proca theory can alleviate the tension.
In Reyes and Escamilla-Rivera (2021), the Horndeski model can describe with significantly good precision the late expansion of the universe thanks to the Hubble parameter data. The same model is considered promising for the solution of the $H_{0}$ tension in Petronikolou et al. (2021).
Ref. Alestas et al. (2021) described how the transition observed in Tully–Fisher data could imply an evolving gravitational strength and explain the tension.
Ref. Ye et al. (2021) explain how the physical models of the pre-recombination era could cause the observed $H_{0}$ values discrepancy and suggest that if the local $H_{0}$ measurements are consistent then a scale-invariant Harrison–Zeldovich spectrum should be considered to solve the $H_{0}$ issue.
The Dynamical Dark Energy (DDE) models are the object of study in Benisty and Staicova (2021); Ó Colgáin et al. (2021): in the former, the DDE is proposed as an alternative to $\Lambda$CDM, while in the latter it is shown how the Chevallier–Polarski–Linder (CPL) parametrization Chevallier and Polarski (2001); Linder (2003) is insensitive to Dark Energy at low redshift scales.
Ref. Aloni et al. (2021); Ghosh et al. (2021) propose Dark Radiation as a new surrogate of the Standard Model.
In Shrivastava et al. (2021), the scalar field cosmological model is used, together with the parametrization of the equation of state, to obtain $H_{0}$ and investigate the nature of Dark Energy. The possibility of a scalar field non minimally coupled to gravity as a probable solution to the $H_{0}$ tension is investigated in Pereira (2021).
Ref. Bag et al. (2021) highlight the advantage of the braneworld models to predict the local higher values of $H_{0}$ and, contemporaneously, respect the CMB constraints.
Another approach is to solve the $H_{0}$ tension by allowing variations in the fundamental constants Franchino-Viñas and Mosquera (2021).
Ref. Palle (2021) propose a non-singular Einstein–Cartan cosmological model with a simple parametrization of spacetime torsion to alleviate the tension, while Liu et al. (2021) propose a model where the Dark Matter is annihilated to produce Dark Radiation.
Ref. Blinov et al. (2021) introduce a hidden sector of atomic Dark Matter in a realistic model that avoids the fine-tuning problem.
The observed weak effect of primordial magnetic fields can create clustering at small scales for baryons and this could explain the $H_{0}$ tension Galli et al. (2021).
Ref. Liu et al. (2021) test the General Relativity at galactic scales through Strong Gravitational Lensing. The Strong Lensing is a promising probe for obtaining new constraints on $H_{0}$, thanks to the next generation DECIGO and B-DECIGO space interferometers Hou et al. (2021).
In Sola (2021), the cosmological constant $\Lambda$ is considered a dynamical quantity in the context of the running vacuum models and this assumption could tackle the $H_{0}$ tension. Cuesta et al. (2021) show the singlet Majoron model to explain the acceleration of the expansion at later times and prove that this is consistent with large-scale data: this model has been subsequently discussed in other works González-López (2021).
The vacuum energy density value is affected by the Hubble tension as well and its measurement may cast more light on this topic Prat et al. (2021).
Ref. Joseph and Saha (2021) discuss the outcomes of the Oscillatory Tracker Model with an $H_{0}$ value that agrees with the CMB measurements.
In Aghababaei et al. (2021), it is explained how the Generalized Uncertainty Principle and the Extended Uncertainty Principle can modify the Hubble parameter.
Bansal et al. (2021) explore the implication of the Mirror Twin Higgs model and the need for future measurements to alleviate the tension.
The artificial neural networks can be applied to reconstruct the behavior of large scale structure cosmological parameters (Dialektopoulos et al., 2021).
Another alternative is given by the gravitational transitions at low redshift which can solve the $H_{0}$ tension better than the late-time $H(z)$ smooth deformations (Marra and Perivolaropoulos, 2021; Alestas et al., 2022).
Another comparison between the late-time gravitational transition models and other models which predict a smooth deformation of the Hubble parameter can be found in Alestas et al. (2021).
Ref. Parnovsky (2021), as modifications to the $\Lambda$CDM model, consider as plausible scenarios or a Dark Matter component with negative pressure or the decay of Dark Energy into Dark Matter.
Ref. Zhang et al. (2021) does not observe the $H_{0}$ tension through the Effective Field Theory of Large Scale Structure and the Baryon Oscillation Spectroscopic Survey (BOSS) Correlation Function.
Considering the Dark Matter particles with two new charges, Ref. Hansen (2021) reproduce a repulsive force which has similar effects to the $\Lambda$ cosmological constant.
Furthermore, the models where interaction between Dark Matter and Dark Energy is present are promising for a solution of the Hubble constant tensions, see Gariazzo et al. (2021).
In Ruiz-Zapatero, Jaime et al. (2021), it is shown how two independent sets of cosmological parameters, the background (geometrical) and the matter density (growth) component parameters, respectively, give consistent results and how the preference for high values of $H_{0}$ is less significant in their analysis.
Ref. Cai et al. (2021) introduce a global parametrization based on the cosmic age which rules out the early-time and the late-time frameworks.
Ref. Mehrabi and Vazirnia (2021) point out, through the use of non-parametric methods, how the cosmological models may induce biases in the cosmological parameters. In the same way, the statistical analysis of galaxies’ redshift value and distance estimations may be affected by biases which could, in turn, affect the estimation of $H_{0}$
Ref. Parnovsky (2021). This consideration holds also for the quadruply lensed quasars which are another method to measure $H_{0}$ Baldwin and Schechter (2021).
Ref. Huber et al. (2021) use the machine learning techniques to measure time delays in lensed SNe Ia, these being an independent method to measure $H_{0}$.
Additionally, in Krishnan et al. (2021) it is explained how an evolution of $H_{0}$ with the redshift is to be expected.
If a statistical approach on the different $H_{0}$ values is used instead, together with the assumption of an alternative cosmology, another solution to the tension could be naturally implied Mercier (2021).
Ref. Ren et al. (2021) use data to reconstruct the $f(T)$ gravity function without assuming any cosmological model: this $f(T)$ could in turn represent a solution to the $H_{0}$ tension.
Ref. Gutiérrez-Luna et al. (2021) discuss how the addition of scalar fields with particle physics motivation to the cosmological model which predicts Dark Matter can retrieve the observed abundances of the Big Bang Nucleosynthesis.
In Hryczuk and Jodłowski (2020), a Dark Matter production mechanism is proposed to alleviate the $H_{0}$ tension.
A general review of the perspectives and proposals concerning the $H_{0}$ tension can be found in Di Valentino et al. (2021) and Perivolaropoulos and Skara (2021); Saridakis et al. (2021).
SNe Ia represents a very good example of standard candles. Here we consider also the contribution of geometrical probes, the so-called standard rulers: while standard candles show a constant intrinsic luminosity (or obey an intrinsic relation between their luminosity and other physical parameters independent of luminosity), standard rulers are characterized by a typical scale dimension.
This property allows estimating their distance according to the apparent angular size.
Among the possible standard rulers, the BAOs assume great importance for cosmological purposes.
We here investigate the $H_{0}$ tension in the Pantheon sample (hereafter PS) from Scolnic et al. (2018) and we add the contribution of BAOs to the cosmological computations to check if the trend of $H_{0}$ found in (Dainotti et al., 2021a) is present also with the addition of other probes. We here point out that the current analysis is not meant to constrain $\Omega_{0m}$ or any other cosmological parameters, but it is focused to study the reliability of the trend of $H_{0}$ as a function of the redshift.
We here point out that this analysis is not meant to constrain $\Omega_{0m}$ or any other cosmological parameters, but it is focused to study the reliability of the trend of $H_{0}$ as a function of the redshift.
The range of redshift in the PS goes from $z=0.01$ to $z=2.26$. We tackle the problem with a redshift binning approach of $H_{0}$, the same used in Dainotti et al. (2021), but here we adopt a starting value of $H_{0}=70$ instead of $73.5$: if a trend with redshift exists, it should be independent on the initial value for $H_{0}$.
The systematic contributions for the PS are calibrated through a reference cosmological model, where $H_{0}$ is $70.0$ Scolnic et al. (2018).
In the current paper, the aforementioned systematic uncertainties are considered for the analysis.
Our approach has a two-fold advantage: on the one hand, it is relatively simple and on the other hand, it avoids the re-estimation of the SNe Ia uncertainties and may be able to highlight a residual dependence on the SNe Ia parameters with redshift.
While a slow varying Einstein constant with the redshift, as it emerges in a modified $f(R)$ gravity, appears as the most natural explanation for a trend $H_{0}(z)$, the analysis of Section 7 seems to indicate that such effect is not necessarily related with the Dark Energy contribution of the late universe. Since the Hu–Sawicki gravity lacks of reproducing the correct profile $H_{0}(z)$ shows that a Dark Energy model in the late Universe may not be enough to explain the observed effect since the scalar mode dynamics can not easily conciliate the Dark Energy contribution with the decreasing trend of $H_{0}(z)$. Thus, it may be necessary a modified gravity scenario more general than a Dark Energy model in the late Universe.
The current paper is composed as expressed in the following: in Section 2 the $\Lambda$CDM and $w_{0}w_{a}$CDM models are briefly introduced together with SNe Ia properties; Section 3 describes the use of BAOs as cosmological rulers; Section 4 contains our binned analysis results, after slicing the PS in 3 redshift bins for the aforementioned models, and assuming locally $H_{0}=70$; in Section 5, we investigate, through simulated events, how the GRBs will be contributing to cosmological investigations by 2030;
in Section 6 we discuss the results; in Section 7 we test the Hu–Sawicki model through a binning approach; in Section 8 we report an overview on the requirements that a suitable $f(R)$ model should have to properly describe the observed trend of $H_{0}$ and in Section 9 our conclusions are reported.
2 SNe Ia Cosmology
SNe Ia are characterized by an intrinsic luminosity that is almost uniform.
Because of this, SNe Ia are considered reliable standard candles.
We compare the theoretical distance moduli $\mu_{th}$ with the observed distance moduli $\mu_{obs}$ of SNe Ia belonging to the PS.
The theoretical distance moduli are defined through the luminosity distance $d_{L}(z)$ which we need to define based on the cosmological model of interest.
We here show the CPL parametrization which describes the $w$ parameter as a function of redshift ($w(z)=w_{0}+w_{a}\times z/(1+z)$) in the $w_{0}w_{a}$CDM model.
In the usual assumptions $w_{0}\sim-1$ and $w_{a}\sim 0$, and $d_{L}(z)$ is defined as the following Weinberg (2008):
$$d_{L}(z,H_{0}...)=\frac{c(1+z)}{H_{0}}\,\int_{0}^{z}\,\frac{dz^{*}}{\sqrt{\Omega_{0m}\,\left(1+z^{*}\right)^{3}+\Omega_{0\Lambda}\,\left(1+z^{*}\right)^{3\,\left(w_{0}+w_{a}+1\right)}\,e^{-3\,w_{a}\,\frac{z^{*}}{1+z^{*}}}}}\,,$$
(1)
where $\Omega_{0\Lambda}$ is the Dark Energy component, $c$ is the speed of light, and $z$ is the redshift.
We stress that in this context the relativistic components are ignored.
Moreover, since in the present universe the radiation density parameter $\Omega_{0r}\approx 10^{-5}$, this contribution can be neglected.
If we substitute $w_{a}=0$, $w_{0}=-1$ in Equation (1) the luminosity distance expression for $\Lambda$CDM model is automatically retrieved.
According to the distance luminosity expression, the theoretical distance modulus can be written in the following form:
$$\mu_{th}=5\hskip 2.15277ptlog_{10}\ d_{L}(z,H_{0},...)+25,$$
(2)
which is usually expressed in Megaparsec (Mpc).
The observed distance modulus, $\mu_{obs}=m^{\prime}_{B}-M$, taken from PS contains the apparent magnitude in the B-band corrected for statistical and systematic effects ($m^{\prime}_{B}$) and the absolute in the B-band for a fiducial SN Ia with a null value of stretch and color corrections ($M$).
Considering the color and stretch population models for SNe Ia, in our approach we average the distance moduli given by the Guy et al. (2010) (G2010) and Chotard et al. (2011) (C2011) models.
We here remind the reader that $H_{0}$ and $M$ are degenerate parameters: in the PS release, $M=-19.35$ such that $H_{0}=70.0$.
Ref. Dainotti et al. (2021) obtain information on $H_{0}$ by comparing $\mu_{obs}$ in Scolnic et al. (2018)\endnotehttps://github.com/dscolnic/Pantheon (accessed on 21 December 2020). with $\mu_{th}$ for each SN.
Moreover, they fix $\Omega_{0m}$ to a fiducial value to better constrain the $H_{0}$ parameter. Furthermore, according to Kenworthy et al. (2019), we consider the correction of the luminosity distance keeping into account the peculiar velocities of the host galaxies which contain the SNe Ia.
To perform our analysis, we define the $\chi^{2}$ for SNe:
$$\chi^{2}_{SN}=\Delta\mu^{T}\cdot\mathcal{C}^{-1}\cdot\Delta\mu.$$
(3)
Here $\Delta\mu=\mu_{obs}-\mu_{th}$, and $\mathcal{C}$ denotes the $1048\times 1048$ covariance matrix, given by Scolnic et al. (2018).
As for the $\mu_{obs}$ values of G2010 and C2011, the systematic uncertainty matrices of the two models have been averaged.
After building the $\mathcal{C}$ total matrix from Equation (16) in Dainotti et al. (2021), we slice the PS in redshift bins, and then we divide $\mathcal{C}$ into submatrices considering the order in redshift.
More in detail, starting from the 1048 SNe Ia redshift-ordering, we divide the SNe Ia into 3 equally populated bins made up of $\approx$349 SNe Ia. Concerning only $D_{stat}$, it is trivial to build its submatrices considering that the statistical matrix is diagonal.
Hence, a single matrix element is related to a given SN of the PS. On the other hand, if the non-diagonal matrix $C_{sys}$ is included, a customized code will be used\endnoteThe code is available upon request. to build the submatrices.
Our code was developed to select only the total covariance matrix elements related to SNe Ia having redshift within the considered bin.
The choice of three bins is justified by the high number of SNe Ia (around hundreds of SNe per bin) that can still constitute statistically illustrative subsamples of the PS and that can properly consider the contribution of systematic uncertainties.
Subsequently to the bins division, we focus on the optimal values of $H_{0}$ to minimize the $\chi^{2}$ in Equation (3).
$H_{0}$ is regarded as a nuisance parameter, which is free to vary, to better analyze a possible redshift function of $H_{0}$.
We follow the assumptions on the fiducial value of $M=-19.35$: while in Dainotti et al. (2021) $M$ was estimated assuming a local ($z=0$) value of $H_{0}=73.5$, we here consider the conventional $H_{0}$ value of the PS release, namely $H_{0}=70.0$ for three bins.
Our choice of a starting value of $H_{0}=70$ is dictated by the presence in the current literature of more than 50 papers that are using the PS in combination with other probes to estimate the value of $H_{0}$, see
Deng and Wei (2018); Akarsu et al. (2019); Shafieloo et al. (2018); Hossienkhani et al. (2019); L’Huillier et al. (2019); Lusso et al. (2019); Ma et al. (2019); Kenworthy et al. (2019); Sadri (2019); Sadri and Khurshudyan (2019); Wagner and Meyer (2019); Zhai and Wang (2019); Zhao et al. (2019); Abdullah et al. (2020); Al Mamon and Saha (2020); Anagnostopoulos et al. (2020); Brout et al. (2021); Cai et al. (2021); Chang et al. (2019); D’Amico et al. (2021); Di Valentino et al. (2020); Gao et al. (2020); Garcia-Quintero et al. (2020); Geng et al. (2020); Ghaffari et al. (2020); Hu et al. (2020); Huang (2020); Çamlıbel et al. (2020); Alestas et al. (2020); Liao et al. (2020); Koo et al. (2021); Li et al. (2020); Linares Cedeño and Nucamendi (2021); Luković et al. (2020); Luongo and Muccino (2020); Micheletti (2020); Mishra and Dua (2020); Nguyen (2020); Odintsov et al. (2021); Prasad et al. (2021); Rezaei et al. (2020); Ringermacher and Mead (2020); Tang et al. (2021); Wang and Chen (2020); Wei and Melia (2020); Zhang and Huang (2020); Baxter and Sherwin (2021); Camarena and Marra (2021); Cao et al. (2021); D’Amico et al. (2021); Heisenberg et al. (2021); Jesus et al. (2021); Kazantzidis et al. (2021); Lee (2021); Montiel et al. (2021); Mukherjee and Banerjee (2021); Wang et al. (2021).
Thus, if an evolutionary effect is present, it is necessary to investigate to which extent this can affect current and future results largely based on the PS sample.
Conversely, we fix $\Omega_{0m}=0.298\pm 0.022$ according to Scolnic et al. (2018) for a standard flat $\Lambda$CDM model.
More specifically, after the minimization of $\chi^{2}$, we extract the $H_{0}$ value in each redshift bin, via the Cobaya code Torrado and Lewis (2021).
To this end, we execute an MCMC using the D’Agostini method to obtain the confidence intervals for $H_{0}$ at the $68\%$ and $95\%$ levels, in three bins.
3 The Contribution of BAOs
The environment of relativistic plasma in the early universe was crossed by the sound waves that were generated by cosmological perturbations.
At redshift $z_{d}\sim 1059.3$, which marks the ending of the drag period Sharov and Vasiliev (2018), the recombination of electrons and protons into a neutral gas interrupted the propagation of the sound waves while the photons were able to propagate further Eisenstein et al. (2005).
In the period between the formation of the perturbations and the recombination, the different modes produced a sequence of peaks and minima in the anisotropy power spectrum.
Given the huge fraction of baryons in the universe, it is expected by cosmological models that the oscillations may affect also the distribution of baryons in the late universe.
As a consequence, the BAOs manifest as a local maximum in the correlation function of the galaxies distribution in correspondence of the comoving sound horizon scale at the given redshift $z_{d}$, namely $r_{s}(z_{d})$: this is associated with the stopping of the propagation of the acoustic waves.
To use the BAOs data for cosmology, we first need to define the following variables:
$$D_{V}(z)=\biggr{[}\frac{czd^{2}_{L}(z)}{(1+z)^{2}H(z)}\biggr{]}^{1/3},\hskip 17.22217ptd_{z}(z)=\frac{r_{s}(z_{d})}{D_{V}(z)}.$$
(4)
The value of the redshift $z_{d}$, which corresponds to the drag era ending and marks the decoupling of the photons, allows estimating the sound horizon scale:
$$(r_{d}\cdot h)_{fid}=104.57\,\textrm{Mpc},\hskip 17.22217ptr_{s}(z_{d})=\frac{(r_{d}\cdot h)_{fid}}{h},$$
(5)
where we use the adimensional ratio $h=H_{0}/100(\textrm{km s}^{-1}$ $\,\textrm{Mpc}^{-1})$.
To estimate $r_{s}$, the following approximated formula Cuceu et al. (2019) can be applied:
$$r_{s}\approx\frac{55.154\cdot e^{-72.3(\omega_{\nu}+0.0006)^{2}}}{\omega_{0m}^{0.25351}\omega_{b}^{0.12807}}\,\textrm{Mpc},$$
(6)
where $\omega_{i}=\Omega_{i}\cdot h^{2}$, and $i=m,\nu,b$ represent matter, neutrino and baryons.
We here assume $\omega_{\nu}=0.00064$ Aubourg et al. (2015) and $\omega_{b}=0.02237$ Aghanim et al. (2020).
Given these quantities, we define the $\chi^{2}$ for BAOs as follows:
$$\chi^{2}_{BAO}=\Delta d^{T}\cdot\mathcal{M}^{-1}\cdot\Delta d,$$
(7)
where $\Delta d=d^{obs}_{z}(z_{i})-d^{theo}_{z}(z_{i})$ and $\mathcal{M}$ is the covariance matrix for the BAO $d^{obs}_{z}(z_{i})$ values.
In this binned analysis, a subset of the 26 BAO observations set available in Sharov and Vasiliev (2018) will be employed.
4 Multidimensional Binned Analysis with SNe Ia and BAOs
To investigate the $H_{0}$ tension through the SNe Ia and BAOs data, we combine the $\chi^{2}$ Equations (3) and (7) to obtain the total $\chi^{2}$
$$\chi^{2}=\frac{1}{2}\chi^{2}_{SN}+\frac{1}{2}\chi^{2}_{BAO}.$$
(8)
In our work, we combine each SNe bin with only 1 BAO data point which has a redshift value within the SNe bin: this approach of using one BAO comes from (Postnikov et al., 2014). In this way we do not have the problem of a different number of BAOs in different bins.
Through Equation (8), we investigate if a redshift evolution of $H_{0}\left(z\right)$ is present, obtaining it from the binning of SNe Ia+BAOs considering three bins with the $\Lambda$CDM and $w_{0}w_{a}$CDM models.
A feasibility study done in (Dainotti et al., 2021) performed with different bins selections has highlighted how the maximum number of bins in which the PS should be divided is 3, otherwise the statistical fluctuations would dominate on a multi-dimensional analysis, leading to relatively large uncertainties which would mask any evolving trend, if present. Furthermore, for the same reason, it is not advisable to leave free to vary more than two parameters at the same time, thus in the current section, we will analyze the behavior of $H_{0}$ in three bins when it is varied together with a second cosmological parameter. The same considerations make necessary the choice of more tight priors since we are basing the current analysis on the prior knowledge, avoiding the degeneracies among the parameter space, and letting the priors have more weight in the process of posteriors estimation. Differently from (Dainotti et al., 2021), for the ${\Lambda}$CDM model, we will let the parameters ${H_{0}}$ and ${\Omega_{0m}}$ vary simultaneously, while in the ${w_{0}w_{a}}$CDM model the varying parameters are ${H_{0}}$ and ${w_{a}}$. We decided to leave ${w_{a}}$ free to vary since, according to the CPL parametrization, ${w_{a}}$ gives direct information about the evolution of the ${w(z)}$ while ${w_{0}}$ is considered a constant in the same model. Concerning the fiducial values and the priors assignment for the MCMC computations, we apply Gaussian priors with mean equal to the central values of ${\Omega_{0m}=0.298\pm 0.022}$ and ${H_{0}=70.393\pm 1.079}$ for ${\Omega_{0m}}$ and ${H_{0}}$, respectively, and with 1 ${\sigma=2*0.022}$ and 1 ${\sigma=2*1.079}$ for ${\Omega_{0m}}$ and ${H_{0}}$, respectively. In summary, to draw the Gaussian priors, we consider the mean value of the parameters as the expected one of the Gaussian distribution and we double the ${\sigma}$ value which is then considered the new standard deviation for the distribution.
Concerning the ${w_{0}w_{a}}$CDM model, we fix ${w_{0}=-0.905}$ and we consider the priors on ${w_{a}}$ with the mean = $-$0.129 taken from Table 13 of (Scolnic et al., 2018), while 1 ${\sigma=}$ is the ${20\%}$ of its central value. Such an assumption with small prior is needed since we need to assume that ${w(z)>-1.168}$ as the value tabulated in Scolnic et al. (2018). Besides, since we are here dealing with standard cosmologies, with this constraint we are avoiding some of the phantom Dark Energy models.
After the $\chi^{2}$ minimization for each bin, we perform a MCMC simulation to draw the mean value of $H_{0}$ and its uncertainty. Once $H_{0}$ is obtained for each bin, we perform a fit of $H_{0}$ using a simple function largely employed to characterize the evolution of many astrophysical objects, such as GRBs and quasars Singal et al. (2011, 2013); Dainotti et al. (2013, 2015b); Dainotti, M. G. et al. (2017); Dainotti et al. (2020a); Petrosian et al. (2015); Lloyd and Petrosian (2000); Dainotti et al. (2021).
More specifically, the fitting of $H_{0}$ is given by
$$f(z)=H_{0}(z)=\frac{\mathcal{H}_{0}}{(1+z)^{\eta}},$$
(9)
in which $\mathcal{H}_{0}$ and $\eta$ are the fitting parameters.
The former $\mathcal{H}_{0}\equiv H_{0}$ at $z=0$, while the latter $\eta$ coefficient describes a possible evolutionary trend of $H_{0}$.
We consider the 68% confidence interval at, namely 1 $\sigma$ uncertainty.
In the current treatment, we consider the calibration of the PS with $H_{0}=70$ as provided by Scolnic et al. (2018).
Results are presented in the panels of Table 1.
We here stress that the fiducial magnitude value is assumed to be $M=-19.35$ for each SNe bin, thus it will not be mentioned in the same Table.
All the uncertainties in the tables in this paper are in 1 ${\sigma}$. As reported in the upper half of Table 1, namely with the ${\Lambda}$CDM model, if we do not include the BAOs then the ${\eta}$ coefficient is compatible with 0 in 2.0 ${\sigma}$ for the three bins case. When we introduce the BAOs within the ${\Lambda}$CDM model, we observe again a reduction of the ${\eta/\sigma_{\eta}}$ ratio for three bins down to 1.2. Concerning the lower half of Table 1 with the ${w_{0}w_{a}}$CDM model, when BAOs are not included we have ${\eta}$ non compatible with 0 in 5.7 ${\sigma}$ and, including the BAOs, the compatibility with 0 is given in 5.8 ${\sigma}$.
The increasing of the ratio ${\eta/\sigma_{\eta}}$ is observed when BAOs are added in the case of ${w_{0}w_{a}}$CDM model in three bins.
The results can be visualized in Figure 1.
Comparing the ${\eta/\sigma_{\eta}}$ ratios with the ones reported in Dainotti et al. (2021) (Table 1) we have that for the ${\Lambda}$CDM model the current ${\eta}$ values are compatible in 1 ${\sigma}$ with the ${\alpha}$ reported in Dainotti et al. (2021), while the ${\eta}$ estimated in the ${w_{0}w_{a}}$CDM model are compatible in 3 ${\sigma}$ with the ${\alpha}$ values in the same reference paper.
5 Perspective of the Future Contribution of GRB-Cosmology in 2030
The discussion of GRBs as possible cosmological tools has been going on for more than two decades (Atteia, 1997, 1998). The best bet is yet to come since we need first to identify the tightest correlation possible with a solid physical grounding. Among the many correlations proposed (Amati, L. et al., 2002; Yonetoku et al., 2004; Liang and Zhang, 2005; Ghirlanda, G. et al., 2010; Ito et al., 2019) we here choose to apply the fundamental plane (or Dainotti relation) (Dainotti et al., 2016; Simone et al., 2021; Cao et al., 2022, 2022), namely the three-dimensional relation between the end of the plateau emission’s luminosity, $L_{a}$, its time in the rest-frame, $T^{*}_{a}$, and the peak luminosity of the GRB, $L_{peak}$: it is possible to estimate how many GRBs are needed to obtain constraints for the cosmological parameters that are comparable with the ones obtained from the other probes, such as SNe Ia and BAOs. After a selection of the best fundamental plane sample through the trimming of GRBs, a simulation of a sample of 1500 and 2000 GRBs according to the properties of the fundamental plane relation has been performed.
The fundamental plane relation can be expressed as the following:
$$log_{10}L_{a}=a\times log_{10}T^{*}_{a}+b\times log_{10}L_{peak}+c,$$
(10)
where $a,b$ are the parameters of the plane and $c$ is the normalization constant. It is important to stress that here the variables $L_{a}$, $T^{*}_{a}$, and $L_{peak}$ have been corrected for evolutionary effects with redshift applying the Efron and Petrosian method Efron and Petrosian (1992). Based on Equation (10), we perform the maximization of the following log-likelihood for the simulated sample of GRBs:
$$\begin{split}ln\mathcal{L}_{GRB}=&-\frac{1}{2}\biggr{(}ln(\sigma^{2}+(a*\delta_{log_{10}T^{*}_{a}})^{2}+(b*\delta_{log_{10}L_{peak}})^{2}+\delta_{log_{10}L_{a}}^{2})\biggr{)}\\
&-\frac{1}{2}\biggr{(}\frac{(log_{10}L_{a,th}-log_{10}L_{a})^{2}}{\sigma^{2}+(a*\delta_{log_{10}T^{*}_{a}})^{2}+(b*\delta_{log_{10}L_{peak}})^{2}+\delta^{2}_{log_{10}L_{a}}}\biggr{)},\end{split}$$
(11)
where $L_{a,th}$ is the theoretical luminosity computed through the fundamental plane in Equation (10), $\sigma$ is the intrinsic scatter of the plane and $\delta_{log_{10}T^{*}_{a}}$, $\delta_{log_{10}L_{peak}}$, and $\delta_{log_{10}L_{a}}$ are the errors on the rest-frame time at the end of the plateau emission, the peak luminosity and the luminosity at the end of the plateau, respectively.
After performing an MCMC analysis using the D’Agostini method D’Agostini (1995) and letting vary the parameters $a,b,c,\sigma,\Omega_{0m}$, the results are shown in Figure 2. Through the simulations of 1000 GRBs, with 9500 steps and keeping the same errors (errors undivided) as the ones observed in the fundamental plane ($n=1$, see the upper left panel of Figure 2) we obtain a value of $\Omega_{0m}=0.310$ with a symmetrized uncertainty of $\sigma_{\Omega_{0m}}=0.078$. In the case of 2000 GRBs with 13,000 steps and $n=1$ (see the upper right panel of Figure 2) instead, we have $\Omega_{0m}=0.300$, $\sigma_{\Omega_{0m}}=0.052$. If we consider the division of the errors on the variables of the fundamental plane by a factor 2 (halved errors, $n=2$) we obtain, in the case of 1500 GRBs with 11100 steps, $\Omega_{0m}=0.300$, $\sigma_{\Omega_{0m}}=0.037$ (see the lower-left panel of Figure 2), while through 2000 simulated GRBs in 14,600 steps (still with $n=2$, see the lower right panel of Figure 2) we have $\Omega_{0m}=0.310$, $\sigma_{\Omega_{0m}}=0.034$. The idea of considering halved errors comes from the prospects for improvement in the fitting procedures of GRB light curves. Through this approach, the GRBs have provided constraints on the value of $\Omega_{0m}$ that are compatible with the ones of previous samples of SNe Ia: in the $n=1$ cases, the values of the uncertainties are comparable with the ones from Conley et al. (2010), while for the $n=2$ cases the values are close to the ones found in Betoule, M. et al. (2014) with 2000 GRBs. Furthermore, the GRBs have proven to be promising standardizable candles and, given the bigger redshift span they can cover if compared with SNe Ia, GRBs will provide more complete information about the structure and the evolution of the early universe after the Big Bang, together with quasars (Giada2021, ; bargiacchi2021quasar, ). After discussing the potentiality of GRBs as future standard candles, we estimate the frequency of GRBs with a plateau emission over the total number of GRBs observed to date. We can expect that by 2030 we will have reached several GRB observations such that these—as standalone probes that respect the properties of the GRB platinum sample Dainotti et al. (2020a)—will give constraints as precise as the ones from Conley et al. (2010) in the case of not halved errors. In case of halved errors, we can reach the level of precision of Conley et al. (2010) even now. In addition, if we consider a machine learning analysis Dainotti et al. (2021, 2019) for which we can double the size of the sample we are able to reach the precision of Conley et al. (2010) now with the case of $n=1$. If we consider the case of reconstructing the light curves and thus we have a sample which has the $47\%$ of cases with halved errors we can reach the limit of Conley et al. (2010) in 2022 if $n=1$ and now if $n=2$.
6 Discussions on the Results
Our results can be interpreted because of astrophysical selection biases or theoretical models alternatives to the standard cosmological models.
6.1 Astrophysical Effects
The main effect that has a stake in the SNe Ia luminosity variation is the presence of metallicity and the difference in stellar ages.
Indeed, ref. Scolnic et al. (2018) correct the PS with a mass-step contribution ($\Delta M$).
Despite this term improving the results, other effects need to be accounted for.
Considering the stretch and the color, ref. Childress et al. (2013) claim that the Hubble residuals, after being properly corrected according to the stretch and color observations, for SNe Ia in low mass and high mass host galaxies show a difference of $0.077\pm 0.014$ mag, compatibly with the result of Scolnic et al. (2018).
SNe Ia age metallicity and age are believed to be responsible for the observed behavior: those can replicate the Hubble residual trends consistent with the ones of Childress et al. (2013).
In the PS, to account for the evolutions of stretch ($\alpha$) and color ($\beta$), the parametrization utilized is the following: $\alpha(z)=\alpha_{0}+(\alpha_{1}\times z)$, $\beta(z)=\beta_{0}+(\beta_{1}\times z)$.
According to Scolnic et al. (2018), there is no clear dependence on the redshift for $\alpha(z)$ and $\beta(z)$, thus $\alpha_{1}$ and $\beta_{1}$ are set to zero.
Only the selection effect for color is noteworthy and Scolnic et al. (2018) consider the uncertainty on $\beta_{1}$ as a statistical contribution.
Concerning the stretch evolution in the PS calibration, it appears to be negligible and is not included at any level.
Conversely, ref. Nicolas et al. (2021) recently studied the SALT2.4 lightcurve stretch and showed that the SN stretch parameter is redshift-dependent.
According to their analysis, the asymmetric Gaussian model assumed by Scolnic et al. (2018) for describing the populations of SNe Ia does not take into consideration the redshift drift of the PS, thus leaving a residual evolutionary trend that manifests at higher redshifts.
Indeed, the simulations performed by Scolnic et al. (2018) for studying the systematics calibration reach redshifts up to $z=0.7$: this threshold is present in the third bin of our analysis.
The effect from $0.7\leq z\leq 2.26$ needs additional investigations.
It is worth noting that this decreasing trend of ${H_{0}}$ (with a given value of ${\eta}$) found in (Dainotti et al., 2021) is consistent in 1 ${\sigma}$ for the ${\Lambda}$CDM both in the cases of SNe Ia only and SNe+BAOs. When we consider the ${w_{0}w_{a}CDM}$, the ${\eta}$ values are compatible in 3 ${\sigma}$ with the ones with SNe Ia only and SNe + BAOs. We here have two cosmological parameters varying at the same time, differently from (Dainotti et al., 2021).
Therefore, one of the possible astrophysical reasons behind the observed trend is the residual stretch evolution with redshift. If so, in our work the effect is simply switched from stretch to $H_{0}$. The forthcoming release of the Pantheon+ data (Brout et al., 2021; Brownsberger et al., 2021; Carr et al., 2021; Popovic et al., 2021; Scolnic et al., 2021; Riess et al., 2022) will give the chance to test if these evolutionary effects may be still visible, but this analysis goes far beyond the scope of the current paper.
The astrophysical interpretation seems to be favored, but also many theoretical explanations may be possible to describe the outcome of these results.
6.2 Theoretical Interpretations
We now investigate possible theoretical explanations for our results, focusing particular attention on modified gravity models. We first discuss a general scalar-tensor formulation and, then, we concentrate our attention on the so-called metric $f(R)$ gravity.
6.2.1 The Scalar Tensor Theory of Gravity
The action of the scalar tensor theories (STTs) of gravity is given by $S=S^{JF}+S_{m}$ Damour and Nordtvedt (1993); Damour and Polyakov (1994); Damour (1996); Boisseau et al. (2000); Esposito-Farèse and Polarski (2001) with the Jordan Frame (JF) action
$$S^{JF}=\frac{1}{16\pi}\int d^{4}x\sqrt{-\tilde{g}}\,\left[\Phi^{2}\tilde{R}\;+4\,\omega(\Phi)\tilde{g}^{\mu\nu}\partial_{\mu}\Phi\partial_{\nu}\Phi-4\tilde{V}(\Phi)\right]\,,$$
(12)
where $\tilde{R}$ is the Ricci scalar obtained with the physical metric $\tilde{g}_{\mu\nu}$, while the matter fields $\Psi_{m}$ couple to the metric tensor $\tilde{g}_{\mu\nu}$ and not to $\Phi$, i.e., $S_{m}=S_{m}[\Psi_{m},\tilde{g}_{\mu\nu}]$.
In this Section we adopt natural units such that $c=1$ and $G=1$.
Different STTs follow with the appropriate choice of the two functions $\omega(\Phi)$ and $\tilde{V}(\Phi)$: e.g., the
Brans–Dicke (BD) theory Jordan (1955); Fierz (1956); Brans and Dicke (1961); Singh and Peracaula (2021) can be obtained for $\omega(\Phi)=\omega$ (const.) and $\tilde{V}(\Phi)=0$, while the metric $f(R)$ gravity, discussed in the next subsection, would correspond to $\omega\equiv 0$.
The action $S^{JF}$ can be rewritten in the Einstein Frame (EF), where one defines
$\tilde{g}_{\mu\nu}\equiv\displaystyle A^{2}(\varphi)g_{\mu\nu}$,
$\Phi^{2}\equiv\displaystyle 8\pi M_{\ast}^{2}A^{-2}(\varphi)$,
$V(\varphi)\equiv\displaystyle\frac{A^{4}(\varphi)}{4\pi}\tilde{V}(\Phi)$,
$\gamma(\varphi)\equiv\displaystyle\,\frac{d\log A(\varphi)}{d\varphi}$,
and $\gamma^{2}(\varphi)=\frac{1}{4\omega(\Phi)+6}$, to get
$$S^{EF}=\frac{M_{\ast}^{2}}{2}\int d^{4}x\sqrt{-{g}}\left[{R}+{g}^{\mu\nu}\partial_{\mu}\varphi\partial_{\nu}\varphi-\frac{2}{M_{\ast}^{2}}V(\varphi)\right]\,.$$
(13)
Matter is coupled to $\varphi$ only through a purely metric coupling, $S_{m}=S_{m}[\Psi_{m},A^{2}(\varphi){g}_{\mu\nu}]$ and $M_{\ast}$ is the Planck mass.
The physical quantities in the Jordan and Einstein frame are related by
$d\tilde{\tau}=A(\varphi)d\tau$, $\;\;\tilde{a}=A(\varphi)a$, $\tilde{\rho}=A(\varphi)^{-4}\rho$,
$\tilde{p}=A(\varphi)^{-4}p$, where $\tau$ is the synchronous time variable.
Defining $N\equiv\log\frac{a}{a_{0}}$, $\lambda\equiv\frac{V(\varphi)}{\rho}$, $w\equiv\frac{p}{\rho}$, and $\varphi^{\prime}=\frac{d\varphi}{dN}=a\frac{d\varphi}{da}$, the combination of cosmological equations allows to write the equation for $\varphi$ in the form (for a flat Friedmann–Robertson–Walker geometry) Catena et al. (2004)
$$\frac{2}{3}\;\frac{1+\lambda}{1-{\varphi^{\prime}}^{2}/6}\ \varphi^{\prime\prime}+[(1-w)+2\lambda]\varphi^{\prime}=-\sqrt{2}\;\gamma(\varphi)\;(1-3w)-2\;\lambda\;\frac{V_{\varphi}(\varphi)}{V}.$$
(14)
Moreover, the Jordan- and Einstein-frame Hubble parameters, $\tilde{H}\equiv d\log\tilde{a}/d\tilde{\tau}$ and $H\equiv d\log a/d\tau$, respectively, are related as
$$\tilde{H}=\frac{1+\gamma(\varphi)\,\varphi^{\prime}}{A(\varphi)}\,H\,.$$
(15)
For our purpose, we consider $A(\varphi)=A_{0}e^{c_{1}\varphi+c_{2}\varphi^{2}/2}$, which implies $\gamma(\varphi)=c_{1}+c_{2}\varphi$, where $c_{1,2}$ are constants.
Under the following conditions $\varphi^{\prime\prime}/\varphi\ll 1$, $\varphi^{\prime\,2}/\varphi^{2}\ll 1$, and $\frac{V_{\varphi}(\varphi)}{\varphi V\rho}\ll 1$, the solution of Equation (14) is $\varphi(z)=C(1+z)^{K}-\frac{c_{1}}{c_{2}}$, where
$K=\frac{1-3w}{1+w}\,\sqrt{2}\,c_{2}$, and $C$ is an integration constant.
We are looking for solutions such that $H=f(\varphi){\tilde{H}}_{0}$, so that ${\tilde{H}}=\frac{\tilde{H}_{0}}{(1+z)^{\eta}}$, where $\tilde{H}_{0}$ is constant.
These relations and (15) allow to derive $f(\varphi)$ (the expression of $f(\varphi)$ is quite involved, and in the case in which $c_{1,2}\ll 1$, it is a polynomial in $\varphi$).
The scalar field $\Phi$ in the (physical) JF can be cast in the form $\Phi(z)=\Phi_{0}(1+z)^{\tilde{K}}$,
where $\Phi_{0}\equiv\frac{\sqrt{8\pi}M_{*}}{A_{0}}\left[1-C\left(c_{1}-\frac{Cc_{2}}{c_{1}}\right)\right]$, ${\tilde{K}}=-\frac{KC(c_{1}+Cc_{2})}{1-C\left(c_{1}+\frac{Cc_{2}}{2}\right)}$, and $z<1$ has been used (note: ${\tilde{K}}$ is positive for $c_{1}$ or $c_{2}$ negative).
The scalar field $\Phi$ reduces to $\phi$
for $\Phi_{0}\to 1$ and ${\tilde{K}}\to 2\eta$.
From the Friedmann Equation Catena et al. (2004)
$$\left(\frac{\dot{a}}{a}\right)^{2}=\frac{1}{3M_{\ast}^{2}}\left[\rho+\frac{M_{\ast}^{2}}{2}\dot{\varphi}^{2}+V(\varphi)\right]\,,$$
(16)
with $\rho$ given by matter ($\rho=\rho_{0m}/a^{3}=\rho_{0m}(1+z)^{3}$), and $c_{1,2}\ll 1$, one infers the effective potential
$$\frac{\tilde{V}}{3m^{2}}=\frac{4\pi M_{*}^{2}}{A_{0}^{2}}\,\left[f_{0}^{2}-\frac{1}{\Omega_{0m}}\left(\frac{\Phi}{\Phi_{0}}\right)^{\frac{3}{2\eta}}-\frac{C^{2}K^{2}\varphi_{0}^{2}}{6\Omega_{0m}}\left(\frac{\Phi}{\Phi_{0}}\right)^{\frac{K-\eta}{\eta}}\right]\,,$$
(17)
where we recall that $\Omega_{0m}=\rho_{0m}/\rho_{cr}$, $\rho_{cr}=3M_{*}^{2}{\tilde{H}}_{0}^{2}$, $f_{0}=f(\varphi=0)$, and $m^{2}=\Omega_{0m}{\tilde{H}}^{2}_{0}$.
For redshift $0\leqslant z<0.3$, to which we are interested, the scalar field varies slowly with $z$, $\Phi\sim\Phi_{0}$, so that the effective potential behaves like a cosmological constant.
We see how the proposed scalar-tensor formulation has the right degrees of freedom to reproduce, in the JF, the required behavior of the (physical) trend of $H_{0}(z)$. In the next subsection, we analyze a sub-case of the general paradigm discussed above, which leads to the well-known $f(R)$ gravity, which is among the most popular modified gravity formulations.
6.2.2 Metric f(R) Gravity in the Jordan Frame
The observed decaying behavior of the Hubble constant $H_{0}$ with the redshift draws significant attention for an explanation and, if it is not due to selection effects or systematics in the sample data, we need to interpret our results from a physical point of view.
As already argued in Kazantzidis and Perivolaropoulos (2020) and Dainotti et al. (2021), the simplest way to account for this unexpected behavior of $H_{0}\left(z\right)$ is that the Einstein constant $\chi=8\pi G$ (where $G$ denotes the gravitational constant), mediating the gravity-matter interaction, is subjected itself to a slow decaying profile with the redshift. In this Section, we consider $c=1$ for the speed of light.
More specifically, since the critical energy density $\rho_{c0}=3\,H_{0}^{2}/\chi$ today must be a constant, we need an evolution for $\chi\sim\left(1+z\right)^{-2\,\eta}$, considering the function $H_{0}\left(z\right)$ given by Equation (9).
The evolution of $\chi\left(z\right)$ is not expected within the cosmological Einsteinian gravity, therefore we are led to think of it as a pure dynamical effect, associated with a modified Lagrangian for the gravitational field beyond the $\Lambda$CDM cosmological model.
Ref. Li et al. (2015) obtained cosmological constraints within the Brans–Dicke theory considering how the evolution of the gravitational constant $G$, contained in $\chi$, affects the SNe Ia peak luminosity.
The most natural extended framework is the $f(R)$-gravity proposal Sotiriou (2006); Nojiri and Odintsov (2007); Sotiriou and Faraoni (2010); Capozziello and de Laurentis (2011) which contains only an additional scalar degree of freedom.
For instance, Ref. Odintsov et al. (2021) try to alleviate the $H_{0}$ tension considering exponential and power-law $f(R)$ models.
The formulation of the $f(R)$ theories in an equivalent scalar-tensor paradigm turns out to be particularly intriguing for our purposes: the function $f(R)$ is restated as a real scalar field $\phi$, which is non-minimally coupled to the metric in the JF.
The information about the function $f$ turns into the expression of the scalar field potential $V\left(\phi\right)$.
The relevance of modified gravity models relies on the possibility that this revised scenario for the gravitational field can account for the physics of the so-called “dark universe”component without the need for a cosmological constant.
Indeed, the observed cosmic acceleration in the late universe via the SNe Ia data is a pure dynamical effect, i.e., associated with a modification of the Einsteinian gravity at very large scales (in the order of the present Hubble length).
According to the standard literature on this field (which includes a large number of proposals), three specific $f(R)$ models, i.e., the Hu–Sawicki Hu and Sawicki (2007), the Starobinsky Starobinsky (2007), and Tsujikawa models Amendola et al. (2007); Tsujikawa (2008), successfully describe the Dark Energy component (say an effective parameter for the Dark Energy $w=w(z)<-1/3$) and overcome all local constraints.
The difference in the form of the Lagrangian densities associated with $f(R)$ models is reflected in the morphology of the potential term governing the dynamics of the scalar field.
For instance, the scalar field potential related to the Hu–Sawicki $f(R)$ proposal, with the power index $n=1$, in the JF is given by
$$V\left(\phi\right)=\frac{m^{2}}{c_{2}}\left[c_{1}+1-\phi-2\sqrt{c_{1}\left(1-\phi\right)}\right],$$
(18)
where we have two free parameters $c_{1}$ and $c_{2}$, while $m^{2}=\chi\,\rho_{0m}/3$.
The scalar-tensor dynamics in the JF for a flat FLRW metric with a matter component is summarized by
$$\displaystyle H^{2}=\frac{\chi\,\rho}{3\,\phi}-H\,\frac{\dot{\phi}}{\phi}+\frac{V\left(\phi\right)}{6\,\phi}$$
(19)
$$\displaystyle\frac{\ddot{a}}{a}=-\frac{\chi\,\rho}{3\,\phi}-\frac{V\left(\phi\right)}{6\,\phi}+\frac{1}{6}\,\frac{dV}{d\phi}+\frac{\dot{a}\,\dot{\phi}}{a\,\phi}$$
(20)
$$\displaystyle 3\ddot{\phi}-2\,V\left(\phi\right)+\phi\,\frac{dV}{d\phi}+9\,H\,\dot{\phi}=\chi\,\rho,$$
(21)
which are the generalized Friedmann equation, the generalized cosmic acceleration equation and the scalar field equation, respectively
Sotiriou and Faraoni (2010).
We recall that $\phi=\phi\left(t\right)$ is a function of the time (or the redshift $z$) only for an isotropic universe.
Considering the first term on the right-hand side of Equation (19), it is possible to recognize that $\phi$ mediates the gravity-matter coupling, and therefore it mimics a space-time varying Einstein constant.
Hence, to account for our observed decay of $H_{0}\left(z\right)$, we have to require that the scalar field assumes a specific behavior with the redshift, i.e.
$$\phi\left(z\right)=\left(1+z\right)^{2\,\eta}.$$
(22)
Moreover, the remaining terms contained in the gravitational field equations must be negligible.
This situation is naturally reached when the potential term is sufficiently slow-varying in a given time interval.
We see that the hypothesis of a near-frozen scalar field evolution is a possible assumption, as far as the potential term should provide a dynamical impact, sufficiently close to a cosmological constant term.
These simple considerations lead us to claim that this scenario is worth to be investigated for the behavior of $H_{0}\left(z\right)$ here observed.
The specific cosmological models affect the expression of the luminosity distance and this should be the starting point of a careful test of a $f(R)$ theory versus the comprehension of the $H_{0}$ tension.
A new binned analysis of the PS, using the corrected luminosity distance obtained through a reliable $f(R)$, may in principle shed new light on the observed decaying trend of $H_{0}\left(z\right)$, testing also new physics.
This analysis is performed in the next Section.
As a preliminary approach, we try to understand which profile we could expect for the scalar field potential, inferred from the behavior of $H_{0}\left(z\right)$.
This is quite different from a standard analysis of $f(R)$ models.
Generally, a specific $f(R)$ function is defined a priori, and then the dynamical equations are studied to obtain constraints on the free parameters.
Here, instead, starting from the observed decreasing trend of $H_{0}\left(z\right)$ and assuming $\phi\left(z\right)$ from Equation (22), we wonder what the scalar field potential would be in a scalar-tensor dynamics.
Eventually, we should have a scalar field in near-frozen dynamics, i.e., a slow-roll of the scalar field potential, mimicking a cosmological constant term ($\phi\rightarrow 1$).
To this end, we rewrite the generalized Friedmann Equation (19) and calculate $V\left(\phi\right)$:
$$V\left(\phi\right)=6(1-2\eta)\,\left(\frac{dz}{dt}\right)^{2}\,\phi^{1-1/\eta}-6m^{2}\,\phi^{3/\,2\eta}\,,$$
(23)
where we have used the standard definition of redshift and the relation (22) for $\phi\left(z\right)$.
Moreover, we recall that for a matter component $\rho\sim(1+z)^{3}$.
As a final step, we need to calculate the term $\frac{dz}{dt}$.
Starting again from the redshift definition, it is well known that
$$\frac{dz}{dt}=-(1+z)\,H(z).$$
(24)
In principle, we would need to compute the Hubble parameter $H(z)$ from the field equations, and then replace $H(z)$ in the term $\frac{dz}{dt}$.
However, this procedure is not viable, since we need to fix a well-defined $V\left(\phi\right)$ to solve the field equations.
Moreover, $H(z)$ appears also in the right-hand-side of Equation (19), because of the non-minimal coupling with the scalar field.
Therefore, we can not calculate exactly $\frac{dz}{dt}$ to get $V\left(\phi\right)$ in the JF.
Then, to obtain ${V\left(\phi\right)}$ inferred from the trend of $H_{0}(z)$, we require that the Hubble function provides the same physical mechanism suggested from our binned analysis in Section 4, i.e., simply replacing ${H_{0}}$ with ${H_{0}(z)}$ given by Equation (9) in the standard Friedmann equation in the ${\Lambda}$CDM model.
With this new definition of ${\mathcal{H}_{0}}$, we write the following condition on the Hubble function:
$$H\left(z\right)=\frac{\mathcal{H}_{0}}{(1+z)^{\eta}}\sqrt{\Omega_{0m}\,\left(1+z\right)^{3}+1-\Omega_{0m}}\,.$$
(25)
In doing so, using Equations (23), (24) and (25), we determine the form of the scalar field potential
$$\frac{V\left(\phi\right)}{m^{2}}=6\,(1-2\eta)\,\left(\frac{1-\Omega_{0m}}{\Omega_{0m}}\right)-12\eta\,\phi^{3/2\eta}\,$$
(26)
inferred from the decreasing trend of ${H_{0}(z)}$. In other words, the potential Equ. 26 might provide an effective Hubble constant that evolves with redshift. In the computation, we have used the expression ${\Omega_{0m}=m^{2}/\mathcal{H}_{0}^{2}}$.
In Figure 3, we plotted this potential profile, observing that, as expected, a flat region consistently appears, validating our guess on the feasibility of $f(R)$-gravity in the JF to account for the observed behavior of $H_{0}(z)$. We set $\eta=0.009$ in Figure 3, according to our binned analysis results for three bins (see Table 1).
We stress that the flatness of the potential does not emerge throughout the Pantheon sample redshift range, $0<z<2.3$, but it appears only in a narrow region for $0<z\lesssim z^{*}$, where $z^{*}=0.3$ is the redshift at the Dark Energy and Matter components equivalence of the universe.
This form of $V\left(\phi\right)$ is reasonable since the Dark Energy contribution, provided by the scalar field in the JF gravity, dominates the matter component only for $0<z\ll z^{*}$.
It is the weak dependence of $H_{0}$ on $z$ that ensures the existence of a flat region of the potential, according to the theoretical scenario argued above.
Finally, we can calculate the form of the $f(R)$ function associated with the potential profile. Recalling the following general relations in the JF
Sotiriou and Faraoni (2010):
$$\displaystyle R=\frac{dV}{d\phi},$$
(27)
$$\displaystyle f(R)=R\,\phi(R)-V\left(\phi(R)\right),$$
(28)
we can obtain:
$$f\left(R\right)=-6\,m^{2}\,\left[\left(1-2\eta\right)\frac{1-\Omega_{0m}}{\Omega_{0m}}+\left(3-2\eta\right)\,\left(-\frac{R}{18\,m^{2}}\right)^{\frac{3}{3-2\eta}}\right]\,.$$
(29)
Note that the formula above provides a generalization of the Einstein theory of gravity, as it should be in the context of a ${f(R)}$ model. Indeed, if ${\eta=0}$, then ${f(R)\equiv R}$ reproduces exactly the Einstein–Hilbert Lagrangian density in GR with a cosmological constant ${\Lambda}$, as soon as you recognize that ${\Lambda=3m^{2}\left(1-\Omega_{0m}\right)\,/\,\Omega_{0m}}$ for a flat geometry, using ${m^{2}=\mathcal{H}^{2}_{0}\,\Omega_{0m}}$.
In particular, expanding the function (29) for ${\eta\sim 0}$, we can see explicitly the deviation from the Einstein–Hilbert term:
$$f\left(R\right)\approx\left(R-6\,m^{2}\,\frac{1-\Omega_{0m}}{\Omega_{0m}}\right)+\frac{2}{3}\eta\,\left[R\,\ln{\left(-\frac{R}{m^{2}}\right)}-\left(1+\ln{18}\right)\,R+18m^{2}\,\frac{1-\Omega_{0m}}{\Omega_{0m}}\right]+O\left(\eta^{2}\right)\,.$$
(30)
The first term at the zero-th order in ${\eta}$ is exactly the Einstein–Hilbert Lagrangian density, while the linear term in ${\eta}$ provides the correction to GR. Therefore, ${\eta}$, in addition to being the physical parameter that describes the evolution of $H_{0}(z)$, also denotes the deviation from GR and the standard cosmological model.
It is worthwhile to remark that the expression above may not be the final form of the underlying modified theory of gravity, associated with the global universe dynamics, but only its asymptotic form in the late Universe, i.e., as the scalar of curvature approaches the value corresponding to the cosmological constant in the $\Lambda$CDM model. In all these computations we do not consider relativistic or radiation components at very high redshifts, but it may be interesting to test this model with other local probes in the late Universe.
In this discussion, we infer that the dependence of $H_{0}$ on the redshift points out the necessity of new physics in the description of the universe dynamics and that such a new framework may be identified in the modified gravity, related to metric theories.
7 The Binned Analysis with Modified f(R) Gravity
To try to explain the observed trend of $H_{0}(z)$, we focus on $f(R)$ theories of gravity, and then we perform the same binned analysis, using the correction for the distance luminosity according to the modified gravity.
We start from the gravitational field action Sotiriou and Faraoni (2010):
$$S_{g}=\frac{1}{2\chi}\int d^{4}x\sqrt{-g}f(R),$$
(31)
where $f(R)$, as a function of the Ricci scalar $R$, is an extra degree of freedom compared to General Relativity.
We rewrite $f(R)=R+F(R)$ to highlight the deviation from the standard gravity.
Varying the total action with respect to the metric, we obtain the flat FLRW metric field equations:
$$H^{2}(1+F_{R})=\frac{\chi\rho}{3}+\biggr{[}\frac{R\,F_{R}-F}{6}-F^{RR}H\dot{R}\biggr{]},$$
(32)
where $F_{R}\equiv\frac{dF(R)}{dR}$. The Ricci scalar $R$ can be cast in the form
$$R=12H^{2}+6HH^{\prime},$$
(33)
where the Hubble parameter $H$ is expressed as a function of $\gamma\equiv ln(a)$, and the prime indicates the derivative with respect to $\gamma$.
Now, we introduce two dimensionless variables Hu and Sawicki (2007)
$$y_{H}=\frac{H^{2}}{m^{2}}-\frac{1}{a^{3}},\hskip 43.05542pty_{R}=\frac{R}{m^{2}}-\frac{3}{a^{3}},$$
(34)
which denote the deviation of $H^{2}$ and $R$ with respect to the matter contribution when compared to the $\Lambda$CDM model.
We rewrite the modified Friedmann Equation (32) and the Ricci scalar relation (33) in terms of $y_{H}$ and $y_{R}$.
Then, we have a set of coupled ordinary differential equations:
$$\displaystyle y_{H}^{\prime}=\frac{1}{3}y_{R}-4y_{H}$$
(35)
$$\displaystyle y_{R}^{\prime}=\frac{9}{a^{3}}-\frac{1}{y_{H}+a^{-3}}\frac{1}{m^{2}F_{RR}}\,\biggr{[}y_{H}-F_{R}\biggr{(}\frac{1}{6}y_{R}-y_{H}-\frac{a^{-3}}{2}\biggr{)}+\frac{1}{6}\frac{F}{m^{2}}\biggr{]}.$$
(36)
The solution of this coupled first-order differential equations system above can not be obtained analytically, but can be numerically calculated. We need initial conditions such that this scenario mimics the $\Lambda$CDM model in the matter dominated universe at initial redshift $z_{i}\gg z^{*}$. Hence, we impose the following conditions for ${y_{H}}$ and ${y_{R}}$ at the redshift ${z_{i}}$:
$$\displaystyle y_{H}(z_{i})=\frac{\Omega_{0\Lambda}}{\Omega_{0m}}$$
(37)
$$\displaystyle y_{R}(z_{i})=12\frac{\Omega_{0\Lambda}}{\Omega_{0m}}.$$
(38)
The standard ${\Lambda}$CDM model is reached for ${z=z_{i}}$ or asymptotically, and we consider a flat geometry, such that ${\Omega_{0\Lambda}=1-\Omega_{0m}}$. Finally, the luminosity distance can be written as
$$d_{L}(z)=\frac{(1+z)}{H_{0}}\int^{z}_{0}\frac{dz^{\prime}}{\sqrt{\Omega_{0m}\biggr{(}y_{H}(z^{\prime})+(1+z^{\prime})^{3}\biggr{)}}}\,,$$
(39)
including the solution $y_{H}(z)$ from Equation (34) Martinelli et al. (2009).
Hu–Sawicki Model
We focus on the Hu–Sawicki model with $n=1$, considering a late-time gravity modification, described by the following function Hu and Sawicki (2007):
$$f(R)=R+F(R)=R-m^{2}\frac{c_{1}\left(R/m^{2}\right)^{n}}{c_{2}\left(R/m^{2}\right)^{n}+1},$$
(40)
corresponding to the potential $V(\phi)$ in Equation (18).
The parameters $c_{1}$ and $c_{2}$ are fixed by the following conditions Hu and Sawicki (2007)
$$\displaystyle\frac{c_{1}}{c_{2}}\approx 6\frac{\Omega_{0\Lambda}}{\Omega_{0m}}$$
(41)
$$\displaystyle F_{R0}\approx-\frac{c_{1}}{c_{2}^{2}}\left(\frac{12}{\Omega_{0m}}-9\right)^{-2},$$
(42)
where ${F_{R0}}$ is the value of the field ${F_{R}\equiv dF/dR}$ at the present time, and ${F(R)}$ is the deviation from the Einstein–Hilbert Lagrangian density.
Cosmological constraints provide $|F_{R0}|\leq 10^{-7}$ from gravitational lensing and $|F_{R0}|\leq 10^{-3}$ from Solar system Burrage and Sakstein (2018); Liu et al. (2018).
We explore several choices of $F_{R0}$.
To simplify the numerical integration of the modified luminosity distance (39), we approximate the numerical solution $y_{H}$, obtained from the system (35), (36), by a polynomial of order 8. This function is an accurate representation of $y_{H}$ when we restrict the solution to the range of PS (see Figure 4).
As a consequence, we obtain constraints on $c_{1}$ and $c_{2}$, according to Equations (41) and (42).
Then, we perform the same binned analysis of Section 4 using the Hu–Sawicki model and the modified luminosity distance (39).
We here run the analysis both for the case of $\Omega_{0m}$ fixed to a fiducial value of ${0.298}$ and for several values of $F_{R0}=-10^{-7},-10^{-6},-10^{-5},-10^{-4}$ (see Table 3 and Figure 5) or we let $\Omega_{0m}$ vary with the two values of $F_{R0}=-10^{-7},-10^{-4}$ (see Table 4 and Figure 7) for the SNe alone and with SNe +BAOs. Note also that the ${\eta}$ parameters are all consistent for the several values of ${F_{R0}}$ in 1 ${\sigma}$, as you can see in Table 3, for both SNe Ia and SNe Ia + BAOs. Moreover, the values of ${\eta}$ are consistent in 1 ${\sigma}$ with the ones obtained from the analysis of the ${\Lambda}$CDM model (see also Table 1).
We consider the cases $F_{R0}=-10^{-7}$, $F_{R0}=-10^{-4}$ and, to study how these results may vary according to the different values of $\Omega_{0m}$ chosen, we tested the model with four values of $\Omega_{0m}=(0.301,0.303,0.305)$ taken from the 1 $\sigma$ from a Gaussian distribution centred around the most probable value of $0.298$, see (Scolnic et al., 2021).
We show in Figure 7 the comparison between the different applications of the Hu–Sawicki model: in the left panels (upper and lower), we consider SNe Ia only, while in the right panels (upper and lower) we combine SNe Ia+BAOs. We here remind that the assumed values for $|F_{R0}|$ of $10^{-4}$ and $10^{-7}$ are well constrained by the $f(T)$ theories. (Hu and Sawicki, 2007; Sotiriou and Faraoni, 2010).
Thus, the existence of this trend is, once again, confirmed, and it remains unexplained also in the modified gravity scenario.
Indeed, a suitable modified gravity model which would be able to predict the observed trend of $H_{0}$, would allow observing a flat profile of $H_{0}(z)$ after a binned analysis.
Further analysis must be carried out with other Dark Energy models or other modified gravity theories to investigate this issue in the future, for instance focusing on the proposed model in Section 6.2.2.
8 Requirements for a Suitable f(R) Model
Since the Hu–Sawicki model seems to be inadequate to account for the observed phenomenon of the decaying $H_{0}(z)$, in what follows, we provide some general properties that an $f(R)$ model in the JF must possess to induce the necessary scenario of a slowly varying Einstein constant.
Now, we consider again the dynamical impact of the scalar field $\phi$, related to the $f(R)$ function.
Let us observe that the following relation holds in the following way:
$$\frac{d\phi}{dz}=-\frac{1}{1+z}\frac{\dot{\phi}}{H}\,.$$
(43)
In order to get the desired behavior $\phi\simeq(1+z)^{2\eta}$, we must deal with a dynamical regime where the following request is satisfied:
$$\frac{\dot{\phi}}{H}=-2\eta\phi\,.$$
(44)
We consider a slow-rolling evolution of the scalar field $\phi$ in the late universe, near enough to $\phi\simeq 1$.
Then, we consider in Equation (19) $\rho\sim 0$, because we are in the Dark Energy dominated phase, and we consider $H_{0}\,\dot{\phi}$ small with respect to the potential term $V(\phi\simeq 1)$.
We neglect, also, the term $\ddot{\phi}$.
Under these conditions, Equations (19) and (21) become
$$H^{2}=\frac{V}{6\phi}\,$$
(45)
and
$$\frac{\dot{\phi}}{H}=\frac{1}{9H^{2}}\left(2V-\phi\frac{dV}{d\phi}\right)\,,$$
(46)
respectively.
Referring to Equation (45) at $z\sim 0$, we make the identification $H_{0}^{2}\equiv V(\phi\simeq 1)/6$.
Hence, in order to reproduce Equation (44), we must require that for $\phi\rightarrow 1$, the following relation holds:
$$\eta=\frac{1}{3V}\left(\phi\frac{dV}{d\phi}-2V\right)\,.$$
(47)
The analysis above states the general features that a $f(R)$ model in the JF has to exhibit to provide a viable candidate to reproduce the observed decay behavior of $H_{0}(z)$ (Equ. 36).
We conclude by observing that the picture depicted above relies on the concept of a slow-rolling phase of the scalar field, when it approaches the value $\phi\simeq 1$ and, in this respect, the potential term should have for such value a limiting dynamics, which remains there confined for a sufficiently long phase. It is just in such a limit that we are reproducing a $\Lambda$CDM model, but with the additional feature of a slowly varying Einstein constant.
As far as the value of $z$ increases, the deviation of the considered model from General Relativity becomes more important, but this effect is observed mainly in the gravity-matter coupling.
In other words, the motion of the photon, as observed in the gravitational lensing, is not directly affected by the considered deviation, since the geodesic trajectories in the space-time do not directly feel the Einstein constant value.
This consideration could allow for a large deviation of $\phi$ from the unity that is expected in studies of the photons’ propagation.
8.1 An Example for Low Redshifts
As a viable example for the Dark Energy dominated Universe (slightly different from the traced above), we consider a potential term (and the associated slow-rolling phase) similar to the one adopted in the so-called chaotic inflation Linde (1983); Riotto (2003), i.e.,:
$$V(\phi)=\delta+6H_{0}^{2}\phi^{2}\,,$$
(48)
where $\delta$ is a positive constant, such that $\delta\ll 6H_{0}^{2}$. From Equation (45), we immediately get
$$H^{2}\simeq H_{0}^{2}\phi\sim H_{0}^{2}\,,$$
(49)
where, we recall that we are considering the slow-rolling phase near $\phi\rightarrow 1$.
Analogously, from Equation (47), we immediately get:
$$\eta\sim-\frac{\delta}{9H_{0}^{2}}\,.$$
(50)
The negative value of $\eta$ is coherent with the behavior $H^{2}\propto\phi$. Hence, we can reproduce the requested behavior of $\phi(z)$ by properly fixing the value of $\delta$ to get $\eta$ as it comes out from the data analysis of Section 4. Specifically, we get $\delta\sim 10^{-3}H_{0}^{2}$ to have $\eta\sim 10^{-2}$.
Furthermore, it is easy to check that, for $\phi\rightarrow 1$, Equation (44) and Equation (49), we find the relation
$$\ddot{\phi}\sim\mid\eta\mid H_{0}\dot{\phi}\ll 3H_{0}\dot{\phi}\,,$$
(51)
which ensures that we are dealing with a real slow-rolling phase.
Finally, we compute the $f(R)$ function corresponding to the potential in Equation (48), recalling the relation (28):
$$f(R)=\frac{R^{2}}{24H_{0}^{2}}-\delta\,.$$
(52)
We conclude by observing that this specific model is reliable only as far the universe matter component is negligible, $z<0.3$. The Einstein constant in front of the matter-energy density $\rho$ would run as $(1+z)^{2\eta}$. The example above confirms that the $f(R)$ gravity in the JF is a possible candidate to account for the observed effect of $H_{0}(z)$, but the accomplishment of a satisfactory model for the whole $\Lambda$CDM phase requires a significant effort in further investigation, especially accounting for the constraints that observations in the local universe provided for modified gravity.
8.2 Discussion
Let us now try to summarize the physical insight that we can get from the analysis above, about the possible theoretical nature of the observed $H_{0}(z)$ behavior.
We can keep as a reliably good starting point the idea that the
origin of a modified scaling of the function $H(z)$ with respect to the standard $\Lambda$CDM model can be identified in a slowly varying Einstein constant with the redshift. Furthermore, it is a comparably good assumption to search, in the framework of a scalar-tensor formulation of gravity, the natural explanation for such a varying Einstein constant. As shown in Section 6.2, a scalar-tensor formulation
can reproduce the required scaling of the function $H(z)$, which we observe as an $H_{0}(z)$ behavior in the
standard $\Lambda$CDM model.
Hence, we naturally explored one of the most interesting and
well-motivated formulations of a scalar-tensor theory, namely the $f(R)$ gravity in the JF. In this respect, in Section 6.2.2, we first evaluated the form of the scalar field potential inferred from the observed decreasing trend of $H_{0}(z)$, and our data analysis suggested a model described in Equations (26) and (29). Then, we investigated if, one of the most reliable models for reproducing the Dark Energy effect with modified gravity, i.e., the Hu–Sawicki proposal, was able to induce the requested luminosity distance to somehow remove the observed effect, thus accounting for its physical nature. The non-positive result of this investigation leads us to explore theoretically the question of reproducing simultaneously the Dark Energy contribution and the observed $H_{0}(z)$ effect, by a single $f(R)$ model of gravity in the JF. In Section 8, it has been addressed this theoretical question, by establishing the conditions that a modified gravity model has to satisfy to reach the simultaneous aims mentioned above. Finally, we considered a specific model for the late universe, based on a slow-rolling picture for the scalar field near its today value $\phi\simeq 1$. This model was successful in explaining the Dark Energy contribution and the necessary variation of the Einstein constant, but it seems hard to be reconciled with the earlier Universe behavior, when the role of the matter contribution becomes relevant.
Thus, based on this systematic analysis, we can conclude that the explanation for $H_{0}(z)$ is probably to be attributed to modified gravity dynamics, but it appears more natural to separate its effect from the existence of a Dark Energy contribution. In other words, we are led to believe that what we discovered about the SNe Ia+BAOs binned analysis must be regarded as a modified gravity physics of the scalar-tensor type, but leaving on a standard Universe, well represented by a $\Lambda$CDM model a priori.
9 Conclusions
We analyzed the PS together with the BAOs in three bins in both the $\Lambda$CDM and $w_{0}w_{a}$CDM models to investigate if an evolutionary trend of $H_{0}$ persists also with the contribution of BAOs and by varying two parameters contemporaneously with ${H_{0}}$ (${\Omega_{0m}}$ and ${w_{a}}$ for the ${\Lambda}$CDM and ${w_{0}w_{a}}$CDM, respectively). The persistence of the trend of $H_{0}$ as a function of redshift is also shown in the case of the Hu–Sawicki model. We here stress that the main goal of the current analysis is to highlight the reliability of the trend of ${H_{0}(z)}$ and not to further constrain ${\Omega_{0m}}$ or any other cosmological parameters.
With the subsequent fitting of $H_{0}$ values through the model $g(z)=\mathcal{H}_{0}/(1+z)^{\eta}$, we obtain $\eta\sim 10^{-2}$,
as in the previous work Dainotti et al. (2021):
those are compatible with zero from 1.2 to 5.8 $\sigma$ (see Table 1). The multidimensional results could reveal a dependence on the redshift of $H_{0}$, assuming that it is observable at any redshift scale.
If this evolution is not caused by statistical effects and other selection biases or hidden evolution of SNe Ia parameters Nicolas et al. (2021), we show how $H_{0}(z)$ could modify the luminosity distance definition within the modified theory of gravity.
If we consider a theoretical interpretation for the observed trend, new cosmological scenarios may explain an evolving Hubble constant with the redshift.
For instance, we test in Section 6.2, and Section 7 a simple class of modified gravity theories given by the $f(R)$ models in the equivalent scalar-tensor formalism.
In principle, this could be due to an effective varying Einstein constant governed by a slow evolution of a scalar field which mediates the gravity-matter interaction.
However, the slow decreasing trend of $H_{0}$ has proven to be independent of the Hu–Sawicki model application.
Indeed, if this theory had worked we would have observed the trend of the $\eta$ parameter to be flattened out and be compatible with 0 in 1 $\sigma$ at any redshift bin.
This is not the case, thus new scenarios must be explored within the modified theories of gravity or slightly alternative approaches (see Sec. 8.2).
We can state that this evolving trend of $H_{0}$ is independent of the starting values of the fitting for $H_{0}$ (we here have considered $H_{0}=70$) and, thus, on the fiducial $M$ and on the redshift bins and even when we consider two cosmological parameters changing contemporaneously (${\Omega_{0m}}$ and ${w_{a}}$ in ${\Lambda}$CDM and ${w_{0}w_{a}}$CDM models, respectively). Thus, we need to further investigate the nature of this trend. In addition, the implementation of GRBs as cosmological probes together with SNe Ia and BAOs has proven to be not only possible in a near future but also necessary since the redshift range that GRBs cover is much larger than the one typical of SNe Ia. This last characteristic will surely allow GRBs to give further information on the nature of the early universe and pose new constraints in the future measurements of $H_{0}$.
\authorcontributions
M. G. D. performed the conceptualization of all project, data curation, formal analysis, methodology, writing original draft, validation, supervision, software.
B. D. S. performed data curation, visualization, formal analysis, methodology, writing original draft, software.
T. S. performed formal analysis, visualization, methodology, writing original draft on f(R) and revised it, G. M. performed a partial conceptualization limited to the theoretical part of the f(R) gravity theory; E. R. edited and review the analysis on f(R) and participated in the general discussion and conceptualization of the paper.
G. L. wrote Sec. 6.2.1 and gave suggestions on the cosmological constraints on $w$.
M. B. performed the conceptualization on the priors to answer the referee report.
S. U. performed a formal analysis on changing the parameters together with $H_{0}$.
\funding
This research received no external funding.
\institutionalreview
Not applicable
\informedconsent
Not applicable
\dataavailability
Not applicable
Acknowledgements.This work made use of Pantheon sample data Scolnic et al. (2018), which can be found in the GitHub repository: https://github.com/dscolnic/Pantheon (accessed on 21 December 2020). This work made use of data supplied by the UK Swift Science Data Centre at the University of Leicester.
We are thankful to V. Nielson, A. Lenart, G. Sarracino, and D. Jyoti for their support on cosmological computations. T. S. is supported in part by INFN under the program TAsP (Theoretical Astroparticle Physics). G. Lambiase and B. De Simone acknowledge the support of INFN. T. S. acknowledges the support of the Department of Physics of the University of Pisa. M. G. Dainotti acknowledges the support from NAOJ and NAOJ—Division of Science.
\conflictsofinterestThe authors declare no conflict of interest.
\printendnotes[custom]
\reftitleReferences
References
Riess et al. (1998)
Riess, A.G.; Filippenko, A.V.; Challis, P.; Clocchiatti, A.; Diercks,
A.; Garnavich, P.M.; Gilliland, R.L.; Hogan, C.J.; Jha, S.;
Kirshner, R.P.; et al.
Observational Evidence from Supernovae for an Accelerating Universe
and a Cosmological Constant.
Astron. J. 1998, 116, 1009–1038,
doi:\changeurlcolorblack10.1086/300499.
Perlmutter et al. (1999)
Perlmutter, S.; Aldering, G.; Goldhaber, G.; Knop, R.A.; Nugent, P.;
Castro, P.G.; Deustua, S.; Fabbro, S.; Goobar, A.; Groom, D.E.;
et al.
Measurements of $\Omega$ and $\Lambda$
from 42 High-Redshift Supernovae.
Astrophys. J. 1999, 517, 565–586,
doi:\changeurlcolorblack10.1086/307221.
Weinberg (1989)
Weinberg, S.
The cosmological constant problem.
Rev. Mod. Phys. 1989, 61, 1–23.
doi:\changeurlcolorblack10.1103/RevModPhys.61.1.
Peebles and Ratra (2003)
Peebles, P.J.; Ratra, B.
The cosmological constant and dark energy.
Rev. Mod. Phys. 2003, 75, 559–606,
doi:\changeurlcolorblack10.1103/RevModPhys.75.559.
Riess et al. (2019)
Riess, A.G.; Casertano, S.; Yuan, W.; Macri, L.M.; Scolnic, D.
Large Magellanic Cloud Cepheid Standards Provide a 1% Foundation
for the Determination of the Hubble Constant and Stronger Evidence for
Physics beyond $\Lambda$CDM.
Astrophys. J. 2019, 876, 85,
doi:\changeurlcolorblack10.3847/1538-4357/ab1422.
Camarena and Marra (2020)
Camarena, D.; Marra, V.
Local determination of the Hubble constant and the deceleration
parameter.
Phys. Rev. Res. 2020, 2, 013028,
doi:\changeurlcolorblack10.1103/PhysRevResearch.2.013028.
Wong et al. (2020)
Wong, K.C.; Suyu, S.H.; Chen, G.C.F.; Rusu, C.E.; Millon, M.;
Sluse, D.; Bonvin, V.; Fassnacht, C.D.; Taubenberger, S.; Auger,
M.W.; et al.
H0LiCOW–XIII. A 2.4 per cent measurement of H${}_{0}$
from lensed quasars: 5.3$\sigma$ tension between early- and
late-Universe probes.
Mon. Not. R. Astron. Soc. 2020,
498, 1420–1439,
doi:\changeurlcolorblack10.1093/mnras/stz3094.
Gómez-Valent and Amendola (2018)
Gómez-Valent, A.; Amendola, L.
H${}_{0}$ from cosmic chronometers and Type Ia supernovae, with
Gaussian Processes and the novel Weighted Polynomial Regression method.
J. Cosmol. Astropart. Phys. 2018, 2018, 051,
doi:\changeurlcolorblack10.1088/1475-7516/2018/04/051.
Reid et al. (2019)
Reid, M.J.; Pesce, D.W.; Riess, A.G.
An Improved Distance to NGC 4258 and Its Implications for the Hubble
Constant.
Astrophys. J. Lett. 2019, 886, L27,
doi:\changeurlcolorblack10.3847/2041-8213/ab552d.
Aghanim et al. (2020)
Collaboration, P.; Akrami, Y.; Arroja, F.; Ashdown, M.; Aumont, J.; Baccigalupi, C.; Ballardini, M.; Banday, A.J.; Barreiro, R.B.; Bartolo, N.; et al. Planck 2018 results. I. Overview and the cosmological legacy of Planck. Astron. Astrophys. 2020.
641, A6,
doi:\changeurlcolorblack10.1051/0004-6361/201833910.
Rodney et al. (2016)
Rodney, S.A.; Riess, A.G.; Scolnic, D.M.; Jones, D.O.; Hemmati, S.; Molino, A.;
McCully, C.; Mobasher, B.; Strolger, L.G.; Graur, O.; et al.
Erratum: “Two SNe Ia at redshift $\sim$2: Improved Classification
and redshift determination with medium-band infrared imaging” (2015, AJ,
150, 156).
Astron. J. 2016, 151, 47.
doi:\changeurlcolorblack10.3847/0004-6256/151/2/47.
Cucchiara et al. (2011)
Cucchiara, A.; Levan, A.J.; Fox, D.B.; Tanvir, N.R.; Ukwatta, T.N.; Berger, E.;
Krühler, T.; Yoldaş, A.K.; Wu, X.F.; Toma, K.; et al.
A photometric redshift OFz$\sim$ 9.4 for grb 090429B. Astrophys. J. 2011,
736, 7.
doi:\changeurlcolorblack10.1088/0004-637x/736/1/7.
Wang et al. (2021)
Wang, F.; Yang, J.; Fan, X.; Hennawi, J.F.; Barth, A.J.; Banados, E.; Bian, F.;
Boutsia, K.; Connor, T.; Davies, F.B.; et al.
A Luminous Quasar at Redshift 7.642. Astrophys. J. Lett. 2021,
907, L1.
doi:\changeurlcolorblack10.3847/2041-8213/abd8c6.
Cardone et al. (2009)
Cardone, V.F.; Capozziello, S.; Dainotti, M.G.
An updated gamma-ray bursts Hubble diagram.
Mon. Not. R. Astron. Soc. 2009,
400, 775–790,
doi:\changeurlcolorblack10.1111/j.1365-2966.2009.15456.x.
Cardone et al. (2010)
Cardone, V.F.; Dainotti, M.G.; Capozziello, S.; Willingale, R.
Constraining cosmological parameters by gamma-ray burst X-ray
afterglow light curves.
Mon. Not. R. Astron. Soc. 2010,
408, 1181–1186,
doi:\changeurlcolorblack10.1111/j.1365-2966.2010.17197.x.
Cardone et al. (2011)
Cardone, V.F.; Perillo, M.; Capozziello, S.
Systematics in the gamma-ray burst Hubble diagram.
Mon. Not. R. Astron. Soc. 2011,
417, 1672–1683,
doi:\changeurlcolorblack10.1111/j.1365-2966.2011.19228.x.
Dainotti et al. (2013)
Dainotti, M.G.; Cardone, V.F.; Piedipalumbo, E.; Capozziello, S.
Slope evolution of GRB correlations and cosmology.
Mon. Not. R. Astron. Soc. 2013,
436, 82–88,
doi:\changeurlcolorblack10.1093/mnras/stt1516.
Postnikov et al. (2014)
Postnikov, S.; Dainotti, M.G.; Hernandez, X.; Capozziello, S.
Nonparametric Study of the Evolution of the
Cosmological Equation of State with Sneia, Bao, and
High-Redshift Grbs. Astrophys. J. 2014,
783, 126.
doi:\changeurlcolorblack10.1088/0004-637x/783/2/126.
Amati, L. et al. (2002)
Amati, L.; Frontera, F.; Tavani, M.; i not Zand, J.J.M..;
Antonelli, A.; Costa, E.; Feroci, M.; Guidorzi, C.; Heise, J.;
Masetti, N.; et al.
Intrinsic spectra and energetics of BeppoSAX Gamma-Ray Bursts with
known redshifts.
Astron. Astrophys. 2002, 390, 81–89.
doi:\changeurlcolorblack10.1051/0004-6361:20020722.
Yonetoku et al. (2004)
Yonetoku, D.; Murakami, T.; Nakamura, T.; Yamazaki, R.; Inoue, A.K.; Ioka, K.
Gamma-Ray Burst Formation Rate Inferred from the Spectral Peak
Energy–Peak Luminosity Relation. Astrophys. J. 2004,
609, 935–951.
doi:\changeurlcolorblack10.1086/421285.
Ito et al. (2019)
Ito, H.; Matsumoto, J.; Nagataki, S.; Warren, D.C.; Barkov, M.V.; Yonetoku, D.
The photospheric origin of the Yonetoku relation in gamma-ray bursts.
Nat. Commun. 2019, 10.
doi:\changeurlcolorblack10.1038/s41467-019-09281-z.
Liang and Zhang (2005)
Liang, E.; Zhang, B.
Model-independent Multivariable Gamma-Ray Burst Luminosity Indicator
and Its Possible Cosmological Implications. Astrophys. J. 2005,
633, 611–623.
doi:\changeurlcolorblack10.1086/491594.
Ghirlanda, G. et al. (2010)
Ghirlanda, G.; Nava, L.; Ghisellini, G..
Spectral-luminosity relation within individual Fermi gamma rays
bursts.
Astron. Astrophys. 2010, 511, A43.
doi:\changeurlcolorblack10.1051/0004-6361/200913134.
Dainotti et al. (2008)
Dainotti, M.G.; Cardone, V.F.; Capozziello, S.
A time-luminosity correlation for $\gamma$-ray bursts in the
X-rays.
Mon. Not. R. Astron. Soc.: Letters 2008, 391, L79–L83,
doi:\changeurlcolorblack10.1111/j.1745-3933.2008.00560.x.
Dainotti et al. (2010)
Dainotti, M.G.; Willingale, R.; Capozziello, S.; Cardone, V.F.; Ostrowski, M.
Discovery of a Tight Correlation for Gamma-Ray Burst
Afterglows with “Canonical”
Light Curves. Astrophys. J. Lett. 2010.
722, L215–L219.
doi:\changeurlcolorblack10.1088/2041-8205/722/2/l215.
Dainotti et al. (2011)
Dainotti, M.G.; Cardone, V.F.; Capozziello, S.; Ostrowski, M.; Willingale, R.
Study of possible systematics in the L*X - Ta* correlation of Gamma
Ray Bursts. Astrophys. J. 2011,
730, 135.
doi:\changeurlcolorblack10.1088/0004-637x/730/2/135.
Dainotti et al. (2013)
Dainotti, M.G.; Petrosian, V.; Singal, J.; Ostrowski, M.
Determination of the intrinsic luminosity time
correlation in the X-ray afterglows of gamma-ray bursts. Astrophys. J.
2013,
774, 157.
doi:\changeurlcolorblack10.1088/0004-637x/774/2/157.
Dainotti et al. (2015a)
Dainotti, M.G.; Vecchio, R.D.; Shigehiro, N.; Capozziello, S.
Selection effects in Gamma-Ray Burst Correlations:
Consequences on the Ratio Between Gamma-Ray Burst And
Star Formation rates. Astrophys. J. 2015,
800, 31.
doi:\changeurlcolorblack10.1088/0004-637x/800/1/31.
Dainotti et al. (2015b)
Dainotti, M.; Petrosian, V.; Willingale, R.; O’Brien, P.; Ostrowski, M.;
Nagataki, S.
Luminosity–time and luminosity–luminosity correlations for GRB
prompt and afterglow plateau emissions.
Mon. Not. R. Astron. Soc. 2015,
451, 3898–3908,
doi:\changeurlcolorblack10.1093/mnras/stv1229.
Dainotti et al. (2016)
Dainotti, M.G.; Postnikov, S.; Hernandez, X.; Ostrowski, M.
A fundamental Plane for Long Gamma-Ray Bursts with
X-Ray plateaus. Astrophys. J. Lett. 2016,
825, L20.
doi:\changeurlcolorblack10.3847/2041-8205/825/2/l20.
Dainotti, M. G. et al. (2017)
Dainotti, M.G.; Nagataki, S.; Maeda, K.; Postnikov, S.; Pian,
E.
A study of gamma ray bursts with afterglow plateau phases associated
with supernovae.
Astron. Astrophys. 2017, 600, A98.
doi:\changeurlcolorblack10.1051/0004-6361/201628384.
Dainotti et al. (2017)
Dainotti, M.G.; Hernandez, X.; Postnikov, S.; Nagataki, S.; O’brien, P.;
Willingale, R.; Striegel, S.
A Study of the Gamma-Ray Burst Fundamental Plane. Astrophys. J. 2017,
848, 88.
doi:\changeurlcolorblack10.3847/1538-4357/aa8a6b.
Dainotti and Amati (2018)
Dainotti, M.G.; Amati, L.
Gamma-ray Burst Prompt Correlations: Selection and Instrumental
Effects. Astrophys. J. 2018,
130, 051001.
doi:\changeurlcolorblack10.1088/1538-3873/aaa8d7.
Dainotti et al. (2018)
Dainotti, M.G.; Del Vecchio, R.; Tarnopolski, M.
Gamma-Ray Burst Prompt Correlations.
Adv. Astron. 2018, 2018, 1–31.
doi:\changeurlcolorblack10.1155/2018/4969503.
Dainotti et al. (2020a)
Dainotti, M.G.; Lenart, A.Ł.; Sarracino, G.; Nagataki, S.; Capozziello, S.;
Fraija, N.
The X-Ray Fundamental Plane of the Platinum Sample, the Kilonovae,
and the SNe Ib/c Associated with GRBs. Astrophys. J. 2020,
904, 97.
doi:\changeurlcolorblack10.3847/1538-4357/abbe8a.
Dainotti et al. (2021a)
Dainotti, M.G.; Omodei, N.; Srinivasaragavan, G.P.; Vianello, G.; Willingale,
R.; O’Brien, P.; Nagataki, S.; Petrosian, V.; Nuygen, Z.; Hernandez, X.;
et al.
On the Existence of the Plateau Emission in High-energy Gamma-Ray
Burst Light Curves Observed by Fermi-LAT.
Astrophys. J. Suppl. Ser. 2021, 255, 13.
doi:\changeurlcolorblack10.3847/1538-4365/abfe17.
Dainotti et al. (2021b)
Dainotti, M.; Levine, D.; Fraija, N.; Chandra, P.
Accounting for Selection Bias and Redshift Evolution in GRB Radio
Afterglow Data.
Galaxies 2021, 9, 95.
doi:\changeurlcolorblack10.3390/galaxies9040095.
Vecchio et al. (2016)
Del Vecchio, R.; Dainotti, M.G.; Ostrowski, M.
Study of Grb Light-Curve Decay Indices in the
Afterglow phase. Astrophys. J. 2016,
828, 36.
doi:\changeurlcolorblack10.3847/0004-637x/828/1/36.
Duncan (2001)
Duncan, R.C.
Gamma-ray bursts from extragalactic Magnetar Flares.
AIP Conf. Proc. 2001.
doi:\changeurlcolorblack10.1063/1.1419599.
Dall’Osso et al. (2012)
Dall’Osso, S.; Granot, J.; Piran, T.
Magnetic field decay in neutron stars: from soft gamma repeaters to
“weak-field magnetars”.
Mon. Not. R. Astron. Soc. 2012,
422, 2878–2903.
doi:\changeurlcolorblack10.1111/j.1365-2966.2012.20612.x.
Rowlinson et al. (2014)
Rowlinson, A.; Gompertz, B.P.; Dainotti, M.; O’Brien, P.T.; Wijers, R.A.M.J.;
van der Horst, A.J.
Constraining properties of GRB magnetar central engines using the
observed plateau luminosity and duration correlation.
Mon. Not. R. Astron. Soc. 2014,
443, 1779–1787,
doi:\changeurlcolorblack10.1093/mnras/stu1277.
Rea et al. (2015)
Rea, N.; Gullón, M.; Pons, J.A.; Perna, R.; Dainotti, M.G.; Miralles,
J.A.; Torres, D.F.
Constraining The Grb-Magnetar Model By Means Of The
Galactic Pulsar PopulatioN. Astrophys. J. 2015,
813, 92.
doi:\changeurlcolorblack10.1088/0004-637x/813/2/92.
Stratta et al. (2018)
Stratta, G.; Dainotti, M.G.; Dall’Osso, S.; Hernandez, X.; Cesare, G.D.
On the Magnetar Origin of the GRBs Presenting X-Ray Afterglow
Plateaus. Astrophys. J. 2018,
869, 155.
doi:\changeurlcolorblack10.3847/1538-4357/aadd8f.
Amati et al. (2019)
Amati, L.; D’Agostino, R.; Luongo, O.; Muccino, M.; Tantalo, M.
Addressing the circularity problem in the $E_{p}$-$E_{iso}$
correlation of gamma-ray bursts.
Mon. Not. R. Astron. Soc. Lett. 2019, 486, L46–L51,
doi:\changeurlcolorblack10.1093/mnrasl/slz056.
Liao et al. (2019)
Liao, K.; Shafieloo, A.; Keeley, R.E.; Linder, E.V.
A Model-independent Determination of the Hubble Constant from Lensed
Quasars and Supernovae Using Gaussian Process Regression.
Astrophys. J. Lett. 2019, 886, L23,
doi:\changeurlcolorblack10.3847/2041-8213/ab5308.
Keeley et al. (2020)
Keeley, R.E.; Shafieloo, A.; Hazra, D.K.; Souradeep, T.
Inflation wars: A new hope.
J. Cosmol. Astropart. Phys. 2020, 2020, 055,
doi:\changeurlcolorblack10.1088/1475-7516/2020/09/055.
Risaliti and Lusso (2019)
Risaliti, G.; Lusso, E.
Cosmological Constraints from the Hubble Diagram of Quasars at High
Redshifts.
Nat. Astron. 2019, 3, 272–277,
doi:\changeurlcolorblack10.1038/s41550-018-0657-z.
Freedman et al. (2019)
Freedman, W.L.; Madore, B.F.; Hatt, D.; Hoyt, T.J.; Jang, I.S.;
Beaton, R.L.; Burns, C.R.; Lee, M.G.; Monson, A.J.; Neeley, J.R.;
et al.
The Carnegie-Chicago Hubble Program. VIII. An Independent
Determination of the Hubble Constant Based on the Tip of the Red Giant
Branch.
Astrophys. J. 2019, 882, 34,
doi:\changeurlcolorblack10.3847/1538-4357/ab2f73.
Dutta et al. (2019)
Dutta, K.; Roy, A.; Ruchika.; Sen, A.A.; Sheikh-Jabbari, M.M.
Cosmology with low-redshift observations: No signal for new
physics.
Phys. Rev. D 2019, 100, 103501,
doi:\changeurlcolorblack10.1103/PhysRevD.100.103501.
Yang et al. (2020)
Yang, T.; Banerjee, A.; Ó Colgáin, E.
Cosmography and flat $\Lambda$CDM tensions at high
redshift.
Phys. Rev. D 2020, 102, 123532,
doi:\changeurlcolorblack10.1103/PhysRevD.102.123532.
Dainotti et al. (2021)
Dainotti, M.G.; De Simone, B.; Schiavone, T.; Montani, G.; Rinaldi,
E.; Lambiase, G.
On the Hubble Constant Tension in the SNe Ia Pantheon Sample.
Astrophys. J. 2021, 912, 150,
doi:\changeurlcolorblack10.3847/1538-4357/abeb73.
Ingram et al. (2021)
Ingram, A.; Mastroserio, G.; van der Klis, M.; Nathan, E.; Connors, R.; Dauser,
T.; García, J.A.; Kara, E.; König, O.; Lucchini, M.; et al.
Agrawal et al. (2019)
Agrawal, P.; Cyr-Racine, F.Y.; Pinner, D.; Randall, L.
Rock ’n’ Roll Solutions to the Hubble Tension.
arXiv 2019, arXiv:1904.01016,
On measuring the Hubble constant with X-ray reverberation mapping of
active galactic nuclei. arXiv 2021, arXiv:2110.15651.
Arjona et al. (2021)
Arjona, R.; Espinosa-Portales, L.; García-Bellido, J.; Nesseris, S.
A GREAT model comparison against the cosmological constant. arXiv 2021,
arXiv:2111.13083.
(55)
Fernandez-Martinez, E., Pierre, M., Pinsard, E. & Rosauro-Alcaraz, S. Inverse Seesaw, dark matter and the Hubble tension. The European Physical Journal C. 81 (2021,10), http://dx.doi.org/10.1140/epjc/s10052-021-09760-y
(56)
Ghose, S. & Bhadra, A. Is non-particle dark matter equation of state parameter evolving with time?. The European Physical Journal C. 81 (2021,8), http://dx.doi.org/10.1140/epjc/s10052-021-09490-1
(57)
Hart, L. & Chluba, J. Varying fundamental constants principal component analysis: additional hints about the Hubble tension. Monthly Notices Of The Royal Astronomical Society. 510, 2206-2227 (2021,10), http://dx.doi.org/10.1093/mnras/stab2777
(58)
Beltrán Jiménez, J., Bettoni, D., Figueruelo, D., Teppa Pannia, F. & Tsujikawa, S. Probing elastic interactions in the dark sector and the role of S8. Physical Review D. 104 (2021,11), http://dx.doi.org/10.1103/PhysRevD.104.103503
(59)
Rezaei, M., Peracaula, J. & Malekjani, M. Cosmographic approach to Running Vacuum dark energy models: New constraints using BAOs and Hubble diagrams at higher redshifts. Monthly Notices Of The Royal Astronomical Society. (2021,11), http://dx.doi.org/10.1093/mnras/stab3117
(60)
Shah, P., Lemos, P. & Lahav, O. A buyer’s guide to the Hubble constant. The Astronomy And Astrophysics Review. 29 (2021,12), http://dx.doi.org/10.1007/s00159-021-00137-4
(61)
Firouzjahi, H. Cosmological constant problem on the horizon. (2022)
Banihashemi et al. (2021)
Banihashemi, A.; Khosravi, N.; Shafieloo, A.
Dark energy as a critical phenomenon: a hint from Hubble tension.
J. Cosmol. Astropart. Phys. 2021, 2021, 003.
doi:\changeurlcolorblack10.1088/1475-7516/2021/06/003.
Ballardini et al. (2021)
Ballardini, M.; Finelli, F.; Sapone, D.
Cosmological constraints on Newton’s gravitational constant. arXiv 2021,
arXiv:2111.09168.
Corona et al. (2021)
Corona, M.A.; Murgia, R.; Cadeddu, M.; Archidiacono, M.; Gariazzo, S.; Giunti,
C.; Hannestad, S.
Pseudoscalar sterile neutrino self-interactions in light of Planck,
SPT and ACT data. arXiv 2021, arXiv:astro-ph.CO/2112.00037.
Cyr-Racine et al. (2021)
Cyr-Racine, F.Y.; Ge, F.; Knox, L.
A Symmetry of Cosmological Observables, and a High Hubble Constant as
an Indicator of a Mirror World Dark Sector. arXiv 2021, arXiv:astro-ph.CO/2107.13000.
Valentino et al. (2021)
Di Valentino, E.; Gariazzo, S.; Giunti, C.; Mena, O.; Pan, S.; Yang, W.
Minimal dark energy: key to sterile neutrino and Hubble constant
tensions? arXiv 2021, arXiv:astro-ph.CO/2110.03990.
Valentino and Melchiorri (2021)
Di Valentino, E.; Melchiorri, A.
Neutrino Mass Bounds in the era of Tension Cosmology. arXiv 2021, arXiv:astro-ph.CO/2112.02993.
Drees and Zhao (2021)
Drees, M.; Zhao, W.
$U(1)_{L_{\mu}-L_{\tau}}$ for Light Dark Matter, $g_{\mu}-2$, the $511$ keV
excess and the Hubble Tension. arXiv 2021, arXiv:hep-ph/2107.14528.
Gu et al. (2021)
Gu, Y.; Wu, L.; Zhu, B.
Axion Dark Radiation and Late Decaying Dark Matter in Neutrino
Experiment. arXiv 2021, arXiv:hep-ph/2105.07232.
Khalifeh and Jimenez (2021)
Khalifeh, A.R.; Jimenez, R.
Using Neutrino Oscillations to Measure $H_{0}$. arXiv 2021, arXiv:astro-ph.CO/2111.15249.
Li et al. (2021)
Li, J.; Zhou, Y.; Xue, X.
Spatial Curvature and Large Scale Lorentz Violation. arXiv 2021, [arXiv:gr-qc/2112.02364.
Lulli et al. (2021)
Lulli, M.; Marciano, A.; Shan, X.
Stochastic Quantization of General Relativity à la Ricci-Flow. arXiv 2021, arXiv:gr-qc/2112.01490.
Mawas et al. (2021)
Mawas, E.; Street, L.; Gass, R.; Wijewardhana, L.C.R.
Interacting dark energy axions in light of the Hubble tension. arXiv 2021, arXiv:astro-ph.CO/2108.13317.
Moreno-Pulido and Peracaula (2021)
Moreno-Pulido, C.; Peracaula, J.S.
Renormalized $\rho_{\rm vac}$ without $m^{4}$ terms. arXiv 2021, arXiv:gr-qc/2110.08070.
Naidoo et al. (2021)
Naidoo, K.; Massara, E.; Lahav, O.
Cosmology and neutrino mass with the Minimum Spanning Tree. arXiv 2021,
arXiv:astro-ph.CO/2111.12088.
Niedermann and Sloth (2021)
Niedermann, F.; Sloth, M.S.
Hot New Early Dark Energy: Towards a Unified Dark Sector of
Neutrinos, Dark Energy and Dark Matter. arXiv 2021,
arXiv:hep-ph/2112.00759.
Nilsson and Park (2021)
Nilsson, N.A.; Park, M.I.
Tests of Standard Cosmology in Horava Gravity. arXiv 2021,
arXiv:hep-th/2108.07986.
Ray et al. (2021)
Ray, P.P.; Tarai, S.; Mishra, B.; Tripathy, S.K.
Cosmological models with Big rip and Pseudo rip Scenarios in extended
theory of gravity. arXiv 2021,
arXiv:gr-qc/2107.04413.
Schöneberg et al. (2021)
Schöneberg, N.; Abellán, G.F.; Sánchez, A.P.; Witte, S.J.; Poulin, V.;
Lesgourgues, J.
The $H_{0}$ Olympics: A fair ranking of proposed models. arXiv 2021,
arXiv:astro-ph.CO/2107.10291.
Trott and Huterer (2021)
Trott, E.; Huterer, D.
Challenges for the statistical gravitational-wave method to measure
the Hubble constant. arXiv 2021,
arXiv:astro-ph.CO/2112.00241.
Ye et al. (2021)
Ye, G.; Zhang, J.; Piao, Y.S.
Resolving both $H_{0}$ and $S_{8}$ tensions with AdS early dark energy
and ultralight axion. arXiv 2021,
arXiv:astro-ph.CO/2107.13391.
Zhou et al. (2021)
Zhou, Z.; Liu, G.; Xu, L.
Can late dark energy restore the Cosmic concordance? arXiv 2021,
arXiv:astro-ph.CO/2105.04258.
Zhu et al. (2021)
Zhu, L.G.; Xie, L.H.; Hu, Y.M.; Liu, S.; Li, E.K.; Napolitano, N.R.; Tang,
B.T.; dong Zhang, J.; Mei, J.
Constraining the Hubble constant to a precision of about 1% using
multi-band dark standard siren detections. arXiv 2021,
arXiv:astro-ph.CO/2110.05224.
Alestas et al. (2022)
Alestas, G.; Perivolaropoulos, L.; Tanidis, K.
Constraining a late time transition of $G_{\rm eff}$ using low-z
galaxy survey data. arXiv 2022,
arXiv:astro-ph.CO/2201.05846.
Cea (2022)
Cea, P.
The Ellipsoidal Universe and the Hubble tension. arXiv 2022,
arXiv:astro-ph.CO/2201.04548.
Gurzadyan et al. (2022)
Gurzadyan, V.G.; Fimin, N.N.; Chechetkin, V.M.
On the origin of cosmic web.
Eur. Phys. J. Plus 2022, 137.
doi:\changeurlcolorblack10.1140/epjp/s13360-022-02373-8.
(87)
Rashkovetskyi, M., Muñoz, J., Eisenstein, D. & Dvorkin, C. Small-scale clumping at recombination and the Hubble tension. Physical Review D. 104 (2021,11), http://dx.doi.org/10.1103/PhysRevD.104.103517
(88)
Das, A. Self-interacting neutrinos as a solution to the Hubble tension?. (2021)
(89)
Karwal, T. & Kamionkowski, M. Dark energy at early times, the Hubble parameter, and the string axiverse. Physical Review D. 94, e103523 (2016,11)
Yang et al. (2018)
Yang, W.; Pan, S.; Di Valentino, E.; Nunes, R.C.; Vagnozzi, S.;
Mota, D.F.
Tale of stable interacting dark energy, observational signatures,
and the H${}_{0}$ tension.
J. Cosmol. Astropart. Phys. 2018, 2018, 019,
doi:\changeurlcolorblack10.1088/1475-7516/2018/09/019.
Di Valentino et al. (2018b)
Di Valentino, E.; Linder, E.V.; Melchiorri, A.
Vacuum phase transition solves the H${}_{0}$ tension.
Phys. Rev. D 2018, 97, 043528,
doi:\changeurlcolorblack10.1103/PhysRevD.97.043528.
Di Valentino et al. (2019)
Di Valentino, E.; Ferreira, R.Z.; Visinelli, L.; Danielsson, U.
Late time transitions in the quintessence field and the H${}_{0}$
tension.
Phys. Dark Univ. 2019, 26, 100385,
doi:\changeurlcolorblack10.1016/j.dark.2019.100385.
Pan et al. (2020)
Pan, S.; Yang, W.; Di Valentino, E.; Shafieloo, A.; Chakraborty, S.
Reconciling H${}_{0}$ tension in a six parameter space?
J. Cosmol. Astropart. Phys. 2020, 2020, 062,
doi:\changeurlcolorblack10.1088/1475-7516/2020/06/062.
Di Valentino et al. (2021a)
Di Valentino, E.; Anchordoqui, L.A.; Akarsu, O.; Ali-Haimoud, Y.; Amendola, L.;
Arendse, N.; Asgari, M.; Ballardini, M.; Basilakos, S.; Battistelli, E.;
et al.
Snowmass2021 - Letter of interest cosmology intertwined I:
Perspectives for the next decade.
Astropart. Phys. 2021, 131, 102606.
doi:\changeurlcolorblack10.1016/j.astropartphys.2021.102606.
Di Valentino et al. (2021b)
Di Valentino, E.; Anchordoqui, L.A.; Akarsu, O.; Ali-Haimoud, Y.; Amendola, L.;
Arendse, N.; Asgari, M.; Ballardini, M.; Basilakos, S.; Battistelli, E.;
et al.
Snowmass2021—Letter of interest cosmology intertwined II: The
hubble constant tension.
Astropart. Phys. 2021, 131, 102605.
doi:\changeurlcolorblack10.1016/j.astropartphys.2021.102605.
Di Valentino et al. (2021c)
Di Valentino, E.; Anchordoqui, L.A.; Akarsu, O.; Ali-Haimoud, Y.; Amendola, L.;
Arendse, N.; Asgari, M.; Ballardini, M.; Basilakos, S.; Battistelli, E.;
et al.
Cosmology Intertwined III: f$\sigma$8 and S8
Astropart. Phys. 2021, 131, 102604.
doi:\changeurlcolorblack10.1016/j.astropartphys.2021.102604.
Di Valentino et al. (2021d)
Di Valentino, E.; Anchordoqui, L.A.; Akarsu, O.; Ali-Haimoud, Y.; Amendola, L.;
Arendse, N.; Asgari, M.; Ballardini, M.; Basilakos, S.; Battistelli, E.;
et al.
Snowmass2021—Letter of interest cosmology intertwined IV: The age
of the universe and its curvature.
Astropart. Phys. 2021, 131, 102607.
doi:\changeurlcolorblack10.1016/j.astropartphys.2021.102607.
Di Valentino et al. (2018)
Di Valentino, E.; Bœhm, C.; Hivon, E.; Bouchet, F.R.
Reducing the H${}_{0}$ and $\sigma$${}_{8}$ tensions with
dark matter-neutrino interactions.
Phys. Rev. D 2018, 97, 043513,
doi:\changeurlcolorblack10.1103/PhysRevD.97.043513.
Di Valentino et al. (2021)
Di Valentino, E.; Pan, S.; Yang, W.; Anchordoqui, L.A.
Touch of neutrinos on the vacuum metamorphosis: Is the H${}_{0}$
solution back?
Phys. Rev. D 2021, 103, 123527,
doi:\changeurlcolorblack10.1103/PhysRevD.103.123527.
Anchordoqui et al. (2021)
Anchordoqui, L.A.; Di Valentino, E.; Pan, S.; Yang, W.
Dissecting the H${}_{0}$ and S${}_{8}$ tensions with Planck + BAO +
supernova type Ia in multi-parameter cosmologies.
J. High Energy Astrophys. 2021, 32, 28–64,
doi:\changeurlcolorblack10.1016/j.jheap.2021.08.001.
Di Valentino et al. (2021)
Di Valentino, E.; Mukherjee, A.; Sen, A.A.
Dark Energy with Phantom Crossing and the H0 Tension.
Entropy 2021, 23, 404,
doi:\changeurlcolorblack10.3390/e23040404.
Di Valentino et al. (2020a)
Di Valentino, E.; Linder, E.V.; Melchiorri, A.
H${}_{0}$ ex machina: Vacuum metamorphosis and beyond H${}_{0}$.
Phys. Dark Univ. 2020, 30, 100733,
doi:\changeurlcolorblack10.1016/j.dark.2020.100733.
Di Valentino et al. (2020b)
Di Valentino, E.; Melchiorri, A.; Silk, J.
Planck evidence for a closed Universe and a possible crisis for
cosmology.
Nat. Astron. 2020, 4, 196–203,
doi:\changeurlcolorblack10.1038/s41550-019-0906-9.
Allali et al. (2021)
Allali, I.J.; Hertzberg, M.P.; Rompineve, F.
Dark sector to restore cosmological concordance.
Phys. Rev. D 2021, 104.
doi:\changeurlcolorblack10.1103/physrevd.104.l081303.
Anderson (2021)
Anderson, R.I.
Relativistic corrections for measuring Hubble’s constant to 1% using
stellar standard candles. arXiv 2021,
arXiv:astro-ph.CO/2108.09067.
Asghari and Sheykhi (2021)
Asghari, M.; Sheykhi, A.
Observational constraints of the modified cosmology through Barrow
entropy. arXiv 2021, arXiv:gr-qc/2110.00059.
Brownsberger et al. (2021)
Brownsberger, S.; Brout, D.; Scolnic, D.; Stubbs, C.W.; Riess, A.G.
The Pantheon+ Analysis: Dependence of Cosmological Constraints on
Photometric-Zeropoint Uncertainties of Supernova Surveys. arXiv 2021,
arXiv:astro-ph.CO/2110.03486.
Cyr-Racine (2021)
Cyr-Racine, F.Y.
Cosmic Expansion: A mini review of the Hubble-Lemaitre tension. arXiv 2021, arXiv:astro-ph.CO/2105.09409.
Khosravi and Farhang (2021)
Khosravi, N.; Farhang, M.
Phenomenological Gravitational Phase Transition: Early and Late
Modifications. arXiv 2021, arXiv:astro-ph.CO/2109.10725.
Mantz et al. (2021)
Mantz, A.B.; Morris, R.G.; Allen, S.W.; Canning, R.E.A.; Baumont, L.; Benson,
B.; Bleem, L.E.; Ehlert, S.R.; Floyd, B.; Herbonnet, R.; et al.
Cosmological constraints from gas mass fractions of massive, relaxed
galaxy clusters.
Mon. Not. R. Astron. Soc. 2021,
510, 131–145.
doi:\changeurlcolorblack10.1093/mnras/stab3390.
Mortsell et al. (2021a)
Mortsell, E.; Goobar, A.; Johansson, J.; Dhawan, S.
The Hubble Tension Bites the Dust: Sensitivity of the Hubble Constant
Determination to Cepheid Color Calibration. arXiv 2021,
arXiv:astro-ph.CO/2105.11461.
Mortsell et al. (2021b)
Mortsell, E.; Goobar, A.; Johansson, J.; Dhawan, S.
The Hubble Tension Revisited: Additional Local Distance Ladder
Uncertainties. arXiv 2021, arXiv:astro-ph.CO/2106.09400.
Theodoropoulos and
Perivolaropoulos (2021)
Theodoropoulos, A.; Perivolaropoulos, L.
The Hubble Tension, the M Crisis of Late Time H(z) Deformation
Models and the Reconstruction of Quintessence Lagrangians.
Universe 2021, 7, 300,
doi:\changeurlcolorblack10.3390/universe7080300.
Gómez-Valent (2022)
Gómez-Valent, A.
Measuring the sound horizon and absolute magnitude of SNIa by
maximizing the consistency between low-redshift data sets. arXiv 2022,
arXiv:astro-ph.CO/2111.15450.
Pol et al. (2022)
Pol, A.R.; Caprini, C.; Neronov, A.; Semikoz, D.
The gravitational wave signal from primordial magnetic fields in the
Pulsar Timing Array frequency band. arXiv 2022,
arXiv:astro-ph.CO/2201.05630.
Wong et al. (2022)
Wong, J.H.W.; Shanks, T.; Metcalfe, N.
The Local Hole: a galaxy under-density covering 90% of sky to 200
Mpc. arXiv 2022, arXiv:astro-ph.CO/2107.08505.
Romaniello et al. (2021)
Romaniello, M.; Riess, A.; Mancino, S.; Anderson, R.I.; Freudling, W.;
Kudritzki, R.P.; Macri, L.; Mucciarelli, A.; Yuan, W.
The iron and oxygen content of LMC Classical Cepheids and its
implications for the Extragalactic Distance Scale and Hubble constant. arXiv 2021,
arXiv:astro-ph.CO/2110.08860.
Luu (2021)
Luu, H.N.
Axi-Higgs cosmology: Cosmic Microwave Background and cosmological
tensions. arXiv 2021, arXiv:astro-ph.CO/2111.01347.
Alestas et al. (2021)
Alestas, G.; Kazantzidis, L.; Perivolaropoulos, L.
w -M phantom transition at z${}_{t}$<0.1 as a resolution of the Hubble
tension.
Phys. Rev. D 2021, 103, 083517,
doi:\changeurlcolorblack10.1103/PhysRevD.103.083517.
Sakr and Sapone (2021)
Sakr, Z.; Sapone, D.
Can varying the gravitational constant alleviate the tensions? arXiv 2021, arXiv:astro-ph.CO/2112.14173.
Wang et al. (2021)
Wang, Y.Y.; Tang, S.P.; Li, X.Y.; Jin, Z.P.; Fan, Y.Z.
Prospects of calibrating afterglow modeling of short GRBs with
gravitational wave inclination angle measurements and resolving the Hubble
constant tension with a GW/GRB association event. arXiv 2021,
arXiv:astro-ph.HE/2111.02027.
Safari et al. (2022)
Safari, Z.; Rezazadeh, K.; Malekolkalami, B.
Structure Formation in Dark Matter Particle Production Cosmology. arXiv 2022, arXiv:astro-ph.CO/2201.05195.
Roth et al. (2021)
Roth, M.M.; Jacoby, G.H.; Ciardullo, R.; Davis, B.D.; Chase, O.;
Weilbacher, P.M.
Towards Precision Cosmology With Improved PNLF Distances Using
VLT-MUSE I. Methodology and Tests.
Astrophys. J. 2021, 916, 21,
doi:\changeurlcolorblack10.3847/1538-4357/ac02ca.
Gutiérrez-Luna et al. (2021)
Gutiérrez-Luna, E.; Carvente, B.; Jaramillo, V.; Barranco, J.;
Escamilla-Rivera, C.; Espinoza, C.; Mondragón, M.; Núñez, D.
Scalar field dark matter with two components: Combined approach from
particle physics and cosmology. arXiv 2021,
arXiv:astro-ph.CO/2110.10258.
(125)
Chang, C. Imprint of early dark energy in stochastic gravitational wave background. Physical Review D. 105 (2022,1), http://dx.doi.org/10.1103/PhysRevD.105.023508
(126)
Liu, S., Zhu, L., Hu, Y., Zhang, J. & Ji, M. Capability for detection of GW190521-like binary black holes with TianQin. Physical Review D. 105 (2022,1), http://dx.doi.org/10.1103/PhysRevD.105.023019
(127)
Farrugia, C., Sultana, J. & Mifsud, J. Spatial curvature in f(R) gravity.Physical Review D. 104 (2021,12),
http://dx.doi.org/10.1103/PhysRevD.104.123503
(128)
Lu, W. & Qin, Y. New constraint of the Hubble constant by proper motions of radio components observed in AGN twin-jets. Research In Astronomy And Astrophysics. 21, 261 (2021,11), http://dx.doi.org/10.1088/1674-4527/21/10/261
(129)
Greene, K. & Cyr-Racine, F. Hubble distancing: Focusing on distance measurements in cosmology, arXiv 2021, arXiv:2112.11567.
(130)
Borghi, N., Moresco, M. & Cimatti, A. Towards a Better Understanding of Cosmic Chronometers: A new measurement of H(z) at $z\sim 0.7.$, arXiv 2021, arXiv:2110.04304.
Asencio et al. (2021)
Asencio, E.; Banik, I.; Kroupa, P.
A massive blow for $\Lambda$CDM - the high redshift,
mass, and collision velocity of the interacting galaxy cluster El Gordo
contradicts concordance cosmology.
Mon. Not. R. Astron. Soc. 2021,
500, 5249–5267,
doi:\changeurlcolorblack10.1093/mnras/staa3441.
Javanmardi et al. (2021)
Javanmardi, B.; Mérand, A.; Kervella, P.; Breuval, L.; Gallenne,
A.; Nardetto, N.; Gieren, W.; Pietrzyński, G.; Hocdé, V.;
Borgniet, S.
Inspecting the Cepheid Distance Ladder: the Hubble Space Telescope
Distance to the SN Ia Host Galaxy NGC 5584.
Astrophys. J. 2021, 911, 12,
doi:\changeurlcolorblack10.3847/1538-4357/abe7e5.
Zhao and Xia (2021)
Zhao, D.; Xia, J.Q.
Constraining the anisotropy of the Universe with the X-ray and UV
fluxes of quasars.
arXiv 2021, arXiv:2105.03965,
Thakur et al. (2021)
Thakur, R.K.; Singh, M.; Gupta, S.; Nigam, R.
Cosmological Analysis using Panstarrs data: Hubble Constant and
Direction Dependence.
arXiv 2021, arXiv:2105.14514,
Sharov and Sinyakov (2020)
Sharov, G.S.; Sinyakov, E.S.
Cosmological models, observational data and tension in Hubble
constant.
arXiv 2020, arXiv:2002.03599.
Vagnozzi et al. (2021)
Vagnozzi, S.; Pacucci, F.; Loeb, A.
Implications for the Hubble tension from the ages of the oldest
astrophysical objects.
arXiv 2021, arXiv:2105.10421.
Staicova (2021)
Staicova, D.
Hints of the $H_{0}-r_{d}$ tension in uncorrelated Baryon Acoustic
Oscillations dataset. arXiv 2021,
arXiv:astro-ph.CO/2111.07907.
Krishnan et al. (2021)
Krishnan, C.; Mohayaee, R.; Colgain, E.O.; Sheikh-Jabbari, M.M.; Yin, L.
Hints of FLRW Breakdown from Supernovae. arXiv 2021,
arXiv:astro-ph.CO/2106.02532.
Li and Shapiro (2021)
Li, B.; Shapiro, P.R.
Precision cosmology and the stiff-amplified gravitational-wave
background from inflation: NANOGrav, Advanced LIGO-Virgo and the Hubble
tension.
J. Cosmol. Astropart. Phys. 2021, 2021, 024.
doi:\changeurlcolorblack10.1088/1475-7516/2021/10/024.
Mozzon et al. (2021)
Mozzon, S.; Ashton, G.; Nuttall, L.K.; Williamson, A.R.
Does non-stationary noise in LIGO and Virgo affect the estimation of
$H_{0}$? arXiv 2021, arXiv:astro-ph.CO/2110.11731.
Abbott et al. (2021)
Abbott, T.C.; Buffaz, E.; Vieira, N.; Cabero, M.; Haggard, D.; Mahabal, A.;
McIver, J.
GWSkyNet-Multi: A Machine Learning Multi-Class Classifier for
LIGO-Virgo Public Alerts. arXiv 2021,
arXiv:astro-ph.IM/2111.04015.
Mehrabi et al. (2021)
Mehrabi, A.; Basilakos, S.; Tsiapi, P.; Plionis, M.; Terlevich, R.; Terlevich,
E.; Moran, A.L.G.; Chavez, R.; Bresolin, F.; Arenas, D.F.; et al.
Using our newest VLT-KMOS HII Galaxies and other cosmic tracers to
test the $\Lambda$CDM tension.
Mon. Not. R. Astron. Soc. 2021,
stab2915,
doi:\changeurlcolorblack10.1093/mnras/stab2915.
Li et al. (2015)
Li, J.X.; Wu, F.Q.; Li, Y.C.; Gong, Y.; Chen, X.L.
Cosmological constraint on Brans–Dicke Model.
Res. Astron. Astrophys. 2015, 15, 2151–2163.
doi:\changeurlcolorblack10.1088/1674-4527/15/12/003.
Freedman (2021)
Freedman, W.L.
Measurements of the Hubble Constant: Tensions in Perspective. Astrophys. J. 2021,
919, 16.
doi:\changeurlcolorblack10.3847/1538-4357/ac0e95.
Wu et al. (2021)
Wu, Q.; Zhang, G.Q.; Wang, F.Y.
An 8% Determination of the Hubble Constant from localized Fast
Radio Bursts. arXiv 2021, arXiv:astro-ph.CO/2108.00581.
Perivolaropoulos and Skara (2021)
Perivolaropoulos, L.; Skara, F.
Hubble tension or a transition of the Cepheid Sn Ia calibrator
parameters? arXiv 2021, arXiv:astro-ph.CO/2109.04406.
Horstmann et al. (2021)
Horstmann, N.; Pietschke, Y.; Schwarz, D.J.
Inference of the cosmic rest-frame from supernovae Ia. arXiv 2021,
arXiv:astro-ph.CO/2111.03055.
Ferree and Bunn (2021)
Ferree, N.C.; Bunn, E.F.
Constraining $H_{0}$ Via Extragalactic Parallax. arXiv 2021,
arXiv:astro-ph.CO/2109.07529.
Luongo et al. (2021)
Luongo, O.; Muccino, M.; Colgáin, E.O.; Sheikh-Jabbari, M.M.; Yin, L.
On Larger $H_{0}$ Values in the CMB Dipole Direction. arXiv 2021,
arXiv:astro-ph.CO/2108.13228.
de Souza et al. (2021)
de Souza, J.M.S.; Sturani, R.; Alcaniz, J.
Cosmography with Standard Sirens from the Einstein Telescope. arXiv 2021,
arXiv:gr-qc/2110.13316.
Fang and Yang (2021)
Fang, Y.; Yang, H.
Orbit Tomography of Binary Supermassive Black Holes with Very Long
Baseline Interferometry. arXiv 2021,
arXiv:gr-qc/2111.00368.
Palmese et al. (2021)
Palmese, A.; Bom, C.R.; Mucesh, S.; Hartley, W.G.
A standard siren measurement of the Hubble constant using
gravitational wave events from the first three LIGO/Virgo observing runs and
the DESI Legacy Survey. arXiv 2021,
arXiv:astro-ph.CO/2111.06445.
Yang et al. (2021)
Yang, T.; Lee, H.M.; Cai, R.G.; gil Choi, H.; Jung, S.
Space-borne Atom Interferometric Gravitational Wave Detections II:
Dark Sirens and Finding the One. arXiv 2021,
arXiv:gr-qc/2110.09967.
Gray et al. (2021)
Gray, R.; Messenger, C.; Veitch, J.
A Pixelated Approach to Galaxy Catalogue Incompleteness: Improving
the Dark Siren Measurement of the Hubble Constant. arXiv 2021,
arXiv:astro-ph.CO/2111.04629.
Moresco et al. (2022)
Moresco, M.; et al.
Unveiling the Universe with Emerging Cosmological Probes. arXiv 2022,
arXiv:astro-ph.CO/2201.07241.
(156)
Collaboration, T.L.S.; the Virgo Collaboration.; the KAGRA Collaboration.;
Abbott, R.; Abe, H.; Acernese, F.; Ackley, K.; Adhikari, N.; et al..
Constraints on the cosmic expansion history from GWTC-3. arXiv 2021, arXiv:2111.03604.
Nunes et al. (2017)
Nunes, R.C.; Pan, S.; Saridakis, E.N.; Abreu, E.M.
New observational constraints onf(R) gravity from cosmic
chronometers.
J. Cosmol. Astropart. Phys. 2017, 2017, 005–005.
doi:\changeurlcolorblack10.1088/1475-7516/2017/01/005.
Nunes (2018)
Nunes, R.C.
Structure formation in f(T) gravity and a solution for H0 tension.
J. Cosmol. Astropart. Phys. 2018, 2018, 052–052.
doi:\changeurlcolorblack10.1088/1475-7516/2018/05/052.
Benetti et al. (2020)
Benetti, M.; Capozziello, S.; Lambiase, G.
Updating constraints on f(T) teleparallel cosmology and the
consistency with big bang nucleosynthesis.
Mon. Not. R. Astron. Soc. 2020,
500, 1795–1805,
doi:\changeurlcolorblack10.1093/mnras/staa3368.
Räsänen (2009)
Räsänen, S.
Light propagation in statistically homogeneous and isotropic dust
universes.
J. Cosmol. Astropart. Phys. 2009, 2009, 011.
doi:\changeurlcolorblack10.1088/1475-7516/2009/02/011.
Räsänen (2010)
Räsänen, S.
Light propagation in statistically homogeneous and isotropic
universes with general matter content.
J. Cosmol. Astropart. Phys. 2010, 2010, 018.
doi:\changeurlcolorblack10.1088/1475-7516/2010/03/018.
Sotiriou (2006)
Sotiriou, T.P.
$f(R)$ gravity and scalar tensor theory.
Class. Quantum Gravity 2006, 23, 5117–5128,
doi:\changeurlcolorblack10.1088/0264-9381/23/17/003.
Nojiri and Odintsov (2007)
Nojiri, S.; Odintsov, S.D.
Introduction to modified gravity and gravitational alternative for
dark energy.
Int. J. Geom. Methods Mod. Phys.
2007, 04, 115–145,
doi:\changeurlcolorblack10.1142/S0219887807001928.
Koksbang (2019b)
Koksbang, S.M.
Towards statistically homogeneous and isotropic perfect fluid
universes with cosmic backreaction.
Class. Quantum Gravity 2019, 36, 185004.
doi:\changeurlcolorblack10.1088/1361-6382/ab376c.
Koksbang (2019c)
Koksbang, S.
Another look at redshift drift and the backreaction conjecture.
J. Cosmol. Astropart. Phys. 2019, 2019, 036.
doi:\changeurlcolorblack10.1088/1475-7516/2019/10/036.
Koksbang (2020)
Koksbang, S.M.
Observations in statistically homogeneous, locally inhomogeneous
cosmological toy-models without FLRW backgrounds. arXiv 2020,
arXiv:astro-ph.CO/2008.07108.
Sotiriou and Faraoni (2010)
Sotiriou, T.P.; Faraoni, V.
$f(R)$ theories of gravity.
Rev. Mod. Phys. 2010, 82, 451–497.
doi:\changeurlcolorblack10.1103/RevModPhys.82.451.
Odderskov et al. (2016)
Odderskov, I.; Koksbang, S.; Hannestad, S.
The local value ofH0in an inhomogeneous universe.
J. Cosmol. Astropart. Phys. 2016, 2016, 001.
doi:\changeurlcolorblack10.1088/1475-7516/2016/02/001.
Nájera and Fajardo (2021)
Nájera, A.; Fajardo, A.
Testing $f(Q,T)$ gravity models that have $\Lambda CDM$ as a
submodel.
arXiv 2021, arXiv:2104.14065.
Nájera and Fajardo (2021a)
Nájera, A.; Fajardo, A.
Fitting $f(Q,T)$ gravity models with a $\Lambda$CDM limit using
$H(z)$ and Pantheon data.
Phys. Dark Univ. 2021, 34, 100889.
doi:\changeurlcolorblack10.1016/j.dark.2021.100889.
Nájera and Fajardo (2021b)
Nájera, A.; Fajardo, A.
Cosmological Perturbation Theory in $f(Q,T)$ Gravity. arXiv 2021,
arXiv:gr-qc/2111.04205.
Linares Cedeño and Nucamendi (2021)
Linares Cedeño, F.X.; Nucamendi, U.
Revisiting cosmological diffusion models in Unimodular Gravity and
the H${}_{0}$ tension.
Phys. Dark Univ. 2021, 32, 100807,
doi:\changeurlcolorblack10.1016/j.dark.2021.100807.
Fung et al. (2021)
Fung, L.W.; Li, L.; Liu, T.; Luu, H.N.; Qiu, Y.C.; Tye, S.H.H.
The Hubble Constant in the Axi-Higgs Universe. arXiv 2021,
arXiv:astro-ph.CO/2105.01631.
Shokri et al. (2021)
Shokri, M.; Sadeghi, J.; Setare, M.R.; Capozziello, S.
Nonminimal coupling inflation with constant slow roll.
arXiv 2021, arXiv:2104.00596.
Castellano et al. (2021)
Castellano, A.; Font, A.; Herraez, A.; Ibáñez, L.E.
A Gravitino Distance Conjecture.
arXiv 2021, arXiv:2104.10181.
Tomita (2017a)
Tomita, K.
Cosmological renormalization of model parameters in second-order
perturbation theory.
Prog. Theor. Exp. Phys. 2017,
2017, 053E01,
doi:\changeurlcolorblack10.1093/ptep/ptx049.
Tomita (2017b)
Tomita, K.
Cosmological models with the energy density of random fluctuations
and the Hubble-constant problem.
Prog. Theor. Exp. Phys. 2017,
2017, 083E04,
doi:\changeurlcolorblack10.1093/ptep/ptx117.
Tomita (2018)
Tomita, K.
Super-horizon second-order perturbations for cosmological random
fluctuations and the Hubble-constant problem.
Prog. Theor. Exp. Phys. 2018,
2018, 021E01,
doi:\changeurlcolorblack10.1093/ptep/pty015.
Tomita (2019)
Tomita, K.
Hubble constants and luminosity distance in the renormalized
cosmological models due to general-relativistic second-order perturbations.
arXiv 2019, arXiv:1906.09519.
Tomita (2020)
Tomita, K.
Cosmological renormalization of model parameters in the second-order
perturbation theory.
Prog. Theor. Exp. Phys. 2020,
2020, 019202.
doi:\changeurlcolorblack10.1093/ptep/ptz162.
Belgacem and Prokopec (2021)
Belgacem, E.; Prokopec, T.
Quantum origin of dark energy and the Hubble tension. arXiv 2021,
arXiv:astro-ph.CO/2111.04803.
Bernardo et al. (2022)
Bernardo, R.C.; Grandón, D.; Said, J.L.; Cárdenas, V.H.
Parametric and nonparametric methods hint dark energy evolution. arXiv 2022, arXiv:astro-ph.CO/2111.08289.
Ambjorn and Watabiki (2021)
Ambjorn, J.; Watabiki, Y.
Easing the Hubble constant tension? arXiv 2021,
arXiv:gr-qc/2111.05087.
Di Bari et al. (2021)
Di Bari, P.; Marfatia, D.; Zhou, Y.L.
Gravitational waves from first-order phase transitions in Majoron
models of neutrino mass.
arXiv 2021, arXiv:2106.00025.
Duan et al. (2021)
Duan, W.F.; Li, S.P.; Li, X.Q.; Yang, Y.D.
Linking $R_{K^{(\ast)}}$ anomalies to $H_{0}$ tension via Dirac
neutrino. arXiv 2021, arXiv:hep-ph/2111.05178.
Ghosh (2017)
Ghosh, D.
Explaining the $R_{K}$ and $R_{K^{*}}$ anomalies.
Eur. Phys. J. C 2017, 77.
doi:\changeurlcolorblack10.1140/epjc/s10052-017-5282-y.
Burgess et al. (2021)
Burgess, C.P.; Dineen, D.; Quevedo, F.
Yoga Dark Energy: Natural Relaxation and Other Dark Implications of a
Supersymmetric Gravity Sector. arXiv 2021,
arXiv:hep-th/2111.07286.
Jiang and Piao (2021)
Jiang, J.Q.; Piao, Y.S.
Testing AdS early dark energy with Planck, SPTpol and LSS data. arXiv
2021, arXiv:astro-ph.CO/2107.07128.
Karwal et al. (2021)
Karwal, T.; Raveri, M.; Jain, B.; Khoury, J.; Trodden, M.
Chameleon Early Dark Energy and the Hubble Tension. arXiv 2021,
arXiv:astro-ph.CO/2106.13290.
Nojiri et al. (2021)
Nojiri, S.; Odintsov, S.D.; Sáez-Chillón Gómez, D.; Sharov, G.S.
Modeling and testing the equation of state for (Early) dark energy.
Phys. Dark Univ. 2021, 32, 100837.
\changeurlcolorblackhttps://doi.org/10.1016/j.dark.2021.100837.
Tian and Zhu (2021)
Tian, S.X.; Zhu, Z.H.
Early dark energy in k-essence.
Phys. Rev. D 2021, 103, 043518,
doi:\changeurlcolorblack10.1103/PhysRevD.103.043518.
Linares Cedeño et al. (2021)
Linares Cedeño, F.X.; Roy, N.; Ureña-López, L.A.
Tracker phantom field and a cosmological constant: dynamics of a
composite dark energy model.
arXiv 2021, arXiv:2105.07103.
Hernández-Almada et al. (2021)
Hernández-Almada, A.; Leon, G.; Magaña, J.; García-Aspeitia, M.A.; Motta,
V.; Saridakis, E.N.; Yesmakhanova, K.
Kaniadakis holographic dark energy: observational constraints and
global dynamics. arXiv 2021,
arXiv:astro-ph.CO/2111.00558.
Hernández-Almada et al. (2021)
Hernández-Almada, A.; García-Aspeitia, M.A.;
Rodríguez-Meza, M.A.; Motta, V.
A hybrid model of viscous and Chaplygin gas to tackle the Universe
acceleration.
Eur. Phys. J. C 2021, 81, 295,
doi:\changeurlcolorblack10.1140/epjc/s10052-021-09104-w.
Abchouyeh and van Putten (2021)
Abchouyeh, M.A.; van Putten, M.H.P.M.
Late-time Universe, ${H}_{0}$-tension, and unparticles.
Phys. Rev. D 2021, 104, 083511.
doi:\changeurlcolorblack10.1103/PhysRevD.104.083511.
Wang (2018)
Wang, D.
Dark energy constraints in light of Pantheon SNe Ia, BAO, cosmic
chronometers and CMB polarization and lensing data.
Phys. Rev. D 2018, 97, 123507,
doi:\changeurlcolorblack10.1103/PhysRevD.97.123507.
Ye et al. (2021)
Ye, G.; Hu, B.; Piao, Y.S.
Implication of the Hubble tension for the primordial Universe in
light of recent cosmological data.
Phys. Rev. D 2021, 104, 063510.
doi:\changeurlcolorblack10.1103/PhysRevD.104.063510.
Nguyen (2020)
Nguyen, H.
Analyzing Pantheon SNeIa data in the context of Barrow’s variable
speed of light.
arXiv 2020, arXiv:2010.10292.
Barrow (1999)
Barrow, J.D.
Cosmologies with varying light speed.
Phys. Rev. D 1999, 59, 043515,
doi:\changeurlcolorblack10.1103/PhysRevD.59.043515.
Artymowski et al. (2021)
Artymowski, M.; Ben-Dayan, I.; Kumar, U.
More on Emergent Dark Energy from Unparticles. arXiv 2021,
arXiv:astro-ph.CO/2111.09946.
Yang et al. (2021)
Yang, W.; Di Valentino, E.; Pan, S.; Shafieloo, A.; Li, X.
Generalized Emergent Dark Energy Model and the Hubble Constant
Tension.
arXiv 2021, arXiv:2103.03815..
Adil et al. (2021)
Adil, S.A.; Gangopadhyay, M.R.; Sami, M.; Sharma, M.K.
Late time acceleration due to generic modification of gravity and
Hubble tension.
arXiv 2021, arXiv:2106.03093..
Vagnozzi (2021)
Vagnozzi, S.
Consistency tests of $\Lambda$CDM from the early ISW effect:
implications for early-time new physics and the Hubble tension.
arXiv 2021, arXiv:2105.10425..
Alestas et al. (2020)
Alestas, G.; Kazantzidis, L.; Perivolaropoulos, L.
H${}_{0}$ tension, phantom dark energy, and cosmological parameter
degeneracies.
Phys. Rev. D 2020, 101, 123516,
doi:\changeurlcolorblack10.1103/PhysRevD.101.123516.
Kazantzidis et al. (2021)
Kazantzidis, L.; Koo, H.; Nesseris, S.; Perivolaropoulos, L.;
Shafieloo, A.
Hints for possible low redshift oscillation around the best-fitting
$\Lambda$CDM model in the expansion history of the Universe.
Mon. Not. R. Astron. Soc. 2021,
501, 3421–3426,
doi:\changeurlcolorblack10.1093/mnras/staa3866.
Martín and Rubio (2021)
Martín, M.S.; Rubio, C.
Hubble tension and matter inhomogeneities: A theoretical perspective. arXiv 2021, arXiv:astro-ph.CO/2107.14377.
Buchert (2000)
Buchert, T.
On Average Properties of Inhomogeneous Fluids in General Relativity:
Dust Cosmologies.
Gen. Relativ. Gravit. 2000, 32, 105–125.
doi:\changeurlcolorblack10.1023/a:1001800617177.
Gasperini et al. (2009)
Gasperini, M.; Marozzi, G.; Veneziano, G.
Gauge invariant averages for the cosmological backreaction.
J. Cosmol. Astropart. Phys. 2009, 2009, 011–011.
doi:\changeurlcolorblack10.1088/1475-7516/2009/03/011.
Gasperini et al. (2011)
Gasperini, M.; Marozzi, G.; Nugier, F.; Veneziano, G.
Light-cone averaging in cosmology: formalism and applications.
J. Cosmol. Astropart. Phys. 2011, 2011, 008–008.
doi:\changeurlcolorblack10.1088/1475-7516/2011/07/008.
Fanizza et al. (2020)
Fanizza, G.; Gasperini, M.; Marozzi, G.; Veneziano, G.
Generalized covariant prescriptions for averaging cosmological
observables.
J. Cosmol. Astropart. Phys. 2020, 2020, 017–017.
doi:\changeurlcolorblack10.1088/1475-7516/2020/02/017.
Ben-Dayan et al. (2013)
Ben-Dayan, I.; Gasperini, M.; Marozzi, G.; Nugier, F.; Veneziano, G.
Average and dispersion of the luminosity-redshift relation in the
concordance model.
J. Cosmol. Astropart. Phys. 2013, 2013, 002–002.
doi:\changeurlcolorblack10.1088/1475-7516/2013/06/002.
Fleury et al. (2017)
Fleury, P.; Clarkson, C.; Maartens, R.
How does the cosmic large-scale structure bias the Hubble diagram?
J. Cosmol. Astropart. Phys. 2017, 2017, 062–062.
doi:\changeurlcolorblack10.1088/1475-7516/2017/03/062.
Adamek et al. (2019)
Adamek, J.; Clarkson, C.; Coates, L.; Durrer, R.; Kunz, M.
Bias and scatter in the Hubble diagram from cosmological large-scale
structure.
Phys. Rev. D 2019, 100, 021301.
doi:\changeurlcolorblack10.1103/PhysRevD.100.021301.
Amendola et al. (2018)
Amendola, L.; Appleby, S.; Avgoustidis, A.; Bacon, D.; Baker, T.; Baldi, M.;
Bartolo, N.; Blanchard, A.; Bonvin, C.; Borgani, S.; et al.
Cosmology and fundamental physics with the Euclid satellite.
Living Rev. Relativ. 2018, 21.
doi:\changeurlcolorblack10.1007/s41114-017-0010-3.
Fanizza (2021)
Fanizza, G.
Precision Cosmology and Hubble tension in the era of LSS survey. arXiv 2021, arXiv:astro-ph.CO/2110.15272.
Andreoni et al. (2021)
Andreoni, I.; Margutti, R.; Salafia, O.S.; Parazin, B.; Villar, V.A.; Coughlin,
M.W.; Yoachim, P.; Mortensen, K.; Brethauer, D.; Smartt, S.J.; et al.
Target of Opportunity Observations of Gravitational Wave Events with
Vera C. Rubin Observatory. arXiv 2021,
arXiv:astro-ph.HE/2111.01945.
Ben-Dayan et al. (2014)
Ben-Dayan, I.; Durrer, R.; Marozzi, G.; Schwarz, D.J.
Value of H${}_{0}$ in the Inhomogeneous Universe.
Phys. Rev. Lett. 2014, 112, 221301,
doi:\changeurlcolorblack10.1103/PhysRevLett.112.221301.
Fanizza et al. (2021)
Fanizza, G.; Fiorini, B.; Marozzi, G.
Cosmic variance of ${H}_{0}$ in light of forthcoming high-redshift
surveys.
Phys. Rev. D 2021, 104, 083506.
doi:\changeurlcolorblack10.1103/PhysRevD.104.083506.
Haslbauer et al. (2020)
Haslbauer, M.; Banik, I.; Kroupa, P.
The KBC void and Hubble tension contradict $\Lambda$CDM
on a Gpc scale—Milgromian dynamics as a possible solution.
Mon. Not. R. Astron. Soc. 2020,
499, 2845–2883,
doi:\changeurlcolorblack10.1093/mnras/staa2348.
Perivolaropoulos (2014)
Perivolaropoulos, L.
Large Scale Cosmological Anomalies and Inhomogeneous Dark Energy. arXiv 2014, arXiv:astro-ph.CO/1401.5044.
Castello et al. (2021)
Castello, S.; Högås, M.; Mörtsell, E.
A Cosmological Underdensity Does Not Solve the Hubble Tension. arXiv 2021, arXiv:astro-ph.CO/2110.04226.
Banik and Zhao (2021)
Banik, I.; Zhao, H.
From galactic bars to the Hubble tension—Weighing up the
astrophysical evidence for Milgromian gravity. arXiv 2021,
arXiv:astro-ph.CO/2110.06936.
Alestas and Perivolaropoulos (2021)
Alestas, G.; Perivolaropoulos, L.
Late-time approaches to the Hubble tension deforming H(z), worsen
the growth tension.
Mon. Not. R. Astron. Soc. 2021,
504, 3956–3962,
doi:\changeurlcolorblack10.1093/mnras/stab1070.
Normann and Brevik (2021)
Normann, B.D.; Brevik, I.H.
Can the Hubble tension be resolved by bulk viscosity?
Mod. Phys. Lett. A 2021, 36, 2150198,
doi:\changeurlcolorblack10.1142/S0217732321501984.
Bernal et al. (2021)
Bernal, J.L.; Verde, L.; Jimenez, R.; Kamionkowski, M.; Valcin, D.;
Wandelt, B.D.
Trouble beyond H${}_{0}$ and the new cosmic triangles.
Phys. Rev. D 2021, 103, 103533,
doi:\changeurlcolorblack10.1103/PhysRevD.103.103533.
Thiele et al. (2021)
Thiele, L.; Guan, Y.; Hill, J.C.; Kosowsky, A.; Spergel, D.N.
Can small-scale baryon inhomogeneities resolve the Hubble tension?
An investigation with ACT DR4.
arXiv 2021, arXiv:2105.03003.
Grande and Perivolaropoulos (2011)
Grande, J.; Perivolaropoulos, L.
Generalized Lemaître-Tolman-Bondi model with inhomogeneous
isotropic dark energy: Observational constraints.
Phys. Rev. D 2011, 84, 023514,
doi:\changeurlcolorblack10.1103/PhysRevD.84.023514.
Dinda (2021)
Dinda, B.R.
Cosmic expansion parametrization: Implication for curvature and
$\text{H}_{0}$ tension.
arXiv 2021, arXiv:2106.02963.
Marra and Perivolaropoulos (2021)
Marra, V.; Perivolaropoulos, L.
Rapid transition of $G_{eff}$ at $z_{t}\approx 0.01$ as a possible
solution of the Hubble and growth tensions.
Phys. Rev. D 2021, 104.
doi:\changeurlcolorblack10.1103/physrevd.104.l021303.
Krishnan et al. (2020)
Krishnan, C.; Colgáin, E.Ó.; Ruchika, Sen, A.A.;
Sheikh-Jabbari, M.M.; Yang, T.
Is there an early Universe solution to Hubble tension?
Phys. Rev. D 2020, 102, 103525,
doi:\changeurlcolorblack10.1103/PhysRevD.102.103525.
Krishnan et al. (2021)
Krishnan, C.; O Colgain, E.; Sheikh-Jabbari, M.M.; Yang, T.
Running Hubble tension and a $H_{0}$ diagnostic.
Phys. Rev. D 2021, 103, 103509.
doi:\changeurlcolorblack10.1103/PhysRevD.103.103509.
Krishnan et al. (2021b)
Krishnan, C.; Mohayaee, R.; Colgáin, E.Ó.; Sheikh-Jabbari,
M.M.; Yin, L.
Does Hubble Tension Signal a Breakdown in FLRW Cosmology?
arXiv 2021, arXiv:2105.09790..
Robertson (1935)
Robertson, H.P.
Kinematics and World-Structure.
Astrophys. J. 1935, 82, 284.
doi:\changeurlcolorblack10.1086/143681.
Gerardi et al. (2021)
Gerardi, F.; Feeney, S.M.; Alsing, J.
Unbiased likelihood-free inference of the Hubble constant from light
standard sirens.
arXiv 2021, arXiv:2104.02728.
Escamilla-Rivera et al. (2021)
Escamilla-Rivera, C.; Levi Said, J.; Mifsud, J.
Performance of Non-Parametric Reconstruction Techniques in the
Late-Time Universe.
arXiv 2021, arXiv:2105.14332.
Sun et al. (2021)
Sun, W.; Jiao, K.; Zhang, T.J.
Influence of the Bounds of the Hyperparameters on the Reconstruction
of Hubble Constant with Gaussian Process.
arXiv 2021, arXiv:astro-ph.CO/2105.12618.
Renzi and Silvestri (2020)
Renzi, F.; Silvestri, A.
A look at the Hubble speed from first principles.
arXiv 2020,
arXiv:astro-ph.CO/2011.10559.
Gurzadyan and Stepanian (2021)
Gurzadyan, V.G.; Stepanian, A.
Hubble tension vs. two flows.
Eur. Phys. J. Plus 2021, 136.
doi:\changeurlcolorblack10.1140/epjp/s13360-021-01229-x.
Geng et al. (2021)
Geng, C.Q.; Hsu, Y.T.; Lu, J.R.; Yin, L.
A Dark Energy model from Generalized Proca Theory.
Phys. Dark Univ. 2021, 32, 100819,
doi:\changeurlcolorblack10.1016/j.dark.2021.100819.
Reyes and Escamilla-Rivera (2021)
Reyes, M.; Escamilla-Rivera, C.
Improving data-driven model-independent reconstructions and updated
constraints on dark energy models from Horndeski cosmology. arXiv 2021, arXiv:2104.04484.
doi:\changeurlcolorblack10.1088/1475-7516/2021/07/048.
Petronikolou et al. (2021)
Petronikolou, M.; Basilakos, S.; Saridakis, E.N.
Alleviating $H_{0}$ tension in Horndeski gravity. arXiv 2021,
arXiv:gr-qc/2110.01338.
Alestas et al. (2021)
Alestas, G.; Antoniou, I.; Perivolaropoulos, L.
Hints for a Gravitational Transition in Tully–Fisher Data.
Universe 2021, 7, 366.
doi:\changeurlcolorblack10.3390/universe7100366.
Benisty and Staicova (2021)
Benisty, D.; Staicova, D.
A preference for Dynamical Dark Energy? arXiv 2021,
arXiv:astro-ph.CO/2107.14129.
Ó Colgáin et al. (2021)
Ó Colgáin, E.; Sheikh-Jabbari, M.; Yin, L.
Can dark energy be dynamical?
Phys. Rev. D 2021, 104.
doi:\changeurlcolorblack10.1103/physrevd.104.023510.
Chevallier and Polarski (2001)
Chevallier, M.; Polarski, D.
Accelerating Universes with Scaling Dark Matter.
Int. J. Mod. Phys. D 2001, 10, 213–223,
doi:\changeurlcolorblack10.1142/S0218271801000822.
Linder (2003)
Linder, E.V.
Exploring the Expansion History of the Universe.
Phys. Rev. Lett. 2003, 90, 091301,
doi:\changeurlcolorblack10.1103/PhysRevLett.90.091301.
Aloni et al. (2021)
Aloni, D.; Berlin, A.; Joseph, M.; Schmaltz, M.; Weiner, N.
A Step in Understanding the Hubble Tension. arXiv 2021,
arXiv:astro-ph.CO/2111.00014.
Ghosh et al. (2021)
Ghosh, S.; Kumar, S.; Tsai, Y.
Free-streaming and Coupled Dark Radiation Isocurvature
Perturbations: Constraints and Application to the Hubble Tension. arXiv 2021,
arXiv:astro-ph.CO/2107.09076.
Shrivastava et al. (2021)
Shrivastava, P.; Khan, A.J.; Goswami, G.K.; Yadav, A.K.; Singh, J.K.
The simplest parametrization of equation of state parameter in the
scalar field Universe. arXiv 2021,
arXiv:astro-ph.CO/2107.05044.
Pereira (2021)
Pereira, S.H.
An unified cosmological model driven by a scalar field nonminimally
coupled to gravity. arXiv 2021,
arXiv:astro-ph.CO/2111.00029.
Bag et al. (2021)
Bag, S.; Sahni, V.; Shafieloo, A.; Shtanov, Y.
Phantom braneworld and the Hubble tension. arXiv 2021,
arXiv:astro-ph.CO/2107.03271.
Franchino-Viñas and Mosquera (2021)
Franchino-Viñas, S.A.; Mosquera, M.E.
The cosmological lithium problem, varying constants and the $H_{0}$
tension. arXiv 2021, arXiv:astro-ph.CO/2107.02243.
Palle (2021)
Palle, D.
Einstein–Cartan cosmology and the high-redshift Universe. arXiv 2021,
arXiv:physics.gen-ph/2106.08136.
Liu et al. (2021)
Liu, W.; Anchordoqui, L.A.; Di Valentino, E.; Pan, S.; Wu, Y.; Yang, W.
Constraints from High-Precision Measurements of the Cosmic Microwave
Background: The Case of Disintegrating Dark Matter with ${\Lambda}$ or
Dynamical Dark Energy. arXiv 2021,
arXiv:astro-ph.CO/2108.04188.
Blinov et al. (2021)
Blinov, N.; Krnjaic, G.; Li, S.W.
Towards a Realistic Model of Dark Atoms to Resolve the Hubble
Tension. arXiv 2021, arXiv:hep-ph/2108.11386.
Galli et al. (2021)
Galli, S.; Pogosian, L.; Jedamzik, K.; Balkenhol, L.
Consistency of Planck, ACT and SPT constraints on magnetically
assisted recombination and forecasts for future experiments. arXiv 2021,
arXiv:astro-ph.CO/2109.03816.
Liu et al. (2021)
Liu, X.H.; Li, Z.H.; Qi, J.Z.; Zhang, X.
Galaxy-Scale Test of General Relativity with Strong Gravitational
Lensing. arXiv 2021, arXiv:astro-ph.CO/2109.02291.
Hou et al. (2021)
Hou, S.; Fan, X.L.; Zhu, Z.H.
Constraining cosmological parameters from strong lensing with DECIGO
and B-DECIGO sources.
Mon. Not. R. Astron. Soc. 2021,
507, 761–771,
doi:\changeurlcolorblack10.1093/mnras/stab2221.
Sola (2021)
Sola, J.
Running vacuum interacting with dark matter or with running
gravitational coupling. Phenomenological implications. arXiv 2021,
arXiv:gr-qc/2109.12086.
Cuesta et al. (2021)
Cuesta, A.J.; Gómez, M.E.; Illana, J.I.; Masip, M.
Cosmology of an Axion-Like Majoron. arXiv 2021,
arXiv:hep-ph/2109.07336.
González-López (2021)
González-López, M.
Neutrino Masses and Hubble Tension via a Majoron in MFV. arXiv 2021,
arXiv:hep-ph/2110.15698.
Prat et al. (2021)
Prat, J.; Hogan, C.; Chang, C.; Frieman, J.
Vacuum Energy Density Measured from Cosmological Data. arXiv 2021,
arXiv:astro-ph.CO/2111.08151.
Joseph and Saha (2021)
Joseph, A.; Saha, R.
Dark energy with oscillatory tracking potential: Observational
Constraints and Perturbative effects. arXiv 2021,
arXiv:astro-ph.CO/2110.00229.
Aghababaei et al. (2021)
Aghababaei, S.; Moradpour, H.; Vagenas, E.C.
Hubble tension bounds the GUP and EUP parameters.
Eur. Phys. J. Plus 2021, 136.
doi:\changeurlcolorblack10.1140/epjp/s13360-021-02007-5.
Bansal et al. (2021)
Bansal, S.; Kim, J.H.; Kolda, C.; Low, M.; Tsai, Y.
Mirror Twin Higgs Cosmology: Constraints and a Possible Resolution
to the $H_{0}$ and $S_{8}$ Tensions. arXiv 2021,
arXiv:hep-ph/2110.04317.
Dialektopoulos et al. (2021)
Dialektopoulos, K.; Said, J.L.; Mifsud, J.; Sultana, J.; Adami, K.Z.
Neural Network Reconstruction of Late-Time Cosmology and Null Tests. arXiv 2021,
arXiv:astro-ph.CO/2111.11462.
Alestas et al. (2021)
Alestas, G.; Camarena, D.; Di Valentino, E.; Kazantzidis, L.; Marra, V.;
Nesseris, S.; Perivolaropoulos, L.
Late-transition vs. smooth $H(z)$ deformation models for the
resolution of the Hubble crisis. arXiv 2021,
arXiv:astro-ph.CO/2110.04336.
Parnovsky (2021)
Parnovsky, S.
Possible Modification of the Standard Cosmological Model to Resolve
a Tension with Hubble Constant Values.
Ukr. J. Phys. 2021, 66, 739.
doi:\changeurlcolorblack10.15407/ujpe66.9.739.
Zhang et al. (2021)
Zhang, P.; D’Amico, G.; Senatore, L.; Zhao, C.; Cai, Y.
BOSS Correlation Function Analysis from the Effective Field Theory
of Large-Scale Structure. arXiv 2021,
arXiv:astro-ph.CO/2110.07539.
Hansen (2021)
Hansen, S.H.
Accelerated expansion induced by dark matter with two charges.
Mon. Not. R. Astron. Soc. Lett. 2021, 508, 22–25,
doi:\changeurlcolorblack10.1093/mnrasl/slab103.
Gariazzo et al. (2021)
Gariazzo, S.; Di Valentino, E.; Mena, O.; Nunes, R.C.
Robustness of non-standard cosmologies solving the Hubble constant
tension. arXiv 2021, arXiv:astro-ph.CO/2111.03152.
Ruiz-Zapatero, Jaime et al. (2021)
Ruiz-Zapatero, J.; Stölzner, B.; Joachimi, B.; Asgari, M.; Bilicki, M.; Dvornik, A.; Giblin, B.; Heymans, C.; Hildebrandt, H.; Kannawadi, A.; et al.
Geometry versus growth—Internal consistency of the flat model with
KiDS-1000.
Astron. Astrophys. 2021, 655, A11.
doi:\changeurlcolorblack10.1051/0004-6361/202141350.
Cai et al. (2021)
Cai, R.G.; Guo, Z.K.; Wang, S.J.; Yu, W.W.; Zhou, Y.
A No-Go guide for the Hubble tension. arXiv 2021,
arXiv:astro-ph.CO/2107.13286.
Mehrabi and Vazirnia (2021)
Mehrabi, A.; Vazirnia, M.
Non-parametric modeling of the cosmological data, base on the
$\chi^{2}$ distribution. arXiv 2021,
arXiv:astro-ph.CO/2107.11539.
Parnovsky (2021)
Parnovsky, S.L.
Bias of the Hubble constant value caused by errors in galactic
distance indicators. arXiv 2021,
arXiv:astro-ph.CO/2109.09645.
Baldwin and Schechter (2021)
Baldwin, D.; Schechter, P.L.
A Malmquist-like bias in the inferred areas of diamond caustics and
the resulting bias in inferred time delays for gravitationally lensed
quasars. arXiv 2021, arXiv:astro-ph.CO/2110.06378.
Huber et al. (2021)
Huber, S.; Suyu, S.H.; Ghoshdastidar, D.; Taubenberger, S.; Bonvin, V.; Chan,
J.H.H.; Kromer, M.; Noebauer, U.M.; Sim, S.A.; Leal-Taixé, L.
HOLISMOKES—VII. Time-delay measurement of strongly lensed SNe Ia
using machine learning. arXiv 2021,
arXiv:astro-ph.CO/2108.02789.
Mercier (2021)
Mercier, C.
A New Physics Would Explain What Looks Like an Irreconcilable Tension
between the Values of Hubble Constants and Allows H0 to Be Calculated
Theoretically Several Ways.
J. Mod. Phys. 2021, 12, 1656–1707.
doi:\changeurlcolorblack10.4236/jmp.2021.1212098.
Ren et al. (2021)
Ren, X.; Wong, T.H.T.; Cai, Y.F.; Saridakis, E.N.
Data-driven reconstruction of the late-time cosmic acceleration with
$f(T)$ gravity.
Phys. Dark Univ. 2021, 32, 100812.
\changeurlcolorblackhttps://doi.org/10.1016/j.dark.2021.100812.
Hryczuk and Jodłowski (2020)
Hryczuk, A.; Jodłowski, K.
Self-interacting dark matter from late decays and the H0 tension.
Phys. Rev. D 2020, 102.
doi:\changeurlcolorblack10.1103/physrevd.102.043024.
Di Valentino et al. (2021)
Di Valentino, E.; Mena, O.; Pan, S.; Visinelli, L.; Yang, W.;
Melchiorri, A.; Mota, D.F.; Riess, A.G.; Silk, J.
In the realm of the Hubble tension—A review of solutions.
Class. Quantum Gravity 2021, 38, 153001,
doi:\changeurlcolorblack10.1088/1361-6382/ac086d.
Perivolaropoulos and Skara (2021)
Perivolaropoulos, L.; Skara, F.
Challenges for $\Lambda$CDM: An update.
arXiv 2021, arXiv:2105.05208.
Saridakis et al. (2021)
Saridakis, E.N.; Lazkoz, R.; Salzano, V.; Vargas Moniz, P.;
Capozziello, S.; Beltrán Jiménez, J.; De Laurentis, M.; Olmo,
G.J.; Akrami, Y.; Bahamonde, S.; et al.
Modified Gravity and Cosmology: An Update by the CANTATA Network.
arXiv 2021, arXiv:2105.12582.
Scolnic et al. (2018)
Scolnic, D.M.; Jones, D.O.; Rest, A.; Pan, Y.C.; Chornock, R.;
Foley, R.J.; Huber, M.E.; Kessler, R.; Narayan, G.; Riess, A.G.;
et al.
The Complete Light-curve Sample of Spectroscopically Confirmed SNe
Ia from Pan-STARRS1 and Cosmological Constraints from the Combined Pantheon
Sample.
Astrophys. J. 2018, 859, 101,
doi:\changeurlcolorblack10.3847/1538-4357/aab9bb.
Weinberg (2008)
Weinberg, S.
Cosmology; Oxford University Press: Oxford, UK, 2008.
Guy et al. (2010)
Guy, J.; Sullivan, M.; Conley, A.; Regnault, N.; Astier, P.;
Balland, C.; Basa, S.; Carlberg, R.G.; Fouchez, D.; Hardin, D.;
et al.
The Supernova Legacy Survey 3-year sample: Type Ia supernovae
photometric distances and cosmological constraints.
Astron. Astrophys. 2010, 523, A7,
doi:\changeurlcolorblack10.1051/0004-6361/201014468.
Chotard et al. (2011)
Chotard, N.; Gangler, E.; Aldering, G.; Antilogus, P.; Aragon, C.;
Bailey, S.; Baltay, C.; Bongard, S.; Buton, C.; Canto, A.; et al.
The reddening law of type Ia supernovae: Separating intrinsic
variability from dust using equivalent widths.
Astron. Astrophys. 2011, 529, L4,
doi:\changeurlcolorblack10.1051/0004-6361/201116723.
Kenworthy et al. (2019)
Kenworthy, W.D.; Scolnic, D.; Riess, A.
The Local Perspective on the Hubble Tension: Local Structure Does
Not Impact Measurement of the Hubble Constant.
Astrophys. J. 2019, 875, 145,
doi:\changeurlcolorblack10.3847/1538-4357/ab0ebf.
Deng and Wei (2018)
Deng, H.K.; Wei, H.
Null signal for the cosmic anisotropy in the Pantheon supernovae
data.
Eur. Phys. J. C 2018, 78, 755,
doi:\changeurlcolorblack10.1140/epjc/s10052-018-6159-4.
Akarsu et al. (2019)
Akarsu, Ö.; Kumar, S.; Sharma, S.; Tedesco, L.
Constraints on a Bianchi type I spacetime extension of the standard
$\Lambda$ CDM model.
Phys. Rev. D 2019, 100, 023532,
doi:\changeurlcolorblack10.1103/PhysRevD.100.023532.
Shafieloo et al. (2018)
Shafieloo, A.; L’Huillier, B.; Starobinsky, A.A.
Falsifying $\Lambda$CDM : Model-independent tests of
the concordance model with eBOSS DR14Q and Pantheon.
Phys. Rev. D 2018, 98, 083526,
doi:\changeurlcolorblack10.1103/PhysRevD.98.083526.
Hossienkhani et al. (2019)
Hossienkhani, H.; Azimi, N.; Zarei, Z.
Probing the anisotropy effects on the CPL parametrizations from
light-curve SNIa, BAO and OHD datasets.
Int. J. Geom. Methods Mod. Phys.
2019, 16, 1950177–L5.
doi:\changeurlcolorblack10.1142/S0219887819501779.
L’Huillier et al. (2019)
L’Huillier, B.; Shafieloo, A.; Linder, E.V.; Kim, A.G.
Model independent expansion history from supernovae: Cosmology
versus systematics.
Mon. Not. R. Astron. Soc. 2019,
485, 2783–2790,
doi:\changeurlcolorblack10.1093/mnras/stz589.
Lusso et al. (2019)
Lusso, E.; Piedipalumbo, E.; Risaliti, G.; Paolillo, M.; Bisogni, S.;
Nardini, E.; Amati, L.
Tension with the flat $\Lambda$CDM model from a
high-redshift Hubble diagram of supernovae, quasars, and gamma-ray bursts.
Astron. Astrophys. 2019, 628, L4,
doi:\changeurlcolorblack10.1051/0004-6361/201936223.
Ma et al. (2019)
Ma, Y.B.; Cao, S.; Zhang, J.; Qi, J.; Liu, T.; Liu, Y.; Geng, S.
Testing Cosmic Opacity with the Combination of Strongly Lensed and
Unlensed Supernova Ia.
Astrophys. J. 2019, 887, 163,
doi:\changeurlcolorblack10.3847/1538-4357/ab50c4.
Sadri (2019)
Sadri, E.
Observational constraints on interacting Tsallis holographic dark
energy model.
Eur. Phys. J. C 2019, 79, 762,
doi:\changeurlcolorblack10.1140/epjc/s10052-019-7263-9.
Sadri and Khurshudyan (2019)
Sadri, E.; Khurshudyan, M.
An interacting new holographic dark energy model: Observational
constraints.
Int. J. Mod. Phys. D 2019, 28, 1950152–26,
doi:\changeurlcolorblack10.1142/S0218271819501529.
Wagner and Meyer (2019)
Wagner, J.; Meyer, S.
Generalized model-independent characterization of strong
gravitational lenses V: reconstructing the lensing distance ratio by
supernovae for a general Friedmann universe.
Mon. Not. R. Astron. Soc. 2019,
490, 1913–1927,
doi:\changeurlcolorblack10.1093/mnras/stz2717.
Zhai and Wang (2019)
Zhai, Z.; Wang, Y.
Robust and model-independent cosmological constraints from distance
measurements.
J. Cosmol. Astropart. Phys. 2019, 2019, 005,
doi:\changeurlcolorblack10.1088/1475-7516/2019/07/005.
Zhao et al. (2019)
Zhao, D.; Zhou, Y.; Chang, Z.
Anisotropy of the Universe via the Pantheon supernovae sample
revisited.
Mon. Not. R. Astron. Soc. 2019,
486, 5679–5689,
doi:\changeurlcolorblack10.1093/mnras/stz1259.
Abdullah et al. (2020)
Abdullah, M.H.; Klypin, A.; Wilson, G.
Cosmological Constraints on $\Omega$${}_{m}$ and
$\sigma$${}_{8}$ from Cluster Abundances Using the GalWCat19
Optical-spectroscopic SDSS Catalog.
Astrophys. J. 2020, 901, 90,
doi:\changeurlcolorblack10.3847/1538-4357/aba619.
Al Mamon and Saha (2020)
Al Mamon, A.; Saha, S.
The logotropic dark fluid: Observational and thermodynamic
constraints.
Int. J. Mod. Phys. D 2020, 29, 2050097,
doi:\changeurlcolorblack10.1142/S0218271820500972.
Anagnostopoulos et al. (2020)
Anagnostopoulos, F.K.; Basilakos, S.; Saridakis, E.N.
Observational constraints on Barrow holographic dark energy.
Eur. Phys. J. C 2020, 80, 826,
doi:\changeurlcolorblack10.1140/epjc/s10052-020-8360-5.
Brout et al. (2021)
Brout, D.; Hinton, S.R.; Scolnic, D.
Binning is Sinning (Supernova Version): The Impact of
Self-calibration in Cosmological Analyses with Type Ia Supernovae.
Astrophys. J. Lett. 2021, 912, L26,
doi:\changeurlcolorblack10.3847/2041-8213/abf4db.
Cai et al. (2021)
Cai, R.G.; Ding, J.F.; Guo, Z.K.; Wang, S.J.; Yu, W.W.
Do the observational data favor a local void?
Phys. Rev. D 2021, 103, 123539,
doi:\changeurlcolorblack10.1103/PhysRevD.103.123539.
Chang et al. (2019)
Chang, Z.; Zhao, D.; Zhou, Y.
Constraining the anisotropy of the Universe with the Pantheon
supernovae sample.
Chin. Phys. C 2019, 43, 125102,
doi:\changeurlcolorblack10.1088/1674-1137/43/12/125102.
D’Amico et al. (2021)
D’Amico, G.; Senatore, L.; Zhang, P.
Limits on wCDM from the EFTofLSS with the PyBird code.
J. Cosmol. Astropart. Phys. 2021, 2021, 006,
doi:\changeurlcolorblack10.1088/1475-7516/2021/01/006.
Di Valentino et al. (2020)
Di Valentino, E.; Gariazzo, S.; Mena, O.; Vagnozzi, S.
Soundness of dark energy properties.
J. Cosmol. Astropart. Phys. 2020, 2020, 045,
doi:\changeurlcolorblack10.1088/1475-7516/2020/07/045.
Gao et al. (2020)
Gao, C.; Chen, Y.; Zheng, J.
Investigating the relationship between cosmic curvature and dark
energy models with the latest supernova sample.
Res. Astron. Astrophys. 2020, 20, 151,
doi:\changeurlcolorblack10.1088/1674-4527/20/9/151.
Garcia-Quintero et al. (2020)
Garcia-Quintero, C.; Ishak, M.; Ning, O.
Current constraints on deviations from General Relativity using
binning in redshift and scale.
J. Cosmol. Astropart. Phys. 2020, 2020, 018,
doi:\changeurlcolorblack10.1088/1475-7516/2020/12/018.
Geng et al. (2020)
Geng, S.; Cao, S.; Liu, T.; Biesiada, M.; Qi, J.; Liu, Y.; Zhu,
Z.H.
Gravitational-wave Constraints on the Cosmic Opacity at z
$\sim$ 5: Forecast from Space Gravitational-wave Antenna
DECIGO.
Astrophys. J. 2020, 905, 54.
doi:\changeurlcolorblack10.3847/1538-4357/abc076.
Ghaffari et al. (2020)
Ghaffari, S.; Sadri, E.; Ziaie, A.H.
Tsallis holographic dark energy in fractal universe.
Mod. Phys. Lett. A 2020, 35, 2050107,
doi:\changeurlcolorblack10.1142/S0217732320501072.
Hu et al. (2020)
Hu, J.P.; Wang, Y.Y.; Wang, F.Y.
Testing cosmic anisotropy with Pantheon sample and quasars at high
redshifts.
Astron. Astrophys. 2020, 643, A93,
doi:\changeurlcolorblack10.1051/0004-6361/202038541.
Huang (2020)
Huang, Z.
Supernova Magnitude Evolution and PAge Approximation.
Astrophys. J. Lett. 2020, 892, L28,
doi:\changeurlcolorblack10.3847/2041-8213/ab8011.
Çamlıbel et al. (2020)
Çamlıbel, A.K.; Semiz, İ.; Feyizoǧlu, M.A.
Pantheon update on a model-independent analysis of cosmological
supernova data.
Class. Quantum Gravity 2020, 37, 235001,
doi:\changeurlcolorblack10.1088/1361-6382/abba48.
Liao et al. (2020)
Liao, K.; Shafieloo, A.; Keeley, R.E.; Linder, E.V.
Determining Model-independent H${}_{0}$ and Consistency Tests.
Astrophys. J. Lett. 2020, 895, L29,
doi:\changeurlcolorblack10.3847/2041-8213/ab8dbb.
Koo et al. (2021)
Koo, H.; Shafieloo, A.; Keeley, R.E.; L’Huillier, B.
Model selection and parameter estimation using the iterative
smoothing method.
J. Cosmol. Astropart. Phys. 2021, 2021, 034,
doi:\changeurlcolorblack10.1088/1475-7516/2021/03/034.
Li et al. (2020)
Li, E.K.; Du, M.; Xu, L.
General cosmography model with spatial curvature.
Mon. Not. R. Astron. Soc. 2020,
491, 4960–4972,
doi:\changeurlcolorblack10.1093/mnras/stz3308.
Luković et al. (2020)
Luković, V.V.; Haridasu, B.S.; Vittorio, N.
Exploring the evidence for a large local void with supernovae Ia
data.
Mon. Not. R. Astron. Soc. 2020,
491, 2075–2087,
doi:\changeurlcolorblack10.1093/mnras/stz3070.
Luongo and Muccino (2020)
Luongo, O.; Muccino, M.
Kinematic constraints beyond $z\simeq 0$ using calibrated GRB
correlations.
Astron. Astrophys. 2020, 641, A174,
doi:\changeurlcolorblack10.1051/0004-6361/202038264.
Micheletti (2020)
Micheletti, S.M.R.
Quintessence and tachyon dark energy in interaction with dark
matter: Observational constraints and model selection.
Int. J. Mod. Phys. D 2020, 29, 2050057,
doi:\changeurlcolorblack10.1142/S0218271820500571.
Mishra and Dua (2020)
Mishra, R.K.; Dua, H.
Phase transition of cosmological model with statistical techniques.
Astrophys. Space Sci. 2020, 365, 131.
doi:\changeurlcolorblack10.1007/s10509-020-03843-0.
Odintsov et al. (2021)
Odintsov, S.D.; Sáez-Chillón Gómez, D.; Sharov, G.S.
Analyzing the $H_{0}$ tension in $F(R)$ gravity models.
Nucl. Phys. B 2021, 966, 115377,
doi:\changeurlcolorblack10.1016/j.nuclphysb.2021.115377.
Prasad et al. (2021)
Prasad, R.; Singh, M.; Yadav, A.K.; Beesham, A.
An exact solution of the observable universe in Bianchi V
space-time.
Int. J. Mod. Phys. A 2021, 36, 2150044,
doi:\changeurlcolorblack10.1142/S0217751X21500445.
Rezaei et al. (2020)
Rezaei, M.; Pour-Ojaghi, S.; Malekjani, M.
A Cosmography Approach to Dark Energy Cosmologies: New Constraints
Using the Hubble Diagrams of Supernovae, Quasars, and Gamma-Ray Bursts.
Astrophys. J. 2020, 900, 70,
doi:\changeurlcolorblack10.3847/1538-4357/aba517.
Ringermacher and Mead (2020)
Ringermacher, H.I.; Mead, L.R.
Reaffirmation of cosmological oscillations in the scale factor from
the Pantheon compilation of 1048 Type Ia supernovae.
Mon. Not. R. Astron. Soc. 2020,
494, 2158–2165,
doi:\changeurlcolorblack10.1093/mnras/staa872.
Tang et al. (2021)
Tang, L.; Li, X.; Lin, H.N.; Liu, L.
Model-independently Calibrating the Luminosity Correlations of
Gamma-Ray Bursts Using Deep Learning.
Astrophys. J. 2021, 907, 121,
doi:\changeurlcolorblack10.3847/1538-4357/abcd92.
Wang and Chen (2020)
Wang, K.; Chen, L.
Constraints on Newton’s constant from cosmological observations.
Eur. Phys. J. C 2020, 80, 570,
doi:\changeurlcolorblack10.1140/epjc/s10052-020-8137-x.
Wei and Melia (2020)
Wei, J.J.; Melia, F.
Cosmology-independent Estimate of the Hubble Constant and Spatial
Curvature using Time-delay Lenses and Quasars.
Astrophys. J. 2020, 897, 127,
doi:\changeurlcolorblack10.3847/1538-4357/ab959b.
Zhang and Huang (2020)
Zhang, X.; Huang, Q.G.
Measuring H${}_{0}$ from low-z datasets.
Sci. China Phys. Mech. Astron. 2020,
63, 290402,
doi:\changeurlcolorblack10.1007/s11433-019-1504-8.
Baxter and Sherwin (2021)
Baxter, E.J.; Sherwin, B.D.
Determining the Hubble constant without the sound horizon scale:
measurements from CMB lensing.
Mon. Not. R. Astron. Soc. 2021,
501, 1823–1835,
doi:\changeurlcolorblack10.1093/mnras/staa3706.
Camarena and Marra (2021)
Camarena, D.; Marra, V.
On the use of the local prior on the absolute magnitude of Type Ia
supernovae in cosmological inference.
Mon. Not. R. Astron. Soc. 2021,
504, 5164–5171,
doi:\changeurlcolorblack10.1093/mnras/stab1200.
Cao et al. (2021)
Cao, S.; Ryan, J.; Ratra, B.
Using Pantheon and DES supernova, baryon acoustic oscillation, and
Hubble parameter data to constrain the Hubble constant, dark energy dynamics,
and spatial curvature.
Mon. Not. R. Astron. Soc. 2021,
504, 300–310,
doi:\changeurlcolorblack10.1093/mnras/stab942.
Heisenberg et al. (2021)
Heisenberg, L.; Bartelmann, M.; Brandenberger, R.; Refregier, A.
Model independent analysis of supernova data, dark energy,
trans-Planckian censorship and the swampland.
Phys. Lett. B 2021, 812, 135990,
doi:\changeurlcolorblack10.1016/j.physletb.2020.135990.
Jesus et al. (2021)
Jesus, J.F.; Valentim, R.; Moraes, P.H.R.S.; Malheiro, M.
Kinematic constraints on spatial curvature from supernovae Ia and
cosmic chronometers.
Mon. Not. R. Astron. Soc. 2021,
500, 2227–2235,
doi:\changeurlcolorblack10.1093/mnras/staa3426.
Lee (2021)
Lee, S.
Constraints on the time variation of the speed of light using
Pantheon dataset.
arXiv 2021, arXiv:2101.09862.
Montiel et al. (2021)
Montiel, A.; Cabrera, J.I.; Hidalgo, J.C.
Improving sampling and calibration of gamma-ray bursts as distance
indicators.
Mon. Not. R. Astron. Soc. 2021,
501, 3515–3526.
doi:\changeurlcolorblack10.1093/mnras/staa3926.
Mukherjee and Banerjee (2021)
Mukherjee, P.; Banerjee, N.
Non-parametric reconstruction of the cosmological jerk parameter.
Eur. Phys. J. C 2021, 81, 36,
doi:\changeurlcolorblack10.1140/epjc/s10052-021-08830-5.
Wang et al. (2021)
Wang, G.J.; Ma, X.J.; Xia, J.Q.
Machine learning the cosmic curvature in a model-independent way.
Mon. Not. R. Astron. Soc. 2021,
501, 5714–5722,
doi:\changeurlcolorblack10.1093/mnras/staa4044.
Torrado and Lewis (2021)
Torrado, J.; Lewis, A.
Cobaya: Code for Bayesian Analysis of hierarchical physical models.
JCAP 2021, 05, 057,
doi:\changeurlcolorblack10.1088/1475-7516/2021/05/057.
Sharov and Vasiliev (2018)
Sharov, G.S.; Vasiliev, V.O.
How predictions of cosmological models depend on Hubble parameter
data sets.
Math. Model. Geom. 2018, 6,
doi:\changeurlcolorblack10.26456/mmg/2018-611.
Eisenstein et al. (2005)
Eisenstein, D.J.; Zehavi, I.; Hogg, D.W.; Scoccimarro, R.; Blanton,
M.R.; Nichol, R.C.; Scranton, R.; Seo, H.J.; Tegmark, M.; Zheng,
Z.; et al.
Detection of the Baryon Acoustic Peak in the Large-Scale Correlation
Function of SDSS Luminous Red Galaxies.
Astrophys. J. 2005, 633, 560–574,
doi:\changeurlcolorblack10.1086/466512.
Cuceu et al. (2019)
Cuceu, A.; Farr, J.; Lemos, P.; Font-Ribera, A.
Baryon Acoustic Oscillations and the Hubble constant: past, present
and future.
J. Cosmol. Astropart. Phys. 2019, 2019, 044,
doi:\changeurlcolorblack10.1088/1475-7516/2019/10/044.
Aubourg et al. (2015)
Aubourg, É.; Bailey, S.; Bautista, J.E.; Beutler, F.; Bhardwaj,
V.; Bizyaev, D.; Blanton, M.; Blomqvist, M.; Bolton, A.S.; Bovy,
J.; et al.
Cosmological implications of baryon acoustic oscillation
measurements.
Phys. Rev. D 2015, 92, 123516,
doi:\changeurlcolorblack10.1103/PhysRevD.92.123516.
Singal et al. (2011)
Singal, J.; Petrosian, V.; Lawrence, A.; Stawarz, Ł.
On the Radio and Optical Luminosity Evolution of Quasars.
Astrophys. J. 2011, 743, 104,
doi:\changeurlcolorblack10.1088/0004-637X/743/2/104.
Singal et al. (2013)
Singal, J.; Petrosian, V.; Stawarz, Ł.; Lawrence, A.
The Radio and Optical Luminosity Evolution of Quasars. II. The SDSS
Sample.
Astrophys. J. 2013, 764, 43,
doi:\changeurlcolorblack10.1088/0004-637X/764/1/43.
Petrosian et al. (2015)
Petrosian, V.; Kitanidis, E.; Kocevski, D.
Cosmological Evolution of Long Gamma-Ray Bursts and the Star
Formation Rate.
Astrophys. J. 2015, 806, 44,
doi:\changeurlcolorblack10.1088/0004-637X/806/1/44.
Lloyd and Petrosian (2000)
Lloyd, N.M.; Petrosian, V.
Synchrotron Radiation as the Source of Gamma-Ray Burst Spectra.
Astrophys. J. 2000, 543, 722–732,
doi:\changeurlcolorblack10.1086/317125.
Dainotti et al. (2021)
Dainotti, M.G.; Petrosian, V.; Bowden, L.
Cosmological Evolution of the Formation Rate of Short Gamma-Ray
Bursts with and without Extended Emission. Astrophys. J. 2021.
914, L40.
doi:\changeurlcolorblack10.3847/2041-8213/abf5e4.
Atteia (1997)
Atteia, J.L.
Gamma-ray bursts: towards a standard candle luminosity.
Astron. Astrophys. 1997, 328, L21–L24,
Atteia (1998)
Atteia, J.L.
Choosing a measure of GRB brightness that approaches a standard
candle.
In Gamma-Ray Bursts, 4th Hunstville Symposium; American
Institute of Physics Conference Series; Meegan, C.A.,
Preece, R.D., Koshut, T.M., Eds.; American Institute of Physics: College Park, MD, USA, 1998; Volume 428, pp. 92–96.
doi:\changeurlcolorblack10.1063/1.55422.
Simone et al. (2021)
De Simone, B.; Nielson, V.; Rinaldi, E.; Dainotti, M.G.
A new perspective on cosmology through Supernovae Ia and Gamma Ray
Bursts. arXiv 2021, arXiv:astro-ph.CO/2110.11930.
Cao et al. (2022)
Cao, S.; Dainotti, M.; Ratra, B.
Standardizing Platinum Dainotti-correlated gamma-ray bursts, and
using them with standardized Amati-correlated gamma-ray bursts to constrain
cosmological model parameters. arXiv 2022,
arXiv:astro-ph.CO/2201.0524.
Cao et al. (2022)
Cao, S.; Khadka, N.; Ratra, B.
Standardizing Dainotti-correlated gamma-ray bursts, and using them with standardized Amati-correlated gamma-ray bursts to constrain cosmological model parameters. MNRAS 2022,
doi:\changeurlcolorblack10.1093/mnras/stab3559.
Efron and Petrosian (1992)
Efron, B.; Petrosian, V.
A simple test of independence for truncated data with applications to
redshift surveys.
Astrophys. J. 1992, 399, 345–352.
D’Agostini (1995)
D’Agostini, G.
A multidimensional unfolding method based on Bayes’ theorem.
Nucl. Instrum. Methods Phys. Res. A 1995, 362, 487–498.
doi:\changeurlcolorblack10.1016/0168-9002(95)00274-X.
Conley et al. (2010)
Conley, A.; Guy, J.; Sullivan, M.; Regnault, N.; Astier, P.; Balland, C.; Basa,
S.; Carlberg, R.G.; Fouchez, D.; Hardin, D.; et al.
Supernova constraints and systematic uncertainties from the first
three years of the supernova legacy survey.
Astrophys. J. Suppl. Ser. 2010, 192, 1.
doi:\changeurlcolorblack10.1088/0067-0049/192/1/1.
Betoule, M. et al. (2014)
Betoule, M.; Kessler, R.; Guy, J.; Mosher, J.; Hardin, D..;
Biswas, R.; Astier, P.; El-Hage, P.; Konig, M.; Kuhlmann, S..;
et al.
Improved cosmological constraints from a joint analysis of the
SDSS-II and SNLS supernova samples.
Astron. Astrophys. 2014, 568, A22.
doi:\changeurlcolorblack10.1051/0004-6361/201423413.
(359)
Bargiacchi, G., Benetti, M., Capozziello, S., Lusso, E., Risaliti, G. & Signorini, M. Quasar cosmology: dark energy evolution and spatial curvature. arXiv 2021, arXiv:2111.02420.
(360)
Bargiacchi, G., Risaliti, G., Benetti, M., Capozziello, S., Lusso, E., Saccardi, A. & Signorini, M. Cosmography by orthogonalized logarithmic polynomials. Astronomy & Astrophysics. 649 pp. A65 (2021,5), http://dx.doi.org/10.1051/0004-6361/202140386
Dainotti et al. (2021)
Dainotti, M.G.; Bogdan, M.; Narendra, A.; Gibson, S.J.; Miasojedow, B.;
Liodakis, I.; Pollo, A.; Nelson, T.; Wozniak, K.; Nguyen, Z.; et al.
Predicting the Redshift of $\upgamma$-Ray-loud AGNs Using
Supervised Machine Learning. Astrophys. J. 2021.
920, 118.
doi:\changeurlcolorblack10.3847/1538-4357/ac1748.
Dainotti et al. (2019)
Dainotti, M.; Petrosian, V.; Bogdan, M.; Miasojedow, B.; Nagataki, S.; Hastie,
T.; Nuyngen, Z.; Gilda, S.; Hernández, X.; Krol, D.
Gamma-ray Bursts as distance indicators through a machine learning
approach. arXiv 2019, arXiv:1907.05074.
Childress et al. (2013)
Childress, M.; Aldering, G.; Antilogus, P.; Aragon, C.; Bailey, S.;
Baltay, C.; Bongard, S.; Buton, C.; Canto, A.; Cellier-Holzem, F.;
et al.
Host Galaxies of Type Ia Supernovae from the Nearby Supernova
Factory.
Astrophys. J. 2013, 770, 107,
doi:\changeurlcolorblack10.1088/0004-637X/770/2/107.
Nicolas et al. (2021)
Nicolas, N.; Rigault, M.; Copin, Y.; Graziani, R.; Aldering, G.;
Briday, M.; Kim, Y.L.; Nordin, J.; Perlmutter, S.; Smith, M.
Redshift evolution of the underlying type Ia supernova stretch
distribution.
Astron. Astrophys. 2021, 649, A74,
doi:\changeurlcolorblack10.1051/0004-6361/202038447.
Brout et al. (2021)
Brout, D.; Taylor, G.; Scolnic, D.; Wood, C.M.; Rose, B.M.; Vincenzi, M.;
Dwomoh, A.; Lidman, C.; Riess, A.; Ali, N.; et al.
The Pantheon+ Analysis: SuperCal-Fragilistic Cross Calibration,
Retrained SALT2 Light Curve Model, and Calibration Systematic Uncertainty. arXiv 2021, arXiv:astro-ph.CO/2112.03864.
Carr et al. (2021)
Carr, A.; Davis, T.M.; Scolnic, D.; Said, K.; Brout, D.; Peterson, E.R.;
Kessler, R.
The Pantheon+ Analysis: Improving the Redshifts and Peculiar
Velocities of Type Ia Supernovae Used in Cosmological Analyses. arXiv 2021,
arXiv:astro-ph.CO/2112.01471.
Popovic et al. (2021)
Popovic, B.; Brout, D.; Kessler, R.; Scolnic, D.
The Pantheon+ Analysis: Forward-Modeling the Dust and Intrinsic
Colour Distributions of Type Ia Supernovae, and Quantifying their Impact on
Cosmological Inferences. arXiv 2021,
arXiv:astro-ph.CO/2112.04456.
Scolnic et al. (2021)
Scolnic, D.; Brout, D.; Carr, A.; Riess, A.G.; Davis, T.M.; Dwomoh, A.; Jones,
D.O.; Ali, N.; Charvu, P.; Chen, R.; et al.
The Pantheon+ Type Ia Supernova Sample: The Full Dataset and
Light-Curve Release. arXiv 2021,
arXiv:astro-ph.CO/2112.03863.
Riess et al. (2022)
Riess, A.G.; Yuan, W.; Macri, L.M.; Scolnic, D.; Brout, D.; Casertano, S.;
Jones, D.O.; Murakami, Y.; Breuval, L.; Brink, T.G.; et al.
A Comprehensive Measurement of the Local Value of the Hubble Constant
with 1 km/s/Mpc Uncertainty from the Hubble Space Telescope and the SH0ES
Team. arXiv 2022, arXiv:astro-ph.CO/2112.04510.
Damour and Nordtvedt (1993)
Damour, T.; Nordtvedt, K.
Tensor-scalar cosmological models and their relaxation toward
general relativity.
Phys. Rev. D 1993, 48, 3436–3450.
doi:\changeurlcolorblack10.1103/PhysRevD.48.3436.
Damour and Polyakov (1994)
Damour, T.; Polyakov, A.M.
The string dilation and a least coupling principle.
Nucl. Phys. B 1994, 423, 532–558,
doi:\changeurlcolorblack10.1016/0550-3213(94)90143-0.
Damour (1996)
Damour, T.
Gravitation, experiment and cosmology.
Les Houches Summer School on Gravitation and Quantizations, Session
57. arXiv 1996, arXiv:gr-qc/gr-qc/9606079.
Boisseau et al. (2000)
Boisseau, B.; Esposito-Farèse, G.; Polarski, D.; Starobinsky, A.A.
Reconstruction of a Scalar-Tensor Theory of Gravity in an
Accelerating Universe.
Phys. Rev. Lett. 2000, 85, 2236–2239,
doi:\changeurlcolorblack10.1103/PhysRevLett.85.2236.
Esposito-Farèse and Polarski (2001)
Esposito-Farèse, G.; Polarski, D.
Scalar-tensor gravity in an accelerating universe.
Phys. Rev. D 2001, 63, 063504,
doi:\changeurlcolorblack10.1103/PhysRevD.63.063504.
Jordan (1955)
Jordan, P.
Schwerkraft und Weltall: Grundlagen der theoretischen Kosmologie;
Die Wissenschaft: Braunschweig, Germany, 1955.
Fierz (1956)
Fierz, M.
On the physical interpretation of P. Jordan’s extended theory of
gravitation.
Helv. Phys. Acta 1956, 29, 128–134.
Brans and Dicke (1961)
Brans, C.; Dicke, R.H.
Mach’s Principle and a Relativistic Theory of Gravitation.
Phys. Rev. 1961, 124, 925–935.
doi:\changeurlcolorblack10.1103/PhysRev.124.925.
Singh and Peracaula (2021)
Singh, C.P.; Peracaula, J.S.
Friedmann cosmology with decaying vacuum density in Brans–Dicke
theory. arXiv 2021, arXiv:gr-qc/2110.12411.
Catena et al. (2004)
Catena, R.; Fornengo, N.; Masiero, A.; Pietroni, M.; Rosati, F.
Dark matter relic abundance and scalar-tensor dark energy.
Phys. Rev. D 2004, 70, 063519,
doi:\changeurlcolorblack10.1103/PhysRevD.70.063519.
Kazantzidis and Perivolaropoulos (2020)
Kazantzidis, L.; Perivolaropoulos, L.
Hints of a local matter underdensity or modified gravity in the low
z Pantheon data.
Phys. Rev. D 2020, 102, 023520,
doi:\changeurlcolorblack10.1103/PhysRevD.102.023520.
Capozziello and de Laurentis (2011)
Capozziello, S.; de Laurentis, M.
Extended Theories of Gravity.
Physics Reports 2011, 509, 167–321,
doi:\changeurlcolorblack10.1016/j.physrep.2011.09.003.
Hu and Sawicki (2007)
Hu, W.; Sawicki, I.
Models of f(R) cosmic acceleration that evade solar system tests.
Phys. Rev. D 2007, 76, 064004,
doi:\changeurlcolorblack10.1103/PhysRevD.76.064004.
Starobinsky (2007)
Starobinsky, A.A.
Disappearing cosmological constant in f( R) gravity.
Soviet Journal of Experimental and Theoretical Physics Letters
2007, 86, 157–163,
doi:\changeurlcolorblack10.1134/S0021364007150027.
Amendola et al. (2007)
Amendola, L.; Gannouji, R.; Polarski, D.; Tsujikawa, S.
Conditions for the cosmological viability of f(R) dark energy
models.
Phys. Rev. D 2007, 75, 083504,
doi:\changeurlcolorblack10.1103/PhysRevD.75.083504.
Tsujikawa (2008)
Tsujikawa, S.
Observational signatures of f(R) dark energy models that satisfy
cosmological and local gravity constraints.
Phys. Rev. D 2008, 77, 023507,
doi:\changeurlcolorblack10.1103/PhysRevD.77.023507.
Martinelli et al. (2009)
Martinelli, M.; Melchiorri, A.; Amendola, L.
Cosmological constraints on the Hu-Sawicki modified gravity
scenario.
Phys. Rev. D 2009, 79, 123516,
doi:\changeurlcolorblack10.1103/PhysRevD.79.123516.
Burrage and Sakstein (2018)
Burrage, C.; Sakstein, J.
Tests of chameleon gravity.
Living Rev. Relativ. 2018, 21, 1,
doi:\changeurlcolorblack10.1007/s41114-018-0011-x.
Liu et al. (2018)
Liu, T.; Zhang, X.; Zhao, W.
Constraining f(R) gravity in solar system, cosmology and binary
pulsar systems.
Phys. Lett. B 2018, 777, 286–293,
doi:\changeurlcolorblack10.1016/j.physletb.2017.12.051.
Linde (1983)
Linde, A.D.
Chaotic inflation.
Phys. Lett. B 1983, 129, 177–181.
doi:\changeurlcolorblack10.1016/0370-2693(83)90837-7.
Riotto (2003)
Riotto, A.
Inflation and the Theory of Cosmological Perturbations.
ICTP Lect. Notes Ser. 2003, 14, 317–413. |
Polaritonic normal modes in Transition State Theory
J. A. Campos-Gonzalez-Angulo
Department of Chemistry and Biochemistry. University of California San Diego. La Jolla, California 92093, USA
J. Yuen-Zhou
joelyuen@ucsd.edu
http://yuenzhougroup.ucsd.edu
Department of Chemistry and Biochemistry. University of California San Diego. La Jolla, California 92093, USA
Abstract
A series of experiments demonstrate that strong light-matter coupling between vibrational excitations in isotropic solutions of molecules and resonant infrared optical microcavity modes leads to modified thermally-activated kinetics. However, Feist and coworkers [Phys. Rev. X., 9, 021057(2019)] have recently demonstrated that, within transition state theory, effects of strong light-matter coupling with reactive modes are electrostatic, and essentially independent of light-matter resonance or even of the formation of vibrational polaritons. To analyze this puzzling theoretical result in further detail, we revisit it under a new light, invoking a normal mode analysis of the transition state and reactant configurations for an ensemble of an arbitrary number of molecules in a cavity, obtaining simple analytical expressions that produce similar conclusions as Feist. While these effects become relevant in optical microcavities if the molecular dipoles are anisotropically aligned, or in cavities with extreme confinement of the photon modes, they become negligible for isotropic solutions in microcavities. It is concluded that further studies are necessary to track the origin of the experimentally observed kinetics.
††preprint: AIP/123-QED
I Introduction
Multiple experimental results show that reactions taking place inside of optical microcavities proceed with different kinetics than outside of them.Thomas et al. (2016); Hiura, Shalabney, and George (2018); Lather et al. (2019); Thomas et al. (2019); Vergauwe et al. (2019); Hirai et al. (2020) Rate modification seems to require that the confined electromagnetic mode couples to one of the varieties of molecular vibrational modes present in the reactive medium.Thomas et al. (2019) For reactions in solution, where molecules are isotropically distributed, this coupling is maximized under resonant conditions, i.e., when the cavity is tuned to a vibrational frequency in the molecules. Also, the effect on the kinetics has been observed to increase as the collective coupling intensifies, as a consequence of the large number of molecules present in a sample.Thomas et al. (2016)
These observations are reminiscent of the description of light-matter coupling in terms of hybrid states known as polaritons,Ebbesen (2016); Ribeiro et al. (2018); Feist, Galego, and Garcia-Vidal (2018); Flick, Rivera, and Narang (2018); Ruggenthaler et al. (2018); Herrera and Owrutsky (2020) which successfully explains the optical properties of these systems.Shalabney et al. (2015); Casey and Sparks (2016); Dunkelberger et al. (2016); Xiang et al. (2018); Erwin, Smotzer, and Coe (2019) Recently, it has been suggested that a class of nonadiabatic charge transfer reactions would experience a catalytic effect from resonant collective coupling between high-frequency modes and infrared cavity modes; the mechanism relies on the formation of vibrational polaritons which feature reduced activation energies compared to the bare molecules.Campos-Gonzalez-Angulo, Ribeiro, and Yuen-Zhou (2019); Phuc, Ishizaki, and Trung (2019)
However, a large class of reactions fall in the adiabatic regime, where the potential energy surfaces of the electronic ground and excited states are well-separated. These reactions should be accurately described by a transition state theory (TST)Truhlar, Garrett, and Klippenstein (1996); Nitzan (2006); Vaillant et al. (2019) that accounts for vibrational strong coupling (VSC). Feist and coworkers have in fact developed a theoretical framework with the essential ingredients to capture the action of a confined electromagnetic field on chemical processes such as nucleophyllic substitution.Galego et al. (2019); Climent et al. (2019) Within this framework, they find that the presence of a cavity mode modifies the reactive potential energy surface, thus predicting conditions for increase and decrease of reaction rates. However, according to their results, resonance is not essential for this modification to take place. Furthermore, the effect depends on the intensity of the single-molecule coupling, and cooperativity can only occur under conditions such as the anisotropic alignment of the permanent dipoles, an unlikely condition for the aforementioned reported experiments.Li, Nitzan, and Subotnik (2020) Remarkably, Feist’s formalism excludes the language of polaritons. In fact, they concede that polaritonic degrees of freedom appear inconsequentially in the form of normal modes near the equilibrium configurations of the system, and that the effects are of the (Casimir-Polder) electrostatic type.Galego et al. (2019) In the present work, we restate their formalism bringing the polaritonic modes into the limelight; we take advantage of the polaritonic framework to expand the formalism and obtain simple and physically intuitive analytical TST expressions that describe the modified collisional prefactors and activation energies in terms of light and matter parameters. Our results are in line with the predictions of Galego et al. (2019); Climent et al. (2019), highlighting that further work must be carried out to understand the difference between experiment and theory in the context of thermally-activated reactions under VSC.
II Theory
According to TST, the rate constant at temperature $T$ is defined asWigner (1938); Hänggi, Talkner, and Borkovec (1990); Pollak and Talkner (2005); Arnaut and Burrows (2006); Henriksen and Hansen (2018)
$$k_{\textrm{TST}}=\frac{k_{B}T}{2\pi\hbar}\frac{Z_{\ddagger}}{Z_{\textrm{eq}}}%
\textrm{e}^{-\frac{E_{a}}{k_{B}T}},$$
(1)
where $k_{B}$ and $\hbar$ are the Boltzmann and reduced Planck constants, respectively. $Z_{\ddagger}$ is the partition function of the transition state (TS) without the contribution of the reactive mode, and $Z_{\textrm{eq}}$ is the total partition function of the reactant state. $E_{a}=V_{\ddagger}+\frac{1}{2}\sum_{i}\hbar\omega_{i,{\ddagger}}-V_{\textrm{eq%
}}-\frac{1}{2}\sum_{j}\hbar\omega_{j,\textrm{eq}}$ is the activation energy, where the frequency $\omega_{i,r}$ corresponds to the square root of the $i$-th positive eigenvalue of the Hessian of the potential energy surface evaluated at the state $r$.
We will determine how the rate constant changes for a thermally-activated process in which the reactant is a heteronuclear diatomic molecule, when it takes place inside an optical microcavity. While the following analysis can be straightforwardly generalized for a multimode system, we will treat only the simplest case for the sake of conceptual clarity. Such a system with $N$ identical reactant molecules can be described by the HamiltonianFlick et al. (2017a); Galego et al. (2019)
$${\hat{H}}={\hat{H}}_{\textrm{EM}}+\sum_{i=1}^{N}\left({\hat{H}}_{\textrm{mol}}%
^{(i)}+\hat{V}_{\textrm{int}}^{(i)}\right),$$
(2)
where ${\hat{H}}_{\textrm{EM}}=\hbar\omega_{0}\left(\hat{a}_{0}^{\dagger}\hat{a}_{0}+%
\tfrac{1}{2}\right)$ characterizes a confined electromagnetic field of frequency $\omega_{0}$, and creation and annihilation operators $\hat{a}_{0}^{\dagger}$ and $\hat{a}_{0}$, respectively. ${\hat{H}}_{\textrm{mol}}^{(i)}=\hat{T}_{\textrm{nuc}}^{(i)}+\hat{V}_{\textrm{%
nuc}}^{(i)}+\hat{T}_{\textrm{elec}}^{(i)}+\hat{V}_{\textrm{elec}}^{(i)}+\hat{V%
}_{\textrm{nuc-elec}}^{(i)}$ is the Hamiltonian of the $i$-th molecule containing the kinetic, $\hat{T}$, and potential, $\hat{V}$, energies of the nuclear and electronic degrees of freedom, as well as their Coulomb interaction. The coupling between light and matter is given by $\hat{V}_{\textrm{int}}^{(i)}=g\omega_{0}\hat{q}_{0}\bm{\epsilon}\cdot\bm{\hat{%
\mu}}_{i}$, where $\hat{q}_{0}=\sqrt{\frac{\hbar}{2\omega_{0}}}\left(\hat{a}_{0}^{\dagger}+\hat{a%
}_{0}\right)$, and $g=-(\mathcal{V}\varepsilon_{0})^{-1/2}$ is the coupling constant, with $\mathcal{V}$ the mode volume and $\varepsilon_{0}$ the vacuum permittivity; $\bm{\epsilon}$ is the polarization vector of the cavity field, and $\bm{\hat{\mu}}_{i}$ is the molecular vibrational electric dipole moment.
In the (cavity) Born-Oppenheimer approximation,Flick et al. (2017b); Ruggenthaler et al. (2018) the ground state potential energy for the electronic Schrödinger equation with Hamiltonian $\hat{H}_{\textrm{elec}}=\hat{H}-\sum_{i=1}^{N}\hat{T}_{\textrm{nuc}}$, can be parameterized in terms of the nuclear coordinates, $\mathbf{R}$, and the photon coordinate $q_{0}$, which is an eigenvalue of the operator $\hat{q}_{0}$. Thus, the potential energy surface governing the nuclear degrees of freedom (Fig. 1) becomes
$$V(\mathbf{R},q_{0})=\sum_{i=1}^{N}V_{\textrm{nuc}}(\mathbf{R}_{i})+\frac{%
\omega_{0}^{2}}{2}q_{0}^{2}+\omega_{0}gq_{0}\bm{\epsilon}\cdot\sum_{i=1}^{N}%
\bm{\mu}(\mathbf{R}_{i}).$$
(3)
In writing Eqs. (2) and (3) we have neglected the diamagnetic term arising from the Power-Zienau-Woolley transformation.Cohen-Tannoudji, Dupont-Roc, and Grynberg (1989) Its relevance for problems in the current context is explored in detail in Refs. Schäfer et al., 2019; Li, Nitzan, and Subotnik, 2020. Nevertheless, since even in the ultrastrong regime, light-matter coupling per molecule is much smaller than the vibrational transition energies,Martínez-Martínez et al. (2018) the inclusion of such term should only account for slight modifications to the formalism that leave the findings unchanged.
In the neighborhood of the equilibrium configuration of the reactants, $\mathbf{R}_{\textrm{eq}}$, the potential is reasonably well described by a second order expansion while the dipole moment can be approximated to first order:
$$\begin{split}\displaystyle V(\mathbf{R}\approx\mathbf{R}_{\textrm{eq}},q_{0})=%
&\displaystyle\sum_{i=1}^{N}V_{\textrm{nuc}}(\mathbf{R}_{i,\textrm{eq}})+\frac%
{\omega_{\textrm{eq}}^{2}}{2}\sum_{i=1}^{N}q_{i}^{2}+\frac{\omega_{0}^{2}}{2}q%
_{0}^{2}\\
&\displaystyle+\omega_{0}gq_{0}\sum_{i=1}^{N}\left(\mu_{i,\textrm{eq}}+\mu_{i,%
\textrm{eq}}^{\prime}q_{i}\right),\end{split}$$
(4)
where $q_{i}$ is the mass-reduced bond elongation with respect to the equilibrium length of the $i$-th molecule, $\omega_{\textrm{eq}}^{2}=\left.\frac{\partial^{2}V_{\textrm{nuc}}^{(i)}}{%
\partial q_{i}^{2}}\right\rvert_{0}$, $\mu_{i,\textrm{eq}}=\bm{\epsilon}\cdot\bm{\mu}(\mathbf{R}_{i,\textrm{eq}})$, and $\mu^{\prime}_{i,\textrm{eq}}=\bm{\epsilon}\cdot\left.\frac{\partial\bm{\mu}(%
\mathbf{R}_{i})}{\partial q_{i}}\right\rvert_{0}$.
We note that this expansion excludes the polarizability term present in the perturbative treatment by Galego et al. (2019); however, as we shall see, this omission does not affect the main conclusions.
Differentiation of Eq. (4) yields
$$\displaystyle\frac{\partial V}{\partial q_{0}}$$
$$\displaystyle=\omega_{0}^{2}q_{0}+\omega_{0}g\sum_{i=1}^{N}\left(\mu_{i,%
\textrm{eq}}+\mu_{i,\textrm{eq}}^{\prime}q_{i}\right)$$
(5a)
$$\displaystyle\frac{\partial V}{\partial q_{j}}$$
$$\displaystyle=\omega_{\textrm{eq}}^{2}q_{j}+\omega_{0}gq_{0}\mu_{j,\textrm{eq}%
}^{\prime}\quad 1\leq j\leq N;$$
(5b)
therefore, at the new minimum, $\mathbf{R}^{\textrm{VSC}}_{\textrm{eq}}$, close to $\mathbf{R}_{\textrm{eq}}$, the coordinates fulfill
$$\displaystyle\begin{pmatrix}\omega_{0}^{2}&\omega_{0}g\sqrt{N\left\langle\mu_{%
\textrm{eq}}^{\prime 2}\right\rangle_{N}}\\
\omega_{0}g\sqrt{N\left\langle\mu_{\textrm{eq}}^{\prime 2}\right\rangle_{N}}&%
\omega_{\textrm{eq}}^{2}\end{pmatrix}\begin{pmatrix}q_{0}\\
q_{\textrm{B}(N)}\end{pmatrix}\\
\displaystyle=-\omega_{0}gN\left\langle\mu_{\textrm{eq}}\right\rangle_{N}%
\begin{pmatrix}1\\
0\end{pmatrix},$$
(6)
where $\langle x\rangle_{N}=\frac{1}{N}\sum_{i=1}^{N}x_{i}$, and the bright molecular mode is given by $q_{\textrm{B}(N)}=\sqrt{\frac{N}{\left\langle\mu_{\textrm{eq}}^{\prime 2}%
\right\rangle_{N}}}\left\langle\mu^{\prime}_{\textrm{eq}}q\right\rangle_{N}$.
The coefficient matrix in Eq. (6) corresponds to the Hopfield-Bogoliubov form of the Dicke model in the normal phaseEmary and Brandes (2003); Bastarrachea-Magnani, Lerma-Hernández, and Hirsch (2014); therefore, its diagonalization gives rise to polariton modes, as shown in Fig. 1. To be specific, Eq. (6) can be rewritten as
$$\begin{pmatrix}\omega_{+(N)}^{2}&0\\
0&\omega_{-(N)}^{2}\end{pmatrix}\begin{pmatrix}q_{+(N)}\\
q_{-(N)}\end{pmatrix}=-\omega_{0}gN\left\langle\mu_{\textrm{eq}}\right\rangle_%
{N}\begin{pmatrix}\cos\theta_{N}\\
\sin\theta_{N}\end{pmatrix},$$
(7)
where $\omega_{\pm(N)}^{2}=\frac{1}{2}\left[\omega_{0}^{2}+\omega_{\textrm{eq}}^{2}%
\pm\sqrt{4\omega_{0}^{2}g^{2}N\left\langle\mu_{\textrm{eq}}^{\prime 2}\right%
\rangle_{N}+\left(\omega_{0}^{2}-\omega_{\textrm{eq}}^{2}\right)^{2}}\right]$ is the frequency squared of the upper(lower) polaritonic mode,
$\begin{pmatrix}q_{+(N)}\\
q_{-(N)}\end{pmatrix}=\begin{pmatrix}\cos\theta_{N}&-\sin\theta_{N}\\
\sin\theta_{N}&\cos\theta_{N}\end{pmatrix}\begin{pmatrix}q_{0}\\
q_{\textrm{B}(N)}\end{pmatrix}$ are the polaritonic mode coordinates, and $\theta_{N}=-\frac{1}{2}\arctan\frac{2\omega_{0}g\sqrt{N\left\langle\mu_{%
\textrm{eq}}^{\prime 2}\right\rangle_{N}}}{\omega_{0}^{2}-\omega_{\textrm{eq}}%
^{2}}$ is the mixing angle.
Equation (4) can be recast using this new set of coordinates in the form
$$\displaystyle V(\mathbf{R}\approx\mathbf{R}_{\textrm{eq}},q_{0})=\sum_{i=1}^{N%
}V_{\textrm{nuc}}(\mathbf{R}_{i,\textrm{eq}})+\frac{\omega_{\textrm{eq}}^{2}}{%
2}\sum_{k=1}^{N-1}q_{\textrm{D}(N)}^{(k)2}\\
\displaystyle+\frac{\omega_{+(N)}^{2}}{2}q_{+(N)}^{2}+\frac{\omega_{-(N)}^{2}}%
{2}q_{-(N)}^{2}\\
\displaystyle+\omega_{0}gN\left\langle\mu_{\textrm{eq}}\right\rangle_{N}\left(%
\cos\theta_{N}q_{+(N)}+\sin\theta_{N}q_{-(N)}\right),$$
(8)
where $q_{\textrm{D}(N)}^{(k)}=\sum_{i=1}^{N}c_{ki}q_{i}$ are the dark vibrational modes, with the coefficients $c_{ki}$ fulfilling $\sum_{i=1}^{N}\mu^{\prime*}_{i,\textrm{eq}}c_{ki}=0$ and $\sum_{i=1}^{N}c_{k^{\prime}i}^{*}c_{ki}=\delta_{k^{\prime}k}$.
Evaluating the potential in Eq. (8) at $\mathbf{R}^{\textrm{VSC}}_{\textrm{eq}}$ yields
$$V_{\textrm{eq}}^{\textrm{VSC}}=\sum_{i=1}^{N}V_{\textrm{nuc}}(\mathbf{R}_{i,%
\textrm{eq}})-\left(\frac{\omega_{0}\omega_{\textrm{eq}}}{\omega_{+(N)}\omega_%
{-(N)}}gN\langle\mu_{\textrm{eq}}\rangle_{N}\right)^{2}.$$
(9)
We note that the modification to the potential is proportional to the ratio of the determinants of the Hessian without and with light-matter coupling, which acts as a measure of the redefinition of the normal modes. Additionally, the presence of the permanent dipole reveals the electrostatic nature of this effect.
Without loss of generality, let us assume that the molecule with label $N$ undergoes a reaction. The potential energy surface in the neighborhood of the TS configuration, $\mathbf{R}_{\ddagger}$, is
$$\displaystyle V(\mathbf{R}\approx\mathbf{R}_{\ddagger},q_{0})=\sum_{i=2}^{N}V_%
{\textrm{nuc}}(\mathbf{R}_{i,\textrm{eq}})+V_{\textrm{nuc}}(\mathbf{R}_{N,{%
\ddagger}})\\
\displaystyle+\frac{\omega_{\textrm{eq}}^{2}}{2}\sum_{i=1}^{N-1}q_{i}^{2}+%
\frac{\omega_{0}^{2}}{2}q_{0}^{2}+\frac{\omega_{\ddagger}^{2}}{2}q_{N}^{2}\\
\displaystyle+\omega_{0}gq_{0}\left[\sum_{i=1}^{N-1}\left(\mu_{i,\textrm{eq}}+%
\mu^{\prime}_{i,\textrm{eq}}q_{i}\right)+\mu_{\ddagger}+\mu^{\prime}_{\ddagger%
}q_{N}\right].$$
(10)
Here, $\omega_{\ddagger}^{2}=\left.\frac{\partial^{2}V_{\textrm{nuc}}^{(N)}}{\partial
q%
_{N}^{2}}\right\rvert_{q_{\ddagger}}<0$ is the squared frequency of the unstable mode, $\mu_{\ddagger}=\bm{\epsilon}\cdot\bm{\mu}(\mathbf{R}_{N,{\ddagger}})$, and $\mu^{\prime}_{\ddagger}=\bm{\epsilon}\cdot\left.\frac{\partial\bm{\mu}(\mathbf%
{R}_{N})}{\partial q_{N}}\right\rvert_{q_{\ddagger}}$.
Applying the previous treatment to the potential energy surface in the saddle point, $\mathbf{R}^{\textrm{VSC}}_{\ddagger}$, the coordinates fulfill
$$\begin{pmatrix}\omega_{0}^{2}&\omega_{0}g\sqrt{(N-1)\left\langle\mu_{\textrm{%
eq}}^{\prime 2}\right\rangle_{N-1}}&\omega_{0}g\mu^{\prime}_{\ddagger}\\
\omega_{0}g\sqrt{(N-1)\left\langle\mu_{\textrm{eq}}^{\prime 2}\right\rangle_{N%
-1}}&\omega_{\textrm{eq}}^{2}&0\\
\omega_{0}g\mu^{\prime}_{\ddagger}&0&\omega_{\ddagger}^{2}\end{pmatrix}\begin{%
pmatrix}q_{0}\\
q_{\textrm{B}(N-1)}\\
q_{N}\end{pmatrix}=-\omega_{0}g\left[(N-1)\left\langle\mu_{\textrm{eq}}\right%
\rangle_{N-1}+\mu_{\ddagger}\right]\begin{pmatrix}1\\
0\\
0\end{pmatrix}.$$
(11)
For typical values of the transition dipole moments, the off-diagonal terms that depend on $N$ remain significant since the number of molecules per cavity mode is estimated between $10^{6}$ and $10^{10}$.del Pino, Feist, and Garcia-Vidal (2015); Daskalakis, Maier, and Kéna-Cohen (2017) The term $g\omega_{0}\mu^{\prime}_{\ddagger}$ is several orders of magnitude smaller, and we can neglect it to recover a polaritonic picture where
$$\displaystyle\begin{pmatrix}\omega_{+(N-1)}^{2}&0&0\\
0&\omega_{-(N-1)}^{2}&0\\
0&0&\omega_{\ddagger}^{2}\end{pmatrix}\begin{pmatrix}q_{+(N-1)}\\
q_{-(N-1)}\\
q_{N}\end{pmatrix}\\
\displaystyle\approx-\omega_{0}g\left[(N-1)\left\langle\mu_{\textrm{eq}}\right%
\rangle_{N-1}+\mu_{\ddagger}\right]\begin{pmatrix}\cos\theta_{N-1}\\
\sin\theta_{N-1}\\
0\end{pmatrix}$$
(12)
at $\mathbf{R}^{\textrm{VSC}}_{\ddagger}$. Thus, the potential at the saddlepoint becomes
$$\begin{split}\displaystyle V_{\ddagger}^{\textrm{VSC}}=&\displaystyle\sum_{i=1%
}^{N-1}V_{\textrm{nuc}}(\mathbf{R}_{i,\textrm{eq}})+V_{\textrm{nuc}}(\mathbf{R%
}_{N,{\ddagger}})\\
&\displaystyle-\left(\frac{\omega_{0}\omega_{\textrm{eq}}}{\omega_{+(N-1)}%
\omega_{-(N-1)}}g\left[(N-1)\left\langle\mu_{\textrm{eq}}\right\rangle_{N-1}+%
\mu_{\ddagger}\right]\right)^{2}.\end{split}$$
(13)
From Eqs. (4), (10) and (12), it follows that the step to the TS can be written as
$$\displaystyle\textrm{UP}_{N}+\textrm{LP}_{N}+\sum_{k=1}^{N-1}D_{N}^{(k)}%
\longrightarrow\\
\displaystyle\textrm{UP}_{N-1}+\textrm{LP}_{N-1}+\sum_{k^{\prime}=1}^{N-2}D_{N%
-1}^{(k^{\prime})}+R_{N}^{\ddagger}$$
(14)
where $R_{N}^{\ddagger}$ represents the reactive molecule in the TS. Therefore, the rate constant should include the partition functions of the whole ensemble of molecules coupled to light; however, as we will see, since only one molecule undergoes the reaction, the ratio of partition functions simplifies to an intelligible expression in terms of the single molecule $k_{\textrm{TST}}$.
Outside of the cavity the rate constant takes the form
$$\begin{split}\displaystyle k_{\textrm{TST}}=&\displaystyle\frac{k_{B}T}{\pi%
\hbar}\frac{Q_{\ddagger}}{Q_{\textrm{eq}}}\sinh\left(\frac{\hbar\omega_{%
\textrm{eq}}}{2k_{B}T}\right)\\
&\displaystyle\times\exp\left(-\frac{V_{\textrm{nuc}}\left(\mathbf{R}_{N,{%
\ddagger}}\right)-V_{\textrm{nuc}}\left(\mathbf{R}_{N,\textrm{eq}}\right)}{k_{%
B}T}\right),\end{split}$$
(15)
where the ratio $Q_{\ddagger}/Q_{\textrm{eq}}$ captures all the information from the translational and rotational degrees of freedom (for a 1D system comprised of the reactive mode only, $Q_{\ddagger}=Q_{\textrm{eq}}$). To characterize the effect of the cavity mode on the kinetics, we define
$$k_{\textrm{TST}}^{\textrm{VSC}}=\kappa_{N}k_{\textrm{TST}},$$
(16)
where the ratio of rate constants is given by
$$\kappa_{N}=A_{\textrm{VSC}}(T)\exp\left(-\frac{\Delta V_{\textrm{VSC}}+\Delta E%
_{0}^{\textrm{VSC}}}{k_{B}T}\right),$$
(17a)
with prefactor
$$A_{\textrm{VSC}}(T)=\frac{\sinh\left(\hbar\omega_{+(N)}/2k_{B}T\right)\sinh%
\left(\hbar\omega_{-(N)}/2k_{B}T\right)}{\sinh\left(\hbar\omega_{+(N-1)}/2k_{B%
}T\right)\sinh\left(\hbar\omega_{-(N-1)}/2k_{B}T\right)},$$
(17b)
cavity-induced potential energy difference
$$\displaystyle\Delta V_{\textrm{VSC}}=\omega_{0}^{2}\omega_{\textrm{eq}}^{2}g^{%
2}\\
\displaystyle\times\left[\left(\frac{N\langle\mu_{\textrm{eq}}\rangle_{N}}{%
\omega_{+(N)}\omega_{-(N)}}\right)^{2}-\left(\frac{(N-1)\left\langle\mu_{%
\textrm{eq}}\right\rangle_{N-1}+\mu_{\ddagger}}{\omega_{+(N-1)}\omega_{-(N-1)}%
}\right)^{2}\right],$$
(17c)
and zero-point-energy difference
$$\Delta E_{0}^{\textrm{VSC}}=\frac{\hbar\omega_{+(N-1)}+\hbar\omega_{-(N-1)}-%
\hbar\omega_{+(N)}-\hbar\omega_{-(N)}}{2}.$$
(17d)
As stated before, $N\gg 1$. In this limit,
$A_{\textrm{VSC}}(T)\approx 1$, $\Delta E_{0}^{\textrm{VSC}}\approx 0$, and
the ratio of rate constants becomes
$$\kappa_{N}\approx\exp\left[\frac{\left(\omega_{\textrm{eq}}g\mu_{\ddagger}%
\right)^{2}}{\left(\omega_{\textrm{eq}}^{2}-g^{2}N\left\langle\mu_{\textrm{eq}%
}^{\prime 2}\right\rangle\right)k_{B}T}\right],$$
(18)
where we have considered that, for typical reactions in liquid solution, the molecular dipoles are isotropically distributed; therefore, $\langle\mu_{\textrm{eq}}\rangle_{N}=0$. Regarding collective effects, in Fig. 2, we show the ratio of rate constants as a function of the collective coupling and the permanent dipole moment of the TS. We can see that the variation of $\kappa_{N}$ throughout the span of the weak and strong light-matter coupling regimes is negligible. Furthermore, even over a huge range of possible values of $\mu_{\ddagger}$, the ratio of rate constants remains too close to 1 to imply any observable change in the reaction rate. In contrast, note that in a sample with perfectly aligned dipoles, $\left\langle\mu_{\textrm{eq}}\right\rangle_{N}\neq 0$, leading to substantial collective $O(N)$ contributions to $\Delta V_{\textrm{VSC}}$ [see Eq. (17c)]. Furthermore, regardless of dipole alignment, it can be shown that $\Delta V_{\textrm{VSC}}$ is independent of $\omega_{0}$, and is therefore unable to describe a resonant effect.
From the previous analysis we reach the same conclusions of Galego et al. (2019): effects of resonance between the cavity and the vibrational modes cannot be captured in a description at the level of TST, and the isotropic distribution of the permanent dipole moments negates the possibility of cooperative light-matter coupling effects. These results contrast with the situation of thermally-activated nonadiabatic charge transfer reactions, where the role of collective light-matter resonance in isotropic media is more evident. While we agree that the role of the polaritonic picture in our present analysis is rather shallow, it undoubtedly simplifies and clarifies the theoretical analysis. In conclusion, our results restate that a TST that takes into account strong coupling of the reactive mode to a resonant optical cavity mode is still insufficient to explain the experimental results involving thermally-activated adiabatic reactions in Refs. Thomas et al., 2016; Hiura, Shalabney, and George, 2018; Lather et al., 2019; Thomas et al., 2019; Vergauwe et al., 2019; Hirai et al., 2020.
Acknowledgements.
The authors thank Raphael F. Ribeiro, Luis A. Martínez-Martínez and Matthew Du for their insightful comments and discussions. This work was partially supported by the Defense Advanced Research Projects Agency under Award No. D19AC00011. JACGA also acknowledges support from UC-MEXUS/CONACYT through scholarship ref. 235273/472318.
Data Availability Statement
Data sharing is not applicable to this article as no new data were created or analyzed in this study.
Appendix A Single-molecule case
When there is a single molecule per cavity mode, the only surviving coupling in Eq. (11) is that between the TS and the photon. In this case, the saddlepoint condition can be recast in terms of the eigenmodes as
$$\begin{pmatrix}\omega_{-{\ddagger}}^{2}&0\\
0&\omega_{+{\ddagger}}^{2}\end{pmatrix}\begin{pmatrix}q_{+{\ddagger}}\\
q_{-{\ddagger}}\end{pmatrix}=-\omega_{0}g\mu_{\ddagger}\begin{pmatrix}\cos%
\theta_{\ddagger}\\
\sin\theta_{\ddagger}\end{pmatrix},$$
(19)
where $\omega_{-{\ddagger}}^{2}<0<\omega_{+{\ddagger}}^{2}$.
The potential energy evaluated at $\mathbf{R}^{\textrm{VSC}}_{\ddagger}$ is
$$V_{\ddagger}^{\textrm{VSC}}=V_{\textrm{nuc}}(\mathbf{R}_{{\ddagger}})-\left(%
\frac{\omega_{0}\omega_{\ddagger}}{\omega_{+{\ddagger}}\omega_{-{\ddagger}}}g%
\mu_{\ddagger}\right)^{2},$$
(20)
which produces
$$\displaystyle A_{\textrm{VSC}}$$
$$\displaystyle=\frac{\sinh\left(\hbar\omega_{+}/2k_{B}T\right)\sinh\left(\hbar%
\omega_{-}/2k_{B}T\right)}{\sinh\left(\hbar\omega_{+{\ddagger}}/2k_{B}T\right)%
\sinh\left(\hbar\omega_{\textrm{eq}}/2k_{B}T\right)},$$
(21a)
$$\displaystyle\Delta V_{\textrm{VSC}}$$
$$\displaystyle=g^{2}\omega_{0}^{2}\left[\left(\frac{\omega_{\textrm{eq}}\mu_{%
\textrm{eq}}}{\omega_{+}\omega_{-}}\right)^{2}-\left(\frac{\omega_{\ddagger}%
\mu_{\ddagger}}{\omega_{+{\ddagger}}\omega_{-{\ddagger}}}\right)^{2}\right],$$
(21b)
$$\displaystyle\Delta E_{0}^{\textrm{VSC}}$$
$$\displaystyle=\frac{\hbar\omega_{+{\ddagger}}-\hbar\omega_{+}-\hbar\omega_{-}+%
\hbar\omega_{\textrm{eq}}}{2}.$$
(21c)
It is worth noting that, despite $A_{\textrm{VSC}}(T)$ and $\Delta E_{0}^{\textrm{VSC}}$ deviating from 1 and 0, respectively, in the single-molecule limit, the effect is still off-resonant, thus reinforcing the findings in Galego et al. (2019). In any case, the mode volumes and transition dipole moments required to modify a reaction rate are unrealistic unless we consider nano- and picocavities.
References
Thomas et al. (2016)
A. Thomas, J. George,
A. Shalabney, M. Dryzhakov, S. J. Varma, J. Moran, T. Chervy, X. Zhong, E. Devaux, C. Genet, J. A. Hutchison, and T. W. Ebbesen, Angew Chem Int Ed 55, 11462 (2016).
Hiura, Shalabney, and George (2018)
H. Hiura, A. Shalabney, and J. George, “Cavity
catalysis –accelerating reactions under vibrational strong coupling–,”
(2018), preprint, posted April 11,
2018.
Lather et al. (2019)
J. Lather, P. Bhatt,
A. Thomas, T. W. Ebbesen, and J. George, Angew
Chem Int Ed 58, 10635
(2019).
Thomas et al. (2019)
A. Thomas, L. Lethuillier-Karl, K. Nagarajan, R. M. A. Vergauwe, J. George,
T. Chervy, A. Shalabney, E. Devaux, C. Genet, J. Moran, and T. W. Ebbesen, Science 363, 615 (2019).
Vergauwe et al. (2019)
R. M. A. Vergauwe, A. Thomas, K. Nagarajan,
A. Shalabney, J. George, T. Chervy, M. Seidel, E. Devaux, V. Torbeev, and T. W. Ebbesen, Angew Chem Int Ed 58, 15324 (2019).
Hirai et al. (2020)
K. Hirai, R. Takeda,
J. A. Hutchison, and H. Uji-i, Angew. Chem. Int. Ed. (2020), 10.1002/anie.201915632.
Ebbesen (2016)
T. W. Ebbesen, Acc Chem Res 49, 2403 (2016).
Ribeiro et al. (2018)
R. F. Ribeiro, L. A. Martínez-Martínez, M. Du, J. Campos-Gonzalez-Angulo, and J. Yuen-Zhou, Chem Sci 9, 6325 (2018).
Feist, Galego, and Garcia-Vidal (2018)
J. Feist, J. Galego, and F. J. Garcia-Vidal, ACS Photonics 5, 205 (2018).
Flick, Rivera, and Narang (2018)
J. Flick, N. Rivera, and P. Narang, Nanophotonics 7, 1479 (2018).
Ruggenthaler et al. (2018)
M. Ruggenthaler, N. Tancogne-Dejean, J. Flick, H. Appel, and A. Rubio, Nat. Rev. Chem. 2, 0118 (2018).
Herrera and Owrutsky (2020)
F. Herrera and J. Owrutsky, J. Chem. Phys. 152, 100902 (2020).
Shalabney et al. (2015)
A. Shalabney, J. George,
J. Hutchison, G. Pupillo, C. Genet, and T. W. Ebbesen, Nat Commun 6, 5981 (2015).
Casey and Sparks (2016)
S. R. Casey and J. R. Sparks, J. Phys. Chem. C 120, 28138 (2016).
Dunkelberger et al. (2016)
A. D. Dunkelberger, B. T. Spann, K. P. Fears,
B. S. Simpkins, and J. C. Owrutsky, Nat Commun 7, 13504
(2016).
Xiang et al. (2018)
B. Xiang, R. F. Ribeiro,
A. D. Dunkelberger,
J. Wang, Y. Li, B. S. Simpkins, J. C. Owrutsky, J. Yuen-Zhou, and W. Xiong, Proc.
Natl. Acad. Sci. U.S.A. 115, 4845 (2018).
Erwin, Smotzer, and Coe (2019)
J. Erwin, M. Smotzer, and J. V. Coe, J Phys Chem B 123, 1302 (2019).
Campos-Gonzalez-Angulo, Ribeiro, and Yuen-Zhou (2019)
J. A. Campos-Gonzalez-Angulo, R. F. Ribeiro, and J. Yuen-Zhou, Nat Commun 10, 4685 (2019).
Phuc, Ishizaki, and Trung (2019)
N. T. Phuc, A. Ishizaki, and P. Q. Trung, “Controlling the
electron-transfer-like reaction rate through molecular-vibration polaritons
in the ultrastrong coupling regime,” (2019), preprint, arXiv:1912.02950 [physics.chem-ph] .
Truhlar, Garrett, and Klippenstein (1996)
D. G. Truhlar, B. C. Garrett, and S. J. Klippenstein, J. Phys. Chem. 100, 12771 (1996).
Nitzan (2006)
A. Nitzan, Chemical Dynamics in
Condensed Phases: Relaxation, Transfer and Reactions in Condensed Molecular
Systems, Oxford Graduate Texts (OUP Oxford, 2006).
Vaillant et al. (2019)
C. L. Vaillant, M. J. Thapa,
J. Vaníček, and J. O. Richardson, J. Chem. Phys. 151, 144111 (2019).
Galego et al. (2019)
J. Galego, C. Climent,
F. J. Garcia-Vidal, and J. Feist, Phys Rev X 9, 021057 (2019).
Climent et al. (2019)
C. Climent, J. Galego,
F. J. Garcia-Vidal, and J. Feist, Angew Chem Int Ed 58, 8698 (2019).
Li, Nitzan, and Subotnik (2020)
T. E. Li, A. Nitzan, and J. E. Subotnik, “On the origin of ground-state
vacuum-field catalysis: Equilibrium consideration,” (2020), preprint, arXiv:2002.09977 [physics.chem-ph]
.
Wigner (1938)
E. Wigner, Trans. Faraday Soc. 34, 29 (1938).
Hänggi, Talkner, and Borkovec (1990)
P. Hänggi, P. Talkner, and M. Borkovec, Rev. Mod. Phys. 62, 251 (1990).
Pollak and Talkner (2005)
E. Pollak and P. Talkner, Chaos: An Interdisciplinary Journal of Nonlinear
Science 15, 026116
(2005).
Arnaut and Burrows (2006)
L. Arnaut and H. Burrows, Chemical Kinetics: From Molecular Structure to Chemical
Reactivity (Elsevier Science, 2006).
Henriksen and Hansen (2018)
N. Henriksen and F. Hansen, Theories of Molecular Reaction Dynamics: The Microscopic
Foundation of Chemical Kinetics, Oxford Graduate Texts (OUP Oxford, 2018).
Flick et al. (2017a)
J. Flick, M. Ruggenthaler,
H. Appel, and A. Rubio, Proc.
Natl. Acad. Sci. U.S.A. 114, 3026 (2017a).
Flick et al. (2017b)
J. Flick, H. Appel,
M. Ruggenthaler, and A. Rubio, J Chem Theory Comput 13, 1616 (2017b).
Cohen-Tannoudji, Dupont-Roc, and Grynberg (1989)
C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Photons and Atoms: Introduction to Quantum
Electrodynamics (Wiley, 1989).
Schäfer et al. (2019)
C. Schäfer, M. Ruggenthaler, V. Rokaj,
and A. Rubio, “Relevance of the quadratic
diamagnetic and self-polarization terms in cavity quantum electrodynamics,” (2019), preprint, arXiv:1911.08427 [quant-ph] .
Martínez-Martínez et al. (2018)
L. A. Martínez-Martínez, R. F. Ribeiro, J. Campos-González-Angulo, and J. Yuen-Zhou, ACS
Photonics 5, 167
(2018).
Emary and Brandes (2003)
C. Emary and T. Brandes, Phys. Rev. E 67, 066203 (2003).
Bastarrachea-Magnani, Lerma-Hernández, and Hirsch (2014)
M. A. Bastarrachea-Magnani, S. Lerma-Hernández, and J. G. Hirsch, Phys. Rev. A 89, 032102 (2014).
del Pino, Feist, and Garcia-Vidal (2015)
J. del
Pino, J. Feist, and F. J. Garcia-Vidal, New J Phys 17, 053040 (2015).
Daskalakis, Maier, and Kéna-Cohen (2017)
K. S. Daskalakis, S. A. Maier, and S. Kéna-Cohen, “Polariton condensation in
organic semiconductors,” in Quantum Plasmonics, edited by S. I. Bozhevolnyi, L. Martin-Moreno, and F. Garcia-Vidal (Springer International
Publishing, Cham, 2017) pp. 151–163. |
Field theoretical approach to spin torques: Slonczewski torques
Junji Fujimoto
Department of Electrical Engineering, Electronics, and Applied Physics, Saitama University, Saitama, 338-8570, Japan
fujimoto.junji@gmail.com
Abstract
The quantum field theoretical approach with the Kubo formula has successfully captured spin torques, such as spin-transfer torques and spin-orbit torques, for continuum systems.
We examine the field theoretical approach to current-induced spin-transfer torques in a magnetic junction system.
We first give a brief overview of the field theoretical approach to spin torques.
Then, we consider a five-layers system consisting of three nonmagnetic metal layers separated by two ferromagnetic metal layers and apply an electric field perpendicular to the layers.
We demonstrate that the Slonczewski-type spin-transfer torque, or shortly the Slonczewski torque, on the magnetizations in ferromagnetic layers is obtained by evaluating nonequilibrium electron spin density, based on the linear response theory with the Green function method.
The obtained coefficient of the Slonczewski torque has a quantum oscillation at absolute zero temperature, which has not been mentioned before.
A field-like torque accompanied by the Slonczewski torque is also evaluated.
I Introduction
The quantum field theory is a powerful tool for investigating various physical phenomena from particle physics to condensed matter physics, and provides us intuitive physical pictures of the phenomena.
In spintronics, the field theoretical approach has succeeded in capturing the spin torques [1, 2, 3], the spin-motive forces [4], the spin pumping [5, 6], and more, whereas many spintronic phenomena remain yet to be studied based on quantum field theory.
Among others, spin torques in various continuum systems have been studied based on the quantum field theory.
Here, we refer to the spin torques as torques of the nonequilibrium conduction electron spin on the magnetization through the $sd$-type exchange interaction between the conduction electron spin and the magnetization.
Current-induced spin-transfer torques are one of the spin torques and arise by applying an electric current in ferromagnetic metals with spatially-varying magnetic textures, such as magnetic domain walls.
The current-induced spin torques have been extensively investigated based on the motivation for magnetization manipulation by electrical means.
Two typical current-induced spin-transfer torques in continuum ferromagnetic systems are known; one is called (adiabatic) spin-transfer torque given as $\bm{\tau}_{\mathrm{stt}}=(\bm{j}_{\mathrm{s}}\cdot\bm{\nabla})\bm{m}$, and the other is called here the $\beta$ torque, which is given as $\bm{\tau}_{\beta}=\beta\bm{m}\times(\bm{j}_{\mathrm{s}}\cdot\bm{\nabla})\bm{m}$, where $\bm{j}_{\mathrm{s}}$ is the spin-polarized current density, $\bm{m}$ is the local unit magnetization vector, and $\beta$ is a coefficient determined by spin relaxation processes and/or nonadiabaticity in the systems.
In continuum systems, the magnetization may have spatial and temporal dependences; $\bm{m}=\bm{m}(\bm{r},t)$, which represents various magnetic textures.
On the other hand, the spin-transfer torque was firstly predicted in a magnetic junction system, based on quantum mechanics [7, 8, 9] and is currently called Slonczewski-type spin-transfer torque, or shortly Slonczewski torque.
Slonczewski considered a five-layer system consisting of three nonmagnetic metals separated by two ferromagnetic metal layers whose magnetizations are noncollinear and apply an electric current perpendicular to the layers, which results in magnetization dynamics due to the spin transfer torque.
Here, we express the magnetizations in two magnetic layers as $\bm{M}_{1}$ and $\bm{M}_{2}$, and then the Slonczewski torque on $\bm{M}_{i}$ with $i=1,2$ is given as $\bm{T}_{i}=c\bm{M}_{i}\times(\bm{m}_{1}\times\bm{m}_{2})$, where $c$ is a coefficient determined by the applied electric current and its spin polarization [7], and $\bm{m}_{i}=\bm{M}_{i}/|\bm{M}_{i}|$ ($i=1,2$).
The actual connection between the (adiabatic) spin-transfer torque $\bm{\tau}_{\mathrm{stt}}$ and the Slonczewski torque $\bm{T}_{i}$ has been unclear, whereas one can expect that both torques should be equivalent.
In the limiting case of the almost collinear magnetizations with the same lengths $|\bm{M}_{i}|=M$ ($i=1,2$), we can see that the Slonczewski torque is reduced to a $\bm{\tau}_{\mathrm{stt}}$-like torque, by setting $\bm{m}_{2}=\bm{m}_{1}+\delta\bm{m}$, where $\delta\bm{m}$ is small compared to $\bm{m}_{1}$ and perpendicular to $\bm{m}_{1}$, which yields $\bm{T}_{1}=c\bm{M}_{1}\times(\bm{m}_{1}\times\bm{m}_{2})=-cM\delta\bm{m}$.
Here, since applied electric current direction is perpendicular to the layers, say the $x$ direction, the (adiabatic) spin-transfer torque is described as $\bm{\tau}_{\mathrm{stt}}=j_{\mathrm{s}}\partial_{x}\bm{m}$ and discretized as $\partial_{x}\bm{m}=\{\bm{m}(x+\delta x)-\bm{m}(x)\}/\delta x$, which results in $\bm{T}_{1}\propto\bm{\tau}_{\mathrm{stt}}$ by setting $\bm{m}(x+\delta x)=\bm{m}_{2}$ and $\bm{m}(x)=\bm{m}_{1}$.
However, the above discussion is not valid in general noncollinear cases with different magnetization lengths, which are of our interest.
Further, the $\beta$ torque is known to be essential for magnetization dynamics in continuum systems, while the torque is sometimes disregarded in junction systems.
It may be worth evaluating the counterpart of the $\beta$ torque in the junction system.
Here, we call $\bm{T}_{i}$ the damping-like Slonczewski torque and the counter part of the $\beta$ torque in the junction systems, $\bm{T}^{\prime}_{1}=c^{\prime}\bm{M}_{1}\times\bm{m}_{2}$ and $\bm{T}^{\prime}_{2}=c^{\prime}\bm{M}_{2}\times\bm{m}_{1}$, the filed-like Slonczewski torque, where $c^{\prime}$ is a coefficient different from $c$.
In this paper, we examine the field theoretical approach to Slonczewski torques.
First, following Slonczewski, we consider the five-layers system consisting of three nonmagnetic metal layers separated by two ferromagnetic metal layers and apply an electric field perpendicular to the layers.
We demonstrate that the Slonczewski torques are obtained in general noncollinear cases by evaluating nonequilibrium electron spin density, based on the linear response theory with the Green function method.
The coefficient of the damping-like Slonczewski torque, $c$, has a spatial quantum oscillation like the Friedel oscillation at absolute zero temperature, which has not been mentioned before.
Moreover, the coefficient $c$ takes a different magnitude depending on the magnetizations in the magnetic layers.
We also discuss the filed-like Slonczewski torque in the junction system.
The coefficient of the field-like torque has the same magnitude with the opposite sign in the magnetic layers.
We find that the field-like torque has the saturation value for large thicknesses of the magnetic layers.
Although recent interest seems to move from the spin-transfer torques to the spin-orbit torques, which arise only in systems with strong spin-orbit coupling (SOC), we discuss current-induced spin-transfer torques in magnetic junction systems without SOC in this paper.
II Overview
We here give a brief overview of magnetization dynamics and the field theoretical approach to spin torques.
II.1 Magnetization dynamics and spin torque
Consider two systems of localized (classical) spins consisting of the magnetization and of conduction electrons, both of which are coupled through the $sd$-type exchange interaction 111Other interactions between the localized spin and the conduction electron spin, such as the RKKY interaction and the Dzyaloshinskii-Moriya interaction, possibly can arise novel types of spin torques, but we focus on the spin torques originating from the $sd$-type exchange interaction..
The total system is described by the following equation;
$$\displaystyle\mathcal{L}$$
$$\displaystyle=\mathcal{L}_{s}+\mathcal{L}_{e}-\mathcal{H}_{sd},$$
(1)
where $\mathcal{L}_{s}$ and $\mathcal{L}_{e}$ are the Lagrangians of the localized spin system and the conduction electron system, and $\mathcal{H}_{sd}$ is the $sd$-type exchange interaction between the localized spin and the conduction electron spin.
Here, we express the $i$-th localized spin degree of freedom as $\bm{S}_{i}$, whose length is fixed as $|\bm{S}_{i}|=S_{i}$, where $S_{i}$ is related to the saturated magnetization $M_{S,i}$ as $M_{S,i}=\gamma_{e}\hbar S_{i}/a^{3}$ with $\gamma_{e}$ being the gyromagnetic ratio and $a$ the length scale of coarse graining.
The dynamics of the localized spin $\bm{S}_{i}$ obeys classical mechanics and determined by the Euler-Lagrange equation with the constrain $\mathcal{C}_{i}=|\bm{S}_{i}|^{2}-S_{i}^{2}=0$ and the phenomenological relaxation function $\mathcal{W}_{s}$;
$$\displaystyle\frac{d}{dt}\left(\frac{\delta\mathcal{L}}{\delta\dot{\bm{S}}_{i}}\right)-\frac{\delta\mathcal{L}}{\delta\bm{S}_{i}}+\lambda\frac{\partial\mathcal{C}_{i}}{\delta\bm{S}_{i}}$$
$$\displaystyle=\frac{\delta\mathcal{W}_{s}}{\delta\dot{\bm{S}}_{i}},$$
(2)
where $\lambda$ is the Lagrange multiplier.
Here, thermal average on the conduction electron system is implied.
Substituting Eq. (1) into Eq. (2), we have
$$\displaystyle\frac{d}{dt}\left(\frac{\delta\mathcal{L}_{s}}{\delta\dot{\bm{S}}_{i}}\right)-\frac{\delta\mathcal{L}_{s}}{\delta\bm{S}_{i}}+\frac{\delta\mathcal{H}_{sd}}{\delta\bm{S}_{i}}+\lambda\bm{S}_{i}$$
$$\displaystyle=\frac{\delta\mathcal{W}_{s}}{\delta\dot{\bm{S}}_{i}},$$
(3)
and then taking $\bm{S}_{i}\times$ to remove the Lagrange multiplier, we obtain
$$\displaystyle\bm{S}_{i}\times\frac{d}{dt}\left(\frac{\delta\mathcal{L}_{s}}{\delta\dot{\bm{S}}_{i}}\right)-\bm{S}_{i}\times\frac{\delta\mathcal{L}_{s}}{\delta\bm{S}_{i}}+\hbar\bm{\tau}_{i}$$
$$\displaystyle=\bm{S}_{i}\times\frac{\delta\mathcal{W}_{s}}{\delta\bm{S}_{i}},$$
(4)
where $\bm{\tau}_{i}$ is the spin torque given by
$$\displaystyle\bm{\tau}_{i}$$
$$\displaystyle=\bm{S}_{i}\times\left\langle\frac{1}{\hbar}\frac{\delta\mathcal{H}_{sd}}{\delta\bm{S}_{i}}\right\rangle.$$
(5)
Here, $\langle\,\cdots\rangle$ means the thermal average on the conduction electorn system.
Note that $\mathcal{L}_{e}$ does not contain $\bm{S}_{i}$, hence $\delta\mathcal{L}_{e}/\delta\bm{S}_{i}=0$, and effects of the conduction electron on the localized spins are only through the $sd$-type exchange interaction.
We also note that the spin torque $\bm{\tau}_{i}$ does not cause the magnetization dynamics in global equilibrium (by definition of the equilibrium), which is understood as that the conduction electron spins align along the localized spins.
The spin torque $\bm{\tau}_{i}$ can induce the magnetization dynamics in nonequilibrium, such as (i) by applying an external electric field, (ii) by a temperature gradient, and (iii) by induced magnetization dynamics.
The spin torques of the case (i) is called the curren-induced spin torques [7, 8, 1, 11, 12, 13, 2, 14, 3, 15], the case (ii) is named the thermal spin torques [16, 17], and the case (iii) is related to the Gilbert damping torque due to the conduction electrons [18, 19, 12, 20].
For the system of ferromagnetically-interacting localized spins, the terms in Eq. (4) read
$$\displaystyle\bm{S}_{i}\times\frac{d}{dt}\left(\frac{\delta\mathcal{L}_{s}}{\delta\dot{\bm{S}}_{i}}\right)=-\hbar\dot{\bm{S}}_{i},$$
$$\displaystyle\quad\bm{S}_{i}\times\frac{\delta\mathcal{L}_{s}}{\delta\bm{S}_{i}}=-\hbar\bm{S}_{i}\times\gamma_{e}\bm{B}_{i},$$
(6)
where $\bm{B}_{i}$ is the effective magnetic field (measured by the unit of Tesla) originating from the localized spin Hamiltonian $\mathcal{H}_{s}$, given as
$$\displaystyle\gamma_{e}\bm{B}_{i}$$
$$\displaystyle=\frac{1}{\hbar}\frac{\partial\mathcal{H}_{s}}{\partial\bm{S}_{i}}.$$
(7)
The relaxation function in ferromagnets is given as $\mathcal{W}_{s}=(\hbar\alpha_{\mathrm{G}}/2S)\sum_{i}\dot{\bm{S}}_{i}^{2}$, which leads to the phenomenological Gilbert damping torque
$$\displaystyle\bm{S}_{i}\times\frac{\delta\mathcal{W}_{s}}{\delta\bm{S}_{i}}$$
$$\displaystyle=\frac{\hbar\alpha_{\mathrm{G}}}{S}\bm{S}_{i}\times\dot{\bm{S}}_{i},$$
(8)
and then Eq. (4) is found equivalent to Landau-Lifshitz-Gilbert equation with the spin torque,
$$\displaystyle\dot{\bm{S}}_{i}$$
$$\displaystyle=\gamma_{e}\bm{S}_{i}\times\bm{B}_{i}-\frac{\alpha_{\mathrm{G}}}{S}\bm{S}_{i}\times\dot{\bm{S}}_{i}+\bm{\tau}_{i}.$$
(9)
By solving this equation of motion, we see the magnetization dynamics.
We here emphasize that the spin torque is given as the effects of the conduction electrons on the magnetization dynamics, as mentioned above, and the above discussion is valid not only for continuum systems but for junction systems, the latter of which is of our interest.
For continuum systems, we usually take the continuum limit $\bm{S}_{i}\to\bm{S}(\bm{r})$.
Note also that we can discuss the order parameters dynamics for the antiferromagnetic and ferrimagnetic systems, but they are out of scope in this paper.
II.2 Spin torque in continuum and junction systems
Next, we discuss the expressions of the spin torque in continuum and junction systems.
The $sd$-type exchange interaction $\mathcal{H}_{sd}$ is given by
$$\displaystyle\mathcal{H}_{sd}$$
$$\displaystyle=-\sum_{i}\int\mathrm{d}\bm{r}\,J_{i}(\bm{r})\bm{S}_{i}\cdot\bm{s}(\bm{r})$$
(10)
in general, where $J_{i}(\bm{r})$ is the interaction strength between the $i$-th localized spin and electron spin, $\bm{s}(\bm{r})$ is the conduction electron spin density (devided by $\hbar/2$).
The definition with Eq. (5) immediately leads to the following expression of the spin torque,
$$\displaystyle\bm{\tau}_{i}$$
$$\displaystyle=-\frac{1}{\hbar}\bm{S}_{i}\times\left\langle\int\mathrm{d}\bm{r}\,J_{i}(\bm{r})\bm{s}(\bm{r})\right\rangle.$$
(11)
This equation indicates that the spin torque is an effective magnetic field due to the conduction electron spins coupling to the localized spin.
In the local interaction case; $J_{i}(\bm{r})=Ja^{3}\delta(\bm{r}-\bm{R}_{i})$, where $\bm{R}_{i}$ is the position of the $i$-th localized spin, and $\delta(\bm{r})$ is the Dirac $\delta$-function, the spin torque is simply given as
$$\displaystyle\bm{\tau}_{i}$$
$$\displaystyle=-\frac{Ja^{3}}{\hbar}\bm{S}_{i}\times\langle\bm{s}(\bm{R}_{i})\rangle.$$
(12)
In the continuum limit, the spin torque is shown as
$$\displaystyle\bm{\tau}_{i}$$
$$\displaystyle\to\bm{\tau}(\bm{r})=\frac{J_{sd}}{\hbar}\bm{m}(\bm{r})\times\langle\bm{s}(\bm{r})\rangle,$$
(13)
where $J_{sd}=SJa^{3}$ is the $sd$ exchange interaction constant, $\bm{m}(\bm{r})$ is the unit magnetization which is antiparallel to the localized spin $\bm{S}(\bm{r})$.
In this paper, we consider magnetic junction systems, where the magnetization of the $i$-th magnetic layer is assumed uniform in the layer.
In the manner of the definition (10), the $sd$-type exchange interaction is treated by the following approximation;
$$\displaystyle J_{i}(\bm{r})$$
$$\displaystyle=J\Theta_{i}(\bm{r})=\left\{\begin{array}[]{l l}J&(\bm{r}\in\Omega_{i})\\
0&\text{(otherwise)},\end{array}\right.$$
(16)
where $\Omega_{i}$ is the space of the $i$-th magnetic layer, and $\Theta_{i}(\bm{r})$ is the step function defined by the above equation.
The spin torque in this case is obtained as
$$\displaystyle\bm{\tau}_{i}$$
$$\displaystyle=-\frac{J}{\hbar}\bm{S}_{i}\times\left\langle\int_{\Omega_{i}}\mathrm{d}\bm{r}\,\bm{s}(\bm{r})\right\rangle.$$
(17)
We use this expression for calculating the Slonczewski torques.
II.3 Linear response theory for spin torque
Then, we see how to evaluate the spin torque in various nonequilibrium cases, based on the linear response theory.
In the previous subsection, we see the spin torque is defined by the effective magnetic field due to the conduction electron spin density [Eq. (11)].
Hence, we can obtain the spin torque by evaluating the conduction spin density in nonequilibirum, and the evaluation is performed based on the linear response theory.
Below are spin torques known in continuum systems.
II.3.1 Current-induced spin torques
For the current-induced spin torque, the spin density is evaluated as
$$\displaystyle\langle s^{\alpha}(\bm{r})\rangle$$
$$\displaystyle=\int\mathrm{d}\bm{r}^{\prime}\,\chi_{i}^{\alpha}(\bm{r},\bm{r}^{\prime})E_{i}(\bm{r}^{\prime}),$$
(18)
where $\alpha$ and $i$ are the spin index and the direction index of the applied electric field, respectively.
From the linear response theory, the response coefficient $\chi_{i}^{\alpha}(\bm{r},\bm{r}^{\prime})$ is found to be obtained as
$$\displaystyle\chi_{i}^{\alpha}(\bm{r},\bm{r}^{\prime})$$
$$\displaystyle=\lim_{\omega\to 0}\frac{K_{i}^{\alpha}(\bm{r},\bm{r}^{\prime};\omega)-K_{i}^{\alpha}(\bm{r},\bm{r}^{\prime};0)}{\mathrm{i}\omega}$$
(19)
with the spin-current correlation function $K_{j}^{\alpha}(\bm{r},\bm{r}^{\prime};\omega)$,
$$\displaystyle K_{i}^{\alpha}(\bm{r},\bm{r}^{\prime};\omega)$$
$$\displaystyle=\frac{\mathrm{i}}{\hbar}\int_{0}^{\infty}\mathrm{d}t\,e^{\mathrm{i}(\omega+\mathrm{i}0)t}\left\langle[s^{\alpha}(\bm{r},t),j_{i}(\bm{r}^{\prime})]\right\rangle,$$
where $[A,B]=AB-BA$ is the communicator, $s^{\alpha}(\bm{r},t)$ is the spin density operator of the Heisenberg picture of the spin density, $j_{i}(\bm{r})$ is the electric current density operator.
The correlation function $K_{j}^{\alpha}(\bm{r},\bm{r}^{\prime};\omega)$ can be evaluated by using some techniques, such as the thermal Green function with the analytic continuation.
Note that it is possible to evaluate the nonequilibrium spin density $\langle s^{\alpha}(\bm{r})\rangle$ based on the Keldysh Green function, by expanding the external force.
For a simple ferromagnetic metal with magnetization texture (without any SOCs), the nonequilibrium spin density is calculated as
$$\displaystyle\langle\bm{s}(\bm{r})\rangle$$
$$\displaystyle=\frac{\hbar}{J_{sd}}\left\{\bm{m}(\bm{r})\times(\bm{j}_{\mathrm{s}}\cdot\bm{\nabla})\bm{m}(\bm{r})+\beta(\bm{j}_{\mathrm{s}}\cdot\bm{\nabla})\bm{m}(\bm{r})\right\},$$
which leads to the adiabatic spin-transfer torque and the $\beta$ torque, $\bm{\tau}=\bm{\tau}_{\mathrm{stt}}+\bm{\tau}_{\beta}$.
Here, $\bm{j}_{\mathrm{s}}=\sigma_{\mathrm{s}}\bm{E}$ is the spin-polarized current with the conductivity $\sigma_{\mathrm{s}}=\sigma_{\uparrow}-\sigma_{\downarrow}$, where $\sigma_{\uparrow}$ and $\sigma_{\downarrow}$ are the spin-resolved conductivities.
Note that, in the alternating current region, another type of spin torque arises [21].
For the two-dimensional (2D) system with the Rashba SOC, the electric current induces the spin polarization, which is known as the Edelstein effect and obtained from Eq. (18);
$$\displaystyle\langle\bm{s}(\bm{r})\rangle$$
$$\displaystyle\sim\lambda_{\mathrm{R}}\hat{z}\times\bm{j}_{e},$$
(20)
where $\hat{z}$ is assumed to be the direction of broken inversion symmetry and perpendicular to the 2D plane, $\lambda_{\mathrm{R}}$ is the Rashba SOC strength, and $\bm{j}_{e}$ is the uniform electric current.
This current-induced spin polarization leads to the Rashba spin-orbit torque [3, 15]
$$\displaystyle\bm{\tau}_{\mathrm{R}}$$
$$\displaystyle\sim\frac{\lambda_{\mathrm{R}}J_{sd}}{\hbar}\bm{m}\times(\hat{z}\times\bm{j}_{e}).$$
(21)
The Rashba spin-orbit torque exists even when the magnetization is uniform and proportional to the SOC strength, hence different from the above spin-transfer torques.
Equation (20) does not depend on the magnetization, but, since we consider the electron system coupling to the magnetization, another type of spin polarization may exist, such as
$$\displaystyle\langle\bm{s}(\bm{r})\rangle$$
$$\displaystyle=C\bm{m}\times(\hat{z}\times\bm{j}_{e}),$$
(22)
which is actually found in the lattice model of 2D Rashba system, while $C=0$ in the continuum model of the 2D Rashba system [22, 23].
We should note that other types of current-induced spin torques are known in the presence both of magnetization textures and of SOC, which can be found in magnetic skyrmion systems [24].
II.3.2 Thermal spin torques
We see the thermal spin torques very briefly, which are spin torques induced by temperature gradients.
The response of the spin density to the temperature gradient is formally given as
$$\displaystyle\langle s^{\alpha}(\bm{r})\rangle$$
$$\displaystyle=\int\mathrm{d}\bm{r}^{\prime}\,\chi_{i}^{\alpha}(\bm{r}-\bm{r}^{\prime})\bm{\nabla}^{\prime}_{i}T(\bm{r}^{\prime}),$$
(23)
where $\bm{\nabla}^{\prime}_{i}$ indicates the gradient of the $i$ direction for the position $\bm{r}^{\prime}$, and we assume the translational symmetry in the system we consider for simplicity.
Phenomenologically, discussions similar to the current-induced spin torques can be done, which leads to the thermal spin-transfer torques [16, 17].
To discuss more rigorously based on the Kubo formula, we have to introduce the fictional gravitational potentials that couple to the heat density or heat current density [25, 26], since the temperature gradient is not a mechanical force but a statistical force, and the Kubo formula is only valid for the mechanical force.
The resultant expressions of the spin torques are same as phenomenologically derived forms [16, 17] for thermal spin-transfer torques.
Thermal spin-orbit toruqes are also discussed in Refs. [27, 22].
II.3.3 Gilbert damping torque due to conduction electrons
At the end of this overview, we would like to mention the Gilbert damping torque due to the conduction electrons.
The physical picture of this torque is as follows: the nonequilibrium spin density is induced by the magnetization dynamics, and conversely, the spin density acts as a spin torque on the magnetization.
The nonequilibrium spin density giving rise to the Gilbert damping torque is obtained from the linear response;
$$\displaystyle\langle s^{\alpha}(\bm{r})\rangle$$
$$\displaystyle=\int\mathrm{d}\bm{r}^{\prime}\,\chi^{\alpha\beta}(\bm{r},\bm{r}^{\prime})\dot{m}^{\alpha}(\bm{r}^{\prime}),$$
(24)
where the response coefficient $\chi^{\alpha\beta}(\bm{r},\bm{r}^{\prime})$ is given as
$$\displaystyle\chi^{\alpha\beta}(\bm{r},\bm{r}^{\prime})$$
$$\displaystyle=\frac{Q^{\alpha\beta}(\bm{r},\bm{r}^{\prime};\omega)-Q^{\alpha\beta}(\bm{r},\bm{r}^{\prime};0)}{\mathrm{i}\omega}$$
(25)
with the spin-spin correlation function
$$\displaystyle Q^{\alpha\beta}(\bm{r},\bm{r}^{\prime};\omega)$$
$$\displaystyle=\frac{\mathrm{i}J_{sd}}{\hbar}\int_{0}^{\infty}\mathrm{d}t\,e^{\mathrm{i}(\omega+\mathrm{i}0)t}\left\langle[s^{\alpha}(\bm{r},t),s^{\beta}(\bm{r}^{\prime})]\right\rangle.$$
In the presence of the spin relaxation of the conduction electrons, the response coefficient is nonzero and
$$\displaystyle\chi^{\alpha\beta}$$
$$\displaystyle=\frac{\hbar}{J_{sd}}\tilde{\alpha}_{\mathrm{G}}\delta^{\alpha\beta},$$
(26)
where $\tilde{\alpha}_{\mathrm{G}}$ is a coefficient determined by the spin relaxation mechanism.
This nonequilibrium spin density arises the Gilbert damping torque as
$$\displaystyle\bm{\tau}_{\mathrm{G}}$$
$$\displaystyle=\tilde{\alpha}_{\mathrm{G}}\bm{m}\times\dot{\bm{m}}.$$
(27)
Phenomenologically, the Gilbert damping torques originating from the strong SOC is captured by the Fermi surface breathing effect [28, 29].
III five-layers system
III.1 Model
Then, we see the correspondence of Slonczewski torque to the (adiabatic) spin-transfer torque.
The framework of this calculation is based on the previous section.
We first present the model which we consider in this paper.
Following Slonczewski, we consider the five-layers system which consists of two ferromagnetic metal layers (denoted by $\mathrm{F1}$ and $\mathrm{F2}$) sandwiched between three nonmagnetic metals (denoted by $\mathrm{N1}$, $\mathrm{N2}$, and $\mathrm{N3}$).
The thicknesses of $\mathrm{F1}$ and $\mathrm{F2}$ are denoted by $L_{1}$ and $L_{2}$, and the magnetizations are $\bm{M}_{1}$ and $\bm{M}_{2}$, respectively.
The cross section of the layers is given by $A$, and the distance between $\mathrm{F1}$ and $\mathrm{F2}$ is shown by $\delta=L-L_{1}$.
Figure 1 depicts the system that we consider here, and the $x$ direction is taken perpendicular to the layers.
The total Lagrangian density is given as
$$\displaystyle\mathcal{L}$$
$$\displaystyle=\mathcal{L}_{\bm{M}_{1}}+\mathcal{L}_{\bm{M}_{2}}-\mathcal{H}_{e}-\mathcal{H}_{\mathrm{ext}},$$
(28)
where $\mathcal{L}_{\bm{M}_{1}}$ and $\mathcal{L}_{\bm{M}_{2}}$ are the Lagrangians of the magnetizations, which are not specified in this paper since we are focusing on the spin torques. The third term of Eq. (28) is the Hamiltonian of the conduction electron in the five-layers system given as
$$\displaystyle\mathcal{H}_{e}$$
$$\displaystyle=-\frac{\hbar^{2}\bm{\nabla}^{2}}{2m_{\mathrm{e}}}-J_{1}\tilde{\bm{m}}_{1}(\bm{r})\cdot\bm{\sigma}-J_{2}\tilde{\bm{m}}_{2}(\bm{r})\cdot\bm{\sigma},$$
(29)
which consists of the kinetic energy and the $sd$-type exchange interaction with the magnetizations.
Here, $m_{\mathrm{e}}$ is the electron mass, $J_{i}$ is the exchange interaction strength, $\bm{\sigma}=(\sigma^{x},\sigma^{y},\sigma^{z})$ is the Pauli matrix, and
$$\displaystyle\tilde{\bm{m}}_{1}(\bm{r})$$
$$\displaystyle=\bm{m}_{1}\Theta(x)\Theta(L_{1}-x),$$
(30a)
$$\displaystyle\tilde{\bm{m}}_{2}(\bm{r})$$
$$\displaystyle=\bm{m}_{2}\Theta(x-L)\Theta(L+L_{2}-x)$$
(30b)
with the Heaviside step function $\Theta(x)$ and the unit magnetization vectors $\bm{m}_{i}=\bm{M}_{i}/|\bm{M}_{i}|$ ($i=1,2$).
The meaning of $sd$-type exchange interction in Eq. (29) is slightly different from that in Eq. (10); the exchange interaction strength does not depend on its position, while Eq. (10) is a position-dependent interaction strength.
However, the difference is not important, since we can rewrite it as $J_{1}\tilde{\bm{m}}_{1}(\bm{r})=J_{1}(\bm{r})\tilde{\bm{m}}_{1}$ with $J_{1}(\bm{r})=J\Theta(x)\Theta(L_{1}-x)$, and so on.
Note also that we treat the conduction electron is described by the single Hamiltonian (29), which may be too much simplification to perform the quantitative evaluation of Slonczewski torques, but the simplification is valid for the qualitative discussion as seen below.
The last term of the Lagrangian (28) indicates that the external electric field induces the electric current, which is given by
$$\displaystyle\mathcal{H}_{\mathrm{ext}}$$
$$\displaystyle=-\bm{J}_{e}\cdot\bm{A}(t),$$
(31)
where $\bm{J}_{e}$ is the electric current, and $\bm{A}=A_{0}\hat{x}e^{-\mathrm{i}\omega t}$ is the vector potential which induces the electric field
$$\displaystyle\bm{E}$$
$$\displaystyle=\mathrm{i}\omega A_{0}\hat{x}e^{-\mathrm{i}\omega t}$$
(32)
with $\omega\to 0$.
In the second quantization representation, the model is given as
$$\displaystyle\mathcal{H}_{e}$$
$$\displaystyle=\sum_{\bm{k}}\frac{\hbar^{2}k^{2}}{2m_{\mathrm{e}}}c^{\dagger}_{\bm{k}}c_{\bm{k}}-V\sum_{\bm{q}}\bm{s}(-\bm{q})\cdot(J_{1}\tilde{\bm{m}}_{1}(\bm{q})+J_{2}\tilde{\bm{m}}_{2}(\bm{q})),$$
(33)
where $c_{\bm{k}}^{(\dagger)}$ is the annihilation (creation) operator of electron with the wavevector $\bm{k}$,
$$\displaystyle\tilde{\bm{m}}_{i}(\bm{q})$$
$$\displaystyle=\frac{1}{V}\int\mathrm{d}\bm{r}\,\tilde{\bm{m}}_{i}(\bm{r})e^{-\mathrm{i}\bm{q}\cdot\bm{r}},$$
(34)
and the spin density $\bm{s}(\bm{q})$ and the charge current $\bm{J}_{e}$ are given by
$$\displaystyle\bm{s}(\bm{q})=\frac{1}{V}\sum_{\bm{k}}c^{\dagger}_{\bm{k}-\bm{q}}\bm{\sigma}c_{\bm{k}},\qquad\bm{J}_{e}=-e\sum_{\bm{k}}\frac{\hbar\bm{k}}{m_{\mathrm{e}}}c^{\dagger}_{\bm{k}}c_{\bm{k}},$$
(35)
where $V=AW$ is the volume of the system with system length $W$, and $e\,(>0)$ is the elementary charge.
III.2 Linear response theory
For the above-mentioned system, we evaluate the Slonczewski torques.
The fundamental way of the evaluation is shown in Sec. II (II.1, II.2, and II.3.1).
Firstly, the magnetization dynamics is described by the Landau-Lifshitz-Gilbert equation,
$$\displaystyle\frac{\mathrm{d}\bm{M}_{i}}{\mathrm{d}t}$$
$$\displaystyle=-\gamma\bm{M}_{i}\times\bm{H}_{i}+\frac{\alpha_{\mathrm{G}}}{M_{S}}\bm{M}_{i}\times\frac{\mathrm{d}\bm{M}_{i}}{\mathrm{d}t}+\bm{T}_{i},$$
(36)
with $i=1,2$, where the first term of the right hand side arises the precession motion, the second term gives rise to the damping motion to the direction of the equilibrium state, and the last term indicates the spin torque.
Note that $\bm{H}_{i}$ is determined by the Lagrangian of the magnetization $\bm{M}_{i}$, and $\alpha_{\mathrm{G}}$ is the Gilbert damping constant.
The spin torque is given by
$$\displaystyle\bm{T}_{i}$$
$$\displaystyle=-\frac{J_{i}}{\hbar}\bm{M}_{i}\times\left\langle\int_{\Omega_{i}}\mathrm{d}\bm{r}\,\bm{s}(\bm{r})\right\rangle_{\mathcal{H}_{e}+\mathcal{H}_{\mathrm{ext}}},$$
(37)
where $\Omega_{i}$ is the space of the magnetic layer ($i=1,2$), and $\bm{s}(\bm{r})$ is the spin density operator defined by $\bm{s}(\bm{r})=\psi^{\dagger}(\bm{r})\bm{\sigma}\psi(\bm{r})$ with the electron field operator $\psi^{(\dagger)}(\bm{r})$, which can be expanded by the annihilation (creation) operator $c_{\bm{k}}^{(\dagger)}$ as
$$\displaystyle\psi(\bm{r})=\frac{1}{\sqrt{V}}\sum_{\bm{k}}c_{\bm{k}}e^{\mathrm{i}\bm{k}\cdot\bm{r}},$$
$$\displaystyle\quad\psi^{\dagger}(\bm{r})=\frac{1}{\sqrt{V}}\sum_{\bm{k}}c_{\bm{k}}^{\dagger}e^{-\mathrm{i}\bm{k}\cdot\bm{r}}.$$
(38)
To evaluate the spin torque, we now consider that the response of the spin to the external field, which is expressed as
$$\displaystyle\langle s^{\alpha}(\bm{r})\rangle_{\mathcal{H}_{e}+\mathcal{H}_{\mathrm{ext}}}$$
$$\displaystyle=\chi^{\alpha}_{j}(\bm{r})E_{j}.$$
(39)
Here, we first consider a general case of applied electric field and calculate the response coefficient $\chi_{j}^{\alpha}(\bm{r})$.
Then, we will assume the specific configuration given in Eq. (32) and obtain the damping-like Slonczewski torque.
The response coefficient $\chi^{\alpha}_{j}(\bm{r})$ is evaluated by using the linear response theory;
$$\displaystyle\chi_{j}^{\alpha}(\bm{r})$$
$$\displaystyle=\lim_{\omega\to 0}\frac{K^{\alpha}_{j}(\omega;\bm{r})-K_{j}^{\alpha}(0;\bm{r})}{\mathrm{i}\omega},$$
(40)
where $K^{\alpha}_{j}(\omega)$ is obtained from the corresponding Matsubara correlation function given by
$$\displaystyle\mathcal{K}^{\alpha}_{j}(\mathrm{i}\omega_{\lambda};\bm{r})$$
$$\displaystyle=\int_{0}^{\beta}\mathrm{d}\tau e^{\mathrm{i}\omega_{\lambda}\tau}\langle\mathrm{T}_{\tau}\{s^{\alpha}(\bm{r},\tau)J_{e,j}\}\rangle_{\mathcal{H}_{e}}$$
(41)
with the analytic continuation $\mathrm{i}\omega_{\lambda}\to\hbar\omega+\mathrm{i}0$;
$$\displaystyle K_{j}^{\alpha}(\omega;\bm{r})$$
$$\displaystyle=\mathcal{K}_{j}^{\alpha}(\hbar\omega+\mathrm{i}0;\bm{r}).$$
(42)
Note that $\beta=1/k_{\mathrm{B}}T$ is the inverse temperature, $\omega_{\lambda}=2\pi\lambda k_{\mathrm{B}}T$ is the bosonic Matsubara frequency with $\lambda$ being integer, $\tau$ is the imaginary time (in the energy unit), $\mathrm{T}_{\tau}$ is the imaginary-time ordering operator, $s^{\alpha}(\bm{r},\tau)$ is the Heisenberg operator in imaginary time, and $\langle\cdots\rangle_{\mathcal{H}_{e}}$ is the thermal average on the Hamiltonian $\mathcal{H}_{e}$.
In the Fourier space, Eq. (39) is written as
$$\displaystyle\langle s^{\alpha}(\bm{q})\rangle_{\mathcal{H}_{e}+\mathcal{H}_{\mathrm{ext}}}$$
$$\displaystyle=\chi^{\alpha}_{j}(\bm{q})E_{j},$$
(43)
where
$$\displaystyle\chi^{\alpha}_{j}(\bm{q})$$
$$\displaystyle=\frac{1}{V}\int\mathrm{d}\bm{r}\,\chi^{\alpha}_{j}(\bm{r})e^{-\mathrm{i}\bm{q}\cdot\bm{r}}.$$
(44)
The Matsubara correlation function in the Fourier space is obtained as
$$\displaystyle\mathcal{K}^{\alpha}_{j}(\mathrm{i}\omega_{\lambda};\bm{q})$$
$$\displaystyle=\int_{0}^{\beta}\mathrm{d}\tau e^{\mathrm{i}\omega_{\lambda}\tau}\langle\mathrm{T}_{\tau}\{s^{\alpha}(\bm{q},\tau)J_{e,j}\}\rangle_{\mathcal{H}_{e}}.$$
(45)
III.3 Damping-like Slonczewski torque
To evaluate the damping-like Slonczewski torque $\bm{T}_{i}$, we consider the uniform component ($\bm{q}=0$) in Eq. (43), because the $\bm{q}\neq 0$ components oscillate spatially and are mainly canceled by integrating in the magnetic layer.
Then, we expand the response coefficient by the $sd$-type exchange interactions up to the first orders of $\bm{m}_{1}$ and $\bm{m}_{2}$ (see Fig. 2 for the corresponding Feynman diagrams), which results in
$$\displaystyle\mathcal{K}^{\alpha}_{j}(\mathrm{i}\omega_{\lambda})$$
$$\displaystyle=\mathcal{K}^{\alpha}_{j}(\mathrm{i}\omega_{\lambda};\bm{q}=0)$$
$$\displaystyle=-\frac{2\mathrm{i}eJ_{1}J_{2}}{\beta}\sum_{n,\bm{q}^{\prime}}\varphi_{j}(\mathrm{i}\epsilon_{n}^{+},\mathrm{i}\epsilon_{n};-\bm{q}^{\prime})\left\{\tilde{\bm{m}}_{1}(\bm{q}^{\prime})\times\tilde{\bm{m}}_{2}(-\bm{q}^{\prime})\right\}^{\alpha},$$
(46)
where $\epsilon_{n}=(2n+1)\pi k_{\mathrm{B}}T$ and $\mathrm{i}\epsilon_{n}^{+}=\mathrm{i}\epsilon_{n}+\mathrm{i}\omega_{\lambda}$ are the fermionic Matsubara frequency with $n$ being integer, and
$$\displaystyle\varphi_{j}(\mathrm{i}\epsilon_{n}^{+},\mathrm{i}\epsilon_{n};-\bm{q})$$
$$\displaystyle=\frac{2}{V}\sum_{\bm{k}}\biggl{(}\frac{\hbar k_{j}}{m_{\mathrm{e}}}g^{+}_{\bm{k}+\bm{q}}(g^{+}_{\bm{k}})^{2}g_{\bm{k}}$$
$$\displaystyle\hskip 30.00005pt+\frac{\hbar k_{j}}{m_{\mathrm{e}}}g^{+}_{\bm{k}}g_{\bm{k}+\bm{q}}(g_{\bm{k}})^{2}$$
$$\displaystyle\hskip 30.00005pt+\frac{\hbar(k_{j}+q_{j})}{m_{\mathrm{e}}}g^{+}_{\bm{k}}g_{\bm{k}}g^{+}_{\bm{k}+\bm{q}}g_{\bm{k}+\bm{q}}\biggr{)}.$$
(47)
Here, $g^{+}_{\bm{k}}=g_{\bm{k}}(\mathrm{i}\epsilon_{n}^{+})$, $g_{\bm{k}}=g_{\bm{k}}(\mathrm{i}\epsilon)$ are the Matsubara Green functions of electrons.
By the inverse Fourier transformation, Eq. (46) reads
$$\displaystyle\mathcal{K}^{\alpha}_{j}(\mathrm{i}\omega_{\lambda})$$
$$\displaystyle=-\frac{2\mathrm{i}eJ_{1}J_{2}}{\beta V^{2}}\sum_{n}\iint\mathrm{d}\bm{r}\,\mathrm{d}\bm{r}^{\prime}\,$$
$$\displaystyle\times\varphi_{j}(\mathrm{i}\epsilon_{n}^{+},\mathrm{i}\epsilon_{n};\bm{r}-\bm{r}^{\prime})\left\{\tilde{\bm{m}}_{1}(\bm{r})\times\tilde{\bm{m}}_{2}(\bm{r}^{\prime})\right\}^{\alpha}$$
(48)
with
$$\displaystyle\varphi_{j}(\mathrm{i}\epsilon_{n}^{+},\mathrm{i}\epsilon_{n};\bm{r})$$
$$\displaystyle=\sum_{\bm{q}}\varphi_{j}(\mathrm{i}\epsilon_{n}^{+},\mathrm{i}\epsilon_{n};\bm{q})e^{\mathrm{i}\bm{q}\cdot\bm{r}}.$$
(49)
By changing the variable $\bm{k}+\bm{q}\to\bm{q}$ in Eq. (49), we have
$$\displaystyle\frac{1}{V}\varphi_{j}(\mathrm{i}\epsilon_{n}^{+},\mathrm{i}\epsilon_{n};\bm{r})$$
$$\displaystyle=2g(\mathrm{i}\epsilon_{n}^{+};\bm{r})\frac{\hbar}{m_{\mathrm{e}}\mathrm{i}}\frac{\partial}{\partial r_{j}}Q(\mathrm{i}\epsilon_{n}^{+},\mathrm{i}\epsilon_{n};\bm{r})$$
$$\displaystyle+2g(\mathrm{i}\epsilon_{n};\bm{r})\frac{\hbar}{m_{\mathrm{e}}\mathrm{i}}\frac{\partial}{\partial r_{j}}Q(\mathrm{i}\epsilon_{n},\mathrm{i}\epsilon_{n}^{+};\bm{r})$$
$$\displaystyle-2R(\mathrm{i}\epsilon_{n}^{+},\mathrm{i}\epsilon_{n};\bm{r})\frac{\hbar}{\mathrm{i}m_{\mathrm{e}}}\frac{\partial}{\partial r_{j}}R(\mathrm{i}\epsilon_{n}^{+},\mathrm{i}\epsilon_{n};\bm{r}),$$
(50)
where we introduced the following notations,
$$\displaystyle g(\mathrm{i}\epsilon_{n};\bm{r})$$
$$\displaystyle\equiv\frac{1}{V}\sum_{\bm{k}}g_{\bm{k}}(\mathrm{i}\epsilon_{n})e^{\mathrm{i}\bm{k}\cdot\bm{r}},$$
(51)
$$\displaystyle Q(\mathrm{i}\epsilon_{n}^{+},\mathrm{i}\epsilon_{n};\bm{r})$$
$$\displaystyle\equiv\frac{1}{V}\sum_{\bm{k}}\left(g_{\bm{k}}(\mathrm{i}\epsilon_{n}^{+})\right)^{2}g_{\bm{k}}(\mathrm{i}\epsilon_{n})e^{\mathrm{i}\bm{k}\cdot\bm{r}},$$
(52)
$$\displaystyle R(\mathrm{i}\epsilon_{n}^{+},\mathrm{i}\epsilon_{n};\bm{r})$$
$$\displaystyle\equiv\frac{1}{V}\sum_{\bm{k}}g_{\bm{k}}(\mathrm{i}\epsilon_{n}^{+})g_{\bm{k}}(\mathrm{i}\epsilon_{n})e^{\mathrm{i}\bm{k}\cdot\bm{r}}.$$
(53)
By taking the analytic continuation $\mathrm{i}\omega_{\lambda}\to\hbar\omega+\mathrm{i}0$ and assuming the absolute zero ($T=0$), we can write down
$$\displaystyle\frac{1}{\beta}\sum_{n}\varphi_{j}(\mathrm{i}\epsilon_{n}^{+},\mathrm{i}\epsilon_{n};\bm{r})$$
$$\displaystyle=\varphi_{j}^{(0)}(\bm{r})+\mathrm{i}\omega\varphi_{j}^{(1)}(\bm{r})+\cdots,$$
(54)
where only the $\omega$-linear term is of our interest.
The calculation detail of $\varphi_{j}^{(1)}(\bm{r})$ is given in Appendix A, the result of the calculation is shown as
$$\displaystyle\frac{1}{V}\varphi_{j}^{(1)}(\bm{r})$$
$$\displaystyle=\frac{\tau r_{j}}{\pi\hbar}\left[\left\{g^{\mathrm{R}}(\bm{r})\right\}^{2}-\left\{g^{\mathrm{A}}(\bm{r})\right\}^{2}\right],$$
(55)
where $\tau$ is the electron lifetime, $r_{j}$ is the $j$ component of the position $\bm{r}$; $r_{x}=x,r_{y}=y,r_{z}=z$, and $g^{\mathrm{R}/\mathrm{A}}(\bm{r})$ is the retarded/advanced Green function in real space, which is given by
$$\displaystyle g^{\mathrm{R}}(\bm{r})$$
$$\displaystyle=-\frac{m_{\mathrm{e}}}{2\pi\hbar^{2}}\frac{e^{\mathrm{i}k_{F+}r}}{r},$$
(56a)
$$\displaystyle g^{\mathrm{A}}(\bm{r})$$
$$\displaystyle=-\frac{m_{\mathrm{e}}}{2\pi\hbar^{2}}\frac{e^{-\mathrm{i}k_{F-}r}}{r}$$
(56b)
with $k_{F\pm}=(\sqrt{2m_{\mathrm{e}}}/\hbar)\sqrt{\mu\pm\mathrm{i}\hbar/2\tau}=k_{F}\sqrt{1\pm\mathrm{i}/k_{F}l}$ with the mean free path $l=v_{F}\tau=\hbar k_{F}\tau/m_{\mathrm{e}}$.
Hence, the response coefficient $\chi_{j}^{\alpha}(\bm{q}=0)$ [Eq. (44)] is obtained as
$$\displaystyle\frac{1}{V}\chi_{j}^{\alpha}(\bm{q}=0)$$
$$\displaystyle=-2\mathrm{i}eJ_{1}J_{2}\int\frac{\mathrm{d}\bm{r}\,}{V}\int\frac{\mathrm{d}\bm{r}^{\prime}\,}{V}$$
$$\displaystyle\times\varphi_{j}^{(1)}(\bm{r}-\bm{r}^{\prime})\left\{\tilde{\bm{m}}_{1}(\bm{r})\times\tilde{\bm{m}}_{2}(\bm{r}^{\prime})\right\}^{\alpha},$$
(57)
which results in our desired expression of the damping-like Slonczewski torque,
$$\displaystyle\bm{T}_{i}$$
$$\displaystyle=cJ_{i}L_{i}\bm{M}_{i}\times(\bm{m}_{1}\times\bm{m}_{2})$$
(58)
with $i=1,2$ and the coefficient $c$ given as
$$\displaystyle c$$
$$\displaystyle=\frac{2\mathrm{i}eJ_{1}J_{2}|\bm{E}|A}{\hbar}\int_{\Omega_{1}}\frac{\mathrm{d}\bm{r}\,}{V}\int_{\Omega_{2}}\frac{\mathrm{d}\bm{r}^{\prime}\,}{V}\varphi_{x}^{(1)}(\bm{r}-\bm{r}^{\prime}),$$
(59)
where we assumed the specific configuration of Eq. (32).
The integrals of $\bm{r}$ and $\bm{r}^{\prime}$ can be done by using the assumption, $|\bm{r}-\bm{r}^{\prime}|\simeq|x-x^{\prime}|$.
The calculation detail is given in Appendix B, and the resultant expression is obtained as
$$\displaystyle cJ_{i}L_{i}$$
$$\displaystyle=\frac{3}{8\pi}\frac{I_{e}}{e}\frac{A}{l_{sd,1}l_{sd,2}}\frac{L_{i}}{W}\frac{\mathrm{Im}\,[F]}{k_{F}l_{sd,i}},$$
(60)
where $I_{e}$ is the electric current,
$$\displaystyle I_{e}$$
$$\displaystyle=A\sigma_{e}|\bm{E}|=G_{e}V_{e}$$
(61)
with the conductance $G_{e}=A\sigma_{e}/W$ and the voltage $V_{e}=W|\bm{E}|$, $k_{F}$ is the Fermi wavenumber, $l_{sd,i}=v_{F}\tau_{sd,i}$ is a typical length of the $sd$ exchange interaction with the typical time scale $\tau_{sd,i}=\hbar/2J_{i}$, and
$$\displaystyle F$$
$$\displaystyle=\frac{-k_{F}}{2\mathrm{i}k_{F+}}e^{2\mathrm{i}k_{F+}\delta}\left(e^{2\mathrm{i}k_{F+}L_{1}}-1\right)\left(e^{2\mathrm{i}k_{F+}L_{2}}-1\right)$$
$$\displaystyle\hskip 10.00002pt+k_{F}(L_{1}+L_{2}+\delta)\mathrm{Ei}(2\mathrm{i}k_{F+}(L_{1}+L_{2}+\delta))$$
$$\displaystyle\hskip 10.00002pt-k_{F}(L_{1}+\delta)\mathrm{Ei}(2\mathrm{i}k_{F+}(L_{1}+\delta))$$
$$\displaystyle\hskip 10.00002pt-k_{F}(L_{2}+\delta)\mathrm{Ei}(2\mathrm{i}k_{F+}(L_{2}+\delta))$$
$$\displaystyle\hskip 10.00002pt+k_{F}\delta\mathrm{Ei}(2\mathrm{i}k_{F+}\delta)$$
(62)
with $L=L_{1}+\delta$ and $\mathrm{Ei}(x)$ being the exponetial integral function.
Note that the electric conductivity $\sigma_{e}$ is given by $\sigma_{e}=e^{2}\rho_{e}\tau/m_{\mathrm{e}}$ with $\rho_{e}=k_{F}^{3}/3\pi^{2}$.
By taking the field theoretical approach, we successfully obtain the Slonczewski torque [Eq. (58)] with the coefficient given by Eq. (60).
We discuss the result in Sec. IV.
III.4 Field-like Slonczewski torque
Now, we evaluate the field-like Slonczewski torque, $\bm{T}^{\prime}_{1}\propto\bm{M}_{1}\times\bm{m}_{2}$ and $\bm{T}^{\prime}_{2}\propto\bm{M}_{2}\times\bm{m}_{1}$.
The field-like torque is also obtained through evaluating the electron spin [Eq. (43)] with Eq. (37).
For the field-like Slonczewski torque, we expand the response coefficient up to the first order with respect to the $sd$-type exchange interactions, whose Feynman diagrams are given in Fig. 3.
The response coefficient in this case reads
$$\displaystyle\mathcal{K}_{j}^{\alpha}(\mathrm{i}\omega_{\lambda};\bm{q})$$
$$\displaystyle=-\frac{2e}{\beta}\sum_{i}J_{i}\sum_{n}\vartheta_{j}(\mathrm{i}\epsilon_{n}^{+},\mathrm{i}\epsilon_{n};\bm{q})\tilde{m}_{i}^{\alpha}(-\bm{q})$$
(63)
with
$$\displaystyle\vartheta_{j}(\mathrm{i}\epsilon_{n}^{+},\mathrm{i}\epsilon_{n};\bm{q})$$
$$\displaystyle=\frac{1}{V}\sum_{\bm{k}}\left(g^{+}_{\bm{k}+\bm{q}}-g_{\bm{k}+\bm{q}}\right)\frac{\hbar k_{j}}{m_{\mathrm{e}}}g^{+}_{\bm{k}}g_{\bm{k}}.$$
(64)
In the real space representation, we get
$$\displaystyle\mathcal{K}_{j}^{\alpha}(\mathrm{i}\omega_{\lambda};\bm{r})$$
$$\displaystyle=-\frac{2e}{\beta V}\sum_{i}J_{i}\int\mathrm{d}\bm{r}^{\prime}\,\sum_{n}\vartheta_{j}(\mathrm{i}\epsilon_{n}^{+},\mathrm{i}\epsilon_{n};\bm{r}-\bm{r}^{\prime})\tilde{m}_{i}^{\alpha}(\bm{r}^{\prime}),$$
(65)
where
$$\displaystyle\vartheta_{j}(\mathrm{i}\epsilon_{n}^{+},\mathrm{i}\epsilon_{n};\bm{r})$$
$$\displaystyle=\sum_{\bm{q}}\vartheta_{j}(\mathrm{i}\epsilon_{n}^{+},\mathrm{i}\epsilon_{n};\bm{q})e^{\mathrm{i}\bm{q}\cdot\bm{r}},$$
(66)
which is evaluated by taking the analytic continuation $\mathrm{i}\omega_{\lambda}\to\hbar\omega+\mathrm{i}0$, and expand it as
$$\displaystyle\frac{1}{\beta}\sum_{n}\vartheta_{j}(\mathrm{i}\epsilon_{n}^{+},\mathrm{i}\epsilon_{n};\bm{r})$$
$$\displaystyle=\vartheta_{j}^{(0)}(\bm{r})+\mathrm{i}\omega\vartheta_{j}^{(1)}(\bm{r})+\cdots,$$
(67)
where the $\omega$-linear term is of our interest and given as
$$\displaystyle\frac{1}{V}\vartheta_{j}^{(1)}(\bm{r})$$
$$\displaystyle=\frac{-\hbar\tau}{2\pi m_{\mathrm{e}}}\frac{\partial}{\partial r_{j}}\left(g^{\mathrm{R}}(\bm{r})-g^{\mathrm{A}}(\bm{r})\right)^{2}.$$
(68)
From the above, we have
$$\displaystyle\bm{T}^{\prime}_{1}=c^{\prime}\bm{M}_{1}\times\bm{m}_{2},\qquad\bm{T}^{\prime}_{2}=-c^{\prime}\bm{M}_{2}\times\bm{m}_{1},$$
(69)
where
$$\displaystyle c^{\prime}$$
$$\displaystyle=\frac{2eJ_{1}J_{2}|\bm{E}|}{\hbar V}\int_{\Omega_{1}}\mathrm{d}\bm{r}\,\int_{\Omega_{2}}\mathrm{d}\bm{r}^{\prime}\,\vartheta_{x}^{(1)}(\bm{r}-\bm{r}^{\prime}).$$
(70)
Here we have used $\vartheta_{j}^{(1)}(\bm{r}^{\prime}-\bm{r})=-\vartheta_{j}^{(1)}(\bm{r}-\bm{r}^{\prime})$.
Substituting Eq. (III.3) into $\vartheta_{j}^{(1)}(\bm{r})$, we finally obtain
$$\displaystyle c^{\prime}$$
$$\displaystyle=\frac{3}{16\pi}\frac{I_{e}}{e}\frac{A}{l_{sd,1}l_{sd,2}}\{h(L+L_{2})-h(L)\},$$
(71)
where we used the approximation $|\bm{r}-\bm{r}^{\prime}|\simeq|x-x^{\prime}|$, and
$$\displaystyle h(y)$$
$$\displaystyle=\frac{1}{k_{F}}\int_{0}^{L_{1}}\mathrm{d}x\,\frac{\left(e^{\mathrm{i}k_{F+}(y-x)}-e^{-\mathrm{i}k_{F-}(y-x)}\right)^{2}}{(y-x)^{2}}$$
$$\displaystyle=P(2k_{F+},y)-2P(k_{F+}-k_{F-},y)+P(-2k_{F-},y)$$
(72)
with
$$\displaystyle P(k,y)$$
$$\displaystyle=\frac{1}{k_{F}}\int_{0}^{L_{1}}\mathrm{d}x\,\frac{e^{\mathrm{i}k(y-x)}}{(y-x)^{2}}$$
$$\displaystyle=\left[\frac{e^{\mathrm{i}k(y-x)}}{k_{F}(y-x)}-\frac{\mathrm{i}k}{k_{F}}\mathrm{Ei}(k(y-x))\right]_{x=0}^{x=L_{1}}$$
$$\displaystyle=\frac{e^{\mathrm{i}k(y-L_{1})}}{k_{F}(y-L_{1})}-\frac{\mathrm{i}k}{k_{F}}\mathrm{Ei}(\mathrm{i}k(y-L_{1}))$$
$$\displaystyle\hskip 10.00002pt-\frac{e^{\mathrm{i}ky}}{k_{F}y}+\frac{\mathrm{i}k}{k_{F}}\mathrm{Ei}(\mathrm{i}ky).$$
(73)
We discuss the results in the following section.
IV Results and Discussion
Here, we discuss the obtained expressions of the damping-like and field-like Slonczewski torques.
Firstly, we successfully obtain the Slonczewski spin-transfer torque (damping-like torque) and the field-like torque corresponding to the $\beta$ torque in continuum systems, which indicates that our field-theoretical approach is valid for the spin-transfer spin torque in the magnetic junction system.
In the present calculation, we took the perturbation method of expanding the $sd$-type exchange interaction with the strength $J$ (to be exact, $J_{1}$ and $J_{2}$), and the damping-like and field-like torques are respectively proportional to $(J/\epsilon_{\mathrm{F}})^{3}$ and $(J/\epsilon_{\mathrm{F}})^{2}$, where $\epsilon_{\mathrm{F}}$ is the Fermi energy.
Since the perturbation method should be valid only for the case of $J/\epsilon_{\mathrm{F}}<1$, the damping-like torque is always small than the field-like torque in our calculation.
For the case of strong $sd$-type exchange interaction, we need to take another method, such as the spin gauge field method.
Then, we see the coefficients of the obtained torques.
The first notable point is that the obtained coefficients seemingly do not contain the spin polarization of the electric current $P$, while Slonczewski showed that the damping-like torque depends on $P$ and vanishes when no spin polarization $P=0$.
Since we expanded the $sd$-type exchange interaction, spin-dependent conductivity $\sigma_{s}$ ($s=\pm$) should be also expanded;
$$\displaystyle\sigma_{s}$$
$$\displaystyle=\frac{e^{2}\epsilon_{\mathrm{F}s}\nu_{s}\tau_{s}}{m_{\mathrm{e}}}\simeq\frac{2sJ_{i}}{\epsilon_{\mathrm{F}}}\sigma_{e},$$
(74)
where $\epsilon_{\mathrm{F}s}=\epsilon_{\mathrm{F}}+sJ_{i}$ is the Fermi energy, $\nu_{s}=\nu(\epsilon_{\mathrm{F}s})\propto\sqrt{\epsilon_{\mathrm{F}s}}$ is the density of states, and the lifetime is given as $\hbar/\tau_{s}=2\pi n_{i}u^{2}\nu_{s}$ with $n_{i}$ the impurity concentration and $u$ the impurity potential.
(Here, we omitted the index $i$ denoting the $i$-th magnetic layer with $i=1,2$ in $\epsilon_{\mathrm{F}s}$, $\nu_{s}$, $\tau_{s}$ and $\sigma_{s}$, for readability.)
Hence, $\sigma_{\mathrm{s}}=\sigma_{+}-\sigma_{-}=P_{i}\sigma_{e}$ with $P_{i}=4J_{i}/\epsilon_{\mathrm{F}}$ ($i=1,2$), which leads to the coefficient of the damping-like Slonczewski torque
$$\displaystyle cJ_{1}L_{1}$$
$$\displaystyle=\frac{3}{24\pi}\frac{P_{2}I_{e}}{e}\frac{A}{l_{sd,1}^{2}}\frac{L_{i}}{W}\mathrm{Im}\,[F],$$
(75)
although an ambiguity on which layer the spin polarization should be used remains.
The coefficient of the damping-like torque depends on the ferromagnetic layer, $\mathrm{F1}$ or $\mathrm{F2}$; $cJ_{1}L_{1}\neq cJ_{2}L_{2}$, if $J_{1}L_{1}\neq J_{2}L_{2}$, as seen in Eq. (58).
It should be noted that the spin torque is given by Eq. (37) and is defined as the integral of the electron spin density over the volume of the magnetic layer.
Since we have evaluated the uniform ($\bm{q}=0$) component of the electron spin for the damping-like torque, the integral over the volume of the magnetic layer is reduced to the volume of the magnetic layer.
On the other hand, the field-like torque has the same magnitude and opposite sign depending on the magnetic layer as seen in Eq. (69).
This critical feature shown in Eq. (69) does not change by considering the spin relaxation.
In continuum systems, the $\beta$ torques corresponding to the field-like torque arise from the spin-nonconserving process and nonadiabaticity.
Since our system is a spin-conserving model, the field-like torque might arise from the nonadiabaticity.
Note that the filed-like torque arises from the nonuniform ($\bm{q}\neq 0$) component of the electron spin density.
In Fig. 4, we depict the dependence of the damping-like and field-like torques on the distance between the magnetic layers $\delta$ with the fixed magnetic layer thicknesses $L_{1}=L_{2}=5~{}\mathrm{nm}$ for various the mean free path $l$.
To compare the damping-like and field-like torques, we set $k_{F}l_{sd,2}=J_{2}/\epsilon_{\mathrm{F}}=0.5$ and $W=L1+L2+\delta$, and measure the coefficients by the unit $c_{0}=(3/8\pi)(I_{e}/e)(A/l_{sd,1}l_{sd,2})$.
We assume that the normal metal layer is made from aluminum; $k_{F}=17.5~{}\mathrm{nm^{-1}}$ [30].
We see that both the torque coefficients decay as the distance between the magnetic layers increases, which is a natural result since the correlation of the magnetic layers is expected to decay.
The coefficient of the damping-like torque on the magnetization $\bm{M}_{2}$ is much smaller than that of the field-like torque in the entire region.
This relation in size is partial because the field-like torque is proportional to $(J/\epsilon_{\mathrm{F}})^{2}$ and the damping-like torque is in the order of $(J/\epsilon_{\mathrm{F}})^{3}$, as already mentioned.
Both the coefficients increase as the mean free path $l$ is larger, but the increments are slight.
For the damping-like torque, the coefficient, $cJ_{i}L_{i}$, takes both the positive and negative values depending on the distance $\delta$.
This feature is shared with the Freidel oscillation and the Ruderman-Kittel-Kasuya-Yosida interaction.
On the other hand, the coefficient of the field-like torque, $c^{\prime}$, takes only the positive values.
In Fig. 5, we show the dependence of the coefficients of the damping-like and field-like torques on $L_{2}$, the thickness of the magnetic layer $\mathrm{F2}$, with the distance $\delta=5~{}\mathrm{nm}$ and the mean free path $l=2~{}\mathrm{nm}$ for various thicknesses of the magnetic layer $\mathrm{F1}$.
We see that the coefficient of the damping-like torque increases slowly as $L_{2}$ increases, while that of the field-like torque increases more rapidly and saturates for $L_{2}\gtrsim 5~{}\mathrm{nm}$.
Comparing the magnitudes of the damping-like and field-like torques, we confirm that the damping-like torque is smaller than the field-like torque, as discussed before.
Moreover, we find that the damping-like torque is smaller as $L_{1}$ is larger, while the field-like torque is larger, although the field-like torque saturates when $L_{1}\gtrsim 10~{}\mathrm{nm}$.
Note that the sign of $cJ_{i}L_{i}$ can change by changing $\delta$, while $c^{\prime}$ is always positive, as seen in Fig. 4.
Here, we estimate the saturation value of $c^{\prime}$ for $L_{1},L_{2}\gg l$.
The main contribution to $c^{\prime}/c_{0}$ is from $-h(L_{1}+\delta)/2$, which is evaluated as
$$\displaystyle c^{\prime}$$
$$\displaystyle=\frac{c_{0}}{k_{F}}\int_{\delta/l}^{\infty}\mathrm{d}t\,\frac{e^{-t}(1-\cos 2k_{F}lt)}{t^{2}},$$
(76)
where we have used $\pm\mathrm{i}k_{F\pm}\simeq-1/2l\pm\mathrm{i}k_{F}$.
The term containing $\cos 2k_{F}lt$ oscillates rapidly so that it cancels out mostly (but not completely).
Hence,
$$\displaystyle c^{\prime}$$
$$\displaystyle\simeq\frac{c_{0}}{k_{F}}\Gamma(-1,\delta/l),$$
(77)
where $\Gamma(a,x)=\int_{a}^{\infty}\mathrm{d}t\,t^{a-1}e^{-t}$ is the incomplete gamma function, which is expanded as
$$\displaystyle\Gamma(-1,x)$$
$$\displaystyle=\left\{\begin{array}[]{c c}1/x+\log x+\gamma-1&(x\ll 1),\\[4.30554pt]
e^{-x}/x^{2}&(x\gg 1).\end{array}\right.$$
(80)
Here, $\gamma$ is Euler’s constant.
We also plot the asymptotic value (77) in Fig. 5, where the slightly difference between the asymptotic and exact values is seen, but the difference arises from the neglected term containing $\cos 2k_{F}lt$ in Eq. (76).
V Conclusion
We have examined the field theoretical approach to the spin-transfer torques in the magnetic junction system composed of two ferromagnetic and three nonmagnetic metal layers.
We successfully obtain the damping-like and field-like Slonczewski torques by evaluating the nonequilibrium spin density due to the electric field.
The coefficient of the damping-like torque takes a different magnitude depending on the magnetizations in the magnetic layers, but that of the field-like torque has the same magnitude with the opposite sign in the magnetic layers.
We find that the coefficent of the damping-like torque has the spatial quantum oscillation like the Friedel oscillation, which has not been mentioned before.
We also find that the field-like torque has the saturation value for large thicknesses of the magnetic layers.
It is better to consider a more sophisticated model, such as a tight-binding model, which consists of the three domains; two ferromagnets and one nonmagnet, to describe the relation in size between the damping-like and field-like Slonczewski torques.
Since the field theoretical approach is valid for spin torques in magnetic junction systems, we are to examine the spin Hall torques based in the same way as we do in this work.
It will also be valuable to estimate the spin-torque ferromagnetic resonance based on the quantum field theory.
Acknowledgements.
The author would like to thank M. Hayashi, G. Tatara, T. Yamaguchi, and Y. Araki.
This work is partially supported by JSPS KAKENHI Grant Number JP22K13997.
Appendix A Calculation of damping-like Slonczewski torque
Here, we give the calculation detail of the damping-like Slonczewski torque.
The $\omega$-linear term in Eq. (54) is obtained as
$$\displaystyle\frac{\pi}{\hbar}\varphi_{i}^{(1)}(\bm{r})$$
$$\displaystyle=g^{\mathrm{R}}(\bm{r})\frac{\hbar}{\mathrm{i}m_{\mathrm{e}}}\frac{\partial}{\partial r_{i}}Q^{\mathrm{R}\mathrm{A}}(\bm{r})$$
$$\displaystyle+g^{\mathrm{A}}(\bm{r})\frac{\hbar}{\mathrm{i}m_{\mathrm{e}}}\frac{\partial}{\partial r_{i}}Q^{\mathrm{A}\mathrm{R}}(\bm{r})$$
$$\displaystyle-R^{\mathrm{R}\mathrm{A}}(\bm{r})\frac{\hbar}{\mathrm{i}m_{\mathrm{e}}}\frac{\partial}{\partial r_{i}}R^{\mathrm{R}\mathrm{A}}(\bm{r}),$$
(81)
where we have defined
$$\displaystyle g^{\mathrm{X}}(\bm{r})$$
$$\displaystyle=\frac{1}{V}\sum_{\bm{k}}g^{\mathrm{X}}_{\bm{k}}e^{\mathrm{i}\bm{k}\cdot\bm{r}}\quad(\mathrm{X}\in\{\mathrm{R},\mathrm{A}\}),$$
(82)
$$\displaystyle Q^{\mathrm{X}\mathrm{Y}}(\bm{r})$$
$$\displaystyle=\frac{1}{V}\sum_{\bm{k}}\left(g_{\bm{k}}^{\mathrm{X}}\right)^{2}g^{\mathrm{Y}}_{\bm{k}}e^{\mathrm{i}\bm{k}\cdot\bm{r}}\quad(\mathrm{X},\mathrm{Y}\in\{\mathrm{R},\mathrm{A}\}),$$
(83)
$$\displaystyle R^{\mathrm{R}\mathrm{A}}(\bm{r})$$
$$\displaystyle=\frac{1}{V}\sum_{\bm{k}}g^{\mathrm{R}}_{\bm{k}}g^{\mathrm{A}}_{\bm{k}}e^{\mathrm{i}\bm{k}\cdot\bm{r}},$$
(84)
and $g^{\mathrm{R}/\mathrm{A}}_{\bm{k}}$ is the retarded/advanced Green function
$$\displaystyle g^{\mathrm{R}/\mathrm{A}}_{\bm{k}}$$
$$\displaystyle=\frac{1}{\mu-\frac{\hbar^{2}k^{2}}{2m_{\mathrm{e}}}\pm\frac{\mathrm{i}\hbar}{2\tau}}.$$
(85)
Note that we introduced the lifetime of the electron $\tau$.
We use the following relation,
$$\displaystyle g^{\mathrm{R}}_{\bm{k}}g^{\mathrm{A}}_{\bm{k}}$$
$$\displaystyle=\frac{1}{\left(\mu+\frac{\mathrm{i}\hbar}{2\tau}-\frac{\hbar^{2}k^{2}}{2m_{\mathrm{e}}}\right)\left(\mu-\frac{\mathrm{i}\hbar}{2\tau}-\frac{\hbar^{2}k^{2}}{2m_{\mathrm{e}}}\right)}$$
$$\displaystyle=-\frac{\tau}{\mathrm{i}\hbar}\left(\frac{1}{\mu+\frac{\mathrm{i}\hbar}{2\tau}-\frac{\hbar^{2}k^{2}}{2m_{\mathrm{e}}}}-\frac{1}{\mu-\frac{\mathrm{i}\hbar}{2\tau}-\frac{\hbar^{2}k^{2}}{2m_{\mathrm{e}}}}\right)$$
$$\displaystyle=-\frac{\tau}{\mathrm{i}\hbar}\left(g^{\mathrm{R}}_{\bm{k}}-g^{\mathrm{A}}_{\bm{k}}\right)$$
(86)
and obtain
$$\displaystyle Q^{\mathrm{R}\mathrm{A}}(\bm{r})$$
$$\displaystyle=-\frac{\tau}{\mathrm{i}\hbar}\frac{1}{V}\sum_{\bm{k}}\left\{\left(g_{\bm{k}}^{\mathrm{R}}\right)^{2}-g^{\mathrm{R}}_{\bm{k}}g^{\mathrm{A}}_{\bm{k}}\right\}e^{\mathrm{i}\bm{k}\cdot\bm{r}}$$
$$\displaystyle=-\frac{\tau}{\mathrm{i}\hbar}\frac{1}{V}\sum_{\bm{k}}\left(g_{\bm{k}}^{\mathrm{R}}\right)^{2}e^{\mathrm{i}\bm{k}\cdot\bm{r}}+\frac{\tau}{\mathrm{i}\hbar}R^{\mathrm{R}\mathrm{A}}(\bm{r}).$$
(87)
Similarly,
$$\displaystyle Q^{\mathrm{A}\mathrm{R}}(\bm{r})$$
$$\displaystyle=-\frac{\tau}{\mathrm{i}\hbar}\frac{1}{V}\sum_{\bm{k}}\left\{g^{\mathrm{R}}_{\bm{k}}g^{\mathrm{A}}_{\bm{k}}-\left(g_{\bm{k}}^{\mathrm{A}}\right)^{2}\right\}e^{\mathrm{i}\bm{k}\cdot\bm{r}}$$
$$\displaystyle=\frac{\tau}{\mathrm{i}\hbar}\frac{1}{V}\sum_{\bm{k}}\left(g_{\bm{k}}^{\mathrm{A}}\right)^{2}e^{\mathrm{i}\bm{k}\cdot\bm{r}}-\frac{\tau}{\mathrm{i}\hbar}R^{\mathrm{R}\mathrm{A}}(\bm{r}),$$
(88)
which results in
$$\displaystyle g^{\mathrm{R}}(\bm{r})\frac{\hbar}{\mathrm{i}m_{\mathrm{e}}}\frac{\partial}{\partial r_{i}}Q^{\mathrm{R}\mathrm{A}}(\bm{r})+g^{\mathrm{A}}(\bm{r})\frac{\hbar}{\mathrm{i}m_{\mathrm{e}}}\frac{\partial}{\partial r_{i}}Q^{\mathrm{A}\mathrm{R}}(\bm{r})$$
$$\displaystyle=g^{\mathrm{R}}(\bm{r})\frac{\hbar}{\mathrm{i}m_{\mathrm{e}}}\frac{\partial}{\partial r_{i}}\left(-\frac{\tau}{\mathrm{i}\hbar}\frac{1}{V}\sum_{\bm{k}}\left(g_{\bm{k}}^{\mathrm{R}}\right)^{2}e^{\mathrm{i}\bm{k}\cdot\bm{r}}\right)$$
$$\displaystyle\hskip 10.00002pt+g^{\mathrm{A}}(\bm{r})\frac{\hbar}{\mathrm{i}m_{\mathrm{e}}}\frac{\partial}{\partial r_{i}}\left(\frac{\tau}{\mathrm{i}\hbar}\frac{1}{V}\sum_{\bm{k}}\left(g_{\bm{k}}^{\mathrm{A}}\right)^{2}e^{\mathrm{i}\bm{k}\cdot\bm{r}}\right)$$
$$\displaystyle\hskip 10.00002pt+\frac{\tau}{\mathrm{i}\hbar}\left\{g^{\mathrm{R}}(\bm{r})-g^{\mathrm{A}}(\bm{r})\right\}\frac{\hbar}{\mathrm{i}m_{\mathrm{e}}}\frac{\partial}{\partial r_{i}}R^{\mathrm{R}\mathrm{A}}(\bm{r}).$$
(89)
Note that the last term can be rewritten as
$$\displaystyle\frac{\tau}{\mathrm{i}\hbar}$$
$$\displaystyle\left\{g^{\mathrm{R}}(\bm{r})-g^{\mathrm{A}}(\bm{r})\right\}\frac{\hbar}{\mathrm{i}m_{\mathrm{e}}}\frac{\partial}{\partial r_{i}}R^{\mathrm{R}\mathrm{A}}(\bm{r})$$
$$\displaystyle=-R^{\mathrm{R}\mathrm{A}}(\bm{r})\frac{\hbar}{\mathrm{i}m_{\mathrm{e}}}\frac{\partial}{\partial r_{i}}R^{\mathrm{R}\mathrm{A}}(\bm{r}).$$
(90)
Hence,
$$\displaystyle\frac{\pi}{\hbar}\varphi_{i}^{(1)}(\bm{r})$$
$$\displaystyle=g^{\mathrm{R}}(\bm{r})\frac{\hbar}{\mathrm{i}m_{\mathrm{e}}}\frac{\partial}{\partial r_{i}}\left(-\frac{\tau}{\mathrm{i}\hbar}\frac{1}{V}\sum_{\bm{k}}\left(g_{\bm{k}}^{\mathrm{R}}\right)^{2}e^{\mathrm{i}\bm{k}\cdot\bm{r}}\right)$$
$$\displaystyle\hskip 10.00002pt+g^{\mathrm{A}}(\bm{r})\frac{\hbar}{\mathrm{i}m_{\mathrm{e}}}\frac{\partial}{\partial r_{i}}\left(\frac{\tau}{\mathrm{i}\hbar}\frac{1}{V}\sum_{\bm{k}}\left(g_{\bm{k}}^{\mathrm{A}}\right)^{2}e^{\mathrm{i}\bm{k}\cdot\bm{r}}\right).$$
(91)
Moreover,
$$\displaystyle\frac{\hbar}{\mathrm{i}m_{\mathrm{e}}}\frac{\partial}{\partial r_{i}}\left(\frac{\tau}{\mathrm{i}\hbar}\frac{1}{V}\sum_{\bm{k}}\left(g_{\bm{k}}^{\mathrm{X}}\right)^{2}e^{\mathrm{i}\bm{k}\cdot\bm{r}}\right)$$
$$\displaystyle=\frac{\tau}{\mathrm{i}\hbar}\frac{1}{V}\sum_{\bm{k}}\frac{\hbar k_{i}}{m_{\mathrm{e}}}\left(g_{\bm{k}}^{\mathrm{X}}\right)^{2}e^{\mathrm{i}\bm{k}\cdot\bm{r}}$$
$$\displaystyle=-\frac{\tau}{\mathrm{i}\hbar^{2}}\frac{1}{V}\sum_{\bm{k}}g_{\bm{k}}^{\mathrm{X}}\frac{\partial}{\partial k_{i}}e^{\mathrm{i}\bm{k}\cdot\bm{r}}$$
$$\displaystyle=-\frac{\tau r_{i}}{\hbar^{2}}g^{\mathrm{X}}(\bm{r}),$$
(92)
where we have used $\partial_{k_{i}}g^{\mathrm{X}}_{\bm{k}}=(\hbar^{2}k_{i}/m_{\mathrm{e}})(g^{\mathrm{X}}_{\bm{k}})^{2}$.
We finally obtain
$$\displaystyle\varphi_{i}^{(1)}(\bm{r})$$
$$\displaystyle=\frac{\hbar}{\pi}\frac{\tau r_{i}}{\hbar^{2}}\left[\left\{g^{\mathrm{R}}(\bm{r})\right\}^{2}-\left\{g^{\mathrm{A}}(\bm{r})\right\}^{2}\right].$$
(93)
Appendix B Integrals in damping-like Slonczewski torque
Here, we show the calculation of the integrals in Eq. (59);
$$\displaystyle c$$
$$\displaystyle=\frac{\mathrm{i}m_{\mathrm{e}}^{2}eJ_{1}J_{2}|\bm{E}|A\tau}{2\pi^{3}\hbar^{6}AW}\int_{\Omega_{1}}\mathrm{d}\bm{r}\,\int_{\Omega_{2}}\mathrm{d}\bm{r}^{\prime}\,$$
$$\displaystyle\hskip 30.00005pt\times\frac{(x-x^{\prime})}{|\bm{r}-\bm{r}^{\prime}|^{2}}\left[e^{2\mathrm{i}k_{F+}|\bm{r}-\bm{r}^{\prime}|}-e^{-2\mathrm{i}k_{F-}|\bm{r}-\bm{r}^{\prime}|}\right].$$
(94)
We presume that the part $|\bm{r}-\bm{r}^{\prime}|\simeq|x-x^{\prime}|$ plays the important role, so that we have
$$\displaystyle cJ_{i}L_{i}$$
$$\displaystyle=\frac{3}{8\pi}\frac{I_{e}}{e}\frac{A}{l_{sd,1}l_{sd,2}}\frac{L_{i}}{W}\frac{\mathrm{Im}\,[F]}{k_{F}l_{sd,i}}$$
(95)
with
$$\displaystyle F$$
$$\displaystyle=k_{F}\int_{0}^{L_{1}}\mathrm{d}x\,\int_{L}^{L+L_{2}}\mathrm{d}x^{\prime}\,\frac{e^{2\mathrm{i}k_{F+}(x^{\prime}-x)}}{x^{\prime}-x}.$$
(96)
We can evaluate $F$ as
$$\displaystyle F$$
$$\displaystyle=k_{F}\int_{0}^{L_{1}}\mathrm{d}x\,\int_{L-x}^{L+L_{2}-x}\mathrm{d}t\,\frac{e^{2\mathrm{i}k_{F+}t}}{t}$$
$$\displaystyle=k_{F}\int_{0}^{L_{1}}\mathrm{d}x\,\left[\mathrm{Ei}(2\mathrm{i}k_{F+}(L+L_{2}-x))-\mathrm{Ei}(2\mathrm{i}k_{F+}(L-x))\right]$$
$$\displaystyle=k_{F}\frac{-1}{2\mathrm{i}k_{F+}}\left(e^{2\mathrm{i}k_{F+}L}-e^{2\mathrm{i}k_{F+}(L-L_{1})}\right)\left(e^{2\mathrm{i}k_{F+}L_{2}}-1\right)$$
$$\displaystyle\hskip 10.00002pt+k_{F}(L+L_{2})\mathrm{Ei}(2\mathrm{i}k_{F+}(L+L_{2}))$$
$$\displaystyle\hskip 10.00002pt-k_{F}L\mathrm{Ei}(2\mathrm{i}k_{F+}L)$$
$$\displaystyle\hskip 10.00002pt-k_{F}(L+L_{2}-L_{1})\mathrm{Ei}(2\mathrm{i}k_{F+}(L+L_{2}-L_{1}))$$
$$\displaystyle\hskip 10.00002pt+k_{F}(L-L_{1})\mathrm{Ei}(2\mathrm{i}k_{F+}(L-L_{1}))),$$
(97)
weher $\mathrm{Ei}(x)$ is the exponetial integral function.
Here, introducing $L=L_{1}+\delta$, we obtain
$$\displaystyle F$$
$$\displaystyle=\frac{-k_{F}}{2\mathrm{i}k_{F+}}e^{2\mathrm{i}k_{F+}\delta}\left(e^{2\mathrm{i}k_{F+}L_{1}}-1\right)\left(e^{2\mathrm{i}k_{F+}L_{2}}-1\right)$$
$$\displaystyle\hskip 10.00002pt+k_{F}(L_{1}+L_{2}+\delta)\mathrm{Ei}(2\mathrm{i}k_{F+}(L_{1}+L_{2}+\delta))$$
$$\displaystyle\hskip 10.00002pt-k_{F}(L_{1}+\delta)\mathrm{Ei}(2\mathrm{i}k_{F+}(L_{1}+\delta))$$
$$\displaystyle\hskip 10.00002pt-k_{F}(L_{2}+\delta)\mathrm{Ei}(2\mathrm{i}k_{F+}(L_{2}+\delta))$$
$$\displaystyle\hskip 10.00002pt+k_{F}\delta\mathrm{Ei}(2\mathrm{i}k_{F+}\delta)$$
(98)
References
Tatara and Kohno [2004]
G. Tatara and H. Kohno, Phys. Rev. Lett. 92, 086601 (2004).
Tatara et al. [2008]
G. Tatara, H. Kohno, and J. Shibata, Phys. Rep. 468, 213 (2008).
Manchon and Zhang [2008]
A. Manchon and S. Zhang, Phys. Rev. B 78, 212405 (2008).
Shibata and Kohno [2011]
J. Shibata and H. Kohno, Phys. Rev. B 84, 184408 (2011).
Ohnuma et al. [2014]
Y. Ohnuma, H. Adachi,
E. Saitoh, and S. Maekawa, Phys.
Rev. B 89, 174417
(2014).
Tatara and Mizukami [2017]
G. Tatara and S. Mizukami, Phys. Rev. B 96, 064423 (2017).
Slonczewski [1996]
J. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996).
Berger [1996]
L. Berger, Phys. Rev. B 54, 9353 (1996).
Bazaliy et al. [1998]
Y. B. Bazaliy, B. A. Jones,
and S.-C. Zhang, Phys. Rev. B 57, R3213 (1998).
Note [1]
Other interactions between the localized spin and the
conduction electron spin, such as the RKKY interaction and the
Dzyaloshinskii-Moriya interaction, possibly can arise novel types of spin
torques, but we focus on the spin torques originating from the $sd$-type
exchange interaction.
Zhang and Li [2004]
S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004).
Duine et al. [2007]
R. A. Duine, A. S. Núñez, J. Sinova, and A. H. MacDonald, Phys. Rev. B 75, 214420 (2007).
Ralph and Stiles [2008]
D. C. Ralph and M. D. Stiles, Journal of Magnetism and Magnetic Materials 320, 1190 (2008).
Brataas et al. [2012]
A. Brataas, A. D. Kent, and H. Ohno, Nature Mater 11, 372
(2012).
Manchon et al. [2019]
A. Manchon, J. Železný, I. M. Miron, T. Jungwirth,
J. Sinova, A. Thiaville, K. Garello, and P. Gambardella, Rev. Mod. Phys. 91, 035004 (2019).
Hatami et al. [2007]
M. Hatami, G. E. W. Bauer, Q. Zhang, and P. J. Kelly, Physical Review Letters 99, 066603 (2007).
Kohno et al. [2016]
H. Kohno, Y. Hiraoka,
M. Hatami, and G. E. W. Bauer, Phys.
Rev. B 94, 104417
(2016), arXiv:1408.4562 .
Kohno et al. [2006]
H. Kohno, G. Tatara, and J. Shibata, J. Phys. Soc. Jpn. 75, 113706 (2006).
Tserkovnyak et al. [2006]
Y. Tserkovnyak, H. J. Skadsem, A. Brataas, and G. E. W. Bauer, Phys. Rev. B 74, 144405 (2006).
Kohno and Shibata [2007]
H. Kohno and J. Shibata, J. Phys. Soc. Jpn. 76, 063710 (2007).
Fujimoto and Matsuo [2019]
J. Fujimoto and M. Matsuo, Phys. Rev. B 100, 220402 (2019).
Fujimoto [2015]
J. Fujimoto, ”Microscopic Theory of Spin-Orbit
Torques Induced by Electric and Heat Currents”, Ph.D.
thesis, Osaka University, Osaka,
Japan (2015).
Ado et al. [2017]
I. A. Ado, O. A. Tretiakov,
and M. Titov, Phys. Rev. B 95, 094401 (2017).
Kurebayashi and Nomura [2019]
D. Kurebayashi and K. Nomura, Sci. Rep. 9, 5365 (2019).
Luttinger [1964]
J. Luttinger, Phys. Rev. 135, A1505 (1964).
Tatara [2015]
G. Tatara, Phys. Rev. Lett. 114, 196601 (2015).
van der Bijl [2014]
E. van
der Bijl, Spin Currents and Magnetization
Dynamics in Multilayer Systems, Ph.D. thesis (2014).
Kamberský [1970]
V. Kamberský, Can. J. Phys. 48, 2906 (1970).
Hillebrands and Ounadjela [2003]
B. Hillebrands and K. Ounadjela, eds., Spin Dynamics in Confined Magnetic
Structures II, Topics in Applied Physics No. v. 87 (Springer, Berlin ; New York, 2003).
Takahashi and Maekawa [2008]
S. Takahashi and S. Maekawa, Sci. Technol. Adv. Mater. 9, 014105 (2008). |
Abstract
We discuss the spatial structure of the Cooper pair in dilute neutron matter and neutron-rich nuclei
by means of the BCS theory and the Skyrme-Hartree-Fock-Bogioliubov model, respectively.
The neutron pairing in dilute neutron matter is close to the region of the
BCS-BEC crossover in a wide density range, giving rise to spatially compact Cooper pair
whose size is smaller than the average interaparticle distance. This behavior extends
to moderate low density ($\sim 10^{-1}$ of the saturation density) where the
Cooper pair size becomes smallerst ($\sim 5$ fm).
The Cooper pair in finite nuclei also exhibits the spatial correlation
favoring the coupling of neutrons at small relative distances $r\mathop{\vbox{
\offinterlineskip\hbox{$<$}\hbox to 7.499886pt{\hss\hbox{$\sim$}\hss}}}3$ fm
with large probability.
Neutron-rich nuclei having small neutron separation
energy may provide us opportunity to probe the spatial correlation since
the neutron pairing and the spatial correlation persists also in an area of
low-density neutron distribution extending from the surface to far outside the
nucleus.
Chapter \thechapter
Spatial structure of Cooper pairs in nuclei
M. Matsuo]Masayuki Matsuo
\body
1 Introduction
The formation and the condensation of the Cooper pairs are the essence of superconductivity
and superfluidity in many-Fermion systems[1]. The binding energy of the Cooper pair is closely
related to
the pairing gap $\Delta$. The spatial size of the Cooper pair is
identified to the coherence length $\xi$ of the superconductors, which
plays important roles in
many aspects, for instance, in distinguishing
the type I and type II
superconductors. What is the size of
the Cooper pair in the superfluidity of nuclear systems? A simple
estimate of the coherence length $\xi$, based on the uncertainty principle
in uniform matter, leads to
$\xi\sim\frac{\hbar v_{F}}{\pi\Delta}$
with $v_{F}$ being the Fermi velocity[1]. If one considers
saturated nuclear matter as a simplification of finite nuclei,
and adopts the typical value of the
pairing gap $\Delta\approx 12/\sqrt{A}\sim 1$ MeV appropriate
for heavy nuclei, the estimate gives $\xi\sim 20$ fm
which is much larger[2, 3] than the radius of nuclei $R\approx 1.2A^{1/3}\sim 3-7$ fm
or interparticle distance $\sim 2.5$ fm in saturated matter. However, if one considers
extreme situations, such as dilute neutron matter and exotic nuclei with large neutron
excess, there appear new features of the
nuclear pairing that can be related to the spatial structure of the
Cooper pair.
It is the aim of this article to illustrate it
using a few examples.
2 Dilute neutron matter
The superfluidity in neutron matter is density
dependent[4, 6, 5].
The pairing gap can be obtained by solving the BCS equations for the bare nuclear force in
the ${}^{1}S$ channel at each neutron density $\rho=k_{F}^{3}/3\pi^{2}$ or the Fermi
momentum $k_{F}$. The gap is small $\Delta\ll 1$ MeV at $k_{F}=1.36$ fm${}^{-1}$
($\rho/\rho_{0}=1$, the neutron density at saturation $\rho_{0}=0.08$ fm${}^{-3}$).
With decreasing the density it first increases, reaching
the maximum $\Delta\approx 3$ MeV around $k_{F}\approx 0.8$ fm${}^{-1}$
($\rho/\rho_{0}\approx 0.2$),
then decreases and approaches to zero
at the low-density limit. Other many-body medium effects
which are beyond the BCS approximation reduce
the gap, but the predictions vary depending on the
theoretical methods[6, 8, 7, 9]. Recent ab initio Monte Carlo calculations[11, 10, 12], on the
other hand, predict rather modest reduction by less than 50%, and
the qualitative features of the density dependence is kept. Having these
reservations in mind, let us consider the structure of the
neutron Cooper pair in the BCS approximation.[13]
The Cooper pair
wave function can be defined, apart from the normalization, as
an expectation value of the pair operator with respect to the BCS state:
$$\Psi_{pair}(\mbox{\boldmath$r$}_{1},\mbox{\boldmath$r$}_{2})=\left<\psi(\mbox{%
\boldmath$r$}_{1}\uparrow)\psi(\mbox{\boldmath$r$}_{2}\downarrow)\right>=\sum_%
{\mbox{\boldmath$k$}}u_{k}v_{k}e^{i\mbox{\boldmath$k$}\cdot\mbox{\boldmath$r$}}.$$
(1)
It is a function of the relative coordinate $\mbox{\boldmath$r$}=\mbox{\boldmath$r$}_{2}-\mbox{\boldmath$r$}_{1}$
of the two neutrons, and in the momentum space it
is a product of the $u$ and $v$ factors.
Examples of the Cooper pair wave functions are shown in Fig. 1
for two different densities[13]. The wave function exhibits
an oscillatory behavior characterized by the Fermi wave length $2\pi/k_{F}$ and
an overall decay profile whose asymptotic form is
exponential $\sim\exp(-r\Delta/\hbar v_{F})$ (
for large relative distance $r=|\mbox{\boldmath$r$}_{2}-\mbox{\boldmath$r$}_{1}|$) whose length scale
is nothing but the coherence length, or the size of the Cooper paper[1].
More precisely, the coherence length can be calculated as the rms radius of the
Cooper pair
$\xi=\sqrt{\left<r^{2}\right>}$ with
$\left<r^{2}\right>=\int d\mbox{\boldmath$r$}r^{2}|\Psi_{pair}(\mbox{\boldmath$%
r$})|^{2}/\int d\mbox{\boldmath$r$}|\Psi_{pair}(\mbox{\boldmath$r$})|^{2}$.
An interesting feature of the neutron Cooper pair in superfluid neutron matter is that its size
also varies significantly with changing the neutron density (See Fig.1(a)). From a very large value
$\xi=46$ fm at $\rho/\rho_{0}=1$, the coherence length $\xi$ decreases sharply with decreasing
the density. The coherence length takes the smallest values $\xi=5-8$ fm for a rather
wide range of the density $\rho/\rho_{0}=0.2-10^{-2}$, and it increases gradually with decreasing
the density.
The Cooper pair wave function at densities where the coherence length is the smallest
is very different from that of of the electron Cooper pair in the traditional metal superconductors.
An example is shown in Fig.1 (c), which is for $\rho/\rho_{0}=1/8$ where
the coherence length $\xi=4.9$ fm is close to the minimum value.
It is seen that
the oscillatory behavior
is strongly suppressed. The probability distribution is
concentrated ($\sim 80\%$) at small relative distances
within the first node $r<\pi/k_{F}\approx 4.5$ fm, and the probability at the second
and third bumps is very small.
This is because
the size of the Cooper pair ($\xi=4.9$ fm) is almost equal to the position of the first node
$\pi/k_{F}$ which is nothing but
the average interparticle distance
$d=\rho^{-1/3}\approx\pi/k_{F}$. The size of the Cooper pair is ”small” in this sense.
This is quite contrasting to the metal superconducters where the Cooper pair size $\xi$
is thousands times larger than the average interparticle distance $d$.
The situation of the ”small” Cooper pair $\xi\mathop{\vbox{
\offinterlineskip\hbox{$<$}\hbox to 7.499886pt{\hss\hbox{$\sim$}\hss}}}d$ is seen in a wide interval of
densities $\rho/\rho_{0}=10^{-4}-10^{-1}$ (Fig.1(a)). The shape of the Cooper
pair wave function at these densities is similar to that of Fig.1(c), and the probability is
even more concentrated in the first bump although the absolute size is larger
at very low densities $\rho/\rho_{0}\approx 10^{-4}-10^{-2}$.
It is noted that the Cooper pair at moderate low densities $\rho/\rho_{0}\approx 10^{-1}-0.5$
exhibits also the strong spatial correlation at small relative distances.
The wave function at $\rho/\rho_{0}=0.5$ is shown in Fig.1(b). In this case
the calculated coherence length $\xi=11$ fm is
a few times larger than the average interparticle distance $d=2.8$ fm.
Nevertheless the concentration of the
probability within the relative distance $r\mathop{\vbox{
\offinterlineskip\hbox{$<$}\hbox to 7.499886pt{\hss\hbox{$\sim$}\hss}}}3$ fm (in the first bump) is
signfinicant, and the probability in $r<3$ fm
reaches as large as $\sim 50\%$.
The situation of the
small Cooper pair $\xi/d\mathop{\vbox{
\offinterlineskip\hbox{$<$}\hbox to 7.499886pt{\hss\hbox{$\sim$}\hss}}}1$ is related to the so-called BCS-BEC
crossover phenomenon[15, 16, 17, 18, 19],
which has been
discussed intensively in ultra-cold Fermi atom gas
in a trap[20, 21]. It is a phenomenon which can occur generally in any kind of many-Fermion
superfuluid systems by changing the strength of the interparticle attractive force or the density.
In a situation of the weak interaction, which the original BCS theory has dealt with,
the bound pair (the Cooper pair) can be formed only in the medium.
However, if the interaction is as strong as to form a bound
pair (a composite boson) even in the free space, the condensed phase is more
close to a condensate of the composite bosons, i.e. the Bose-Einstein condensate (BEC).
The BCS-BEC crossover is characterized by
the ratio $\xi/d$ of the coherence length and the average interparticle distance
and the ratio $\Delta/e_{F}$ of the pairing gap and the Fermi energy.
The weak-coupling BCS and the BEC limits correspond to $\xi/d\gg 1,\ \Delta/e_{F}\ll 1$
and $\xi/d\ll 1,\ \Delta/e_{F}\gg 1$, respectively while the region of
the crossover may be related to $0.2\mathop{\vbox{
\offinterlineskip\hbox{$<$}\hbox to 7.499886pt{\hss\hbox{$\sim$}\hss}}}\xi/d%
\mathop{\vbox{
\offinterlineskip\hbox{$<$}\hbox to 7.499886pt{\hss\hbox{$\sim$}\hss}}}1.2$ and $0.2\mathop{\vbox{
\offinterlineskip\hbox{$<$}\hbox to 7.499886pt{\hss\hbox{$\sim$}\hss}}}\Delta/%
e_{F}\mathop{\vbox{
\offinterlineskip\hbox{$<$}\hbox to 7.499886pt{\hss\hbox{$\sim$}\hss}}}1.3$.
[17, 18, 19]
At the midway of the crossover, called the
unitarity limit, the interaction strength is on the threshold to
form the isolated two-particle bound state, and the values are $\xi/d=0.36,\Delta/e_{F}=0.69$.
In the BCS calculation discussed above[13], small $\xi/d$ ratio $0.7-1.2$ and large
$\Delta/e_{F}$ ratio $0.2-0.4$ is realized at $\rho/\rho_{0}\sim 10^{-4}-10^{-1}$.
(Note that also in an ab initio calculation[11], the large gap ratio
$\Delta/e_{F}\sim 0.2-0.3$ is obtained in approximately the same
but slightly small density region.) We can regard dilute neutron matter in
the wide low-density interval
$\rho/\rho_{0}=10^{-4}-10^{-1}$ (or in slightly narrower interval)
as being in the crossover region. We note here that the nuclear force in the ${}^{1}S$ channel
has a large scattering length $a=-18$ fm,
indicating that the interaction strength is very close to the threshold to
form a two-neutron bound state. The small Cooper pair
$\xi/d\mathop{\vbox{
\offinterlineskip\hbox{$<$}\hbox to 7.499886pt{\hss\hbox{$\sim$}\hss}}}1$ at low densities originates from the nature of the nuclear force.
3 Cooper pair in neutron-rich nuclei
Let us consider the spatial structure of the Cooper pair in
finite nuclei.
The spatial structure of the correlated two neutrons has been discussed
intensively
for two neutrons in the light two-neutron halo nuclei ${}^{11}$Li and ${}^{6}$He
in (inert or active) core plus two neutron models
[23, 22, 24, 25, 26, 27, 28, 29, 30].
A common prediction is
that the valence halo neutrons exhibit a spatial
correlation favoring the ’di-neutron’ configuration with two neutrons coupled at small relative
distances. The spatial correlation is also discussed in stable heavy nuclei with
closed-shell core plus two neutrons, e.g.
${}^{206,210}$Pb, by means of shell model approaches.[31, 32, 33, 34, 35]
One can generalize these findings by using the Hartree-Fock-Bogoliubov (HFB) method,
which can be applied to a wide class of open shell nuclei including isotopes
very close to the drip-line and also to non-uniform matter.
Let us start defining the wave function of the Cooper pair in finite nuclei. It may be given by
$$\Psi_{pair}(\mbox{\boldmath$r$}_{1},\mbox{\boldmath$r$}_{2})=\left<\Phi_{A-2}|%
\psi(\mbox{\boldmath$r$}_{1}\uparrow)\psi(\mbox{\boldmath$r$}_{2}\downarrow)|%
\Phi_{A}\right>$$
(2)
using the pair correlated ground states $\Phi_{A}$ and $\Phi_{A-2}$. This
represents the probability amplitude of removing two neutrons (positioned at
$\mbox{\boldmath$r$}_{1}$ and $\mbox{\boldmath$r$}_{2}$) from the ground state $\Phi_{A}$, and
leaving the remaining system in the ground state $\Phi_{A-2}$.
Provided that the ground state is described
within the HFB framework, where the ground states with different
nucleon numbers are represented by a single HFB state $\Phi_{{\rm HFB}}$,
the definition Eq.(2) can be replaced with the expectation value
as in Eq.(1). Then, since the HFB state
is a generalized Slater determinant consisting of the Bogolviubov
quasiparticle states, this quantity is evaluated[36]
as a sum over all quasiparticle states $i$
$$\Psi_{pair}(\mbox{\boldmath$r$}_{1},\mbox{\boldmath$r$}_{2})=\left<\Phi_{{\rm
HFB%
}}|\psi(\mbox{\boldmath$r$}_{1}\uparrow)\psi(\mbox{\boldmath$r$}_{2}\downarrow%
)|\Phi_{{\rm HFB}}\right>=\sum_{i}\varphi_{i}^{(1)}(\mbox{\boldmath$r$}_{1}%
\uparrow)\varphi_{i}^{(2)*}(\mbox{\boldmath$r$}_{2}\downarrow)$$
(3)
using the first and the second components of
the quasiparticle wave function
$\phi_{i}(\mbox{\boldmath$r$}\sigma)=(\varphi_{i}^{(1)}(\mbox{\boldmath$r$}%
\sigma),\varphi_{i}^{(2)}(\mbox{\boldmath$r$}\sigma))$.
In the following we show the results of our HFB calculation, which adopts
the Skryme functional and the density-dependent contact
interaction as a phenomenological pairing force[38, 37]. The parameter set of the
pairing interaction is such that it reproduces the scattering length $a=-18$ fm
in the low-density limit, and reproduces the average pairing gap in known nuclei[38, 37].
An example calculated for ${}^{142}$Sn is shown in Fig.2.
Here one neutron is
fixed at the position slightly outside the nucleus $r_{1}=7$ fm
and the probability distribution $|\Psi_{pair}(\mbox{\boldmath$r$}_{1},\mbox{\boldmath$r$}_{2})|^{2}$ is
plotted as a function of $\mbox{\boldmath$r$}_{2}$. It shows that
the second neutron has a large probability ($\sim 50\%$) to be correlated at small
relative distances
$|\mbox{\boldmath$r$}_{1}-\mbox{\boldmath$r$}_{2}|\mathop{\vbox{
\offinterlineskip\hbox{$<$}\hbox to 7.499886pt{\hss\hbox{$\sim$}\hss}}}3$ fm to the partner neutron.
The spatial correlation seen here is generic in a sense that it is seen
systematically in Ca, Ni, and Sn isotopes including both stable and neutron-rich nuclei[36].
The strong spatial correlation is also seen in other HFB calculations which adopt the
finite-range Gogny force as the effective pairing force.[39, 40]
I emphasize here that a large single-particle space is necessary
in describing the spatial correlation[36].
In order to describe the correlation with the length scale
$D\sim 3$ fm, the single-particle basis needs to cover a momentum range up to
$p_{max}\sim h/D$, which corresponds to a maximal energy
$e_{max}\sim p_{max}^{2}/2m\sim 80$ MeV, or a maximal angular momentum
$l_{max}\sim Rp_{max}\sim 10\hbar$ (for the nuclear radius $R\sim 5$ fm).
This is demonstrated in Fig.2(right), where the summation over the quasiparticle
states $i$ in Eq.(3) is truncated by introducing a cut-off with respect to
the orbital angular momentum $l$. Single-particle orbits with large angular
momentum up to $l_{max}\sim 10$ have sizable contributions. Note that
in ${}^{142}$Sn with $N=92$ the Fermi energy is around the
$3p_{3/2}$ orbit, and the maximal orbital angular momentum of the orbits
occupied in the independent particle limit is $l=5$. The single-particle states
with $l=5-10$ lie high above
the Fermi energy.
If one uses the harmonic
oscillator basis, it should include $\sim$ 10 oscillator quanta. In fact, all the
HFB calculations[36, 39, 40] where the strong spatial correlation in the Cooper pair
wave functions is demonstrated adopt such a large single-particle space.
Equivalently, a small single-particle space is
insufficient. If
we restrict ourselves to a single-$j$ shell $(nlj)$, i.e.,
the sum in Eq.(3) is restricted to the magnetic substates of the orbit $(nlj)$,
we obtain the angular correlation[41, 42] $P_{l}(\theta_{12})$
for small relative angles $\theta_{12}\mathop{\vbox{
\offinterlineskip\hbox{$<$}\hbox to 7.499886pt{\hss\hbox{$\sim$}\hss}}}1/l$, but the correlation with respect to
the radial direction is not produced. Inclusion of all the orbits
in one oscillator shell still has deficiency[39, 40]. The Cooper pair wave function in
this case exhibits an artificial symmetry
$\Psi_{pair}(\mbox{\boldmath$r$}_{1},\mbox{\boldmath$r$}_{2})=\pm\Psi_{pair}(-%
\mbox{\boldmath$r$}_{1},\mbox{\boldmath$r$}_{2})$ because of
the common single-particle parity, and the probability appears not only
around $\mbox{\boldmath$r$}_{2}\sim\mbox{\boldmath$r$}_{1}$, but also around the mirror reflected
position $\mbox{\boldmath$r$}_{2}\sim-\mbox{\boldmath$r$}_{1}$.
The spatial correlation of neutron Cooper pairs plays an important role if
we consider neutron-rich nuclei with small neutron separation energy. Nuclei of
this kind often accompany low-density distribution of neutrons, called skin or
halo, extending from the nuclear surface toward the outside.
Figure 3(a) is an example of the pair potential $\Delta(r)$ for the very neutron rich
nucleus ${}^{142}$Sn obtained in the same Skyrme-HFB calculation as in Fig.2. The
pair potential $\Delta(r)$
exhibits significant enhancement around $r\sim 5-8$ fm, which is slightly
outside the nuclear surface (the corresponding neutron density there is
about 1/2-1/10 of the central density). The pair potential decreases rather
slowly with moving outside the surface region,
and it is about to diminish only at very large distances $r\mathop{\vbox{
\offinterlineskip\hbox{$>$}\hbox to 7.499886pt{\hss\hbox{$\sim$}\hss}}}12$ fm.
It is much more extended than the neutron density. Furthermore
the spatial correlation persists in this low density region as shown
in Fig.3(b). We note that the spatial correlation is present also in
stable open shell nuclei[36, 39, 40], and it is enhanced around
the nuclear surface. However the nucleons (and hence the Cooper pairs) do not
penetrate far outside the surface in stable isotopes (cf bottom panel of Fig.3(b)).
The pair correlations
in the dilute surrounding is a unique feature of weakly bound nuclei.
4 Probing the spatially correlated Cooper pair
4.1 Soft modes
If spatially correlated di-neutrons exist in nuclei, especially in the low-density skin/halo
region, there may emerge new modes of excitation
reflecting the motion of di-neutron(s). This simple idea[22, 24] has been a focus of theoretical
and experimental studies of the soft dipole excitation in two-neutron
halo nuclei. Although the reality is not that simple,
the core+n+n models[27, 30, 26] of ${}^{11}$Li explain
the observed large E1 strength of the soft dipole excitation[43]
in terms of the pairing and the spatial correlation of the valence halo neutrons.
It is interesting to explore possibility of similar excitation modes in heavier mass neutron-rich
nuclei, where more than two weakly bound neutrons contribute to the pair correlation.
A useful scheme to describe excitation modes built on the
pair correlated ground state is the quasiparticle random phase approximation (QRPA).
Let us take the formulation based on the same Skyrme-HFB model that is used for the
description of the ground state.[44, 37, 45] Having a QRPA
excited state $\left|n,LM\right\rangle$, one can calculate the two-particle amplitude
$\left<n,LM|\psi^{\dagger}(\mbox{\boldmath$r$}_{1}\uparrow)\psi^{\dagger}(\mbox%
{\boldmath$r$}_{2}\downarrow)|0_{gs}\right>$
which tells us how two particles move in the excited state $\left|n,LM\right>$
in reference to the ground state (of the $N-2$ system). For simplicity
let us look at the zero-range part at $\mbox{\boldmath$r$}_{1}=\mbox{\boldmath$r$}_{2}$ of the amplitude:
$$P_{n}^{pair}(\mbox{\boldmath$r$})=\left<n,LM|\psi^{\dagger}(\mbox{\boldmath$r$%
}\uparrow)\psi^{\dagger}(\mbox{\boldmath$r$}\downarrow)|0_{gs}\right>,$$
(4)
which is called the pair transition density.
Figure 4 is an example of soft dipole excitation
which suggests motion of the spatially correlated di-neutrons[44].
The soft dipole excitation is seen here as a bump of the E1 strength which
lies just above the neutron separation energies
($S_{1n},S_{2n}=1.9,2.4$ MeV). In neutron-rich Ni isotopes beyond the $N=50$ shell
closure
both of the one- and the two-neutron separation energies are calculated to be very low
$S_{n},S_{2n}\approx 1-3$ MeV. In such weakly bound nuclei, the low-lying
dipole modes appear just above the separation energy since
it is possible to excite a bound neutron to unbound orbits in the continuum,
letting the neutron escape from the nucleus. If the pair correlation is
taken into account, however, the mode is dominated by the pair motion
rather than by a simple particle-hole (or independent two-quasiparticle excitation).
Consequently the pair transition density
$P_{n}^{pair}(\mbox{\boldmath$r$})$ has larger amplitude, especially for $r>R_{surf}$, as
seen in Fig.4(b).
It is not explicit in this figure whether the neutron pair in the excited state is
spatially correlated, but we can infer it from the observation that
a large number of orbital angular momenta $l$ reaching more than 10$\hbar$
have significant and coherent contributions to the pair transition density. As we discussed
above (cf. Fig.2), large $l$ implies a spatial correlation at small distances
between the two neutrons.
A similar mode of excitation having the character of di-neutron motion is
predicted also in the octupole response in the same istopes ${}^{>80}$Ni beyond $N=50$.[44]
It is a smooth distribution of neutron strength
lying just above the threshold energy (like the soft dipole mode), and it coexists
from
the octupole surface vibrational mode of the isoscalar character seen in many of
stable nuclei.
In contrast to the light two-neutron halo nuclei, the presence of the spatial correlation does not
influence strongly the E1 strength of soft dipole excitation
in heavy neutron-rich nuclei such as ${}^{84}$Ni. We need
other probes which are directly connected to the pair transition density.
Since the soft dipole excitation in ${}^{11}$Li and
in ${}^{>80}$Ni is located above the two-neutron separation energy, one can expect that
momentum distribution/correlation of
two neutrons emitted from the
soft mode may carry information on the spatial correlation of the neutron pair.
Quantitative theoretical description of the two-neutron correlation is achieved only for the
core+n+n models[46, 47] for ${}^{11}$Li and ${}^{6}$He, and experimental information
is very scarce so far[49, 48]. It is possible to describe the
two-neutron correlation in the continuum also in the framework of the QRPA since
the information on the directions of two neutrons are contained
in the pair transition density
$\left<n,LM|\psi^{\dagger}(\mbox{\boldmath$r$}_{1}\uparrow)\psi^{\dagger}(\mbox%
{\boldmath$r$}_{2}\downarrow)|0_{gs}\right>$
especially in its asymptotic form at $|\mbox{\boldmath$r$}_{1}|,|\mbox{\boldmath$r$}_{2}|\rightarrow\infty$.
It is an interesting future subject to study
in heavier neutron-rich nuclei such as ${}^{>80}$Ni using the HFB+QRPA formalism.
4.2 Two-neutron transfer
The two-neutron transfer reactions such as (p,t)
and (t,p) are known as a good probe
to the pair correlation in the ground state[50, 51, 2, 3]. More precisely
it can be regarded as a probe of the Cooper pair
wave function, especially its behavior at small relative distances
between the paired neutrons. Consider the (p,t) reaction
populating the ground state of the neighboring
$N-2$ nucleus in the single-step DWBA and the zero-range approximation. Then
the transition matrix elements involves the form factor[51, 52]
$$F(\mbox{\boldmath$R$})=\int d\mbox{\boldmath$r$}\left<0_{gs,N-2}|\psi(\mbox{%
\boldmath$R$}+\mbox{\boldmath$r$}/2\uparrow)\psi(\mbox{\boldmath$R$}-\mbox{%
\boldmath$r$}/2\downarrow)|0_{gs,N}\right>\phi(\mbox{\boldmath$r$})$$
which is the convolution of
the Cooper pair wave function $\Psi_{pair}(\mbox{\boldmath$R$}+\mbox{\boldmath$r$}/2,\mbox{\boldmath$R$}-%
\mbox{\boldmath$r$}/2)$
with the two-particle wave function $\phi(\mbox{\boldmath$r$})$ in the triton.
Noting the small radius of the triton $\sim 2$ fm, we see immediately
the form factor picks up the correlation
at small relative distances in the Cooper pair wave functions.
It is then not a very bad approximation to utilize
the Cooper pair wave function at zero relative distance $\mbox{\boldmath$r$}=0$, i.e.
$\Psi_{pair}(\mbox{\boldmath$R$},\mbox{\boldmath$R$})=\left<\psi(\mbox{%
\boldmath$R$}\uparrow)\psi(\mbox{\boldmath$R$}\downarrow)\right>\equiv P_{pair%
}(\mbox{\boldmath$R$})$
as a substitute of the form factor assuming $F(\mbox{\boldmath$R$})\propto P_{pair}(\mbox{\boldmath$R$})$.
$P_{pair}(\mbox{\boldmath$R$})$ is nothing but the pair density $\tilde{\rho}(\mbox{\boldmath$R$})$ implemented automatically in
the Skyrme-HFB model using the pairing force of the contact type[53].
An example[45] of the calculated pair transition density $P_{pair}(\mbox{\boldmath$R$})$ is shown in Fig.5(a)
for Sn isotopes covering from stable isotopes to very neutron-rich ${}^{150}$Sn.
It is seen that the radial dependence of
$P_{pair}(\mbox{\boldmath$R$})$ suddenly changes at the $N=82$ shell closure (at ${}^{132}$Sn).
In neutron-rich isotopes beyond $N=82$, the amplitude extends far outside the nuclear
surface $r>R_{surf}+3$ fm ($\mathop{\vbox{
\offinterlineskip\hbox{$>$}\hbox to 7.499886pt{\hss\hbox{$\sim$}\hss}}}9$ fm). This happens because neutron single-particle
orbits above the $N=82$ shell gap are bound only weakly, and the weakly
bound neutrons have density distributions extended far outside the nuclear surface.
(The one-neutron
separation energy is the order of $\sim 2-3$ MeV for $A>132$, while it is more than
8 MeV in isotopes with $A\leq 132$.)
Consequently both the pair potential $\Delta(r)$ and the Cooper pair
wave function keep non-negligible magnitude even far outside (See also Fig.3).
As seen in the figure the amplitude $P_{pair}(\mbox{\boldmath$R$})$ extends up to
$r\sim 12$ fm for the isotopes $A>140$.
The above observation leads to an expectation that the (p,t) and (t,p) cross sections may
be enhanced considerably as the neutron separation energy becomes small[53].
An estimate of the isotopic trend, much simpler than the DWBA calculation,
is shown in Fig.5(b). Here is plotted the
’strength’ which is defined by
$B(P0)=\left|\int d\mbox{\boldmath$R$}P_{pair}(\mbox{\boldmath$R$})\right|^{2}.$
It is illuminating to compare it with the isotopic trends of
the pairing gap $\Delta$ squared ($\Delta$ being an average value of
the pair potential $\Delta(r)$). If the pair potential and the pair
transition density are confined in the nuclear volume, a proportionality relation
$B(P0)\propto\Delta^{2}$ is expected[50] in analogy with the
$B(E2)$ of the deformed rotor since the pair gap is a deformation parameter[51, 2].
We see in Fig.5(b) that the proportionality $B(P0)\propto\Delta^{2}$ valid
for $100<A<132$ is violated
for $A>132$ and especially $A>140$, where the strength $B(P0)$ significantly
increases.
The two-neutron transfer reaction on the neutron-rich isotopes (e.g.
the Sn isotopes with $A>132$) thus provides us a tool to probe the Cooper pair
wave function in the low-density region far outside the nuclear surface.
It is predicted[45] also that the isotopes ${}^{134-140}$Sn with $A=134-140$ exhibit
a precursor phenomenon, i.e., an anomalously large two-neutron transfer strength
of (t,p) type for the transitions to the excited $0^{+}$ states (Fig.5(b)).
Recently two-neutron transfer experiment on the halo nucleus ${}^{11}$Li has become available,
and the crucial role of the pair correlation is demonstrated.[54, 55]
We wait for a future experiment using the radioactive beams of neutron-rich
Sn isotopes with $A>132$.
This work was supported by
the Grant-in-Aid for Scientific Research
(Nos. 21105507, 21340073 and 23540294) from the Japan
Society for the Promotion of Science.
References
1.
J. Bardeen, L. N. Cooper, and
J. R. Schrieffer, Phys. Rev. 108, 1175 (1957).
P. G. de Gennes, Superconductivity of Metals and Alloys,
Benjamin (1966).
M. Tinkham, Introduction to Superconductivity,
McGraw-Hill (1975).
2.
A. Bohr and B. R. Mottelson, Nuclear
Structure, vol. II, Benjamin (1975).
3.
D. M. Brink and R. A. Broglia, Nuclear Superfluidity: Pairing in Finite Systems,
Cambridge University Press, Cambridge (2005).
4.
T. Takatsuka and R. Tamagaki, Prog. Theor. Phys. Suppl.
No. 112, 27 (1993).
5.
D. J. Dean and M. Hjorth-Jensen,
Rev. Mod. Phys. 75, 607 (2003).
6.
U. Lombardo and H.-J. Schulze, In
Lecture Notes in Physics, Vol.578, p.30, Springer (2001).
7.
A. Schwenk, B. Friman, G. E. Brown,
Nucl. Phys. A713, 191 (2003).
8.
L. G. Cao, U. Lombardo, and P. Schuck,
Phys. Rev. C 74, 064301 (2006).
9.
A. Fabrocini, S. Fantoni, A. Y. Illarionov, and K. E. Schmidt,
Phys. Rev. Lett. 95, 192501 (2005).
10.
T. Abe and R. Seki,
Phys. Rev. C 79, 054002 (2009).
11.
A. Gezerlis and J. Carlson,
Phys. Rev. C 81, 025803 (2010).
12.
S. Gandolfi, A. Yu. Illarionov, F. Pederiva, K. E. Schmidt, S. Fantoni,
Phys. Rev. C 80, 045802 (2009).
13.
M. Matsuo,
Phys. Rev. C 73, 044309 (2006).
14.
R. Tamagaki,
Prog. Theor. Phys. 39, 91 (1968).
15.
A. J. Leggett,
In eds. A. Pekalski and R. Przystawa
Modern Trends in the Theory of
Condensed Matter,
Lecture Note in Physics 115,
Springer-Verlag, Berlin, (1980);
A. J. Leggett, J. de Phys. 41, C7-19 (1980).
16.
P. Nozières and S. Schmitt-Rink,
J. Low Temp. Phys.
59, 195 (1985).
17.
C. A. R. Sá de Melo, M. Randeria, and J. R. Engelbrecht,
Phys. Rev. Lett. 71, 3202 (1993).
18.
J. R. Engelbrecht, M. Randeria, and C. A. R. Sá de Melo,
Phys. Rev. B 55, 15153 (1997).
19.
M. Randeria, In eds. A. Griffin, D. Snoke, and S. Stringari,
Bose-Einstein Condensation,
Cambridge Univ. Press, Cambridge, (1995).
20.
C. A. Regal, M. Greiner, and D. S. Jin,
Phys. Rev. Lett. 92, 040403 (2004).
21.
S. Giorgini, L. V. Pitaevski, S. Stringari,
Rev. Mod. Phys. 80, 1215 (2008).
22.
P. G. Hansen and B. Jonson,
Europhys. Lett. 4, 409 (1987).
23.
G. F. Bertsch and H. Esbensen, Ann. Phys. 209, 327 (1991);
H. Esbensen and G. F. Bertsch, Nucl. Phys. A542, 310 (1992).
24.
K. Ikeda, Nucl. Phys. A538, 355c (1992).
25.
M. V. Zhukov, B. V. Danilin, D. V. Fedorov, J. M. Bang,
I. J. Thompson, and J. S. Vaagen,
Phys. Rep. 231, 151 (1993).
26.
F. Barranco, P. F. Bortignon, R. A. Broglia, G. Coló, and E. Vigezzi,
Eur. Phys. J. A11, 385 (2001).
27.
K. Hagino and H. Sagawa,
Phys. Rev. C 72, 044321 (2005).
28.
K. Hagino, H. Sagawa, J. Carbonell, and P. Schuck,
Phys. Rev. Lett. 99, 022506 (2007).
29.
K. Hagino, H. Sagawa, and P. Schuck,
J. Phys. G 37, 064040 (2010).
30.
T. Myo, Y. Kikuchi, K. Katō, H. Toki and K. Ikeda,
Prog. Theor. Phys.
119, 561 (2008).
31.
G. F. Bertsch, R. A. Broglia, and C. Riedel,
Nucl. Phys. A91, 123 (1967).
32.
R. H. Ibarra, N. Austern, M. Vallieres, and D. H. Feng,
Nucl. Phys. A288, 397 (1977).
33.
F. A. Janouch and R. J. Liotta,
Phys. Rev. C 27, 896 (1983).
34.
F. Catara, A. Insolia, E. Maglione, and A. Vitturi,
Phys. Rev. C 29, 1091 (1984).
35.
L. Ferreira, R. Liotta, C. H. Dasso, R. A. Broglia, and A. Winther,
Nucl. Phys. A426, 276 (1984).
36.
M. Matsuo, K. Mizuyama, and Y. Serizawa,
Phys. Rev. C 71, 064326 (2005).
37.
M. Matsuo and Y. Serizawa,
Phys. Rev. C 82, 024318 (2010).
38.
M. Matsuo, Y. Serizawa, and K. Mizuyama,
Nucl. Phys. A788, 307c (2007).
39.
N. Pillet, N. Sandulescu, and P. Schuck,
Phys. Rev. C 76, 024310 (2007).
40.
N. Pillet, N. Sandulescu, P. Schuck, and J. -F. Berger,
Phys. Rev. C 81, 034307 (2010).
41.
B. Mottelson,
In eds.
H. Nifenecker, J.-P. Blaizot, G. F. Bertsch, W. Weise,
and F. David,
Trends in Nuclear Physics, 100 Years
Later, (Les Houches, Session LXVI, 1996), Elsevier, (1998).
42.
P. Ring and P. Schuck,
The Nuclear Many-Body Problem,
Springer-Verlag, (1980).
43.
T. Nakamura et al.,
Phys. Rev. Lett. 96, 252502 (2006).
44.
Y. Serizawa and M. Matsuo,
Prog. Theor. Phys. 121, 97 (2009).
45.
H. Shimoyama and M. Matsuo,
Phys. Rev. C
84, 044317 (2011).
46.
K. Hagino, H. Sagawa, T. Nakamura, and S. Shimoura,
Phys. Rev. C 80, 032301(R) (2009).
47.
Y. Kikuchi, K. Kato, T. Myo, M. Takashina, and K. Ikeda,
Phys. Rev. C 81, 044308 (2010).
48.
T. Nakamura, private communication
49.
L. V. Chulkov, et al.,
Nucl. Phys. A759, 23 (2005).
50.
S. Yoshida,
Nucl. Phys. 33, 685 (1962).
51.
R. A. Broglia, O. Hansen, and C. Riedel, In eds.
M. Baranger and E. Vogt,
Advances in Nuclear Physics vol.6, pp.287-457,
Plenum, New York, (1973)
52.
N. K. Glendenning, Direct Nuclear Reactions,
Academic Press, (1983).
53.
J. Dobaczewski, W. Nazarewicz, T. R. Werner, J. F. Berger, C. R. Chinn,
and J. Dechargé,
Phys. Rev. C 53, 2809 (1996).
54.
I. Tanihata et al.,
Phys. Rev. Lett. 100, 192502 (2008).
55.
G. Potel, F. Brancco, E. Vigezzi, and R. A. Broglia,
Phys. Rev. Lett. 105, 172502 (2010). |
Characterization of trace spaces on regular trees
via dyadic norms
00footnotetext:
$2010$ Mathematics Subject classfication: 46E35, 30L05
Key words and phases: regular tree, trace space, dyadic norm, Orlicz-Sobolev space
The author has been supported by the Academy of Finland grant 323960.
Zhuang Wang
( )
Abstract
In this paper, we study the traces of Orlicz-Sobolev spaces on a regular rooted tree.
After giving a dyadic decomposition of the boundary of the regular tree, we present a characterization on the trace spaces of those first order Orlicz-Sobolev spaces whose Young function is of the form $t^{p}\log^{\lambda}(e+t)$, based on integral averages on dyadic elements of the dyadic decomposition.
1 Introduction
The problem of the characterization of the trace spaces (on the boundary of a domain) of Sobolev spaces has a long history. It was first studied in the Euclidean setting by Gagliardo [12], who proved that the trace operator $T:W^{1,p}({\mathbb{R}}^{n+1}_{+})\rightarrow B^{1-1/p}_{p,p}({\mathbb{R}}^{n})$, where $B^{1-1/p}_{p,p}({\mathbb{R}}^{n})$ stands for the classical Besov space, is linear and bounded for every $p>1$ and that there exists a bounded linear extension operator that acts as a right inverse of $T$. Moreover, he proved that the trace operator $T:W^{1,1}({\mathbb{R}}^{n+1}_{+})\rightarrow L^{1}({\mathbb{R}}^{n})$ is a bounded linear surjective operator with a non-linear right inverse. Peetre [37] showed that one can not find a bounded linear extension operator that acts as a right inverse of $T:W^{1,1}({\mathbb{R}}^{n+1}_{+})\rightarrow L^{1}({\mathbb{R}}^{n})$. We refer to the seminal monographs by Peetre [38] and Triebel [44, 45] for extensive treatments of the Besov spaces and related smoothness spaces.
In potential theory, certain types of Dirichlet problem are guaranteed to have solutions when the boundary data belongs to a trace space corresponding to the Sobolev class on the domain. In the Euclidean setting, we refer to [1, 30, 33, 42, 47, 48] for more information on the traces of (weighted) Sobolev spaces and [9, 10, 11, 28, 35, 8, 29, 36] for results on traces of (weighted) Orlicz-Sobolev spaces.
Over the past two decades, analysis in general metric measure spaces has attracted a lot of attention, e.g., [2, 4, 15, 16, 17, 18, 19]. The trace theory in the metric setting has been under development. Malý [31] proved that the trace space of the Newtonian space $N^{1,p}(\Omega)$ is the Besov space $B^{1-\theta/p}_{p,p}(\partial\Omega)$ provided that $\Omega$ is a John domain for $p>1$ (uniform domain for $p\geq 1$) that admits a $p$-Poincaré inequality and whose boundary $\partial\Omega$ is endowed with a codimensional-$\theta$ Ahlfors regular measure with $\theta<p$. We also refer to the paper [40] for studies on the traces of Hajłasz-Sobolev functions to porous Ahlfors regular closed subsets via a method based on hyperbolic fillings of a metric space, see [6, 43].
The recent paper [3] dealt with geometric analysis on Cantor-type sets which are uniformly perfect totally disconnected metric measure spaces, including various types of Cantor sets. Cantor sets embedded in Euclidean spaces support a fractional Sobolev space theory based on Besov spaces. Indeed, suitable Besov functions on such a set are traces of the classical Sobolev functions on the ambient Euclidean spaces, see Jonsson-Wallin [20, 21]. The paper [3, 24] obtained similar trace and extension theorems for Sobolev and Besov spaces on regular trees and their Cantor-type boundaries. Indeed, for a regular $K$-ary tree $X$ with $K\geq 2$ and its Cantor-type boundary ${{\partial X}}$, if we give the uniformizing metric (see (2.1))
$$d_{X}(x,y)=\int_{[x,y]}e^{-\epsilon|z|}\,d\,|z|$$
and the weighted measure (see (2.2) )
(1.1)
$$d{{\mu_{\lambda}}}(x)=e^{-\beta|x|}(|x|+C)^{\lambda}\,d\,|x|$$
on $X$, then the Besov space ${\mathcal{B}^{\theta,\lambda}_{p}(\partial X)}$ in Definition 2.4 below is exactly the trace of the Newton-Sobolev space $N^{1,p}(X,{{\mu_{\lambda}}})$ defined in Section 2.3, see [24, Theorem 1.1] and [3, Theorem 6.5]. Here the smoothness exponent of the Besov space is
$$\theta=1-\frac{\beta/\epsilon-Q}{p},\ \ 0<\theta<1,$$
where $Q=\log K/\epsilon$ is the Hausdorff dimension of the Cantor-type boundary and $\beta/\epsilon-Q$ is a “codimension” determined by the uniformizing metric $d_{X}$ and the measure $\mu$ on the tree.
In Euclidean spaces, the classical Besov norm is equivalent to a dyadic norm, and the trace spaces of the Sobolev spaces can be characterized by the Besov spaces defined via dyadic norms, see e.g. [23, Theorem 1.1]. Inspired by this, we give a dyadic decomposition of the boundary ${{\partial X}}$ and define a Besov space ${\mathcal{B}^{\theta}_{p}(\partial X)}$ on the boundary ${{\partial X}}$ by using a dyadic norm, see Section 2.4 and Definition 2.5. We show in Proposition 2.7 that the dyadic Besov spaces ${\mathcal{B}^{\theta}_{p}(\partial X)}$ coincide with the Besov space $B^{\theta}_{p,p}({{\partial X}})$ and the Hajłasz-Besov space $N^{\theta}_{p,p}({{\partial X}})$, see Definition 2.3 and Definition 2.6 for definitions of $B^{\theta}_{p,p}({{\partial X}})$ and $N^{\theta}_{p,p}({{\partial X}})$. We refer to [3, 14, 22, 13, 25, 26] for more information about
Besov spaces $B^{\theta}_{p,p}(\cdot)$ and Hajłasz-Besov spaces $N^{\theta}_{p,p}(\cdot)$ on metric measure spaces.
By relying on dyadic norms, we define the Orlicz-Besov space ${\mathcal{B}^{\theta,\lambda_{2}}_{\Phi}(\partial X)}$, $\lambda_{2}\in\mathbb{R}$ for the Young function $\Phi(t)=t^{p}\log^{\lambda_{1}}(e+t)$ with $p>1,\lambda_{1}\in{\mathbb{R}}$ or $p=1,\lambda_{1}\geq 0$, see Definition 2.8. Our first result shows that the Orlicz-Besov space ${\mathcal{B}^{\theta,\lambda_{2}}_{\Phi}(\partial X)}$ is the trace space of the Orlicz-Sobolev space $N^{1,\Phi}(X,\mu_{\lambda_{2}})$ defined in Section 2.3.
Theorem 1.1.
Let $X$ be a $K$-ary tree with $K\geq 2$ and let $\Phi(t)=t^{p}\log^{\lambda_{1}}(e+t)$ with $p>1,\lambda_{1}\in{\mathbb{R}}$ or $p=1,\lambda_{1}\geq 0$. Fix $\lambda_{2}\in{\mathbb{R}}$ and let $\mu_{\lambda_{2}}$ be the weighted measure given by (1.1). Assume that $p>(\beta-\log K)/\epsilon>0$. Then the trace space of $N^{1,\Phi}(X,\mu_{\lambda_{2}})$ is the space ${\mathcal{B}^{\theta,\lambda_{2}}_{\Phi}(\partial X)}$ where $\theta=1-(\beta-\log K)/\epsilon p$.
Here and throughout this paper, for given Banach spaces $\mathbb{X}({{\partial X}})$ and $\mathbb{Y}(X)$, we say that the space $\mathbb{X}({{\partial X}})$ is a trace space of $\mathbb{Y}(X)$ if and only if there is a bounded linear operator $T:\mathbb{Y}(X)\rightarrow\mathbb{X}({{\partial X}})$ and there exists a bounded linear extension operator $E:\mathbb{X}({{\partial X}})\rightarrow\mathbb{Y}(X)$ that acts as a right inverse of $T$, i.e., $T\circ E={\rm Id}\,$ on the space $\mathbb{X}({{\partial X}})$.
Our next result identifies the Orlicz-Besov space ${\mathcal{B}^{\theta,\lambda_{2}}_{\Phi}(\partial X)}$ as the Besov space ${\mathcal{B}^{\theta,\lambda}_{p}(\partial X)}$.
Proposition 1.2.
Let $\lambda,\lambda_{1},\lambda_{2}\in{\mathbb{R}}$.
Let $\Phi(t)=t^{p}\log^{\lambda_{1}}(e+t)$ with $p>1,\lambda_{1}\in{\mathbb{R}}$ or $p=1,\lambda_{1}\geq 0$. Assume that $\lambda_{1}+\lambda_{2}=\lambda$. Then the Banach spaces ${\mathcal{B}^{\theta,\lambda}_{p}(\partial X)}$ and ${\mathcal{B}^{\theta,\lambda_{2}}_{\Phi}(\partial X)}$ coincide, i.e., ${\mathcal{B}^{\theta,\lambda}_{p}(\partial X)}={\mathcal{B}^{\theta,\lambda_{2%
}}_{\Phi}(\partial X)}$.
By combining Theorem 1.1 and Proposition 1.2, we obtain the following result.
Corollary 1.3.
Let $X$ be a $K$-ary tree with $K\geq 2$. Let $\lambda,\lambda_{1},\lambda_{2}\in{\mathbb{R}}$.
Assume that $p>(\beta-\log K)/\epsilon>0$ and let $\theta=1-(\beta-\log K)/\epsilon p$. Let $\Phi(t)=t^{p}\log^{\lambda_{1}}(e+t)$ with $p>1,\lambda_{1}\in{\mathbb{R}}$ or $p=1,\lambda_{1}\geq 0$. Then the Besov-type space ${\mathcal{B}^{\theta,\lambda}_{p}(\partial X)}$ is the trace space of $N^{1,\Phi}(X,\mu_{\lambda_{2}})$ whenever $\lambda_{1}+\lambda_{2}=\lambda$.
When $\lambda_{1}=0$ and $\lambda_{2}=\lambda$, the above result coincides with [24, Theorem 1.1], which states that the Besov-type space ${\mathcal{B}^{\theta,\lambda}_{p}(\partial X)}$ is the trace space of $N^{1,p}(X,\mu_{\lambda})$ for a suitable $\theta$. The above result shows that the Besov-type space ${\mathcal{B}^{\theta,\lambda}_{p}(\partial X)}$ is not only the trace space of $N^{1,p}(X,\mu_{\lambda})$ but actually the trace spaces of all these Orlicz-Sobolev spaces $N^{1,\Phi}(X,\mu_{\lambda_{2}})$ (including $N^{1,p}(X,\mu_{\lambda})$) for suitable $\theta,\lambda_{2}$ and $\Phi$. It maybe worth to point out here that these Orlicz-Sobolev spaces $N^{1,\Phi}(X,\mu_{\lambda_{2}})$ are different from each other.
The paper is organized as follows. In Section 2, we give all the necessary preliminaries. More precisely, we introduce regular trees in Section 2.1 and we consider a doubling property of the measure $\mu$ on a regular tree $X$ and the Ahlfors regularity of its boundary ${{\partial X}}$. The definition of Young functions is given in Section 2.2. We introduce the Newtonian and Orlicz-Sobolev spaces on $X$ and the Besov-type spaces on ${{\partial X}}$ in Section 2.3 and Section 2.4, respectively. In Section 3, we give the proofs of Theorem 1.1 and Proposition 1.2.
In what follows, the letter $C$ denotes a constant that may change at different occurrences. The notation $A\approx B$ means that there is a constant $C$ such that $1/C\cdot A\leq B\leq C\cdot A$. The notation $A\lesssim B$ ($A\gtrsim B$) means that there is a constant $C$ such that $A\leq C\cdot B$ ($A\geq C\cdot B$).
2 Preliminaries
2.1 Regular trees and their boundaries
A graph $G$ is a pair $(V,E)$, where $V$ is a set of vertices and $E$ is a set of edges. We call a pair of vertices $x,y\in V$ neighbors if $x$ is connected to $y$ by an edge. The degree of a vertex is the number of its neighbors. The graph structure gives rise to a natural connectivity structure. A tree $G$ is a connected graph without cycles.
We call a tree $G$ a rooted tree if it has a distinguished vertex called the root, which we will denote by $0$. The neighbors of a vertex $x\in V$ are of two types: the neighbors that are closer to the root are called parents of $x$ and all other neighbors are called children of $x$. Each vertex has a unique parent, except for the root itself that has none.
A $K$-ary tree $G$ is a rooted tree such that each vertex has exactly $K$ children. Then all vertices except the root of $G$ have degree $K+1$, and the root has degree $K$. We say that a tree $G$ is $K$-regular if it is a $K$-ary tree for some $K\geq 1$.
Let $G$ be a $K$-regular tree with a set of vertices $V$ and a set of edges $E$ for some $K\geq 1$. For simplicity of notation, we let $X=V\cup E$ and call it a $K$-regular tree. A $K$-regular tree $X$ is made into a metric graph by considering each edge as a geodesic of length one.
For $x\in X$, let $|x|$ be the distance from the root $0$ to $x$, that is, the length of the geodesic from $0$ to $x$, where the length of every edge is $1$ and we consider each edge to be an isometric copy of the unit interval. The geodesic connecting $x,y\in V$ is denoted by $[x,y]$, and its length is denoted $|x-y|$. If $|x|<|y|$ and $x$ lies on the geodesic connecting $0$ to $y$, we write $x<y$ and call the vertex $y$ a descendant of the vertex $x$. More generally, we write $x\leq y$ if the geodesic from $0$ to $y$ passes through $x$, and in this case $|x-y|=|y|-|x|$.
Let $\epsilon>0$ be fixed. We introduce a uniformizing metric (in the sense of Bonk-Heinonen-Koskela [5], see also [3] ) on $X$ by setting
(2.1)
$$d_{X}(x,y)=\int_{[x,y]}e^{-\epsilon|z|}\,d\,|z|.$$
Here $d\,|z|$ is the measure which gives each edge Lebesgue measure $1$, as we consider each edge to be an isometric copy of the unit interval and the vertices are the end points of this interval. In this metric, $\text{\rm\,diam}X=2/\epsilon$ if $X$ is a $K$-ary tree with $K\geq 2$.
Next we construct the boundary of the regular $K$-ary tree by following the arguments in [3, Section 5]. We define the boundary of a tree $X$, denoted ${{\partial X}}$, by completing $X$ with respect to the metric $d_{X}$. An equivalent construction of ${{\partial X}}$ is as follows. An element $\xi$ in ${{\partial X}}$ is identified with an infinite geodesic in $X$ starting at the root $0$. Then we may denote $\xi=0x_{1}x_{2}\cdots$, where $x_{i}$ is a vertex in $X$ with $|x_{i}|=i$, and $x_{i+1}$ is a child of $x_{i}$. Given two points $\xi,\zeta\in{{\partial X}}$, there is an infinite geodesic $[\xi,\zeta]$ connecting $\xi$ and $\zeta$. Then the distance of $\xi$ and $\zeta$ is the length (with respect to the metric $d_{X}$) of the infinite geodesic $[\xi,\zeta]$. More precisely, if $\xi=0x_{1}x_{2}\cdots$ and $\zeta=0y_{1}y_{2}\cdots$, let $k$ be an integer with $x_{k}=y_{k}$ and $x_{k+1}\not=y_{k+1}$. Then by (2.1)
$$d_{X}(\xi,\zeta)=2\int_{k}^{+\infty}e^{-\epsilon t}\,dt=\frac{2}{\epsilon}e^{-%
\epsilon k}.$$
The restriction of $d_{X}$ to ${{\partial X}}$ is called the visual metric on ${{\partial X}}$ in Bridson-Haefliger [7].
The metric $d_{X}$ is thus defined on $\bar{X}$. To avoid confusion, points in $X$ are denoted by Latin letters such as $x,y$ and $z$, while for points in ${{\partial X}}$ we use Greek letters such as $\xi,\zeta$ and $\omega$. Moreover, balls in $X$ will be denoted $B(x,r)$, while $B(\xi,r)$ stands for a ball in ${{\partial X}}$.
On the regular $K$-ary tree $X$, we use the weighted measure ${{\mu_{\lambda}}}$ introduced in [24, Section 2.2], defined by
(2.2)
$$d{{\mu_{\lambda}}}(x)=e^{-\beta|x|}(|x|+C)^{\lambda}\,d\,|x|,$$
where $\beta>\log K$, $\lambda\in\mathbb{R}$ and $C\geq\max\{2|\lambda|/(\beta-\log K),2(\log 4)/\epsilon\}$. If $\lambda=0$, then
$$d\mu_{0}(x)=e^{-\beta|x|}\,d|x|=d\mu(x),$$
which coincides with the measure used in [3].
The following proposition gives the doubling property of the measure ${{\mu_{\lambda}}}$, see [24, Corollary 2.9].
Proposition 2.1.
For any $\lambda\in\mathbb{R}$,
the measure $\mu_{\lambda}$ is doubling, i.e., ${{\mu_{\lambda}}}(B(x,2r))\lesssim{{\mu_{\lambda}}}(B(x,r))$.
The result in [3, Lemma 5.2] shows that the boundary ${{\partial X}}$ of the regular $K$-ary tree $X$ is Ahlfors regular with the regularity exponent depending only on $K$ and on the metric density exponent $\epsilon$ of the tree.
Proposition 2.2.
The boundary ${{\partial X}}$ is an Ahlfors $Q$-regular space with Hausdorff dimension
$$Q=\frac{\log K}{\epsilon}.$$
Hence we have an Ahlfors $Q$-regular measure $\nu$ on ${{\partial X}}$:
$$\nu(B(\xi,r))\approx r^{Q}=r^{\log K/\epsilon},$$
for any $\xi\in{{\partial X}}$ and $0<r\leq\text{\rm\,diam}{{\partial X}}.$
Throughout the paper we assume that $1\leq p<+\infty$ and that $X$ is a $K$-ary tree with $K\geq 2$.
2.2 Young functions and Orlicz spaces
In the standard definition of an Orlicz space, the function $t^{p}$ of an $L^{p}$-space is replaced with a more general convex function, a Young function. We recall the definition of a Young function. We refer to [46, section 2.2] and [39] for more details about Young functions and we also warn the reader of slight differences between the definitions in various references.
A function $\Phi:[0,\infty)\rightarrow[0,\infty)$ is a Young function if
it is a continuous, increasing and convex function satisfying $\Phi(0)=0$,
$$\lim_{t\rightarrow 0+}\frac{\Phi(t)}{t}=0\ \ \text{and}\ \ \ \lim_{t%
\rightarrow+\infty}\frac{\Phi(t)}{t}=+\infty.$$
A Young function $\Phi$ can be expressed as
$$\Phi(t)=\int_{0}^{t}\phi(s)\,ds,$$
where $\phi:[0,\infty)\rightarrow[0,\infty)$ is an increasing, right-continuous function with $\phi(0)=0$ and $\underset{t\rightarrow+\infty}{\lim}\phi(t)=+\infty.$
A Young function $\Phi$ is said to satisfy the $\Delta_{2}-$condition if there is a constant $C_{\Phi}>0$, called a doubling constant of $\Phi$, such that
$$\Phi(2t)\leq C_{\Phi}\Phi(t),\ \forall\ \ t\geq 0.$$
If Young function $\Phi$ satisfies the $\Delta_{2}-$condition, then for any constant $c>0$, there exist $c_{1},c_{2}>0$ such that
$$c_{1}\Phi(t)\leq\Phi(ct)\leq c_{2}\Phi(t)\ \ \ \ {\rm for\ all}\ \ \ t\geq 0,$$
where $c_{1}$ and $c_{2}$ depend only on $c$ and the doubling constant $C_{\Phi}$. Therefore, we obtain that if $A\approx B$, then $\Phi(A)\approx\Phi(B)$. This property will be used frequently in the rest of this paper.
Let $\Phi_{1},\Phi_{2}$ be two Young functions. If there exist two constants $k>0$ and $C\geq 0$ such that
$$\Phi_{1}(t)\leq\Phi_{2}(kt)\ \ \ \ {\rm for}\ \ \ t\geq C,$$
we write
$$\Phi_{1}\prec\Phi_{2}.$$
The function $\Phi(t)=t^{p}\log^{\lambda}(e+t)$ with $p>1,\lambda\in{\mathbb{R}}$ or $p=1,\lambda\geq 0$ is a Young function and it satisfies the $\Delta_{2}-$condition. Moreover, it also satisfies that
(2.3)
$$t^{\max\{p-\delta,1\}}\prec\Phi(t)\prec t^{p+\delta}$$
for any $\delta>0$.
Let $\Phi$ be a Young function. Then the Orlicz space $L^{\Phi}(X)$ is defined by setting
$$L^{\Phi}(X,{{\mu_{\lambda}}})=\left\{u:X\rightarrow{\mathbb{R}}:u\ {\rm
measurable%
,}\ \int_{X}\Phi(\alpha|u|)\,d{{\mu_{\lambda}}}<+\infty\ {\rm for\ some}\ %
\alpha>0\right\}.$$
As in the theory of $L^{p}$-spaces, the elements in $L^{\Phi}(X,{{\mu_{\lambda}}})$ are actually equivalence classes consisting of functions that differ only on a set of measure zero. The Orlicz space $L^{\Phi}(X,{{\mu_{\lambda}}})$ is a vector space and, equipped with the Luxemburg norm
$$\|u\|_{L^{\Phi}(X,{{\mu_{\lambda}}})}=\inf\left\{k>0:\int_{X}\Phi(|u|/k)\,d{{%
\mu_{\lambda}}}\leq 1\right\},$$
a Banach space, see [39, Theorem 3.3.10]. If $\Phi(t)=t^{p}$ with $p\geq 1$, then $L^{\Phi}(X,{{\mu_{\lambda}}})=L^{p}(X,{{\mu_{\lambda}}})$. We refer to [34, 39, 46] for more detailed discussions and properties of Orlicz spaces.
2.3 Newtonian spaces and Orlicz-Sobolev spaces on $X$
Let $u\in L_{\rm loc}^{1}(X,{{\mu_{\lambda}}})$. We say that a Borel function $g:X\rightarrow[0,\infty]$ is an upper gradient of $u$ if
(2.4)
$$|u(z)-u(y)|\leq\int_{\gamma}g\,ds_{X}$$
whenever $z,y\in X$ and $\gamma$ is the geodesic from $z$ to $y$, where $ds_{X}$ denotes the arc length measure with respect to the metric $d_{X}$. In the setting of a tree any rectifiable curve with end points $z$ and $y$ contains the geodesic connecting $z$ and $y$, therefore the upper gradient defined above is equivalent to the definition which requires that inequality (2.4) holds for all rectifiable curves with end points $z$ and $y$.
The notion of upper gradients is due to Heinonen and Koskela [18]; we refer interested readers to [2, 15, 19, 41] for a more detailed discussion on upper gradients.
The Newtonian space $N^{1,p}(X,{{\mu_{\lambda}}})$, $1\leq p<\infty$, is defined as the collection of all functions $u$ for which the norm of $u$ defined as
$$\|u\|_{N^{1,p}(X,{{\mu_{\lambda}}})}:=\left(\int_{X}|u|^{p}\,d{{\mu_{\lambda}}%
}+\inf_{g}\int_{X}g^{p}\,d{{\mu_{\lambda}}}\right)^{1/p}$$
is finite,
where the infimum is taken over all upper gradients of $u$.
For any Young function $\Phi$, the Orlicz-Sobolev space $N^{1,\Phi}(X,{{\mu_{\lambda}}})$ is defined as the collection of all functions $u$ for which the norm of $u$ defined as
$$\|u\|_{N^{1,\Phi}(X,{{\mu_{\lambda}}})}=\|u\|_{L^{\Phi}(X,{{\mu_{\lambda}}})}+%
\inf_{g}\|g\|_{L^{\Phi}(X,{{\mu_{\lambda}}})}$$
is finite,
where the infimum is taken over all upper gradients of $u$.
For the Young function $\Phi(t)=t^{p}$, $1\leq p<\infty$, the Orlicz-Sobolev space $N^{1,\Phi}(X,{{\mu_{\lambda}}})$ is exactly the Newtonian space $N^{1,p}(X,{{\mu_{\lambda}}})$. We refer to [46] for further results on Orlicz-Sobolev spaces on metric measure spaces. If $u\in N^{1,p}(X,{{\mu_{\lambda}}})$ ($u\in N^{1,\Phi}(X,{{\mu_{\lambda}}})$ with $\Phi$ doubling), then it has a minimal $p$-weak upper gradient ($\Phi$-weak upper gradient) $g_{u}$, which in our case is an upper gradient. The minimal upper gradient is minimal in the sense that if $g\in L^{p}(X,{{\mu_{\lambda}}})$ ($g\in L^{\Phi}(X,{{\mu_{\lambda}}})$) is any upper gradient of $u$, then $g_{u}\leq g$ a.e. We refer the interested reader to [15, Theorem 7.16] ($p\geq 1$) and [46, Corollary 6.9]($\Phi$ doubling) for proofs of the existence of such a minimal upper gradient.
2.4 Besov-type spaces on ${{\partial X}}$
We first recall the Besov space ${B^{\theta}_{p,p}(\partial X)}$ defined in [3].
Definition 2.3.
For $0<\theta<1$ and $p\geq 1$, The Besov space ${B^{\theta}_{p,p}(\partial X)}$ consists of all functions $f\in L^{p}({{\partial X}})$ for which the seminorm $\|f\|_{{\dot{B}^{\theta}_{p}(\partial X)}}$ defined as
$$\|f\|^{p}_{{\dot{B}^{\theta}_{p}(\partial X)}}:=\int_{{{\partial X}}}\int_{{{%
\partial X}}}\frac{|f(\zeta)|-f(\xi)|^{p}}{d_{X}(\zeta,\xi)^{\theta p}\nu(B(%
\zeta,d_{X}(\zeta,\xi)))}d\nu(\xi)\,d\nu(\zeta)$$
is finite. The corresponding norm for ${B^{\theta}_{p,p}(\partial X)}$ is
$$\|f\|_{{B^{\theta}_{p,p}(\partial X)}}:=\|f\|_{{L^{p}(\partial X)}}+\|f\|_{{%
\dot{B}^{\theta}_{p}(\partial X)}}.$$
Next, we give a dyadic decomposition on the boundary ${{\partial X}}$ of the $K$-ary tree $X$, see also [24, Section 2.4].
Let $V_{n}=\{x_{j}^{n}:j=1,2,\cdots,K^{n}\}$ be the set of all $n$-level vertices of the tree $X$ for each $n\in{\mathbb{N}}$, where a vertex $x$ is of $n$-level if $|x|=n$. Then we have that
$$V=\bigcup_{n\in{\mathbb{N}}}V_{n}$$
is the set containing all the vertices of the tree $X$.
For any vertex $x\in V$, denote by $I_{x}$ the set
$$\{\xi\in{{\partial X}}:\text{the geodesic $[0,\xi)$ passes through $x$}\}.$$
We denote by $\mathscr{Q}$ the set $\{I_{x}:x\in V\}$ and $\mathscr{Q}_{n}$ the set $\{I_{x}:x\in V_{n}\}$ for each $n\in{\mathbb{N}}$. Then $\mathscr{Q}_{0}=\{\partial X\}$ and we have
$$\mathscr{Q}=\bigcup_{n\in{\mathbb{N}}}\mathscr{Q}_{n}.$$
Then the set $\mathscr{Q}$ is called a dyadic decomposition of ${{\partial X}}$. Moreover, for any $n\in{\mathbb{N}}$ and $I\in\mathscr{Q}_{n}$, there is a unique element $\widehat{I}$ in $\mathscr{Q}_{n-1}$ such that $I$ is a subset of $\widehat{I}$. It is easy to see that if $I=I_{x}$ for some $x\in V_{n}$, then $\widehat{I}=I_{y}$ with $y$ the unique parent of $x$ in the tree $X$. Hence the structure of the dyadic decomposition of $\partial X$ is uniquely determined by the structure of the $K$-ary tree $X$.
Definition 2.4.
For $0\leq\theta<1$ and $p\geq 1$, the Besov-type space ${\mathcal{B}^{\theta,\lambda}_{p}(\partial X)}$ consists of all functions $f\in L^{p}({{\partial X}})$ for which the ${\dot{\mathcal{B}}^{\theta,\lambda}_{p}}$-dyadic energy of $f$ defined as
$$\|f\|^{p}_{{\dot{\mathcal{B}}^{\theta,\lambda}_{p}(\partial X)}}:=\sum_{n=1}^{%
\infty}e^{\epsilon n\theta p}n^{\lambda}\sum_{I\in\mathscr{Q}_{n}}\nu(I)\left|%
f_{I}-f_{\widehat{I}}\right|^{p}$$
is finite. The norm on ${\mathcal{B}^{\theta,\lambda}_{p}(\partial X)}$ is
$$\|f\|_{{\mathcal{B}^{\theta,\lambda}_{p}(\partial X)}}:=\|f\|_{{L^{p}(\partial
X%
)}}+\|f\|_{{\dot{\mathcal{B}}^{\theta,\lambda}_{p}(\partial X)}}.$$
Here and throughout this paper, the measure $\nu$ on the boundary ${{\partial X}}$ is the Ahlfors regular measure in Proposition 2.2 and $f_{I}:=\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-13.499794pt}}{{\vbox{%
\hbox{$\scriptstyle-$ }}\kern-12.149815pt}}{{\vbox{\hbox{$\scriptscriptstyle-$%
}}\kern-9.899849pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-8.999863pt}}%
\!\int_{I}f\,d\nu=\frac{1}{\nu(I)}\int_{I}f\,d\nu$ is the usual mean value.
Definition 2.5.
For $0<\theta<1$ and $p\geq 1$, The Besov space ${\mathcal{B}^{\theta}_{p}(\partial X)}$ consists of all the functions $f\in L^{p}({{\partial X}})$ for which the ${\dot{\mathcal{B}}^{\theta}_{p}}$-dyadic energy of $f$ defined as
$$\|f\|^{p}_{{\dot{\mathcal{B}}^{\theta}_{p}(\partial X)}}:=\sum_{n=1}^{\infty}e%
^{\epsilon n\theta p}\sum_{I\in\mathscr{Q}_{n}}\nu(I)\left|f_{I}-f_{\widehat{I%
}}\right|^{p}$$
is finite. The norm of ${\mathcal{B}^{\theta}_{p}(\partial X)}$ is
$$\|f\|_{{\mathcal{B}^{\theta}_{p}(\partial X)}}:=\|f\|_{{L^{p}(\partial X)}}+\|%
f\|_{{\dot{\mathcal{B}}^{\theta}_{p}(\partial X)}}.$$
Notice that ${\mathcal{B}^{\theta}_{p}(\partial X)}$ coincides with ${\mathcal{B}^{\theta,\lambda}_{p}(\partial X)}$ when $\lambda=0$.
Next we introduce the Hajłasz-Besov spaces $N^{s}_{p,p}({{\partial X}})$ on the boundary ${{\partial X}}$.
Definition 2.6.
(i) Let $0<\theta<\infty$ and let $u$ be a measurable function on ${{\partial X}}$. A sequence of nonnegative measurable functions, $\vec{g}=\{g_{k}\}_{k\in\mathbb{Z}}$, is called a fractional $\theta$-Hajłasz gradient of $u$ if there exists $Z\subset{{\partial X}}$ with $\nu(Z)=0$ such that for all $k\in\mathbb{Z}$ and $\zeta,\xi\in{{\partial X}}\setminus Z$ satisfying $2^{-k-1}\leq d_{X}(\zeta,\xi)<2^{-k}$,
$$|u(\zeta)-u(\xi)|\leq[d_{X}(\zeta,\xi)]^{\theta}[g_{k}(\zeta)+g_{k}(\xi)].$$
Denote by $\mathbb{D}^{\theta}(u)$ the collection of all fractional $\theta$-Hajłasz gradients of $u$.
(ii) Let $0<\theta<\infty$ and $0<p<\infty$. The Hajłasz-Besov space $N^{\theta}_{p,p}({{\partial X}})$ consists of all functions $u\in L^{p}({{\partial X}})$ for which the seminorm $\|u\|_{\dot{N}^{\theta}_{p,p}({{\partial X}})}$ defined as
$$\|u\|_{\dot{N}^{\theta}_{p,p}({{\partial X}})}:=\inf_{\vec{g}\in\mathbb{D}^{%
\theta}(u)}\|(\|g_{k}\|_{L^{p}({{\partial X}})})_{k\in\mathbb{Z}}\|_{l^{p}}=%
\inf_{\vec{g}\in\mathbb{D}^{\theta}(u)}\left(\sum_{k\in\mathbb{Z}}\int_{{{%
\partial X}}}[g_{k}(\xi)]^{p}\,d\nu(\xi)\right)^{1/p}$$
is finite. The norm of $N^{\theta}_{p,p}({{\partial X}})$ is
$$\|u\|_{N^{\theta}_{p,p}({{\partial X}})}:=\|u\|_{L^{p}({{\partial X}})}+\|u\|_%
{\dot{N}^{\theta}_{p,p}({{\partial X}})}.$$
The following proposition states that these three Besov-type spaces ${\mathcal{B}^{\theta}_{p}(\partial X)}$, ${B^{\theta}_{p,p}(\partial X)}$ and $N^{\theta}_{p,p}({{\partial X}})$ coincide with each other.
Proposition 2.7.
Let $0<\theta<1$ and $p\geq 1$. For any $f\in L^{1}_{\rm loc}({{\partial X}})$, we have
$$\|f\|_{{\dot{B}^{\theta}_{p}(\partial X)}}\approx\|f\|_{{\dot{\mathcal{B}}^{%
\theta}_{p}(\partial X)}}\approx\|f\|_{\dot{N}^{\theta}_{p,p}({{\partial X}})}.$$
Proof.
Notice that $\text{\rm\,diam}({{\partial X}})\approx 1$. The first part $\|f\|_{{\dot{B}^{\theta}_{p}(\partial X)}}\approx\|f\|_{{\dot{\mathcal{B}}^{%
\theta}_{p}(\partial X)}}$ follows by using [3, Lemma 5.4] and a modification of the proof of [23, Proposition A.1]. We omit the details.
The second part $\|f\|_{{\dot{B}^{\theta}_{p}(\partial X)}}\approx\|f\|_{\dot{N}^{s}_{p,p}({{%
\partial X}})}$ is given by [3, Lemma 5.4] and [14, Theorem 1.2].
∎
The dyadic norms give an easy way to introduce Orlicz-Besov spaces by replacing $t^{p}$ with some Orlicz function $\Phi(t)$.
Definition 2.8.
Let $\Phi$ be the Young function $\Phi(t)=t^{p}\log^{\lambda_{1}}(e+t)$ with $p>1,\lambda_{1}\in{\mathbb{R}}$ or $p=1,\lambda_{1}\geq 0$. Then the Orlicz-Besov space ${\mathcal{B}^{\theta,\lambda_{2}}_{\Phi}(\partial X)}$ consists of all $f\in L^{\Phi}({{\partial X}})$ whose norm generally defined as
$$\|f\|_{{\mathcal{B}^{\theta,\lambda_{2}}_{\Phi}(\partial X)}}:=\|f\|_{L^{\Phi}%
({{\partial X}})}+\inf\left\{k>0:|f/k|_{{\dot{\mathcal{B}}^{\theta,\lambda_{2}%
}_{\Phi}(\partial X)}}\leq 1\right\}$$
is finite, where for any $g\in L^{1}_{\rm loc}({{\partial X}})$, the ${\dot{\mathcal{B}}^{\theta,\lambda_{2}}_{\Phi}}$-dyadic energy is defined as
$$|g|_{{\dot{\mathcal{B}}^{\theta,\lambda_{2}}_{\Phi}(\partial X)}}:=\sum_{n=1}^%
{\infty}e^{\epsilon n(\theta-1)p}n^{\lambda_{2}}\sum_{I\in\mathscr{Q}_{n}}\nu(%
I)\Phi\left(\frac{\left|g_{I}-g_{\widehat{I}}\right|}{e^{-\epsilon n}}\right).$$
In this paper, we are only interested in the Young functions in the above definition. Hence in the rest of this paper, we always assume that the Young function is $\Phi(t)=t^{p}\log^{\lambda_{1}}(e+t)$ with $p>1,\lambda_{1}\in{\mathbb{R}}$ or $p=1,\lambda_{1}\geq 0$.
3 Proofs
3.1 Proof of Theorem 1.1
Proof.
Trace Part:
Let $f\in N^{1,\Phi}(X)$. We first define the trace operator as
(3.1)
$${\rm Tr}\,f(\xi):=\tilde{f}(\xi)=\lim_{[0,\xi)\ni x\rightarrow\xi}f(x),\ \ \xi%
\in{{\partial X}},$$
where the limit is taken along the geodesic ray $[0,\xi)$. Then our task is to show that the above limit exists for $\nu$-a.e. $\xi\in{{\partial X}}$ and that the trace ${\rm Tr}\,f$ satisfies the norm estimates.
Let $\xi\in{{\partial X}}$ be arbitrary and let $x_{j}=x_{j}(\xi)$ be the ancestor of $\xi$ with $|x_{j}|=j$.
To show that the limit in (3.1) exists for $\nu$-a.e. $\xi\in{{\partial X}}$, it suffices to show that the function
(3.2)
$${\tilde{f}}^{*}(\xi)=|f(0)|+\int_{[0,\xi)}g_{f}\,ds$$
is in $L^{p}({{\partial X}})$, where $[0,\xi)$ is the geodesic ray from $0$ to $\xi$ and $g_{f}$ is an upper gradient of $f$. To be more precise, if $\tilde{f}^{*}\in L^{p}({{\partial X}})$, we have $|\tilde{f}^{*}|<\infty$ for $\nu$-a.e. $\xi\in{{\partial X}}$, and hence the limit in $\eqref{trace-operator}$ exists for $\nu$-a.e. $\xi\in{{\partial X}}$.
Set $r_{j}=2e^{-j\epsilon}/\epsilon$. Recall that on the edge $[x_{j},x_{j+1}]$ we have
(3.3)
$$ds\approx e^{(\beta-\epsilon)j}j^{-\lambda_{2}}\,d{{\mu_{\lambda_{2}}}}\approx
r%
_{j}^{1-\beta/\epsilon}j^{-\lambda_{2}}\,d\mu\ \ \ {\rm and}\ \ {{\mu_{\lambda%
_{2}}}}([x_{j},x_{j+1}])\approx r_{j}^{\beta/\epsilon}j^{\lambda_{2}}.$$
Then we obtain that
$$\displaystyle{\tilde{f}}^{*}(\xi)$$
$$\displaystyle=|f(0)|+\int_{[0,\xi)}g_{f}\,ds\leq|f(0)|+\sum_{j=0}^{+\infty}%
\int_{[x_{j},x_{j+1}]}g_{f}\,ds$$
(3.4)
$$\displaystyle\approx|f(0)|+\sum_{j=0}^{+\infty}{r_{j}^{1-\beta/\epsilon}}j^{-%
\lambda_{2}}\int_{[x_{j},x_{j+1}]}g_{f}\,d{{\mu_{\lambda_{2}}}}\approx|f(0)|+%
\sum_{j=0}^{+\infty}{r_{j}}\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-13.4%
99794pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-12.149815pt}}{{\vbox{\hbox{$%
\scriptscriptstyle-$ }}\kern-9.899849pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }%
}\kern-8.999863pt}}\!\int_{[x_{j},x_{j+1}]}g_{f}\,d{{\mu_{\lambda_{2}}}}.$$
Since $\theta=1-(\beta-\log K)/(p\epsilon)>0$, we may choose $1\leq q<\infty$ such that $\max\{(\beta-\log K)/\epsilon,1\}<q<p$ if $p>1$ or $q=1=p$. Let $\Psi(t)=t^{p/q}\log^{\lambda/q}(e+t)$. Then $\Psi^{q}=\Phi$ and $\Psi$ is a doubling Young function. By the Jensen inequality and the doubling property of $\Psi$, since $\sum_{j=0}^{+\infty}r_{j}\approx 1$, we have that
$$\displaystyle\Psi({\tilde{f}}^{*}(\xi))$$
$$\displaystyle\lesssim\Psi(|f(0)|)+\Psi\left(\sum_{j=0}^{+\infty}{r_{j}}%
\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-13.499794pt}}{{\vbox{\hbox{$%
\scriptstyle-$ }}\kern-12.149815pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}%
\kern-9.899849pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-8.999863pt}}\!%
\int_{[x_{j},x_{j+1}]}g_{f}\,d{{\mu_{\lambda_{2}}}}\right)$$
$$\displaystyle\lesssim\Psi(|f(0)|)+\sum_{j=0}^{+\infty}r_{j}\mathchoice{{\vbox{%
\hbox{$\textstyle-$ }}\kern-13.499794pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern%
-12.149815pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-9.899849pt}}{{\vbox{%
\hbox{$\scriptscriptstyle-$ }}\kern-8.999863pt}}\!\int_{[x_{j},x_{j+1}]}\Psi(g%
_{f})\,d{{\mu_{\lambda_{2}}}}.$$
Choose $0<\kappa<1-(\beta-\log K)/(q\epsilon)$. If $q>1$, by the Hölder inequality, we obtain the estimate
$$\displaystyle\Phi({\tilde{f}}^{*}(\xi))=\Psi({\tilde{f}}^{*}(\xi))^{q}$$
$$\displaystyle\lesssim\Phi(|f(0)|)+\left(\sum_{j=0}^{+\infty}r_{j}^{\kappa}\,r_%
{j}^{(1-\kappa)}\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-13.499794pt}}{{%
\vbox{\hbox{$\scriptstyle-$ }}\kern-12.149815pt}}{{\vbox{\hbox{$%
\scriptscriptstyle-$ }}\kern-9.899849pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }%
}\kern-8.999863pt}}\!\int_{[x_{j},x_{j+1}]}\Psi(g_{f})\,d{{\mu_{\lambda_{2}}}}%
\right)^{q}$$
$$\displaystyle\lesssim\Phi(|f(0)|)+\sum_{j=0}^{+\infty}r_{j}^{(1-\kappa)q}\left%
(\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-13.499794pt}}{{\vbox{\hbox{$%
\scriptstyle-$ }}\kern-12.149815pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}%
\kern-9.899849pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-8.999863pt}}\!%
\int_{[x_{j},x_{j+1}]}\Psi(g_{f})\,d{{\mu_{\lambda_{2}}}}\right)^{q}$$
$$\displaystyle\lesssim\Phi(|f(0)|)+\sum_{j=0}^{+\infty}r_{j}^{q-\kappa q-\beta/%
\epsilon}j^{-\lambda_{2}}\int_{[x_{j},x_{j+1}]}\Phi(g_{f})\,d{{\mu_{\lambda_{2%
}}}},$$
since
$$\sum_{j=0}^{+\infty}r_{j}^{kq/(q-1)}\approx 1.$$
If $q=1$, then $\Psi=\Phi$, and hence the Hölder inequality is not needed in the estimate. It follows that
$$\Phi({\tilde{f}}^{*}(\xi))\lesssim\Phi(|f(0)|)+\sum_{j=0}^{+\infty}r_{j}^{q-%
\kappa q-\beta/\epsilon}j^{-\lambda_{2}}\int_{[x_{j},x_{j+1}]}\Phi(g_{f})\,d\mu.$$
Integrating over all $\xi\in{{\partial X}}$, since $\nu({{\partial X}})\approx 1$, we obtain by means of Fubini’s theorem that
$$\displaystyle\int_{{\partial X}}\Phi({\tilde{f}}^{*}(\xi))\,d\nu$$
$$\displaystyle\lesssim\Phi(|f(0)|)+\int_{{\partial X}}\sum_{j=0}^{+\infty}r_{j}%
^{q-\kappa q-\beta/\epsilon}j^{-\lambda_{2}}\int_{[x_{j},x_{j+1}]}\Phi(g_{f})%
\,d{{\mu_{\lambda_{2}}}}\,d\nu(\xi)$$
$$\displaystyle=\Phi(|f(0)|)+\int_{X}\Phi(g_{f}(x))\int_{{\partial X}}\sum_{j=0}%
^{+\infty}r_{j}^{q-\kappa q-\beta/\epsilon}j^{-\lambda_{2}}\chi_{[x_{j},x_{j+1%
}]}(x)\,d\nu(\xi)\,d{{\mu_{\lambda_{2}}}}(x).$$
Note that $\chi_{[x_{j},x_{j+1}]}(x)$ is nonzero only if $j\leq|x|\leq j+1$ and $x<\xi$. Thus the last estimate can be rewritten as
$$\int_{{\partial X}}\Phi({\tilde{f}}^{*}(\xi))\,d\nu\lesssim\Phi(|f(0)|)+\int_{%
X}\Phi(g_{f}(x))r_{j(x)}^{q-\kappa q-\beta/\epsilon}j(x)^{-\lambda_{2}}\nu(E(x%
))\,d\mu(x),$$
where $E(x)=\{\xi\in{{\partial X}}:x<\xi\}$ and $j(x)$ is the largest integer such that $j(x)\leq|x|$. Since $\nu(E(x))\lesssim r_{j(x)}^{Q}$ and $q-\kappa q-\beta/\epsilon+Q>0$,
then for any $j(x)\in\mathbb{N}$, we have that
$$r_{j(x)}^{p(1-\kappa)-\beta/\epsilon+Q}j(x)^{-\lambda_{2}}\lesssim 1,$$
which induces the estimate
$$\displaystyle\int_{{\partial X}}\Phi({\tilde{f}}^{*}(\xi))\,d\nu$$
$$\displaystyle\lesssim\Phi(|f(0)|)+\int_{X}\Phi(g_{f}(x))r_{j(x)}^{q-\kappa q-%
\beta/\epsilon+Q}j(x)^{-\lambda_{2}}\,d{{\mu_{\lambda_{2}}}}(x)$$
$$\displaystyle\lesssim\Phi(|f(0)|)+\int_{X}\Phi(g_{f}(x))\,d{{\mu_{\lambda_{2}}%
}}(x).$$
Actually, the value $|f(0)|$ is not essential. For any $y\in\{x\in X:|x|<1\}$, a neighborhood of $0$, we could modify the definition of ${\tilde{f}}^{*}(\xi)$ as
$${\tilde{f}}^{*}(\xi)=|f(y)|+|f(y)-f(0)|+\sum_{j=0}^{+\infty}|f(x_{j+1})-f(x_{j%
})|.$$
Since ${{\mu_{\lambda_{2}}}}(X)\approx 1$, we have that
$$\Phi(|f(y)-f(0)|)\leq\Phi\left(\int_{[0,y]}g_{f}\,ds\right)\leq\Phi\left(\int_%
{X}g_{f}\,ds\right)\lesssim\int_{X}\Phi(g_{f})\,d{{\mu_{\lambda_{2}}}}.$$
By the same argument as above, we obtain the estimate
$$\int_{{\partial X}}\Phi({\tilde{f}}^{*}(\xi))\,d\nu(\xi)\lesssim\Phi(|f(y)|)+%
\int_{X}\Phi(g_{f})\,d{{\mu_{\lambda_{2}}}},$$
for any $y\in\{x\in X:|x|<1\}$. The fact that $f\in L^{\Phi}(X,{{\mu_{\lambda_{2}}}})$ gives us that $\Phi(|f(y)|)<\infty$ for ${{\mu_{\lambda_{2}}}}$-a.e. $y\in X$. This shows that ${\tilde{f}}^{*}(\xi)$ is $\Phi$-integrable on ${{\partial X}}$, which finishes the proof of the existence of the limit in (3.1). Moreover, since $|\tilde{f}|\leq{\tilde{f}}^{*}$ for any modified ${\tilde{f}}^{*}$, the above arguments also show that for any $y\in\{x\in X:|x|<1\}$, we have that
$$\int_{{\partial X}}\Phi(\tilde{f}(\xi))\,d\nu(\xi)\lesssim\Phi(|f(y)|)+\int_{X%
}\Phi(g_{f})\,d{{\mu_{\lambda_{2}}}}.$$
Integrating over all $y\in\{x\in X:|x|<1\}$, since ${{\mu_{\lambda_{2}}}}(\{x\in X:|x|<1\})\approx 1$, we finally arrive at the estimate
(3.5)
$$\int_{{\partial X}}\Phi(\tilde{f}(\xi))\,d\nu(\xi)\lesssim\int_{X}\Phi(|f|)\,d%
{{\mu_{\lambda_{2}}}}+\int_{X}\Phi(g_{f})\,d{{\mu_{\lambda_{2}}}}.$$
Assume that $\|f\|_{L^{\Phi}(X,{{\mu_{\lambda_{2}}}})}=t_{1}$ and $\|g_{f}\|_{L^{\Phi}(X,{{\mu_{\lambda_{2}}}})}=t_{2}$. By the definition of Luxemburg norms, we know that
$$\int_{X}\Phi(f/t_{1})\,d{{\mu_{\lambda_{2}}}}\leq 1\ \ \ {\rm and}\ \ \ \int_{%
X}\Phi(g_{f}/t_{2})\,d{{\mu_{\lambda_{2}}}}\leq 1.$$
By estimate (3.5), there exists a constant $C>0$ such that
$$\int_{{\partial X}}\Phi(\tilde{f}(\xi))\,d\nu(\xi)\lesssim C\left(\int_{X}\Phi%
(|f|)\,d{{\mu_{\lambda_{2}}}}+\int_{X}\Phi(g_{f})\,d{{\mu_{\lambda_{2}}}}%
\right).$$
We may assume $C\geq 1$, since if $C<1$, we choose $C=1$. Then we obtain that
$$\displaystyle\int_{{\partial X}}\Phi\left(\frac{\tilde{f}(\xi)}{2C(t_{1}+t_{2}%
)}\right)\,d\nu$$
$$\displaystyle\leq C\left(\int_{X}\Phi\left(\frac{f}{2Ct_{1}}\right)\,d{{\mu_{%
\lambda_{2}}}}+\int_{X}\Phi\left(\frac{g_{f}}{2Ct_{2}}\right)\,d{{\mu_{\lambda%
_{2}}}}\right)$$
$$\displaystyle\leq\frac{1}{2}\left(\int_{X}\Phi(f/t_{1})\,d{{\mu_{\lambda_{2}}}%
}+\int_{X}\Phi(g_{f}/t_{2})\,d{{\mu_{\lambda_{2}}}}\right)\leq 1,$$
which implies
(3.6)
$$\|\tilde{f}(\xi)\|_{L^{\Phi}({{\partial X}})}\leq 2C(t_{1}+t_{2})\approx\|f\|_%
{L^{\Phi}(X,{{\mu_{\lambda_{2}}}})}+\|g_{f}\|_{L^{\Phi}(X,{{\mu_{\lambda_{2}}}%
})}=\|f\|_{N^{1,\phi}(X,{{\mu_{\lambda_{2}}}})}.$$
To estimate the dyadic energy $|\tilde{f}|_{{\dot{\mathcal{B}}^{\theta,\lambda_{2}}_{\Phi}(\partial X)}}$, for any $I\in\mathscr{Q}_{n}$, $\xi\in I$ and $\zeta\in\widehat{I}$, we have that
$$|\tilde{f}(\xi)-\tilde{f}(\zeta)|\leq\sum_{j=n-1}^{+\infty}|f(x_{j})-f(x_{j+1}%
)|+\sum_{j=n-1}^{+\infty}|f(y_{j})-f(y_{j+1})|,$$
where $x_{j}=x_{j}(\xi)$ and $y_{j}=y_{j}(\zeta)$ are the ancestors of $\xi$ and $\zeta$ with $|x_{j}|=|y_{j}|=j$, respectively. In the above inequality, we used the fact that $x_{n-1}=y_{n-1}$. By using (3.3) and an argument similar to (3.4), we obtain that
$$|\tilde{f}(\xi)-\tilde{f}(\zeta)|\lesssim\sum_{j=n-1}^{+\infty}r_{j}%
\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-13.499794pt}}{{\vbox{\hbox{$%
\scriptstyle-$ }}\kern-12.149815pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}%
\kern-9.899849pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-8.999863pt}}\!%
\int_{[x_{j},x_{j+1}]}g_{f}\,d{{\mu_{\lambda_{2}}}}+\sum_{j=n-1}^{+\infty}r_{j%
}\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-13.499794pt}}{{\vbox{\hbox{$%
\scriptstyle-$ }}\kern-12.149815pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}%
\kern-9.899849pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-8.999863pt}}\!%
\int_{[y_{j},y_{j+1}]}g_{f}\,d{{\mu_{\lambda_{2}}}}.$$
It follows from the Jensen inequality that
$$\displaystyle\Psi\left(\frac{|\tilde{f}(\xi)-\tilde{f}(\zeta)|}{e^{-\epsilon n%
}}\right)$$
$$\displaystyle\lesssim{\sum_{j=n-1}^{+\infty}r_{n-1}^{-1}r_{j}\mathchoice{{%
\vbox{\hbox{$\textstyle-$ }}\kern-13.499794pt}}{{\vbox{\hbox{$\scriptstyle-$ }%
}\kern-12.149815pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-9.899849pt}}{{%
\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-8.999863pt}}\!\int_{[x_{j},x_{j+1}]}%
\Psi(g_{f})\,d{{\mu_{\lambda_{2}}}}}+{\sum_{j=n-1}^{+\infty}r_{n-1}^{-1}r_{j}%
\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-13.499794pt}}{{\vbox{\hbox{$%
\scriptstyle-$ }}\kern-12.149815pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}%
\kern-9.899849pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-8.999863pt}}\!%
\int_{[y_{j},y_{j+1}]}\Psi(g_{f})\,d{{\mu_{\lambda_{2}}}}},$$
since we have the estimate
$$r_{n-1}\approx e^{-\epsilon n}\approx\sum_{j=n-1}^{+\infty}r_{j}.$$
By using the fact $\Phi=\Psi^{q}$ and the Hölder inequality if $q>1$ (if $q=1$, the Hölder inequality is not needed), we get that
$$\displaystyle\Phi\left(\frac{|\tilde{f}(\xi)-\tilde{f}(\zeta)|}{e^{-\epsilon n%
}}\right)$$
$$\displaystyle=\Psi\left(\frac{|\tilde{f}(\xi)-\tilde{f}(\zeta)|}{e^{-\epsilon n%
}}\right)^{q}$$
$$\displaystyle\lesssim r_{n-1}^{-q+\kappa q}\sum_{j=n-1}^{+\infty}r_{j}^{q-%
\beta/\epsilon-\kappa q}j^{-\lambda_{2}}\left(\int_{[x_{j},x_{j+1}]}\Phi(g_{f}%
)\,d{{\mu_{\lambda_{2}}}}+\int_{[y_{j},y_{j+1}]}\Phi(g_{f})\,d{{\mu_{\lambda_{%
2}}}}\right).$$
Since $\nu(I)\approx\nu(\widehat{I})$ and every $\widehat{I}$ is the parent of $I$, it follows from Fubini’s theorem that
$$\displaystyle\sum_{I\in\mathscr{Q}_{n}}\nu(I)$$
$$\displaystyle\Phi\left(\frac{|\tilde{f}_{I}-\tilde{f}_{\widehat{I}}|}{e^{-%
\epsilon n}}\right)\leq\sum_{I\in\mathscr{Q}_{n}}\nu(I)\mathchoice{{\vbox{%
\hbox{$\textstyle-$ }}\kern-13.499794pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern%
-12.149815pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-9.899849pt}}{{\vbox{%
\hbox{$\scriptscriptstyle-$ }}\kern-8.999863pt}}\!\int_{I}\mathchoice{{\vbox{%
\hbox{$\textstyle-$ }}\kern-13.499794pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern%
-12.149815pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-9.899849pt}}{{\vbox{%
\hbox{$\scriptscriptstyle-$ }}\kern-8.999863pt}}\!\int_{\widehat{I}}\Phi\left(%
\frac{|\tilde{f}(\xi)-\tilde{f}(\zeta)|}{e^{-\epsilon n}}\right)\,d\nu(\zeta)%
\,d\nu(\xi)$$
$$\displaystyle\lesssim\int_{{{\partial X}}}r_{n-1}^{-q+\kappa q}\sum_{j=n-1}^{+%
\infty}r_{j}^{q-\beta/\epsilon-\kappa q}j^{-\lambda_{2}}\int_{[x_{j},x_{j+1}]}%
\Phi(g_{f})\,d{{\mu_{\lambda_{2}}}}\,d\nu(\xi)$$
$$\displaystyle=\int_{X\cap\{|x|\geq n-1\}}\Phi(g_{f})r_{n-1}^{-q+\kappa q}\int_%
{{{\partial X}}}\sum_{j=n-1}^{+\infty}r_{j}^{q-\beta/\epsilon-\kappa q}j^{-%
\lambda_{2}}\chi_{[x_{j},x_{j+1}]}(x)\,d\nu(\xi)\,d{{\mu_{\lambda_{2}}}}(x).$$
Note that $\chi_{[x_{j},x_{j+1}]}(x)$ is nonzero only if $j\leq|x|\leq j+1$ and $x<\xi$. Thus the last estimate can be rewritten as
$$\displaystyle\sum_{I\in\mathscr{Q}_{n}}\nu(I)\Phi\left(\frac{|\tilde{f}_{I}-%
\tilde{f}_{\widehat{I}}|}{e^{-\epsilon n}}\right)$$
$$\displaystyle\lesssim\int_{X\cap\{|x|\geq n-1\}}\Phi(g_{f})r_{n-1}^{-q+\kappa q%
}r_{j(x)}^{q-\beta/\epsilon-\kappa q}j(x)^{-\lambda_{2}}\nu(E(x))\,d{{\mu_{%
\lambda_{2}}}}(x)$$
$$\displaystyle\lesssim\int_{X\cap\{|x|\geq n-1\}}\Phi(g_{f})r_{n-1}^{-q+\kappa q%
}r_{j(x)}^{q-\beta/\epsilon-\kappa q+Q}j(x)^{-\lambda_{2}}\,d{{\mu_{\lambda_{2%
}}}}(x),$$
where $E(x)=\{\xi\in{{\partial X}}:x<\xi\}$ and $j(x)$ is the largest integer such that $j(x)\leq|x|$. Here in the last inequality, we used the fact that $\nu(E(x))\lesssim r_{j(x)}^{Q}$. Since $e^{-\epsilon n}\approx r_{n-1}$, we obtain the estimate
$$\displaystyle|\tilde{f}|_{{\dot{\mathcal{B}}^{\theta,\lambda_{2}}_{\Phi}(%
\partial X)}}$$
$$\displaystyle\lesssim\sum_{n=1}^{+\infty}r_{n-1}^{(1-\theta)p-q+\kappa q}n^{%
\lambda_{2}}\int_{X\cap\{|x|\geq n-1\}}\Phi(g_{f})r_{j(x)}^{q-\beta/\epsilon-%
\kappa q+Q}j(x)^{-\lambda_{2}}\,d{{\mu_{\lambda_{2}}}}(x)$$
$$\displaystyle=\sum_{n=0}^{+\infty}r_{n}^{(1-\theta)p-q+\kappa q}(n+1)^{\lambda%
_{2}}\sum_{j=n}^{+\infty}\int_{X\cap\{j\leq|x|<j+1\}}\Phi(g_{f})r_{j}^{q-\beta%
/\epsilon-\kappa q+Q}j^{-\lambda_{2}}\,d{{\mu_{\lambda_{2}}}}(x)$$
$$\displaystyle=\sum_{j=0}^{+\infty}\int_{X\cap\{j\leq|x|<j+1\}}\Phi(g_{f})r_{j}%
^{q-\beta/\epsilon-\kappa q+Q}j^{-\lambda_{2}}\,d{{\mu_{\lambda_{2}}}}(x)\left%
(\sum_{n=0}^{j}r_{n}^{(1-\theta)p-q+\kappa q}(n+1)^{\lambda_{2}}\right).$$
Recall that $r_{n}=2e^{-n\epsilon}/\epsilon$ and
$$(1-\theta)p-q+\kappa q=\kappa q-(q-(\beta-\log K)/\epsilon)=\kappa q+\beta/%
\epsilon-q-\log K/\epsilon<0.$$
Hence we obtain that
$$\sum_{n=0}^{j}r_{n}^{(1-\theta)p-q+\kappa q}(n+1)^{\lambda_{2}}\approx r_{j}^{%
\kappa q+\beta/\epsilon-q-\log K/\epsilon}(j+1)^{\lambda_{2}}=r_{j}^{\kappa q+%
\beta/\epsilon-q-Q}j^{\lambda_{2}}.$$
Therefore, our estimate above for the dyadic energy can be rewritten as
$$|\tilde{f}|_{{\dot{\mathcal{B}}^{\theta,\lambda_{2}}_{\Phi}(\partial X)}}%
\lesssim\sum_{j=0}^{+\infty}\int_{X\cap\{j\leq|x|<j+1\}}\Phi(g_{f})\,d{{\mu_{%
\lambda_{2}}}}(x)=\int_{X}\Phi(g_{f})\,d{{\mu_{\lambda_{2}}}}(x).$$
By an argument similar to the one that we used to prove (3.6) after getting (3.5), we have that
$$\inf\left\{k>0:|\tilde{f}/k|_{{\dot{\mathcal{B}}^{\theta,\lambda_{2}}_{\Phi}(%
\partial X)}}\leq 1\right\}\lesssim\|g_{f}\|_{L^{\Phi}(X,{{\mu_{\lambda_{2}}}}%
)},$$
which together with (3.6) gives the norm estimate
$$\|\tilde{f}\|_{{\mathcal{B}^{\theta,\lambda_{2}}_{\Phi}(\partial X)}}\lesssim%
\|f\|_{N^{1,\Phi}(X,{{\mu_{\lambda_{2}}}})}.$$
Extension Part: Let $u\in{\mathcal{B}^{\theta,\lambda_{2}}_{\Phi}(\partial X)}$. For $x\in X$ with $|x|=n\in\mathbb{N}$, let
(3.7)
$$\tilde{u}(x)=\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-13.499794pt}}{{%
\vbox{\hbox{$\scriptstyle-$ }}\kern-12.149815pt}}{{\vbox{\hbox{$%
\scriptscriptstyle-$ }}\kern-9.899849pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }%
}\kern-8.999863pt}}\!\int_{I_{x}}u\,d\nu,$$
where $I_{x}\in\mathscr{Q}_{n}$ is the set of all the points $\xi\in{{\partial X}}$ such that the geodesic $[0,\xi)$ passes through $x$, that is, $I_{x}$ consists of all the points in ${{\partial X}}$ that have $x$ as an ancestor. Then (3.1) and (3.7) imply that ${\rm Tr}\,\tilde{u}(\xi)=u(\xi)$ whenever $\xi\in{{\partial X}}$ is a Lebesgue point of $u$.
If $y$ is a child of $x$, then $|y|=n+1$ and $I_{x}$ is the parent of $I_{y}$. Hence we extend $\tilde{u}$ to the edge $[x,y]$ as follows: For each $t\in[x,y]$, set
(3.8)
$$g_{\tilde{u}}(t)=\frac{\tilde{u}(y)-\tilde{u}(x)}{d_{X}(x,y)}=\frac{\epsilon(u%
_{I_{y}}-u_{I_{x}})}{(1-e^{-\epsilon})e^{-\epsilon n}}=\frac{\epsilon(u_{I_{y}%
}-u_{\widehat{I}_{y}})}{(1-e^{-\epsilon})e^{-\epsilon n}}$$
and
(3.9)
$$\tilde{u}(t)=\tilde{u}(x)+g_{\tilde{u}}(t)d_{X}(x,t).$$
Then we define the extension of $u$ to be $\tilde{u}$.
Since $g_{\tilde{u}}$ is a constant and $\tilde{u}$ is linear with respect to the metric $d_{X}$ on the edge $[x,y]$, it follows that $|g_{\tilde{u}}|$ is a minimal upper gradient of $\tilde{u}$ on the edge $[x,y]$. Then we get the estimate
$$\displaystyle\int_{[x,y]}\Phi(|g_{\tilde{u}}|)\,d{{\mu_{\lambda_{2}}}}$$
$$\displaystyle\approx\int_{n}^{n+1}\Phi\left(\frac{|u_{I_{y}}-u_{\widehat{I}_{y%
}}|}{e^{-\epsilon(n+1)}}\right)e^{-\beta\tau}(\tau+C)^{\lambda_{2}}\,d\tau$$
$$\displaystyle\approx e^{-\beta(n+1)}(\tau+1)^{\lambda_{2}}\Phi\left(\frac{|u_{%
I_{y}}-u_{\widehat{I}_{y}}|}{e^{-\epsilon(n+1)}}\right).$$
Now sum up the above integrals for all the edges of $X$ to obtain that
(3.10)
$$\int_{X}\Phi(|g_{\tilde{u}}|)\,d{{\mu_{\lambda_{2}}}}\approx\sum_{n=1}^{+%
\infty}\sum_{I\in\mathscr{Q}_{n}}e^{-\beta n}n^{\lambda_{2}}\Phi\left(\frac{|u%
_{I}-u_{\widehat{I}}|}{e^{-\epsilon n}}\right).$$
For any $I\in\mathscr{Q}_{n}$, we have that
$$\nu(I)\approx e^{-\epsilon nQ},$$
which implies that
(3.11)
$$e^{\epsilon n(\theta-1)p}\nu(I)\approx e^{-\epsilon n((\beta-\log K)/\epsilon+%
Q)}\approx e^{-\beta n}.$$
The above estimates (3.10) and (3.11) give
(3.12)
$$\int_{X}\Phi(|g_{\tilde{u}}|)\,d{{\mu_{\lambda_{2}}}}\approx\sum_{n=1}^{\infty%
}e^{\epsilon n(\theta-1)p}n^{\lambda_{2}}\sum_{I\in\mathscr{Q}_{n}}\nu(I)\Phi%
\left(\frac{\left|u_{I}-u_{\widehat{I}}\right|}{e^{-\epsilon n}}\right)=|u|_{{%
\dot{\mathcal{B}}^{\theta,\lambda_{2}}_{\Phi}(\partial X)}}.$$
For the $L^{\Phi}$-estimate of $\tilde{u}$, we first observe that
(3.13)
$$|\tilde{u}(t)|\leq|\tilde{u}(x)|+|g_{\tilde{u}}|d_{X}(x,y)=|\tilde{u}(x)|+|%
\tilde{u}(y)-\tilde{u}(x)|\lesssim|u_{I_{x}}|+|u_{I_{y}}|$$
for any $t\in[x,y]$. Then we get the estimate
$$\int_{[x,y]}\Phi(|\tilde{u}(t)|)\,d{{\mu_{\lambda_{2}}}}\lesssim{{\mu_{\lambda%
_{2}}}}([x,y])\big{(}\Phi(|u_{I_{x}}|)+\Phi(|u_{I_{y}}|)\big{)}\lesssim e^{-%
\beta n+\epsilon nQ}n^{\lambda_{2}}\int_{I_{x}}\Phi(|u|)\,d\nu.$$
Here in the last inequality, we used the facts ${{\mu_{\lambda_{2}}}}([x,y])\approx e^{-\beta n}n^{\lambda_{2}}$ and $\nu(I_{x})\approx\nu(I_{y})\approx e^{-\epsilon nQ}$. Now sum up the above integrals for all the edges of $X$ to obtain that
$$\displaystyle\int_{X}\Phi(|\tilde{u}(t)|)\,d{{\mu_{\lambda_{2}}}}$$
$$\displaystyle\lesssim\sum_{n=0}^{+\infty}\sum_{I\in\mathscr{Q}_{n}}e^{-\beta n%
+\epsilon nQ}n^{\lambda_{2}}\int_{I}\Phi(|u|)\,d\nu$$
$$\displaystyle=\sum_{n=0}^{+\infty}e^{-\beta n+\epsilon nQ}n^{\lambda_{2}}\int_%
{{{\partial X}}}\Phi(|u|)\,d\nu.$$
Since $\beta-\epsilon Q=\beta-\log K>0$, the sum of $e^{-\beta n+\epsilon nQ}n^{-\lambda_{2}}$ converges. Hence we obtain the estimate
(3.14)
$$\int_{X}\Phi(|\tilde{u}(t)|)\,d{{\mu_{\lambda_{2}}}}\lesssim\int_{{{\partial X%
}}}\Phi(|u|)\,d\nu.$$
Applying the very same arguments that we used in proving (3.6) after getting (3.5) to (3.12) and (3.14), we finally arrive at the norm estimate
$$\|\tilde{u}\|_{N^{1,\Phi}(X,{{\mu_{\lambda_{2}}}})}\lesssim\|u\|_{{\mathcal{B}%
^{\theta,\lambda_{2}}_{\Phi}(\partial X)}}.$$
∎
3.2 Proof of proposition 1.2
In this section, we always assume that $\Phi(t)=t^{p}\log^{\lambda_{1}}(e+t)$ with $p>1,\lambda_{1}\in{\mathbb{R}}$ or $p=1,\lambda_{1}\geq 0$.
Lemma 3.1.
Let $\lambda,\lambda_{1},\lambda_{2}\in{\mathbb{R}}$.
Assume that $\lambda_{1}+\lambda_{2}=\lambda$. For any $f\in L^{1}({{\partial X}})$, we have that
$\|f\|_{{\dot{\mathcal{B}}^{\theta,\lambda}_{p}(\partial X)}}<\infty$ is equivalent to $|f|_{{\dot{\mathcal{B}}^{\theta,\lambda_{2}}_{\Phi}(\partial X)}}<\infty$ whenever $0<\theta<1$.
Proof.
When $\lambda_{1}=0$, then the result is obvious since $\|f\|^{p}_{{\dot{\mathcal{B}}^{\theta,\lambda}_{p}(\partial X)}}=|f|_{{\dot{%
\mathcal{B}}^{\theta,\lambda_{2}}_{\Phi}(\partial X)}}$.
When $\lambda_{1}>0$, first we estimate the logarithmic term from above. Since $f\in L^{1}({{\partial X}})$, for any $I\in\mathscr{Q}_{n}$, it follows from $\nu(I)\approx\nu(\widehat{I})\approx e^{-n\log K}$ that
$$\displaystyle\log^{\lambda_{1}}\left(e+\frac{|f_{I}-f_{\widehat{I}}|}{e^{-%
\epsilon n}}\right)\leq\log^{\lambda_{1}}\left(e+\frac{|f_{I}|+|f_{\widehat{I}%
}|}{e^{-\epsilon n}}\right)\lesssim\log^{\lambda_{1}}\left(e+\frac{\|f\|_{L^{1%
}({{\partial X}})}}{e^{-(\epsilon+\log K)n}}\right)\leq Cn^{\lambda_{1}},$$
where $C=C(\|f\|_{L^{1}({{\partial X}})},\lambda_{1},\epsilon,K)$. Hence we can estimate $|f|_{{\dot{\mathcal{B}}^{\theta,\lambda_{2}}_{\Phi}(\partial X)}}$ as follows:
$$\displaystyle|f|_{{\dot{\mathcal{B}}^{\theta,\lambda_{2}}_{\Phi}(\partial X)}}$$
$$\displaystyle=\sum_{n=1}^{\infty}e^{\epsilon n(\theta-1)p}n^{\lambda_{2}}\sum_%
{I\in\mathscr{Q}_{n}}\nu(I)\Phi\left(\frac{\left|g_{I}-g_{\widehat{I}}\right|}%
{e^{-\epsilon n}}\right)$$
$$\displaystyle=\sum_{n=1}^{\infty}e^{\epsilon n\theta p}n^{\lambda_{2}}\sum_{I%
\in\mathscr{Q}_{n}}\nu(I)|f_{I}-f_{\widehat{I}}|^{p}\log^{\lambda_{1}}\left(e+%
\frac{|f_{I}-f_{\widehat{I}}|}{e^{-\epsilon n}}\right)$$
$$\displaystyle\leq C\sum_{n=1}^{\infty}e^{\epsilon n\theta p}n^{\lambda_{2}+%
\lambda_{1}}\sum_{I\in\mathscr{Q}_{n}}\nu(I)|f_{I}-f_{\widehat{I}}|^{p}=C\|f\|%
^{p}_{{\dot{\mathcal{B}}^{\theta,\lambda}_{p}(\partial X)}},$$
where $C=C(\|f\|_{L^{1}({{\partial X}})},\lambda_{1},\epsilon,K)$.
In order to estimate the logarithmic term from below, for any $I\in\mathscr{Q}_{n}$, we define
(3.15)
$$\chi(n,I)=\left\{\begin{array}[]{cc}1,&\ \mathrm{if}\ \ |f_{I}-f_{\widehat{I}}%
|>e^{-\epsilon n(\theta+1)/2}\\
0,&\mathrm{otherwise}.\end{array}\right.$$
Then we have that
$$\displaystyle\|f\|^{p}_{{\dot{\mathcal{B}}^{\theta,\lambda}_{p}(\partial X)}}$$
$$\displaystyle=\sum_{n=1}^{\infty}e^{\epsilon n\theta p}n^{\lambda}\sum_{I\in%
\mathscr{Q}_{n}}\nu(I)|f_{I}-f_{\widehat{I}}|^{p}$$
$$\displaystyle=\sum_{n=1}^{\infty}e^{\epsilon n\theta p}n^{\lambda}\sum_{I\in%
\mathscr{Q}_{n}}\nu(I)\chi(n,I)|f_{I}-f_{\widehat{I}}|^{p}$$
$$\displaystyle\ \ \ \ +\sum_{n=1}^{\infty}e^{\epsilon n\theta p}n^{\lambda}\sum%
_{I\in\mathscr{Q}_{n}}\nu(I)(1-\chi(n,I))|f_{I}-f_{\widehat{I}}|^{p}$$
$$\displaystyle=:P_{1}+P_{2}.$$
If $|f_{I}-f_{\widehat{I}}|>e^{-\epsilon n(\theta+1)/2}$, since $\theta<1$ and $\lambda_{1}>0$, we obtain that
$$\log^{\lambda_{1}}\left(e+\frac{|f_{I}-f_{\widehat{I}}|}{e^{-\epsilon n}}%
\right)>\log^{\lambda_{1}}\left(e+e^{\epsilon n(1-\theta)/2}\right)\geq Cn^{%
\lambda_{1}},$$
where $C=C(\epsilon,\theta,\lambda_{1})$. Hence we have the estimate
$$P_{1}\leq C\sum_{n=1}^{\infty}e^{\epsilon n\theta p}n^{\lambda_{2}}\sum_{I\in%
\mathscr{Q}_{n}}|f_{I}-f_{\widehat{I}}|^{p}\log^{\lambda_{1}}\left(e+\frac{|f_%
{I}-f_{\widehat{I}}|}{e^{-\epsilon n}}\right)=C|f|_{{\dot{\mathcal{B}}^{\theta%
,\lambda_{2}}_{\Phi}(\partial X)}}.$$
For $P_{2}$, since $\sum_{I\in\mathscr{Q}_{n}}\nu(I)\approx 1$, we have that
$$P_{2}\leq\sum_{n=1}^{\infty}e^{\epsilon n\theta p}n^{\lambda}\sum_{I\in%
\mathscr{Q}_{n}}\nu(I)e^{-\epsilon np(\theta+1)/2}\approx\sum_{n=1}^{\infty}e^%
{\epsilon np(\theta-1)/2}n^{\lambda}=C^{\prime}<+\infty,$$
where $C^{\prime}=C^{\prime}(\theta,p,\lambda)$. Therefore, we obtain
(3.16)
$$\frac{1}{C}|f|_{{\dot{\mathcal{B}}^{\theta,\lambda_{2}}_{\Phi}(\partial X)}}%
\leq\|f\|^{p}_{{\dot{\mathcal{B}}^{\theta,\lambda}_{p}(\partial X)}}=P_{1}+P_{%
2}\leq C|f|_{{\dot{\mathcal{B}}^{\theta,\lambda_{2}}_{\Phi}(\partial X)}}+C^{%
\prime},$$
where $C$ and $C^{\prime}$ are constants depending only on $\epsilon,\theta,\lambda_{1},\lambda,p$ znd $\|f\|_{L^{1}({{\partial X}})}$.
When $\lambda_{1}<0$, in order to estimate the logarithmic term from above, using definition (3.15), we obtain that
$$\displaystyle|f|_{{\dot{\mathcal{B}}^{\theta,\lambda_{2}}_{\Phi}(\partial X)}}$$
$$\displaystyle=\sum_{n=1}^{\infty}e^{\epsilon n\theta p}n^{\lambda_{2}}\sum_{I%
\in\mathscr{Q}_{n}}\nu(I)|f_{I}-f_{\widehat{I}}|^{p}\log^{\lambda_{1}}\left(e+%
\frac{|f_{I}-f_{\widehat{I}}|}{e^{-\epsilon n}}\right)$$
$$\displaystyle=\sum_{n=1}^{\infty}e^{\epsilon n\theta p}n^{\lambda_{2}}\sum_{I%
\in\mathscr{Q}_{n}}\nu(I)\chi(n,I)|f_{I}-f_{\widehat{I}}|^{p}\log^{\lambda_{1}%
}\left(e+\frac{|f_{I}-f_{\widehat{I}}|}{e^{-\epsilon n}}\right)$$
$$\displaystyle\ \ \ \ \ \ +\sum_{n=1}^{\infty}e^{\epsilon n\theta p}n^{\lambda_%
{2}}\sum_{I\in\mathscr{Q}_{n}}\nu(I)(1-\chi(n,I))|f_{I}-f_{\widehat{I}}|^{p}%
\log^{\lambda_{1}}\left(e+\frac{|f_{I}-f_{\widehat{I}}|}{e^{-\epsilon n}}\right)$$
$$\displaystyle=:P_{1}^{\prime}+P_{2}^{\prime}.$$
If $|f_{I}-f_{\widehat{I}}|>e^{-\epsilon n(\theta+1)/2}$, since $\theta<1$ and $\lambda_{1}<0$, we have that
$$\log^{\lambda_{1}}\left(e+\frac{|f_{I}-f_{\widehat{I}}|}{e^{-\epsilon n}}%
\right)<\log^{\lambda_{1}}\left(e+e^{\epsilon n(1-\theta)/2}\right)\leq Cn^{%
\lambda_{1}},$$
where $C=C(\epsilon,\theta,\lambda_{1})$. Hence we have the estimate
$$P_{1}^{\prime}\leq C\sum_{n=1}^{\infty}e^{\epsilon n\theta p}n^{\lambda_{2}+%
\lambda_{1}}\sum_{I\in\mathscr{Q}_{n}}\nu(I)|f_{I}-f_{\widehat{I}}|^{p}=C\|f\|%
^{p}_{{\dot{\mathcal{B}}^{\theta,\lambda}_{p}(\partial X)}}.$$
For $P_{2}^{\prime}$, since $\log^{\lambda_{1}}(e+t)\leq 1$ for any $t\geq 0$ and $\sum_{I\in\mathscr{Q}_{n}}\nu(I)\approx 1$, we obtain that
$$P_{2}^{\prime}\leq\sum_{n=1}^{\infty}e^{\epsilon n\theta p}n^{\lambda_{2}}\sum%
_{I\in\mathscr{Q}_{n}}\nu(I)e^{-\epsilon n(\theta+1)/2}=\sum_{n=1}^{\infty}e^{%
\epsilon np(\theta-1)/2}n^{\lambda_{2}}=C^{\prime}<+\infty,$$
where $C^{\prime}=C(\epsilon,\theta,\lambda_{2})$.
Next, we estimate the logarithmic term from below. Since $f\in L^{1}({{\partial X}})$ and $\lambda_{1}<0$, for any $I\in\mathscr{Q}_{n}$, it follows from $\nu(I)\approx\nu(\widehat{I})\approx e^{-n\log K}$ that
$$\displaystyle\log^{\lambda_{1}}\left(e+\frac{|f_{I}-f_{\widehat{I}}|}{e^{-%
\epsilon n}}\right)\geq\log^{\lambda_{1}}\left(e+\frac{|f_{I}|+|f_{\widehat{I}%
}|}{e^{-\epsilon n}}\right)\gtrsim\log^{\lambda_{1}}\left(e+\frac{\|f\|_{L^{1}%
({{\partial X}})}}{e^{-(\epsilon+\log K)n}}\right)\geq Cn^{\lambda_{1}},$$
where $C=C(\|f\|_{L^{1}({{\partial X}})},\lambda_{1},\epsilon,K)$. Now we get the estimate
$$\displaystyle\|f\|^{p}_{{\dot{\mathcal{B}}^{\theta,\lambda}_{p}(\partial X)}}$$
$$\displaystyle=\sum_{n=1}^{\infty}e^{\epsilon n\theta p}n^{\lambda_{2}+\lambda_%
{1}}\sum_{I\in\mathscr{Q}_{n}}\nu(I)|f_{I}-f_{\widehat{I}}|^{p}$$
$$\displaystyle\leq C\sum_{n=1}^{\infty}e^{\epsilon n\theta p}n^{\lambda_{2}}%
\sum_{I\in\mathscr{Q}_{n}}\nu(I)|f_{I}-f_{\widehat{I}}|^{p}\log^{\lambda_{1}}%
\left(e+\frac{|f_{I}-f_{\widehat{I}}|}{e^{-\epsilon n}}\right)$$
$$\displaystyle=C|f|_{{\dot{\mathcal{B}}^{\theta,\lambda_{2}}_{\Phi}(\partial X)%
}}.$$
Therefore, we obtain the estimate
(3.17)
$$\frac{1}{C}\|f\|^{p}_{{\dot{\mathcal{B}}^{\theta,\lambda}_{p}(\partial X)}}%
\leq|f|_{{\dot{\mathcal{B}}^{\theta,\lambda_{2}}_{\Phi}(\partial X)}}=P_{1}^{%
\prime}+P_{2}^{\prime}\leq C\|f\|^{p}_{{\dot{\mathcal{B}}^{\theta,\lambda}_{p}%
(\partial X)}}+C^{\prime},$$
where $C$ and $C^{\prime}$ are constants depending only on $\epsilon,\theta,\lambda_{1},\lambda_{2}$ and $\|f\|_{L^{1}({{\partial X}})}$.
Combining the inequalities (3.16) and (3.17) which are respect to $\lambda_{1}>0$ and $\lambda_{1}<0$ with the case $\lambda_{1}=0$, we obtain that $\|f\|^{p}_{{\dot{\mathcal{B}}^{\theta,\lambda}_{p}(\partial X)}}<+\infty$ is equivalent to $|f|_{{\dot{\mathcal{B}}^{\theta,\lambda_{2}}_{\Phi}(\partial X)}}<+\infty$.
∎
We need a result from functional ananlysis.
Lemma 3.2 (Closed graph theorem).
Let $X,Y$ be Banach spaces and let $T:X\rightarrow Y$ be a linear operator. Then $T$ is continuous if and only if the graph $\sum:=\{(x,T(x)):x\in X\}$ is closed in $X\times Y$ with the product topology.
Let $L^{\Phi}({{\partial X}})\cap{\dot{\mathcal{B}}^{\theta,\lambda}_{p}(\partial X)}$ be the Banach space equipped with the norm
$$\|f\|_{L^{\Phi}({{\partial X}})\cap{\dot{\mathcal{B}}^{\theta,\lambda}_{p}(%
\partial X)}}:=\|f\|_{L^{\Phi}({{\partial X}})}+\|f\|_{{\dot{\mathcal{B}}^{%
\theta,\lambda}_{p}(\partial X)}}.$$
Using the same manner, we could define the space $X\cap Y$ for any two spaces $X$ and $Y$.
Corollary 3.3.
Let $\lambda,\lambda_{1},\lambda_{2}$ and $\Phi$ be as in Lemma 3.1.
Then we have
$$L^{\Phi}({{\partial X}})\cap{\dot{\mathcal{B}}^{\theta,\lambda}_{p}(\partial X%
)}={\mathcal{B}^{\theta,\lambda_{2}}_{\Phi}(\partial X)}$$
with equivalent norms.
Proof.
It directly follows from Lemma 3.1 that $L^{\Phi}({{\partial X}})\cap{\dot{\mathcal{B}}^{\theta,\lambda}_{p}(\partial X)}$ and ${\mathcal{B}^{\theta,\lambda_{2}}_{\Phi}(\partial X)}$ are the same vector spaces. Next we use Lemma 3.2 (Closed graph theorem) to show that they are the same Banach spaces with equivalent norms.
Consider the identity map ${\rm Id}\,:L^{\Phi}({{\partial X}})\cap{\dot{\mathcal{B}}^{\theta,\lambda}_{p}%
(\partial X)}\rightarrow{\mathcal{B}^{\theta,\lambda_{2}}_{\Phi}(\partial X)}$, i.e., ${\rm Id}\,(x)=x$ for any $x\in L^{\Phi}({{\partial X}})\cap{\dot{\mathcal{B}}^{\theta,\lambda}_{p}(%
\partial X)}$. Then the graph of ${\rm Id}\,$ is closed. Indeed, if $(x_{n},x_{n})$ is a sequence in this graph that converges to $(x,y)$ in $(L^{\Phi}({{\partial X}})\cap{\dot{\mathcal{B}}^{\theta,\lambda}_{p}(\partial X%
)})\times(L^{p}({{\partial X}})\cap{\dot{\mathcal{B}}^{\theta,\lambda_{2}}_{%
\Phi}(\partial X)})$ with product topology, then $x_{n}$ converges to $x$ in $\|\cdot\|_{L^{\Phi}({{\partial X}})\cap{\dot{\mathcal{B}}^{\theta,\lambda}_{p}%
(\partial X)}}$ norm and hence in $L^{\Phi}({{\partial X}})$. In the same manner, $x_{n}$ converges to $y$ in $\|\cdot\|_{{\mathcal{B}^{\theta,\lambda_{2}}_{\Phi}(\partial X)}}$ and hence in $L^{\Phi}({{\partial X}})$. But the limits are unique in $L^{\Phi}({{\partial X}})$, so $x=y$.
Applying Lemma 3.2 (Closed graph theorem), we see that the map ${\rm Id}\,$ is continuous from $L^{\Phi}({{\partial X}})\cap{\dot{\mathcal{B}}^{\theta,\lambda}_{p}(\partial X)}$ to ${\mathcal{B}^{\theta,\lambda_{2}}_{\Phi}(\partial X)}$; similarly for the inverse. Thus the norms $\|\cdot\|_{L^{\Phi}({{\partial X}})\cap{\dot{\mathcal{B}}^{\theta,\lambda}_{p}%
(\partial X)}}$ and $\|\cdot\|_{{\mathcal{B}^{\theta,\lambda_{2}}_{\Phi}(\partial X)}}$ are equivalent and the claim follows.
∎
There is a slightly difference between the results in Corollary 3.3 and Proposition 1.2, since ${\mathcal{B}^{\theta,\lambda}_{p}(\partial X)}=L^{p}({{\partial X}})\cap{\dot{%
\mathcal{B}}^{\theta,\lambda}_{p}(\partial X)}$. To get Proposition 1.2 from Corollary 3.3, we need some estimates between the $L^{p}$-norm and $L^{\Phi}$-norm. Since $\nu({{\partial X}})=1$, we have the following lemma, see [27, Theorem 3.17.1 and Theorem 3.17.5].
Lemma 3.4.
Let $\Phi_{1},\Phi_{2}$ be two Young functions.
If $\Phi_{2}\prec\Phi_{1}$, then
$$\|u\|_{L^{\Phi_{2}}({{\partial X}})}\lesssim\|u\|_{L^{\Phi_{1}}({{\partial X}})}$$
for all $u\in L^{\Phi_{1}}({{\partial X}})$.
By the relation (2.3), for any $\delta>0$, we have
(3.18)
$$\|u\|_{L^{\max\{p-\delta,1\}}({{\partial X}})}\lesssim\|u\|_{L^{\Phi}({{%
\partial X}})}\lesssim\|u\|_{L^{p+\delta}({{\partial X}})}$$
for all $u\in L^{p+\delta}({{\partial X}})$.
Recall that $\nu({{\partial X}})=1$ and $\text{\rm\,diam}({{\partial X}})\approx 1$. Since ${{\partial X}}$ is Ahlfors $Q$-regular where $Q=\frac{\log K}{\epsilon}$, we obtain the following lemma immediately from [22, Theorem 4.2]
Lemma 3.5.
Let $0<s<1$ and $p\geq 1$. Let $u\in\dot{N}^{s}_{p,p}({{\partial X}})$. If $0<sp<Q=\frac{\log K}{\epsilon}$, then $u\in L^{p^{*}}({{\partial X}})$, $p^{*}=\frac{Qp}{Q-sp}$ and
$$\inf_{c\in\mathbb{R}}\left(\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-13.4%
99794pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-12.149815pt}}{{\vbox{\hbox{$%
\scriptscriptstyle-$ }}\kern-9.899849pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }%
}\kern-8.999863pt}}\!\int_{{{\partial X}}}|u-c|^{p^{*}}\,d\nu\right)^{1/{p^{*}%
}}\lesssim\|u\|_{\dot{N}^{s}_{p,p}({{\partial X}})}$$
Proof of Proposition 1.2.
Let $s=\min\{\frac{\theta}{2},\frac{Q}{2p}\}$, where $Q=\frac{\log K}{\epsilon}$. Let $p^{*}=\frac{Qp}{Q-sp}$ and $\delta=p^{*}-p$. It follows from the definitions of our Besov-type spaces and Proposition 2.7 that
$${\dot{\mathcal{B}}^{\theta,\lambda}_{p}(\partial X)}\subset\dot{\mathcal{B}}^{%
s}_{p}({{\partial X}})=\dot{N}^{s}_{p,p}({{\partial X}}).$$
By Lemma 3.5 and triangle inequality, we obtain that
$$\displaystyle\left(\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-13.499794pt}%
}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-12.149815pt}}{{\vbox{\hbox{$%
\scriptscriptstyle-$ }}\kern-9.899849pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }%
}\kern-8.999863pt}}\!\int_{{{\partial X}}}|u-u_{{{\partial X}}}|^{p^{*}}\,d\nu%
\right)^{1/{p^{*}}}$$
$$\displaystyle\leq 2\inf_{c\in\mathbb{R}}\left(\mathchoice{{\vbox{\hbox{$%
\textstyle-$ }}\kern-13.499794pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-12.149%
815pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-9.899849pt}}{{\vbox{\hbox{$%
\scriptscriptstyle-$ }}\kern-8.999863pt}}\!\int_{{{\partial X}}}|u-c|^{p^{*}}%
\,d\nu\right)^{1/{p^{*}}}$$
$$\displaystyle\lesssim\|u\|_{\dot{N}^{s}_{p,p}({{\partial X}})}\lesssim\|u\|_{{%
\dot{\mathcal{B}}^{\theta,\lambda}_{p}(\partial X)}},$$
for any $u\in{\dot{\mathcal{B}}^{\theta,\lambda}_{p}(\partial X)}$, where $u_{{{\partial X}}}=\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-13.499794pt}%
}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-12.149815pt}}{{\vbox{\hbox{$%
\scriptscriptstyle-$ }}\kern-9.899849pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }%
}\kern-8.999863pt}}\!\int_{{{\partial X}}}u\,d\nu$. Since $|u|\leq|u-u_{{{\partial X}}}|+|u_{{{\partial X}}}|$ and $\nu({{\partial X}})=1$, it follows from the Minkowski inequality that
$$\displaystyle\|u\|_{L^{p^{*}}({{\partial X}})}$$
$$\displaystyle\leq\|u-u_{{{\partial X}}}\|_{L^{p^{*}}({{\partial X}})}+\|u_{{{%
\partial X}}}\|_{L^{p^{*}}({{\partial X}})}$$
$$\displaystyle=\left(\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-13.499794pt%
}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-12.149815pt}}{{\vbox{\hbox{$%
\scriptscriptstyle-$ }}\kern-9.899849pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }%
}\kern-8.999863pt}}\!\int_{{{\partial X}}}|u-u_{{{\partial X}}}|^{p^{*}}\,d\nu%
\right)^{1/{p^{*}}}+\left|\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-13.49%
9794pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-12.149815pt}}{{\vbox{\hbox{$%
\scriptscriptstyle-$ }}\kern-9.899849pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }%
}\kern-8.999863pt}}\!\int_{{{\partial X}}}u\,d\nu\right|$$
$$\displaystyle\lesssim\|u\|_{L^{1}({{\partial X}})}+\|u\|_{{\dot{\mathcal{B}}^{%
\theta,\lambda}_{p}(\partial X)}},$$
for any $u\in{\dot{\mathcal{B}}^{\theta,\lambda}_{p}(\partial X)}$. Since $\|\cdot\|_{L^{1}({{\partial X}})}\leq\|\cdot\|_{L^{p}({{\partial X}})}\leq\|%
\cdot\|_{L^{p^{*}}({{\partial X}})}$ is trivial, we have that
$$L^{1}({{\partial X}})\cap{\dot{\mathcal{B}}^{\theta,\lambda}_{p}(\partial X)}=%
{\mathcal{B}^{\theta,\lambda}_{p}(\partial X)}=L^{p^{*}}({{\partial X}})\cap{%
\dot{\mathcal{B}}^{\theta,\lambda}_{p}(\partial X)}.$$
Recall the relation (3.18) and $\delta=p^{*}-p$. Hence we have that
$$\|\cdot\|_{L^{1}}({{\partial X}})\lesssim\|\cdot\|_{L^{\Phi}({{\partial X}})}%
\lesssim\|\cdot\|_{L^{p^{*}}({{\partial X}})}.$$
Thus,
$${\mathcal{B}^{\theta,\lambda}_{p}(\partial X)}=L^{\Phi}({{\partial X}})\cap{%
\dot{\mathcal{B}}^{\theta,\lambda}_{p}(\partial X)}.$$
Combining with Corollary 3.3, i.e.,
$$L^{\Phi}({{\partial X}})\cap{\dot{\mathcal{B}}^{\theta,\lambda}_{p}(\partial X%
)}=L^{\Phi}({{\partial X}})\cap{\dot{\mathcal{B}}^{\theta,\lambda_{2}}_{\Phi}(%
\partial X)}={\mathcal{B}^{\theta,\lambda_{2}}_{\Phi}(\partial X)},$$
we finally arrive at
$${\mathcal{B}^{\theta,\lambda}_{p}(\partial X)}={\mathcal{B}^{\theta,\lambda_{2%
}}_{\Phi}(\partial X)}.$$
∎
Acknowledgement. The author would like to thank his advisor Professor Pekka Koskela for helpful comments and suggestions.
References
[1]
N. Aronszajn: Boundary values of functions with finite Dirichlet integral, Techn. Report 14, University of Kansas, 1955.
[2]
A. Björn and J. Björn: Nonlinear potential theory on metric spaces, EMS Tracts Math. 17, European Mathematical Society, Zurich 2011.
[3]
A. Björn, J. Björn, J. T. Gill and N. Shanmugalingam: Geometric analysis on Cantor sets and trees. J. Reine Angew. Math. 725 (2017), 63-114.
[4]
A. Björn, J. Björn and N. Shanmugalingam: The Dirichlet problem for p-harmonic functions on metric spaces, J. Reine Angew. Math. 556 (2003), 173-203.
[5]
M. Bonk, J. Heinonen and P. Koskela: Uniformizing Gromov hyperbolic spaces, Astérisque No. 270 (2001), viii+99 pp.
[6]
M. Bonk and E. Saksman: Sobolev spaces and hyperbolic fillings, J. Reine Angew. Math. 737 (2018), 161-187.
[7]
M. Bridson and A. Haefliger: Metric spaces of non-positive curvature, Grundlehren Math. Wiss. 319, Springer-Verlag, Berlin 1999.
[8]
A. Cianchi:
Orlicz-Sobolev boundary trace embeddings,
Math. Z. 266 (2010), no. 2, 431-449.
[9]
R. N. Dhara and A. Kałamajska: On one extension theorem dealing with weighted Orlicz-Slobodetskii space. Analysis on cube, Math. Inequal. Appl. 18 (2015), no. 1, 61–89.
[10]
R. N. Dhara and A. Kałamajska: On one extension theorem dealing with weighted Orlicz-Slobodetskii space. Analysis on Lipschitz subgraph and Lipschitz domain, Math. Inequal. Appl. 19 (2016), no. 2, 451–488.
[11]
A. Fougères: Théorèmes de trace et de prolongement dans les espaces de Sobolev et Sobolev-Orlicz, C. R. Acad. Sci. Paris Sér. A–B 274 (1972), A181–A184.
[12]
E. Gagliardo: Caratterizzazioni delle tracce sulla frontiera relative ad alcune classi di funzioni in $n$ variabili, Rend. Sem. Mat. Univ. Padova 27 (1957), 284–305.
[13]
A. Gogatishvili, P. Koskela and N. Shanmugalingam: Interpolation properties of Besov spaces defined on metric spaces, Math. Nachr. 283 (2010), no. 2, 215-231.
[14]
A. Gogatishvili, P. Koskela and Y. Zhou: Characterizations of Besov and Triebel-Lizorkin spaces on metric measure spaces, Forum Math. 25 (2013), no. 4, 787-819.
[15]
P. Hajłasz: Sobolev space on metric-measure spaces, in Heat kernels and analysis on manifolds, graphs and metric spaces (Paris 2002), Contemp. Math. 338, American Mathematical Society, Providence (2003), 173-218.
[16]
P. Hajłasz and P. Koskela: Sobolev met Poincaré, Mem. Amer. Math. Soc. (2000), no. 688, x+101 pp.
[17]
J. Heinonen: Lectures on analysis on metric spaces, Universitext, Springer-Verlag, New York 2001. x+140 pp.
[18]
J. Heinonen and P. Koskela: Quasiconformal mappings in metric spaces with controlled geometry, Acta Math. 181 (1998), 1-61.
[19]
J. Heinonen, P. Koskela, N. Shanmugalingam and J. Tyson: Sobolev Spaces on Metric Measure Spaces: An Approach Based on Upper Gradients. Cambridge: Cambridge University Press, 2015.
[20]
A. Jonsson and H. Wallin: The trace to subsets of ${\mathbb{R}}^{n}$ of Besov spaces in the general case, Anal. Math. 6 (1980), 223-254.
[21]
A. Jonsson and H. Wallin: Function spaces on subsets of ${\mathbb{R}}^{n}$, Math. Rep. 2 (1984), no. 1, xiv+221 pp.
[22]
N. Karak: Measure density and embeddings of Hajłasz-Besov and Hajłasz-Triebel-Lizorkin spaces, J. Math. Anal. Appl. 475 (2019), no. 1, 966-984.
[23]
P. Koskela, T. Soto and Z. Wang: Traces of weighted function spaces: dyadic norms and Whitney extensions, Sci. China Math. 60 (2017), no. 11, 1981-2010.
[24]
P. Koskela and Z. Wang: Dyadic norm Besov-type spaces as trace spaces on regular trees, to appear in Potential Anal. arXiv:1908.06937.
[25]
P. Koskela, D. Yang and Y. Zhou: A characterization of Hajłasz-Sobolev and Triebel-Lizorkin spaces via grand Littlewood-Paley functions, J. Funct. Anal. 258 (2010), no. 8, 2637-2661.
[26]
P. Koskela, D. Yang and Y. Zhou: Pointwise characterizations of Besov and Triebel-Lizorkin spaces and quasiconformal mappings, Adv. Math. 226 (2011), no. 4, 3579-3621.
[27]
A. Kufner, O. John and S. Fucik: Function spaces. Monographs and Textbooks on Mechanics of Solids and Fluids; Mechanics: Analysis, Noordhoff International Publishing, Leyden; Academia, Prague, 1977. xv+454 pp.
[28]
M.-Th. Lacroix:
Caractérisation des traces dans les espaces de Sobolev-Orlicz,
C. R. Acad. Sci. Paris Sér. A-B 274 (1972), A1813-A1816.
[29]
M.-Th. Lacroix:
Espaces de traces des espaces de Sobolev-Orlicz, J. Math. Pures Appl. (9) 53 (1974), 439–458.
[30]
P. I. Lizorkin: Boundary properties of functions from “weight” classes (Russian), Dokl. Akad. Nauk SSSR 132 (1960), 514–517; translated as Soviet Math. Dokl. 1 (1960), 589–593.
[31]
L. Malý: Trace and extension theorems for Sobolev-type functions in metric spaces, arXiv:1704.06344.
[32]
L. Malý, N. Shanmugalingam and M. Snipes: Trace and extension theorems for functions of bounded variation, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 18 (2018), no. 1, 313-341.
[33]
P. Mironescu and E. Russ: Traces of weighted Sobolev spaces. Old and new, Nonlinear Anal. 119 (2015), 354-381.
[34]
W. Orlicz:
On certain properties of $\varphi$-functions,
Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 8 1960 439â443.
[35]
G. Palmieri:
An approach to the theory of some trace spaces related to the Orlicz-Sobolev spaces,
Boll. Un. Mat. Ital. B (5) 16 (1979), no. 1, 100-119.
[36]
G. Palmieri: The traces of functions in a class of Sobolev-Orlicz spaces with weight, Boll. Un. Mat. Ital. B (5) 18 (1981), no. 1, 87–117.
[37]
J. Peetre: A counterexample connected with Gagliardo’s trace theorem, Comment. Math. 2 (1979), 277-282.
[38]
J. Peetre: New thoughts on Besov spaces, Duke University Mathematics Series, No. 1. Mathematics Department, Duke University, Durham, N.C., 1976.
[39]
M. M. Rao and Z. D. Ren: Theory of Orlicz spaces Monographs and Textbooks in Pure and Applied Mathematics 146, Marcel Dekker, Inc., New York, 1991.
[40]
E. Saksman and T. Soto: Traces of Besov, Triebel-Lizorkin and Sobolev spaces on metric spaces, Anal. Geom. Metr. Spaces 5 (2017), 98-115.
[41]
N. Shanmugalingam, Newtonian spaces: An extension of Sobolev spaces to metric measure spaces, Rev. Mat. Iberoam., 243-279 ,Vol. 16, 2000.
[42]
L. N. Slobodetskii and V. M. Babich: On boundedness of the Dirichlet integrals (Russian), Dokl. Akad. Nauk SSSR (N.S.) 106 (1956), 604–606.
[43]
T. Soto: Besov spaces on metric spaces via hyperbolic fillings, arXiv:1606.08082.
[44]
H. Triebel: Theory of function spaces, Monographs in Mathematics, 78. Birkhäuser Verlag, Basel, 1983.
[45]
H. Triebel: The structure of functions, Monographs in Mathematics, 97. Birkhäuser Verlag, Basel, 2001.
[46]
H. Tuominen: Orlicz-Sobolev spaces on metric measure spaces. Dissertation, University of Jyväskylä, Jyväskylä, 2004. Ann. Acad. Sci. Fenn. Math. Diss. No. 135 (2004), 86 pp.
[47]
A. I. Tyulenev: Description of traces of functions in the Sobolev space with a Muckenhoupt weight, Proc. Steklov Inst. Math. 284 (2014), no. 1, 280-295.
[48]
A. I. Tyulenev: Traces of weighted Sobolev spaces with Muckenhoupt weight. The case $p=1$, Nonlinear Anal. 128 (2015), 248-272.
Zhuang Wang
Department of Mathematics and Statistics, University of Jyväskylä, PO Box 35, FI-40014 Jyväskylä, Finland
E-mail address: zhuang.z.wang@jyu.fi |
High-spin low-spin transition
[
[
[
[
[
Department of Chemistry, University of Odense, DK-5230 Odense,
Denmark
trautwein@physik.mu-luebeck.de
Institut für Physik, Medizinische Universität Lübeck,
D-23538 Lübeck, Germany
Abstract
Temperature dependent nuclear inelastic-scattering (NIS) of synchrotron radiation was applied to investigate both spin states of the spin-crossover complex [Fe(tpa)(NCS)${}_{2}$] (tpa=tris(2-pyridylmethyl)amine).
A remarkable increase of the iron-ligand bond stretching upon spin crossover has unambiguously been identified by comparing the measured NIS spectra of with theoretical simulations based on density-functional calculations.
:
This is an unedited preprint. The original publication is available at
http://www.springerlink.com
http://www.doi.org/10.1023/A:1017056831002
A]H. Grünsteudel B]H. Paulsen
B]H. Winkler
B]A. X. Trautwein††thanks: Corresponding author
A]H. Toftlund
The iron(II) complex [Fe(tpa)(NCS)${}_{2}$] (Fig. 1, tpa =
tris(2-pyridylmethyl) amine) belongs to the family of thermally
driven spin-crossover complexes, which exhibit a transition from a low-spin
(LS) to a high-spin (HS) state by increasing the temperature. These complexes
are very promising materials for optical information storage
[1].
IR measurements on several spin-crossover complexes with a central [FeN${}_{6}$]
octahedron indicate a remarkable increase of the Fe-N bond stretching
frequencies from about 25 to 30 meV in the HS state to about 50 to 60 meV in
the LS state[2]. In this case nuclear inelastic scattering (NIS)
turns out a very valuable alternative to IR and Raman spectroscopy because the
Fe-N stretching modes can be definitely identified in the NIS spectra, whereas
the IR and Raman spectra are rather complex in this frequency region
[3] making an unambiguous assignment of these modes very
difficult.
NIS spectra were recorded at the Nuclear Resonance Beamline ID 18 of the
European Synchrotron Radiation Facility (ESRF) in Grenoble, France
[4]. The 6 GeV electron storage ring was operated in 16 bunch mode
and the purity of the filling (population of parasitic bunches compared to the
single bunch) was better than $10^{-9}$. The incident beam was monochromatized
by a double-crystal Si(111) premonochromator to the bandwidth of 2.5 eV at the
energy of 14.413 keV. A further decrease of bandwidth down to 6 meV was
obtained with a ”nested” high-resolution monochromator [5]. The
sample was mounted in a closed-cycle cryostat to allow measurements at
different temperatures. An avalanche photo-diode with $10\times 10$ mm${}^{2}$
area, 100 $\mu$m active thickness, and a time resolution of less than 1 ns has
been used as a detector [6] to count the 14.413 keV $\gamma$-quanta
and, mainly, the K-fluorescence photons ($\approx 6.4$ keV). The data were
collected during several energy scans with 140 steps on average, each with 2
meV stepsize and 10 s measuring time. All individual scans were corrected for
the approximately 9% decrease of beam intensity of the storage ring during
the 1500 s required for each scan and added up afterwards.
The NIS spectra of the HS and LS isomers of [Fe(tpa)(NCS)${}_{2}$] exhibit central
peaks of 12 and 7 meV linewidth, respectively, and a pronounced inelastic peak
at 30 meV in the HS state and at 50 meV in the LS state
(Fig. 2)[7, 8, 9]. Comparing the
intensity of the pronounced inelastic peaks in the HS and the LS spectrum it
should be kept in mind, that the HS peak at 30 meV is located on the shoulder
of the corresponding central peak. The linewidth of the LS peak observed at 50
meV is significantly larger than the linewidth of the corresponding central
peak. The LS peak should, therefore, be regarded as a superposition of two or
more individual peaks. The LS spectrum exhibits another, rather small peak at
66 meV, which is invisible in the HS spectrum.
The first momentum $\overline{E}$ of the measured absorption probability
density $S_{\rm meas}(E)$ of the HS isomer (1.8 meV) is in reasonable
agreement with the recoil energy $E_{R}$ = 1.96 meV of the free ${}^{57}$Fe
nucleus,
as expected according to Lipkin’s rule, [10] whereas the first
momentum of the LS spectrum amounts to 4 meV, which is about twice as large as
$E_{R}$. This large measured $\overline{E}$ is due to the increased attenuation
of the incident radiation at nuclear resonance [11]. This
phenomenon can be neglected if the elastic peak has a small weight $[f_{\rm LM}^{\rm HS}=0.21(1)]$. If however a large fraction of the spectrum belongs to
the elastic peak $[f_{\rm LM}^{\rm LS}=0.68(1)]$ the measured absorption
probability density has to be corrected. For this reason the NIS spectrum of
the LS isomer was normalized according to
$$S(E)=S_{\rm meas}(E)\frac{E_{R}}{\overline{E}}+\left(1-\frac{E_{R}}{\overline{%
E}}\right)R(E)$$
(1)
The resolution function $R(E)$ describes the energy distribution of the
incident radiation and is assumed to be an even function.
For the HS and the LS isomers of [Fe(tpa)(NCS)${}_{2}$] electronic structure
calculations were performed using the density functional theory (DFT) method
B3LYP, [12] implemented in the gaussian94 program system
[13] together with the split valence 3-21G* basis set
[14]. The geometries were fully optimized applying the Berny
algorithm to redundant internal coordinates [15]. Force constants were
calculated analytically at the same level of theory using the optimized
geometries, and the resulting vibrational frequencies were corrected by the
scaling factor 0.9613 as has been proposed by Wong [16] for the
6-31G* basis set. The calculated normal modes for both isomers have been used
to simulate the absorption probability density $S(E)$ according to the
procedure described in Ref[9].
No X-ray structures are available that can be compared with the calculated geometries, but the calculated bond lengths of the HS and LS isomers qualitatively resemble the increase of the Fe-N bond distances upon spin crossover of about 10 to 20 pm observed in various spin-crossover complexes with a central [FeN${}_{6}$] octahedron [2].
The vibrational spectra of the HS and the LS isomers of [Fe(tpa)(NCS)${}_{2}$] consist of 135 normal modes and are, in the following discussion, subdivided into a high-frequency region above 75 meV and a low-frequency region below 75 meV.
The high-frequency region is of minor interest for the purposes of this study, since the vibrational modes in this frequency region do not contribute to the mean-square displacement (msd) of the iron nucleus and, therefore, can not be observed by NIS.
Among the 41 normal modes of the low-frequency region the iron-ligand bond stretching vibrations are of special interest here.
Due to the almost octahedral environment of the iron center three out of six Fe-N stretching modes are invisible in NIS and IR spectra.
Those modes that transform according to the A${}_{1}$ and E${}_{\rm g}$ representations of the ideal octahedron do not contribute to the msd of the iron nucleus or to the variation of the electric dipole moment.
Only the remaining three modes, that transform according to the T${}_{\rm 1u}$ representations can be observed in NIS and IR spectra.
These three modes, with calculated frequencies of 29.9, 30.1, and 35.3 meV for the HS state and 42.8, 46.6, and 52.6 meV for the LS state, give rise to prominent peaks in the simulated NIS spectra of both isomers of [Fe(tpa)(NCS)${}_{2}$].
Considerable contributions to the calculated absoption probability also arise from N-Fe-N bending modes in the range from 3 to 20 meV.
These modes can not be identified in the experimental spectra because they are superimposed by the much larger contributions to the NIS spectra originating from the acoustical phonons.
By IR spectroscopy [17] Fe-N bond stretching frequencies of 59.5
and 66.0 meV have been found for the LS isomer, while for the region below
35 meV, that is difficult to reach, no frequencies are reported. The Fe-N bond
stretching frequencies calculated for the LS isomer are about 12.4 meV
smaller than the IR values given above; however, they are in good agreement
with the frequencies obtained from NIS. The broad peak at 50 meV observed in
the measured NIS spectrum of the LS isomer (Fig.2) represents the
envelope of the three Fe-N stretching modes in the range of 45 to 55 meV.
The pronounced peak at 30 meV in the NIS spectrum of the HS isomer is
assigned to
the same modes (Fig.2). These modes reflect, according to the
intensity of the peaks, the substantial contributions to the msd of the iron
nucleus that is associated with the three T${}_{\rm 1u}$ Fe-N stretching modes.
According to the normal mode analysis the low-intensity peak at 66 meV in the
measured NIS spectrum as well as the line at 65.7 meV in the IR spectrum must
be assigned to a mode which has predominantly N-C-S bending character and to
some extent Fe-N stretching character. The mixed character of this mode is due
to interactions between Fe-N stretching and N-C-S bending modes, which are
close in energy in the LS isomer.
The calculated N-C-S bending modes of the HS isomer do not show any admixture
of Fe-N stretching modes because of the relatively large energy gap of about
30 meV between these modes. Correspondingly the NIS spectrum of the HS isomer
does not exhibit a peak at the respective energy. In summary, the NIS spectra
of the LS isomer as well as the DFT calculations suggest, that the IR line
attributed previously to an Fe-N bond stretching mode of the LS isomer should
be assigned to a bending mode of the NCS group instead. As a result the
frequency shift of the Fe-N stretching mode upon spin crossover is about 40% smaller than assumed earlier. DFT calculations for another spin-crossover
complex with NCS groups, i.e., [Fe(phen)${}_{2}$(NCS)${}_{2}$] (phen =
1,10-phenanthroline), lead to a similar conclusion.
The measured Lamb-Mössbauer factor of [Fe(tpa)(NCS)${}_{2}$] is decreasing
from $f_{\rm LM}^{\rm LS}=0.68(1)$ for the LS state at 34 K to $f_{\rm LM}^{\rm HS}=0.21(1)$ for the HS state at 200 K
[7]. Comparison of these values with the calculated
molecular Lamb-Mössbauer factors ($f_{\rm mol}^{\rm LS}=92$ and $f_{\rm mol}^{\rm HS}=0.52$) indicates, that for both spin states the major part of
the iron msd is due to inter-molecular vibrations. However, the msd of
the HS state contains, according to the calculations, also significant
contributions from intra-molecular vibrations.
Due to its ability to focus on few modes out of a rather complex vibrational
spectrum NIS can be a complementary or, for certain problems, even a superior
alternative to conventional methods like IR and Raman spectroscopy. A good
example is the investigation of iron(II) spin-crossover complexes as presented
here. IR and Raman spectra are rather complex in the frequency range of the
Fe-N bond stretching modes (20 - 60 meV). Even if the isotope technique is
used, the assignment of these modes to the observed bands often remains
doubtful as has been demonstrated for [Fe(tpa)(NCS)${}_{2}$]. In the NIS spectra,
however, the Fe-N stretching modes could be unambiguously identified.
Acknowledgements.The authors acknowledge the support by A.I. Chumakov, R. Rüffer, and
H. F. Grünsteudel and by the relevant ESRF services during these
measurements, and the financial support by the European Union
(ERB-FMRX-CT-0199) via the TMR-TOSS-network, by the German Research Foundation
(DFG) and by the German Federal Ministery for Education, Science, Research and
Technology (BMBF).
References
[1]
P. Gütlich, A. Hauser, and H. Spiering,
Angew. Chem. 106 (1994), 2109.
[2]
E. König, Struct. Bonding 76 (Berlin, 1991), 51.
[3]
J. H. Takemoto and B. Hutchinson, Inorg. Chem. 12 (1973),
705.
[4]
R. Rüffer and A. I. Chumakov,
Hyperfine. Interact. 97/98 (1996), 589.
[5]
T. Ishikawa, Y. Yoda, K. Izumi, C. K. Suzuki,
X. W. Zhang, M. Ando, and S. Kikuta, Rev. Sci. Instr. 63 (1992), 1015;
T. Toellner, T. Mooney, S. Shastri, and E. E. Alp, in SPIE Proceedings
1740 (1992), p. 218.
[6]
A. Q. R. Baron, Nucl. Instrum. Methods Phys. Res. A 352
(1995), 665.
[7]
H. Grünsteudel, PhD thesis, Lübeck (1998).
[8]
H. Grünsteudel, H. Paulsen, W. Meyer-Klaucke,
H. Winkler, A. X. Trautwein, H. F. Grünsteudel, A. Q. R. Baron,
A. I. Chumakov, R. Rüffer, and H. Toftlund, Hyperfine Interact. 113
(1998), 311.
[9]
H. Paulsen, H. Winkler, A. X. Trautwein,
H. Grünsteudel, V. Rusanov, and H. Toftlund, Phys. Rev. B 59 (1999), 975.
[10]
H. J. Lipkin, Ann. Phys. 9 (Paris, 1960), 332.
[11]
W. Sturhahn, T. S. Toellner, E. E. Alp, X. Zhang,
M. Ando, Y. Yoda, S. Kikuta, M. Seto, C.W. Kimball, and B. Dabrowski,
Phys. Rev. Lett. 74 (1995), 3832.
[12]
A. D. Becke, J. Chem. Phys. 98 (1993), 5648; C. Lee,
W. Yang, and R. G. Parr, Phys. Rev. B 37 (1988), 785.
[13]
M. J. Frisch et al., Gaussian 94, Revision C.3,
Gaussian, Inc. (Pittsburgh, PA, 1995).
[14]
J. S. Binkley, J. A. Pople, and W. J. Hehre,
J. Amer. Chem. Soc. 102 (1980), 939; M. S. Gordon, J. S. Binkley, J. A. Pople,
W. J. Pietro, and W. J. Hehre, ibid. 104 (1982), 2797; W. J. Pietro,
M. M. Francl, W. J. Hehre, D. J. Defrees, J. A. Pople, and J. S. Binkley, ibid. 104 (1982), 5039.
[15]
C. Peng and H. B. Schlegel, J. Comp. Chem. 17 (1996), 49.
[16]
M. W. Wong, Chem. Phys. Lett. 256 (1996), 391.
[17]
F. Højland, H. Toftlund, and S. Yde-Andersen, Acta
Chem. Scand. A 37 (1983), 251. |
Rings with an elementary abelian $p$-group of units
Sunil Chebolu
Department of Mathematics
Illinois State University
Normal, IL 61790, USA
schebol@ilstu.edu, jacorry@ilstu.edu, evgrimm@ilstu.edu, abhatfi@ilstu.edu
,
Jeremy Corry
,
Elizabeth Grimm
and
Andrew Hatfield
Abstract.
What are all rings $R$ for which $R^{*}$ (the group of invertible elements of $R$ under multiplication) is an elementary abelian $p$-group? We answer this question for finite-dimensional commutative $k$-algebras, finite commutative rings, modular group algebras, and path algebras. Two interesting byproducts of this work are a characterization of Mersenne primes and a connection to Dedekind’s problem.
Key words and phrases: Commutative rings, group of units, group algebras, Wedderburn-Artin, local rings
2000 Mathematics Subject Classification: Primary — 11T06, 16U60
The first author is supported by Simons Foundation: Collaboration Grant for Mathematicians (516354).
1. Introduction
Robert Gilmer [7] classified all finite commutative rings $R$ with identity such that the group of units of $R$ is cyclic. He showed that these rings are isomorphic to a finite product of rings $R_{i}$ from the list below such that for $i\neq j$, $\gcd(|R_{i}^{*}|,|R_{j}^{*}|)=1$:
•
The finite field $\mathbb{F}_{p^{k}}$,
•
$\mathbb{Z}/p^{m}$ where $p$ is an odd prime and $m>1$,
•
$\mathbb{Z}/4$,
•
$\mathbb{F}_{p}[x]/(x^{2})$ where $p$ is any prime,
•
$\mathbb{F}_{2}[x]/(x^{3})$, and
•
$\mathbb{Z}[x]/(4,2x,x^{2}-2)$.
The unit groups of the above-mentioned rings are well-known. Therefore, given any positive integer $m$, using Gilmer’s theorem, one can write down all finite commutative rings whose unit group is $C_{m}$, the cyclic group of order $m$.
After cyclic groups, an important family of finite groups is the class of elementary abelian $p$-groups - a direct sum of copies of the cyclic group $C_{p}$, where $p$ is a fixed prime. These groups feature in many fields, including topology, group cohomology, and representation theory. This paper obtains a parallel to Gilmer’s result by classifying all finite commutative rings whose unit group is an elementary abelian $p$-group. In addition, we also obtain a classification of all finite-dimensional $k$-algebras, modular group algebras, and path algebras for which the unit group is an elementary abelian $p$-group.
The first author studied this problem for the families $\mathbb{Z}_{n}$ [3], $\mathbb{Z}_{n}[x_{1},x_{2},\cdots x_{m}]$ [6], and group algebras over fields [5] under the name of diagonal property. A ring $R$ is said to have the diagonal property if all units in the multiplication table for $R$ fall on the main diagonal. For any prime $p$, a ring $R$ is said to be a $\Delta_{p}$-ring if all units $u$ in $R$ satisfy $u^{p}=1$. It is easy to see that $R$ has the diagonal property if and only if it is a $\Delta_{2}$-ring. Furthermore, these conditions are equivalent to $R^{*}$ being an elementary abelian $2$-group; see Proposition 2.1. We now state our main results.
Theorem 1.1.
Classifying rings whose unit group is an elementary abelian $2$-group.
(1)
Let $R$ be a finite dimensional commutative $k$-algebra where $k$ is a field. $R$ is a $\Delta_{2}$-ring if and only if $R$ is isomorphic to a finite product of rings that are either $\mathbb{F}_{3}$ or a quotient of a truncated polynomial ring of the form $\mathbb{F}_{2}[x_{1},x_{2},\cdots,x_{l}]/(x_{1}^{2},x_{2}^{2},\cdots,x_{l}^{2})$, $l\geq 1$.
(2)
Let $R$ be a finite commutative ring. $R$ is a $\Delta_{2}$-ring if and only if $R$ is isomorphic to any quotient of a finite product of rings $R_{i}$, where each $R_{i}$ is one of the following rings:
•
$\mathbb{F}_{2}[x_{1},x_{2},\cdots,x_{l}]/(x_{1}^{2},x_{2}^{2},\cdots,x_{l}^{2})$, $l\geq 1$.
•
$\mathbb{F}_{3}$
•
$\mathbb{Z}_{4}[x_{1},x_{2},\cdots,x_{l}]/(r(r+2)\colon r\in\mathcal{P})$ where $\mathcal{P}=(2,x_{1}-1,x_{2}-1,\cdots x_{l}-1)\subseteq\mathbb{Z}_{4}[x_{1},x_{2},\cdots,x_{l}]$.
•
$\mathbb{Z}_{8}[x_{1},x_{2},\cdots,x_{l}]/(r(r+2)\colon r\in\mathcal{P})$ where $\mathcal{P}=(2,x_{1}-1,x_{2}-1,\cdots x_{l}-1)\subseteq\mathbb{Z}_{8}[x_{1},x_{2},\cdots,x_{l}]$.
(3)
Let $n$ be a positive integer and $G$ be a finite group. $\mathbb{Z}_{n}G$ is a $\Delta_{2}$-ring if and only if $n=2,3,6$ and $G$ is an elementary abelian $2$-group or $n=4,12$ and $G=C_{2}$.
(4)
Let $Q$ be a finite and acyclic quiver. The path algebra $kQ$ is a $\Delta_{2}$-ring if and only if $k=\mathbb{F}_{3}$ and $Q$ is trivial or $k=\mathbb{F}_{2}$ and $Q$ has no directed path of length $2$.
Analyzing quotients mentioned in the above classification led us to Dedekind’s problem, which is wide open; see Remark 3.2. Also, note that the so-called exceptional primes (2 and 3) feature in parts 1 and 4 of the above theorem. These primes play a special role in many places. For instance, they appear in work on generating hypothesis for the stable module categories. Combining this theorem with earlier work done in Tate cohomology [2, Theorem 1.1] gives the following intriguing characterization of these exceptional primes.
Corollary 1.2.
Let $p$ be a prime number. Then the following are equivalent.
(1)
$p=2$ or $3$
(2)
There is a finite-dimensional $\mathbb{F}_{p}$-algebra whose unit group is an elementary abelian $2$-group.
(3)
The Tate cohomology functor $\hat{H}(C_{p},-)$ is faithful on the stable module category of finitely generated $\mathbb{F}_{p}C_{p}$-modules.
When looking for rings whose unit group is an elementary abelian $p$-group, with $p$ odd, it is natural to ask which elementary abelian $p$-groups occur as unit groups of rings. That is Fuchs’ problem for elementary abelian $p$-groups and was answered in [4]. An abelian $p$-group occurs as a unit group of a ring if and only if $p=2$ or a Mersenne prime (a prime of the form $2^{m}-1$ for some $m$). That is the reason why Mersenne primes feature in the next theorem.
Moreover, for odd primes $p$, the $\Delta_{p}$ condition and $R^{*}$ being an elementary abelian $p$-group are equivalent for commutative rings; see Proposition 2.1.
Theorem 1.3.
Classifying rings whose unit group is an elementary $p$-abelian group, where $p$ is an odd prime.
(1)
Let $R$ be a finite dimensional commutative $k$-algebra. $R$ is a $\Delta_{p}$-ring if and only if $p$ is a Mersenne prime and $R=(\mathbb{F}_{2})^{a}\times(\mathbb{F}_{p+1})^{b}$ for some nonnegative integers $a$ and $b$.
(2)
A finite commutative ring $R$ is is a $\Delta_{p}$-ring if and only if $p$ is Mersenne prime and $R$ is isomorphic to $R=(\mathbb{F}_{2})^{a}\times(\mathbb{F}_{p+1})^{b}$ for some nonnegative integers $a$ and $b$.
(3)
$\mathbb{Z}_{n}G$ ($G$ abelian) is a $\Delta_{p}$-ring if and only if $n=2$, $G$ is an elementary abelian $p$-group, and $p$ is some Mersenne prime.
(4)
Let $Q$ be a finite acyclic quiver. The path algebra $kQ$ is $\Delta_{p}$ if and only if $k=\mathbb{F}_{2}$ or $\mathbb{F}_{p+1}$ where $p$ is a Mersenne prime and $Q$ is trivial.
It is worth noting that the above theorem gives some characterizations of Mersenne primes; see Theorem 1.1 [5] for similar characterizations.
Corollary 1.4.
Let $p$ be an odd prime. Then the following are equivalent.
(1)
$p$ is a Mersenne prime.
(2)
There exists a finite dimensional commutative $k$-algebra that is $\Delta_{p}$.
(3)
There exists a finite commutative $\Delta_{p}$-ring.
(4)
There exists a finite acyclic quiver $Q$ and a field $k$ such that the path algebra $kQ$ is $\Delta_{p}$-ring.
We see from the above results that $\Delta_{p}$-rings are rare. For instance, working over a field $k$, it is easy to see that for $n>1$, $M_{n}(k)$ is never a $\Delta_{p}$ ring.
The paper is organized as follows. We begin in Section 2 with some preliminaries. We then classify the $\Delta_{p}$-rings for finite-dimensional commutative $k$-algebras (Section 3), finite commutative rings (Section 4), modular group algebras (Section 5), and path algebras of quivers (Section 6).
We thank Dave Benson, Jon Carlson, Srikanth Iyengar, and Richard Stanley for the discussions related to this paper. This led to a better understanding of the lattice of ideals in truncated polynomial rings and its relation to the Dedekind problem; see Remark 3.2.
2. Preliminaries
This section collects some background material and lemmas that we will need to prove our main results. We begin with the relationship between $\Delta_{p}$ and $R^{*}$ being an elementary abelian $p$-group.
Proposition 2.1.
Let $R$ be a unital ring. Consider the following statements.
(1)
$R$ is a $\Delta_{p}$-ring.
(2)
$R$ is a ring such that $R^{*}$ is an elementary abelian $p$-group
When $p=2$ the above two statements are equivalent. When $p>2$, $2$ implies $1$, and $1$ implies $2$ provided $R^{*}$ is commutative.
Proof.
$(2)\implies(1)$ with $p=2$ is the only non-trivial part, and that follows from an exercise in group theory: any group $G$ such that $g^{2}=e$ for all $g$ in $G$ is abelian.
If $G$ is abelian such that $g^{2}=e$ for all $g$ in $G$, then $G$ is a vector space over $\mathbb{F}_{2}$. If we let
$r$ be the dimension of $G/\mathbb{F}_{2}$, then we see that $G\cong\mathbb{F}_{2}^{r}\cong C_{2}^{r}$,
so $G$ is an elementary abelian 2-group.
∎
We now give a summary of results from [3, 6, 5] that are related to the $\Delta_{p}$ condition.
Theorem 2.2.
Examples of $\Delta_{2}$ and $\Delta_{p}$ rings.
(1)
[3] $\mathbb{Z}_{n}$ is a $\Delta_{2}$ ring if and only if $n$ divides 24.
(2)
[6] $\mathbb{Z}_{n}[x_{1},\cdots x_{m}]$ is a $\Delta_{2}$ ring if and only of $n$ divides 12 and $m\geq 1$.
(3)
[5] For a field $k$ and group $G$, $kG$ is $\Delta_{2}$ if and only if $k=\mathbb{F}_{2}$ or $\mathbb{F}_{3}$ and $G$ is an elementary abelian $2$-group of possibly infinite rank.
(4)
[5] Let $p$ be an odd prime and let a field $k$ and $G$ be an abelian group. $kG$ is $\Delta_{p}$ if and only $p$ is Mersenne and $kG$ is either $\mathbb{F}_{2}C_{p}^{r}$ or $\mathbb{F}_{p+1}C_{p}^{r}$ where $0<r\leq\infty$.
Lemma 2.3.
A subring of a $\Delta_{p}$-ring is again a $\Delta_{p}$-ring. A direct product of $\Delta_{p}$-rings is $\Delta_{p}$ if and only if each factor is a $\Delta_{p}$-ring.
Lemma 2.4.
Let $R$ be a ring of characteristic $n\,(>1)$. If $R$ is a $\Delta_{2}$-ring, then $n$ divides $24$.
Proof.
For a $\Delta_{2}$-ring $R$ with characteristic $n$, $\mathbb{Z}_{n}$ is a subring of $R$, and therefore a $\Delta_{2}$-ring. It was shown in [3] that $\mathbb{Z}_{n}$ is a $\Delta_{2}$-ring if and only if $n$ divides $24$.
∎
Lemma 2.5.
[5]
Let $p$ be a prime. A field $k$ is $\Delta_{p}$ if and only if $k=\mathbb{F}_{2}$ or $k=\mathbb{F}_{3}$ with
$p=2$, or $k=\mathbb{F}_{2}$ or $\mathbb{F}_{p+1}$ with $p$ a Mersenne prime. In particular, these conditions on $k$ hold for any $k$-algebra that is $\Delta_{p}$.
An Artinian ring is a ring which satisfies the descending chain condition on its ideals. That is, for every descending chain of ideals $I_{1}\supseteq I_{2}\supseteq\cdots\supseteq I_{k}\supseteq\cdots$ in $R$, there is an integer $k$ such that $I_{n}=I_{n+1}$ for all $n>k$. It is clear that finite commutative rings and finite-dimensional $k$-algebras are Artinian. Artinian rings have the following Artin decompositon [1]: every Artinian ring $R$ is a direct product of Artin local rings $R_{i}$
$$R=R_{1}\times R_{2}\times\cdots R_{n}.$$
Moreover, an Artinian local ring has a unique prime ideal. The following lemma will be used in our analysis.
Lemma 2.6.
Let $R$ be an Artinian ring with a unique prime ideal $P$. Then $P$ is the set of all nilpotent elements, and $R\setminus P$ is the set of all units. Moreover, $R$ is generated as a ring by its units. In particular, when $R^{*}$ is an elementary abelian $p$-group of rank $t$, $R$ is a quotient of the group ring $\mathbb{Z}_{n}[C_{p}^{t}]$, where $n$ is the characteristic of $R$.
Proof.
The first statement is well-known. To see that $R\setminus P$ generates $R$, note that for every nilpotent $\eta$, the element $u:=1+\eta$ is a unit. This shows that $\eta=u-1$, proving that the units generate all elements of the ring. Let $n\,(\geq 0)$ be the characteristic of $R$. When $R^{*}=R\setminus P=C_{p}^{t}$ generates $R$, there is a surjective ring homomorphism
$$Z_{n}[C_{p}^{l}]\rightarrow R.$$
This shows that $R$ is a quotient of the group ring $\mathbb{Z}_{n}[C_{p}^{t}]$.
∎
Theorem 2.7 (Wedderburn-Artin).
Any semisimple ring $R$ is isomorphic to a product of finitely many matrix
rings over division rings $D_{i}$ for some dimension $n_{i}$, both of which are uniquely determined
by permutation of $i$, i.e.
$$\displaystyle R=\prod M_{n_{i}}(D_{i}).$$
The most famous semisimple rings come from Maschke’s theorem, which states that the group algebra $kG$ is semisimple when the characteristic of the field $k$ is relatively prime to the order of the finite group $G$.
Corollary 2.8.
Every finite commutative semisimple ring is a direct product of finite fields. In particular, any quotient $\mathbb{F}_{p}G/\!\!\sim$ of a group algebra $kG$, where $G$ is a group whose order is relatively prime to $p$, is a finite product of finite fields of characteristic $p$.
Proof.
By Wedderburn-Artin theorem, every semisimple ring $R$ is a direct product of matrix rings over division rings. If $R$ is commutative, these matrix rings have order $1\times 1$. That is, $R$ is a product of division rings. Furthermore, when $R$ is finite, by Wedderburn’s little theorem, we know that finite division rings are fields. This shows that $R$ is a product of finite fields. When $p$ and $G$ are as given, Maschke’s theorem implies that $\mathbb{F}_{p}G$ is a product of finite fields of characteristic $p$. Since a quotient of a product of fields is again a product (with possibly fewer factors), the second statement follows.
∎
3. Finite dimensional commutative $k$-algebras
We want to classify all finite-dimensional $k$-algebras that are $\Delta_{p}$. We begin with case $p=2$.
For any ring, $A$, $A/\!\!\sim$ will denote an arbitrary quotient of $A$.
Theorem 3.1.
Let $R$ be a finite dimensional commutative $k$-algebra. Then $R$ is $\Delta_{2}$ if and only if $R\cong\prod\mathbb{F}_{3}$ or a quotient of $\prod\mathbb{F}_{2}[x_{1},...,x_{l}]/(x_{1}^{2},...,x_{l}^{2})$.
Proof.
Let $R$ be as given and assume that it is $\Delta_{2}$. Then $R$ has an Artin decomposition:
$$R=R_{1}\times\cdots R_{l},$$
where each $R_{i}$ is a local ring. By Lemma 2.3, we know that $R$ is $\Delta_{2}$ if and only if each $R_{i}$ is $\Delta_{2}$. So, we can assume that $R$ is a finite dimensional local $k$-algebra. Moreover, by Lemma 2.5, it is enough to assume that $k=\mathbb{F}_{2}$ or $\mathbb{F}_{3}$. Artin local rings have a unique prime ideal. Therefore, Lemma 2.6 implies that $R\cong\mathbb{F}_{2}[C_{2}^{r}]/\!\!\sim$ or $R\cong\mathbb{F}_{3}[C_{2}^{r}]/\!\!\sim$.
Let us first consider the case $R\cong\mathbb{F}_{2}[C_{2}^{r}]/\!\!\sim$. Note that
$$\mathbb{F}_{2}[C_{2}^{r}]\cong\mathbb{F}_{2}[x_{1},\cdots,x_{r}]/(x_{1}^{2}-1,\cdots,x_{r}^{2}-1)\cong\mathbb{F}_{2}[t_{1},\cdots,t_{r}]/(t_{1}^{2},\cdots,t_{r}^{2}),$$
The first isomorphism sends a generator in the $i$th factor in $C_{2}^{r}$ to $x_{i}$. The second isomorphism is obtained using a change of variables: $t_{i}=x_{i}-1$. Note that $t_{i}^{2}=(x_{i}-1)^{2}=x_{i}^{2}-2x_{i}+1=x_{i}^{2}-1$ in a field with characteristic 2. Because $R$ is isomorphic to a quotient of ${F}_{2}[C_{2}^{r}]$, we see that
$R\cong\mathbb{F}_{2}[t_{1},\cdots,t_{r}]/I$, where $\langle t_{1}^{2},\cdots,t_{r}^{2}\rangle\subseteq I$. Now let $x$ be a unit in $R$. Using Lemma 2.6, we can write $x$ as $\overline{1+\eta}$ for some $\eta$ in $(t_{1},t_{2},\cdots t_{r})$. Then $x^{2}=\overline{1^{2}+\eta^{2}}=\overline{1}$ because characteristic of $R$ is $2$ and $\eta^{2}=0$ in $R$. This shows that $R$ is a $\Delta_{2}$-ring.
The second case is $R\cong\mathbb{F}_{3}[C_{2}^{r}]/\!\!\sim$. In this case, Corollary 2.8 implies that $R\cong\prod\mathbb{F}_{3^{r_{i}}}$. Taking units, we see that
$$R^{*}\cong\prod(\mathbb{F}_{3^{r_{i}}})^{*}\cong\prod C_{3^{r_{i}}-1}.$$
This is an elementary abelian $2$-group if and only if $3^{r_{i}}-1=2$, or $r_{i}=1$ for all $i$. This shows that $R\cong\prod\mathbb{F}_{3}$, which is clearly $\Delta_{2}$.
∎
Remark 3.2.
Note that in the above theorem, we classified the finite-dimensional $\mathbb{F}_{2}$-algebras that are $\Delta_{2}$ in terms of the quotients of the ring $\mathbb{F}_{2}[x_{1},...,x_{l}]/(x_{1}^{2},...,x_{l}^{2})$. So, a natural question is: what are all quotients of this ring?
The quotients of this ring correspond to the ideals of the polynomial ring $\mathbb{F}_{2}[x_{1},...,x_{l}]$ that contain the ideal $(x_{1}^{2},...,x_{l}^{2})$. These ideals form a partially ordered set under inclusion.
A complete description of this lattice seems hopeless. Even the sublattice of ideals generated by sets of monomials in $\mathbb{F}_{2}[x_{1},...,x_{l}]$ is not well-understood. The latter is the free distributive lattice ([8]) $\text{FD}(l)$ on $l$ generators, and computing its cardinality is the Dedekind problem, which is open. The exact values are known only for $1\leq l\leq 8$: 3, 6, 20, 168, 7581, 7828354, 2414682040998, 56130437228687557907788 (sequence A000372 in the OEIS).
Here are the Hasse diagrams for the poset of ideals for $l=2$. The left diagram corresponds to the lattice of all ideals of the quotient ring $\mathbb{F}_{2}[x_{1},x_{2}]/(x_{1}^{2},x_{2}^{2})$, the right one to those ideals that are generated by monomials.
$\textstyle{(1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(x_{1},x_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(x_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(x_{1}+x_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(x_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(x_{1}x_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(0)}$
$\textstyle{(1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(x_{1},x_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(x_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(x_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(x_{1}x_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(0)}$
The actual number of quotients of the ring $\mathbb{F}_{2}[x_{1},...,x_{l}]/(x_{1}^{2},...,x_{l}^{2})$ is at least as big as $|\text{FD}(l)|$ and it can be computed using GAP for small values of $l$. This gave $3,7,47,4979$ for $1\leq l\leq 4$.
Theorem 3.3.
Let $R$ be a finite dimensional commutative $k$-algebra and let $p$ be an odd prime. $R$ is a $\Delta_{p}$-ring if and only if $p$ is a Mersenne prime and $R=(\mathbb{F}_{2})^{a}\times(\mathbb{F}_{p+1})^{b}$ for some nonnegative integers $a$ and $b$.
Proof.
Let $R$ be as stated in the theorem and assume that $R$ is $\Delta_{p}$ for some odd prime $p$. As argued in the proof of the above theorem, we may assume that $R$ is a local Artin $k$-algebra.
Then by Lemma 2.5, $k$ is either $\mathbb{F}_{2}$ or $\mathbb{F}_{p+1}$ where $p$ is a Mersenne prime. Then Lemma 2.6 implies that that
$R\cong\mathbb{F}_{2}[C_{p}^{r}]/\!\!\sim$ or $R\cong\mathbb{F}_{p+1}[C_{p}^{r}]/\!\!\sim$.
Taking products of local algebras and using Lemma 2.8 we get the classification stated in the theorem because the only finite fields that are $\Delta_{p}$ are $\mathbb{F}_{2}$ and $\mathbb{F}_{p+1}$ with $p$ Mersenne.
∎
Corollary 3.4.
An odd prime $p$ is Mersenne if and only if there exists a finite-dimensional commutative $k$-algebra that is a $\Delta_{p}$-ring.
4. Finite commutative rings
Let $R$ be a finite commutative ring. Note that $R$ is Artinian, and therefore it can be written as a product of Artin local rings: $R=R_{1}\times R_{2}\cdots\times R_{l}$. Recall that $R$ is $\Delta_{2}$ if and only if $R_{i}$ is $\Delta_{2}$ for all $i$.
So, we may assume that $R$ is finite and has only one prime ideal.
Let $n$ be the characteristic of $R$. Then Lemma 2.4 implies that $n$ divides $24$.
If $n=2$ or $3$, then $R$ is a finite-dimensional $\mathbb{F}_{2}$ or $\mathbb{F}_{3}$-algebra. By the previous section, we know that $R$ is either a product of $\mathbb{F}_{3}$’s or a quotient of a product of rings of the form $\mathbb{F}_{2}[x_{1},x_{2},\cdots x_{l}]$. By the Chinese Remainder Theorem, every ring of characteristic 6 is a product of two rings, one of characteristic 2 and one of characteristic 3. So this also takes care of rings of characteristic 6.
It is enough to consider rings of characteristics 4 and 8. The $\Delta_{2}$ rings in these two cases, when combined with products of $\mathbb{F}_{3}$, will complete the cases of characteristics 12 and 24 using the Chinese Remainder Theorem.
By Lemma 2.6, any local Artin ring of characteristic $4$ that is $\Delta_{2}$ must be a quotient of
$$\mathbb{Z}_{4}[C_{2}^{l}]\cong\mathbb{Z}_{4}[x_{1},x_{2},\cdots x_{l}]/(x_{1}^{2}-1,x_{2}^{2}-1,\cdots x_{l}^{2}-1).$$
So, what are all the $\Delta_{2}$-quotients? We begin by determining the unique prime ideal of this ring.
Lemma 4.1.
The ring $R=\mathbb{Z}_{4}[x_{1},x_{2},\cdots x_{l}]/(x_{1}^{2}-1,x_{2}^{2}-1,\cdots x_{l}^{2}-1)$ has only prime ideal $\mathcal{P}=(2,x_{1}-1,x_{2}-1,\cdots x_{l}-1)$.
Proof.
Let $P$ be a prime ideal of $R$. Since we are working in characteristic $4$, we have $2^{2}=4=0\in P$. Since $P$ is a prime ideal, this means $2$ belongs to $P$. For all $i$, we have $x_{i}^{2}-1$ and $2$ belong to $P$. This means $(x_{i}^{2}-1)+2=x_{i}^{2}+1$, and hence $x_{i}^{2}+1-2x_{i}=(x_{i}-1)^{2}$ belong to $P$. This shows that $x_{i}-1$ is in $P$ for all $i$. The ideal generated by $(2,x_{1}-1,\cdots x_{l}-1)$ is maximal in $\mathbb{Z}_{4}[x_{1},x_{2},\cdots x_{l}]$ because the quotient is the residue field $\mathbb{F}_{2}$. This shows that $P=(2,x_{1}-1,\cdots x_{l}-1)$.
∎
The next proposition characterizes the quotients of $\mathbb{Z}_{4}[x_{1},x_{2},\cdots x_{l}]/(x_{1}^{2}-1,x_{2}^{2}-1,\cdots x_{l}^{2}-1)$ that are $\Delta_{2}$.
Proposition 4.2.
Let $R=\mathbb{Z}_{4}[x_{1},x_{2},\cdots x_{l}]/(x_{1}^{2}-1,x_{2}^{2}-1,\cdots x_{l}^{2}-1)$. $S$, a quotient of $R$, is a $\Delta_{2}$-ring if and only if
$S=R/J$ where $J$ contains $(\eta(\eta+2)\colon\eta\in P)$.
Proof.
Note that any quotient of $R$ is of the form $S=R/J$ where $(x_{1}^{2}-1,\cdots x_{l}^{2}-1)\subseteq J\subseteq P$. Let $u$ be a unit of $S$. Then since $S$ has a unique prime ideal $P$ and has characteristic $4$, the residue field is $\mathbb{F}_{2}$. We conclude that $u=\overline{1+\eta}$ for some $\eta$ in $P$. $u^{2}=1$ in $S$, forces $\eta^{2}+2\eta=\eta(\eta+2)$ in $J$.
Conversely, let $S$ be a quotient of $R$ as given. Since $S$ is local, every unit $u$ in $S$ is of the form $u=\overline{1+\eta}$ where $\eta\in P$. $u^{2}=\overline{1+\eta^{2}+2\eta}=\overline{1+\eta(\eta+2)}$. This last expression is equal to $\overline{1}$ in $S$, by construction. This shows that $S$ is $\Delta_{2}$.
∎
In characteristic 8, we have the following result similar to the previous with almost identical proof. We use the fact that $2^{3}=0$ in $\mathbb{Z}_{8}$.
Proposition 4.3.
$R=\mathbb{Z}_{8}[x_{1},x_{2},\cdots x_{l}]/(x_{1}^{2}-1,x_{2}^{2}-1,\cdots x_{l}^{2}-1)$. $S$, a quotient of $R$, is a $\Delta_{2}$-ring if and only if
$S=R/J$ where $J$ contains $(\eta(\eta+2)\colon\eta\in P)$. (Note: $P$ is the unique prime ideal of $R$.)
Packing all the above lemmas and propositions, we get the following theorem that gives a complete characterization of finite commutative rings that are $\Delta_{2}$.
Theorem 4.4.
Let $R$ be a finite commutative ring that is $\Delta_{2}$. Then the characteristic of $R$ is a divisor of $24$. In each characteristic $n$ that divides $24$, the finite commutative rings that are $\Delta_{2}$ are all quotients (with characteristic $n$) of the rings shown in Table 1.
Note: $\mathcal{P}$ is the prime ideal $(2,x_{1}-1,x_{2}-1,\cdots,x_{l}-1)$ in the corresponding polynomial ring. $\prod A_{l}$ denotes a finite product of rings that are of the form $A_{l}$.
Theorem 4.5.
A finite commutative ring $R$ is is a $\Delta_{p}$-ring if and only if
Let $R$ be a finite commutative ring and let $p$ be an odd prime. $R$ is a $\Delta_{p}$-ring if and only if
$p$ is Mersenne prime and $R$ is isomorphic to $(\mathbb{F}_{2})^{a}\times(\mathbb{F}_{p+1})^{b}$ for some $a$ and $b\geq 0$.
Proof.
First note that $\Delta_{p}$, for $p$ odd, can occur only when $p$ is a Mersenne prime. So, let $p=2^{l}-1$ for some $l$. As before, we may assume without loss of generality that $R$ is a local Artin ring. Let $n$ be the characteristic of $R$. Then $\mathbb{Z}_{n}$ is a $\Delta_{p}$-ring. That is $u^{p}=1$ for all $u$ in $\mathbb{Z}_{n}^{*}$. If $n\neq 2$, $\mathbb{Z}_{n}^{*}$ will have an element of order $2$, which is impossible in a $\Delta_{p}$-ring. This shows that $n=2$. By Lemma 2.6, $R$ then has to be to be a quotient of $\mathbb{F}_{2}[C_{p}^{l}]$.
By 2.8, a quotient of $\mathbb{F}_{2}[C_{p}^{l}]$ must be a product of fields of characteristic $2$. Since the multiplicative group must be a elementary abelian $p$-group, the finite fields can be either $\mathbb{F}_{2}$ or $\mathbb{F}_{p+1}$ with $p$ Mersenne. This shows that $R$ must have the form stated in the theorem. Conversely, it is easy to see that these rings have elementary abelian $2$-groups as their unit groups.
∎
5. Modular group algebras: $\mathbb{Z}_{n}G$
Theorem 5.1.
The group algebra $\mathbb{Z}_{n}G$ is $\Delta_{2}$ if and only if $n\in\{2,3,6\}$ and
$G=C_{2}^{r}$ or $n\in\{4,12\}$ and $G=C_{2}$.
Proof.
Let $\mathbb{Z}_{n}G$ be a $\Delta_{2}$-ring. Then $\mathbb{Z}_{n}$ must be $\Delta_{2}$, as it is
a subring of $\mathbb{Z}_{n}G$. Then by Lemma 2.4, we know that $n$ divides 24.
Note that every element of $G$ is a unit of $\mathbb{Z}_{n}G$, thus we have $g^{2}=e$ for any $g\in G$. Lemma 2.1 implies that $G\cong C_{2}^{r}$, an elementary abelian $2$-group.
Let $C_{2}=\{e,\sigma\}$, and denote the
$i$th generator of $C_{2}^{r}$ by $\sigma_{i}$. We proceed by checking each divisor of 24. We use the fact that the sum of a unit and a nilpotent element in any commutative ring is a unit.
•
$\mathbb{Z}_{2}C_{2}^{r}$ and $\mathbb{Z}_{3}C_{2}^{r}$ are $\Delta_{2}$-rings from [5, Theorem 2.3].
•
Note that $\mathbb{Z}_{8}C_{2}^{r}$ is not $\Delta_{2}$, as $u=e+2\sigma$ is a unit,
but $u^{2}=5e+4\sigma\neq e$.
•
Similarly, $\mathbb{Z}_{24}C_{2}^{r}$ is not $\Delta_{2}$, as $u=e+6\sigma$ is a unit,
but $u^{2}=13e+12\sigma\neq e$.
•
$\mathbb{Z}_{6}C_{2}^{r}$ is isomorphic to $\mathbb{Z}_{2}C_{2}^{r}\times\mathbb{Z}_{3}C_{2}^{r}$ via the
Chinese Remainder Theorem, thus because each of $\mathbb{Z}_{2}C_{2}^{r}$ and $\mathbb{Z}_{3}C_{2}^{r}$
are $\Delta_{2}$, we know that $\mathbb{Z}_{6}C_{2}^{r}$ is $\Delta_{2}$.
•
To show that $\mathbb{Z}_{4}C_{2}$ is $\Delta_{2}$, consider the augmentation
map $\epsilon:\mathbb{Z}_{4}C_{2}\to\mathbb{Z}_{4}$ defined by $\sum\alpha_{g}g\mapsto\sum\alpha_{g}$.
Then $\epsilon$ is a ring homomorphism, so it must map unit elements of $\mathbb{Z}_{4}C_{2}$ to
unit elements of $\mathbb{Z}_{4}$. The only unit elements of $\mathbb{Z}_{4}$ are
$\pm 1$, and it can be verified that each of the elements in $\{e,\sigma,3\sigma,e+2\sigma,2e+\sigma,2e+3\sigma,3e,3e+2\sigma\}$ are units of $\mathbb{Z}_{4}C_{2}$ satisfying
$u^{2}=e$.
However,
for $\mathbb{Z}_{4}C_{2}^{r}$ where $r>1$, note that $\sigma_{1}+\sigma_{2}$ is a nilpotent
as $(\sigma_{1}+\sigma_{2})^{4}=0$. Then $u=e+\sigma_{1}+\sigma_{2}$ is a unit in
$\mathbb{Z}_{4}C_{2}^{r}$, but $u^{2}=3e+2\sigma_{1}+2\sigma_{2}+2\sigma_{1}\sigma_{2}\neq e$.
•
We know by the Chinese Remainder Theorem that for all $r\geq 1$, $\mathbb{Z}_{12}C_{2}^{r}\cong\mathbb{Z}_{4}C_{2}^{r}\times\mathbb{Z}_{3}C_{2}^{r}$ (because $4$ and $3$ are relatively prime). When $r=1$, we know from previous cases that both $\mathbb{Z}_{4}C_{2}$ and $\mathbb{Z}_{3}C_{2}$ are $\Delta_{2}$. It follows that that $\mathbb{Z}_{12}C_{2}$ is $\Delta_{2}$ as well.
However, when $r>1$, $\mathbb{Z}_{4}C_{2}^{r}$ was shown to be not $\Delta_{2}$. Therefore, $\mathbb{Z}_{12}C_{2}^{r}$ also can’t be $\Delta_{2}$.
∎
Theorem 5.2.
Let $p$ be an odd prime.
$\mathbb{Z}_{n}G$ ($G$ abelian) is a $\Delta_{p}$-ring if and only if $n=2$, $G$ is an elementary abelian $p$-group, and $p$ is some Mersenne prime.
Proof.
Let $\mathbb{Z}_{n}G$ with $G$ abelian be a $\Delta_{p}$-ring.
Since $G$ is abelian and $\mathbb{Z}_{n}G$ is $\Delta_{p}$, every non-trivial element in $G$ has order $p$. Clearly, this means $G$ has to be an elementary abelian $p$-group. Since the subring $\mathbb{Z}_{n}$ must also be a $\Delta_{p}$-ring, the characteristic $n$ must be $2$, because otherwise we will have $-1(\neq 1)$, a unit element in the ring of order $2$, impossible in a $\Delta_{p}$-ring with $p$ odd. So our group ring is $\mathbb{F}_{2}C_{p}^{r}$. This is $\Delta_{p}$ if and only if $p$ is Mersenne; [5].
∎
The next proposition gives an ideal-theoretic explanation for why $\mathbb{Z}_{4}[C_{2}]$ is $\Delta_{2}$, but $\mathbb{Z}_{4}[C_{2}^{l}]$ is not $\Delta_{2}$ when $l>1$. In the ring $\mathbb{Z}_{4}[x_{1},\cdots,x_{l}]$, consider the following ideals.
(1)
$P_{l}:=(2,x_{1}-1,x_{2}-1,\cdots x_{l}-1)$
(2)
$J_{l}:=(n(n+2):n\in P_{l})$
(3)
$I_{l}:=(x_{1}^{2}-1,\cdots x_{l}^{2}-1)$
Proposition 5.3.
Let $l$ be a positive integer. The following are equivalent.
(1)
$l=1$.
(2)
$\mathbb{Z}_{4}[C_{2}^{l}]$ is a $\Delta_{2}$-ring.
(3)
$I_{l}=J_{l}$.
Proof.
We have already seen the equivalence of (1) and (2). We will show that (1) and (3) are equivalent.
To this end, we have to show that, in the ring $\mathbb{Z}_{4}[x]$, $(x^{2}-1)=(n(n+2)\colon\colon n\in P)$ where $P=(2,x-1)$. The inclusion $(x^{2}-1)\subseteq(n(n+2)\colon n\in P)$ is obvious because $x^{2}-1=(x-1)(x+1)=(x-1)((x-1)+2)$. For the other inclusion, it is enough to show that for all $n$ in $P$, the element $n(n+2)$ is $0$ in the quotient ring $\mathbb{Z}_{4}[x]/(x^{2}-1)$. Note that all multiples of 4 will be 0 and $x^{2}=1$ in the quotient ring. Keeping this in mind, consider an arbitrary element $n(n+2)$ in this quotient ring, where $n=2(c+bx)+(x-1)(a+bx)$ is in $P$. Then we have the following equations.
$$\displaystyle n(n+2)$$
$$\displaystyle=$$
$$\displaystyle n^{2}+2n$$
$$\displaystyle=$$
$$\displaystyle(2(c+dx)+(x-1)(a+bx))^{2}+2(2(c+dx)+(x-1)(a+bx))$$
$$\displaystyle=$$
$$\displaystyle(x-1)^{2}(a+bx)^{2}+2(x-1)(a+bx)$$
$$\displaystyle=$$
$$\displaystyle 2(1-x)(a^{2}+b^{2})+2(x-1)(a+bx)$$
$$\displaystyle=$$
$$\displaystyle 2(1-x)(a^{2}+b^{2}-a-bx)$$
The last expression is directly seen to be zero in our quotient ring for any choice of $a$ and $b$ in $\mathbb{Z}_{4}$. For $l>1$, we claim that $I_{l}\subsetneq J_{l}$. It is clear that $I_{l}\subseteq J_{l}$. To see that the inclusion is strict, note that, for $l>1$, from the above results we have $\mathbb{Z}_{4}[x_{1},\cdots x_{l}]/J_{l}$ is a $\Delta_{2}$-ring but $\mathbb{Z}_{4}[x_{1},\cdots x_{l}]/I_{l}$ is not a $\Delta_{2}$-ring. This shows that $I_{l}\subsetneq J_{l}$ for $l>1$.
∎
6. Path algebras: $kQ$
Let $Q$ be a quiver (a directed, not necessarily simple graph), and $k$ be a field.
We define the path algebra $kQ$ to be a vector space
over $k$ with basis given by paths in $Q$, including the trivial paths of length $0$ starting
and ending at the same vertex (we will denote the trivial path of length $0$ starting and ending
at a vertex $i$ by $e_{i}$). For any two paths $p,q$, we define the multiplication $pq$ to be
the concatenation of $p$ and $q$ if $t(q)=s(p)$, and 0 otherwise; here $t(q)$ is the tail of $q$ and $s(p)$ is the head of $p$.
The existence of an identity element is guaranteed in any finite quiver. Moreover, the path algebra $kQ$ of a quiver $Q$ will be finite-dimensional if and only if $Q$ is acyclic. So, we will assume that our quivers are finite and acyclic.
Lemma 6.1 (Karthika-Viji [9]).
Let $k$ be a field and $Q$ be a finite acyclic quiver. Then the identity element of
the path algebra $kQ$ is given by $\sum_{i\in V(Q)}e_{i}$. We denote this element by $e$.
The following theorem gives a useful characterization of units in path algebras.
Theorem 6.2 (Karthika-Viji [9]).
Let $k$ be a field, and $Q$ be a finite acyclic quiver.
Then an element $a\in kQ$ is a unit if and only if the coefficient of $e_{i}$ is nonzero for all vertices $i\in Q$.
The following two results are a complete classification of all $\Delta_{p}$-path algebras.
Theorem 6.3.
Let $k$ be a field and $Q$ be a finite acyclic quiver. Then $kQ$ is $\Delta_{2}$ if and only if
$k=\mathbb{F}_{3}$ and $Q$ contains no edges, or $k=\mathbb{F}_{2}$ and $Q$ has no directed paths of length $2$.
Proof.
Let $k=\mathbb{F}_{3}$ and $Q$ be a quiver containing no nontrivial path. Then by Theorem 6.2,
the only unit in $kQ$ is the identity $e=\sum e_{i}$. Then we see that $e^{2}=e$, as
$e$ is the identity.
Let $k=\mathbb{F}_{3}$ and $Q$ be a quiver containing a directed edge $p$. By Theorem 6.2, $e+p$ is a unit, but $(e+p)^{2}=e^{2}+ep+pe+p^{2}=e+p+p+0=e+2p\neq e$. Thus $kQ$ is not $\Delta_{p}$.
Let $k=\mathbb{F}_{2}$ and $Q$ be a quiver containing a directed path $\beta\alpha$ comprised of edges $\alpha,\beta$. Then
by Theorem 6.2, $e+\alpha+\beta$ is a unit, but $(e+\alpha+\beta)^{2}=e^{2}+2\alpha+2\beta+\beta\alpha\neq e$.
If $k=\mathbb{F}_{2}$ and $Q$ is a quiver with no directed paths of length two, then for any two paths $\alpha,\beta$,
we have $\alpha\beta=0$. Let $p$ be the sum of any paths. Then $e+p$ is a unit, but
$(e+p)^{2}=e+2p=e$ as desired.
∎
Example 6.4.
Here is an example of a quiver on $5$ vertices for which the path algebra over $\mathbb{F}_{2}$ or $\mathbb{F}_{3}$ is a $\Delta_{2}$-ring.
$$\begin{tikzcd}$$
Note that this quiver has no directed path of length $2$ or more.
Theorem 6.5.
Let $p$ be an odd prime, $k$ be a field, and $Q$ be a finite acyclic quiver. The path algebra
$kQ$ is $\Delta_{p}$ if and only if $Q$ is trivial and $k=\mathbb{F}_{2}$ or $k=\mathbb{F}_{p+1}$ where $p$ is a Mersenne prime.
Proof.
Let $kQ$ be a $\Delta_{p}$-ring for some odd prime $p$. Note that $kQ$ is a $k$-algebra. Then by Corollary 2.5, $k$ must be $\mathbb{F}_{2}$ or $k=\mathbb{F}_{p+1}$ where $p$ is Mersenne. We claim that $Q$ cannot have any directed edges. Suppose to the contrary, $Q$ has some directed edge $\alpha$. Consider the element $u=e+\alpha$. Then
$u$ is a unit by Lemma 6.2, and $u^{2}=(e+\alpha)^{2}=e+2\alpha$.
Since the characteristic of $k$ is 2, $2\alpha=0$, thus $u^{2}=e$ and we see that $u$ has order 2. This contradicts the fact that $kQ^{*}=C_{p}^{r}$, as every element of $kQ^{*}$ should have order $p$. This proves one direction. The other direction is obvious because the path algebra $kQ$ of a trivial quiver is a product of copies of $k$, and the direct product of fields stated in the theorem is a $\Delta_{p}$-ring.
∎
References
[1]
M. F. Atiyah and I. G. Macdonald.
Introduction to commutative algebra.
Addison-Wesley Series in Mathematics. Westview Press, Boulder, CO,
economy edition, 2016.
[2]
Jon F. Carlson, Sunil K. Chebolu, and Ján Mináč.
Freyd’s generating hypothesis with almost split sequences.
Proc. Amer. Math. Soc., 137(8):2575–2580, 2009.
[3]
Sunil K. Chebolu.
What is special about the divisors of 24?
Math. Mag., 85(5):366–372, 2012.
[4]
Sunil K. Chebolu and Keir Lockridge.
How many units can a commutative ring have?
Amer. Math. Monthly, 124(10):960–965, 2017.
[5]
Sunil K. Chebolu, Keir Lockridge, and Gaywalee Yamskulna.
Characterizations of Mersenne and 2-rooted primes.
Finite Fields Appl., 35:330–351, 2015.
[6]
Sunil K. Chebolu and Michael Mayers.
What is special about the divisors of 12?
Math. Mag., (2), 2013.
[7]
Robert W. Gilmer, Jr.
Finite rings having a cyclic multiplicative group of units.
Amer. J. Math., 85:447–452, 1963.
[8]
George Grätzer.
Lattice theory: foundation.
Birkhäuser/Springer Basel AG, Basel, 2011.
[9]
S. Karthika and M. Viji.
Unit elements in the path algebra of an acyclic quiver.
Indian J. Pure Appl. Math., 52(1):138–140, 2021. |
Acceleration of cosmic rays at supernova remnant shocks: constraints from gamma-ray observations
M. Lemoine-Goumard
Centre d’Études Nucléaires de Bordeaux Gradignan
Université Bordeaux 1, CNRS/IN2P3
33175 Gradignan, France
E-mail: lemoine@cenbg.in2p3.fr
Funded by contract ERC-StG-259391 from the European Community
Abstract
In the past few years, gamma-ray astronomy has entered a golden age. At TeV energies, only a handful
of sources were known a decade ago, but the current generation of ground-based imaging
atmospheric Cherenkov telescopes has increased this number to more than one hundred. At GeV
energies, the Fermi Gamma-ray Space Telescope has increased the number of known sources by
nearly an order of magnitude in its first 2 years of operation. The recent detection and
unprecedented morphological studies of gamma-ray emission from shell-type supernova remnants
is of great interest, as these analyses are directly linked to the long standing issue of the origin of the
cosmic-rays. However, these detections still do not constitute a conclusive proof that supernova remnants
accelerate the bulk of Galactic cosmic-rays, mainly due to the difficulty of disentangling the hadronic
and leptonic contributions to the observed gamma-ray emission. In this talk, I will review the most
relevant cosmic ray related results of gamma ray astronomy concerning supernova remnants.
keywords: cosmic-rays; supernova remnants
\bodymatter
1 The cosmic-ray mystery
1.1 The link between cosmic-rays and supernova remnants
The association between supernova remnants (SNRs) and Galactic cosmic rays (CRs) is very popular since 1934,
when Baade and Zwicky argued that this class of astrophysical objects can account for the required CR energetics [\refcitebaade].
Indeed, in order to maintain the cosmic-ray energy density in the Galaxy, about 3 supernovae per century should transform 10 percent of their kinetic energy in cosmic-ray energy. This argument has also been supported by E. Fermi’s proposal of a very general mechanism for particle acceleration, which is very efficient if applied at SNR shocks [\refcitebell]. The extremely interesting point of the diffusive shock acceleration (DSA) mechanism is that it naturally yields power-law spectra for the energy distribution of accelerated particles. However, until recently there were absolutely no observational evidence concerning the
acceleration of protons and nuclei in SNRs. Indeed, through their interaction with the interstellar magnetic fields, the charged particles arriving on Earth have lost all directional information and cannot be used to pinpoint the sources. That is why, almost 100 years after their discovery by V. Hess, the origins of the cosmic-rays and their cosmic accelerators remain
unknown.
Astronomy with gamma-rays provides a means to study these sources of high energy particles. Indeed,
cosmic rays (ionized nuclei of all species, but mostly protons, plus a small fraction of electrons) can interact with ambient matter and photons producing
gamma-rays via two different channels. One mechanism invokes the interaction of accelerated protons at supernova remnants shocks with interstellar material generating neutral pions which in turn decay into gamma rays. We call this mechanism the hadronic scenario. A second competing channel exists in the inverse Compton scattering of the photon fields in the surroundings of the SNR by the same relativistic electrons that generate the synchrotron X-ray emission. This is the leptonic scenario. Being of leptonic or hadronic origin, these gamma-rays are not affected while they travel to Earth and can therefore be used to pinpoint the cosmic accelerators in our Galaxy.
1.2 Gamma-ray experiments
Two major breakthroughs in gamma-ray astronomy occurred in recently.
Firstly, after more
than 20 years of development, the first source of very high energy gamma-rays, the Crab Nebula, was
discovered in 1989 by the Whipple telescope. Since this date the technical progresses in this field have led to
important scientific results, especially by the Cherenkov telescopes H.E.S.S., VERITAS and MAGIC. These ground-based experiments for gamma-ray astronomy rely
on the development of cascades (air-showers) initiated by astrophysical gamma-rays. Such cascades only persist to ground-level above 1 TeV and only produce significant Cherenkov light above a few GeV, setting a fundamental threshold to the range
of this technique. Today, more than 120 gamma-ray sources have been detected with high significance, 17 being associated to supernova remnants or molecular clouds.
Second, in space, the Large Area Telescope (LAT) onboard the Fermi satellite has considerably improved our knowledge of the 0.1-100 GeV gamma-ray sky with 1873 objects detected in only two years of observation [\refcite2FGLcat]. It has moved the field from the detection of a small number of sources to the detailed study of several classes of Galactic and extragalactic objects. A complete study of association of the 1873 sources detected show that $\sim 4$% of them are associated to supernova remnants [\refcite2FGLcat].
There is no doubt today that supernova remnants can accelerate efficiently particles up to $10^{14}$ eV. The question is whether these particles are protons or electrons and if they can be accelerated up to the knee of the cosmic-ray spectrum ($10^{15}$ eV).
1.3 First evidence of efficient particle acceleration in supernova remnants with X-ray satellites
Accelerated electrons producing gamma-ray emission through inverse Compton scattering also radiate through synchrotron
emission when spiraling in a magnetic field. This emission extends from the radio to the X-ray domain. While radio synchrotron emission is observed in most SNRs (in 203 over the 217 observed Galactic SNRs, [\refcitegreen]), X-ray synchrotron emission is observed only in a few remnants up to now. In some of these X-ray detected SNRs, the X-ray synchrotron
emission exhibits a filamentary emission just behind the blast wave. One plausible explanation is that the magnetic field is
large enough ($\sim 100\,\mu$G) to induce strong radiative losses in the high energy electrons [\refcitevink, \refciteballet]. If the magnetic field is indeed amplified at the limbs, the maximum energy at which particles can be accelerated is much larger there ($>$ 1000 TeV) than outside the limbs (E $\approx$ 25 TeV if B $\approx 10\,\mu$G).
Recently, a discovery of the brightening and decay of X-ray hot spots in the shell of the SNR RX J1713.7-3946 on a one-year timescale has been reported by Uchiyama and collaborators [\refciteuchiyama]. This rapid variability implies that electron acceleration needs to take place in a strongly magnetized environment, indicating amplification of the magnetic field by a factor of more than 100.
A last evidence of very efficient particle acceleration in supernova remnants is provided by the postshock plasma temperatures observed in SNRs 1E 0102.2-7219 and RCW 86, that are
lower than expected for their measured shock velocities [\refcitehughes, \refcitehelder]. For the first time, by comparing the measured post-shock proton temperature with the one determined using the shock velocity, the authors presented the evidence that $>50$% of the post-shock pressure is produced by cosmic rays.
There are strong indirect arguments confirming that electrons and protons are accelerated up to at
least TeV energies (maybe even PeV) in supernova remnants. A direct signature of accelerated protons is
expected through pion decay emission in the GeV-TeV gamma ray range.
2 Detection of supernova remnants in gamma-rays
The sample of supernova remnants detected in gamma-rays is now extremely large: it goes from evolved supernova remnants interacting with molecular clouds (MC) up to young shell-type supernova remnants and historical supernova remnants. The Fermi-LAT even detected one evolved supernova remnants without MC interaction, Cygnus loop. This section will review the main characteristics of the detected SNRs.
2.1 Supernova remnants interacting with molecular clouds
The Fermi LAT Collaboration has so far reported the discoveries of five middle aged ($\sim 10^{4}$ yrs) remnants interacting with molecular clouds: W51C [\refcitefermi_w51], W44 [\refcitefermi_w44], IC 443 [\refcitefermi_ic443], W49 [\refcitefermi_w49b] and W28 [\refcitefermi_w28]. Apart from W44, they have all been detected in the TeV regime as well. These SNRs are generally much brighter in GeV than in TeV in terms of energy flux (due to a spectral steepening arising at a few GeV), which emphasizes the importance of the GeV observations. The interaction with a molecular cloud provides the target material that allows one to enhance the gamma-ray emission, either through bremsstrahlung by relativistic electrons or by pion-decay gamma-rays produced by high-energy protons. The observed large luminosity
of the GeV gamma-ray emission precludes the inverse-Compton scattering off the CMB and interstellar
radiation fields as the main emission mechanism since it would require an extremely low density (to suppress the bremsstrahlung and proton-proton interaction), a low magnetic field to enhance the gamma/X-ray flux ratio and an unrealistically large energy injected into protons. In addition, the break in the electron spectrum corresponding to the gamma-ray spectrum directly appears in the radio data leading to a bad modeling of the radio data and therefore disfavours the bremsstrahlung process. A model in which gamma-rays are produced via proton-proton interaction gives the most satisfactory explanation for the GeV gamma-rays observed in SNRs interacting with molecular gas as seen in Figure 1 for the case of W51C.
There are two different types of hadronic scenarios to explain the GeV gamma-ray emission arising from such SNRs: the ”Runaway CR” model [\refciteaha_escape, \refciteohira] and the ”Crushed Cloud” model [\refciteuchiyama_CR]. The Runaway CR model considers gamma-ray emission from molecular clouds illuminated by runaway CRs that have escaped from their accelerators, whereas the Crushed Cloud model invokes a shocked molecular cloud into which cosmic-ray particles are adiabatically compressed and accelerated resulting in enhanced synchrotron and pion-decay gamma-ray emissions.
2.2 Young shell-type supernova remnants
Four young shell-like SNRs with clear shell-type morphology resolved in VHE gamma-rays have been detected by H.E.S.S.: RX J1713.7-3946 [\refciteaharonian_rxj1, \refciteaharonian_rxj2], RX J0852.04622 - also known as Vela Junior - [\refciteaharonian_velajr1],
SN 1006 [\refciteacero_sn1006] and HESS J1731-347 [\refciteacero_1731]. A fifth case, RCW 86 [\refciteaharonian_rcw86], might be added to this list although the TeV shell morphology has not yet been clearly proved. Two of them, RX J1713.7-946 [\refcitefermi_rxj] and Vela Junior [\refcitefermi_velajr], have been detected by Fermi-LAT allowing direct investigation of young shell-type SNRs as sources of cosmic rays. Concerning RX J1713.7-3946, the Fermi-LAT spectrum is well described by a very hard power-law with a photon index of $\Gamma=1.5\pm 0.1$ that coincides in normalization with the steeper H.E.S.S.-detected gamma-ray spectrum at higher energies. The GeV measurements with Fermi-LAT do not agree with the expected fluxes around 1 GeV in most hadronic models published so far (e.g., Berezhko & Voelk 2010 [\refciteberezhko]) and requires an unrealistically large density of the medium. The agreement with the expected IC spectrum is better (as can be seen in Figure 2) but requires a very low magnetic field of $\sim 10\,\mu$G in comparison to the one measured in the thin filaments by X-ray observations. It is possible to reconcile a high magnetic field with the leptonic model if GeV gamma rays are radiated not only from the filamentary structures seen by Chandra, but also from other regions in the SNR where the magnetic field may be weaker. Similar conclusions are reported for Vela Junior supernova remnant even though in this case the hadronic scenario can not be ruled out. However, being of hadronic or leptonic origin, the GeV-TeV gamma-ray detections imply a low maximal energy for the accelerated particles of $\sim 100$ TeV, well below the knee of the cosmic-ray spectrum.
2.3 Historical supernova remnants
Two historical SNRs have been detected both at GeV and TeV energies: Cassiopeia A (Cas A) [\refcitefermi_casa, \refcitemagic_casa, \refciteveritas_casa] and Tycho [\refcitefermi_tycho, \refciteveritas_tycho].
Cas A is the remnant of SN 1680. It is the brightest radio source in our Galaxy and its overall brightness across the electromagnetic spectrum makes it a unique laboratory for studying high-energy phenomena in SNRs. A multiwavelength modeling of Cas A does not allow a discrimination between the hadronic and leptonic scenarios. However, regardless of the origin of the observed gamma rays, this modeling implies that the total content of CRs accelerated in Cas A is $\sim$(1 – 2)$\times 10^{49}$ erg, and the magnetic field amplified at the shock can be constrained as B $\approx$ 0.12 mG. Even though Cas A is considered to have entered the Sedov phase, the total amount of CRs accelerated in the remnant constitutes only a minor fraction ($\sim 2$%) of the total kinetic energy of the supernova, which is well below the $\sim 10$% commonly used to maintain the cosmic-ray energy density in the Galaxy.
Tycho’s SNR (SN 1572) is classified as a Type Ia (thermonuclear explosion of a white dwarf) based on observations of the light-echo spectrum. Thanks to the large amount of data available at various
wave bands, this remnant can be considered one of the most promising object where to test the shock acceleration theory and
hence the CR – SNR connection. First, using the precise radio and X-ray observations of this SNR, Morlino & Caprioli (2011) [\refcitemorlino] have shown that the magnetic field at the shock has to be $>200\mu$G to reproduce the data. Then, using multiwavenlength data, especially the GeV and TeV detections, they could infer that the gamma-ray emission detected from Tycho cannot be of leptonic origin, but has to be due to accelerated protons (this result is consistent with another modeling proposed in \refcitefermi_tycho). These protons are accelerated up to energies as large as $\sim$500 TeV, with a total energy converted into CRs estimated to be about 12 per cent of the forward shock bulk kinetic energy. This is much more reasonable in the context of acceleration of Galactic cosmic-rays in SNRs.
3 Where are the PeVatrons ?
The recent GeV and TeV detections of supernova remnants confirm the theoretical predictions that supernova remnants can operate as powerful cosmic ray accelerators. However, if these objects are responsible for the bulk of galactic
cosmic rays, they should be able to accelerate protons and nuclei at least up to $10^{15}$ eV and therefore act as PeVatrons. Gabici and Aharonian (2007) [\refcitegabici] have shown that the spectrum of nonthermal particles extends to PeV energies only during a relatively short period of the evolution of the remnant since high energy particles are the first to escape from the supernova remnant shock. For this reason one may expect spectra of secondary gamma-rays extending to energies beyond 10 TeV only from less than 1 kyr old supernova remnants. In this respect, Tycho could be considered as a half-PeVatron at least, since
there is no evidence of a cut-off in the VERITAS data. One may wonder how many PeVatrons are expected to be detectable in our Galaxy. A simple estimate has been provided by Gabici and Aharonian (2007): assuming a rate of $\sim$3 supernovae per century in our Galaxy, this directly implies that only a dozen of PeVatrons are present in the Galaxy on average and hence that they are likely to be distant and weak. This emphasizes the importance of TeV observations by the future generation of Cherenkov telescopes such as the Cherenkov Telescope Array (CTA) which will have a better effective area in the energy range already covered but that will also allow the observation up to 100 TeV of sources such as Tycho, therefore constraining the maximal energy at which protons are being accelerated in young SNRs.
Acknowledgements
I thank all the members of the Fermi GALACTIC and HESS SNR-PWN working groups for valuable discussion. I gratefully acknowledge funding from the European Community (contract ERC-StG-259391).
References |
Superconductivity in the ternary compound SrPt${}_{10}$P${}_{4}$ with complex new structure
Bing Lv${}^{1{\star}}$, BenMaan I. Jawdat${}^{2}$, Zheng Wu${}^{2}$, Sheng Li${}^{1}$, and Ching-Wu Chu${}^{3,4{\star}}$
${}^{1}$Department of Physics, University of Texas at Dallas,Richardson, TX 75080, USA
${}^{2}$Air Force Research Laboratory, Kirtland Air Force Base, Albuquerque, NM 87123, USA
${}^{3}$TcSUH and Department of Physics, University of Houston, TX 77204, USA
${}^{4}$Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA
Abstract
We report superconductivity at 1.4K in the ternary SrPt${}_{10}$P${}_{4}$ with a complex new structure. SrPt${}_{10}$P${}_{4}$ crystallizes in a monoclinic space-group C2/c (#15) with lattice parameters a= 22.9151(9)${\AA}$, b= 13.1664(5)${\AA}$, c=13.4131(5)${\AA}$, and $\beta$= 90.0270(5)${{}^{\circ}}$. Bulk superconductivity in the samples has been demonstrated through resistivity, ac susceptibility, and heat capacity measurements. High pressure measurements have shown that the superconducting T${}_{C}$ is systematically suppressed upon application of pressure, with a dT${}_{C}$/dP coefficient of -0.016 K/GPa.
pacs: 74.20.Rp, 74.70.Dd, 74.62.Dh, 65.40.Ba
I Introduction
The discovery of Fe-pnictide superconductors with T${}_{C}$ up to 57K1 ; 2 has stimulated many research efforts to search for new superconductors in other transition metal pnictide compounds and in their Fe-based layered analogs with related or unrelated structures. Among different types of Fe-pnictide discovered to date, the so-called ”122” materials3 ; 4 ; 5 with ThCr${}_{2}$Si${}_{2}$-type structures are of particular interest. More than 700 compounds of AM${}_{2}$X${}_{2}$ adopt this structures6 , and a variety of interesting physics, such as valence fluctuations7 ; 8 , heavy fermion behavior9 ; 10 , and magnetic properties11 ; 12 ; 13 , have been discussed before. Questions, therefore, are raised that whether superconductivity could be found in other materials which adopts similar type structures. In fact, many new superconducting compounds with related structures have been discovered, including BaNi${}_{2}$P${}_{2}$ (T${}_{C}$=3K)14 , BaNi${}_{2}$As${}_{2}$ (T${}_{C}$= 0.7K)15 and BaIr${}_{2}$As${}_{2}$ (T${}_{C}$=2.45K)16 , with a ThCr${}_{2}$Si${}_{2}$-type structure, SrPt${}_{2}$As${}_{2}$ (T${}_{C}$=5.2 K)17 and BaPt${}_{2}$Sb${}_{2}$ (T${}_{C}$=1.8K)18 with a CaBe${}_{2}$Ge${}_{2}$-type structure, and SrPtAs (T${}_{C}$=2.4K) with a MgB${}_{2}$-type structure19 . On the other hand, many Pt-based superconductors with unrelated structures have been discovered which include noncentrosymmetric BaPtSi${}_{3}$20 , Li${}_{2}$Pt${}_{3}$B21 ; 22 , heavy Fermion CePt${}_{3}$Si23 , and the recently discovered centrosymmetric SrPt${}_{3}$P (T${}_{C}$=8.4 K)24 . SrPt${}_{3}$P crystallizes in an antiperovskite structure that is very similar to that of CePt${}_{3}$Si but displays centrosymmetry. Further calorimetric studies suggested that it is a conventional strong electron-phonon coupling superconductor, but further theoretical calculations have brought a new perspective25 ; 26 , i.e. one may be able to tune the structure from centrosymmetric to noncentrosymmetric through chemical doping or application of pressure25 and to induce possible unconventional superconductivity in this compound. Various experimental efforts27 ; 28 ; 29 have been carried out, but unfortunately have not shown significant change of the structures. In the course of chemical doping studies of SrPt${}_{3}$P, we have discovered several new compounds, namely SrPt${}_{6}$P${}_{2}$ (T${}_{C}$=0.6K)30 and SrPt${}_{10}$P${}_{4}$. SrPt${}_{10}$P${}_{4}$ crystallizes in a very complex structure (with a total of 240 atoms in one unit cell). Resistivity, ac susceptibility, and heat capacity measurements have demonstrated bulk superconductivity at 1.4K in this new compound. High pressure measurements have shown that the superconducting T${}_{c}$ is systematically suppressed upon application of pressure, with a dT${}_{C}$/dP coefficient of -0.016 K/GPa.
II Experimental section
The polycrystalline samples were prepared by high temperature reactions of stoichiometric Sr pieces (Alfa Aesar, 99.95$\%$), Pt powder
(Alfa Aesar, $>$99.95$\%$), and prereacted PtP${}_{2}$ from Pt powder and P powder (Sigma Aldrich, $>$99.99$\%$) within an alumina crucible that was sealed inside a clean and dried quartz tube under vacuum. The tube was placed in a furnace, heated slowly up to 1000${{}^{\circ}}$C overnight, and maintained at 1000${{}^{\circ}}$C for four days before being slowly cooled down to 400${{}^{\circ}}$C with a rate of 0.5${{}^{\circ}}$C/min. The assembly was finally quenched in ice water from 400${{}^{\circ}}$C to avoid the possible formation of white phosphorus caused by any unreacted P. To improve the homogeneity, the sample was reground, pelletized, and reheated following the previously described temperature profile. All synthesis procedures were carried out within a purified Ar-atmosphere glovebox with total O${}_{2}$ and H${}_{2}$O levels $<$0.1 ppm. Single crystal XRD data was collected using a Bruker SMART APEX diffractometer equipped with 1K CCD area detector using graphite-monochromated Mo K${}_{\alpha}$ radiation. The electrical resistivity $\rho$(T, H) was measured by employing a standard 4-probe method, and heat capacity data were collected using a relaxation method down to 0.5 K under magnetic field up to 1 T using a ${}_{3}$He-attachment in a Quantum Design Physical Property Measurement System. The ac magnetic susceptibility at 15.9 Hz as a function of temperature, $\chi$(T), was measured by employing a compensated dual coil for mutual inductance measurement using the Linear Research LR 400 Bridge. High-pressure resistivity measurements up to 18 kbar were conducted using a BeCu piston-cylinder cell with Fluorinert77 as the quasihydrostatic pressure medium. A lead manometer was used to measure the pressure in situ with the LR 400 Inductance Bridge31 ; 32 .
III Results and Discussion
The as-synthesized pellet has a dark grey color, and is stable in air. On its top, many small shiny crystals with metallic luster are found, indicating that the material is a congruently melting compound, and that larger size crystals could be obtained through slow cooling process from melt. Indeed, we obtained large crystals with size up to 0.7 mm (as shown in the inset to Fig. 1), which were subsequently used for electrical resistivity and magnetic susceptibility measurements. X-ray diffraction studies reveal a c-axis preferred orientation, as shown in Fig. 1. The (004) peak is too weak to be observed, consistent with the theoretical pattern generated from crystal structures determined by X-ray single-crystal diffraction. Small blocks of shiny single crystals with a typical size of 0.04 x 0.04 x 0.06 mm${}^{3}$ were isolated for single-crystal X-ray diffraction to determine the crystal structure. The refined composition is SrPt${}_{10}$P${}_{4}$, which is consistent with SEM-WDS results of Sr:Pt:P=1:9.66(1):3.67(2). The compound crystallizes in a monoclinic space-group C2/c (#15) with lattice parameters a= 22.9151(9)${\AA}$, b= 13.1664(5)${\AA}$, c=13.4131(5)${\AA}$, $\beta$= 90.0270(5)${{}^{\circ}}$ and z=16. It should be noted that we have carried out symmetry tests on all of the refined atom positions by using the program PLATON and concluded that C2/c is the correct space group and that no higher symmetry can be found. Arbitrarily merging data to the orthorhombic Laue group mmm results in an unusually high R(int) = 0.427, as compared to the actual monoclinic Laue group 2/m with R(int) = 0.039. This further confirms that the compound indeed crystallizes in a monoclinic, not orthorhombic, structure. The detailed crystallographic information, including the Wyckoff position of individual atoms, is listed in Table I.
The structure of SrPt${}_{10}$P${}_{4}$ is rather complicated with its unit cell shown in Fig. 2a. The fundamental build unit is a 6-coordinated P-centered highly distorted Pt${}_{6}$P octahedral or trigonal prism, as seen previously in the SrPt${}_{3}$P11 and SrPt${}_{6}$P${}_{2}$15 structures. The crystal structure could be considered as two distinct types of layers intertwined with one another, shown in Fig. 2a (projection view along b axis), forming a complex three-dimensional structure. One layer consists of distorted trigonal prismatic Pt${}_{6}$P units that are edge-shared with each other and form a honeycomb-like network as seen in Fig. 2b (projection view along c axis). Sr atoms occupy the centers of the honeycomb-like network (crystallographic 4e site). The other layer is composed of a network of highly distorted Pt${}_{6}$P octahedral building blocks. These Pt${}_{6}$P octahedra are edge-shared first and forms pairs. These pairs of edge-shared octahedra are then corner shared with one another and form a network in which the Pt-Pt distance of neighboring Pt${}_{6}$P octahedra is too short ($\sim$2.668${\AA}$) to accommodate any Sr atom, as seen in Fig. 2c (projection view along c axis).
The distorted octahedral Pt${}_{6}$P is reminiscent of the distorted anti-perovskite Pt${}_{6}$P building blocks in SrPt${}_{3}$P. In SrPt${}_{3}$P, the Pt${}_{6}$P octahedra are corner-shared, arranged antipolar, and thus formed stoichiometry as ”Pt${}_{6/2}$P=Pt${}_{3}$P”. But in StPt${}_{10}$P${}_{4}$, these Pt${}_{6}$P octahedra are arranged completely differently, where they are edge-shared forming “Pt${}_{10}$P${}_{2}$”, and then corner-shared. To facilitate such an arrangement, the Pt${}_{6}$P octahedr is much more distorted than those in SrPt${}_{3}$P. Similarly, the trigonal prismatic Pt6P building blocks in SrPt${}_{10}$P${}_{4}$ are very similar to those found in SrPt${}_{6}$P${}_{2}$, but less distorted to accommodate the edge-sharing coordination in SrPt${}_{10}$P${}_{4}$, rather than corner-shared feature in SrPt${}_{6}$P${}_{2}$.
The electrical resistivity $\rho$(T) of SrPt${}_{10}$P${}_{4}$ from 270K down to 0.5K is shown in Fig. 3. The room temperature resistivity of SrPt${}_{10}$P${}_{4}$ is $\sim$ 300$\mu\Omega\cdot$cm, smaller than that of SrPt${}_{3}$P and SrPt${}_{6}$P${}_{2}$, and consistent with the expected trend toward increased metallicity with increased Pt:P ratio in the compound. The temperature dependence of the resistivity has a typical metallic behavior, with a much stronger negative curvature in the normal state than is seen in SrPt${}_{3}$P and SrPt${}_{6}$P${}_{2}$, indicating the stronger electron correlations in this material. The relatively high value of residual resistivity ratio (RRR), $\rho$(270 K)/$\rho$(1.5 K)=32, suggests that the sample is of high quality. The resistivity drops sharply to zero below 1.4 K at zero field, characteristic of a superconducting transition. The narrow width of the superconducting transition (less than 0.1 K) indicates the high quality of the sample. In the presence of a magnetic field, the superconducting transition is systematically broadened and shifted to lower temperature, and is suppressed to below 0.5K upon the application of a magnetic field of 1T, as shown in the inset to Fig. 3.
Fig. 4 shows the ac susceptibility data of StPt${}_{10}$P${}_{4}$, where a large and narrow diamagnetic shift starting from 1.4K is clearly visible. This magnetic susceptibility is shifted toward the left upon applying magnetic field as expected, and also reveals the suppression of T${}_{C}$ with increasing field.
In order to further verify the bulk nature of the superconductivity in SrPt${}_{10}$P${}_{4}$, we carried out specific heat measurements. Because the mass of the largest crystal we have is $\sim$0.1mg, which falls below the critical mass needed for the specific heat measurement, we decided to measure a pure, bulk, polycrystalline sample. Figure 5 (a) shows the raw data of this measurement under different applied magnetic fields up to 4000 Oe. The jump at T${}_{C}$ observed in the specific heat data clearly demonstrates the bulk nature of the superconductivity in the sample.
It can also be clearly seen that the superconducting and normal state specific heat data deviate from one another, which may be caused by the Schottky anomaly at low temperature range. Using C${}_{Total}$=C${}_{El}$+C${}_{P}$+C${}_{Sch}$=$\gamma_{N}$T+$\beta$T${}^{3}$ + nR($\frac{\Delta}{k_{B}T}$)${}^{2}$$\exp$($\frac{\Delta}{k_{B}T}$)/(1+$\exp$($\frac{\Delta}{k_{B}T}$))${}^{2}$, where $\Delta$ is the energy between two levels by considering spin J=1/233 , we can obtain the contributions of electrons (C${}_{El}$) and phonons (C${}_{P}$) to the total specific heat (C${}_{Total}$) by subtracting that associated with the Schottky anomaly (C${}_{Sch}$). The normal state electron and phonon contribution (Fig. 5b, line with red circles) was obtained by subtracting C${}_{Sch}$(0.4T) from the normal state C${}_{Total}$(T) at 0.4T. By fitting C${}_{Total}$(T, 0.4T) through the Debye model, we obtained a Sommerfeld coefficient $\gamma_{N}$ = 20.6 mJ/mol K${}^{2}$ and $\beta$ = 2.64 mJ/mol K${}^{4}$, which correspond to the electronic and lattice contributions to the specific heat, respectively. The Debye temperature can be deduced from the $\beta$ value through the relationship $\Theta_{D}$= (12$\pi^{4}$k${}_{B}$N${}_{A}$Z/5$\beta$)${}^{1/3}$ and the obtained Debye temperature $\Theta_{D}$= 223K. Through a similar approach, we can also obtain the zero field electron and phonon specific heat (Fig. 5b, line with black squares) by subtracting the zero field Schottky contribution C${}_{Sch}$(T, 0T). By subtracting the normal state specific heat from the superconducting one, we can get $\Delta$C${}_{El}$/T${}_{C}$ $\sim$ 21 mJ/mol K${}^{2}$, which yields a $\Delta$C${}_{El}$/$\gamma_{N}$T${}_{C}$ of about 1.02, as seen in the inset to Fig. 5b. This value is comparable to the 1.2 for SrPt${}_{6}$P${}_{2}$34 but much smaller than the 2 for SrPt${}_{3}$P. Such a small value of $\Delta$C${}_{El}$/$\gamma_{N}$T${}_{C}$ indicates weak coupling in this compound in comparison with SrPt${}_{3}$P. Our preliminary specific heat fitting and critical field analysis strongly suggest multiple gap superconductivity feature for this compound, which will be addressed separately.
The Kadowaki-Woods ratio R${}_{KW}$=A/$\gamma_{N}^{2}$ has been used to judge the correlation of a metal where A is the quadratic term of resistivity of a Fermi liquid, and $\gamma_{N}$ is the Sommerfeld coefficient of the specific heat35 . The R${}_{KW}$ is found to be a constant value for transition metals($\sim 10^{-6}\mu\Omega cmK^{2}mol^{2}/mJ^{2}$) and heavy fermion compounds($\sim 10^{-5}\mu\Omega cmK^{2}mol^{2}/mJ^{2}$)36 . We, therefore, have fitted the low temperature resistivity data using $\rho=\rho_{0}+AT^{2}$ and get A = 0.032$\mu\Omega$ cm /K${}^{2}$. Combined with the $\gamma_{N}$ value obtained above, the calculated R${}_{KW}$ is 7.5*10${}^{-5}$$\mu\Omega$ cm K${}^{2}$ mol${}^{2}$/mJ${}^{2}$, which indicates strong correlation in this material.
To probe the effects of high pressure on SrPt${}_{10}$P${}_{4}$, we applied high physical pressure using a BeCu piston-cylinder type pressure cell. Fig. 6 shows the resistivity data of SrPt${}_{10}$P${}_{4}$ under different applied pressure. The superconducting transition is slightly lower and is relatively broader with the onset of the resistivity drop at 1.33K and zero resistance at $\sim$1.25 K. The broader superconducting transition may be related to the polycrystalline nature of the samples used for this measurement (as well as possible sample degradation
during the preparation) and grain-grain coherence within the sample. A systematic suppression of the transition temperature with increasing pressure is
clearly visible. Taking a 50$\%$ drop of resistivity as the criteria of T${}_{C}$, one can obtain a linear fit of T${}_{C}$ vs. pressure, shown in the inset to Fig. 6,
which yields a pressure coefficient dT${}_{C}$/dP = -0.016 K /GPa. The systematic suppression of T${}_{C}$ suggests that there is no significant peak in the density of states near the Fermi level. The relatively small change in T${}_{C}$ with pressure is comparable to that of many elemental superconductors, which exhibit a linear suppression of T${}_{C}$ with pressure near ambient, and is very close to the value of dT${}_{C}$/dP = -0.02 K/GPa for pure niobium metal37 . The suppression of T${}_{C}$ with pressure in SrPt${}_{10}$P${}_{4}$ can therefore be explained as the result of a stiffening of the lattice induced by the pressure, which results in a weakening of the electron-phonon coupling or slightly decreasing of the density of state(DOS) at the Fermi surface.
In summary, a new ternary compound SrPt${}_{10}$P${}_{4}$ with a new structure type is synthesized through high-temperature solid state reactions, and its crystal structure is determined by X-ray single crystal diffraction. The compound crystallizes in a very complex three-dimensional structure that consists of two distinct layers based on P-centered highly distorted Pt${}_{6}$P octahedral or trigonal prismatic building units. We have carried out systematic magnetization, electrical resistivity, and specific heat measurements, and demonstrated the bulk superconductivity with T${}_{C}$ at 1.4K in this material. High pressure measurements have shown that the superconducting T${}_{C}$ is systematically suppressed upon applying pressure, with a dT${}_{C}$/dP coefficient of -0.016 K/GPa.
Acknowledgements.
The authors would like to thank X. Q. Wang for the help with single crystal diffraction measurement. This work in Houston is supported in part by US Air Force Office of Scientific Research Grant No. FA9550-15-1-0236, the T.L.L. Temple Foundation, the John J. and Rebecca
Moores Endowment, and the State of Texas through the Texas Center for Superconductivity at the University of Houston. B. Lv and S. Li also would like to acknowledge the funding support from US Air Force Office of Scientific Research and the Start-Up funds from University of Texas at Dallas.
${}^{\star}$ To whom correspondence should be addressed:
blv@utdallas.edu, cwchu@uh.edu
References
(1)
Y. Kamihara, T. Watanabe, M. Hirano, and H. Hosono, J. Am. Chem. Soc., 130, 3296 (2008).
(2)
Z. Wei, H. Li, W. L. Hong, Z. Lv, H. Wu, X. Guo, and K. Ruan, J. Supercond. Nov. Magn. 21, 213(2008).
(3)
M. Rotter, M. Tegel, and D. Johrendt, Phys. Rev. Lett. 101, 107006 (2008).
(4)
K. Sasmal, B. Lv, B. Lorenz, A. M Guloy, F. Chen, Y. Y. Xue, and C. W. Chu, Phys. Rev. Lett., 101, 107007(2008).
(5)
N. Ni, S. Nandi, A. Kreyssig, A. I. Goldman, E. D. Mun, S. L. Bud’ko, and P. C. Canfield, Phys. Rev. B 78, 014523(2008).
(6)
R. Hoffman, and C. Zheng, J. Phys. Chem., 89, 4175(1985).
(7)
E. R. Bauminger, D. Froindlich, I. Nowik, S. Ofer, I. Felner, and I. Mayer, Phys. Rev. Lett. 30, 1053 (1973).
(8)
J. R. Neilson, T. M. McQueen, A. Llobet, J. Wen, and M. R. Suchomel, Phys. Rev. B 87, 045124 (2013).
(9)
F. Steglich, J. Aarts, C. D. Bredl, W. Lieke, D. Meschede, W. Franz, and H. Schäfer, Phys. Rev. Lett. 43, 1892(1979).
(10)
Z. Hossain, C. Geibel, F. Weickert, T. Radu, Y. Tokiwa, H. Jeevan, P. Gegenwart, and F. Steglich, Phys. Rev. B 72, 094411 (2005).
(11)
U. B. Paramanik, R. Prasad, C. Geibel, Z. Hossain, Phys. Rev. B 89, 144423 (2014).
(12)
V. K. Anand and D. C. Johnston, Phys. Rev. B 91, 184403 (2015).
(13)
M. Imai, C. Michioka, H. Ueda, and K. Yoshimura, Phys. Rev. B 91, 184414 (2015).
(14)
T. Mine, H. Yanagi, T. Kamiya, Y. Kamihara, M. Hirano, and H. Hosono. Solid State Commun. 147, 111(2008).
(15)
F. Ronning, N. Kurita, E. D. Bauer, B. L. Scott, T. Park, T. Klimczuk, R. Movshovich, and J. D. Thompson: J. Phys.: Condens. Matter 20, 342203(2008).
(16)
X. C. Wang, B. B. Ruan, J. Yu, B. J. Pan, Q. G. Mu, T.Liu, G. F. Chen, Z. A. Ren, Supercond. Sci. Technol., 30, 035007(2017).
(17)
K. Kudo, Y. Nishikubo, and M. Nohara: J. Phys. Soc. Jpn. 79, 123710(2010).
(18)
M. Imai, S. Ibuka, N. Kikugawa, T. Terashima, S. Uji, T. Yajima, H. Kageyama, and I. Hase Phys. Rev. B 91, 014513(2015).
(19)
Y. Nishikubo, K. Kudo, M. Nohara, J. Phys. Soc. Jpn. 80, 055002 (2011).
(20)
E. Bauer, R. T. Khan, H. Michor, E. Royanian, A. Grytsiv, N. Melnychenko-Koblyuk, P. Rogl, D. Reith, R. Podloucky, E.-W. Scheidt, W. Wolf, and M. Marsman, Phys. Rev. B 80, 064504 (2009).
(21)
K. Togano, P. Badica, Y. Nakamori, S. Orimo, H. Takeya, and K. Hirata, Phys. Rev. Lett. 93, 247004 (2004).
(22)
P. Badica, T. Kondo, and K. Togano, J. Phys. Soc. Jpn. 74, 1014(2005).
(23)
E. Bauer, G. Hilscher, H. Michor, C. Paul, E. W. Scheidt, A. Gribanov, Y. Seropegin, H. Noel, M. Sigrist, and P. Rogl, Phys. Rev. Lett. 92, 027003 (2004).
(24)
T. Takayama, K. Kuwano, D. Hirai, Y. Katsura, A. Yamamoto, and H. Takagi, Phys. Rev. Lett. 108, 237001 (2012).
(25)
A. Subedi, L. Ortenzi, and L. Boeri, Phys. Rev. B 87, 144504 (2013).
(26)
H. Chen, X. Xu, C. Cao, and J. Dai, Phys. Rev. B 86, 125116 (2012).
(27)
B. I. Jawdat, B. Lv, X. Y. Zhu, Y. Y. Xue, and C. W. Chu, Phys. Rev. B 91, 094514 (2015).
(28)
K. Hu, B. Gao, Q. Ji, Y. Ma, W. Li, X. Xu, H. Zhang, G. Mu, F. Huang, C. Cai, X. Xie, and M. Jiang, Phys. Rev. B 93 214510 (2016).
(29)
K. Hu, B. Gao, Q. Ji, Y. Ma, H. Zhang, G. Mu, F. Huang, C. Cai, and X. Xie, Front. Phys. , 11, 117403 (2016)
(30)
B. Lv. B. I. Jawdat, Z. Wu, M. Sorolla, M. Gooch, K. Zhao, L. Deng, Y. Y. Xue, B. Lorenz. A. M. Guloy, and C. W. Chu, Inorg. Chem, 54, 1049(2015).
(31)
T. F. Smith, C. W. Chu, and M. B. Maple, Cryogenics 9, 53 (1969).
(32)
C. W. Chu, Phys. Rev. Lett. 33, 1283 (1974).
(33)
A. Tari, ”The Specific Heat of Master at Low Temperatures”, Imperial College Press, London, p. 150, (2003).
(34)
B. M. Jawdat, B. Lv, Z. Wu, and C. W. Chu, unpublished data.
(35)
K. Kadowaki, and S. B. Woods, Solid State Commun. 58, 507-509 (1986).
(36)
A. C. Jacko, J. O. Fjærestad, B. J. Powell Nat. Phys. 5, 422-425 (2009).
(37)
T. Smith, Physics Letters 33A, 465 (1970). |
Statistical mechanics of inference
Jonathan Landy
landy@mrl.ucsb.edu
Materials Department, University of California, Santa Barbara
(December 6, 2020)
Abstract
Statistical modeling often involves identifying an optimal estimate to some underlying probability distribution known to satisfy some given constraints. I show here that choosing as estimate the centroid, or center of mass, of the set consistent with the constraints formally minimizes an objective measure of the expected error. Further, I obtain a useful approximation to this point, valid in the thermodynamic limit, that immediately provides much information relating to the full solution set’s geometry. For weak constraints, the centroid is close to the popular maximum entropy solution, whereas for strong constraints the two are far apart. Because of this, centroid inference is often substantially more accurate. The results I present allow for its straightforward application.
pacs: 82.20.Pm, 05.40.-a, 89.70.Cf, 02.50.Tt
One is sometimes confronted with the challenge of estimating probabilities from partial information. For example, given a stochastic system that transitions between a very large number of distinct states, the sampling time required to directly obtain a statistically significant estimate – through binning, say – to the occupation probability of some particular state may be prohibitively long. This is often the case in neuroscience experiments, because the number of distinct states that a neural network can access grows exponentially with network size Shlens et al. (2006); Schneidman et al. (2006); Cocco et al. (2009).
Although rigorous distribution identification is not possible in such situations, inference strategies that intelligently make use of available data can provide good estimates Pressé et al. (2013).
Here, I consider probabilistic inference in the uniform ensemble, where all distributions consistent with a given set of constraints are supposed equally likely. Using methods of statistical mechanics, I obtain a simple approximation to the centroid of the solution set, defined by equations (11)-(13), below. This, in turn, leads to useful results characterizing the full solution set’s geometry, and it also allows for comparison to the maximum entropy solution. I find that the centroid is sometimes expected to be substantially more accurate.
I consider here the following general scenario: It is given that a desired, underlying distribution $\textbf{p}^{*}\equiv(p_{1},p_{2},\ldots,p_{N})$ on $N$ states, with $p_{i}^{*}\in[0,1]$ the probability of state $i$,
satisfies a set of $\mathcal{C}\ll N$ linear constraints of the form
$$\displaystyle\sum_{i}p_{i}^{*}f_{ji}=1,\ \ \ j\in\{1,2,\ldots,\mathcal{C}\},$$
(1)
with the real-valued $\left\{f_{ji}\right\}$ specified. The distribution is normalized to
$$\displaystyle\sum_{i}p_{i}^{*}=1.$$
(2)
I refer to the set $\mathcal{S}$ of distributions p satisfying (1) and (2) as the solution set. We are to select from $\mathcal{S}$ one distribution that is optimal: Here, I consider the case where the selected distribution is supposed to be a good approximation to the unknown $\textbf{p}^{*}$. I take as a measure of error in estimate p the quantity
$$\displaystyle E\left(\textbf{p}^{*},\textbf{p}\right)\equiv\left|\textbf{p}^{*%
}-\textbf{p}\right|^{2}=\sum_{i}\left(p_{i}^{*}-p_{i}\right)^{2},$$
(3)
the squared distance between the underlying and the estimated distributions.
I stress that (3) is not necessarily the only appropriate measure of error in an inference problem. However, it does represent an objective measure that is both familiar and useful to consider.
In certain situations, it may be appropriate to consider certain members of $\mathcal{S}$ more likely to be $\textbf{p}^{*}$ than others. For example, if we know that $p^{*}$ was generated by a process more likely to generate sparse distributions, sparse members of $\mathcal{S}$ should be weighted more heavily Albanna et al. (2012). However, in the absence of such information, an axiom of equal probability is appropriate: Every state consistent with (1) and (2) should be considered equally likely to be the underlying distribution 111This is an application of Laplace’s rule of indifference.. I work under this axiom here. In this case, the expected error in an estimate p is obtained by averaging (3) over $\textbf{p}^{*}$,
$$\displaystyle\left\langle E(\textbf{p}^{*},\textbf{p})\right\rangle_{\textbf{p%
}^{*}\in\mathcal{S}}=\sum_{i}p_{i}^{2}-2p_{i}\left\langle p_{i}^{*}\right%
\rangle_{\mathcal{S}}+\left\langle p_{i}^{*2}\right\rangle_{\mathcal{S}}.$$
(4)
The solution $\textbf{p}^{c}$ that minimizes the expected error is obtained by setting the derivative of (4), with respect to $p_{i}$, to zero. This gives,
$$\displaystyle\textbf{p}\to\textbf{p}^{c}=\left\langle\textbf{p}^{*}\right%
\rangle_{\mathcal{S}},$$
(5)
the centroid of the solution set 222It is a simple matter to prove that $\mathcal{S}$ is convex. Thus, $\textbf{p}^{c}\in\mathcal{S}$: The centroid is a valid solution.. This represents the formal solution to a particular, well-defined inference problem. Namely, this returns the distribution in $\mathcal{S}$ minimizing (4). Unfortunately, a simple, general formula for $\textbf{p}^{c}$ does not exist Rademacher (2007). However, in the following, I obtain an estimate to $\textbf{p}^{c}$ that is easy to evaluate. Comparison to this $\textbf{p}^{c}$ estimate then provides a simple method for testing the expected performance of other solutions: For p close to $\textbf{p}^{c}$, the expected error (4) is nearly minimized. On the other hand, the expected error (4) is relatively large for p far from $\textbf{p}^{c}$.
In order to characterize the solution set, I consider the configuration partition sum associated with a free particle, with position p, moving through $\mathcal{S}$. This is
$$\displaystyle\mathcal{Z}=\int_{0}^{\infty}\delta\left(\sum p_{i}-s\right)\prod%
_{j=1}^{\mathcal{C}}\delta\left(\sum_{i}p_{i}f_{ji}-t_{j}\right)\prod_{i=1}^{N%
}dp_{i},$$
(6)
where I have generalized slightly the constraint equations (1) and (2), now requiring the sum over probabilities to be equal to $s$ and the dot product of p along $\textbf{f}_{j}$ to be $t_{j}$. As defined, $\mathcal{Z}$ is simply equal to the volume of the solution set $\mathcal{S}$. The Laplace transform of $\mathcal{Z}$ is
$$\displaystyle\tilde{Z}\left(m,\{\lambda_{j}\}\right)$$
$$\displaystyle\equiv$$
$$\displaystyle\int_{0}^{\infty}\mathcal{Z}\left(s,\{t_{j}\}\right)e^{-sm-\sum_{%
j}t_{j}\lambda_{j}}ds\prod_{j}dt_{j}$$
(7)
$$\displaystyle=$$
$$\displaystyle\int_{0}^{\infty}e^{-m\sum p_{i}-\sum_{i}p_{i}\sum_{j}\lambda_{j}%
f_{ji}}\prod_{i=1}^{N}dp_{i}.$$
Here, I have used (6) to obtain the second line. The integrals over the $\{p_{i}\}$ are now decoupled, and $\tilde{Z}$ can be evaluated in closed form as
$$\displaystyle\tilde{Z}\left(m,\{\lambda_{j}\}\right)=\prod_{i=1}^{N}\frac{1}{m%
+\sum_{j}\lambda_{j}f_{ji}}.$$
(8)
The solution set volume is given formally by the inverse Laplace transform of this quantity,
$$\displaystyle\mathcal{Z}\left(s,\{t_{j}\}\right)\equiv\oint\tilde{Z}\left(m,\{%
\lambda_{j}\}\right)e^{sm+\sum_{j}t_{j}\lambda_{j}}dm\prod_{j}d\lambda_{j},$$
(9)
where the indicated contours are parallel to the imaginary axis Jones (1943).
In order to evaluate the integral (9), I now assume that $N$, the number of accessible states, or components of p, is large. In this case, the integrand in (9) will be highly peaked, and an asymptotic series for $\log\mathcal{Z}$ can be obtained, the first term being the saddle point value Jones (1943). Setting $s$ and the $\{t_{j}\}$ to their common, physical value, one, we have
$$\displaystyle\mathcal{Z}$$
$$\displaystyle=$$
$$\displaystyle\oint_{m,\{\lambda_{j}\}}e^{m+\sum_{j}\lambda_{j}-\sum_{i}\log%
\left(m+\sum_{j}\lambda_{j}f_{ji}\right)}$$
(10)
$$\displaystyle\approx$$
$$\displaystyle e^{m^{*}+\sum_{j}\lambda_{j}^{*}-\sum_{i}\log\left(m^{*}+\sum_{j%
}\lambda_{j}^{*}f_{ji}\right)},$$
where the saddle point $m^{*}$ and $\{\lambda_{j}^{*}\}$ values are those that leave the derivative of $\log\mathcal{Z}$ stationary. That is, they satisfy the following equations, obtained by setting the derivatives of the exponent in (10), with respect to $m$ and the $\{\lambda_{j}\}$, individually to zero:
$$\displaystyle 1-\sum_{i=1}^{N}\frac{1}{m^{*}+\sum_{j}\lambda_{j}^{*}f_{ji}}$$
$$\displaystyle=$$
$$\displaystyle 0$$
(11)
$$\displaystyle 1-\sum_{i=1}^{N}\frac{f_{ji}}{m^{*}+\sum_{j}\lambda_{j}^{*}f_{ji}}$$
$$\displaystyle=$$
$$\displaystyle 0,\ \ \ j\in\{1,2,\ldots\mathcal{C}\}.$$
We can solve directly for one unknown: Summing over the second line above, multiplied by $\lambda_{j}^{*}$, and adding to this $m^{*}$ times the first line, gives
$$\displaystyle m^{*}=N-\sum_{j}\lambda_{j}^{*},$$
(12)
a simple relationship.
The remaining $\mathcal{C}$ unknowns, the $\left\{\lambda_{j}^{*}\right\}$, must be solved for using (1), or, equivalently, the latter conditions of (11).
Notice that if we define
$$\displaystyle p_{i}^{c,1}\equiv\frac{1}{m^{*}+\sum_{j}\lambda_{j}^{*}f_{ji}},$$
(13)
the saddle point conditions (11) imply that the distribution $\textbf{p}^{c,1}$ satisfies both (1) and (2). In fact, $\textbf{p}^{c,1}$ is the first-order, saddle point estimate to the centroid of $\mathcal{S}$. This is most easily proven by introducing a field $h_{i}$ in (6) coupled to $p_{i}$. Following steps similar to those shown above, this gives
$$\displaystyle\mathcal{Z}$$
$$\displaystyle\to$$
$$\displaystyle\int\delta\left(\sum p_{i}-s\right)\prod_{j}\delta\left(\sum_{i}p%
_{i}f_{ji}-t_{j}\right)e^{-\sum_{i}h_{i}p_{i}}$$
(14)
$$\displaystyle=$$
$$\displaystyle\oint_{m,\{\lambda_{j}\}}e^{m+\sum_{j}\lambda_{j}-\sum_{i}\log%
\left(m+\sum_{j}\lambda_{j}f_{ji}+h_{i}\right)}.$$
From the first line above, we obtain
$$\displaystyle p^{c}_{i}\equiv\langle p_{i}\rangle_{\mathcal{S}}=-\left.%
\partial_{h_{i}}\log\mathcal{Z}\right|_{\{h_{j}\}=0},$$
(15)
an exact identity.
Applying the saddle point approximation to (14) gives the analog of (10), with the $h_{i}$ field included. Plugging in to (15) then gives $\textbf{p}^{c}\sim\textbf{p}^{c,1}$, the value in (13). More accurate estimates are obtained through expansion about the saddle point. For example, writing $m=m^{*}+\delta m$ and $\lambda_{j}=\lambda_{j}^{*}+\delta\lambda_{j}$, evaluation of the Gaussian fluctuations about the saddle point gives
$$\displaystyle\log\mathcal{Z}$$
$$\displaystyle\sim$$
$$\displaystyle\sum_{j=0}^{\mathcal{C}}\lambda_{j}^{*}-\sum_{i}\log\left(h_{i}+%
\sum_{j=0}^{\mathcal{C}}\lambda_{j}^{*}f_{ji}\right)$$
(16)
$$\displaystyle-\frac{1}{2}\log\det\mathcal{M},$$
where $\mathcal{M}$ is the $(\mathcal{C}+1)\times(\mathcal{C}+1)$ matrix with components
$$\displaystyle\mathcal{M}_{\alpha\beta}\equiv\sum_{i}\frac{f_{\alpha i}f_{\beta
i%
}}{\left(h_{i}+\sum_{j=0}^{\mathcal{C}}\lambda_{j}^{*}f_{ji}\right)^{2}}.$$
(17)
Here, I have written $m^{*}\equiv\lambda^{*}_{0}$ and $f_{0i}\equiv 1$, in order to briefly simplify notation.
Combining (15) and (16) gives the second order centroid estimate $\textbf{p}^{c,2}$, a refinement to $\textbf{p}^{c,1}$. In order to carry out the implied variation of (16) with respect to $h_{i}$ here, the field dependences of the $\left\{\lambda_{j}^{*}\right\}$ are needed within $\mathcal{M}$ 333In evaluating $\textbf{p}^{c,1}$ this is not necessary because the $\left\{\lambda_{j}^{*}\right\}$ leave the exponent stationary at the saddle point level.. Differentiating the saddle point equations, $\sum_{i}f_{ji}\left(h_{i}+\sum_{k}\lambda_{k}^{*}f_{ki}\right)^{-1}=1$, gives the matrix equation
$$\displaystyle\mathcal{M}_{\alpha\beta}\partial_{h_{i}}\lambda_{\beta}^{*}=-f_{%
\alpha i}\left(p^{c,1}_{i}\right)^{2},$$
(18)
which can be inverted to solve for the necessary derivatives.
Carrying out this procedure is useful for small $N$. However, for $N\gtrsim 100$, $\textbf{p}^{c,1}$ already provides an accurate approximation to $\textbf{p}^{c}$.
Once $\textbf{p}^{c,1}$ has been evaluated, one can immediately characterize, approximately, the solution set’s geometry. For example, from (14), the variance of $p_{i}$ is
$$\displaystyle\sigma^{2}_{p_{i}}\equiv\left\langle p_{i}^{2}\right\rangle_{%
\mathcal{S}}-\left\langle p_{i}\right\rangle_{\mathcal{S}}^{2}=\left.\partial^%
{2}_{h_{i}}\log\mathcal{Z}\right|_{\{h_{j}\}=0}.$$
(19)
Plugging in the saddle point estimate for $\mathcal{Z}$ gives
$$\displaystyle\sigma_{p_{i}}\sim p_{i}^{c,1}.$$
(20)
That is, the width of the solution set in the $\hat{\textbf{e}}_{i}$ direction is approximately equal to $p_{i}^{c}$, the $i^{\text{th}}$ component of the solution set’s centroid. Higher-order cumulant averages also immediately follow. Further, at the saddle point level, from (10), (12), and (13),
$$\displaystyle\mathcal{Z}\sim e^{N}\prod_{i=1}^{N}\frac{1}{m^{*}+\sum_{j}%
\lambda_{j}^{*}f_{ji}}=e^{N}\prod_{i=1}^{N}p^{c,1}_{i}.$$
(21)
We see that the solution set volume is proportional to the product of the centroid’s components. This provides a simple, qualitative means for determining whether a given set of constraints (1) is strong or weak: By the arithmetic-geometric mean inequality, we have
$$\displaystyle\mathcal{Z}\sim e^{N}\prod_{i=1}^{N}p^{c,1}_{i}$$
$$\displaystyle\leq$$
$$\displaystyle e^{N}\times\left\{\frac{1}{N}\sum_{i}p^{c}_{i}\right\}^{N}$$
(22)
$$\displaystyle=$$
$$\displaystyle\exp\left[N-N\log N\right],$$
where I have made use of the normalization condition (2) to obtain the second line.
Equality holds here if and only if each of the $\left\{p_{i}^{c,1}\right\}$ are equal to $\frac{1}{N}$, which is the case only in the absence of constraints. If constraints are applied, and the $\{p_{i}^{c,1}\}$ are substantially different in magnitude, the upper bound in (22) is far from strict. In this limit, the solution set volume is significantly diminished, and the constraints can be considered strong. On the other hand, if the $\{p_{i}^{c,1}\}$ are all similar in magnitude, $\log\mathcal{Z}$ is only slightly diminished, and the constraints can be considered weak.
We are now in a position to compare the centroid solution $\textbf{p}^{c}$, which minimizes the expected error (4), to the maximum entropy solution $\textbf{p}^{ME}$, which maximizes
$$\displaystyle S\equiv-\sum_{i}p_{i}\log p_{i},$$
(23)
the Shannon entropy Shannon (1948).
In a sense, $\textbf{p}^{ME}$ is the member of $\mathcal{S}$ having the smoothest distribution. Objective criteria for its success are of great value, as maximum entropy inference is applied in many contexts.
Using Lagrange multipliers, it is easy to show that $\textbf{p}^{ME}$ is given formally by Jaynes (1957); Pressé et al. (2013)
$$\displaystyle p_{i}^{ME}=\exp\left[-m-\sum_{j}\lambda_{j}f_{ji}\right],$$
(24)
where $m$ and the $\left\{\lambda_{j}\right\}$ must now be chosen so that (24) satisfies the constraints (1) and (2). As in the $\textbf{p}^{c,1}$ analysis, the normalization condition provides a simple solution for one of the unknowns:
$$\displaystyle e^{m}=\sum_{i}e^{-\sum_{j}\lambda_{j}f_{ji}}.$$
(25)
The $\left\{\lambda_{j}\right\}$ must again be solved for using (1).
The distance between $\textbf{p}^{ME}$ and the exact $\textbf{p}^{c}$ can be estimated analytically by comparing (24) to (13), which take a very similar form. Assuming the $\{f_{ji}\}$ are Gaussian distributed, with $P(f_{ji})\propto\exp\left[-f_{ji}^{2}/2\sigma^{2}\right]$, expanding either $\textbf{p}^{c,1}$ or $\textbf{p}^{ME}$ to first order in the $\{f_{ji}\}$ results in the following solution:
$$\displaystyle p_{i}\sim\alpha+\sum_{j}\beta_{j}f_{ji}+\ldots,$$
(26)
where $\alpha$ and the $\{\beta_{j}\}$ are given by
$$\displaystyle\alpha$$
$$\displaystyle\sim$$
$$\displaystyle\frac{1}{N}$$
$$\displaystyle\beta_{j}$$
$$\displaystyle\sim$$
$$\displaystyle\frac{1}{\sum_{k}f_{jk}^{2}}\times\left\{1-\frac{\sum_{i}f_{ji}}{%
N}\right\},$$
(27)
the values needed for (26) to satisfy (1) and (2) to leading order in $N$. The leading form (26), (Statistical mechanics of inference) is common to both $\textbf{p}^{c,1}$ and $\textbf{p}^{ME}$ because they both take the form of functions having arguments linear in the $\{f_{ji}\}$. If $\sigma\gg\sqrt{N}$, the condition formally defining the weak constraint limit for Gaussian-distributed $\{f_{ji}\}$ 444 For random p and for Gaussian distributed $\{f_{ji}\}$, $\textbf{p}\cdot\textbf{f}_{j}\sim O\left(\frac{\sigma}{N^{1/2}}\right)$. If this typical dot product value is very large, the constraint (1) resembles one of orthogonality, $\textbf{p}\cdot\textbf{f}_{j}\approx 0$. On the other hand, if $\frac{\sigma}{N^{1/2}}\ll O(1)$, satisfaction of (1) requires p nearly parallel to $\textbf{f}_{j}$, a much stronger condition. Thus, the ratio $\sigma/N^{1/2}$ determines whether the constraints are strong or weak., the term proportional to $\sum_{i}f_{ji}$ dominates $\beta_{j}$ in (Statistical mechanics of inference), and the second term in (26) is of order $O\left(\frac{\mathcal{C}^{1/2}}{N^{3/2}}\right)$. This is smaller than the leading $\alpha$ contribution in (26), which is $O(N^{-1})$. In this case, the distance between $\textbf{p}^{ME}$ and $\textbf{p}^{c,1}$ can be estimated by considering expansion up to second order in the $\{f_{ji}\}$, where the two solutions have differing Taylor series coefficients: $e^{x}\sim 1+x+\frac{x^{2}}{2}+\ldots$, while $\frac{1}{1-x}\sim 1+x+x^{2}+\ldots$. This gives
$$\displaystyle p_{i}^{c,1}-p_{i}^{ME}\sim O\left(\sum_{j}\frac{\beta_{j}^{2}}{%
\alpha}f_{ji}^{2}\right)\sim O\left(\frac{\mathcal{C}}{N^{2}}\right),$$
(28)
much smaller than the width of the solution space, which, from (20) and (26), is given by $\sigma_{p_{i}}\sim O\left(\frac{1}{N}\right)$. The maximum entropy and centroid solutions are very close in the large $N$, weak constraint limit.
In the strong constraint limit, $\sigma\ll N^{1/2}$, the first term in (Statistical mechanics of inference) dominates $\beta_{j}$, and the second term in (26) is $\sum_{j}\beta_{j}f_{ji}\sim O\left(\frac{1}{N\sigma}\right)$. As $\sigma\to O(1)$, this is no longer smaller than $\alpha$, signaling the breakdown of the asymptotic expansion. Empirically, I find that in this case
$$\displaystyle p_{i}^{c,1}-p_{i}^{ME}\sim O\left(\sigma_{p_{i}}\right)\sim O%
\left(p_{i}^{c,1}\right).$$
(29)
That is, in the strong constraint limit, the two solutions are distant, with component separations comparable to the solution space widths. A typical example illustrating this is shown in Fig. 1. Here, as expected, $\textbf{p}^{c,1}$ is much closer to the exact $\textbf{p}^{c}$ (obtained via averaging over a random walk through $\mathcal{S}$) than is $\textbf{p}^{ME}$. The discrepancy between the two is largest when $p_{i}^{c}$ (which sets the width $\sigma_{p_{i}}$) is large. Further, $p^{ME}_{i}<p^{c}_{i}$ for all components taking relatively large or relatively small values, whereas $p_{i}^{ME}>p_{i}^{c}$ for all $i$ taking intermediate values. This qualitative observation appears to hold quite generally, with entropy maximization occurring at a point whose intermediate weight components are substantially bolstered relative to those of the centroid, while all other components are relatively diminished.
In summary, then, I have shown that the centroid $\textbf{p}^{c}$ of $\mathcal{S}$ provides the formal solution to (1) and (2) that minimizes (4). Although other variational score functions could be employed – e.g., the entropy – (4) represents a useful one to consider, in that it provides an objective measure for the expected error. By comparing the popular maximum entropy solution $\textbf{p}^{ME}$ to the centroid’s saddle point approximation – $\textbf{p}^{c,1}$, given by equations (11)-(13), I have shown that $\textbf{p}^{ME}$ actually performs quite well, in general, in the weak constraint limit. This is a very useful result, as most prior tests of the maximum entropy principle have relied upon particular, testable examples. In the strong constraint limit, the centroid and maximum entropy solutions are distant, and $\textbf{p}^{ME}$ is expected to perform poorly, by measure (4). In this limit, centroid inference is typically much more accurate.
Like maximum entropy inference, centroid inference has the benefit of being free from any bias associated with fitting to a particular, model form. In practice, the centroid estimate can be obtained through averaging over a random walk through $\mathcal{S}$. However, the walk time required increases relatively quickly with $N$. Alternatively, successive analytic approximations to $\textbf{p}^{c}$ can be obtained using the method I outline here. The saddle point approximation $\textbf{p}^{c,1}$ provides a simple, first estimate, very similar in form to $\textbf{p}^{ME}$, that is accurate in the large $N$ limit. Evaluation of $\textbf{p}^{c,1}$ provides substantial value, even when not working within the uniform ensemble, as it immediately provides much information relating to the solution set’s geometry, as well as to the strength of the applied constraints.
Acknowledgements.
I thank Mike DeWeese for helpful discussions, Frank Brown and Phil Pincus for helpful comments, Jonathan Bergknoff for computer programming assistance, and the USA NSF for support through grant No. DMR-1101900.
References
Shlens et al. (2006)
J. Shlens,
G. D. Field,
J. L. Gauthier,
M. I. Grivich,
D. Petrusca,
A. Sher,
A. M. Litke, and
E. J. Chichilnisky,
J. Neurosci. 26,
8254 (2006).
Schneidman et al. (2006)
E. Schneidman,
M. J. I. Berry,
R. Segev, and
W. Bialek,
Nature 440,
1007 (2006).
Cocco et al. (2009)
S. Cocco,
S. Leibler, and
R. Monasson,
Proc. Natl. Acad. Sc. 106,
14058 (2009).
Pressé et al. (2013)
S. Pressé,
K. Ghosh,
J. Lee, and
K. A. Dill,
Rev. Mod. Phys. 85,
1115 (2013).
Albanna et al. (2012)
B. F. Albanna,
C. Hillar,
J. Sohl-Dickstein,
and M. R.
DeWeese, arXiv preprint arXiv:1209.3744
(2012).
Rademacher (2007)
L. A. Rademacher, in
Proceedings of the twenty-third annual symposium on
Computational geometry (ACM, 2007),
pp. 302–305.
Jones (1943)
L. M. Jones,
An Introduction to Mathematical Methods of Physics
(Benjamin Cummings, 1943).
Shannon (1948)
C. E. Shannon,
Bell System Tech. J. 27,
379 (1948).
Jaynes (1957)
E. T. Jaynes,
Phys. Rev. 106,
620 (1957). |
Imaging Se diffusion across the FeSe/SrTiO${}_{3}$ interface
Samantha O’Sullivan
Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
Ruizhe Kang
School of Engineering & Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA
Jules A. Gardener
Center for Nanoscale Systems, Harvard University, Cambridge, MA, USA
Austin J. Akey
Center for Nanoscale Systems, Harvard University, Cambridge, MA, USA
Christian E. Matt
christian.matt87@gmail.com
Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
Jennifer E. Hoffman
jhoffman@physics.harvard.edu
Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
School of Engineering & Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA
Abstract
Monolayer FeSe on SrTiO${}_{3}$ superconducts with reported $T_{c}$ as high as 100 K, but the dramatic interfacial $T_{c}$ enhancement remains poorly understood. Oxygen vacancies in SrTiO${}_{3}$ are known to enhance the interfacial electron doping, electron-phonon coupling, and superconducting gap, but the detailed mechanism is unclear. Here we apply scanning transmission electron microscopy (STEM) and electron energy loss spectroscopy (EELS) to FeSe/SrTiO${}_{3}$ to image the diffusion of selenium into SrTiO${}_{3}$ to an unexpected depth of several unit cells, consistent with the simultaneously observed depth profile of oxygen vacancies. Our density functional theory (DFT) calculations support the crucial role of oxygen vacancies in facilitating the thermally driven Se diffusion. In contrast to excess Se in the FeSe monolayer or FeSe/SrTiO${}_{3}$ interface that is typically removed during post-growth annealing, the diffused Se remains in the top few unit cells of the SrTiO${}_{3}$ bulk after the extended post-growth annealing that is necessary to achieve superconductivity. Thus, the unexpected Se in SrTiO${}_{3}$ may contribute to the interfacial electron doping and electron-phonon coupling that enhance $T_{c}$, suggesting another important role for oxygen vacancies as facilitators of Se diffusion.
I Introduction
1
Monolayer FeSe grown on SrTiO${}_{3}$ (STO) superconducts with a transition temperature $T_{c}$ as high as 100 K [1, 2, 3], an order of magnitude higher than bulk FeSe ($T_{c}\sim 8.8$ K [4]).
While there is general consensus that the interface plays a crucial role in the enhanced superconductivity [5, 6, 7, 8, 9, 10, 11, 12, 13, 14], the specific mechanism remains controversial. Angle-resolved photoemission spectroscopy (ARPES), electron energy loss spectroscopy (EELS), and scanning tunneling microscopy (STM) found evidence for a cooperative interplay of two effects: substrate-induced electron doping [6, 7, 8, 9, 10] and interfacial electron-phonon coupling [11, 12, 13, 14, 15].
But the wide range of measured $T_{c}$ in nominally similar samples suggests that both effects are strongly influenced by the detailed atomic structure and chemical composition of the interface.
2
Oxygen plays a key role in both electron doping and electron-phonon coupling at the STO interface. Oxygen vacancies directly donate charge carriers [5, 7, 6, 8, 16, 17], or indirectly alter the STO work function and associated charge transfer induced by band bending [10].
On the other hand, STO surface oxygen and its substitutions control the energy and form of the phonon modes that couple to the FeSe electrons [11, 14, 15]. Such electron-phonon coupling strongly influences $T_{c}$ [11, 12, 9], but could be screened by excess Se at the interface
[17]. Finally, the pronounced dependence of electron-phonon coupling on oxygen vacancy concentration [18, 19] complicates the interplay between the electron doping and electron-phonon coupling contributions to $T_{c}$. The fact that enhanced superconductivity has been found in monolayer FeSe grown on various oxides, including anatase TiO${}_{2}$ [20], BaTiO${}_{3}$ [21], LaTiO${}_{3}$ [22], NdGaO${}_{3}$ [23], and MgO [24] – while absent in non-oxide systems [25, 13] – further emphasizes the importance of oxygen chemistry on FeSe superconductivity.
3
Selenium belongs to the same chemical family as oxygen, which suggests that Se atoms might fill the O vacancies that typically form during high temperature vacuum annealing [26, 27, 28, 29]. Indeed such a scenario has been theoretically predicted for oxygen vacancies in the top TiO${}_{2-x}$ layer [16] and experimentally supported by ARPES [7] and scanning transmission electron microscopy (STEM) [17, 15].
Furthermore, several groups employ high-temperature annealing under high Se pressure to prepare the STO surface prior to FeSe growth [1, 30], which might enhance Se diffusion into STO as more oxygen vacancies are created and the formation energy for Se substitution is lowered. Often, excess Se in the FeSe film and at the interface is removed during post-growth annealing [17], which might not be possible for Se diffused deeper into the STO subsurface.
Although accurate knowledge of the interface chemical composition is of profound importance for exact modeling of the superconductivity enhancement in the FeSe/STO heterostructure, no experiment has investigated Se diffusion into subsurface layers of STO.
4
Here we use STEM and EELS to reveal the diffusion of Se several unit cells deep into STO that occurs during the monolayer FeSe film growth and annealing, both performed at temperatures below $\sim 520^{\circ}$C. We find that the excess Se decays exponentially into STO, as predicted by Fick’s law of thermally activated elemental diffusion [31]. Furthermore, we observe a similar line profile and decay length of oxygen vacancies at the STO surface which, in combination with density functional theory (DFT) calculations, suggests that oxygen vacancies play a pivotal role for Se diffusion. The role of oxygen vacancies in facilitating Se diffusion is further supported by the contrast between the Se and Fe line profiles and the negligible diffusion of Fe, which belongs to a different chemical family and therefore does not substitute for oxygen.
II Methods
5
Monolayer FeSe was grown by molecular beam epitaxy (MBE) on a Nb-doped (0.05%) STO(001) substrate from Crystek. The STO substrate was etched with buffered HF (NH${}_{4}$F : HF = 7 : 1, diluted with equal volume of deionized water) for 30 seconds, then annealed in O${}_{2}$ at 950 ${}^{\circ}$C for 1 hour. The substrate was transferred into the MBE chamber (base pressure $<5\times 10^{-10}$ Torr) and degassed for 3 h at 500${}^{\circ}$C. Importantly, no high-temperature Se molecular beam etching was performed prior to growth [1]. FeSe was deposited in three rounds by co-evaporating Fe (99.995%) and Se (99.999%) with a molar flux ratio of 1:30 and substrate temperatures between 400${}^{\circ}$C and 520${}^{\circ}$C, followed by post-growth annealing at 450${}^{\circ}$C - 520${}^{\circ}$C
(First round: 0.95 unit cells FeSe deposited at a substrate temperature of $400^{\circ}$C and 3 h post-growth annealing at $450^{\circ}$C.
Second round: 0.3 unit cells FeSe at $400^{\circ}$C with 3 h post-growth annealing at $450^{\circ}$C and 4 h at $520^{\circ}$C.
Third round: $\sim 0.2$ unit cells FeSe at $520^{\circ}$C, post-growth annealed at the same temperature for 4 h.)
The final annealing step in ultrahigh vacuum (UHV, $<5\times 10^{-10}$ Torr) was performed at $\sim 510^{\circ}$C for 10 h. After each growth step the sample was transferred through UHV to a home-built scanning tunneling microscope (STM) for imaging at $\sim 77$ K. The final STM scan confirmed that the high annealing temperature $\gtrsim 500^{\circ}$C was effective in removing all 2-unit-cell islands [9]. Finally, the film was capped with a $\sim 40$ nm Te layer [32] at room temperature to prepare for cross-sectional STEM and EELS measurements. A lamella of thickness $30\pm 6$ nm was prepared using focused ion beam milling (FEI Helios 660). A JEOL ARM 200F operated at 200 kV was used to record room temperature STEM (JEOL HAADF detector) and EELS measurements at six different locations of the lamella. EELS data was acquired with STEM probe settings of 197 pA current and 22.4 mrad convergence angle, using a Gatan Enfinium EELS spectrometer. We grew a second sample for low-temperature STM imaging, and confirmed a superconducting gap of $\sim 15$ meV at $T=4.7$ K.
We performed DFT calculations using the open-source Quantum Espresso (QE) software package [33, 29]. We constructed a $3\times 3\times 3$ STO supercell, terminated by double TiO${}_{2-x}$ layer and added $20\;\mathrm{\AA}$ of vacuum spacing along the (001) axis to simulate the two dimensional surface structure using periodic boundary conditions. We used ultra-soft pseudo-potentials for Sr, Ti and Se atoms and projector augmented-wave pseudo-potential for O atoms. We set the kinetic energy cutoff to be 40 Ry and the charge density cutoff to 400 Ry. We used a Gaussian smearing of 0.01 Ry to improve the convergence during the relaxation. We relaxed the entire structure until both the forces and total energy for ionic minimization was smaller than $1\times 10^{-4}$ Hartree/Bohr and $1\times 10^{-4}$ Hartree, respectively. The energy convergence threshold for self-consistency was $1\times 10^{-6}$ Hartree. We sampled the first Brillouin zone by a $4\times 4\times 1$ $k$-grid. When relaxing the structure, we allowed only the top three atomic layers (top TiO${}_{2-x}$, second TiO${}_{2-y}$, top SrO) to move, while the rest was fixed.
III Results
6
The high crystalline quality of the FeSe film is apparent in the STM topography in Fig. 1(a) showing a uniform monolayer coverage and atomically smooth surface areas (see inset). Reflection high-energy electron diffraction (RHEED) images of the STO surface in Fig. 1(b) show sharp diffraction spots, indicating a non-reconstructed ($1\times 1$) termination. The post-growth RHEED image in Fig. 1(c) depicts the typical pattern for epitaxial monolayer FeSe [2]. Figure 1(d) shows our atomic resolution cross-sectional STEM measurement in which we can identify the Te capping layer, the monolayer FeSe, and an atomically sharp FeSe/STO interface.
We measure the inter-atomic distance between the bottom Se and top Ti layers to be $3.35\pm 0.21$ Å, consistent with previous STEM measurement [17]. We also observe the double layer TiO${}_{2-x}$ termination of the STO, which is commonly seen [17, 34, 35, 36, 10].
While we don’t observe any ordered Se layer between the FeSe and the top TiO${}_{2-x}$ layer [10] we note that the top Ti atoms appear slightly elongated along the (001) direction, which has been interpreted as a sign of additional Se at the interface [17].
7
To identify the chemical composition of the interface, we analyze the EELS measurement over a wide energy range covering Ti, Fe, Se, O, and Sr absorption edges. We average the absorption spectra along the (100) direction of the scan window and subtract a power law background, as shown in Figs. 2(a-e) (see also Appendix A and Fig. 5).
Figures 2(f-j) show a resolution-limited cutoff for Ti, O, and Sr above the top TiO${}_{2-x}$ layer and for Fe below the TiO${}_{2-x}$ layer, as expected for an atomically sharp interface. In contrast, we find that the Se intensity has a longer tail below the top TiO${}_{2-x}$ layer shown in Fig. 2(j), suggesting that Se diffused into the STO substrate.
This observation is confirmed in the energy-integrated linecuts, shown in Figs. 2(k-o). The intensity drop of Ti, O above the topmost TiO${}_{2-x}$ layer and Sr above the SrO layer are determined by the beam shape of the STEM probe (see Appendix B and Fig. 6). The Fe linecut follows the same expected resolution-limited intensity profile, dropping just above the TiO${}_{2-x}$ line. However, the Se linecut deviates significantly from the expected profile and extends at higher intensities for several STO subsurface layers, indicating a significant concentration of Se below the top TiO${}_{2-x}$ layer.
8
The contrast between Se and Fe downward diffusion is shown by the differing deviations of their measured linecuts from their expected resolution-broadened intensity profiles, in Figs. 2(n-o). While the excess Fe signal below the Fe layer in Fig. 2(n) is within the instrument broadening and noise level, the excess Se, marked by the blue shaded area in Fig. 2(o), is significant and extends deep into the subsurface layer of STO. The Fe intensity peak above the FeSe monolayer may indicate the presence of excess Fe that formed FeTe islands during the Te capping process, as previously suggested by Refs. [37, 38].
9
To investigate the origin of Se diffusion into STO and a possible connection with preformed O vacancies, we analyze their spatial profile across the interface in Fig. 3. The excess Se signal peaks just above the TiO${}_{2-x}$ layer and falls exponentially along the (00$\bar{1}$) direction with decay length $\xi_{\mathrm{Se}}=0.74\pm 0.05$ nm (vertical black arrow). The peak position above the TiO${}_{2-x}$ layer demonstrates excess Se between STO and the FeSe layer, consistent with Fig. 1(d) and previous STEM studies [10, 17]. The exponential profile is a solution of Fick’s diffusion law [31], which points towards thermally-driven diffusion along an element concentration gradient, as is often observed at interfaces [39].
However, the contrasting absence of Fe diffusion into the STO suggests that thermal activation alone is not sufficient, and a second mechanism must contribute to the Se diffusion into the STO substrate. Selenium belongs to the same chemical family as oxygen, suggesting that oxygen vacancies may be partially filled with Se, similar to predictions for the top TiO${}_{2-x}$ layer [16, 17].
Fig. 3 shows the concentration of O vacancies extracted from the spatial dependence of the O${}_{2}$/Ti ratio of the EELS linecuts in Figs. 2(k-l). We find an exponential decay length of $\xi_{\mathrm{O_{\mathrm{vac}}}}=0.57\pm 0.30$ nm (red arrow), corresponding to an O vacancy concentration of $11\pm 3\%$ for the top TiO${}_{2-x}$ layer and $6\pm 3\%$ for the second TiO${}_{2-y}$ layer. The consistency between $\xi_{\mathrm{Se}}$ and $\xi_{\mathrm{O_{\mathrm{vac}}}}$ supports the hypothesis that oxygen vacancies are crucial in facilitating the Se diffusion into STO.
10
To further investigate the role of oxygen vacancies on Se diffusing into the STO surface, we use DFT to calculate the formation energies for various vacancy configurations (for more details, see Methods and Appendix C). In our calculations we assume that oxygen vacancies form during vacuum annealing prior to FeSe growth. We investigate the following two configurations:
$(i)$ One vacancy per supercell in the top TiO${}_{2-x}$ layer, corresponding to 5.5% O vacancies, and $(ii)$ two vacancies (11%) in the top TiO${}_{2-x}$ layer and one vacancy (5.5%) in the second TiO${}_{2-y}$ layer, corresponding to the measured amount of oxygen vacancies in Fig. 3 (see Appendix C and Fig. 7). Figures 4(a-d) show the final relaxed structures after we filled each oxygen vacancy with Se (Se${}_{\mathrm{O}}$). Our calculation shows that Se${}_{\mathrm{O}}$ in the top TiO${}_{2-x}$ layer protrudes slightly from the layer, consistent with our experimental observation of apparent Ti atom elongation in Fig. 1(d), and the excess Se peak just above the TiO${}_{2-x}$ layer in Fig. 3. In contrast, the Se${}_{\mathrm{O}}$ in the second TiO${}_{2-y}$ layer remains at its initial location, suggesting that diffused Se predominantly occupies the oxygen vacancy sites instead of interstitial locations.
We next calculate the formation energies for Se atoms filling the vacancies,
$$E_{\mathrm{form}}=E^{\mathrm{DFT}}_{n\mathrm{Se}}-E^{\mathrm{DFT}}_{n\mathrm{O_{vac}}}-n\mu_{\mathrm{Se}}.$$
(1)
Here, $E^{\mathrm{DFT}}_{n\mathrm{O_{vac}}}$ and $E^{\mathrm{DFT}}_{n\mathrm{Se}}$ are the energies of the fully relaxed structure with $n$ oxygen vacancies and $n$ Se substitutions, respectively, and $\mu_{\mathrm{Se}}$ is the temperature and pressure dependent chemical potential of a single Se atom. In Fig. 4(e), we find that $E_{\mathrm{form}}$ is negative for a range of $\mu_{\mathrm{Se}}$ corresponding to experimental substrate temperatures and Se partial pressures, marked by the purple shaded area (see Appendix C and D). Our calculations thus suggest that Se diffusion below the top TiO${}_{2-x}$ layer is energetically favorable in presence of oxygen vacancies.
11
We consider the implications of Se diffusion for the charge carrier concentration in STO.
While O vacancies create free electrons at the STO surface, which likely dope the monolayer FeSe [5, 7], excess Se has been theoretically [40, 41] and experimentally [7, 5] shown to act as hole dopants.
However, as the electronegativity of Se (2.55) is lower than that of O (3.44) we expect that even in the extreme case of all O vacancies being filled with Se, there will remain excess free electrons.
Furthermore, the excess Se could influence the STO work function and the associated interfacial band bending, altering the electron transfer into FeSe [10]. The STO charge carrier concentration also modifies the electron-phonon coupling [18, 19]. Additional theoretical and experimental study is required to understand the detailed effects of subsurface Se on the interfacial electron-phonon coupling, charge transfer and superconductivity enhancement.
IV Conclusion
12
To conclude, we imaged the monolayer FeSe/STO interface using atomic-resolution STEM and EELS, and we observed Se diffusion several unit cells deep into the STO. Our EELS measurements further revealed oxygen vacancies in the surface and subsurface layers of STO which, in combination with our DFT calculations, supports the scenario that oxygen vacancies are crucial to facilitate the Se diffusion. Surprisingly, the diffused Se persisted in the STO even after extended ($\sim 10$ h) post-growth UHV annealing above 500${}^{\circ}$C, which has been shown to remove excess Se from the FeSe layer and the immediate interface between FeSe and STO [17]. The post-growth anneal is a crucial step to obtain the high-temperature superconductivity in the FeSe/STO heterostructure [7, 17]. Our findings call for future experiments to measure the relation between Se diffusion depth and superconducting $T_{c}$, and future theoretical models to calculate the effects of Se diffusion on electron-phonon coupling and interfacial doping. Our observation may also help to resolve the inconsistency between the calculated [42] and experimentally measured band structure of the monolayer FeSe/STO heterostructure [6, 11].
Acknowledgements
We thank Yu Xie, Boris Kozinsky, Dennis Huang, and Jason Hoffman for insightful discussions.
This work was performed in part at the Center for Nanoscale Systems (CNS), a member of the National Nanotechnology Coordinated Infrastructure Network (NNCI), which is supported by the National Science Foundation under NSF award no. 1541959. CNS is part of Harvard University.
C.E.M. was supported by the Swiss National Science Foundation under fellowships P2EZP2_175155 and P400P2_183890, and by the Office of Naval Research grant N00014-18-1-2691.
Appendix A Selenium absorption edge
Figure 5 shows the evolution of the Se absorption spectra across the FeSe/STO interface, within one representative region. The location of the lower Se layer is marked by a blue line in all three panels. The Se $L$ edge after background subtraction is shown in false color in Fig. 5(b) and as a linecut in Fig. 5(c). The total Se signal for each position along the (001) direction is the integrated green area under each curve. We reproduced this data in six distinct regions along the FeSe/STO lamella, and we show the averaged results in Fig. 2(o).
Appendix B STEM spatial resolution
To determine spatial broadening due to the finite STEM electron beam width, we fit Gaussian curves to the tails of the Ti and Sr EELS signals that extend above the top Ti and Sr layers in Figs. 2(k,m). In Fig. 6, we show both signals and fits, and we also shift the Sr fit to overlap with the Ti fit, which demonstrates a similar lineshape (Sr is slightly broader). The similarity indicates a spatial resolution that is almost independent of energy (absorption energies: Ti $L_{3}$ edge: 456 eV, Sr $L_{3}$ edge: 1940 eV). We then compare the Sr fit tail (representing pure instrument broadening) to our Fe and Se line profiles in Figs. 2(n) and 2(o) to determine the Fe and Se excess signal that corresponds to real element diffusion into the STO bulk. Since the monolayer FeSe consists of only one Fe layer and two Se layers, we expect the peak amplitude of these element profiles to be reduced due to the finite spatial resolution. We therefore normalize the Fe and Se profiles in Fig. 2(n,o) such that the intensity of the Fe (Se) profile at the location of the Fe (Se) layer equals the Sr signal at the top Sr layer (which is less than the Sr signal in the bulk of STO).
Appendix C Formation energy of selenium diffusion
Electrical [27], magnetic resonance [29, 28], and optical studies [26] have shown that oxygen vacancies occur during heat treatment in vacuum near the surface of STO substrates. Thus our DFT calculations start from supercell models that have oxygen vacancies on the top two layers of TiO${}_{2-x}$.
We define the formation energy using
$$E_{\mathrm{form}}=E^{\mathrm{DFT}}_{n\mathrm{Se}}-E^{\mathrm{DFT}}_{n\mathrm{O_{vac}}}-n\mu_{\mathrm{Se}}$$
(2)
where $E^{\mathrm{DFT}}_{n\mathrm{Se}}$ is the energy of the fully relaxed structure with Se implemented in either the top or the second TiO${}_{2-x}$ layer. $E^{\mathrm{DFT}}_{n\mathrm{O_{vac}}}$ is the energy of the fully relaxed STO supercell with vacancies in either the top or the second TiO${}_{2-x}$ layer. Both of them are calculated by DFT. $\mu_{\mathrm{Se}}$ is the chemical potential of a single Se atom, which is a function of temperature $T$ and pressure $p$. $\mu_{\mathrm{Se}}$ can be written as
$$\mu_{\mathrm{Se}}(T,p)=\frac{1}{2}\mu_{\mathrm{Se_{2}}}=\frac{1}{2}(E_{\mathrm{Se_{2}}}^{\mathrm{DFT}}+\mu_{\mathrm{Se_{2}}}(T,p))$$
(3)
where $E_{\mathrm{Se_{2}}}^{\mathrm{DFT}}$ is the energy of an isolated Se dimer molecule as calculated by DFT. Since it is well known that DFT tends to overbind the molecule [43], we then use Eq. 4 to finally determine the energy of the Se${}_{2}$ molecule
$$E_{\mathrm{Se_{2}}}^{\mathrm{DFT}}=2E_{\mathrm{Se}}^{\mathrm{DFT}}-E_{\mathrm{bond}}$$
(4)
where $E_{\mathrm{Se}}^{\mathrm{DFT}}$ is the energy of an isolated single Se atom determined by a self-consistent DFT calculation. $E_{\mathrm{bond}}$ is the bond energy of the $\mathrm{Se_{2}}$ molecules obtained from Ref. [44], and $\mu_{\mathrm{Se_{2}}}(T,p)$ in Eq. 3 is the chemical potential for the selenium dimer molecule, which depends on temperature and pressure, as derived in Appendix D.
To calculate the formation energy, we first fully relaxed the pristine STO supercell with a double-layer TiO${}_{2-x}$ termination.
The calculated distance between the double TiO${}_{2-x}$ layers is 2.19 Å, which is very close to our experimental value of $\sim 1.9\pm 0.3$ Å, and confirms the validity of our relaxed structure.
We then calculated various oxygen vacancy configurations and their relaxed crystal structures and energies $E^{\mathrm{DFT}}_{n\mathrm{O_{vac}}}$ for each case. Consecutively we replaced each oxygen vacancy with a selenium atom and again relaxed the supercell to obtain $E^{\mathrm{DFT}}_{n\mathrm{Se}}$ and the final structures presented in Figs. 4(a-d) and Fig. 7.
Appendix D Estimation of the chemical potential of a single selenium atom
To estimate the chemical potential of a single Se atom, we determine the chemical potential for a selenium dimer molecule. Considering the Se${}_{2}$ molecule as ideal diatomic gas, its partition function has contributions from translation, vibration and rotation, which can be written as
$$Z=Z_{\mathrm{trans}}Z_{\mathrm{vib}}Z_{\mathrm{rot}}$$
(5)
Here we ignore the contribution from the electronic levels since they will contribute to the thermodynamic properties only at high temperature or if unpaired electrons are present [45].
Using the rigid rotor-harmonic oscillator approximation [45, 46], one can explicitly evaluate all the thermodynamic quantities. The chemical potential can be expressed in terms of a reference pressure as shown in [47]
$$\mu_{\mathrm{Se_{2}}}(T,p)=\mu^{0}_{\mathrm{Se_{2}}}(T,p^{0})+k_{\mathrm{B}}T\mathrm{log}\left(\frac{p}{p^{0}}\right)$$
(6)
where $\mu^{0}_{\mathrm{Se_{2}}}$ is the chemical potential at reference pressure $p^{0}$, which is usually taken as 1 atm; $k_{B}$ is the Boltzmann constant.
Table 1 shows some of the calculated values of thermodynamical quantities within the temperature range that is close to our experimental condition. The enthalpy $H$ and entropy $S$ were adapted from Ref. [45], and the chemical potential $\mu^{0}_{\mathrm{Se_{2}}}$ is defined as Gibbs free energy ($G$) per molecule,
$$\mu^{0}_{\mathrm{Se_{2}}}=\frac{G}{N_{A}}=\frac{H-TS}{N_{A}}$$
(7)
If we insert the values from Table 1
into Eq. 6, we will obtain the chemical potential for a single Se${}_{2}$ molecule at any given temperature and pressure. Given our experimental conditions, here we consider two extreme cases:
(1) For $T=800$ K and pressure $p=10^{-10}$ Torr, $\mu_{\mathrm{Se_{2}}}=-4.20$ eV, which is the lower limit.
(2) For $T=700$ K and pressure $p=10^{-9}$ Torr, $\mu_{\mathrm{Se_{2}}}=-3.50$ eV, which is the upper limit.
Plugging these values into Eq. 3, we can estimate that under our experimental conditions, the range of the chemical potential for a single Se atom (for simplicity, here we set $E^{\mathrm{DFT}}_{\mathrm{Se}}=0$) is
$$\mu_{\mathrm{Se}}(T,p)\in[\ -2.10\mathrm{\ eV},-1.75\mathrm{\ eV}]\ $$
(8)
References
Wang et al. [2012]
Q.-Y. Wang, Z. Li, W.-H. Zhang, Z.-C. Zhang, J.-S. Zhang, W. Li, H. Ding, Y.-B. Ou, P. Deng, K. Chang,
J. Wen, C.-L. Song, K. He, J.-F. Jia, S.-H. Ji, Y.-Y. Wang, L.-L. Wang,
X. Chen, X.-C. Ma, and Q.-K. Xue, Interface-induced high-temperature superconductivity in single
unit-cell FeSe films on SrTiO${}_{3}$, Chinese Physics Letters 29, 037402 (2012).
Huang and Hoffman [2017]
D. Huang and J. E. Hoffman, Monolayer FeSe on
SrTiO${}_{3}$, Annual Review of Condensed Matter Physics 8, 311 (2017).
Ge et al. [2015]
J.-F. Ge, Z.-L. Liu,
C. Liu, C.-L. Gao, D. Qian, Q.-K. Xue, Y. Liu, and J.-F. Jia, Superconductivity above 100 K in
single-layer FeSe films on doped SrTiO${}_{3}$, Nature Materials 14, 285 (2015).
Böhmer et al. [2016]
A. E. Böhmer, V. Taufour,
W. E. Straszheim,
T. Wolf, and P. C. Canfield, Variation of transition temperatures and residual
resistivity ratio in vapor-grown FeSe, Physical Review B 94, 024526 (2016).
Zhang et al. [2014]
W. Zhang, Z. Li, F. Li, H. Zhang, J. Peng, C. Tang, Q. Wang, K. He, X. Chen, L. Wang, X. Ma, and Q.-K. Xue, Interface charge doping effects on
superconductivity of single-unit-cell FeSe films on SrTiO${}_{3}$
substrates, Physical Review B 89, 060506 (2014).
Liu et al. [2012]
D. Liu, W. Zhang, D. Mou, J. He, Y.-B. Ou, Q.-Y. Wang, Z. Li, L. Wang, L. Zhao, S. He, Y. Peng, X. Liu, C. Chen, L. Yu, G. Liu, X. Dong, J. Zhang, C. Chen,
Z. Xu, J. Hu, X. Chen, X. Ma, Q. Xue, and X. Zhou, Electronic origin of high-temperature
superconductivity in single-layer FeSe superconductor, Nature Communications 3, 931 (2012).
He et al. [2013]
S. He, J. He, W. Zhang, L. Zhao, D. Liu, X. Liu, D. Mou, Y.-B. Ou, Q.-Y. Wang, Z. Li, L. Wang, Y. Peng, Y. Liu, C. Chen, L. Yu, G. Liu, X. Dong, J. Zhang,
C. Chen, Z. Xu, X. Chen, X. Ma, Q. Xue, and X. J. Zhou, Phase diagram and electronic indication of
high-temperature superconductivity at 65 K in single-layer FeSe films, Nature Materials 12, 605 (2013).
Tan et al. [2013]
S. Tan, Y. Zhang, M. Xia, Z. Ye, F. Chen, X. Xie, R. Peng, D. Xu, Q. Fan, H. Xu, J. Jiang, T. Zhang,
X. Lai, T. Xiang, J. Hu, B. Xie, and D. Feng, Interface-induced superconductivity
and strain-dependent spin density waves in FeSe/SrTiO${}_{3}$ thin films, Nature Materials 12, 634 (2013).
Zhang et al. [2017]
H. Zhang, D. Zhang,
X. Lu, C. Liu, G. Zhou, X. Ma, L. Wang, P. Jiang, Q.-K. Xue, and X. Bao, Origin of charge transfer and enhanced electron – phonon coupling
in single unit-cell FeSe films on SrTiO${}_{3}$, Nature Communications 8, 214 (2017).
Zhao et al. [2018]
W. Zhao, M. Li, C.-Z. Chang, J. Jiang, L. Wu, C. Liu, J. S. Moodera,
Y. Zhu, and M. H. W. Chan, Direct imaging of electron transfer and its
influence on superconducting pairing at FeSe/SrTiO${}_{3}$ interface, Science Advances 4, eaao2682 (2018).
Lee et al. [2014]
J. J. Lee, F. T. Schmitt,
R. G. Moore, S. Johnston, Y.-T. Cui, W. Li, M. Yi, Z. K. Liu,
M. Hashimoto, Y. Zhang, D. H. Lu, T. P. Devereaux, D.-H. Lee, and Z.-X. Shen, Interfacial
mode coupling as the origin of the enhancement of $T_{c}$ in FeSe films on
SrTiO${}_{3}$, Nature 515, 245 (2014).
Song et al. [2019]
Q. Song, T. L. Yu,
X. Lou, B. P. Xie, H. C. Xu, C. H. P. Wen, Q. Yao, S. Y. Zhang,
X. T. Zhu, J. D. Guo, R. Peng, and D. L. Feng, Evidence of Cooperative Effect on the Enhanced Superconducting
Transition Temperature at the FeSe/SrTiO${}_{3}$ Interface, Nature Communications 10, 758 (2019).
Zhang et al. [2016a]
W. H. Zhang, X. Liu, C. H. P. Wen, R. Peng, S. Y. Tan, B. P. Xie, T. Zhang, and D. L. Feng, Effects of surface electron
doping and substrate on the superconductivity of epitaxial FeSe films, Nano Letters 16, 1969 (2016a).
Zhang et al. [2016b]
S. Zhang, J. Guan,
X. Jia, B. Liu, W. Wang, F. Li, L. Wang, X. Ma, Q. Xue, J. Zhang, E. W. Plummer, X. Zhu, and J. Guo, Role of SrTiO${}_{3}$ phonon penetrating
into thin FeSe films in the enhancement of superconductivity, Physical Review B 94, 081116 (2016b).
Gong et al. [2019]
G. Gong, H. Yang, Q. Zhang, C. Ding, J. Zhou, Y. Chen, F. Meng, Z. Zhang, W. Dong, F. Zheng, P. Zhang, L. Yang, L. Gu, Q.-K. Xue, and L. Wang, Oxygen vacancy modulated superconductivity in
monolayer FeSe on SrTiO${}_{3-\delta}$, Physical Review B 100, 224504 (2019).
Bang et al. [2013]
J. Bang, Z. Li, Y. Y. Sun, A. Samanta, Y. Y. Zhang, W. Zhang, L. Wang, X. Chen, X. Ma, Q.-K. Xue, and S. B. Zhang, Atomic and electronic structures of single-layer FeSe on
SrTiO${}_{3}$(001): The role of oxygen deficiency, Physical Review B 87, 220503 (2013).
Li et al. [2016]
F. Li, Q. Zhang, C. Tang, C. Liu, J. Shi, C. Nie, G. Zhou, Z. Li, W. Zhang, C.-L. Song, K. He, S. Ji, S. Zhang, L. Gu, L. Wang, X.-C. Ma, and Q.-K. Xue, Atomically
resolved FeSe/SrTiO${}_{3}$(001) interface structure by scanning transmission
electron microscopy, 2D Materials 3, 024002 (2016).
Chen et al. [2015]
C. Chen, J. Avila,
E. Frantzeskakis, A. Levy, and M. C. Asensio, Observation of a Two-Dimensional Liquid of Fröhlich
Polarons at the Bare SrTiO${}_{3}$ Surface, Nature Communications 6, 8585 (2015).
Wang et al. [2016]
Z. Wang, S. M. Walker,
A. Tamai, Y. Wang, Z. Ristic, F. Y. Bruno, A. de la Torre, S. Riccò, N. C. Plumb, M. Shi, P. Hlawenka,
J. Sánchez-Barriga,
A. Varykhalov, T. K. Kim, M. Hoesch, P. D. C. King, W. Meevasana, U. Diebold, J. Mesot, B. Moritz, T. P. Devereaux, M. Radovic, and F. Baumberger, Tailoring the Nature
and Strength of Electron – Phonon Interactions in the SrTiO${}_{3}$(001) 2D
Electron Liquid, Nature Materials 15, 835 (2016).
Ding et al. [2016]
H. Ding, Y.-F. Lv,
K. Zhao, W.-L. Wang, L. Wang, C.-L. Song, X. Chen, X.-C. Ma, and Q.-K. Xue, High-Temperature Superconductivity in Single-Unit-Cell FeSe Films on
Anatase TiO${}_{2}$, Phys. Rev. Lett. 117, 067001 (2016).
Peng et al. [2014]
R. Peng, H. C. Xu,
S. Y. Tan, H. Y. Cao, M. Xia, X. P. Shen, Z. C. Huang, C. H. P. Wen, Q. Song, T. Zhang,
B. P. Xie, X. G. Gong, and D. L. Feng, Tuning the band structure and superconductivity in
single-layer FeSe by interface engineering, Nature Communications 5, 1 (2014).
Jia et al. [2021]
T. Jia, Z. Chen, S. N. Rebec, M. Hashimoto, D. Lu, T. P. Devereaux, D.-H. Lee, R. G. Moore, and Z.-X. Shen, Magic Doping and
Robust Superconductivity in Monolayer FeSe on Titanates, Advanced Science , 2003454 (2021).
Yang et al. [2019]
H. Yang, G. Zhou, Y. Zhu, G.-M. Gong, Q. Zhang, M. Liao, Z. Li, C. Ding, F. Meng, M. Rafique, H. Wang, L. Gu, D. Zhang, L. Wang, and Q.-K. Xue, Superconductivity above 28 K in Single Unit Cell FeSe Films
Interfaced with GaO${}_{2-\delta}$ Layer on NdGaO${}_{3}$(110), Science Bulletin 64, 490 (2019).
Zhou et al. [2018]
G. Zhou, Q. Zhang,
F. Zheng, D. Zhang, C. Liu, X. Wang, C.-L. Song, K. He, X.-C. Ma, L. Gu, P. Zhang, L. Wang, and Q.-K. Xue, Interface
enhanced superconductivity in monolayer FeSe films on MgO(001): Charge
transfer with atomic substitution, Science Bulletin 63, 747 (2018).
Song et al. [2011]
C.-L. Song, Y.-L. Wang,
Y.-P. Jiang, Z. Li, L. Wang, K. He, X. Chen, X.-C. Ma, and Q.-K. Xue, Molecular-beam epitaxy and robust
superconductivity of stoichiometric FeSe crystalline films on bilayer
graphene, Phys. Rev. B 84, 020503 (2011).
Tenne et al. [2007]
D. A. Tenne, I. E. Gonenli,
A. Soukiassian, D. G. Schlom, S. M. Nakhmanson, K. M. Rabe, and X. X. Xi, Raman study of oxygen reduced and re-oxidized strontium
titanate, Physical Review B 76, 024303 (2007).
Szot et al. [2002]
K. Szot, W. Speier,
R. Carius, U. Zastrow, and W. Beyer, Localized metallic conductivity and self-healing during thermal
reduction of SrTiO${}_{3}$, Physical Review Letters 88, 075508 (2002).
Merkle and Maier [2003]
R. Merkle and J. Maier, Defect Association in Acceptor-Doped
SrTiO${}_{3}$: Case study for Fe${}^{\prime}_{\mathrm{Ti}}$V${}^{{}^{\cdot\cdot}}_{\mathrm{O}}$
and Mn${}^{\prime\prime}_{\mathrm{Ti}}$V${}^{{}^{\cdot\cdot}}_{\mathrm{O}}$, Physical Chemistry Chemical Physics 5, 2297 (2003).
Zvanut et al. [2008]
M. E. Zvanut, S. Jeddy,
E. Towett, G. M. Janowski, C. Brooks, and D. Schlom, An annealing study of an oxygen vacancy related defect in
SrTiO${}_{3}$ substrates, Journal of Applied Physics 104, 064122 (2008).
Xu et al. [2017]
M. Xu, X. Song, and H. Wang, Substrate and band bending effects on monolayer
FeSe on SrTiO${}_{3}$(001), Phys. Chem. Chem. Phys. 19, 7964 (2017).
Kasap and Capper [2017]
S. Kasap and P. Capper, eds., Springer Handbook of Electronic and Photonic Materials (Springer International Publishing, Cham, 2017).
Li et al. [2021]
Y. Li, Z. Wang, R. Xiao, Q. Li, K. Wang, A. Richardella, J. Wang, and N. Samarth, Capping layer influence and isotropic
in-plane upper critical field of the superconductivity at the
FeSe/SrTiO${}_{3}$ interface, Physical Review Materials 5, 034802 (2021).
Hensling et al. [2017]
F. V. E. Hensling, C. Xu, F. Gunkel, and R. Dittmann, Unraveling the enhanced oxygen vacancy formation
in complex oxides during annealing and growth, Scientific Reports 7, 39953 (2017).
Erdman et al. [2002]
N. Erdman, K. R. Poeppelmeier, M. Asta,
O. Warschkow, D. E. Ellis, and L. D. Marks, The structure and chemistry of the TiO${}_{2}$-rich surface
of SrTiO${}_{3}$(001), Nature 419, 55 (2002).
Zhu et al. [2012]
G.-Z. Zhu, G. Radtke, and G. A. Botton, Bonding and structure of a reconstructed (001)
surface of SrTiO${}_{3}$ from TEM, Nature 490, 384 (2012).
Marshall et al. [2015]
M. S. J. Marshall, A. E. Becerra-Toledo, L. D. Marks, and M. R. Castell, Defects on
strontium titanate, in Defects at Oxide Surfaces, Springer Series
in Surface Sciences, edited by J. Jupille and G. Thornton (Springer, Cham, Cham, 2015) pp. 327–349.
Hu et al. [2018]
Y. Hu, Y. Xu, Q. Wang, L. Zhao, S. He, J. Huang, C. Li, G. Liu, and X. J. Zhou, Identification of a large amount of excess Fe in superconducting
single-layer FeSe/SrTiO${}_{3}$ films, Physical Review B 97, 224512 (2018).
Hu et al. [2020]
Y. Hu, Y. Xu, Y.-M. Zhang, Q. Wang, S. He, D. Liu, A. Liang, J. Huang, C. Li, Y. Cai, D. Wu, G. Liu, F.-S. Li, J.-Q. Fan, G.-Y. Zhou, L. Wang, C.-L. Song, X.-C. Ma, Q.-K. Xue,
Z. Xu, L. Zhao, and X. J. Zhou, Discovery of an insulating parent phase in single-layer
FeSe/SrTiO${}_{3}$ films, Physical Review B 102, 115144 (2020).
Casey and Pearson [1975]
H. C. Casey and G. L. Pearson, Diffusion in
Semiconductors, in Point Defects in Solids: Volume 2 Semiconductors and
Molecular Crystals, edited by J. H. Crawford and L. M. Slifkin (Springer US, Boston, MA, 1975) pp. 163–255.
Berlijn et al. [2014]
T. Berlijn, H.-P. Cheng,
P. J. Hirschfeld, and W. Ku, Doping Effects of Se Vacancies in Monolayer
FeSe, Physical Review B 89, 020501 (2014).
Shanavas and Singh [2015]
K. V. Shanavas and D. J. Singh, Doping SrTiO${}_{3}$ supported
FeSe by excess atoms and oxygen vacancies, Physical Review B 92, 035144 (2015).
Tikhonova and Korshunov [2020]
L. V. Tikhonova and M. M. Korshunov, Effect of the
additional Se layer on the electronic structure of iron-based
superconductor FeSe/SrTiO${}_{3}$, Journal of Superconductivity and Novel Magnetism 33, 171 (2020).
Lindman et al. [2015]
A. Lindman, P. Erhart, and G. Wahnström, Implications of the band gap problem
on oxidation and hydration in acceptor-doped barium zirconate, Phys. Rev. B 91, 245114 (2015).
James E. Huheey and Keiter [1993]
E. A. K. James E. Huheey and R. L. Keiter, eds., Inorganic chemistry: principal
of structure and reactivity, 4th ed. (Harper Collins College Publishers, 1993).
Tiwari et al. [2012]
S. Tiwari, P. Tandon, and K. N. Uttam, Thermodynamical quantities of
chalcogenide dimers (O${}_{2}$, S${}_{2}$, Se${}_{2}$, and Te${}_{2}$) from
spectroscopic data, Journal of Spectroscopy 2013, 956581 (2012).
McQuarrie [1976]
D. A. McQuarrie, Statistical
Mechanics (Harper & Row, New York, 1976).
Otgonbaatar et al. [2014]
U. Otgonbaatar, W. Ma,
M. Youssef, and B. Yildiz, Effect of niobium on the defect chemistry and oxidation
kinetics of tetragonal ZrO${}_{2}$, Journal of Physical Chemistry C 118, 20122 (2014). |
Scalar $f_{0}(980)$ meson effect in radiative $\phi\rightarrow\pi^{+}\pi^{-}\gamma$ decay
Ayşe Küçükarslan
kucukarslan@metu.edu.tr
Middle East Technical University, Physics Department. (06531), Ankara/Turkey
Saime Solmaz
skerman@balikesir.edu.tr
Balikesir University, Physics Department.(10100),
Balikesir/Turkey
(November 21, 2020)
Abstract
We study the effect of scalar-isoscalar $f_{0}(980)$ meson in the
mechanism of the radiative $\phi\rightarrow\pi^{+}\pi^{-}\gamma$
decay. A phenomenological approach is used to study this decay by
considering the contributions of $\sigma$-meson, $\rho$-meson and
$f_{0}(980)$-meson. The interference effects between different
contributions are analyzed and the branching ratio for this decay
is calculated. We observe that $f_{0}$ meson contribution is much
larger than the contributions of the other terms.
I Introduction
Radiative decays of vector mesons offer the possibility of
investigating new physics features about the interesting mechanism
involved in these decays. One particular mechanism involves the
exchange of scalar mesons. The scalar mesons, isoscalar $\sigma$
and $f_{0}(980)$ and isovector $a_{0}(980)$, with vacuum quantum
numbers $J^{PC}=0^{++}$ are known to be crucial for a full
understanding of the low energy QCD phenomenology and the symmetry
breaking mechanisms in QCD. The scalar mesons have been a
persistent problem in hadron spectroscopy. In addition to the
identification of their nature, the role of scalar mesons in
hadronic processes is of extreme importance and the study of
radiative decays of vector mesons may provide insights about their
role.
In particular, radiative $\phi$ meson decays,
$\phi\rightarrow\pi\pi\gamma$ and
$\phi\rightarrow\pi^{0}\eta\gamma$, can play a crucial role in the
clarification of the structure and properties of scalar
$f_{0}(980)$ and $a_{0}(980)$ mesons since these decays primarily
proceed through processes involving scalar resonances such as
$\phi\rightarrow f_{0}(980)\gamma$ and $\phi\rightarrow a_{0}(980)\gamma$, with the subsequent decays into $\pi\pi\gamma$
and $\pi^{0}\eta\gamma$ R1 ; R2 . Achasov and Ivanchenko
R1 showed that if the $f_{0}(980)$ and $a_{0}(980)$
resonances are four-quark $(q^{2}\bar{q}^{2})$ states the
processes $\phi\rightarrow f_{0}(980)\gamma$ and $\phi\rightarrow a_{0}(980)\gamma$ are dominant and enhance the decays
$\phi\rightarrow\pi\pi\gamma$ and
$\phi\rightarrow\pi^{0}\eta\gamma$ by at least an order of
magnitude over the results predicted by the Wess-Zumino terms.
Then Close et al. R2 noted that the study of the scalar
states in $\phi\rightarrow S\gamma$, where $S=f_{0}~{}or~{}a_{0}$,
may offer unique insights into the nature of the scalar mesons.
They have shown that although the transition rates
$\Gamma(\phi\rightarrow f_{0}\gamma)$ and $\Gamma(\phi\rightarrow a_{0}\gamma)$ depend on the unknown dynamics, the ratio of the
decay rates $\Gamma(\phi\rightarrow a_{0}\gamma)/\Gamma(\phi\rightarrow f_{0}\gamma)$ provides an
experimental test which distinguishes between alternative
explanations of their structure. On the experimental side, the
Novosibirsk CMD-2 R3 ; R4 and SND R5 collaborations
give the following branching ratios for
$\phi\rightarrow\pi^{+}\pi^{-}\gamma$ and
$\phi\rightarrow\pi^{0}\eta\gamma$ decays:
$BR(\phi\rightarrow\pi^{+}\pi^{-}\gamma)=(0.41\pm 0.12\pm 0.04)\times 10^{-4}$
R3 ,
$BR(\phi\rightarrow\pi^{0}\eta\gamma)=(0.90\pm 0.24\pm 0.10)\times 10^{-4}$
R4 ,
$BR(\phi\rightarrow\pi^{0}\eta\gamma)=(0.88\pm 0.14\pm 0.09)\times 10^{-4}$
R5 , where the first error is statistical and the second one
is systematic.
Theoretically, the role of $f_{0}(980)$-meson in the radiative
decay processes $\phi\rightarrow\pi\pi\gamma$ was also
investigated by Achasov et al. R6 . They calculated the
branching ratio for this decay by considering only
$f_{0}(980)$-meson contribution. They used two different models of
$f_{0}(980)$-meson: the four-quark model and $K\bar{K}$ molecular
model. In the four-quark model they obtained the value for the
branching ratio as $BR(\phi\rightarrow f_{0}\gamma\rightarrow\pi\pi\gamma)=2.3\times 10^{-4}$ and in case
of the $K\bar{K}$ molecular model, the branching ratio was
$BR(\phi\rightarrow f_{0}\gamma\rightarrow\pi\pi\gamma)=1.7\times 10^{-5}$. Later,
Marco et al. considered the radiative $\phi$ meson decays
R7 as well as other radiative vector meson decays within
the framework of chiral unitary theory developed earlier by Oller
R8 . They obtained the result
$BR(\phi\rightarrow\pi^{+}\pi^{-}\gamma)=1.6\times 10^{-4}$ for the
branching ratio of the $\phi\rightarrow\pi^{+}\pi^{-}\gamma$ decay
and emphasized that the branching ratio for
$\phi\rightarrow\pi^{+}\pi^{-}\gamma$ decay is twice the one for
$\phi\rightarrow\pi^{0}\pi^{0}\gamma$ decay. Recently, the
radiative $\phi\rightarrow\pi^{0}\pi^{0}\gamma$ decay, where the
scalar $f_{0}(980)$-meson plays an important role was studied by
Gökalp and Yılmaz R9 within the framework of a
phenomenological approach in which the contributions of
$\sigma$-meson, $\rho$-meson and $f_{0}$-meson are considered.
They analyzed the interference effects between different
contributions. Their analysis showed that $f_{0}(980)$-meson
amplitude makes a substantial contribution to the branching ratio
of this decay. Furthermore, recently Escribano has been studied
the scalar meson exchange in $V\rightarrow\pi^{0}\pi^{0}\gamma$
decays R10 . He discussed the scalar contributions to the
$\phi\rightarrow\pi^{0}\pi^{0}\gamma$,
$\phi\rightarrow\pi^{0}\eta\gamma$ and
$\rho^{0}\rightarrow\pi^{0}\pi^{0}\gamma$ decays in the framework
of the linear sigma model ($L\sigma M$). He obtained the result
$BR(\phi\rightarrow\pi^{0}\pi^{0}\gamma)=1.16\times 10^{-4}$ for
the branching ratio of the $\phi\rightarrow\pi^{0}\pi^{0}\gamma$
decay and noted that, the branching ratio for this decay is
dominated by $f_{0}(980)$ meson amplitude.
In this work, we study the radiative vector meson decay
$\phi\rightarrow\pi^{+}\pi^{-}\gamma$ to investigate the role of
the scalar $f_{0}(980)$ meson and to extract the relevant
information on the properties of this scalar meson. Theoretically,
the radiative $\phi\rightarrow\pi^{+}\pi^{-}\gamma$ decay has not
been studied extensively up to now. One of the rare studies of
this decay was by Marco et al. R7 who neglected the
contributions coming from intermediate vector meson states.
Therefore, this decay should be reconsidered and the VMD amplitude
should be added to the $f_{0}$-meson and $\sigma$-meson
amplitudes.
II Formalism
We study the radiative decay $\phi\rightarrow\pi^{+}\pi^{-}\gamma$ within the framework of a phenomenological
approach in which the contributions of $\sigma$-meson,
$\rho$-meson and $f_{0}$-meson are considered and we do not make
any assumption about the structure of the $f_{0}$ meson. In our
phenomenological approach we describe the $\phi KK$-vertex by the
effective Lagrangian
$$\displaystyle{\cal L}^{eff.}_{\phi KK}=-ig_{\phi KK}\phi^{\mu}\left(K^{+}%
\partial_{\mu}K^{-}-K^{-}\partial_{\mu}K^{+}\right)~{}~{},$$
(1)
and for the $f_{0}KK$-vertex we use the phenomenological
Lagrangian
$$\displaystyle{\cal L}^{eff.}_{f_{0}KK}=g_{f_{0}KK}M_{f_{0}}K^{+}K^{-}f_{0}~{}~%
{}.$$
(2)
The effective Lagrangians for the $\phi KK$- and $f_{0}KK$-vertices also serve to define the coupling constants $g_{\phi KK}$ and $g_{f_{0}KK}$ respectively. The decay width for the
$\phi\rightarrow K^{+}K^{-}$ decay is obtained from the Lagrangian
given in Eq. 1 and this decay width is
$$\displaystyle\Gamma(\phi\rightarrow K^{+}K^{-})=\frac{g^{2}_{\phi KK}}{48\pi}M%
_{\phi}\left[1-\left(\frac{2M_{K}}{M_{\phi}}\right)^{2}\right]^{3/2}~{}~{}.$$
(3)
We then obtain the coupling constant $g_{\phi KK}$ from the
experimental partial width R11 of the radiative decay
$\phi\rightarrow K^{+}K^{-}$ as $g_{\phi KK}=(4.59\pm 0.05)$. The
amplitude of the radiative decay $\phi\rightarrow f_{0}\gamma$ is
obtained from the diagrams shown in Fig. 1 where the last diagram
assures gauge invariance R1 ; R12 . This amplitude is
$$\displaystyle{\cal M}\left(\phi\rightarrow f_{0}\gamma\right)=-\frac{1}{2\pi^{%
2}M_{K}^{2}}\left(g_{f_{0}KK}M_{f_{0}}\right)\left(eg_{\phi KK}\right)I(a,b)%
\left[\epsilon\cdot u~{}k\cdot p-\epsilon\cdot p~{}k\cdot u\right]$$
(4)
where $(u,p)$ and $(\epsilon,k)$ are the polarizations and
four-momenta of the $\phi$ meson and the photon respectively, and
also $a=M_{\phi}^{2}/M_{K}^{2}$, $b=M_{f_{0}}^{2}/M_{K}^{2}$. The
$I(a,b)$ function has been calculated in different contexts
R2 ; R8 ; R13 and is defined as
$$\displaystyle I(a,b)=\frac{1}{2(a-b)}-\frac{2}{(a-b)^{2}}\left[f(\frac{1}{b})-%
f(\frac{1}{a})\right]+\frac{a}{(a-b)^{2}}\left[g(\frac{1}{b})-g(\frac{1}{a})%
\right]~{}~{},$$
(5)
where
$$\displaystyle f(x)=\left\{\begin{array}[]{rr}-\left[\arcsin(\frac{1}{2\sqrt{x}%
})\right]^{2}~{},&~{}~{}x>\frac{1}{4}\\
\frac{1}{4}\left[\ln(\frac{\eta_{+}}{\eta_{-}})-i\pi\right]^{2}~{},&~{}~{}x<%
\frac{1}{4}\end{array}\right.$$
$$\displaystyle g(x)=\left\{\begin{array}[]{rr}(4x-1)^{\frac{1}{2}}\arcsin(\frac%
{1}{2\sqrt{x}})~{},&~{}~{}x>\frac{1}{4}\\
\frac{1}{2}(1-4x)^{\frac{1}{2}}\left[\ln(\frac{\eta_{+}}{\eta_{-}})-i\pi\right%
]~{},&~{}~{}x<\frac{1}{4}\end{array}\right.$$
$$\displaystyle\eta_{\pm}=\frac{1}{2x}\left[1\pm(1-4x)^{\frac{1}{2}}\right]~{}.$$
(6)
Then, the decay rate for the $\phi\rightarrow f_{0}\gamma$ decay
is
$$\displaystyle\Gamma(\phi\rightarrow f_{0}\gamma)=\frac{\alpha}{6(2\pi)^{4}}%
\frac{M_{\phi}^{2}-M_{f_{0}}^{2}}{M_{\phi}^{3}}g^{2}_{\phi KK}\left(g_{f_{0}KK%
}M_{f_{0}}\right)^{2}\left|(a-b)I(a,b)\right|^{2}~{}~{}.$$
(7)
Utilizing the experimental value for the branching ratio
$BR(\phi\rightarrow f_{0}\gamma)=(3.4\pm 0.4)\times 10^{-4}$ for
the decay $\phi\rightarrow f_{0}\gamma$ R11 , we determine
the coupling constant $g_{f_{0}KK}$ as $g_{f_{0}KK}=(4.13\pm 1.42)$. In our calculation, we assume that the radiative decay
$\phi\rightarrow\pi^{+}\pi^{-}\gamma$ proceeds through the
reactions $\phi\rightarrow\sigma\gamma\rightarrow\pi^{+}\pi^{-}\gamma$,
$\phi\rightarrow\rho^{\mp}\pi^{\pm}\rightarrow\pi^{+}\pi^{-}\gamma$ and $\phi\rightarrow f_{0}\gamma\rightarrow\pi^{+}\pi^{-}\gamma$. Therefore, our calculation is based on the
Feynman diagrams shown in Fig. 2. For the
$\phi\sigma\gamma$-vertex, we use the effective Lagrangian
$$\displaystyle{\cal L}^{eff.}_{\phi\sigma\gamma}=\frac{e}{M_{\phi}}g_{\phi%
\sigma\gamma}[\partial^{\alpha}\phi^{\beta}\partial_{\alpha}A_{\beta}-\partial%
^{\alpha}\phi^{\beta}\partial_{\beta}A_{\alpha}]\sigma~{}~{},$$
(8)
which also defines the coupling constant $g_{\phi\sigma\gamma}$.
The coupling constant $g_{\phi\sigma\gamma}$ is determined by
Gökalp and Yılmaz R9 as $g_{\phi\sigma\gamma}=(0.025\pm 0.009)$ using the experimental value of the branching ratio for
the radiative decay $\phi\rightarrow\pi^{0}\pi^{0}\gamma$
R14 . For the $\sigma\pi\pi$-vertex we use the effective
Lagrangian
$$\displaystyle{\cal L}^{eff.}_{\sigma\pi\pi}=\frac{1}{2}g_{\sigma\pi\pi}M_{%
\sigma}\vec{\pi}\cdot\vec{\pi}\sigma~{}~{}.$$
(9)
The decay width of the $\sigma$-meson that results from this
effective Lagrangian is given as
$$\displaystyle\Gamma_{\sigma}\equiv\Gamma(\sigma\rightarrow\pi\pi)=\frac{g^{2}_%
{\sigma\pi\pi}}{4\pi}\frac{3M_{\sigma}}{8}\left[1-\left(\frac{2M_{\pi}}{M_{%
\sigma}}\right)^{2}\right]^{1/2}~{}~{}.$$
(10)
For given values of $M_{\sigma}$ and $\Gamma_{\sigma}$, we use
this expression to determine the coupling constant
$g_{\sigma\pi\pi}$. Therefore, using the experimental values for
$M_{\sigma}$ and $\Gamma_{\sigma}$ R15 , given as
$M_{\sigma}=(478\pm 17)~{}MeV$ and $\Gamma_{\sigma}=(324\pm 21)~{}MeV$, we obtain the coupling constant
$g_{\sigma\pi\pi}=(5.25\pm 0.32)$. The $\phi\rho\pi$-vertex is
conventionally described by the effective Lagrangian
$$\displaystyle{\cal L}^{eff.}_{\phi\rho\pi}=\frac{g_{\phi\rho\pi}}{M_{\phi}}%
\epsilon^{\mu\nu\alpha\beta}\partial_{\mu}\phi_{\nu}\partial_{\alpha}\vec{\rho%
_{\beta}}\cdot\vec{\pi}~{}~{}.$$
(11)
The coupling constant $g_{\phi\rho\pi}$ is calculated as
$g_{\phi\rho\pi}=(0.811\pm 0.081)~{}GeV^{-1}$ by Achasov and Gubin
R16 using the data on the decay
$\phi\rightarrow\rho\pi\rightarrow\pi^{+}\pi^{-}\pi^{0}$
R11 . For the $\rho\pi\gamma$-vertex the effective
Lagrangian
$$\displaystyle{\cal L}^{eff.}_{\rho\pi\gamma}=\frac{e}{M_{\rho}}g_{\rho\pi%
\gamma}\epsilon^{\mu\nu\alpha\beta}\partial_{\mu}\vec{\rho_{\nu}}\cdot\vec{\pi%
}~{}\partial_{\alpha}A_{\beta}~{}~{},$$
(12)
is used. At present there is a discrepancy between the
experimental widths of the $\rho^{0}\rightarrow\pi^{0}\gamma$ and
$\rho^{+}\rightarrow\pi^{+}\gamma$ decays. We use the experimental
rate for the decay $\rho^{0}\rightarrow\pi^{0}\gamma$ R11 to
extract the coupling constant $g_{\rho\pi\gamma}$ as
$g_{\rho\pi\gamma}=(0.69\pm 0.35)$ since the experimental value
for the decay rate of $\phi\rightarrow\pi^{0}\pi^{0}\gamma$ was
used by Gökalp and Yılmaz R9 to estimate the coupling
constant $g_{\phi\sigma\gamma}$. Finally, the $f_{0}\pi\pi$-vertex
is described conventionally by the effective Lagrangian
$$\displaystyle{\cal L}^{eff.}_{f_{0}\pi\pi}=\frac{1}{2}g_{f_{0}\pi\pi}M_{f_{0}}%
\vec{\pi}\cdot\vec{\pi}f_{0}~{}~{}.$$
(13)
In our calculation of the invariant amplitude, we make the
replacement $q^{2}-M^{2}\rightarrow q^{2}-M^{2}+iM\Gamma$, where
$q$ and $M$ are four-momentum and mass of the virtual particles
respectively, in $\rho$-, $\sigma$- and $f_{0}$- propagators in
order to take into account the finite widths of these unstable
particles and use the experimental value $\Gamma_{\rho}=(150.2\pm 0.8)~{}MeV$ R11 for $\rho$-meson. However, since the mass
$M_{f_{0}}=980~{}MeV$ of $f_{0}$-meson is very close to the
$K^{+}K^{-}$ threshold this gives rise to a strong energy
dependence on the width of the $f_{0}$-meson and to include this
energy dependence different expressions for $\Gamma_{f_{0}}$ can
be used. First option is to use an energy dependent width for
$f_{0}$
$$\displaystyle\Gamma_{f_{0}}(q^{2})=\Gamma_{\pi\pi}^{f_{0}}(q^{2})~{}\theta%
\left(\sqrt{q^{2}}-2M_{\pi}\right)+\Gamma_{K\overline{K}}^{f_{0}}(q^{2})~{}%
\theta\left(\sqrt{q^{2}}-2M_{K}\right)~{}~{},$$
(14)
where $q^{2}$ is the four-momentum square of the virtual
$f_{0}$-meson and the width $\Gamma_{\pi\pi}^{f_{0}}(q^{2})$ is
given as
$$\displaystyle\Gamma_{\pi\pi}^{f_{0}}(q^{2})=\Gamma_{\pi\pi}^{f_{0}}~{}\frac{M_%
{f_{0}}^{2}}{q^{2}}\sqrt{\frac{q^{2}-4M_{\pi}^{2}}{M_{f_{0}}^{2}-4M_{\pi}^{2}}%
}~{}~{}.$$
(15)
We use the experimental value for $\Gamma_{\pi\pi}^{f_{0}}$ as
$\Gamma_{\pi\pi}^{f_{0}}=40-100~{}MeV$ R11 . The width
$\Gamma_{K\overline{K}}^{f_{0}}(q^{2})$ is given by a similar
expression as for $\Gamma_{\pi\pi}^{f_{0}}(q^{2})$. Another and
widely accepted option is the work of Flatté R17 . In
his work, the expression for
$\Gamma_{K\overline{K}}^{f_{0}}(q^{2})$ is extended below the
$K\overline{K}$ threshold where $\sqrt{q^{2}-4M_{K}^{2}}$ is
replaced by $i\sqrt{4M_{K}^{2}-q^{2}}$ so
$\Gamma_{K\overline{K}}^{f_{0}}(q^{2})$ becomes purely imaginary.
However in our work, we take into account both options. The
invariant amplitude ${\cal M}(E_{\gamma},E_{1})$ is expressed as
${\cal M}(E_{\gamma},E_{1})={\cal M}_{a}+{\cal M}_{b}+{\cal M}_{c}+{\cal M}_{d}$ where ${\cal M}_{a}$, ${\cal M}_{b}$, ${\cal M}_{c}$ and ${\cal M}_{d}$ are the invariant amplitudes resulting
from the diagrams $(a)$, $(b)$, $(c)$ and $(d)$ in Fig. 2
respectively. Therefore, the interference between different
reactions contributing to the decay $\phi\rightarrow\pi^{+}\pi^{-}\gamma$ is taken into account. The differential
decay probability for an unpolarized $\phi$-meson at rest is given
as
$$\displaystyle\frac{d\Gamma}{dE_{\gamma}dE_{1}}=\frac{1}{(2\pi)^{3}}~{}\frac{1}%
{8M_{\phi}}~{}\mid{\cal M}\mid^{2}~{}~{},$$
(16)
where E${}_{\gamma}$ and E${}_{1}$ are the photon and pion energies
respectively. We perform an average over the spin states of
$\phi$-meson and a sum over the polarization states of the photon.
The decay width $\Gamma(\phi\rightarrow\pi^{+}\pi^{-}\gamma)$ is
then obtained by integration
$$\displaystyle\Gamma=\int_{E_{\gamma,min.}}^{E_{\gamma,max.}}dE_{\gamma}\int_{E%
_{1,min.}}^{E_{1,max.}}dE_{1}\frac{d\Gamma}{dE_{\gamma}dE_{1}}~{}~{},$$
(17)
where the minimum photon energy is E${}_{\gamma,min.}=0$ and the
maximum photon energy is given as
$E_{\gamma,max.}=(M_{\phi}^{2}-4M_{\pi}^{2})/2M_{\phi}=471.8~{}MeV$.
The maximum and minimum values for the pion energy E${}_{1}$ are
given by
$$\displaystyle\frac{1}{2(2E_{\gamma}M_{\phi}-M_{\phi}^{2})}[-2E_{\gamma}^{2}M_{%
\phi}+3E_{\gamma}M_{\phi}^{2}-M_{\phi}^{3}$$
$$\displaystyle\pm E_{\gamma}\sqrt{(-2E_{\gamma}M_{\phi}+M_{\phi}^{2})(-2E_{%
\gamma}M_{\phi}+M_{\phi}^{2}-4M_{\pi}^{2})}~{}]~{}~{}.$$
(18)
III Results and Discussion
In order to determine the coupling constant $g_{f_{0}\pi\pi}$, we
choose for the $f_{0}$-meson parameters the values
$M_{f_{0}}=980~{}MeV$ and $\Gamma_{f_{0}}=(70\pm 30)~{}MeV$. Therefore,
through the decay rate that results from the effective Lagrangian
given in Eq. 13 we obtain the coupling constant $g_{f_{0}\pi\pi}$
as $g_{f_{0}\pi\pi}=(1.58\pm 0.30)$. If we use the form for
$\Gamma^{f_{0}}_{K\bar{K}}(q^{2})$, proposed by Flatté
R17 , the desired enhancement in the invariant mass spectrum
appears in its central part rather than around the $f_{0}$ pole.
Bramon et al. R18 also encountered a similar problem in
their study of the effects of the $a_{0}(980)$ meson in the
$\phi\rightarrow\pi^{0}\eta\gamma$ decay. Therefore, in the
analysis which we present below for $\Gamma_{f_{0}}(q^{2})$ we use
the form given in Eq. 14. The invariant mass distribution
$dB/dM_{\pi\pi}=(M_{\pi\pi}/M_{\phi})dB/dE_{\gamma}$ for the
radiative decay $\phi\rightarrow\pi^{+}\pi^{-}\gamma$ is plotted in
Fig. 3 as a function of the invariant mass $M_{\pi\pi}$ of
$\pi^{+}\pi^{-}$ system. In this figure we indicate the contributions
coming from different reactions
$\phi\rightarrow\sigma\gamma\rightarrow\pi^{+}\pi^{-}\gamma$,
$\phi\rightarrow\rho^{\mp}\pi^{\pm}\rightarrow\pi^{+}\pi^{-}\gamma$
and $\phi\rightarrow f_{0}\gamma\rightarrow\pi^{+}\pi^{-}\gamma$ as
well as the contribution of the total amplitude which includes the
interference terms as well. It is clearly seen from Fig. 3 that
the spectrum for the decay $\phi\rightarrow\pi^{+}\pi^{-}\gamma$ is
dominated by the $f_{0}$-amplitude. On the other hand the
contribution coming from $\sigma$-amplitude can only be noticed in
the region $M_{\pi\pi}<0.7~{}GeV$ through interference effects.
Likewise $\rho$-meson contribution can be seen in the region
$M_{\pi\pi}<0.8~{}GeV$ so we can say that the contribution of the
$f_{0}$-term is much larger than the contributions of the
$\sigma$-term and $\rho$-term as well as the contribution of the
total interference term having opposite sign.
The dominant $f_{0}$-term characterizes the invariant mass
distribution in the region where $M_{\pi\pi}>0.7~{}GeV$. In our
study contributions of different amplitudes to the branching ratio
of the decay $\phi\rightarrow\pi^{+}\pi^{-}\gamma$ are
$BR(\phi\rightarrow f_{0}\gamma\rightarrow\pi^{+}\pi^{-}\gamma)=2.54\times 10^{%
-4}$,
$BR(\phi\rightarrow\sigma\gamma\rightarrow\pi^{+}\pi^{-}\gamma)=0.07\times 10^{%
-4}$,
$BR(\phi\rightarrow\rho^{\mp}\pi^{\pm}\rightarrow\pi^{+}\pi^{-}\gamma)=0.26%
\times 10^{-4}$,
$BR(\phi\rightarrow(f_{0}\gamma+\pi^{\pm}\rho^{\mp})\rightarrow\pi^{+}\pi^{-}%
\gamma)=2.74\times 10^{-4}$,
$BR(\phi\rightarrow(f_{0}\gamma+\sigma\gamma)\rightarrow\pi^{+}\pi^{-}\gamma)=2%
.29\times 10^{-4}$ and for the total interference term
$BR(\textrm{interference})=-0.29\times 10^{-4}$. We then calculate
the total branching ratio as $BR(\phi\rightarrow\pi^{+}\pi^{-}\gamma)=2.57\times 10^{-4}$. Our result is twice the
theoretical result for $\phi\rightarrow\pi^{0}\pi^{0}\gamma$ decay,
obtained by Gökalp and Yılmaz R9 , as it should be. They
obtained the following values: $BR(\phi\rightarrow f_{0}\gamma\rightarrow\pi^{0}\pi^{0}\gamma)=1.29\times 10^{%
-4}$,
$BR(\phi\rightarrow\sigma\gamma\rightarrow\pi^{0}\pi^{0}\gamma)=0.04\times 10^{%
-4}$,
$BR(\phi\rightarrow\rho^{0}\pi^{0}\rightarrow\pi^{0}\pi^{0}\gamma)=0.14\times 1%
0^{-4}$,
$BR(\phi\rightarrow(f_{0}\gamma+\pi^{0}\rho^{0})\rightarrow\pi^{0}\pi^{0}\gamma%
)=1.34\times 10^{-4}$,
$BR(\phi\rightarrow(f_{0}\gamma+\sigma\gamma)\rightarrow\pi^{0}\pi^{0}\gamma)=1%
.16\times 10^{-4}$ and $BR\textrm{(interference)}=-0.25\times 10^{-4}$.
Moreover, our calculation for the branching ratio of the radiative
decay $\phi\rightarrow\pi^{+}\pi^{-}\gamma$ is nearly twice the value
for the branching ratio of the radiative decay
$\phi\rightarrow\pi^{0}\pi^{0}\gamma$ that was obtained by Achasov and
Gubin R16 . Besides, $\phi\rightarrow\pi^{+}\pi^{-}\gamma$
decay was considered by Marco et al. R7 in the framework of
unitarized chiral perturbation theory. The branching ratio for
$\phi\rightarrow\pi^{+}\pi^{-}\gamma$, they obtained, was
$BR(\phi\rightarrow\pi^{+}\pi^{-}\gamma)=1.6\times 10^{-4}$ and
for $\phi\rightarrow\pi^{0}\pi^{0}\gamma$ was
$BR(\phi\rightarrow\pi^{0}\pi^{0}\gamma)=0.8\times 10^{-4}$. As we
mentioned above, they noted that the branching ratio for
$\phi\rightarrow\pi^{0}\pi^{0}\gamma$ is one half of
$\phi\rightarrow\pi^{+}\pi^{-}\gamma$. Therefore our calculation
for the branching ratio of $\phi\rightarrow\pi^{+}\pi^{-}\gamma$
decay is in accordance with the theoretical expectations. A
similar relation can be seen between the decay rates of
$\omega\rightarrow\pi^{+}\pi^{-}\gamma$ and
$\omega\rightarrow\pi^{0}\pi^{0}\gamma$ R19 . It was noticed
that
$\Gamma(\omega\rightarrow\pi^{0}\pi^{0}\gamma)=1/2\Gamma(\omega\rightarrow\pi^{%
+}\pi^{-}\gamma)$
and the factor $1/2$ is a result of charge conjugation invariance
to order $\alpha$ which imposes pion pairs of even angular
momentum. The experimental value of the branching ratio for
$\phi\rightarrow\pi^{+}\pi^{-}\gamma$, measured by Akhmetshin et
al., is
$BR(\phi\rightarrow\pi^{+}\pi^{-}\gamma)=(0.41\pm 0.12\pm 0.04)\times 10^{-4}$ R3 . So the value of the branching ratio that we
obtained is approximately six times larger than the value of the
measured branching ratio. As it was stated by Marco et al.
R7 , we should not compare our calculation for the branching
ratio of the radiative decay $\phi\rightarrow\pi^{+}\pi^{-}\gamma$
directly with experiment since the experiment is done using the
reaction $e^{+}e^{-}\rightarrow\phi\rightarrow\pi^{+}\pi^{-}\gamma$,
which interferes with the off-shell $\rho$ dominated amplitude
coming from the reaction
$e^{+}e^{-}\rightarrow\rho\rightarrow\pi^{+}\pi^{-}\gamma$ R20 .
Also the result in R3 is based on model dependent
assumptions.
IV Acknowledgement
We thank A. Gökalp and O. Yılmaz for their invaluable comments
and suggestions during this work.
References
(1)
N. N. Achasov, V. N. Ivanchenko, Nucl. Phys. B315, 465 (1989).
(2)
F. E. Close, N. Isgur, S. Kumona, Nucl. Phys. B389, 513 (1993).
(3)
R. R. Akhmetshin et al., Phys. Lett. B462, 371 (1999).
(4)
R. R. Akhmetshin et al., Phys. Lett. B462, 380 (1999).
(5)
M. N. Achasov et al., Phys. Lett. B479, 53 (2000).
(6)
N. N. Achasov, V. V. Gubin and E. P. Solodov, Phys. Rev. D55, 2672 (1997).
(7)
E. Marco, S. Hirenzaki, E. Oset and H. Toki, Phys. Lett. B470, 20 (1999).
(8)
J. A. Oller, Phys. Lett. B426, 7 (1998).
(9)
A. Gökalp and O. Yılmaz, Phys. Rev. D64, 053017 (2001).
(10)
R. Escribano, Talk presented at the 9th International High-Energy Physics Conference in
Quantum Chromodynamics (QCD 2002), Montpellier, France, 2-9 July
2002, hep-ph/0209375 (2002).
(11)
Particle Data Group, D. E. Groom et al., Eur. Phys. J.C15, 1 (2000).
(12)
V. E. Markushin, Eur. Phys. J. A8, 389 (2000).
(13)
J. L. Lucio M., J. Pestieau, Phys. Rev. D42, 3253 (1990); D43, 2447 (1991).
(14)
M. N. Achasov et al., Phys. Lett. B485, 349 (2000).
(15)
E791 Collaboration, E. M. Aitala et al., Phys. Rev. Lett. 86, 770 (2001).
(16)
N. N. Achasov and V. V. Gubin, Phys. Rev. D63, 094007 (2001).
(17)
S. M. Flatté, Phys. Lett. B63, 224 (1976).
(18)
A. Bramon, R. Escribano, J. L. Lucio M., M. Napsuciale, G. Pancheri, Phys. Lett. B494, 221 (2000).
(19)
P. Singer, Phys. Rev. 128, 2789 (1962); 130, 2441 (1963); 161, 1694 (1967).
(20)
A. Bramon, G. Colangelo and M. Greco, Phys. Lett. B287, 263 (1992). |
Resonant Production of Light Sterile Neutrinos in Compact Binary Merger Remnants
Garðar Sigurðarson
ID
Niels Bohr International Academy & DARK, Niels Bohr Institute,
University of Copenhagen, Blegdamsvej 17, 2100 Copenhagen, Denmark
Irene Tamborra
ID
Niels Bohr International Academy & DARK, Niels Bohr Institute,
University of Copenhagen, Blegdamsvej 17, 2100 Copenhagen, Denmark
Meng-Ru Wu
ID
Institute of Physics, Academia Sinica, Taipei, 11529, Taiwan
Institute of Astronomy and Astrophysics, Academia Sinica, Taipei, 10617, Taiwan
Abstract
The existence of eV-mass sterile neutrinos is not ruled out because of persistent experimental anomalies. Upcoming multi-messenger detections of neutron-star merger remnants could provide indirect constraints on the existence of these particles. We explore the active-sterile flavor conversion phenomenology in a two-flavor scenario ($1$ active $+1$ sterile species) as a function of the sterile neutrino mixing parameters, neutrino emission angle from the accretion torus, and temporal evolution of the merger remnant. The torus geometry and the neutron richness of the remnant are responsible for the occurrence of multiple resonant active-sterile conversions. The number of resonances strongly depends on the neutrino emission direction above or inside the remnant torus and leads to large production of sterile neutrinos (and no antineutrinos) in the proximity of the polar axis as well as more sterile antineutrinos than neutrinos in the equatorial region.
As the black hole torus evolves in time, the shallower baryon density is responsible for more adiabatic flavor conversion, leading to larger regions of the mass-mixing parameter space being affected by flavor mixing.
Our findings imply that the production of sterile states can have indirect implications on the disk cooling rate, its outflows, and related electromagnetic observables which remain to be assessed.
I Introduction
The coalescence of a neutron star (NS) with another NS or a black hole (BH) leads to the formation of a compact binary merger. Compact binary mergers lose angular momentum through the emission of gravitational waves. This conjecture was recently confirmed through the detection of the gravitational-wave event GW170817 Margutti and Chornock (2021); Abbott et al. (2017a, b, c); Goldstein et al. (2017); Savchenko et al. (2017); Margutti et al. (2017); Troja et al. (2017). Electromagnetic follow-up observations across multiple wavebands of GW170817 confirmed that NS merger remnants are factories of the elements heavier than iron and harbor short gamma-ray bursts Kasen et al. (2017); Drout et al. (2017); Cowperthwaite et al. (2017); Villar et al. (2017); Shibata et al. (2017); Metzger (2020).
While no neutrino has been observed from gravitational wave sources yet Veske et al. (2020); Aartsen et al. (2020); Hayato et al. (2018), thermal neutrinos are copiously produced in binary NS mergers, with the neutrino luminosities reaching up to $10^{54}$ erg/s within $\mathcal{O}(100)$ ms Foucart et al. (2015); Ruffert et al. (1997). Neutrinos dominate the cooling of the NS merger remnant and affect the ejecta composition, while neutrino pair annihilation above the BH accretion disk contributes to power the short gamma-ray burst jet Ruffert et al. (1997); Metzger and Fernández (2014); Foucart et al. (2015); Perego et al. (2014); Wanajo et al. (2014); Just et al. (2015); Fujibayashi et al. (2020); Kullmann et al. (2022); Narayan et al. (1992); Berger (2014); Just et al. (2016).
Despite intense work, the treatment of neutrino transport in hydrodynamical simulations of binary NS mergers is still approximated because of the technical challenges linked to the fully three-dimensional general-relativistic magnetohydrodynamical modeling of the source. In addition, neutrinos are treated as radiation, neglecting the occurrence of flavor conversion. However, the protonization of the merger remnant (i.e., the excess of electron antineutrinos with respect to electron neutrinos) presumably leads to the occurrence of the matter-neutrino resonance due to the cancellation of the matter potential involving interactions of neutrinos with electrons and the neutrino-neutrino potential Malkus et al. (2014, 2012); Wu et al. (2016); Zhu et al. (2016); Frensel et al. (2017); Tian et al. (2017); Shalgar (2018). In addition, recent work has focused on exploring the implications of $\nu$–$\nu$ interactions on the synthesis of the elements heavier than iron in the neutrino-driven outflow and the physics of neutrino-cooled accretion disks Tamborra and Shalgar (2021); Wu and Tamborra (2017); Wu et al. (2017); George et al. (2020); Just et al. (2022); Li and Siegel (2021); Fernández et al. (2022).
The expected large number of binary NS merger remnant observations will offer unprecedented opportunities to characterize the population of binary NS mergers as well as the physics of NSs and their nuclear equation of state Alves Batista et al. (2021); Burns (2020); Burns et al. (2019); Aggarwal et al. (2021). At the same time, upcoming multi-messenger observations of binary NS merger remnants and short gamma-ray bursts could provide constraints on physics beyond the Standard Model, see e.g. Refs. Diamond and Marques-Tavares (2022); Berezhiani and Drago (2000); Berezhiani et al. (2003) for some examples. An interesting and unexplored scenario in this regard concerns extra sterile neutrino families with eV mass Dasgupta and Kopp (2021); Abazajian et al. (2012).
The existence of sterile families of neutrinos has not been confirmed yet. However, to date, it is challenging to interpret a number of experimental results within the standard three neutrino flavor framework Böser et al. (2020); Acero et al. (2022).
Earlier hints on the existence of a fourth sterile neutrino family were provided by the LSND experiment and partly confirmed by MiniBooNE Aguilar-Arevalo et al. (2001, 2021). Along the same direction, reactor neutrino data could have been explained by invoking the existence of an eV-mass sterile neutrino; these puzzling effects concerning reactor neutrino fluxes seem to be now fully understood Mueller et al. (2011); Huber (2011); Giunti et al. (2022a); Kopeikin et al. (2021); Berryman and Huber (2021), despite remaining uncertainties on the reactor energy spectra Andriamirado et al. (2020); Berryman et al. (2022). Additional anomalies were also found by the Gallium experiments GALLEX and SAGE Giunti and Laveder (2011), and recently confirmed by BEST Barinov et al. (2022); Barinov and Gorbunov (2022). As a consequence, global fits invoking the existence of extra sterile neutrino families easily accommodate some datasets, but are somewhat in tension with others Dentler et al. (2018, 2018); Argüelles et al. (2022); Giunti et al. (2022b). Cosmological data do not rule out the existence of light sterile neutrinos Hagstotz et al. (2021); Hannestad et al. (2012); Chu et al. (2018); Archidiacono et al. (2020).
The phenomenology of light sterile neutrinos in core-collapse supernovae (SNe) has been widely investigated; these particles could have an impact on the synthesis of the elements heavier than iron as well as on shock revival Nunokawa et al. (1997); Tamborra et al. (2012); Wu et al. (2014); Pllumbi et al. (2015); Xiong et al. (2019).
Their existence could also strongly affect the expected neutrino signal from the next galactic SN Esmaili et al. (2014); Tang et al. (2020).
However, despite similarities in terms of neutrino number densities and energetics, the active-sterile flavor conversion physics and its indirect consequences on the multi-messenger emission have not been explored in the context of binary NS mergers.
In this paper, we rely on the output from one of the binary remnant simulations of Ref. Just et al. (2015) and, for the first time, investigate the production of sterile particles from active states through resonant neutrino-matter interactions, and eventual reconversion into active states. Our work is organized as follows. In Sec. II, we introduce our benchmark binary NS merger remnant model and its main features. Section III focuses on the physics of active-sterile flavor conversions in binary NS merger remnants. The flavor conversion phenomenology is explored in Sec. IV as a function of the active-sterile mixing parameters, while the production of sterile particles as the BH torus evolves as a function of time is outlined in Sec. V.
Finally, a summary of our findings is reported in Sec. VI.
II Binary neutron star merger remnant model
We rely on outputs of the hydrodynamical simulation of the BH accretion torus formed in the post-merger phase of a compact binary merger; specifically, we adopt the model M3A8m3a5 presented in Ref. Just et al. (2015). This simulation has a central BH of $3\ M_{\odot}$, dimensionless BH spin parameter $0.8$, and torus of $0.3\ M_{\odot}$. The total mass of the neutrino-driven ejecta is $1.47\times 10^{-3}\ M_{\odot}$, while the total outflow mass is $66.2\times 10^{-3}\ M_{\odot}$. We refer the interested reader to Ref. Just et al. (2015) for details on the simulation setup.
In this model, as the accretion torus forms, it starts to lose mass while accreting onto the central BH. During the first $\mathcal{O}(10)$ ms, the environment is optically thick and neutrino cooling is less efficient. As the density drops, it follows a phase of neutrino-dominated accretion flow, during which neutrino cooling balances viscous heating.
As the mass and density of the torus decrease, the neutrino production rate is also reduced, until neutrino cooling is not anymore efficient and the torus enters a phase dominated by advection, during which the viscous heating drives the expansion of the torus and launches outflows.
Figure 1 illustrates the characteristic properties of our BH torus remnant model and displays the baryon mass density, the electron fraction, as well as the $\nu_{e}$ number density, and the relative difference between the $\nu_{e}$ and $\bar{\nu}_{e}$ number densities in the region above the disk. All quantities have been extracted at $25$ ms for representative purposes and are shown in the $x$–$z$ plane, under the assumption of cylindrical symmetry around the $z$ axis. Note that, in the following, we track the flavor conversion physics along a radial direction $r$, defined such that $x=r\cos\theta$ and $z=r\sin\theta$, with $\theta$ being the polar angle measured with respect to the $z$ axis.
One can see that $n_{\bar{\nu}_{e}}\simeq n_{\nu_{e}}$ in the polar region. However, as a function of time, the BH torus evolves from a configuration where $n_{\bar{\nu}_{e}}>n_{\nu_{e}}$ to one with $n_{\bar{\nu}_{e}}<n_{\nu_{e}}$ in the proximity of the polar axis Just et al. (2015); Wu et al. (2017).
On the other hand, non-electron flavors of neutrinos and antineutrinos are
thermally produced in small amounts, but can be
generated through flavor conversion Just et al. (2015).
At high densities, neutrinos are coupled to the matter background. As the matter density decreases, neutrinos decouple from matter and start to free stream. The neutrino energy distributions for the electron flavors follow Fermi-Dirac distributions with non-zero chemical potential in the trapping regime and then tend to become pinched in the free-streaming regime. In the numerical computations, we rely on the numerical energy densities provided as output of our benchmark NS merger model Just et al. (2015). In order to assess whether the production of sterile particles occurs while the active neutrinos free stream, we estimate the location of the decoupling surfaces by requiring that the following condition is satisfied for the flux factor Wu et al. (2017)
$$\frac{|\textbf{F}_{\nu_{e},\bar{\nu}_{e}}|}{n_{\nu_{e},\bar{\nu}_{e}}}=\frac{1}{3}\ ,$$
(1)
where $\textbf{F}_{\nu_{e},\bar{\nu}_{e}}$ is the number flux and $n_{\nu_{e},\bar{\nu}_{e}}$ is the number density of $\nu_{e}$ or $\bar{\nu}_{e}$.
Sterile particles could be produced in the collisional regime (see Sec. III), hence we compute the mean-free path for the main neutral current (NC) and charged current (CC) interactions (i.e., scattering of neutrinos on nucleons, neutrino-(anti)neutrino scattering, Bremsstrahlung processes, and beta reactions) following Refs. Leitner et al. (2006); Hannestad and Raffelt (1998); Strumia and Vissani (2003); Suliga et al. (2020, 2019):
$$\lambda_{\nu_{e},\bar{\nu}_{e}}(E)=\sum_{\mathrm{CC,NC}}\frac{1}{n_{t}\sigma(E)}\ ,$$
(2)
where $\sigma(E)$ is the interaction cross section and $n_{t}$ the number density of targets. We assume that Pauli blocking effects are negligible, because the torus has a mass density much lower than
the nuclear saturation density ($\rho_{B}\ll\mathcal{O}(10^{14})$ g$/$cm${}^{3}$, see Fig. 1) and is only moderately degenerate for electrons (see, e.g., Fig. 1 of Ref. Wu et al. (2017)).
III Active sterile flavor conversion physics
In this section, we introduce the equations of motion describing the production of sterile particles. We then investigate the resonant production of sterile particles in NS merger remnants.
III.1 Neutrino equations of motion
For simplicity, in this paper, we work in the two-flavor basis $(\nu_{e},\nu_{s})$ and focus on flavor conversion between electron and sterile flavors.
In fact, the non-electron flavors are produced through flavor mixing; however, we neglect flavor conversion among the active flavors. The latter is an approximation, in light of recent hints supporting evidence for the development of non-negligible fast neutrino conversion at high densities Tamborra and Shalgar (2021); Richers and Sen (2022); Shalgar and Tamborra (2022a); Nagakura and Zaizen (2022); Shalgar and Tamborra (2022b).
Similar to what shown in Ref. Tamborra et al. (2012), the production of sterile flavors may further trigger flavor transformation in the active sector, repopulating it. Nevertheless, because of the numerical challenges involved in the modeling of neutrino self-interaction and since we rely on mass and mixing angles between the active and the sterile sectors that are larger than the active sector mixing parameters, we aim to provide a first explorative glimpse on the production of sterile states in NS merger remnants. An improved modeling of the flavor conversion physics in the presence of sterile neutrinos is left to future work.
Under the assumption of stationarity, the evolution of the neutrino field in the flavor space is described by the Liouville equation Sigl and Raffelt (1993):
$$\displaystyle\partial_{r}\rho_{E}$$
$$\displaystyle=$$
$$\displaystyle-i[H_{E},\rho_{E}]+\mathcal{C}(\rho_{E},\bar{\rho}_{E})\ ,$$
(3)
$$\displaystyle\partial_{r}\bar{\rho}_{E}$$
$$\displaystyle=$$
$$\displaystyle-i[\bar{H}_{E},\bar{\rho}_{E}]+\bar{\mathcal{C}}(\rho_{E},\bar{\rho}_{E})\ ,$$
(4)
where, for each energy mode $E$, $\rho_{E}$ is a $2\times 2$ density matrix, whose diagonal terms are the neutrino number densities for each flavor: $(n_{\nu_{e}},n_{\nu_{s}})$. The bar denotes antineutrino quantities.
We assume that sterile neutrinos are generated through flavor conversion, i.e. the initial conditions of our ensemble are such that $\rho_{E}=\mathrm{diag}(n_{\nu_{e}}^{0},0)$ and $\bar{\rho}_{E}=\mathrm{diag}(n_{\bar{\nu}_{e}}^{0},0)$. The Hamiltonian is
$$H_{E}=H_{v,E}+H_{m}\ .$$
(5)
The vacuum term is a function of the active-sterile mixing angle $\theta_{v}$ and the vacuum frequency $\omega=\Delta m^{2}/2E$ (with $\Delta m^{2}>0$ being the mass-squared difference):
$$\displaystyle H_{v,E}=\omega\begin{pmatrix}-\cos 2\theta_{v}&\sin 2\theta_{v}\\
\sin 2\theta_{v}&\cos 2\theta_{v}\end{pmatrix}\ .$$
(8)
The vacuum term has opposite sign for neutrinos and antineutrinos.
The matter term of the Hamiltonian takes into account the coherent forward scattering on matter
$$\displaystyle H_{m}=\begin{pmatrix}\lambda&0\\
0&-\lambda\end{pmatrix}\ .$$
(11)
The effective matter potential is given by Nunokawa et al. (1997):
$$\lambda=\frac{\sqrt{2}G_{F}\rho_{B}}{2m_{N}}(3Y_{e}-1)\ ,$$
(12)
where $G_{F}$ is the Fermi constant, $\rho_{B}$ is the baryon mass density, $m_{N}$ is the nucleon mass, and $Y_{e}=(n_{e^{-}}-n_{e^{+}})/n_{B}$ is the electron fraction.
The terms $\mathcal{C}$ and $\bar{\mathcal{C}}$ in Eqs. 3 and 4 represent the collision terms due to the incoherent part of the scattering on the matter background.
Equations 3 and 4 assume
that neutrinos propagate along radial directions ($r$) for simplicity. In fact, while the contribution to the flavor conversion history from neutrinos traveling along non-radial directions should not be negligible, for this explorative work we expect that the behavior along the radial direction is representative of the flavor transformation phenomenology.
In dense regions, where the electron flavors are thermally produced, neutrino flavor conversion is suppressed because $\lambda\gg\omega$. As the matter density decreases, sterile flavors can be resonantly produced if the Mikheyev–Smirnov–Wolfenstein (MSW) resonance condition is satisfied Mikheev and Smirnov (1986); Wolfenstein (1978); Mikheev and Smirnov :
$$\lambda_{\text{res}}=\pm\omega\cos 2\theta_{v}\ ,$$
(13)
where $+$ applies to neutrinos and $-$ to antineutrinos.
To quantify the amount of flavor conversion at each resonance, we calculate the adiabaticity parameter $\gamma$ at the resonance Kim et al. (1988):
$$\gamma=\frac{\omega}{\pi}\frac{\sin^{2}2\theta_{v}}{\cos 2\theta_{v}}\left|\frac{{d\lambda/dr}}{\lambda}\right|^{-1}.$$
(14)
The corresponding transition probability at the resonance energy $E_{\mathrm{res}}$ is approximated by the Landau-Zener formula Kim et al. (1988); Parke (1986); Blennow and Smirnov (2013) 111Note that, for our cases of interest, $|\lambda|\gg|\omega|$; however, if $\lambda\rightarrow 0$, Eq. 15 holds for $\theta_{v}\ll 1$ Parke (1986).:
$$P_{\nu_{e}\rightarrow\nu_{s}}(E_{\mathrm{res}})\approx 1-\exp\left(-\frac{\pi^{2}}{2}\gamma\right)\ .$$
(15)
Resonant conversion between $\overline{\nu}_{e}$ and $\overline{\nu}_{s}$ occurs when $\lambda<0$, i.e. $Y_{e}\lesssim 1/3$ (see Eq. 13). From Fig. 1, it thus becomes evident that $\bar{\nu}_{s}$’s cannot be produced around the polar region, since $Y_{e}\gtrsim 1/3$ there.
Sterile particles can also be produced collisionally Raffelt and Sigl (1993).
However,
for our benchmark NS merger remnant model and sterile mass-mixing parameters, we have verified that the collisional production of sterile (anti)neutrinos is always negligible. An analogous situation occurs in the SN context, where however keV-mass sterile states can be produced collisionally Suliga et al. (2019, 2020).
III.2 Active-sterile flavor conversion
We now intend to investigate the active-sterile flavor conversion phenomenology for our benchmark NS merger remnant model. Figure 2 shows the radial neutrino-matter forward scattering potential at $t=25$ ms for representative radial directions with emission angles: $\theta=1^{\circ}$, $40^{\circ}$, $48^{\circ}$, and $90^{\circ}$. The $\theta=1^{\circ}$ direction allows to investigate the flavor conversion physics in the proximity of the polar region and the corresponding $\lambda$ is always positive. $\theta=40^{\circ}$ is representative of intermediate directions between the pole and the equator where $\lambda$ is both positive and negative. The $\theta=48^{\circ}$ potential represents intermediate directions along which $\lambda$ changes sign multiple times, while the $\theta=90^{\circ}$ potential shows the typical radial evolution of $\lambda$ in the proximity of the equator. All other directions in between the ones we select as representative have similar features.
The magnitude of $\lambda$ is the highest around the equatorial plane (see bottom right panel) and drops towards the polar axis (see top left panel), see also Eq. 12 and Fig. 1.
For $(\sin^{2}\theta_{v},\Delta m^{2})=(10^{-2},10^{-1}$ eV${}^{2})$ and $E\in[0.1,300]$ MeV, we can see from Fig. 2 that (anti)neutrinos undergo multiple resonances because of the spatial variations of $Y_{e}$ and $\rho_{B}$ as shown in Fig. 1.
In order for the MSW resonance to occur, the neutrino mean free path (Eq. 2) has to be larger than the resonance width,
$$\Delta_{\mathrm{res}}=\tan 2\theta_{v}\left|\frac{d\lambda/dr}{\lambda}\right|^{-1}\ .$$
(16)
We verified that this is always the case for all cases considered in this work.
For the emission directions with $\theta=1^{\circ}$, $40^{\circ}$ and $\theta=48^{\circ}$, the first resonance occurs outside the decoupling surfaces (see Fig. 2), where the production of $\nu_{e}$’s and $\bar{\nu}_{e}$’s has essentially stopped and the active flavors have entered the free-streaming regime.
This implies that a large production of sterile particles may deplete the active sector (which is not repopulated through thermal processes) with implications on the electron abundance.
On the other hand, for the $\theta=90^{\circ}$ direction, the first resonance occurs deep inside in the torus, where $\nu_{e}$’s and $\bar{\nu}_{e}$’s are still being thermally produced.
As a consequence, although flavor conversion at the first resonance may not significantly impact the local number density of $\nu_{e}$ and $\bar{\nu}_{e}$, quickly replenished through thermal processes, it can potentially affect the evolution of the disk.
In order to evaluate the amount of flavor transformation numerically, we introduce the $\nu_{e}$ survival probability at the resonance radius $r_{i}$:
$$P_{\nu_{e}\rightarrow\nu_{e}}(E,r_{i})=\frac{n_{\nu_{e}}(E,r_{i})}{n_{\nu_{e}}^{0}(E,r_{i})}\ ,$$
(17)
where the index $0$ denotes quantities before flavor transformation and $P_{\nu_{e}\rightarrow\nu_{s}}(E)=1-P_{\nu_{e}\rightarrow\nu_{e}}(E)$. A similar expression holds for $P_{\bar{\nu}_{e}\rightarrow\bar{\nu}_{e}}$.
Figure 3 shows the survival probability of $\nu_{e}$’s and $\bar{\nu}_{e}$’s for $(\sin^{2}\theta_{v},\Delta m^{2})=(10^{-2},10^{-1}$ eV${}^{2})$
and a selected neutrino energy $E=20$ MeV, obtained by solving Eqs. 3 and 4 numerically and applying Eq. 17.
The same computation can be carried out analytically; in this case, the neutrino number density at each resonance radius $r_{i}$, occurring after neutrino decoupling, is:
$$\displaystyle n_{\nu_{e}}(E,r_{i})$$
$$\displaystyle=$$
$$\displaystyle P_{\nu_{e}\rightarrow\nu_{e}}(E,r_{i})n_{\nu_{e}}(E,r_{i-1})\left(\frac{r_{i-1}}{r_{i}}\right)^{2}+$$
(18)
$$\displaystyle P_{\nu_{s}\rightarrow\nu_{e}}(E,r_{i})n_{\nu_{s}}(E,r_{i-1})\left(\frac{r_{i-1}}{r_{i}}\right)^{2}\ ,$$
with the survival probability being computed trough Eq. 15, and $i-1$ being the former resonance in the event that multiple resonances take place (see Fig. 2). A special case occurs for the first resonance, where $n_{\nu_{e}}(E,r_{1})=P_{\nu_{e}\rightarrow\nu_{e}}(E,r_{1})n_{\nu_{e}}(E,r_{1})$ with $n_{\nu_{e}}(E,r_{1})$ being extracted from our benchmark remnant simulation model. Moreover, for $\theta=90^{\circ}$, the first MSW resonance occurs before neutrino decoupling, hence $n_{\nu_{e}}(E,r_{2})=P_{\nu_{e}\rightarrow\nu_{e}}(E,r_{2})n_{\nu_{e}}(E,r_{2})+P_{\nu_{s}\rightarrow\nu_{e}}(E,r_{1})n_{\nu_{s}}(E,r_{1})(r_{1}/r_{2})^{2}$, where $n_{\nu_{e}}(E,r_{2})$ is extracted from our remnant simulation model. Analogous expressions hold for $n_{\nu_{s}}(E,r_{i})$. We find that our analytical computations are in agreement with the numerical ones (results not shown here).
We can see that the flavor conversion physics is highly dependent on the emission direction and more than two resonances could occur for some directions, as already noticeable from Fig. 2 (see, e.g., $\theta=48^{\circ}$). As expected, according to the emission direction and for our fixed neutrino energy, the adiabaticity of flavor conversion changes. This has the effect that $\nu_{s}$’s are minimally produced in the equatorial plane (see $\theta=90^{\circ}$), whereas in the polar region, no $\bar{\nu}_{s}$’s are produced since no $\bar{\nu}_{e}$–$\bar{\nu}_{s}$ resonances can occur (see $\theta=1^{\circ}$).
Interestingly, due to the torus geometry,
the flavor conversion phenomenology in NS merger remnants differs from the SN one Nunokawa et al. (1997); Tamborra et al. (2012); Wu et al. (2014); Xiong et al. (2019); Pllumbi et al. (2015), where at most two MSW resonances were observed. However, similar to the SN case, the innermost resonances are less adiabatic than the outer ones because of the steep radial profile of $\lambda$.
Another difference with respect to the SN case is that $\bar{\nu}_{e}$’s are naturally more abundant in the compact merger scenario.
Figure 4 shows the active and sterile differential number densities as functions of the energy, for the same representative mixing parameters and emission directions shown in Fig. 3. We can see how the effects of the varying adiabaticity and the occurrence of MSW resonances distort the shape of the sterile distributions at $1000$ km, creating energy-dependent features. In addition, $\bar{\nu}_{s}$’s are not produced in the sourroundings of the polar region ($\theta=1^{\circ}$, top left panel) while more $\bar{\nu}_{s}$’s than $\nu_{s}$’s are produced in the proximity of the equatorial region ($\theta=90^{\circ}$, bottom right panel).
IV Dependence of the flavor conversion phenomenology on the sterile mass and mixing parameters
In this section, we investigate the physics of flavor conversion and the production of sterile particles as functions of the sterile mixing parameters.
IV.1 Occurrence of multiple MSW resonances
In order to compute the average amount of flavor mixing across multiple resonances, we introduce the energy averaged survival probability for neutrinos at each resonance:
$$\langle P_{\nu_{e}\rightarrow\nu_{e}}(r_{i})\rangle=\frac{\int dEP_{\nu_{e}\rightarrow\nu_{e}}(E,r_{i})n_{\nu_{e}}(E,r_{i-1})}{\int dEn_{\nu_{e}}(E,r_{i-1})}\ ,$$
(19)
with $P_{\nu_{e}\rightarrow\nu_{e}}(E,r_{i})$ defined as in Eq. 15.
An analogous expression holds for the survival probability of antineutrinos, $\langle P_{\bar{\nu}_{e}\rightarrow\bar{\nu}_{e}}(r_{i})\rangle$.
Figures 5–8 show contour plots of $\langle P_{\nu_{e}\rightarrow\nu_{e}}(r_{i})\rangle$ ($\langle P_{\bar{\nu}_{e}\rightarrow\bar{\nu}_{e}}(r_{i})\rangle$) in the plane spanned by $(\sin^{2}\theta_{v},\Delta m^{2}$) on top (bottom). Each resonance is identified through the number of times $d{\lambda}/dr$ changes sign, as shown in Fig. 2.
The amount of flavor transformation that occurs within each resonance region depends on $\lambda$, which controls which $(\sin^{2}\theta_{v},\Delta m^{2})$ undergo MSW resonances. On the other hand, $d{\lambda}/dr$ controls the adiabaticity of each resonance.
There are general features which most resonance regions display, independently of $\theta$. The region of partial or full flavor conversion into sterile states has a triangular shape. This is because the adidabaticity of the MSW resonance increases as $\sin^{2}\theta_{v}$ and/or $\Delta m^{2}$ increase (see Eq. 14). These triangular regions are bounded from above in most cases since a high $\Delta m^{2}$ causes $\lambda_{\mathrm{res}}$ to exceed the maximum value of the potential $\lambda$ within that region (or minimum value for antineutrinos), resulting in the absence of resonances for the upper part of the parameter space.
The bottom panels of Fig. 8 represent the survival probability of antineutrinos for $\theta=90^{\circ}$. We see how the first resonance is much less adiabatic than the second one. This is due to how rapidly $Y_{e}$ crosses $1/3$, while $\rho_{B}$ is still very large, causing ${d\lambda/dr}\gg 1$; i.e., $\lambda$ is very steep as it changes sign for the first time (see Fig. 2). A similar trend can be seen in the bottom panels of Figs. 6 and 7 for $\theta=40$ and $48^{\circ}$, though not as prominently.
Otherwise, the same general features in the neutrino survival probability, also present themsleves in the antineutrino plots.
IV.2 Overall production of sterile neutrinos and antineutrinos
Figures 9 and 10 show the overall number densities of sterile neutrinos and antineutrinos at $r=1000$ km, respectively, produced through multiple MSW resonances.
The lowest number density of resonantly produced sterile particles is visible in the bottom left corner of the $(\sin^{2}\theta_{v},\Delta m^{2})$ plane for all panels of Figs. 9 and 10. In general, as
$\sin^{2}\theta_{v}$ increases, the number density of sterile neutrinos in Fig. 9 increases.
The shape of the countours closely follows the patterns shown in Figs. 5–8 and is a direct consequence of multiple resonances.
As expected, sterile neutrinos are abundantly produced for a larger region of the mass-mixing parameter space for $\theta=1^{\circ}$.
In the top panels of Fig. 10
(for $\theta=40^{\circ},48^{\circ}$), the combined effect of all MSW resonances can be observed in the asymptotic emission of $\overline{\nu}_{s}$’s.
On the other hand, in the bottom panel of Fig. 10 ($\theta=90^{\circ}$), the effect of the first resonance is not visible in the top right part of the parameter space. This is because $\bar{\nu}_{s}$’s generated adiabatically at the first resonance are reconverted back to $\bar{\nu}_{e}$’s at the second resonance. Thus $n_{\bar{\nu}_{s}}$ in our plot is determined by the conversions into sterile states occurring at the second resonance; on the other hand, flavor conversion at the first and second resonances leads to an enhanced $n_{\bar{\nu}_{e}}$ in the top right corner of the parameter space.
V Active-sterile flavor conversion as a function of the torus evolution
In this section, we explore the active-sterile flavor conversion physics for three snapshots of the disk evolution ($t=10$, $25$, and $50$ ms) and a fixed emission angle $\theta=90^{\circ}$. The matter potential, $\lambda$, is shown in Fig. 11. One can see that the MSW resonance patterns are similar to the ones investigated in Fig. 2 for $t=25$ ms. Similar features to the ones illustrated in Fig. 2 can be found for the radial profiles of $\lambda$ along the polar region and for some intermediate values of $\theta$. However, in the proximity of the equatorial plane ($\theta=90^{\circ}$), the electron fraction drops causing $\lambda$ to change sign in the innermost regions for $t>25$ ms.
As a consequence, no MSW resonances occur for $\nu_{e}$ at $t=50$ ms and one less MSW resonance takes place for $\bar{\nu}_{e}$ with respect to the $t=25$ ms case. Nevertheless, since the innermost resonances were mainly non-adiabatic, we do not find large changes in the overall flavor conversion physics, as shown in Fig. 12 (see Fig. 8 for comparison).
As $t$ increases, the matter gradient along the radial directions becomes gentler, because of the drop in baryon density, resulting in more adiabatic resonances and a larger region of the mass-mixing parameter space affected by flavor conversion at $t=50$ ms than at $t=10$ ms as can be seen from Fig. 12.
VI Outlook
By relying on a two-flavor framework ($1$ active $+1$ sterile species), we have explored the active-sterile flavor conversion phenomenology in compact binary merger remnants for the first time. We have investigated the production of sterile states as a function of the sterile neutrino mixing parameters, representative radial directions of neutrino emission
from the accretion torus, and temporal evolution of the merger remnant.
Because of the torus geometry and the neutron richness of the environment, large flavor conversion occurs for antineutrinos. In particular, differently from the SN case, we find that multiple (up to six, see Fig. 2) MSW resonances can take place according to the neutrino emission direction. The torus
geometry is responsible for a large production of sterile neutrinos (and no antineutrinos) in the proximity of the polar region and more sterile antineutrinos than neutrinos in the equatorial region. While we rely on the output of one hydrodynamical simulation of a BH accretion torus Just et al. (2015), our main conclusions should be generic as they are fundamentally linked to the characteristic properties and geometry of binary merger remnants.
As the BH torus evolves, the active-sterile oscillation phenomenology remains unchanged overall. However, the shallower baryon density at later times is responsible for more adiabatic flavor conversion that leads to a larger region of the mass-mixing parameter space being affected by active-sterile flavor conversion.
Our findings rely on a simplified framework for what concerns the modeling of the flavor conversion physics. We neglect any impact of neutrino-neutrino interaction on the active-sterile neutrino conversion, because of the uncertainties currently involved in our understanding of this phenomenon and the related numerical challenges. Yet, within a simplified framework, it has been proven that neutrino self-interaction could further affect the active-sterile conversion physics in the SN context Tamborra et al. (2012); Wu et al. (2014); Pllumbi et al. (2015); Xiong et al. (2019).
Despite the caveats of our modeling, our results robustly suggest that the non-trivial active-sterile flavor phenomenology occurring in merger remnants can have indirect implications on the disk cooling rate and its outflows.
For instance, the adiabaticity of $\nu_{e}$ and $\bar{\nu}_{e}$ flavor conversion into sterile states inside the neutrino sphere (see, e.g., the 1st resonance panel of Fig. 8 for $\theta=90^{\circ}$) could potentially accelerate the cooling of the remnant disk and lower $Y_{e}$ in the disk in a similar fashion to what was discussed in Ref. Just et al. (2022).
In addition, flavor conversion occurring in the polar region at radii of $\mathcal{O}(100)$ km (see, e.g., Fig. 2 or 3) would also reduce the neutrino capture rates by nucleons in the polar outflows where $Y_{e}$ and the nucleosynthesis outcomes are sensitive to the abundance of the electron flavors, like in the scenario considered in Ref. Wu et al. (2017).
A robust assessment of the impact of the active-sterile flavor conversion physics on the electromagnetic observables as well as on the disk cooling rate are left to future work, once a reliable modeling of the active-active conversion physics in the presence of neutrino self-interactions will be available. In order to place robust constraints on the sterile mixing parameters through future multi-messenger observations, a survey of the flavor conversion phenomenology for various compact binary merger models and related feedback on the observables will be required. This work proves the unexplored potential of upcoming multi-messenger observations of compact binary merger remnants to unveil the existence of sterile neutrinos.
Acknowledgements.We would like to thank Oliver Just for insightful discussions.
We acknowledge support from the Villum Foundation (Project No. 13164), the Carlsberg Foundation (CF18-0183), the Deutsche Forschungsgemeinschaft through Sonderforschungbereich
SFB 1258 “Neutrinos and Dark Matter in Astro- and
Particle Physics” (NDM), the National Science and Technology Council, Taiwan under Grant Nos. 110-2112-M-001-050, 111-2628-M-001-003-MY4, the Academia Sinica (Project No. AS-CDA-109-M11), and the Physics Division of the National Center for Theoretical Sciences, Taiwan.
References
Margutti and Chornock (2021)
Raffaella Margutti and Ryan Chornock, “First Multimessenger Observations of a Neutron Star Merger,” Ann. Rev. Astron. Astrophys. 59, 155–202 (2021), arXiv:2012.04810 [astro-ph.HE]
.
Abbott et al. (2017a)
Benjamin P. Abbott et al. (LIGO
Scientific, Virgo), “GW170817: Observation of Gravitational Waves from a Binary Neutron Star
Inspiral,” Phys. Rev. Lett. 119, 161101 (2017a), arXiv:1710.05832 [gr-qc] .
Abbott et al. (2017b)
Benjamin P. Abbott et al. (LIGO
Scientific, Virgo, Fermi-GBM, INTEGRAL), “Gravitational Waves and Gamma-rays from a Binary Neutron
Star Merger: GW170817 and GRB 170817A,” Astrophys. J. Lett. 848, L13 (2017b), arXiv:1710.05834 [astro-ph.HE]
.
Abbott et al. (2017c)
Benjamin P. Abbott et al. (LIGO
Scientific, Virgo, Fermi GBM, INTEGRAL, IceCube, AstroSat Cadmium Zinc
Telluride Imager Team, IPN, Insight-Hxmt, ANTARES, Swift, AGILE Team, 1M2H
Team, Dark Energy Camera GW-EM, DES, DLT40, GRAWITA, Fermi-LAT, ATCA, ASKAP,
Las Cumbres Observatory Group, OzGrav, DWF (Deeper Wider Faster Program),
AST3, CAASTRO, VINROUGE, MASTER, J-GEM, GROWTH, JAGWAR, CaltechNRAO,
TTU-NRAO, NuSTAR, Pan-STARRS, MAXI Team, TZAC Consortium, KU, Nordic Optical
Telescope, ePESSTO, GROND, Texas Tech University, SALT Group, TOROS, BOOTES,
MWA, CALET, IKI-GW Follow-up, H.E.S.S., LOFAR, LWA, HAWC, Pierre Auger, ALMA,
Euro VLBI Team, Pi of Sky, Chandra Team at McGill University, DFN, ATLAS
Telescopes, High Time Resolution Universe Survey, RIMAS, RATIR, SKA South
Africa/MeerKAT), “Multi-messenger Observations of a Binary Neutron Star Merger,” Astrophys. J. Lett. 848, L12 (2017c), arXiv:1710.05833 [astro-ph.HE]
.
Goldstein et al. (2017)
Adam Goldstein et al., “An Ordinary Short Gamma-Ray Burst with Extraordinary Implications:
Fermi-GBM Detection of GRB 170817A,” Astrophys. J. Lett. 848, L14 (2017), arXiv:1710.05446 [astro-ph.HE] .
Savchenko et al. (2017)
Volodymyr Savchenko et al., “INTEGRAL Detection of the First Prompt Gamma-Ray Signal
Coincident with the Gravitational-wave Event GW170817,” Astrophys. J. Lett. 848, L15 (2017), arXiv:1710.05449 [astro-ph.HE] .
Margutti et al. (2017)
Raffaella Margutti et al., “The Electromagnetic Counterpart of the Binary Neutron
Star Merger LIGO/VIRGO GW170817. V. Rising X-ray Emission from an Off-Axis
Jet,” Astrophys. J. Lett. 848, L20 (2017), arXiv:1710.05431 [astro-ph.HE]
.
Troja et al. (2017)
Eleonora Troja et al., “The X-ray counterpart to the gravitational wave event GW
170817,” Nature 551, 71–74 (2017), arXiv:1710.05433 [astro-ph.HE]
.
Kasen et al. (2017)
Daniel Kasen, Brian Metzger,
Jennifer Barnes, Eliot Quataert, and Enrico Ramirez-Ruiz, “Origin of the heavy elements in binary
neutron-star mergers from a gravitational wave event,” Nature 551, 80 (2017), arXiv:1710.05463
[astro-ph.HE] .
Drout et al. (2017)
Maria R. Drout et al., “Light Curves of the Neutron Star Merger GW170817/SSS17a:
Implications for R-Process Nucleosynthesis,” Science 358, 1570–1574
(2017), arXiv:1710.05443 [astro-ph.HE] .
Cowperthwaite et al. (2017)
Philip S. Cowperthwaite et al., “The Electromagnetic Counterpart of the Binary Neutron
Star Merger LIGO/Virgo GW170817. II. UV, Optical, and Near-infrared Light
Curves and Comparison to Kilonova Models,” Astrophys. J. Lett. 848, L17 (2017), arXiv:1710.05840 [astro-ph.HE] .
Villar et al. (2017)
V. Ashley Villar et al., “The Combined Ultraviolet, Optical, and Near-Infrared
Light Curves of the Kilonova Associated with the Binary Neutron Star Merger
GW170817: Unified Data Set, Analytic Models, and Physical Implications,” Astrophys. J. Lett. 851, L21 (2017), arXiv:1710.11576 [astro-ph.HE]
.
Shibata et al. (2017)
Masaru Shibata, Sho Fujibayashi, Kenta Hotokezaka, Kenta Kiuchi, Koutarou Kyutoku, Yuichiro Sekiguchi, and Masaomi Tanaka, “Modeling
GW170817 based on numerical relativity and its implications,” Phys. Rev. D 96, 123012 (2017), arXiv:1710.07579 [astro-ph.HE]
.
Metzger (2020)
Brian D. Metzger, “Kilonovae,” Living Rev. Rel. 23, 1 (2020), arXiv:1910.01617 [astro-ph.HE]
.
Veske et al. (2020)
Doğa Veske, Zsuzsa Márka, Imre Bartos, and Szabolcs Márka, “Neutrino emission upper limits with maximum likelihood estimators for joint
astrophysical neutrino searches with large sky localizations,” JCAP 05, 016
(2020), arXiv:2001.00566 [astro-ph.HE] .
Aartsen et al. (2020)
Mark G. Aartsen et al. (IceCube), “IceCube Search for Neutrinos Coincident
with Compact Binary Mergers from LIGO-Virgo’s First
Gravitational-wave Transient Catalog,” Astrophys. J. Lett. 898, L10 (2020), arXiv:2004.02910 [astro-ph.HE] .
Hayato et al. (2018)
Yoshinari Hayato et al. (Super-Kamiokande), “Search for Neutrinos in Super-Kamiokande Associated with the GW170817
Neutron-star Merger,” Astrophys. J. Lett. 857, L4 (2018), arXiv:1802.04379 [astro-ph.HE]
.
Foucart et al. (2015)
Francois Foucart, Evan O’Connor, Luke Roberts, Matthew D. Duez, Roland Haas, Lawrence E. Kidder, Christian D. Ott, Harald P. Pfeiffer, Mark A. Scheel, and Bela Szilagyi, “Post-merger
evolution of a neutron star-black hole binary with neutrino transport,” Phys. Rev. D 91, 124021 (2015), arXiv:1502.04146 [astro-ph.HE]
.
Ruffert et al. (1997)
Maximilian Ruffert, H.-Thomas Janka, Kazuya Takahashi, and Gerhard Schaefer, “Coalescing neutron stars: A Step towards physical models. 2. Neutrino
emission, neutron tori, and gamma-ray bursts,” Astron. Astrophys. 319, 122–153 (1997), arXiv:astro-ph/9606181
.
Metzger and Fernández (2014)
Brian D. Metzger and Rodrigo Fernández, “Red or blue? A potential kilonova imprint of the delay until black hole
formation following a neutron star merger,” Mon. Not.
Roy. Astron. Soc. 441, 3444–3453 (2014), arXiv:1402.4803 [astro-ph.HE] .
Perego et al. (2014)
Albino Perego, Stephan Rosswog, Ruben M. Cabezón, Oleg Korobkin, Roger Käppeli, Almudena Arcones, and Matthias Liebendörfer, “Neutrino-driven winds from neutron star merger remnants,” Mon. Not. Roy. Astron. Soc. 443, 3134–3156 (2014), arXiv:1405.6730 [astro-ph.HE] .
Wanajo et al. (2014)
Shinya Wanajo, Yuichiro Sekiguchi, Nobuya Nishimura, Kenta Kiuchi, Koutarou Kyutoku, and Masaru Shibata, “Production of
all the $r$-process nuclides in the dynamical ejecta of neutron star
mergers,” Astrophys. J. Lett. 789, L39 (2014), arXiv:1402.7317 [astro-ph.SR] .
Just et al. (2015)
Oliver Just, Andreas Bauswein, Ricard Ardevol Pulpillo, Stephane Goriely, and H.-Thomas Janka, ‘‘Comprehensive
nucleosynthesis analysis for ejecta of compact binary mergers,” Mon. Not. Roy. Astron. Soc. 448, 541–567 (2015), arXiv:1406.2687 [astro-ph.SR] .
Fujibayashi et al. (2020)
Sho Fujibayashi, Masaru Shibata, Shinya Wanajo, Kenta Kiuchi,
Koutarou Kyutoku, and Yuichiro Sekiguchi, “Mass ejection from disks
surrounding a low-mass black hole: Viscous neutrino-radiation hydrodynamics
simulation in full general relativity,” Phys. Rev. D 101, 083029 (2020), arXiv:2001.04467 [astro-ph.HE]
.
Kullmann et al. (2022)
Ina Kullmann, Stephan Goriely, Oliver Just,
Ricard Ardevol-Pulpillo,
Andreas Bauswein, and H.-Thomas Janka, “Dynamical ejecta of neutron
star mergers with nucleonic weak processes I: nucleosynthesis,” Mon. Not. Roy. Astron. Soc. 510, 2804–2819 (2022), arXiv:2109.02509 [astro-ph.HE]
.
Narayan et al. (1992)
Ramesh Narayan, Bohdan Paczynski, and Tsvi Piran, “Gamma-ray bursts
as the death throes of massive binary stars,” Astrophys. J.
Lett. 395, L83–L86
(1992), arXiv:astro-ph/9204001 .
Berger (2014)
Edo Berger, “Short-Duration
Gamma-Ray Bursts,” Ann. Rev. Astron. Astrophys. 52, 43–105 (2014), arXiv:1311.2603
[astro-ph.HE] .
Just et al. (2016)
Oliver Just, Martin Obergaulinger, H.-Thomas Janka, Andreas Bauswein, and Nicole Schwarz, ‘‘Neutron-star
merger ejecta as obstacles to neutrino-powered jets of gamma-ray bursts,” Astrophys. J. Lett. 816, L30 (2016), arXiv:1510.04288 [astro-ph.HE]
.
Malkus et al. (2014)
Annelise Malkus, Alexander Friedland, and Gail C. McLaughlin, “Matter-Neutrino Resonance Above Merging Compact Objects,” (2014), arXiv:1403.5797 [hep-ph] .
Malkus et al. (2012)
Annelise Malkus, James P. Kneller, Gail C. McLaughlin, and Rebecca Surman, “Neutrino oscillations above black hole accretion disks: disks with
electron-flavor emission,” Phys. Rev. D 86, 085015 (2012), arXiv:1207.6648 [hep-ph]
.
Wu et al. (2016)
Meng-Ru Wu, Huaiyu Duan, and Yong-Zhong Qian, “Physics of neutrino flavor
transformation through matter–neutrino resonances,” Phys. Lett. B 752, 89–94 (2016), arXiv:1509.08975 [hep-ph] .
Zhu et al. (2016)
Yong-Lin Zhu, Albino Perego, and Gail C. McLaughlin, “Matter
Neutrino Resonance Transitions above a Neutron Star Merger Remnant,” Phys. Rev. D 94, 105006 (2016), arXiv:1607.04671 [hep-ph] .
Frensel et al. (2017)
Maik Frensel, Meng-Ru Wu,
Cristina Volpe, and Albino Perego, “Neutrino Flavor Evolution
in Binary Neutron Star Merger Remnants,” Phys.
Rev. D 95, 023011
(2017), arXiv:1607.05938 [astro-ph.HE] .
Tian et al. (2017)
James Y. Tian, Amol V. Patwardhan, and George M. Fuller, “Neutrino Flavor Evolution in Neutron Star Mergers,” Phys. Rev. D 96, 043001 (2017), arXiv:1703.03039 [astro-ph.HE]
.
Shalgar (2018)
Shashank Shalgar, “Multi-angle calculation of the matter-neutrino resonance near an accretion
disk,” JCAP 02, 010 (2018), arXiv:1707.07692 [hep-ph] .
Tamborra and Shalgar (2021)
Irene Tamborra and Shashank Shalgar, “New
Developments in Flavor Evolution of a Dense Neutrino Gas,” Ann. Rev. Nucl. Part. Sci. 71, 165–188 (2021), arXiv:2011.01948 [astro-ph.HE]
.
Wu and Tamborra (2017)
Meng-Ru Wu and Irene Tamborra, “Fast neutrino
conversions: Ubiquitous in compact binary merger remnants,” Phys. Rev. D 95, 103007 (2017), arXiv:1701.06580 [astro-ph.HE]
.
Wu et al. (2017)
Meng-Ru Wu, Irene Tamborra,
Oliver Just, and H.-Thomas Janka, “Imprints of neutrino-pair
flavor conversions on nucleosynthesis in ejecta from neutron-star merger
remnants,” Phys. Rev. D 96, 123015 (2017), arXiv:1711.00477 [astro-ph.HE]
.
George et al. (2020)
Manu George, Meng-Ru Wu,
Irene Tamborra, Ricard Ardevol-Pulpillo, and H.-Thomas Janka, “Fast neutrino flavor
conversion, ejecta properties, and nucleosynthesis in newly-formed
hypermassive remnants of neutron-star mergers,” Phys. Rev. D 102, 103015 (2020), arXiv:2009.04046 [astro-ph.HE]
.
Just et al. (2022)
Oliver Just, Sajad Abbar,
Meng-Ru Wu, Irene Tamborra, H.-Thomas Janka, and Francesco Capozzi, “Fast neutrino conversion in hydrodynamic
simulations of neutrino-cooled accretion disks,” Phys. Rev. D 105, 083024 (2022), arXiv:2203.16559 [astro-ph.HE]
.
Li and Siegel (2021)
Xinyu Li and Daniel M. Siegel, “Neutrino Fast
Flavor Conversions in Neutron-Star Postmerger Accretion Disks,” Phys. Rev. Lett. 126, 251101 (2021), arXiv:2103.02616 [astro-ph.HE]
.
Fernández et al. (2022)
Rodrigo Fernández, Sherwood Richers, Nicole Mulyk,
and Steven Fahlman, “The Fast Flavor Instability
in Hypermassive Neutron Star Disk Outflows,” (2022), arXiv:2207.10680
[astro-ph.HE] .
Alves Batista et al. (2021)
Rafael Alves Batista et al., “EuCAPT White Paper: Opportunities and Challenges for Theoretical
Astroparticle Physics in the Next Decade,” (2021), arXiv:2110.10074
[astro-ph.HE] .
Burns (2020)
Eric Burns, “Neutron Star
Mergers and How to Study Them,” Living Rev. Rel. 23, 4
(2020), arXiv:1909.06085 [astro-ph.HE] .
Burns et al. (2019)
Eric Burns et al., “A
Summary of Multimessenger Science with Neutron Star Mergers,” (2019), arXiv:1903.03582 [astro-ph.HE] .
Aggarwal et al. (2021)
Nancy Aggarwal et al., “Challenges and opportunities of gravitational-wave searches at MHz to GHz
frequencies,” Living Rev. Rel. 24, 4 (2021), arXiv:2011.12414 [gr-qc] .
Diamond and Marques-Tavares (2022)
Melissa D. Diamond and Gustavo Marques-Tavares, “$\gamma$-Ray Flashes from Dark Photons in Neutron Star
Mergers,” Phys. Rev. Lett. 128, 211101 (2022), arXiv:2106.03879 [hep-ph]
.
Berezhiani and Drago (2000)
Zurab Berezhiani and Alessandro Drago, “Gamma-ray bursts via emission of axion - like particles,” Phys. Lett. B 473, 281–290 (2000), arXiv:hep-ph/9911333 .
Berezhiani et al. (2003)
Zurab Berezhiani, Ignazio Bombaci, Alessandro Drago, Filippo Frontera, and Andrea Lavagno, “Gamma-ray
bursts from delayed collapse of neutron stars to quark matter stars,” Astrophys. J. 586, 1250–1253 (2003), arXiv:astro-ph/0209257 .
Dasgupta and Kopp (2021)
Basudeb Dasgupta and Joachim Kopp, “Sterile
Neutrinos,” Phys. Rept. 928, 1–63 (2021), arXiv:2106.05913 [hep-ph] .
Abazajian et al. (2012)
Kevork N. Abazajian et al., “Light Sterile Neutrinos: A White Paper,” (2012), arXiv:1204.5379 [hep-ph] .
Böser et al. (2020)
Sebastian Böser, Christian Buck, Carlo Giunti, Julien Lesgourgues, Livia Ludhova, Susanne Mertens, Anne Schukraft, and Michael Wurm, “Status of Light
Sterile Neutrino Searches,” Prog. Part. Nucl. Phys. 111, 103736 (2020), arXiv:1906.01739
[hep-ex] .
Acero et al. (2022)
Mario A. Acero et al., “White Paper on Light Sterile Neutrino Searches and
Related Phenomenology,” (2022), arXiv:2203.07323 [hep-ex]
.
Aguilar-Arevalo et al. (2001)
Alexis A. Aguilar-Arevalo et al. (LSND), “Evidence for
neutrino oscillations from the observation of $\bar{\nu}_{e}$ appearance in a
$\bar{\nu}_{\mu}$ beam,” Phys. Rev. D 64, 112007 (2001), arXiv:hep-ex/0104049
.
Aguilar-Arevalo et al. (2021)
Alexis A. Aguilar-Arevalo et al. (MiniBooNE), “Updated
MiniBooNE neutrino oscillation results with increased data and new background
studies,” Phys. Rev. D 103, 052002 (2021), arXiv:2006.16883 [hep-ex] .
Mueller et al. (2011)
Thomas A. Mueller et al., “Improved Predictions of Reactor Antineutrino Spectra,” Phys. Rev. C 83, 054615 (2011), arXiv:1101.2663 [hep-ex] .
Huber (2011)
Patrick Huber, “On the
determination of anti-neutrino spectra from nuclear reactors,” Phys. Rev. C 84, 024617 (2011), [Erratum:
Phys.Rev.C 85, 029901 (2012)], arXiv:1106.0687 [hep-ph] .
Giunti et al. (2022a)
Carlo Giunti, Yufeng Li,
Christoph A. Ternes, and Zhao Xin, “Reactor antineutrino
anomaly in light of recent flux model refinements,” Phys. Lett. B 829, 137054 (2022a), arXiv:2110.06820 [hep-ph] .
Kopeikin et al. (2021)
V. Kopeikin, M. Skorokhvatov, and O. Titov, “Reevaluating
reactor antineutrino spectra with new measurements of the ratio between U235
and Pu239 $\beta$ spectra,” Phys. Rev. D 104, L071301 (2021), arXiv:2103.01684 [nucl-ex] .
Berryman and Huber (2021)
Jeffrey M. Berryman and Patrick Huber, “Sterile Neutrinos and the Global Reactor Antineutrino Dataset,” JHEP 01, 167 (2021), arXiv:2005.01756
[hep-ph] .
Andriamirado et al. (2020)
Manoa Andriamirado et al. (PROSPECT,
STEREO), “Note on
arXiv:2005.05301, ’Preparation of the Neutrino-4 experiment on search for
sterile neutrino and the obtained results of measurements’,” (2020), arXiv:2006.13147 [hep-ex] .
Berryman et al. (2022)
Jeffrey M. Berryman, Pilar Coloma, Patrick Huber, Thomas Schwetz, and Albert Zhou, “Statistical significance of the sterile-neutrino hypothesis in the context
of reactor and gallium data,” JHEP 02, 055 (2022), arXiv:2111.12530 [hep-ph] .
Giunti and Laveder (2011)
Carlo Giunti and Marco Laveder, “Statistical
Significance of the Gallium Anomaly,” Phys.
Rev. C 83, 065504
(2011), arXiv:1006.3244 [hep-ph] .
Barinov et al. (2022)
Vladislav V. Barinov et al., “Results from the Baksan Experiment on Sterile Transitions
(BEST),” Phys. Rev. Lett. 128, 232501 (2022), arXiv:2109.11482 [nucl-ex]
.
Barinov and Gorbunov (2022)
Vladislav V. Barinov and Dmitry Gorbunov, “BEST impact on sterile neutrino hypothesis,” Phys. Rev. D 105, L051703 (2022), arXiv:2109.14654 [hep-ph] .
Dentler et al. (2018)
Mona Dentler, Álvaro Hernández-Cabezudo, Joachim Kopp, Pedro A. N. Machado, Michele Maltoni, Ivan Martinez-Soler, and Thomas Schwetz, ‘‘Updated Global Analysis of Neutrino Oscillations in the Presence of
eV-Scale Sterile Neutrinos,” JHEP 08, 010 (2018), arXiv:1803.10661 [hep-ph] .
Argüelles et al. (2022)
Carlos A. Argüelles, Ivan Esteban, Matheus Hostert, Kevin J. Kelly, Joachim Kopp, Pedro A. N. Machado, Ivan Martinez-Soler, and Yuber F. Perez-Gonzalez, “MicroBooNE and the $\nu$e Interpretation of
the MiniBooNE Low-Energy Excess,” Phys. Rev. Lett. 128, 241802 (2022), arXiv:2111.10359 [hep-ph] .
Giunti et al. (2022b)
C. Giunti, Y. F. Li,
C. A. Ternes, O. Tyagi, and Z. Xin, “Gallium Anomaly: Critical View from the Global Picture of
$\nu_{e}$ and $\bar{\nu}_{e}$ Disappearance,” (2022b), arXiv:2209.00916 [hep-ph] .
Hagstotz et al. (2021)
Steffen Hagstotz, Pablo F. de Salas, Stefano Gariazzo, Martina Gerbino, Massimiliano Lattanzi, Sunny Vagnozzi, Katherine Freese, and Sergio Pastor, “Bounds on light
sterile neutrino mass and mixing from cosmology and laboratory searches,” Phys. Rev. D 104, 123524 (2021), arXiv:2003.02289 [astro-ph.CO]
.
Hannestad et al. (2012)
Steen Hannestad, Irene Tamborra, and Thomas Tram, “Thermalisation of
light sterile neutrinos in the early universe,” JCAP 07, 025 (2012), arXiv:1204.5861
[astro-ph.CO] .
Chu et al. (2018)
Xiaoyong Chu, Basudeb Dasgupta, Mona Dentler, Joachim Kopp,
and Ninetta Saviano, “Sterile neutrinos with
secret interactions—cosmological discord?” JCAP 11, 049 (2018), arXiv:1806.10629
[hep-ph] .
Archidiacono et al. (2020)
Maria Archidiacono, Stefano Gariazzo, Carlo Giunti, Steen Hannestad, and Thomas Tram, ‘‘Sterile neutrino
self-interactions: $H_{0}$ tension and short-baseline anomalies,” JCAP 12, 029
(2020), arXiv:2006.12885 [astro-ph.CO] .
Nunokawa et al. (1997)
Hiroshi Nunokawa, Juha T. Peltoniemi, Anna Rossi, and Jose W. F. Valle, “Supernova bounds
on resonant active sterile neutrino conversions,” Phys.
Rev. D 56, 1704–1713
(1997), arXiv:hep-ph/9702372 .
Tamborra et al. (2012)
Irene Tamborra, Georg G. Raffelt, Lorenz Hüdepohl, and H.-Thomas Janka, “Impact of eV-mass sterile neutrinos on neutrino-driven supernova
outflows,” JCAP 01, 013 (2012), arXiv:1110.2104 [astro-ph.SR] .
Wu et al. (2014)
Meng-Ru Wu, Tobias Fischer,
Lutz Huther, Gabriel Martínez-Pinedo, and Yong-Zhong Qian, “Impact of active-sterile
neutrino mixing on supernova explosion and nucleosynthesis,” Phys. Rev. D 89, 061303 (2014), arXiv:1305.2382 [astro-ph.HE] .
Pllumbi et al. (2015)
Else Pllumbi, Irene Tamborra, Shinya Wanajo, H.-Thomas Janka, and Lorenz Hüdepohl, “Impact of
neutrino flavor oscillations on the neutrino-driven wind nucleosynthesis of
an electron-capture supernova,” Astrophys. J. 808, 188
(2015), arXiv:1406.2596 [astro-ph.SR] .
Xiong et al. (2019)
Zewei Xiong, Meng-Ru Wu,
and Yong-Zhong Qian, ‘‘Active-Sterile
Neutrino Oscillations in Neutrino-driven Winds: Implications for
Nucleosynthesis,” Astrophys. J. 880, 81 (2019), arXiv:1904.09371 [astro-ph.HE]
.
Esmaili et al. (2014)
Arman Esmaili, Orlando L. G. Peres, and Pasquale Dario Serpico, “Impact of sterile neutrinos on the early time flux from a galactic
supernova,” Phys. Rev. D 90, 033013 (2014), arXiv:1402.1453 [hep-ph] .
Tang et al. (2020)
Jian Tang, Tsechun Wang, and Meng-Ru Wu, “Constraining sterile
neutrinos by core-collapse supernovae with multiple detectors,” JCAP 10, 038
(2020), arXiv:2005.09168 [hep-ph] .
Leitner et al. (2006)
Tina Leitner, Luis Alvarez-Ruso, and Ulrich Mosel, “Neutral current neutrino-nucleus interactions at intermediate energies,” Phys. Rev. C 74, 065502 (2006), arXiv:nucl-th/0606058 .
Hannestad and Raffelt (1998)
Steen Hannestad and Georg G. Raffelt, “Supernova
neutrino opacity from nucleon-nucleon Bremsstrahlung and related
processes,” Astrophys. J. 507, 339–352 (1998), arXiv:astro-ph/9711132 .
Strumia and Vissani (2003)
Alessandro Strumia and Francesco Vissani, “Precise quasielastic neutrino/nucleon cross-section,” Phys. Lett. B 564, 42–54 (2003), arXiv:astro-ph/0302055 .
Suliga et al. (2020)
Anna M. Suliga, Irene Tamborra, and Meng-Ru Wu, “Lifting the
core-collapse supernova bounds on keV-mass sterile neutrinos,” JCAP 08, 018
(2020), arXiv:2004.11389 [astro-ph.HE] .
Suliga et al. (2019)
Anna M. Suliga, Irene Tamborra, and Meng-Ru Wu, “Tau lepton
asymmetry by sterile neutrino emission – Moving beyond one-zone supernova
models,” JCAP 12, 019 (2019), arXiv:1908.11382 [astro-ph.HE]
.
Richers and Sen (2022)
Sherwood Richers and Manibrata Sen, ‘‘Fast
Flavor Transformations,” (2022), arXiv:2207.03561 [astro-ph.HE]
.
Shalgar and Tamborra (2022a)
Shashank Shalgar and Irene Tamborra, “Supernova Neutrino Decoupling Is Altered by Flavor Conversion,” (2022a), arXiv:2206.00676 [astro-ph.HE]
.
Nagakura and Zaizen (2022)
Hiroki Nagakura and Masamichi Zaizen, “Time-dependent,
quasi-steady, and global features of fast neutrino-flavor conversion,” (2022), arXiv:2206.04097 [astro-ph.HE]
.
Shalgar and Tamborra (2022b)
Shashank Shalgar and Irene Tamborra, “Neutrino Flavor Conversion, Advection, and Collisions: The Full
Solution,” (2022b), arXiv:2207.04058 [astro-ph.HE]
.
Sigl and Raffelt (1993)
Günther Sigl and Georg G. Raffelt, “General kinetic description of relativistic mixed neutrinos,” Nucl. Phys. B 406, 423–451 (1993).
Mikheev and Smirnov (1986)
S. P. Mikheev and A. Yu. Smirnov, ‘‘Neutrino
Oscillations in a Variable Density Medium and Neutrino Bursts Due to the
Gravitational Collapse of Stars,” Sov. Phys. JETP 64, 4–7 (1986), arXiv:0706.0454 [hep-ph] .
Wolfenstein (1978)
L. Wolfenstein, “Neutrino
oscillations in matter,” Phys. Rev. D 17, 2369–2374 (1978).
(92)
S P Mikheev and A Y Smirnov, “Resonance
enhancement of oscillations in matter and solar neutrino spectroscopy,” Sov. J.
Nucl. Phys. (Engl. Transl.); (United States) .
Kim et al. (1988)
C. W. Kim, Jewan Kim, and W. K. Sze, “On
the Geometrical Representation of Neutrino Oscillations in Vacuum and
Matter,” Phys. Rev. D 37, 1072 (1988).
Parke (1986)
Stephen J. Parke, “Nonadiabatic Level Crossing in Resonant Neutrino Oscillations,” Phys. Rev. Lett. 57, 1275–1278 (1986).
Blennow and Smirnov (2013)
Mattias Blennow and Alexei Yu. Smirnov, “Neutrino
propagation in matter,” Adv. High Energy Phys. 2013, 972485 (2013), arXiv:1306.2903 [hep-ph]
.
Raffelt and Sigl (1993)
Georg G. Raffelt and Günther Sigl, “Neutrino flavor conversion in a supernova core,” Astropart. Phys. 1, 165–184 (1993), arXiv:astro-ph/9209005 . |
Minimum Rényi and Wehrl entropies at the output of bosonic
channels
Vittorio Giovannetti,${}^{1}$111Now with
NEST-INFM & Scuola
Normale Superiore, Piazza dei Cavalieri 7, I-56126, Pisa, Italy. Seth
Lloyd,${}^{1,2}$
Lorenzo Maccone,${}^{1}$222Now with QUIT - Quantum Information
Theory Group,
Dipartimento di Fisica “A. Volta”
Universita’ di Pavia, via A. Bassi 6
I-27100, Pavia, Italy. Jeffrey H. Shapiro,${}^{1}$ and Brent J.
Yen${}^{1}$
${}^{1}$Massachusetts Institute of Technology –
Research Laboratory of Electronics
${}^{2}$Massachusetts Institute of
Technology – Department of Mechanical Engineering
77
Massachusetts Ave., Cambridge, MA 02139, USA
Abstract
The minimum Rényi and Wehrl output entropies are found for bosonic
channels in which the signal photons are either randomly displaced
by a Gaussian distribution (classical-noise channel), or in which
they are coupled to a thermal environment through lossy propagation
(thermal-noise channel). It is shown that the Rényi output
entropies of integer orders $z\geqslant 2$ and the output Wehrl
entropy are minimized when the channel input is a coherent state.
pacs: 03.67.Hk,03.67.-a,03.65.Db,42.50.-p
A principal aim of the quantum theory of information is to determine
the ultimate limits on communicating classical
information, i.e., limits arising from quantum
physics chuang ; shor . Among
the various figures of merit employed in this undertaking, one of the
most basic
is the minimum output entropy shorequiv . It measures the amount
of noise accumulated during the transmission, and may be used to
derive important
properties, such as the additivity, of other figures of merit, e.g.,
the channel
capacity. Here we will focus on the Rényi and Wehrl output
entropies for a class
of Gaussian bosonic channels in which the input field undergoes a random
displacement. The Rényi entropies
$\{\,S_{z}(\rho):0<z<\infty,z\neq 1\,\}$ are a family of functions that
describe the purity of a state asomov . In particular,
the von Neumann entropy $S(\rho)$ can be found from this family, because
$S(\rho)=\lim_{z\rightarrow 1}S_{z}(\rho)$. So too can the
linearized entropy,
because it is a monotonic function of the second-order Rényi
entropy zyc . On the other hand, the Wehrl entropy characterizes the
phase-space localization of a bosonic state: its minimum value is realized by
coherent states, whose quadratures have minimum uncertainty product and minimum
uncertainty sum. In this respect, the Wehrl output entropy can be used to
quantify the channel noise by measuring the phase-space “spread”
of the output
state (see also anderson for a previous analysis of Wehrl output
entropy). For the classical-noise and thermal-noise channels that we will
consider, we show that coherent-state inputs minimize the Rényi
output entropies
of integer orders $z\geqslant 2$, and the Wehrl output entropy. The results
presented in this paper are connected with the study of the von Neumann output
entropies of the classical-noise and thermal-noise channels given
in von ,
and with the analysis of these channels’ additivity properties given
in futuro .
In Sec. I we introduce the classical-noise channel map. In
Sec. II we analyze the Rényi entropy at the output of
this channel. We
first show that a coherent-state input minimizes $S_{z}(\rho)$ for
$z\geqslant 2$ an
integer, and that it minimizes $S_{z}(\rho)$ for all $z$ when the input is
restricted to be a Gaussian state (Sec. II.1). We then provide lower
bounds, for arbitrary input states, that are consistent with
coherent-state inputs
minimizing Rényi output entropies of all orders (Sec. II.2). In
Sec. III, we analyze the Wehrl output entropy, proving that it too is
minimized by coherent-state inputs. Moreover, in Sec.III.1, we
introduce the
Rényi-Wehrl entropies, and show that here as well coherent-state inputs yield
minimum-entropy outputs. The preceding results will all be developed for the
classical-noise channel; in Sec. IV we show that they also apply to the
thermal-noise channel.
I Classical-noise channel
The classical-noise channel is a unital Gaussian map, i.e., it
transforms Gaussian input states into Gaussian output states while leaving the
identity operator unaffected. It is given by the completely-positive (CP) map
$$\displaystyle{\cal N}_{n}(\rho)=\int{\rm d}^{2}\mu\;P_{n}(\mu)\;D(\mu)\rho D^{%
\dagger}(\mu)$$
(1)
where
$$\displaystyle P_{n}(\mu)=\frac{e^{-|\mu|^{2}/n}}{\pi n},$$
(2)
and $D(\mu)\equiv\exp(\mu a^{\dagger}-\mu^{*}a)$ is the displacement
operator of the electromagnetic mode $a$ used for the communication.
This map describes a bosonic field that picks up noise through
random displacement by a Gaussian probability distribution $P_{n}(\mu)$. It is
useful, among other things, to study the fidelity obtainable in
continuous-variable teleportation with finite two-mode
squeezing caves1 . Moreover, this simple one-parameter map can
be used to derive properties of more complicated channels, such as the
thermal-noise CP map of Sec. IV. When ${\cal N}_{n}$ acts on a
vacuum-state input
it produces the thermal-state output
$$\displaystyle\rho^{\prime}_{0}\equiv{\cal N}_{n}(|0\rangle\langle 0|)=\frac{1}%
{n+1}\left(\frac{n}{n+1}\right)^{a^{\dagger}a}\;.$$
(3)
The covariance property of ${\cal N}_{n}$ under
displacement implies that a coherent-state input $|\alpha\rangle$
produces the output state
$\rho_{\alpha}^{\prime}=D(\alpha)\rho_{0}^{\prime}D^{\dagger}(\alpha)$. See hal1l ; von ; futuro
for a more
detailed description of the classical-noise map.
II Rényi entropies
The quantum Rényi
entropy $S_{z}(\rho)$ is defined as follows zyc ,
$$\displaystyle S_{z}(\rho)\equiv-\frac{\ln\mbox{Tr}[\rho^{z}]}{z-1}\quad\mbox{%
for $0<z<\infty,$ $z\neq 1$}\;,$$
(4)
It is a monotonic function of the “${z}$-purity” Tr$[\rho^{z}]$,
and it reduces to the von Neumann entropy in the limit ${z}\to 1$, viz.,
$$\lim_{z\rightarrow 1}S_{z}(\rho)=S(\rho)\equiv-\mbox{Tr}[\rho\ln\rho].$$
(5)
For
$z=2$, the Rényi entropy is a monotonic function of the linearized
entropy $S_{\rm lin}(\rho)\equiv 1-$Tr$[\rho^{2}]$.
We are interested in the
minimum value that $S_{z}(\rho)$ achieves at the output of the
classical-noise channel, i.e.,
$$\displaystyle{\mathbb{S}}_{z}({\cal N}_{n})\equiv\min_{\rho\in{\cal H}}\>S_{z}%
({\cal N}_{n}(\rho))\;,$$
(6)
where the minimization is performed over all states in the
Hilbert space $\cal H$ associated with the channel’s input. The concavity of
$S_{z}$ implies that the minimum in Eq. (6) is achieved by
a pure-state
input, $\rho=|\psi\rangle\langle\psi|$. Our working hypothesis is that
$S_{z}({\cal N}_{n}(\rho))$ achieves its minimum value when the input is a coherent
state $|\alpha\rangle$, in which case we find that
$$\displaystyle S_{z}({\cal N}_{n}(|\alpha\rangle\langle\alpha|))=\frac{\ln[(n+1%
)^{z}-n^{z}]}{{z}-1}\;.$$
(7)
[Note that this quantity does not depend on $\alpha$, thanks to the
invariance of the Rényi entropy under unitary transformations.]
Clearly, Eq. (7) provides an upper bound on ${\mathbb{S}}_{z}({\cal N}_{n})$. We
conjecture that it is also a lower bound, whence
$$\displaystyle{\mathbb{S}}_{z}({\cal N}_{n})=\frac{\ln[(n+1)^{z}-n^{z}]}{{z}-1}\;.$$
(8)
The monotonicity of $S_{z}(\rho)$ with respect to the
${z}$-purity permits restating the conjecture (8) as follows,
$$\displaystyle\mbox{Tr}\left\{[{\cal N}_{n}(\rho)]^{z}\right\}\leqslant\frac{1}%
{(n+1)^{z}-n^{z}}\;,$$
(9)
where the right-hand side of the inequality is the ${z}$-purity at
the output of
the classical-noise channel when its input is a coherent state. In
Sec. II.1 we will show that this relation is true for integer
${z}\geqslant 2$, thus proving the conjecture (8) in this
case newnota . There we also show that (9) holds for all
$0<z<\infty$, $z\neq 1$ when the input is restricted to be a Gaussian
state. In
Sec. II.2 we will present some lower bounds on the Rényi
output entropy
of arbitrary order.
II.1 Integer-${z}$ Rényi entropy
From the definition of the
classical-noise channel, we see that
$$\displaystyle\mbox{Tr}\left\{[{\cal N}_{n}(\rho)]^{k}\right\}=\int{\rm d}^{2}%
\mu_{1}\cdots{\rm d}^{2}\mu_{k}\,P_{n}(\mu_{1})\cdots P_{n}(\mu_{k})$$
$$\displaystyle\times\mbox{Tr}[D(\mu_{1})\rho D^{\dagger}(\mu_{1})D(\mu_{2})\rho
D%
^{\dagger}(\mu_{2})\cdots D^{\dagger}(\mu_{k})]\;,$$
(10)
with $k\geqslant 1$ an integer. For a pure-state input
$|\psi\rangle$, the trace
can be expressed as
$$\displaystyle\mbox{Tr}[D(\mu_{1})\rho D^{\dagger}(\mu_{1})\cdots D^{\dagger}(%
\mu_{k})]$$
$$\displaystyle=$$
$$\displaystyle\langle\psi|D^{\dagger}(\mu_{1})D(\mu_{2})|\psi\rangle\langle\psi%
|D^{\dagger}(\mu_{2})D(\mu_{3})|\psi\rangle\cdots\langle\psi|D^{\dagger}(\mu_{%
k})D(\mu_{1})|\psi\rangle$$
$$\displaystyle=$$
$$\displaystyle\mbox{Tr}\left\{(\rho\otimes\rho\otimes\cdots\otimes\rho)\left[D_%
{1}^{\dagger}(\mu_{1})D_{1}(\mu_{2})\otimes D_{2}^{\dagger}(\mu_{2})D_{2}(\mu_%
{3})\otimes\cdots\otimes D_{k}^{\dagger}(\mu_{k})D_{k}(\mu_{1})\right]\right\}\;,$$
where the ${k}$ scalar products in the input Hilbert space $\cal H$ in
the first line were replaced with a single expectation value on the
tensor-product
Hilbert space
${\cal H}^{\otimes{k}}$ in the second line. Here
$D_{j}(\mu)$ is a displacement operator that acts on the $j$th
annihilation operator $a_{j}$ of this enlarged Hilbert space. With this
replacement, Eq. (10), which is nonlinear in $\rho$,
can be evaluated as the linear expectation value of an operator $\Theta$ on
${\cal H}^{\otimes{k}}$, i.e.,
$$\displaystyle\mbox{Tr}\left\{[{\cal N}_{n}(\rho)]^{k}\right\}=\mbox{Tr}[(\rho%
\otimes\cdots\otimes\rho)\Theta]\;,$$
(12)
with $\Theta$ being a convolution of tensor products of the displacements
$\{D_{j}\}$, namely
$$\displaystyle\Theta=\int\frac{{\rm d}^{2}\vec{\mu}}{(\pi n)^{k}}\>{e^{-\vec{%
\mu}\cdot C\cdot\vec{\mu}\,^{\dagger}+\vec{\mu}\cdot G^{\dagger}\cdot\vec{a}\,%
^{\dagger}-\vec{a}\cdot G\cdot\vec{\mu}\,^{\dagger}}}\;,$$
(13)
where $\vec{\mu}$ is the complex vector $(\mu_{1},\cdots,\mu_{k})$ and
$\vec{a}\equiv(a_{1},\cdots,a_{k})$. In Eq. (13),
$C\equiv\frac{\openone}{n}+\frac{A}{2}$ and $G$ are ${k}\times{k}$
real matrices, with $\openone$ being the identity and
$$\displaystyle A$$
$$\displaystyle\equiv$$
$$\displaystyle\left[\begin{array}[]{rrrrrr}0&-1&0&\cdots&0&1\cr 1&0&-1&\cdots&0%
&0\cr 0&1&0&\cdots&0&0\cr\vdots&&&\ddots&&\cr 0&0&0&\cdots&0&-1\cr-1&0&0&%
\cdots&1&0\cr\end{array}\right]\;,$$
(14)
$$\displaystyle G$$
$$\displaystyle\equiv$$
$$\displaystyle\left[\begin{array}[]{rrrrrr}-1&1&0&\cdots&0&0\cr 0&-1&1&\cdots&0%
&0\cr 0&0&-1&\cdots&0&0\cr\vdots&&&\ddots&&\cr 0&0&0&\cdots&-1&1\cr 1&0&0&%
\cdots&0&-1\cr\end{array}\right]\;.$$
(15)
[The matrix $A$ is null when $k=2$.] $A$ and $G$ are commuting
circulant matrices circulant , hence they possess a common
basis of orthogonal eigenvectors. This means that there exists a
unitary matrix $Y$ such that $D\equiv Y\>C\>Y^{\dagger}$ and $E\equiv Y\>G\>Y^{\dagger}$ are diagonal. Rewriting $\Theta$ from
Eq. (13) in factored form by performing the change of
integration variables $\vec{\nu}\equiv\vec{\mu}\cdot Y^{\dagger}$, and then
introducing the new annihilation operators $\vec{b}\equiv\vec{a}\cdot Y^{\dagger}$, we find that
$$\displaystyle\Theta=\bigotimes_{j=1}^{k}\Theta_{j}\;,$$
(16)
with
$$\displaystyle\Theta_{j}\equiv\frac{1}{n|e_{j}|^{2}}\int\frac{{\rm d}^{2}\nu}{%
\pi}\>{e^{-d_{j}|\nu|^{2}/|e_{j}|^{2}}}\>D_{b_{j}}(\nu)\;,$$
(17)
where $D_{b_{j}}(\nu)\equiv\exp[\nu b_{j}^{\dagger}-\nu^{*}b_{j}]$ is the
displacement operator associated with $b_{j}$, while $d_{j}$ and $e_{j}$ are
the $j$th diagonal elements of the matrices $D$ and $E$, respectively
(i.e., they are the $j$th eigenvalues of $C$ and $G$). As discussed in
App. A, the operator $\Theta_{j}$ is diagonal in the Fock
basis of the mode $b_{j}$ and takes the thermal-like form caves1
$$\displaystyle\Theta_{j}=\frac{2/n}{2d_{j}+|e_{j}|^{2}}\left(\frac{2d_{j}-|e_{j%
}|^{2}}{2d_{j}+|e_{j}|^{2}}\right)^{b_{j}^{\dagger}b_{j}}\;.$$
(18)
Because the $\{d_{j}\}$ have positive real parts equal to $1/n$ [see
Eq. (59)], the vacuum state of $b_{j}$ is the $\Theta_{j}$-eigenvector
whose associated eigenvalue has the maximum absolute value,
${2/[n(2d_{j}+|e_{j}|^{2})]}$. It then follows from Eq. (16)
that for any state $R\in{\cal H}^{\otimes{k}}$ we have
$$\displaystyle\Big{|}\mbox{Tr}[R\>\Theta]\Big{|}$$
$$\displaystyle\leqslant$$
$$\displaystyle\prod_{j=1}^{k}\frac{2/n}{2d_{j}+|e_{j}|^{2}}=\frac{1/n^{k}}{\det%
[C+G^{\dagger}G/2]}$$
(19)
$$\displaystyle=$$
$$\displaystyle\frac{1}{(n+1)^{k}-n^{k}}\;,$$
where in deriving the first equality we have used the invariance of
the determinant under the unitary transformation $Y$. Because
inequality (9) now follows directly from Eq. (12),
this completes the proof: for integer $k\geqslant 2$ the maximum
$k$-purity (or, equivalently, the minimum Rényi entropy
${\mathbb{S}}_{k}({\cal N}_{n})$) is provided by a coherent state input (see also
App. B).
Gaussian-state inputs:
Suppose that the channel input is restricted to be a Gaussian state,
$\rho_{G}$. It
is easy to show that a coherent-state input minimizes $S_{z}(\rho_{G})$ for all
$0<z<\infty$, $z\neq 1$. A Gaussian state is completely characterized by its
mean $\langle a\rangle$ and its covariance matrix,
$$\displaystyle\Gamma\equiv\left[\begin{array}[]{cc}\langle\{\Delta a,\Delta a^{%
\dagger}\}\rangle/2&\langle(\Delta a)^{2}\rangle\cr\langle(\Delta a)^{2}%
\rangle&\langle\{\Delta a,\Delta a^{\dagger}\}\rangle/2\end{array}\right]\;,$$
(20)
where $\langle\;\cdot\;\rangle\equiv\mbox{ Tr}[\;\cdot\;\rho_{G}]$ is
expectation with respect to $\rho_{G}$, $\Delta a\equiv a-\langle a\rangle$, and
$\{\;\cdot\;,\;\cdot\;\}$ denotes the anticommutator. As shown in von ,
the classical-noise channel’s output state $\rho^{\prime}_{G}$, when its input
is $\rho_{G}$,
is also Gaussian. The mean, $\langle a\rangle$, is unaffected by the CP map
${\cal N}_{n}$, but the covariance matrix is modified by the presence of
classical noise, viz., $\Gamma\rightarrow\Gamma^{\prime}=\Gamma+n\openone$.
By concatenating two unitary
transformations—a displacement to drive
$\langle a\rangle$ to zero, and a squeeze operator to symmetrize the quadrature
uncertainties—$\rho^{\prime}_{G}$ can be converted into the thermal state
$$\displaystyle\tau^{\prime}_{G}=\frac{1}{n^{\prime}+1}\left(\frac{n^{\prime}}{n%
^{\prime}+1}\right)^{a^{\dagger}a}\;,$$
(21)
where $n^{\prime}=\sqrt{\det\Gamma^{\prime}}-1/2$. The state (21) has Rényi
entropy
$$S_{z}(\tau^{\prime}_{G})=\frac{\ln[(n^{\prime}+1)^{z}-{n^{\prime}}^{z}]}{{z}-1%
}\quad\mbox{for $0<z<\infty$, $z\neq 1$}.$$
(22)
Moreover, because Rényi entropy is
invariant under unitary transformations, we have $S_{z}(\rho^{\prime}_{G})=S_{z}(\tau^{\prime}_{G})$.
Equation (22) thus shows that $S_{z}(\rho^{\prime}_{G})$ is
monotonically increasing
with increasing $n^{\prime}=\sqrt{\det\Gamma^{\prime}}-1/2$, and in von we showed that
$\min_{\rho_{G}}(\sqrt{\det\Gamma^{\prime}}-1/2)=n$ is achieved by coherent-state
inputs. It follows that $S_{z}({\cal N}_{n}(\rho_{G}))$ is minimized, for all $0<z<\infty$,
$z\neq 1$, when the channel input is a coherent state. The corresponding
Gaussian-state result for the von Neumann entropy at the
classical-noise channel’s
output was derived in von .
Comments:
The most interesting cases for integer-order Rényi output entropy are
$k=2$ and
$k\to\infty$, where we have
$$\displaystyle{\mathbb{S}}_{2}$$
$$\displaystyle=$$
$$\displaystyle\ln(2n+1)\;,$$
(23)
$$\displaystyle{\mathbb{S}}_{\infty}$$
$$\displaystyle=$$
$$\displaystyle\ln(n+1)\;.$$
(24)
Equation (23) has been used in von to derive lower
bounds for the von Neumann entropy at the output of the
classical-noise channel. On
the other hand, Eq. (24) establishes an upper bound on the
maximum eigenvalue $\lambda_{\rm max}$ of any output state ${\cal N}_{n}(\rho)$
of the channel. This is so because the Rényi
entropy becomes $S_{\infty}({\cal N}_{n}(\rho))=-\ln(\lambda_{\rm max})$ in the limit
${k}\to\infty$ zyc , and Eq. (24) requires that
$\lambda_{\rm max}\leqslant 1/(n+1)$.
II.2 Rényi entropy lower bounds
In this section we develop four lower bounds on ${\mathbb{S}}_{z}$ for arbitrary
$z$, which
support the conjecture (8).
Lower bound 1):
The Rényi entropy $S_{z}(\rho)$ is a decreasing function of ${z}$ zyc .
So, using our knowledge of ${\mathbb{S}}_{k}({\cal N}_{n})$ for integers $k\geqslant 2$, we have that
$$\displaystyle{\mathbb{S}}_{z}({\cal N}_{n})\geqslant{\mathbb{S}}_{k}({\cal N}_%
{n})=\frac{\ln((n+1)^{k}-n^{k})}{k-1}\;,$$
(25)
for all ${z}\leqslant k$.
For ${z}\leqslant 1$, we can employ the best of the von Neumann output entropy
lower bounds that we established in von to
derive a tighter lower bound on the Rényi entropy. Together with
Eq. (25), this additional bound produces the staircase function 1)
shown in Figs. 1 and 2.
Lower bound 2):
The definition of the Rényi entropy leads to the following
monotonicity property zyc ,
$$\displaystyle\frac{{z}-1}{{z}}S_{{z}}(\rho)\geqslant\frac{{z}^{\prime}-1}{{z}^%
{\prime}}S_{{z}^{\prime}}(\rho)\;,$$
(26)
for any ${z}\geqslant{z}^{\prime}$ and for all $\rho$. Allowing $\rho$
to be an arbitrary output state from the channel ${\cal N}_{n}$, and minimizing both
sides of (26) over all the possible inputs, we obtain
$$\displaystyle\frac{{z}-1}{{z}}{\mathbb{S}}_{{z}}({\cal N}_{n})\geqslant\frac{{%
z}^{\prime}-1}{{z}^{\prime}}{\mathbb{S}}_{{z}^{\prime}}({\cal N}_{n})\;.$$
(27)
When ${z}^{\prime}=k\geqslant 2$ is an integer, this relation provides the lower
bound
$$\displaystyle{\mathbb{S}}_{z}({\cal N}_{n})\geqslant\frac{z}{{z}-1}\frac{\ln((%
n+1)^{k}-n^{k})}{k}\;,$$
(28)
for $z\geqslant k$, which is shown as curve 2) in Figs. 1
and 2.
Lower bound 3):
Using the relation between different measures of entropy established
in renyi ; wang , the following inequality can be derived
(see App. C):
$$\displaystyle{\mathbb{S}}_{z}({\cal N}_{n})\geqslant-\frac{1}{z-1}\ln\left\{h_%
{z}\left[h_{k}^{-1}\left(\frac{1}{(n+1)^{k}-n^{k}}\right)\right]\right\}\;,$$
(29)
for all $z\leqslant k$ and integers $k\geqslant 2$. Here, $h_{z}(x)$ is
the function defined in (68) and $h^{-1}_{z}(x)$ its inverse.
For $z\leqslant 1$ a further lower bound can be obtained from
$\bar{\mathbb{S}}({\cal N}_{n})$, the best of the lower bounds on the von Neumann
output entropy
given in von :
$$\displaystyle{\mathbb{S}}_{z}({\cal N}_{n})\geqslant-\frac{1}{z-1}\ln\left\{h_%
{z}\left[v^{-1}\left(\bar{\mathbb{S}}({\cal N}_{n})\right)\right]\right\}\;,$$
(30)
where $v^{-1}(x)$ is the inverse of the function $v(x)$ defined in
Eq. (LABEL:vdef). Curve 3) of Figs. 1
and 2 has been obtained by considering the maximum of all
the functions on the right-hand sides of (29)
and (30).
Lower bound 4):
Our final lower bound can be derived from the inequality von
$$\displaystyle\mbox{Tr}\{[{\cal N}_{n}(\rho)]^{z}\}\leqslant\frac{\mbox{Tr}[{%
\cal N}_{n/{z}}(\rho)]}{{z}\>n^{{z}-1}}=\frac{1}{{z}\>n^{{z}-1}}\;,$$
(31)
for ${z}\geqslant 1$,
which implies
$$\displaystyle S_{z}({\cal N}_{n}(\rho))\geqslant\frac{\ln{z}}{{z}-1}+\ln n\;,$$
(32)
for any input $\rho$ and ${z}\geqslant 1$. Inequality (31) was derived
in von from the convexity of $x^{z}$ for $z\geqslant 1$. For
${z}\leqslant 1$, the function $x^{z}$ is concave and we obtain
$$\displaystyle\mbox{Tr}\{[{\cal N}_{n}(\rho)]^{z}\}\geqslant\frac{\mbox{Tr}[{%
\cal N}_{n/{z}}(\rho)]}{{z}\>n^{{z}-1}}=\frac{1}{{z}\>n^{{z}-1}}\;.$$
(33)
The sign change associated with the
$1/({z}-1)$ factor in the Rényi entropy definition then shows that
(32)
also applies for ${z}\leqslant 1$. Lower bound (32) is plotted as curve
4) in Figs. 1 and 2.
III Wehrl entropy
The Wehrl entropy is the continuous Boltzmann-Gibbs entropy of the
Husimi probability function for the state $\rho$ wehrl ,
$$\displaystyle{W}(\rho)\equiv-\int{{\rm d}^{2}\mu}\>Q(\mu)\ln[\pi Q(\mu)]\;,$$
(34)
where $Q(\mu)\equiv\langle\mu|\rho|\mu\rangle/\pi$ with $|\mu\rangle$
a coherent state. The Wehrl entropy provides a measurement of the
“localization” of the state $\rho$ in the phase space: its minimum
value is achieved on coherent states wehrl ; lieb . It is also
useful in characterizing the statistics associated with heterodyne
detection heter . Here we study this minimum restricted to the
output states from the classical-noise channel, i.e.,
$$\displaystyle{\mathbb{W}}({\cal N}_{n})\equiv\min_{\rho\in{\cal H}}\>W({\cal N%
}_{n}(\rho))\;.$$
(35)
We will show that coherent-state inputs achieve this minimum, which is then
given by
$$\displaystyle{\mathbb{W}}({\cal N}_{n})=1+\ln(n+1)\;.$$
(36)
The output-state Husimi function $Q^{\prime}(\mu)$ for the channel map
${\cal N}_{n}$ is the convolution of the input-state Husimi function
$Q(\mu)$ with the Gaussian probability distribution $P_{n}$ from Eq. (2),
$$\displaystyle Q^{\prime}(\mu)=(P_{n}*Q)(\mu)=\int{\rm d}^{2}\nu\>P_{n}(\nu)\>Q%
(\mu-\nu)\;.$$
(37)
This property can be used to show that the right-hand side of
Eq. (36) is an upper bound for ${\mathbb{W}}$, because it is the value
achieved by a coherent-state input. In particular, the Husimi function of the
coherent state
$|\alpha\rangle$ is
$Q_{\alpha}(\mu)\equiv\left|\langle\alpha|\mu\rangle\right|^{2}/\pi=\exp(-|\mu-%
\alpha|^{2})/\pi$, which evolves into
$$\displaystyle Q^{\prime}_{\alpha}(\mu)=\frac{\exp\left[-\frac{|\mu-\alpha|^{2}%
}{n+1}\right]}{\pi(n+1)}\;,$$
(38)
under (37).
The resulting Wehrl output entropy is then
$$\displaystyle W({\cal N}_{n}(|\alpha\rangle\langle\alpha|))$$
$$\displaystyle=$$
$$\displaystyle\int{\rm d}^{2}\mu\>Q^{\prime}_{\alpha}(\mu)\>\frac{|\mu-\alpha|^%
{2}}{n+1}+\ln(n+1)$$
(39)
$$\displaystyle=$$
$$\displaystyle 1+\ln(n+1)\;.$$
(An analogous result was also given in hall .)
To show that this quantity is also a lower bound for $\mathbb{W}$, we
use Theorem 6 of lieb , which states that for two
probability distributions $f(\mu)$ and $h(\mu)$ on ${\mathbb{C}}$ we have
$$\displaystyle W((f*h)(\mu))$$
$$\displaystyle\geqslant$$
$$\displaystyle\lambda\;W(f(\mu))+(1-\lambda)W(h(\mu))$$
(40)
$$\displaystyle-\lambda\ln\lambda-(1-\lambda)\ln(1-\lambda)$$
for all $\lambda\in[0,1]$,
where $f*h$ is the convolution of $f$ and $h$ and where the Wehrl
entropy of a probability distribution is found from Eq. (34)
by replacing $Q(\mu)$ with the given distribution. Choosing $f={P_{n}}$ and
$h={Q}$ makes $f*h$ the classical-noise channel’s output-state
Husimi function,
$Q^{\prime}$. Hence,
inequality (40) implies that
$$\displaystyle W({\cal N}_{n}(\rho))$$
$$\displaystyle\geqslant$$
$$\displaystyle\lambda\>W(P_{n})+(1-\lambda)\>W(\rho)$$
(41)
$$\displaystyle-\lambda\ln\lambda-(1-\lambda)\ln(1-\lambda)\;,$$
where $W(P_{n})=1+\ln n$ is the Wehrl entropy of the distribution $P_{n}$.
Because $W(\rho)\geqslant 1$ for any $\rho$ lieb , Eq. (41)
gives
$$\displaystyle W({\cal N}_{n}(\rho))\geqslant\lambda\ln n+1-\lambda\ln\lambda-(%
1-\lambda)\ln(1-\lambda),$$
(42)
which for $\lambda=n/(n+1)$ becomes
$$\displaystyle W({\cal N}_{n}(\rho))$$
$$\displaystyle\geqslant$$
$$\displaystyle 1+\ln(n+1)\;.$$
(43)
Inasmuch as this relation applies for all $\rho$, Eq. (36) then follows.
III.1 Rényi-Wehrl entropies
The ${z}$–Rényi-Wehrl entropies are defined by rwen
$$\displaystyle W_{z}(\rho)$$
$$\displaystyle\equiv$$
$$\displaystyle-\frac{1}{{z}-1}\ln(m_{z}(\rho))\;,$$
(44)
$$\displaystyle m_{z}(\rho)$$
$$\displaystyle\equiv$$
$$\displaystyle\int\frac{{\rm d}^{2}\mu}{\pi}[\pi Q(\mu)]^{z}\;,$$
(45)
where $Q(\mu)$ is the Husimi function of $\rho$ and ${z}\geqslant 1$.
Thus, the Wehrl entropy $W(\rho)$ is the limit as ${z}\to 1$ of
$W_{z}(\rho)$, and
$W_{z}(\rho)$ achieves its minimum value, $\ln(z)/({z}-1)$, when $\rho$ is a
coherent state $|\alpha\rangle$, for which
$m_{z}(|\alpha\rangle\langle\alpha|)=1/{z}$. For arbitrary
$\rho$, Theorem 3 of lieb implies
$$\displaystyle m_{{z}}(\rho)=\int\frac{{\rm d}^{2}\mu}{\pi}[\pi\>Q(\mu)]^{{z}}%
\leqslant\frac{1}{z}\;.$$
(46)
We now show that ${\mathbb{W}}_{z}({\cal N}_{n})\equiv\min_{\rho}(W_{z}({\cal N}_{n}(\rho))$ is achieved
by coherent-state inputs. From Eq. (38), the classical-noise channel’s
Rényi-Wehrl output entropy for the coherent-state input
$|\alpha\rangle$ can be
shown to be
$$\displaystyle W_{z}({\cal N}_{n}(|\alpha\rangle\langle\alpha|))=\frac{\ln{z}}{%
{z}-1}+\ln(n+1)\;.$$
(47)
To show that the right-hand side of this equation is the global
minimum, we observe that, for an arbitrary state $\rho$ and for all
$p,q\geqslant 1$ such that $1/p+1/q=1+1/{z}$, the sharp form of
Young’s inequality (Lemma 5 of Ref. lieb ) together
Eq. (37) give
$$\displaystyle m_{z}({\cal N}_{n}(\rho))=\int\frac{{\rm d}^{2}\mu}{\pi}[\pi\>Q^%
{\prime}(\mu)]^{z}\leqslant\left(\frac{C_{p}C_{q}}{C_{z}}\right)^{2{z}}$$
$$\displaystyle\times\left[\int\frac{{\rm d}^{2}\mu}{\pi}[\pi\>Q(\mu)]^{p}\right%
]^{{z}/p}\left[\int\frac{{\rm d}^{2}\mu}{\pi}\frac{e^{-q|\mu|^{2}/n}}{n^{q}}%
\right]^{{z}/q}$$
$$\displaystyle=\left(\frac{C_{p}C_{q}}{C_{z}}\right)^{2{z}}\left[m_{p}(\rho)%
\right]^{{z}/p}\left[\frac{n}{qn^{q}}\right]^{{z}/q}\;,$$
(48)
where $C_{p}$, $C_{q}$, and $C_{z}$ are the Young’s inequality constants,
$$\displaystyle C_{x}\equiv\left[\frac{x^{1/x}}{(x^{\prime})^{1/x^{\prime}}}%
\right]^{1/2}\;\qquad x^{\prime}\equiv x/(x-1)\;.$$
(49)
Choosing $p=(n+1){z}/(n{z}+1)$ and, hence, $q=(n+1){z}/({z}+n)$, we
then obtain
$$\displaystyle m_{z}({\cal N}_{n}(\rho))\leqslant\frac{1}{{z}(n+1)^{{z}-1}}\;,$$
(50)
which, via Eq. (44), completes the proof.
IV Thermal-noise channel
Thus far we have limited our attention to the CP map ${\cal N}_{n}$
associated with the
classical-noise channel. This channel is a limiting case of the
thermal-noise channel, in which the
signal mode
$a$ and a thermal-reservoir mode $b$ couple to the
channel output through a beam splitter von ; futuro . The thermal-noise
channel’s CP map ${\cal E}_{\eta}^{N}$ is obtained by tracing away the noise
mode—which
initially is in a thermal state with average photon number $N$—from the
evolution
$$\displaystyle a\longrightarrow\sqrt{\eta}\;a+\sqrt{1-\eta}\;b\;,$$
(51)
where $\eta$ is the coupling parameter (the channel’s quantum
efficiency).
A detailed characterization of the two maps ${\cal N}_{n}$ and
${\cal E}_{\eta}^{N}$ is given in von , where, in particular, it is shown
that they are related through the composition rule
$$\displaystyle{\cal E}_{\eta}^{N}(\rho)=\left({\cal N}_{(1-\eta)N}\circ{\cal E}%
_{\eta}^{0}\right)(\rho)\equiv{\cal N}_{(1-\eta)N}\left({\cal E}_{\eta}^{0}(%
\rho)\right)\;.$$
(52)
This means that the thermal-noise channel ${\cal E}_{\eta}^{N}$ can be regarded
as the application of the map ${\cal N}_{n}$ to the output of the pure-loss
channel ${\cal E}_{\eta}^{0}$, with the latter being a zero-temperature
($N=0$) thermal-noise channel.
We can use (52) to extend all the analyses from the previous
sections to the thermal-noise channel. Specifically, the minimum
$z$-Rényi output entropy of the thermal-noise channel, obeys
$$\displaystyle{\mathbb{S}}_{z}({\cal E}_{\eta}^{N})={\mathbb{S}}_{z}({\cal N}_{%
(1-\eta)N}\circ{\cal E}_{\eta}^{0})\geqslant{\mathbb{S}}_{z}({\cal N}_{(1-\eta%
)N})\;,$$
(53)
because the implicit minimization on the left is performed over a
subset of the states considered in the implicit minimization on the
right. Replacing $n$ with $(1-\eta)N$ in this inequality, we
immediately find that the lower bounds from Sec. II.2 also
apply to the thermal-noise channel ${\cal E}_{\eta}^{N}$. Moreover, for
$z\geqslant 2$ an integer, (53) becomes an equality, because the
implicit minimum on the left is achieved by the vacuum-state input
$|0\rangle$, for which, according to Eq. (52),
$$\displaystyle{\cal E}_{\eta}^{N}(|0\rangle\langle 0|)={\cal N}_{(1-\eta)N}(|0%
\rangle\langle 0|)\;.$$
(54)
This proves that for integers $k\geqslant 2$ the minimum Rényi
entropy at the
output of the thermal-noise channel is
$$\displaystyle{\mathbb{S}}_{k}({\cal E}_{\eta}^{N})=\frac{\ln\{[(1-\eta)N+1]^{k%
}-[(1-\eta)N]^{k}\}}{k-1}\;.$$
(55)
Some preliminary results in this regard were obtained in paz ,
where it was shown that the linearized entropy of the thermal-noise
channel—i.e., $S_{2}({\cal E}_{\eta}^{N}(\rho))$—is minimized by the
vacuum input in
the limit of low coupling ($\eta\ll 1)$ and high temperature ($N\gg 1$).
When the input to the thermal-noise channel is a Gaussian state
$\rho_{G}$ with covariance matrix $\Gamma$, the output state will be
Gaussian with covariance matrix $\Gamma^{\prime}=\eta\Gamma+(1-\eta)(N+1/2)\openone$ von . We have previously shown that
$\min_{\rho_{G}}(\sqrt{\det\Gamma^{\prime}}-1/2)=(1-\eta)N$ is achieved when
the input is a coherent state, which shows that coherent-state inputs
minimize $S_{z}({\cal E}_{\eta}^{N}(\rho_{G}))$ for all $0<z<\infty$, $z\neq 1$.
Finally, arguments identical to the ones given earlier for the
minimum Wehrl and
Rényi-Wehrl entropies at the output of the classical-noise channel also
apply to the minimum Wehrl and Rényi-Wehrl entropies at the output of
the thermal-noise channel. Because the minimum values
${\mathbb{W}}({\cal N}_{n})$ and
${\mathbb{W}}_{z}({\cal N}_{n})$ are achieved by coherent-state inputs, such as the vacuum,
Eqs. (52) and (54) imply that
$$\displaystyle{\mathbb{W}}({\cal E}_{\eta}^{N})$$
$$\displaystyle=$$
$$\displaystyle 1+\ln[(1-\eta)N+1]$$
$$\displaystyle{\mathbb{W}}_{z}({\cal E}_{\eta}^{N})$$
$$\displaystyle=$$
$$\displaystyle\frac{\ln z}{z-1}+\ln[(1-\eta)N+1]\;.$$
(56)
with these minima being realized by coherent-state inputs.
V Conclusion
The minimum Rényi and Wehrl output entropies have been analyzed for bosonic
channels in which the signal photons are disturbed by classical
additive Gaussian
noise, or by a combination of propagation loss and Gaussian noise.
We conjectured
that the Rényi output entropy is minimized by coherent-state inputs. Some
arguments were provided to place this conjecture on solid ground. In
particular,
we have shown that it is true for integer orders greater than
one, and it is true when the input state is restricted to being
Gaussian. For the
general case—non-integer orders and arbitrary input states—we have provided
entropic lower bounds that are compatible with the upper bound implied by the
conjecture. In addition, we have shown that coherent-state inputs minimize
the Wehrl and the Rényi-Wehrl output entropies for these two channels.
Appendix A Derivation of Eq. (18)
In this appendix we show that the operator $\Theta_{j}$ defined in
Eq. (17) coincides with the right-hand side of
Eq. (18). The easiest way to prove this assertion is to show that
these operators have the same characteristic function. We take
advantage of the interesting analysis in caves1 , where the
maximal-entanglement teleportation fidelity is calculated for the
classical-noise channel, and $k=2$ version of (18) was implicitly
demonstrated.
From Eq. (17), we immediately see that the symmetrical
characteristic function walls of the operator $\Theta_{j}$ is
$$\displaystyle\chi_{j}(\nu)\equiv\mbox{Tr}[\Theta_{j}\>D_{j}(\nu)]=\frac{\exp(-%
{d_{j}|\nu|^{2}}/{|e_{j}|^{2}})}{n|e_{j}|^{2}}\;.$$
(57)
On the other hand, the characteristic function of the right-hand side
of Eq. (18) is given by
$$\displaystyle\chi^{\prime}_{j}(\nu)=\frac{2/n}{2d_{j}+|e_{j}|^{2}}\sum_{m=0}^{%
\infty}\left(\frac{2d_{j}-|e_{j}|^{2}}{2d_{j}+|e_{j}|^{2}}\right)^{m}\langle m%
|D_{b_{j}}|m\rangle$$
$$\displaystyle=\frac{2/n}{2d_{j}+|e_{j}|^{2}}\sum_{m=0}^{\infty}\left(\frac{2d_%
{j}-|e_{j}|^{2}}{2d_{j}+|e_{j}|^{2}}\right)^{m}e^{-|\nu|^{2}/2}L_{m}(|\nu|^{2}%
)\;,$$
where $\{|m\rangle\}$ are the Fock states of the $b_{j}$ mode and $L_{m}$ is the
Laguerre polynomial of order $m$. From the definition of the matrix
$C$ [see Eq. (13)] we know that
$$\displaystyle d_{j}=1/n+i\xi_{j}\;,$$
(59)
where $\{i\xi_{j}\}$
are the imaginary eigenvalues of the real anti-symmetric matrix $A$ from
Eq. (14). This implies that the $\{d_{j}\}$ have positive real parts, so
that the absolute value of the parenthetical term in Eq. (LABEL:llag),
$({2d_{j}-|e_{j}|^{2}})/({2d_{j}+|e_{j}|^{2}})$, is less than
one. The summation in Eq. (LABEL:llag) can thus be performed using the
formula grad
$$\displaystyle\sum_{m=0}^{\infty}z^{m}L_{m}(x)=\frac{\exp[xz/(z-1)]}{1-z}\qquad%
\mbox{for }|z|<1\;.$$
(60)
With this relation Eq. (LABEL:llag) yields $\chi_{j}$, concluding the
derivation.
Examples
Here, for the sake of clarity, we carry out calculations of the
$\{\Theta_{j}\}$ for the cases $k=2$ and $k=3$.
When $k=2$, the matrix $A$ is null and $G$ has eigenvalues
$e_{1}=0$ and $e_{2}=2$. The unitary transformation that diagonalizes
$A$ and $G$ is then
$$Y=\frac{1}{\sqrt{2}}\left[\begin{array}[]{rr}1&1\cr-1&1\end{array}\right]\;,$$
(61)
so
that $\Theta_{1}=\openone$ on the mode $b_{1}=(a_{1}+a_{2})/\sqrt{2}$, and
$$\displaystyle\Theta_{2}=\frac{1}{1+2n}\left(\frac{1-2n}{1+2n}\right)^{b_{2}^{%
\dagger}b_{2}}\;,$$
(62)
on the mode $b_{2}=(a_{2}-a_{1})/\sqrt{2}$.
When $k=3$, the matrix $A$ has eigenvalues $i\xi_{1}=0$,
$i\xi_{2}=i\sqrt{3}/2$, and $i\xi_{3}=-i\sqrt{3}/2$. On the other hand,
$G$ has eigenvalues $e_{1}=0$, $e_{2}=i\sqrt{3}e^{i2\pi/3}$, and
$e_{3}=-i\sqrt{3}e^{i2\pi/3}$. Now the unitary matrix $Y$ is
$$\displaystyle Y=\frac{1}{\sqrt{3}}\left[\begin{array}[]{ccc}1&1&1\cr e^{2i\pi/%
3}&e^{4i\pi/3}&1\cr e^{4i\pi/3}&e^{2i\pi/3}&1\end{array}\right]\;,$$
(63)
so that $\Theta_{1}=\openone$ on the mode $b_{1}=(a_{1}+a_{2}+a_{3})/\sqrt{3}$,
$$\displaystyle\Theta_{2}=\frac{2}{2+(3+i\sqrt{3})n}\left(\frac{{2+(-3+i\sqrt{3}%
)n}}{{2+(3+i\sqrt{3})n}}\right)^{b_{2}^{\dagger}b_{2}}\;,$$
(64)
on the mode $b_{2}=(e^{4i\pi/3}a_{1}+e^{2i\pi/3}a_{2}+a_{3})/\sqrt{3}$, and
$$\displaystyle\Theta_{3}=\frac{2}{2+(3-i\sqrt{3})n}\left(\frac{{2-(3+i\sqrt{3})%
n}}{{2+(3-i\sqrt{3})n}}\right)^{b_{3}^{\dagger}b_{3}}\;,$$
(65)
on the mode $b_{3}=(e^{2i\pi/3}a_{1}+e^{4i\pi/3}a_{2}+a_{3})/\sqrt{3}$.
Appendix B Entropy-minimizing input states
Even though it was already proven in Sec. II [see
Eqs. (7) and (9)], it is instructive to use a
different method to explicitly show that the upper bound (19)
on the integer-order Rényi output entropy can be achieved by
employing a vacuum-state input, $\rho=|0\rangle\langle 0|$. By
construction, the vacuum state for the $b_{j}$ modes,
$R_{0}=|0\rangle_{b_{1}}\langle 0|\otimes\cdots\otimes|0\rangle_{b_{k}}\langle 0|$, saturates this
bound. Because $\vec{a}$ is obtained from $\vec{b}$ through the unitary
matrix $Y$, the state $R_{0}$ is also the vacuum state of the $\vec{a}$
modes. Indeed, from the symmetric characteristic function
decomposition, we find
$$\displaystyle R_{0}$$
$$\displaystyle=$$
$$\displaystyle\int\frac{{\rm d}^{2}\vec{\nu}}{\pi^{k}}\>\exp[-|\vec{\nu}|^{2}/2%
+\vec{\nu}\cdot\vec{b}\,^{\dagger}-\vec{b}\cdot\vec{\nu}\,^{\dagger}]$$
(66)
$$\displaystyle=$$
$$\displaystyle\int\frac{{\rm d}^{2}\vec{\mu}}{\pi^{k}}\>\exp[-|\vec{\mu}|^{2}/2%
+\vec{\mu}\cdot\vec{a}\,^{\dagger}-\vec{a}\cdot\vec{\mu}\,^{\dagger}]$$
$$\displaystyle=$$
$$\displaystyle|0\rangle_{a_{1}}\langle 0|\otimes\cdots\otimes|0\rangle_{a_{k}}%
\langle 0|\;,$$
where $\vec{\nu}=\vec{\mu}\cdot Y^{\dagger}$. From Eq. (7) we know
that all coherent-state inputs produce the same Rényi output
entropy. This means
that every coherent state
$|\beta\rangle_{a_{1}}\langle\beta|\otimes\cdots\otimes|\beta\rangle_{a_{k}}%
\langle\beta|$
must saturate the bound (19). To show that this is so, we
note that for any integer ${k}$ the matrices $G$ and $A$ have a null
eigenvalue (say for $j=1$), associated with the common eigenvector
$(1,1,\cdots,1)$. In this case $e_{1}=0$ and $d_{1}=1/n$, so that
$\Theta_{1}=\openone_{j=1}$. This means that for arbitrary
$|\varphi\rangle_{b_{1}}$, any state of the form
$R_{\varphi}\equiv|\varphi\rangle_{b_{1}}\langle\varphi|\otimes|0\rangle_{b_{2}%
}\langle 0|\otimes\cdots\otimes|0\rangle_{b_{k}}\langle 0|$ saturates the
bound (19). If $|\varphi\rangle$ is not a coherent state, then
it corresponds to an entangled state of the $a_{j}$ modes, so it cannot be
written in the form
$\rho\otimes\cdots\otimes\rho$. Thus Tr$[R_{\varphi}\Theta]$
cannot be an output $k$-purity of the classical-noise channel. If, instead, we
repeat the same analysis of Eq. (66) with
$|\varphi\rangle=|\sqrt{k}\beta\rangle$ being a coherent state, we
find that the
resulting $R_{\varphi}$ is a tensor product of coherent states $|\beta\rangle$ in
the
$a_{j}$ modes, so that Tr$[R_{\varphi}\Theta]$ is the classical-noise
channel’s output $k$-purity relative to the coherent-state input
$|\beta\rangle$.
Appendix C Derivation of lower bound 3)
In this appendix we derive the lower bound 3), given by
(29) and (30).
The $z$-purity Tr$[\rho^{z}]$ for $z\neq 1$ belongs to the class of
entropic measures defined in renyi . Hence, for
$1<z^{\prime}\leqslant z$, the state that minimizes Tr$[\rho^{z}]$ over the
family of states
having constant Tr$[\rho^{z^{\prime}}]=c$ is
known renyi to have a $q$-times degenerate eigenvalue
$\lambda_{1}$, and a nondegenerate eigenvalue
$\lambda_{0}=1-q\lambda_{1}\leqslant\lambda_{1}$. The value of the
parameters $\lambda_{1}$ and $q$ are determined by the constraint
$$\displaystyle\lambda_{0}^{z^{\prime}}+q\lambda_{1}^{z^{\prime}}=c\;,$$
(67)
which, for $1\geqslant\lambda_{1}\geqslant\lambda_{0}\geqslant 0$, gives
$q=\lfloor 1/\lambda_{1}\rfloor$, and can be written as
$$\displaystyle h_{z^{\prime}}(\lambda_{1})\equiv\left(1-\left\lfloor\frac{1}{%
\lambda_{1}}\right\rfloor\lambda_{1}\right)^{z^{\prime}}+\left\lfloor\frac{1}{%
\lambda_{1}}\right\rfloor\lambda_{1}^{z^{\prime}}=c\;,$$
(68)
where $\lfloor x\rfloor$ is the integer part of $x$. The function
$h_{z}(x)$ can be shown to be continuous and monotonically increasing
(see Fig. 3), so that Eq. (68) has only one
solution in the range $c\in[0,1]$. Hence, following renyi ,
we can establish the inequality,
$$\displaystyle\mbox{Tr}[\rho^{z}]\geqslant h_{z}\left[h^{-1}_{z^{\prime}}\left(%
\mbox{Tr}[\rho^{z^{\prime}}]\right)\right]\;,$$
(69)
which applies for all $\rho$ and $z\geqslant z^{\prime}>1$ ($h^{-1}$ being the
inverse of the function $h$). Because $h_{z}(h^{-1}_{z^{\prime}}(x))$ is
monotonically increasing, Eq. (69) can be recast
as
$$\displaystyle S_{z^{\prime}}(\rho)\geqslant-\frac{\ln\left[h_{z^{\prime}}\left%
(h_{z}^{-1}(\mbox{Tr}[\rho^{z}])\right)\right]}{z^{\prime}-1}\;.$$
(70)
Evaluating this expression on the output states ${\cal N}_{n}(\rho)$, we can
obtain a lower bound for ${\mathbb{S}}_{z^{\prime}}({\cal N}_{n})$ by minimizing
both terms. Moreover, we can replace the term Tr$[\rho^{z}]$ in
Eq. (70) with its maximum value, because it is the argument
of a decreasing function. For $z=k$ an integer, we can then use the
results of Sec. II.1 (where the maximum value of
Tr$\{[{\cal N}_{n}(\rho)]^{k}\}$ was calculated) to derive (29)
from (70).
The same analysis can be repeated for $z^{\prime}<1$; in this case
$h_{z^{\prime}}(x)$ is monotonically decreasing, which is compensated by
the sign change of the factor $1/(z^{\prime}-1)$ in
Eq. (70).
In order to derive (30), we apply
the analysis of renyi to the von Neumann entropy $S(\rho)$
and Tr$[\rho^{z^{\prime}}]$ with $z^{\prime}<1$. Maximizing $S(\rho)$
over the family of states that have constant Tr$[\rho^{z^{\prime}}]=c$, we
find that the optimal state has the same eigenvalue structure
$\{\lambda_{0},\lambda_{1}\}$ encountered above. Equation (69)
is thus replaced by
$$\displaystyle S(\rho)\leqslant v\left[h^{-1}_{z^{\prime}}\left(\mbox{Tr}[\rho^%
{z^{\prime}}]\right)\right]\;,$$
(71)
where
$$\displaystyle v(x)\equiv-\left(1-\left\lfloor\frac{1}{x}\right\rfloor x\right)%
\ln\left(1-\left\lfloor\frac{1}{x}\right\rfloor x\right)-\left\lfloor\frac{1}{%
x}\right\rfloor x\ln x$$
is the decreasing function plotted in Fig. 3. Because
$v\left[h^{-1}_{z^{\prime}}\left(x\right)\right]$ is monotonically increasing,
Eq. (71) can be used to derive (30).
Acknowledgments:
the Authors thank P. W. Shor, H. P. Yuen and P. Zanardi for useful
discussions. This work was funded by the ARDA, NRO, NSF, and by ARO
under a MURI program.
References
(1)
M. A. Nielsen and I. L. Chuang, Quantum
Computation and Quantum Information (Cambridge University Press,
Cambridge, 2000).
(2)
C. H. Bennett and P. W. Shor, IEEE Trans. Info. Theory
44, 2724 (1998).
(3)
P. W. Shor, eprint quant-ph/0305035; A. S. Holevo,
eprint quant-ph/0306196.
(4)
G. G. Amosov, A. S. Holevo, and R. F. Werner,
Problems Inform. Trans. 36, 305 (2000), eprint
math-ph/0003002; C. King and M. B. Ruskai, IEEE Trans. Info. Theory
47, 192 (2001); C. King, IEEE Trans. Info. Theory, 49
221 (2003).
(5)
K. Życzkowski, Open Syst. and Inf. Dyn. 10, 297
(2003); C. Beck and F. Schlögl, Thermodynamics of Chaotic
Systems (Cambridge University Press, Cambridge, 1993).
(6)
A. Anderson and J. J. Halliwell, Phys. Rev. D 48, 2753 (1993).
(7)
V. Giovannetti, S. Guha, S. Lloyd, L. Maccone, and J.
H. Shapiro, eprint quant-ph/0404005.
(8)
V. Giovannetti and S. Lloyd, Phys. Rev. A, accepted
for publication, eprint quant-ph/0403075.
(9)
C. Caves, “Hidden-variable model for
continuous-variable teleportation,” in
http://info.phys.unm.edu/$\sim$caves/reports/cvteleportation.pdf .
(10)
M. J. W. Hall and M. J. O’Rourke, Quantum Opt. 5,
161 (1993); M. J. W. Hall, Phys. Rev. A 50, 3295 (1994).
(11)
In the case ${z}=1$, Eq. (9) is
trivially verified. However, this does not imply the
conjecture in (8) owing to the presence of the $1/({z}-1)$
factor in the Rényi entropy.
(12)
U. Grenander and G. Szegö, Toeplitz Forms
and Their Applications, (Univ. of Calif. Press, Berkeley, 1958).
(13)
X. Wang, B. C. Sanders, and D. W. Berry, Phys. Rev. A
67, 042323 (2003); V. M. Kendon, K. Życzkowski, and
W. J. Munro, Phys. Rev. A 66, 062310 (2002).
(14)
D. W. Berry and B. C. Sanders, J. Phys. A: Math. Gen.
36, 12255 (2003); P. Harremoës and F. Topsøe, IEEE Trans.
Info. Theory 47, 2944 (2001).
(15)
A. Wehrl, Rev. Mod. Phys. 50, 221 (1978); Rep.
Math. Phys. 16, 353 (1979).
(16)
E. H. Lieb, Commun. Math. Phys. 62, 35 (1978).
(17)
H. P. Yuen, and J. H. Shapiro, IEEE Trans. Info. Theory, 26, 78 (1980).
(18)
M. J. W. Hall, Phys. Rev. A 55, 100 (1997).
(19)
F. Mintert, K. Życzkowski, eprint quant-ph/0307169
(2003).
(20)
W. H. Zurek, S. Habib, and J. P. Paz, Phys. Rev. Lett.
70, 1187 (1993).
(21)
D. F. Walls and G. J. Milburn, Quantum Optics
(Springer Verlag, Berlin, 1994).
(22)
I. S. Gradshteyn, I. M. Ryzhik , Table of
Integrals, Series, and Products (Academic Press, San Diego,
2000), see Chap. 8.957. |
Non-Newtonian hydrodynamic modes in two-dimensional electron fluids
Serhii Kryhin, Leonid Levitov
Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139
Abstract
Two-dimensional Fermi systems is an appealing platform to explore exceptionally long-lived excitations arising due to collinear scattering governed by phase-space constraints. Recently it was shown that the lifetimes of these excitations surpass the fundamental bound set by Landau’s Fermi-liquid theory by a factor as large as $(T_{F}/T)^{\alpha}$ with $\alpha\approx 2$. As always, long-lived degrees of freedom can amplify the response to a weak perturbation, producing long-lasting collective memory effects. This leads to new hydrodynamics in 2D electron fluids, which includes several viscous modes with non-newtonian viscosity not anticipated by previous work. Here we present a detailed analysis of these modes and and discuss their experimental implications.
Recent years have seen a surge of interest in Gurzhi’s electron hydrodynamics [1]
as a framework to describe
transport in quantum materials at diverse length and time scales
[7, 12, 2, 4, 5, 8, 9, 10, 11, 3, 4, 6, 13, 14, 15, 16, 17, 18, 19, 20].
However, despite this interest, the fundamental question of how an orderly hydrodynamic behavior on macroscales stems from a chaotic dynamics due to interactions and collisions on microscales, in particular the role of the quantum effects, has received relatively little attention. The situation is well understood for classical gases, where all moments of momentum distribution not protected by conservation laws are extremely fragile, being quickly erased after just a few ($\sim 1$) collisions [21]. To the contrary, as shown below, quantum gases and liquids
feature surprising collective memory effects occurring over the span of $N\gg 1$ collisions with $N$ rapidly diverging at low temperatures. The long-time and long-wavelength dynamics in such systems cannot be captured by a conventional hydrodynamic description that relies on a closed system of equations for local flow velocity, particle density and temperature. Instead, a full description must account for memory effects due to certain quantities that are not protected by microscopic conservation laws.
Besides Gurzhi’ pioneering work on hydrodynamic effects in solids[1], his landmark 1995 paper [29] provided one of the first indications that
collective memory effects are particularly striking in two-dimensional (2D)
systems. These results attracted considerable attention recently as the 2D systems are at the center of ongoing efforts to achieve electron hydrodynamics [22, 23, 24, 25, 26]. The properties of 2D systems lie somewhere between those of 3D and 1D systems and are sharply distinct from both.
For 3D systems, the Fermi-liquid theory confirms Boltzmann’s short-time memory picture with the onset of hydrodynamics occurring after $\sim 1$ quasiparticle collisions [27].
The 1D systems feature manifestly non-Boltzmann behavior, described by the Luttinger-liquid theory that predicts integrable non-ergodic behavior that extends to arbitrarily large times and distances [28].
The unique behavior in 2D Fermi systems, which is
due to the dominant role of head-on collisions
[29, 30, 31], deviates strongly from that in both 3D and 1D systems.
The new long-lived excitations and collective modes are associated with the odd-$m$ harmonics of the Fermi surface (FS) modulation. Our microscopic analysis of quasiparticle scattering at a circular FS predicts quenching of the Landau $T^{2}$ damping for such modes. Low-lying excitations in this system are the FS modulations evolving in space and time as
$\delta f({\boldsymbol{p}},{\boldsymbol{x}},t)\sim{\textstyle\sum_{m}}\alpha_{m}(\epsilon,{\boldsymbol{x}},t)\cos m\theta+\beta_{m}(\epsilon,{\boldsymbol{x}},t)\sin m\theta$,
where $\theta$ is the angle parameterizing the FS. The microscopic decay rates, illustrated in Fig.1, govern dynamics of spatially-uniform excitations, $\alpha_{m}$, $\beta_{m}\sim e^{-\gamma_{m}t}$. As illustrated in Fig.1, at low temperatures $T\ll T_{F}$ the lifetimes of these modes greatly exceed the even-$m$ ones and show strong departure from conventional scaling. The decay rates in Fig.1 are obtained by a direct calculation that treats quasiparticle scattering exactly, using a method that does not rely on the small parameter $T/T_{F}\ll 1$. The odd-$m$ decay rates display scaling $\gamma\sim T^{\alpha}$ with super-Fermi-liquid exponents $\alpha>2$. In our analysis we find $\alpha$ values close to $4$, i.e. the odd-$m$ rates are suppressed strongly compared to the even-$m$ rates, $\gamma_{\rm odd}/\gamma_{\rm even}\sim(T/T_{F})^{2}$.
This defines a new hierarchy of lifetimes for collective modes, leading to hydrodynamics with non-newtonian (scale-dependent) viscosity. To illustrate these properties we retain the two harmonics with longest lifetimes, $m=1$ and $3$, and suppress those with other $m$. The velocity mode $m=1$ is genuinely undamped owing to momentum conservation in two-body collisions, whereas the $m=3$ mode represents the longest-lived odd-parity excitation (see Fig.1).
Here we focus on the shear modes relevant for hydrodynamics, $\beta_{1}({\boldsymbol{x}},t)\sin\theta+\beta_{3}({\boldsymbol{x}},t)\sin 3\theta$,
with
$\theta$ measured from ${\boldsymbol{k}}$ direction, with a harmonic dependence $\beta_{m}({\boldsymbol{x}},t)\sim e^{i{\boldsymbol{k}}{\boldsymbol{x}}-i\omega t}$.
Integrating out the fast-relaxing modes with $m\neq 1,3$, as discussed below,
yields a pair of coupled diffusion equations
$$\partial_{t}\beta_{1}=-\nu k^{2}\beta_{1}-\nu k^{2}\beta_{3},\quad\partial_{t}\beta_{3}=-(2\nu k^{2}+\gamma_{3})\beta_{3}-\nu k^{2}\beta_{1}$$
(1)
where $\nu=v_{F}^{2}/4\gamma_{2}$ is the ordinary ‘newtonian’ viscosity. These relations describe behavior at times and distances
$$\omega\ {\rm and}\ kv_{F}\ll\gamma_{m\neq 1,3}.$$
(2)
Since $\gamma_{3}\ll\gamma_{m\neq 1,3}$, two different hydrodynamic regimes arise: the short-time regime $\omega,\ \nu k^{2}\gg\gamma_{3}$ and the long-time regime $\omega,\ \nu k^{2}\ll\gamma_{3}$. In the first regime, i.e. at relatively short times $t\gamma_{3}\ll 1$, these coupled modes yield two separate viscous modes found by diagonalizing the $2\times 2$ problem, with viscosity taking universal values
$$\nu_{1}=\nu(3+\sqrt{5})/2,\quad\nu_{2}=\nu(3-\sqrt{5})/2$$
(3)
with a large ratio $\nu_{1}/\nu_{2}=(3+\sqrt{5})/(3-\sqrt{5})\approx 6.9$.
To gain insight in the behavior at long times, for which the memory effects matter,
we eliminate the $\beta_{3}$ variable in Eq.(1) and write the dynamics in a closed form for the velocity mode. This yields Stokes equation with a scale-dependent (non-newtonian) viscosity:
$$-i\omega\beta_{1}=\Xi_{k,\omega}k^{2}\beta_{1},\quad\Xi_{k,\omega}=\nu\frac{\nu k^{2}+\gamma_{3}-i\omega}{2\nu k^{2}+\gamma_{3}-i\omega}.$$
(4)
As expected, the newtonian viscosity $\Xi=\nu$ is recovered in the long-wavelength limit $k\to 0$, whereas for $\nu k^{2}\gg\gamma_{3}$ Eq.(4) yields the two viscous modes found in Eq.(3).
The conventional (newtonian) hydrodynamics is restored at distances such that $\nu k^{2}$ is smaller than $\gamma_{3}$, i.e.
$$L>v_{F}/\sqrt{2\gamma_{2}\gamma_{3}}.$$
(5)
These lengthscales are reached by a particle only after
$$N=\sqrt{\frac{\gamma_{2}}{2\gamma_{3}}}\sim\frac{T_{F}}{T}\gg 1$$
(6)
collisions. This is consistent with the picture discussed above: particles collide at a normal Landau $T^{2}$ rate (here estimated as $\gamma_{2}$) and diffuse in space with the diffusivity $\nu$. However, since collisions are collinear, only the even part of the angular distribution relaxes at times $\sim\gamma_{2}^{-1}$, whereas the odd part remains unrelaxed producing memory effects manifested in the second hydrodynamic mode. The number $N$ of collisions required for the memory of the microstate to be erased diverges at $T\to 0$, indicating a sharply non-Boltzmann behavior manifested in the non-newtonian hydrodynamics due to long-lived modes.
Turning to the analysis, we consider electron momentum distribution obeying kinetic equation linearized near the equilibrium state,
$$\left(\partial_{t}+{\boldsymbol{v}}\cdot\nabla\right)\delta f-I[\delta f]=-e{\boldsymbol{E}}(r)\cdot\frac{\partial f_{0}}{\partial{\boldsymbol{p}}}.$$
(7)
Here $I$ is a linearized collision integral, ${\boldsymbol{E}}$ is the electric field, and $\delta f(p,r,t)=f-f_{0}$ describes a state weakly perturbed away from the Fermi-Dirac equilibrium state $f_{0}(p)=1/(e^{\beta(\epsilon(p)-\mu)}+1)$.
Carrying out the Fourier transform gives
$$\delta f(p,r,t)=\sum_{k,\omega}\delta f(p)e^{i{\boldsymbol{k}}{\boldsymbol{r}}-i\omega t},$$
where $\omega$ and ${\boldsymbol{k}}$ are the perturbation frequency and wavenumber, and the Fourier harmonics $\delta f(p)$ are in general functions of $\omega$ and ${\boldsymbol{k}}$ determined as discussed below.
Taking into account that the momentum distribution $\delta f(p)$ is concentrated near the Fermi surface, we expand it in the angular harmonics basis:
$$\delta f(p)=-\frac{\partial f_{0}}{\partial p}\sum_{m}e^{-im\theta}\delta f_{m}.$$
(8)
where $\theta$ is the angle parameterizing the Fermi surface measured from the direction of ${\boldsymbol{k}}$. The angular harmonics diagonalize the collision integral $I[\delta f_{m}]=-\gamma_{m}\,\delta f_{m}$.
The electric field ${\boldsymbol{E}}(r)$ in Eq.(7) can describe several different settings. It can be either applied externally, as required, for instance, for calculating conductivity, or can be internal to the system, describing the electron-electron interactions in the Fermi liquid. Focusing on the latter case, here we consider the electric field induced by the density perturbed away from equilibrium. As we will see, in this case the behavior is quite different for the transverse and longitudinal modes. For longitudinal modes, current has a finite divergence that leads to a density perturbation describing collective plasma waves. For transverse modes, current has zero divergence and density remains constant and equal to that in equilibrium, leading to hydrodynamic modes.
This can be seen most easily by considering Fourier harmonics, this gives
$${\boldsymbol{E}}(k)=-i{\boldsymbol{k}}U({\boldsymbol{k}})\sum_{p}\delta f(p),\quad\quad U({\boldsymbol{k}})=\frac{2\pi e^{2}}{\kappa|{\boldsymbol{k}}|},$$
(9)
where $\kappa$ is the dielectric constant. Plugging ${\boldsymbol{E}}(k)$ in the Eq. (7) yields a system of coupled equations for different harmonics $\delta f_{m}$. This problem describe both longitudinal and transverse modes, where $\delta f_{m}=\delta f_{-m}$ and $\delta f_{m}=-\delta f_{-m}$, respectively. Here we focus on the transverse modes (therefore, assume that $\delta f_{m}=-\delta f_{-m}$) and show that the hybridization of the $m=\pm 1$ and $m=\pm 3$ harmonics results in modes with a scale-dependent (non-newtonian) viscosity. Since the condition for transverse modes, $\delta f_{m}=-\delta f_{-m}$, implies $\delta f_{0}=0$, the transverse modes are not accompanied by charge buildup. Accordingly, in this case the electric field drops out. The electric field term, however, is important for the longitudinal modes such as plasmons.
With this in mind, we will write the kinetic equation in a general form which includes the ${\boldsymbol{E}}$ term:
$$\displaystyle\left(\gamma_{m}-i\omega\right)\delta f_{m}+\frac{ikv}{2}\left(\delta f_{m-1}+\delta f_{m+1}\right)$$
$$\displaystyle=-\frac{ievk}{2m}\frac{\partial f_{0}}{\partial\varepsilon}U(k)\delta f_{0}\left(\delta_{m,1}+\delta_{m,-1}\right),$$
(10)
where mode decay rates $\gamma_{m}$ obey relations (i) $\gamma_{0}=0$ due to particle conservation, (ii) $\gamma_{m}=\gamma$ for $m\neq 1,3$, and $\gamma_{3}=\gamma^{\prime}\ll\gamma$ for long-lived excitations. At sufficiently long times such that $\omega\ll\gamma$, the even-$m$ harmonics are mostly relaxed, giving an expression that links the even harmonics to the odd harmonics:
$$\delta f_{2m}=-\frac{ikv}{2\gamma}\left(\delta f_{2m+1}+\delta f_{2m-1}\right).$$
(11)
For the odd-$m$ harmonics we write separately the expressions for the $m=\pm 1$ and $m=\pm 3$ harmonics,
$$\displaystyle\left(\gamma_{p}-i\omega\right)\delta f_{\pm 1}+\frac{ikv}{2}\left(\delta f_{0}+\delta f_{\pm 2}\right)=-\frac{ievk}{2m}U(k)\frac{\partial f_{0}}{\partial\varepsilon}\delta f_{0}$$
(12)
$$\displaystyle(\gamma^{\prime}-i\omega)\delta f_{\pm 3}+\frac{ikv}{2}\left(\delta f_{\pm 2}+\delta f_{\pm 4}\right)=0$$
(13)
The above equations are true for both the transverse and longitudinal modes.
From now on we specialize to transverse modes. In this case, as noted above,
density remains unperturbed, $\delta f_{0}=0$, and therefore the electric field ${\boldsymbol{E}}$ induced by density variation, Eq.(9), vanishes. Introducing notation $\nu=v^{2}/4\gamma$ and plugging Eqs.(11) and (13) into Eq. (12), and solving for the velocity mode $m=1$, we obtain obtain a collective mode dispersion relation
$$\gamma_{p}-i\omega+\nu k^{2}-\frac{\left(\nu k^{2}\right)^{2}}{\gamma^{\prime}-i\omega+2\nu k^{2}}=0.$$
(14)
This relation, compared to the one defined by the Stokes equation, $-i\omega+\gamma_{p}+\nu k^{2}=0$, indicates that our system is described by a scale-dependent viscosity,
$$\Xi_{\omega,k}=\nu\frac{\gamma^{\prime}-i\omega+\nu k^{2}}{\gamma^{\prime}-i\omega+2\nu k^{2}},$$
which agrees with Eq.(4) in the limit $\gamma^{\prime}=0$.
In the limit $\gamma_{p}=0$ and $|\omega|\ll\gamma^{\prime}$, this predicts a non-newtonian viscous mode of the form $i\omega=\tilde{\nu}k^{2}$, where
$$\tilde{\nu}=\nu\left(1-\frac{\nu k^{2}}{\gamma^{\prime}}\right),$$
(15)
and one slowly-decaying mode with $i\omega\sim\gamma^{\prime}$.
In the limit $\gamma^{\prime}\ll|\omega|\ll\gamma$, we obtain two viscous modes with different viscosities
$$\tilde{\nu}_{1,2}=\frac{3\pm\sqrt{5}}{2}\nu$$
(16)
with a universal ratio $\nu_{1}/\nu_{2}=(3+\sqrt{5})/(3-\sqrt{5})$.
As a sanity check, in the limit when the long-lived mode becomes short-lived, $\gamma^{\prime}\approx\gamma$, only one (the ordinary newtonian) viscous mode survives.
Since the number of the viscous modes changes with the length scale, it is instructive to consider the $k$ dependence without making any simplifying approximations.
Eq.(14) yields a quadratic equation that can be solved to obtain two distinct dispersing modes:
$$i\omega=\frac{1}{2}\left(\gamma_{p}+\gamma^{\prime}+3\nu k^{2}\right)\pm\sqrt{\left(\nu k^{2}\right)^{2}+\frac{1}{4}\left(\gamma^{\prime}-\gamma_{p}+\nu k^{2}\right)^{2}}.$$
(17)
From this result we can recover all asymptotic regimes discussed above. In the long-wavelength limit $\nu k^{2}\ll\gamma^{\prime}$, one of the modes indeed remains damped, with $i\omega$ proportional to $\gamma^{\prime}$ (assuming $\gamma_{p}\ll\gamma^{\prime}$). The second mode in this limit is viscous and has viscosity described by Eq.(15). The viscous mode in Eq.(15) upon varying $k$ transforms into the mode $\tilde{\nu}_{2}$ in Eq.(16) with a minus sign.
In case when more then one long-lived mode is present, determining the change of viscosity with scale becomes more complicated. We consider a toy model, where $n\gg 1$ such modes are present simultaneously, for simplicity taking their decay rates to be negligibly small. As above, we eliminate the fast-relaxing even-$m$ harmonics from consideration by substituting Eq. (11) into Eq. (Non-Newtonian hydrodynamic modes in two-dimensional electron fluids), which yields a closed-form system of equations of motion for odd-$m$ excitations. For the first $n$ modes these equations are
$$-i\omega\delta f_{2m+1}=-\nu k^{2}\left(2\delta f_{2m+1}+\delta f_{2m-1}+\delta f_{2m+3}\right)$$
(18)
for $0<m\leq n$.
From the form of the equation it is evident that $n$ hydrodynamic modes are present in this case, since all the solutions that satisfy Eq. (18) have the relaxation rates proportional to $\nu k^{2}$. Determining the spectrum of viscosities in the system therefore is equivalent to diagonalizing an $n\times n$ matrix
$$\begin{bmatrix}1&1&0&\dots&&&\\
1&2&1&&&&\\
0&1&2&&&&\\
\vdots&&&\ddots&&&\\
&&&&2&1&0\\
&&&&1&2&1\\
&&&&0&1&2\\
\end{bmatrix}.$$
(19)
The eigenvectors and eigenvalues for this matrix are readily found, allowing one to
obtain viscosities explicitly as
$$\nu_{j}=2\nu\left[1+\cos\left(\frac{2\pi j}{2n-1}\right)\right]$$
(20)
The expression for $\nu_{j}$ implies that the largest viscosity $\nu_{j}$ converges to a $n$-independent constant as $n$ tends to infinity. In fact, in this limit $\max{\nu_{j}}$ approaches $4\nu$. In contrast, the smallest viscosity scales inversely with $n^{2}$: $\min\nu_{i}\sim\nu/n^{2}$. Therefore all $n$ long-lived modes will be present, but only a subset of those, having exceptionally low viscosities, will dominate the hydrodynamic behavior. We note that a more realistic model can be constructed by letting
different odd-$m$ harmonics to have different decay rates, and their number $n$ to scale with temperature. This generalized toy model can be used to understand
the interplay between $\gamma^{\prime}/\gamma$ and $n$.
The conventional hydrodynamics corresponds to length scales $\nu k^{2}\ll\gamma^{\prime}$. From Eq. (17) it follows that the hydrodynamic mode frequencies are of the order $\nu k^{2}$. Therefore, the time scale of transition between one-mode regime and two-mode regime corresponds to $\nu k^{2}\sim\gamma^{\prime}$. This defines a characteristic length scale $L=1/k\sim v_{F}/\sqrt{\gamma\gamma^{\prime}}$. As a reminder, $\gamma^{\prime}$ corresponds to $\gamma_{3}$ in the main text.
This provides a simple illustration of a relation between the long-lived excitations and novel hydrodynamic effects, such as scale-dependent viscosity and, at intermediate times and distances, multiple viscous modes with different viscosity values. It is all but natural to expect that this behavior persists when more long-lived excitation modes are present in the system.
The effects of the long-lived modes manifest in additional hydrodynamic modes in the infinite material. In the material with the diffusive boundaries these extra hydrodynamic modes express themselves through changing the effective conductivity of the sample.
As an example, we will show how the new modes manifest themselves in a stripe geometry. We consider an infinite length stripe with width $L$. To compute the electrical conductivity governed by Eq. (Non-Newtonian hydrodynamic modes in two-dimensional electron fluids), we add extra terms that correspond to external electric field coupling to $\delta f_{m}$ modes. External electric field ${\boldsymbol{E}}_{\mathrm{ext}}$ with components $E_{\parallel}$ along ${\boldsymbol{k}}$ and $E_{\perp}$ perpendicular to ${\boldsymbol{k}}$. The presence of these fields results in the extra term in the RHS of the form
$$-\frac{ev_{F}}{2}\frac{\partial f_{0}}{\partial\varepsilon}\left[E_{\parallel}\left(\delta_{m,1}+\delta_{m,-1}\right)+iE_{\perp}\left(\delta_{n,-1}-\delta_{n,+1}\right)\right].$$
For DC responce the current cannot vary along the stripe, and therefore varies only in the perpendicular direction due to translational invariance along the stripe. Therefore, present in the solution modes will all have ${\boldsymbol{k}}$ perpendicular to the stripe and we can set $E_{\parallel}=0$, and $E_{\perp}=E_{0}$. Along with the transverse mode condition, equations Eq. (Non-Newtonian hydrodynamic modes in two-dimensional electron fluids) allows to express the conductivity as
$$\sigma_{\perp}(k)=\frac{e^{2}v_{F}^{2}m}{4\pi\hbar^{2}}\frac{1}{\gamma_{p}+\frac{v_{F}^{2}k^{2}/4}{\gamma_{2}+\frac{v_{F}^{2}k^{2}/4}{\gamma_{3}+...}}}.$$
(21)
In the expression above we set $\omega=0$ for the DC responce. if we consider again the previously defined toy model with first $n$ starting from $\gamma_{3}$ odd relaxation rates to be all $\gamma^{\prime}\ll\gamma$, where $\gamma$ are all the other relaxation rates’ values.
For $i>2n1$, the tail of the infinite fraction is resummable exactly and yields
$$\frac{v_{F}^{2}k^{2}/4}{\gamma+\frac{v_{F}^{2}k^{2}/4}{\gamma+...}}=\frac{\gamma}{2}\left(\sqrt{1+\frac{v_{F}^{2}k^{2}}{\gamma^{2}}}-1\right)\approx\frac{v_{F}^{2}k^{2}}{4\gamma}.$$
(22)
In the last line we asumed that $v_{F}k\ll\gamma$, which corresponds to the physical regime away from the ballistic transport. We are the most interested in exploring the transport regime $\gamma^{\prime}\ll\nu k^{2}\ll\gamma$ that is transient between normal hydrodynamic an ballistic regime to see the effects of additional viscous modes.
Expression for $\sigma_{\perp}(k)$ along with substitution from Eq. (22) greatly simplifies in the regime of interest, where $\gamma^{\prime}$ terms can be neglected in Eq. (21):
$$\sigma_{\perp}(k)=\frac{e^{2}v_{F}^{2}m}{4\pi\hbar^{2}}\frac{n+1}{\nu k^{2}}.$$
(23)
The expression above clearly shows that the extra long-lived modes manifest themselves in conductivity by carrying the same amount of charge as the conventional hydrodynamic mode.
We use phenomenological method defined in [32] to construct a connection between $\sigma_{\perp}(k)$ and the actual effective conductivity of a stripe. The method is based on introducing an extra field source such that it fixes the current on the boundary of the stripe to be 0. Since our expression for $\sigma_{\perp}(k)$ in Eq. (23) has the same $k$ scaling dependence as a normal hydrodynamic limit, the results of the work [32] can be directly applied in the case of our interest and lead to a current density $j(x)$ in the stripe to have a Poiseuille-like flow:
$$j(x)=\sigma_{\mathrm{eff}}E_{0}\frac{x}{L}\left(1-\frac{x}{L}\right),\quad\sigma_{\mathrm{eff}}=\frac{\gamma e^{2}mL^{2}}{12\pi\hbar^{2}}(n+1).$$
(24)
where $x$ is a coordinate perpendicular to the stripe with stripe borders located at $x=0$ and $x=L$.
In the absence of odd long-lived modes, the conventional result for hydrodynamic regime can be extracted by taking $n=0$.
We will denote that effective conductivity as $\sigma_{H}\equiv\sigma_{\mathrm{eff}}(n=0)$. The amount of long-lived modes grows as temperature of the system becomes smaller as can be seen from Fig. 1. According to the results of [33], the scaling of $\gamma_{m}$ for even and odd $m$ is
$$\gamma_{2m}\sim\frac{T^{2}}{T_{F}^{2}}\ln m,\quad\gamma_{2m+1}\sim\frac{T^{4}}{T_{F}^{4}}m^{4}\ln m.$$
(25)
As we can see from Fig. 1, the odd modes initially exhibit $T^{2}$ behavior, and smaller $m$ modes start exhibiting $T^{4}$ behavior earlier then higher $m$ modes. Therefore, for $m_{\mathrm{max}}\sim\sqrt{T_{F}/T}$ the odd mode will be of the order of the corresponding to it even mode. Therefore, the amount of long-lived modes $n\sim\sqrt{T_{F}/T}$ grows as a power law with the inverse temperature. This claim is also supported by the low-temperature regime collision integral diagonalization in [35] that repeats the calculation from Ref. [34]
with higher precision.
Therefore, in the transient regime between the conventional hydrodynamic and ballistic transport regimes shows the presence of extra long-lived hydrodynamic modes. These modes manifest themselves through various geometry dependent effects. One of the effects shown above is a modification of the temperature scaling of the conductivity of an infinite stripe. To summarize, the effective conductivity $\sigma_{\mathrm{eff}}$ is enhanced in comparison to a hydrodynamic regime by
$$\sigma_{\mathrm{eff}}\sim\sqrt{\frac{T_{F}}{T}}\sigma_{H}$$
(26)
for the regime where the width of the stripe $L$ satisfies
$$\frac{T}{T_{F}}\ll\frac{v_{F}}{2\gamma L}\ll 1.$$
(27)
In summary, restricted phase space renders quasiparticle scattering a highly collinear process even when the microscopic interactions have a weak angular dependence. The unusual kinetics originating in this regime, is relevant for a variety of 2D systems, in particular those where small carrier density and small kinetic energy make electron-electron collisions a dominant scattering mechanism that overwhelms other carrier relaxation pathways. These long-lived excitations manifest themselves through new hydrodynamic modes with non-newtonian (scale-dependent) viscosity. This leads to
multiple viscous modes impossible in conventional fluids, providing a clear testable signature of the unique behavior originating from long-lived excitations in electron fluids.
This work was supported by the Science and Technology Center for Integrated Quantum Materials, NSF Grant No. DMR1231319; Army Research Office Grant W911NF-18-1-0116; US-Israel Binational Science Foundation Grant No. 2018033; and Bose Foundation Research fellowship.
References
[1]
R. N. Gurzhi, Sov. Phys. Usp. 11, 255 (1968).
DOI 10.1070/PU1968v011n02ABEH003815
[2]
Müller, M., Schmalian, J., Fritz, L. Graphene: a nearly perfect fluid. Phys. Rev. Lett. 103,
2–5 (2009).
[3]
A. Tomadin, G. Vignale, M. Polini,
A Corbino disk viscometer for 2D quantum electron liquids
Phys. Rev. Lett. 113, 235901 (2014).
[4]
A. Principi, G. Vignale, M. Carrega, M. Polini,
Bulk and shear viscosities of the two-dimensional electron liquid in a doped graphene sheet
Phys. Rev. B 93, 125410 (2016)
[5]
Scaffidi, T., Nandi, N., Schmidt, B., Mackenzie, A. P., Moore, J. E. Hydrodynamic electron
flow and Hall viscosity. Phys. Rev. Lett. 118, 226601 (2017).
[6]
A. Lucas, K. C. Fong,
Hydrodynamics of electrons in graphene,
J. Phys.: Condens. Matter 30 053001 (2018).
[7]
Guerrero-Becerra, K. A., Pellegrino, F. M. D., Polini, M. Magnetic hallmarks of viscous
electron flow in graphene. Phys. Rev. B 99, 041407 (2019).
[8]
Narozhny, B. N., Sch’́utt, M. Magnetohydrodynamics in graphene: Shear and Hall
viscosities. Phys. Rev. B 100, 035125 (2019).
[9]
Alekseev, P. S., Dmitriev, A. P. Viscosity of two-dimensional electrons. Phys. Rev. B 102,
241409 (2020)
[10]
Toshio, R., Takasan, K., Kawakami, N. Anomalous hydrodynamic transport in interacting
noncentrosymmetric metals. Phys. Rev. Res. 2, 032021 (2020).
[11]
Narozhny, B. N., Gornyi, I. V., Titov, M. Hydrodynamic collective modes in graphene.
Phys. Rev. B 103, 115402 (2021).
[12]
Hasdeo, E. H., Ekström, J., Idrisov, E. G., Schmidt, T. L. Electron hydrodynamics of
two-dimensional anomalous Hall materials. Phys. Rev. B 103, 125106 (2021).
[13]
M. Qi, A. Lucas,
Distinguishing viscous, ballistic, and diffusive current flows in anisotropic metals,
Phys. Rev. B 104 (19), 195106 (2021)
[14]
C. Q. Cook, A. Lucas, Viscometry of electron fluids from symmetry,
Phys. Rev. Lett. 127 (17), 176603 (2021)
[15]
D. Valentinis, J. Zaanen, D. van der Marel
Propagation of shear stress in strongly interacting metallic Fermi liquids enhances transmission of terahertz radiation
Sci. Rep. 11, 7105 (2021)
[16]
D. Valentinis,
Optical signatures of shear collective modes in strongly interacting Fermi liquids
Phys. Rev. Research 3, 023076 (2021)
[17]
J. Hofmann and S. Das Sarma, Collective modes in interacting two-dimensional tomographic Fermi liquids,
Phys. Rev. B 106, 205412 (2022)
[18]
H. Guo, E. Ilseven, G. Falkovich, and L. Levitov, Higher-than-ballistic conduction of viscous electron flows,
Proc. Natl. Acad. Sci. U.S.A. 114, 3068 (2017).
[19]
A. V. Shytov, J. F. Kong, G. Falkovich, and L. S. Levitov, Particle Collisions and Negative Nonlocal Response of Ballistic Electrons,
Phys. Rev. Lett. 121, 176805 (2018).
[20]
K. G. Nazaryan, L. S. Levitov, Robustness of vorticity in electron fluids,
arXiv:2111.09878.
[21]
R. L. Liboff, Kinetic Theory: Classical, Quantum, and Relativistic Descriptions (Springer-Verlag, New York, 2003)
[22]
J. A. Sulpizio, L. Ella, A. Rozen, J. Birkbeck, D. J. Perello, D. Dutta, M. Ben-Shalom, T. Taniguchi, K. Watanabe, T. Holder, R. Queiroz, A. Principi, A. Stern, T. Scaffidi, A. K. Geim, S. Ilani,
Visualizing Poiseuille flow of hydrodynamic electrons
Nature 576, 75-79 (2019)
[23]
M. J. H. Ku, T. X. Zhou, Q. Li, Y. J. Shin, J. K. Shi, C. Burch, L. E. Anderson, A. T. Pierce, Y. Xie, A. Hamo, U. Vool, H. Zhang, F. Casola, T. Taniguchi, K. Watanabe, M. M. Fogler, P. Kim, A. Yacoby, R. L. Walsworth,
Imaging viscous flow of the Dirac fluid in graphene
Nature 583, 537-541 (2020)
[24]
B. A. Braem, F. M. D. Pellegrino, A. Principi, M. Roosli, C. Gold, S. Hennel, J. V. Koski, M. Berl, W. Dietsche, W. Wegscheider, M. Polini, T. Ihn, and K. Ensslin,
Scanning gate microscopy in a viscous electron fluid
Phys. Rev. B 98, 241304(R) (2018).
[25]
U. Vool, A. Hamo, G. Varnavides, Y. Wang, T. X. Zhou, N. Kumar, Y. Dovzhenko, Z. Qiu, C. A. C. Garcia, A. T. Pierce, J. Gooth, P. Anikeeva, C. Felser, P. Narang, A. Yacoby,
Imaging phonon-mediated hydrodynamic flow in WTe2
Nature Physics 17, 1216-1220 (2021).
[26]
Direct observation of vortices in an electron fluid
A. Aharon-Steinberg, T. V’́olkl, A. Kaplan, A. K. Pariari, I. Roy, T. Holder, Y. Wolf, A. Y. Meltzer, Y. Myasoedov, M. E. Huber, B. Yan, G. Falkovich, L. S. Levitov, M. H’́ucker, E. Zeldov
Nature 607, 74–80 (2022).
[27]
G. Baym, C. Pethick,
Landau Fermi‐Liquid Theory: Concepts and Applications
(Wiley, 1991)
[28]
T. Giamarchi, Quantum Physics in One Dimension, Clarendon Press, Oxford, 2004.
[29]
R. N. Gurzhi, A. N. Kalinenko, and A. I. Kopeliovich, Electron-Electron Collisions and a New Hydrodynamic Effect in Two-Dimensional Electron Gas, Phys. Rev. Lett. 74, 3872 (1995)
[30]
H. Buhmann, L. W. Molenkamp, 1D diffusion: a
novel transport regime in narrow 2DEG channels,
Physica E 12, 715-718 (2002)
[31]
P. J. Ledwith, H. Guo, L. Levitov,
Angular Superdiffusion and Directional Memory in Two-Dimensional Electron Fluids,
arXiv:1708.01915
[32]
P. Ledwith, H. Guo, A. Shytov, L. Levitov
Tomographic Dynamics and Scale-Dependent Viscosity in 2D Electron
Systems, Phys. Rev. Lett. 123, 116601 (2019)
[33]
P. J. Ledwith, H. Guo, L. Levitov, The Hierarchy of Excitation Lifetimes in Two-Dimensional Fermi Gases, Ann. Phys. 411, 167913 (2019)
[34]
S. Kryhin, L. Levitov,
Collinear scattering and long-lived excitations in two-dimensional
electron fluids,
arXiv:2112.05076
[35]
Hofmann, Johannes and Gran, Ulf,
Anomalously long lifetimes in two-dimensional Fermi liquids,
arXiv preprint arXiv:2210.16300 (2022) |
The Banana project. IV. Two aligned stellar rotation axes in the young
eccentric binary system EP Crucis: primordial orientation and tidal alignment${}^{\star}$
Simon Albrecht11affiliation: Department of Physics, and Kavli Institute for
Astrophysics and Space Research, Massachusetts Institute of
Technology, Cambridge, MA 02139, USA ,
Johny Setiawan22affiliation: Max-Planck-Institut für Astronomie,
Königstuhl 17, 69117 Heidelberg, Germany 33affiliation: Embassy of the Republic of Indonesia, Lehrter Str. 16-17, 10557 Berlin, Germany ,
Guillermo Torres44affiliation: Harvard-Smithsonian Center for Astrophysics,
Cambridge, MA 02138, USA ,
Daniel C. Fabrycky55affiliation: Department of Astronomy and Astrophysics, University
of California, Santa Cruz, Santa Cruz, CA 95064, USA ,
Joshua N. Winn11affiliation: Department of Physics, and Kavli Institute for
Astrophysics and Space Research, Massachusetts Institute of
Technology, Cambridge, MA 02139, USA ,
Abstract
With observations of the EP Cru system, we continue our series
of measurements of spin-orbit angles in eclipsing binary
star systems, the BANANA project (Binaries Are Not Always Neatly
Aligned). We find a close alignment between the sky projections of
the rotational and orbital angular momentum vectors for both stars
($\beta_{\rm p}=-1.8\pm 1.6^{\circ}$ and $|\beta_{\rm s}|<17^{\circ}$). We also
derive precise absolute dimensions and stellar ages for this system.
The EP Cru and DI Her systems provide an interesting comparison: they
have similar stellar types and orbital properties, but DI Her is
younger and has major spin-orbit misalignments, raising the question
of whether EP Cru also had a large misalignment at an earlier phase of
evolution. We show that tidal dissipation is an unlikely
explanation for the good alignment observed today, because
realignment happens on the same timescale as spin-orbit
synchronization, and the stars in EP Cru are far from
syncrhonization (they are spinning 9 times too quickly). Therefore
it seems that some binaries form with aligned axes, while other
superficially similar binaries are formed with misaligned axes.
Subject headings:stars: kinematics and
dynamics –stars: early-type – stars: rotation – stars:
formation – binaries: eclipsing – stars: individual (EP Crucis)
– stars: individual (DI Herculis) – techniques: spectroscopic
$\star$$\star$affiliationtext: Based on observations made with ESOs $2.2$ m
Telescopes at the La Silla Paranal Observatory under programme ID
084.C-1008 ($12.5$%) and under MPIA guaranteed time ($87.5$%).
1. Introduction
One might expect star-planet and close binary star systems to have
well-aligned orbital and rotational angular momenta, since they
originate from the same portion of a molecular cloud. However, there
are also reasons to expect misaligned systems. Star formation is a
chaotic process, with accretion from different directions at different
times possibly leading to misalignment between the stellar and orbit
rotation axes (e.g. Bate et al., 2010; Thies et al., 2011). There are also
processes that could alter the stellar and orbital spin directions
after their formation. For example a third body orbiting a close pair
on a highly inclined orbit can introduce large oscillations in the
orbital inclination and eccentricity of the close pair
(Kozai, 1962), thereby introducing large angles between the
stellar spins and orbital angular momentum of the close pair. Close
encounters and possible exchange of members in binary systems
(e.g. Gualandris et al., 2004) would leave, among other clues, a
fingerprint in the form of misalignment between the components. Tidal
forces will over time erase these clues, because dissipation will tend
to bring the axes into alignment while also synchronizing the
rotational and orbital periods
(e.g. Zahn, 1977; Hut, 1981; Eggleton & Kiseleva-Eggleton, 2001). Thus the degree of
alignment between the stellar rotation axes and the orbital axis
depends on its particular history of formation and evolution.
Therefore measurements of stellar obliquities – the angle between
stellar equator and orbital plane – allow us to test theories of
formation and evolution in close star-planet and star-star systems.
For example the formation of star-star systems with orbital distances
of only a few stellar radii is not completely understood. It seems
unlikely that the stars formed at these orbital distances because they
would have overlapped during their pre-main sequence phase, when they
had larger sizes. Therefore the orbital distance likely decreased
after formation. A possible mechanism is KCTF – Kozai Cycles with
Tidal Friction (Eggleton & Kiseleva-Eggleton, 2001; Fabrycky & Tremaine, 2007), which requires a
third body on a wide orbit around the close pair. Tokovinin et al. (2006)
found that $96\%$ of binary stars with orbital periods less than
$3$ days have a third companion on a wide orbit while only $34\%$ of
binaries with orbital periods larger than $12$ days have a third
companion. Additional evidence for KCTF would be a misalignment
between the stellar spin axes and the orbital spin, assuming that
close binaries have aligned axes at birth, and that tides have not had
enough time to align the spin axes. Thus measurements of stellar
obliquities in close binary systems together with a good understanding
of tidal dissipation in these systems might lead to a better
understanding of binary formation.
For the case of close star-planet (hot-Jupiter) systems such an
approach has been fruitful. Hot-Jupiters are thought to have formed
much further from the star than their current orbital distances,
mainly because not enough material would have been available so close
to the star. Different processes which could have transported the
planet inward would lead to different spin-orbit angles, and indeed
systems with both small and large spin-orbit angles have been found
(see, e.g., Winn et al., 2005; Hébrard et al., 2008; Johnson et al., 2009; Albrecht et al., 2012a; Brown et al., 2012). In addition Winn et al. (2010) and
Albrecht et al. (2012b) presented evidence that all hot-Jupiter systems
once had high obliquities, and that tides are responsible for the
frequently observed low obliquities. This suggests that the inward
migration of hot Jupiters involves changes of the orbital planes of the
planets.
With the BANANA project (Binaries Are Not Always Neatly Aligned) we
aim to get a better understanding of the formation of close binaries
as well as their tidal spin evolution. Here we study the EP Cru
binary system. This is the fourth system which we study as part of the
BANANA project (Albrecht et al., 2007, 2009, 2011, Papers
I–III). We also refer the
reader to Triaud et al. (2012), for a description of a similar project by
other investigators. While most of the stars in our sample are of
early spectral types, their EBLM project focuses on eclipsing systems
harboring low-mass stars.
EP Cru was only recently characterized by Clausen et al. (2007).
Table 1 gives some system parameters. We
selected this system because Clausen et al. (2007) found it to be similar
to DI Her, for which we already found the spin-orbit angles to be
very large (Albrecht et al., 2009). In particular the orbital parameters,
the stellar masses, and the projected stellar rotation speeds ($v\sin i_{\star}$), are similar in these two systems. Here $v$ indicates the
equatorial rotation speed and $i_{\star}$ the inclination of the
stellar rotation axis along the line of sight (LOS). There is one
important difference between the two systems: the age of the stars.
Clausen et al. (2007) estimated an age of $\approx 50$ Myr for the two
stars in the EP Cru system while DI Her is essentially a Zero Age
Main Sequence (ZAMS) system with an estimated age of $4.5\pm 2.5$ Myr
(Claret et al., 2010). Therefore by studying EP Cru we have the
opportunity to learn if according to our current understanding of
binary evolution one system is simply an older version of the other,
or if they had different childhoods altogether.
The plan of this paper is as follows. In the following section we
describe our observations. The analysis of the spectroscopic
observations during eclipses and out of eclipses is presented in
Section 3. We present the results on the absolute
dimensions, the orientations of the stellar rotation axes, and the
derived age in section 4. For the remainder of the
paper we focus on the interpretation of the obliquity measurements
before we summarize our findings in the conclusions.
2. Spectroscopic Observations
We observed the EP Cru system with the FEROS spectrograph
(Kaufer et al., 1999) on the $2.2$ m telescope at ESO’s La Silla
observatory. We obtained 48 observations on multiple nights between
December 2009 and March 2011. Table 2 gives an
observation log. The observations had a typical integration time of
10 min. On each night, ThAr exposures were taken to calculate a
wavelength solution and monitor any changes in the spectrograph. For
all observations we used the MIDAS FEROS package installed on the
observatory computers to reduce the raw 2D CCD images and to obtain
stellar flux as a function of wavelength. The uncertainty in the
wavelength solution, expressed in velocity, is few m s${}^{-1}$, and is
negligible for our purposes. The resulting spectra have a resolution
of $\approx 50\,000$ around $4481$ Å (the wavelength area of the
spectra we analyze). We corrected for the radial-velocity (RV) of the
observatory, performed initial flat fielding with the nightly blaze
function, and flagged and omitted bad pixels.
3. Analysis
In this section we outline the analysis of the spectra with the aim of
deriving absolute dimensions of the system and learning about the
projected obliquities of both stars via measurements of the
Rossiter-McLaughlin (RM) effect, which occurs during eclipses. We
describe which part of spectrum we analyze and briefly introduce the
model to which we compare the data and the algorithm used to extract
system parameters. Our approach for EP Cru is similar to the approach
employed in Papers I–III.
Spectral region
We focus on the Mg ii line at $4481$ Å, as this line is
relatively deep and chiefly broadened by stellar rotation. It is
located in the red wing of the pressure-broadened He i line at
$4471$ Å. While this line might also be included in the analysis
(Albrecht et al., 2011), we decided here to exclude it as there is enough
signal in the Mg ii line and modeling the pressure broadening
in the He i line represents an additional complication. Thus we
fitted a Lorentzian model to the encroaching wing of the He I line and
subtracted it before modeling the Mg ii line. For this fit we
used the spectral regions $4472$ Å – $4476$ Å and
$4486$ Å – $4498$ Å, thereby avoiding the influence of the
Mg ii line. Each spectrum was binned to a resolution of about
12 km s${}^{-1}$, to speed up subsequent computations. Because the
stellar rotation speeds are an order of magnitude larger, there is no
significant loss of information due to the binning; we verified this
by experimenting with higher resolutions.
Model
The measured spectra show absorption lines of both stars in the
system. Before, during, and after eclipses the RVs of both stars are
similar, leading to a substantial overlap of the two absorption lines.
Hence, light emitted from both stars has to be accounted for when
analyzing the RM effect. We used the numerical code from
Albrecht et al. (2007) which simulates the spectra of both stars in a
system.
The stellar disks are discretized with $\sim 30\,000$ pixels in a
Cartesian coordinate system. We assume the stars to be spherical
because they are well separated with rotation speeds much slower than
the breakup velocity; Clausen et al. (2007) estimates an oblateness of
about $0.0008$. We further assume uniform rotation and quadratic
limb darkening. Stellar surface velocity fields are parametrized
adopting the macro-turbulence model by Gray (2005).111Here
we do not need to take the point spread function (PSF) of the
spectrograph into account as our binning ($12$ km s${}^{-1}$) is
larger than the width of the PSF ($6$ km s${}^{-1}$). The
coordinates of both stars projected on the sky are calculated and
light from visible parts of the stellar hemispheres is integrated. The
resulting absorption line kernels are shifted in wavelength
corresponding to the line they represent and weighted according to the
light contribution of the respective star.
Parameter choices
Having the model in place we can now learn about the EP Cru system by
specifying a number of parameters. The Keplerian orbit of the two
stars can be described with the following $6$ parameters: The orbital
period ($P$), a specific time of minimum light during primary eclipse
($T_{\rm min,I}$), the orbital eccentricity ($e$), the argument of
periastron ($\omega$), and the velocity semi-amplitudes of the primary
and secondary stars ($K_{\rm i}$). Here the subscript ’i’ stands for
either ’p’ indicating the primary star or ’s’ indicating the secondary
(slightly less massive) star. In addition, velocity offsets
($\gamma_{\rm i}$) are needed. For $e$ and $\omega$ we use the
stepping parameters $\sqrt{e}\cos\omega$ and $\sqrt{e}\sin\omega$,
as they are less correlated than $e$ and $\omega$ themselves. The
results from the photometric study by Clausen et al. (2007) can be used
to constrain some of these orbital parameters. However the photometry
used by Clausen et al. (2007) was gathered about 20 years ago and the
system is expected to have an apsidal motion period of a few thousand
years. The change in $\omega$ could be of order $1^{\circ}$ over the
last two decades. In addition the apsidal motion is not measured yet
and we cannot calculate it from the known system parameters as it
depends on the true stellar obliquity and not only the sky projection
(Shakura, 1985; Albrecht et al., 2009). Thus we do not use the photometric
values on $T_{\rm minI,}$ and $\omega$ as priors and we only use the
measurements of $P$ and $e$ by Clausen et al. (2007) as prior
constraints. We revisit this subject in section 4.3.
Additional parameters are needed to describe the projected equatorial
rotation speeds ($v\sin i_{\rm i}$), the Gaussian width of the
macro-turbulence ($\zeta_{\rm i}$), and the parameters of greatest
interest for this study, the sky-projected spin-orbit angles
($\beta_{\rm i}$). The angle is defined according to the convention of
Hosokawa (1953).
The photometric character of the eclipses are specified by another set
of parameters: the light ratio between the two stars at the wavelength
of interest ($L_{\rm s}/L_{\rm p}$ at 4480 Å), the quadratic limb
darkening parameters ($u1_{\rm i}$ and $u2_{\rm i}$), the fractional
radii of the stars ($r_{\rm i}$), and the orbital inclination ($i_{o}$),
for which we step in $\cos i_{o}$. For the fractional radii and the
orbital inclination we use prior information from Clausen et al. (2007).
For $L_{\rm s}/L_{\rm p}$ we use their results in the b band. To
constrain the limb darkening parameters we used the
’jktld’222http://www.astro.keele.ac.uk/jkt/codes/jktld.html tool to query
the ATLAS atmospheres (Claret, 2000) and placed a Gaussian prior
on $u1_{\rm i}+u2_{\rm i}$ with a width of 0.1 and held the difference
$u1_{\rm i}-u2_{\rm i}$ fixed at the tabulated value.
An additional parameter is needed for each star to describe the
relative depth of the Mg ii lines. The Mg ii line
consists of a doublet, given the close spacing ($0.2$ Å) we
model it here as single line.
The two components in the EP Cru system are very similar to each
other (see Table 1). We therefore decided to
use the same limb darkening parameters and macro-turbulence
velocities, for both stars, thereby reducing the number of free
parameters to 20. Of these, 7 are further constrained by Gaussian
priors as explained above. Table 3 summarizes all of
the prior constraints.
There is always a small residual uncertainty in the initial
normalization of the spectra. To propagate this into the uncertainty
intervals of the final parameters we added for each of the $48$
observation $3$ free parameters which describe a quadratic function
used to normalize the continuum level. The values of the normalization
parameters were optimized using a separate 3-parameter minimization
for each observation, each time a set of system parameters is
evaluated. This process is similar to the “Hyperplane Least Squares”
method that was described and tested by Bakos et al. (2010).
Parameter estimation
A MCMC code was used to obtain
uncertainty intervals. The chains consisted of $0.5$ million
calculations of $\chi^{2}$. The results reported below are the median
values of the posterior distribution and the uncertainties intervals
are the values which exclude $15.85$ % of the values at each side of
the posterior and encompassing $68.3$ % of all values.
4. Results
The results for the model parameters are given in
Table 3. Figure 1 shows the
spectra in the vicinity of the Mg ii line and the corresponding
model for the out-of-eclipse observations.
Figures 2 and 3 show
the same for the spectra obtained during primary and secondary
eclipses. The apparent radial velocities in the EP Cru system are
shown in Figure 4 as well as a pole-on view of the
orbit.
4.1. Stellar Rotation and Projected Obliquities
The main result of our analysis is that the sky projections of the two
stellar rotation axes $\beta_{\rm p}=-1.8\pm 1.6^{\circ}$ and
$\beta_{\rm s}=-13\pm 4^{\circ}$ indicate close alignment between the
stellar rotation axes and the orbital angular momentum. However while
the value for $\beta_{\rm p}$ is consistent with prefect alignment $\beta_{\rm s}$ seems to indicate a small but significant misalignment. How
robust is this finding of a small misalignment? We note that we have
somewhat lower SNR observations during the secondary eclipse than
during the primary eclipse and fewer observations directly before,
during and after the eclipse (See Table 2, and
Figures 2, 3, and
4). To test the robustness of the result we reran
the MCMC chain with different model assumptions. For example we
constrained the model more, by leaving $\gamma_{\rm i}$ and the line
depths of both stars tied to each other, or we left $\zeta_{\rm i}$
and the limb darkening parameters completely free. We also excluded
some observations to test if a small number of observations are having
a disproportionate influence on the result. For all these runs we
found a negative $\beta_{\rm s}$. The result closest to alignment was
$\beta_{\rm s}=-9\pm 5^{\circ}$. At the same time the result for
$\beta_{\rm p}$ did not change by more than $0.4^{\circ}$ during these
tests. There are, however, two peculiarities about our result for
$\beta_{\rm s}$. The posteriors for all the other parameters have only
a single peak, while the posterior of $\beta_{\rm s}$ has a small (two
orders of magnitude lower) secondary peak at positive angles around
$\beta_{\rm s}=13^{\circ}$ (Figure 5, upper panel). In
addition there is a correlation between $\beta_{\rm s}$ and $r_{\rm p}$ (Figure 5, lower panel). $r_{\rm p}$ is the only
parameter for which we do find a more than $1$–$\sigma$ displacement
between the prior constraints and results (Table 3).
Taken the above mentioned points into consideration we are confident
that $|\beta_{\rm s}|<17^{\circ}$ but we cannot exclude a small
projected obliquity for the secondary star, with the data at hand.
For the projected rotation speeds we find $v\sin i_{\rm p}=141.4\pm 1.2$ km s${}^{-1}$ and $v\sin i_{\rm s}=137.8\pm 1.1$ km s${}^{-1}$. We consider the formal uncertainties
in the $v\sin i_{\rm i}$ to be too low for the following reasons: We
tested different limb darkening laws and found for example that for a
linear limb darkening law the best fitting $v\sin i$ values are lower
by about $4$ km s${}^{-1}$. Also we suspect that our particular choice
of parametrization of the stellar surface velocity fields will
influence the values we find for the projected rotation. We therefore
estimate that a uncertainty of $5$ km s${}^{-1}$ is more realistic and
also indicated that uncertainty in Table 3. As
mentioned above, normalization for each observed spectrum is included
in our routine, hence any uncertainty in normalization is already
incorporated in the formal uncertainty.
We note that the projected rotation speeds are similar to the average
rotation speed for B stars ($v\sin i=130$ km s${}^{-1}$), as analyzed
by Abt et al. (2002). However the stars might have undergone a change of
$v$ due to tidal interactions (Section 5). Therefore we
can not conclude from the similarity of the measured $v\sin i$ to the
expected $v$ that $\sin i$ is close to unity. Nevertheless it seems
unlikely that the stars have large inclinations relative to the line of
sight and at the same time their projected axes on the plane of the
sky are both aligned. In what follows we assume that not only the
projections of the rotation axes are small, but that the axes
themselves are aligned too ($\sin i\approx 1$).
Concerning the values for the macro-turbulence, we expect that the
value we find does not have a simple physical interpretation. This is
because we assume equal brightness of raising and falling material as
well as equal surface coverage of movement tangential and radial to
the stellar surface, both assumptions do not need to be fulfilled in
reality. We did test if there is a strong dependence of the measured
values for projected obliquities on our adopted model for
macro-turbulence, and found none.
4.2. Absolute Dimensions and Age
From the posterior of our MCMC chain we find $K_{\rm p}=102.2\pm 1.5$ km s${}^{-1}$ and $K_{\rm s}=106.2\pm 1.4$ km s${}^{-1}$ in agreement with values from the
literature (Table 3). We also calculated the $K_{\rm i}$ values only using out of eclipse data, making them less
dependent on any assumption included in our eclipse model. With
$K_{\rm p}=101.9\pm 1.5$ km s${}^{-1}$ and $K_{\rm s}=106.2\pm 1.6$ km s${}^{-1}$ we obtain consistent
results.333John Southworth provided us with the 5 out of
eclipse spectra used
in the Clausen et al. (2007) study and we found that these are
consistent with our data set. Because of the potential small change
in the argument of periastron over the last 20 yr they have not been
included in this study.
With the new spectroscopic data we not only obtain precise mass
estimates for both stars but also improve on the absolute radii. This
is because the accurate scaled radii $r_{i}$ obtained via photometry
need to be multiplied by the absolute scale of the system, the
semi-major axis ($a$).
With the new values for the stellar masses, surface gravities ($\log g_{i}$), and the effective temperatures $T_{\rm eff\,i}$ measured by
(Clausen et al., 2007, see also Table 1) we
can estimate the stellar ages using stellar evolution models. Here we
employ the Yonsei-Yale evolutionary tracks
(Yi et al., 2001; Demarque et al., 2004). We find a good fit for solar metallicity
and an age of $57\pm 5$ Myr (Figure 6). Another good check
is the temperature difference between the stars, since the difference
is probably better determined than the absolute temperatures. Indeed
the temperature difference predicted by the models (i.e., the
separation between the evolutionary tracks) is in good agreement with
the temperature difference measured by Clausen et al. (2007).
4.3. Apsidal Motion
Now that the stellar rotation is known we can calculate the expected
apsidal motion in the EP Cru system. We use the apsidal motion
constant $\log(k_{2})=-2.3$ from Claret (2004) for both stars in
the system. We assign an uncertainty of $0.1$ in $\log$ space to this
constant. From the results in Table 4 we can see that
we expect a shift of $\approx 1.6\pm 0.2^{\circ}$ over the last $20$
years (which approximately have elapsed since the photometric
measurements). Most of this shift is expected because of deformation
of the stars by their rotation. That we find a small increase in the
argument of the periastron and a ($\approx 22$ min) earlier primary
eclipse than expected from linear ephemeris seems to indicate apsidal
motion of the order of magnitude as expected. However spectroscopic
data is not very good at determining $\omega$ and we find it difficult
to estimate the significance of the measured value for $\omega$.
Therefore to make a meaningful comparison between the measured and
expected apsidal motion, new photometric eclipse timings should be
undertaken.
5. The Alignment in Context
Having established the absolute dimensions, age, and state of rotation
in the EP Cru system we can now compare EP Cru to its apparently
younger sibling DI Her. In Table 5 we reprint some of
the values from EP Cru. According to these values the two systems are
similar, apart from two characteristics: 1) Their ages, EP Cru is
about an order of magnitude older, 2) EP Cru appears to have aligned
axes which is definitely not the case for DI Her. We would like to
find a picture in which the misalignment in the young DI Her system
can be explained as well as the alignment in the older EP Cru system.
Knowing that the scaled radii are large enough in these systems to
allow for substantial tides, we might suspect that the difference in
spin-orbit alignment is a result of observing these systems at
different stages in their evolution rather than them having two
different formation and evolution paths. The hypothesis would be that
both stars had misaligned axes, and we see EP Cru with aligned axes
only because it is older and tides have had enough time to align the
axes.
The large eccentricities seen in both DI Her and EP Cru is
consistent with this hypothesis, because tides first align and
synchronize rotation and only on a longer timescale do they
circularize the orbit. This is mainly due to the higher amount of
angular momentum stored in the orbital motion compared to the stellar
rotation, and for systems with a low-mass secondary this is not
necessary the case. However another finding makes the hypothesis
difficult to reconcile with current tidal theories. The stars rotate
at $\sim 9$ times the speeds expected for synchronized or
pseudosynchronized states (Table 5). Thus tides have
not yet synchronized the stellar rotation speeds in the EP Cru
system. Formulations of tidal interactions predict that damping of any
significant spin-orbit misalignment should occur on the same time
scale as synchronization of the rotation
(Hut, 1981; Eggleton & Kiseleva-Eggleton, 2001).444The timescales are not exactly
equal, and which is faster depends on the ratio between the orbital
and rotational angular momentum in the equilibrium states. For a
system like EP Cru the timescale for pseudo synchronization is
about twice the timescale for alignment (Hut, 1981). This is
because in these tidal models, a single coefficient describes the
coupling between tides and rotation. When stellar rotation is much
faster than the synchronized value rotation around any axis is damped
by about the same amount. Thus the angle between the overall angular
momentum and stellar spin does not change: only the rotation speed is
reduced. When the stellar rotation around a axis parallel to the
orbital angular momentum approaches the synchronized value than
rotation around this axis couples less to tides. Rotation around any
other axis is still damped by tides, which only ceases when the
rotation around these axes stops. The stellar rotation aligns to the
orbital axis.
To illustrate this point we used the TOPPLE tidal-evolution code
developed by Eggleton & Kiseleva-Eggleton (2001). For this simulation we used the
EP Cru parameters from Table 5 but with initial
obliquities taken from DI Her, and an initial faster stellar rotation
speed at zero-age main sequence. The results are shown in
Figure 7. The stellar obliquities remain large until the
stellar rotation speeds approach synchronization, at which point
obliquities are damped. This suggests that EP Cru had aligned axes
when it was as young as DI Her, implying in turn that DI Her and
EP Cru do not represent different stages of one evolution, but rather
two different evolution paths.
At the moment it is not possible to make more general statements as
only a few measurements of obliquities have been carried out in close
double-star systems. Furthermore most of these have been conducted in
Algol systems which have undergone mass transfer (see Table 1
of Albrecht et al., 2011). Obliquity observations should be carried out in a
variety of systems. Of particular interest would be young systems with
short orbital periods with and without a third star. The systems
should be young to minimize the influence of tides, they should have
orbital periods ranging from few days to few tens of days. Obliquity
measurements in these systems would be helpful in testing predictions
of KCTF and thereby of close binary formation. Measurements of
obliquities in wider systems would probe the length scale over which
the primordial angular momentum was influential. Conducting such
measurements is the aim of the BANANA project.
6. Summary
We have analyzed high resolution spectra of the eclipsing close double
star system EP Cru. We obtained absolute dimensions and
showed that the rotation axes of both stars are aligned with each
other and the orbital rotation ($\beta_{\rm p}=-1.8\pm 1.6^{\circ}$
and $|\beta_{\rm s}|<17^{\circ}$).
EP Cru is similar in its orbital and stellar characteristics to
DI Her. The two exceptions are that DI Her is younger and has two
strongly misaligned stellar rotation axes. We have been unable to show
that both systems represent different stages of one evolution
path. This is because the stars in EP Cru rotate at a few times their
synchronized value and tidal theory predicts that synchronization
occurs around the same time as alignment. Therefore the two systems
likely represent two different formation scenarios rather then two
different evolutionary stages. The sample of close double star systems
for which the obliquities are measured remains small. We plan to
ratify this situation by measuring obliquities in more close double
star systems in the framework of the BANANA project.
The authors are grateful to Peter Eggleton for insightful discussions
on binary evolution and for making his TOPPLE code available to us. We
thank John Southworth for providing us with the spectra used in the
study by Clausen et al. (2007). S.A. acknowledges support during part of
this project by a Rubicon fellowship from the Netherlands Organisation
for Scientific Research (NWO). Work by S.A. and J.N.W. was supported
by NASA Origins award NNX09AB33G and NSF grant no. 1108595. D.C.F.
acknowledges NASA support through Hubble Fellowship grant
HF-51272.01-A, awarded by STScI, operated by AURA under contract NAS
5-26555. This research has made use of the following web resources:
simbad.u-strasbg.fr,
adswww.harvard.edu,arxiv.org, http://arxiv.org
References
Abt et al. (2002)
Abt, H. A., Levato, H., & Grosso, M. 2002, ApJ, 573, 359
Albrecht et al. (2007)
Albrecht, S., Reffert, S., Snellen, I., Quirrenbach, A., & Mitchell,
D. S. 2007, A&A, 474, 565
Albrecht et al. (2009)
Albrecht, S., Reffert, S., Snellen, I. A. G., & Winn, J. N. 2009,
Nature, 461, 373
Albrecht et al. (2012a)
Albrecht, S., Winn, J. N., Butler, R. P., et al. 2012a,
ApJ, 744, 189
Albrecht et al. (2011)
Albrecht, S., Winn, J. N., Carter, J. A., Snellen, I. A. G., & de
Mooij, E. J. W. 2011, ApJ, 726, 68
Albrecht et al. (2012b)
Albrecht, S., Winn, J. N., Johnson, J. A., et al. 2012b,
ApJ, 757, 18
Bakos et al. (2010)
Bakos, G. Á., Torres, G., Pál, A., et al. 2010, ApJ, 710, 1724
Bate et al. (2010)
Bate, M. R., Lodato, G., & Pringle, J. E. 2010, MNRAS, 401, 1505
Brown et al. (2012)
Brown, D. J. A., Cameron, A. C., Anderson, D. R., et al. 2012, MNRAS,
423, 1503
Claret (2000)
Claret, A. 2000, A&A, 363, 1081
Claret (2004)
—. 2004, A&A, 424, 919
Claret et al. (2010)
Claret, A., Torres, G., & Wolf, M. 2010, A&A, 515, A4
Clausen et al. (2007)
Clausen, J. V., Helt, B. E., Giménez, A., et al. 2007, A&A, 461,
1065
Demarque et al. (2004)
Demarque, P., Woo, J.-H., Kim, Y.-C., & Yi, S. K. 2004, ApJS, 155,
667
Eggleton & Kiseleva-Eggleton (2001)
Eggleton, P. P., & Kiseleva-Eggleton, L. 2001, ApJ, 562, 1012
ESA (1997)
ESA. 1997, ESA SP-1200
Fabrycky & Tremaine (2007)
Fabrycky, D., & Tremaine, S. 2007, ApJ, 669, 1298
Gray (2005)
Gray, D. F. 2005, The Observation and Analysis of Stellar Photospheres,
3${}^{\rm rd}$ Ed. (ISBN 0521851866, Cambridge University Press)
Gualandris et al. (2004)
Gualandris, A., Portegies Zwart, S., & Eggleton, P. P. 2004, MNRAS,
350, 615
Hébrard et al. (2008)
Hébrard, G., Bouchy, F., Pont, F., et al. 2008, A&A, 488, 763
Hosokawa (1953)
Hosokawa, Y. 1953, PASJ, 5, 88
Hut (1981)
Hut, P. 1981, A&A, 99, 126
Johnson et al. (2009)
Johnson, J. A., Winn, J. N., Albrecht, S., et al. 2009, PASP, 121,
1104
Kaufer et al. (1999)
Kaufer, A., Stahl, O., Tubbesing, S., et al. 1999, The Messenger, 95, 8
Kozai (1962)
Kozai, Y. 1962, AJ, 67, 591
Shakura (1985)
Shakura, N. I. 1985, Soviet Astronomy Letters, 11, 224
Thies et al. (2011)
Thies, I., Kroupa, P., Goodwin, S. P., Stamatellos, D., & Whitworth,
A. P. 2011, MNRAS, 417, 1817
Tokovinin et al. (2006)
Tokovinin, A., Thomas, S., Sterzik, M., & Udry, S. 2006, A&A, 450,
681
Torres et al. (2010)
Torres, G., Andersen, J., & Giménez, A. 2010, A&A Rev., 18, 67
Triaud et al. (2012)
Triaud, A. H. M. J., Hebb, L., Anderson, D. R., et al. 2012, ArXiv
Winn et al. (2010)
Winn, J. N., Fabrycky, D., Albrecht, S., & Johnson, J. A. 2010, ApJ,
718, L145
Winn et al. (2005)
Winn, J. N., Noyes, R. W., Holman, M. J., et al. 2005, ApJ, 631, 1215
Yi et al. (2001)
Yi, S., Demarque, P., Kim, Y.-C., et al. 2001, ApJS, 136, 417
Zahn (1977)
Zahn, J.-P. 1977, A&A, 57, 383 |
RUNHETC-2002-18
Long-distance asymptotics of spin-spin correlation functions
for the XXZ spin chain
Sergei Lukyanov${}^{a,b}$ and
Véronique Terras${}^{a,c}$
${}^{a}$NHETC, Department of Physics and Astronomy
Rutgers University
Piscataway, NJ 08854, USA
${}^{b}$L.D. Landau Institute for Theoretical Physics
Kosygina 2, Moscow, Russia
${}^{c}$LPMT, UMR 5825 du CNRS
Université Montpellier II,
Montpellier, France
Abstract
We study asymptotic expansions of
spin-spin correlation functions for the XXZ Heisenberg chain in the
critical regime. We use the fact that
the long-distance effects can be described by the Gaussian conformal
field theory.
Comparing exact results for form factors in the XYZ and sine-Gordon
models, we determine correlation amplitudes for the leading and main
sub-leading terms in the asymptotic expansions of spin-spin correlation
functions. We also study the isotropic (XXX) limit of these expansions.
1 Introduction
In the domain of two-dimensional integrable models,
it is, in general, still a challenging
problem to compute
correlation functions in the form of compact and manageable expressions.
For lattice systems,
a few methods of computation have been developed
[1, 2, 3, 4]: in particular, it is possible
in some cases to obtain exact integral representations
of correlation functions.
However it is still quite difficult to analyse those
expressions, and especially to extract their long-distance asymptotic
behavior.
On the other hand, at the critical point,
when the gap in the spectrum of
the lattice Hamiltonian ${\boldsymbol{\mathrm{H}}}_{\text{latt}}$ vanishes,
the correlation length becomes
infinite in units of the lattice spacing. As a result,
the leading long-distance effects can be described by
Conformal Field Theory (CFT) with the Hamiltonian ${\boldsymbol{\mathrm{H}}}_{\text{CFT}}$
[5].
In general, it is expected
that the critical lattice Hamiltonian admits
an asymptotic power series expansion
involving an infinite set of local scaling fields (see e.g.
[6, 7]):
$${\boldsymbol{\mathrm{H}}}_{\text{latt}}\sim{\boldsymbol{\mathrm{H}}}_{\text{%
CFT}}+\lambda_{1}\,\varepsilon^{d_{1}-2}\int dx\,\mathcal{O}_{1}+\lambda_{2}\,%
\varepsilon^{{d_{2}}-2}\int dx\,\mathcal{O}_{2}+\dots\ .$$
(1.1)
Here $d_{k}$ denotes the scaling dimension of the field
$\mathcal{O}_{k}$, and
the explicit dependence on the lattice spacing
$\varepsilon$ is used to show the relative smallness of various terms.
All fields occurring in
(1.1) are irrelevant and
the corresponding exponents of $\varepsilon$
are positive.
Such an expansion also involves a set of non-universal
coupling constants $\lambda_{k}$ depending on the microscopic properties
of the model, and whose values rely on the chosen normalization for
$\mathcal{O}_{k}$.
The asymptotic expansion (1.1) is a powerful
tool to study the low energy spectrum of lattice theories.
At the same time, in order to analyse
lattice correlation functions, one should also know how
local lattice operators $\mathcal{O}_{\text{latt}}$ are
represented in terms of scaling fields.
Just as the Hamiltonian density, they
can be expressed as formal series in powers of the lattice spacing $\varepsilon$,
$$\mathcal{O}_{\text{latt}}\sim C_{m_{1}}\,\varepsilon^{d_{m_{1}}}\mathcal{O}_{m%
_{1}}+C_{m_{2}}\,\varepsilon^{d_{m_{2}}}\mathcal{O}_{m_{2}}+\dots\ ,$$
(1.2)
where again the non-universal constants $C_{m}$ depend on the
normalization of the scaling fields.
The knowledge of the explicit form of (1.1) and (1.2)
enables one to obtain the long-distance asymptotic expansion of
lattice correlation functions in powers
of lattice distances.
For example, in the case of a vacuum correlator,
the leading term
is given by the CFT correlation function of the
operator occurring in (1.2) at the lowest order in $\varepsilon$,
whereas subleading asymptotics come from both higher order terms
in (1.2) and from the perturbative
corrections
to the CFT vacuum toward the lattice ground state.
In this expansion, the exact values of the exponents follow
from the scaling dimensions of the CFT fields,
while the corresponding amplitudes come from the
values of the constants appearing in (1.1)
and (1.2).
Thus, to investigate quantitatively the long-distance behavior of lattice
correlation functions,
one needs
to determine (for a fixed suitable normalization of the scaling fields)
exact analytical
expressions for the non-universal constants $\lambda_{k}$
and $C_{m}$.
The aim of this article is to study expansions of the type (1.2)
in the case of
the XXZ spin 1/2 Heisenberg chain.
In particular, we calculate the first constants
$C_{m}$ occurring in the expansion (1.2) of local spin operators.
The article is organized as follows.
In Section 2, we recall the definition
of the XXZ spin chain (see e.g. [8] for details),
and discuss
its continuous limit, the Gaussian CFT [9, 10, 11].
We review in particular how, from the analysis of the
global symmetries of the model,
one can derive selection rules for the expansions (1.2)
of the lattice operators in terms of the scaling fields of the
Gaussian model. This enables one to predict the structure of the
long-distance asymptotic expansions of the spin-spin correlation functions.
The problem of computing the constants occurring
in the expansions (1.2) of spin operators is the subject of
Section 3. There we explain how, moving slightly away from
criticality, and comparing exact results obtained
for the XYZ and sine-Gordon models,
one can quantitatively connect lattice spin operators to scaling
fields.
This gives us access
to
the correlation amplitudes
of the spin-spin correlation functions.
Our predictions are gathered in Section 4, where they are
compared to existing numerical data.
In Section 5, we study these spin-spin correlation
functions in the isotropic (XXX) limit by means of the exact
Renormalization Group (RG) approach.
This section contains an erratum of Section 5
of [12].
Finally, we conclude this article with several remarks.
2 XXZ spin chain in the continuous limit
2.1 Preliminaries[8, 11]
To illustrate the problem of the
determination of correlation amplitudes,
we consider an example,
the XXZ Heisenberg chain of spins 1/2.
It is defined by the Hamiltonian
$${\boldsymbol{\mathrm{H}}}_{\mathrm{XXZ}}=-\frac{J}{2}\sum\limits_{l=-\infty}^{%
\infty}\Bigl{\{}\sigma_{l}^{x}\sigma_{l+1}^{x}+\sigma_{l}^{y}\sigma_{l+1}^{y}+%
\Delta\,\sigma_{l}^{z}\sigma_{l+1}^{z}\Bigr{\}}\,,$$
(2.1)
where the spin operators $\sigma_{l}^{a}$ ($a=x,y,z$) denote
the conventional Pauli matrices associated with the $l$-th site of the
infinite lattice,
and $\Delta$ is an anisotropy parameter.
As ${\boldsymbol{\mathrm{H}}}_{\mathrm{XXZ}}(-J,-\Delta)$ can be obtained from
${\boldsymbol{\mathrm{H}}}_{\mathrm{XXZ}}(J,\Delta)$ by a unitary transformation,
we choose in the following the coupling constant $J$ to be positive.
The nature of the spectrum of the infinite chain depends on
the value of the anisotropy parameter $\Delta$.
The study we present here concerns the critical regime of
the chain, which corresponds to the domain $-1\leqslant\Delta<1$.
We shall use the parameterizations
$$\Delta=\cos(\pi\eta)\ \qquad\ (0<\eta\leq 1)\,,$$
(2.2)
and
$$J=\frac{1-\eta}{\varepsilon\sin(\pi\eta)}\qquad(\varepsilon>0)\,.$$
(2.3)
To describe leading long-distance (low-energy)
effects,
it is useful to consider a continuous limit of the lattice model.
This limit can be obtained from the representation of (2.1)
in terms of lattice fermions,
$${\boldsymbol{\mathrm{H}}}_{\mathrm{XXZ}}=-J\ \sum_{l}\Bigl{\{}\,\psi_{l}^{%
\dagger}\psi_{l+1}+\psi_{l+1}^{\dagger}\psi_{l}+\frac{\Delta}{2}\ \big{(}1-2%
\psi^{\dagger}_{l}\psi_{l}\big{)}\,\big{(}1-2\psi^{\dagger}_{l+1}\psi_{l+1}%
\big{)}\,\Bigr{\}}\,,$$
(2.4)
where the fermionic operators are
related to the spin operators through the Jordan-Wigner
transformation,
$$\sigma_{l}^{z}=1-2\psi_{l}^{\dagger}\psi_{l}\,,\qquad\psi_{l}^{\dagger}=\prod_%
{j<l}\sigma^{z}_{j}\cdot\sigma_{l}^{-}\,,\qquad\psi_{l}=\prod_{j<l}\sigma^{z}_%
{j}\cdot\sigma_{l}^{+}\,,$$
(2.5)
and $\sigma_{l}^{\pm}=(\sigma_{l}^{x}\pm i\sigma_{l}^{y})/2\,.$
With the parameterization (2.3) of
the constant $J$,
the spin-1/2 spin wave
dispersion relation has the form [13, 14]
$${\cal E}(k)=-\frac{\cos(k)}{\varepsilon}\,.$$
(2.6)
The ground state of the chain has all levels filled with $|k|<\pi/2$.
Linearizing the dispersion relation in the vicinity of the
Fermi points $k_{F}=\pm\pi/2$,
one can take the continuous limit of the model in which
the lattice operators $\psi_{l}$ are replaced by two
fields, $\psi_{R}(x)$ and $\psi_{L}(x)$, varying slowly
on the lattice scale:
$$\psi_{l}\propto e^{\frac{i\pi l}{2}}\,\psi_{R}(x)+e^{-\frac{i\pi l}{2}}\,\psi_%
{L}(x)\,.$$
(2.7)
Here $x=l\,\varepsilon$, and thus
the parameter $\varepsilon$ can be interpreted as a
lattice spacing.
The continuous
Fermi fields are governed by the Hamiltonian
$${\bf H}_{\mathrm{Thirring}}=\int\frac{dx}{2\pi}\,\Bigl{\{}-i\psi^{\dagger}_{R}%
\partial_{x}\psi_{R}+i\psi^{\dagger}_{L}\partial_{x}\psi_{L}+{\tt g}\,\psi^{%
\dagger}_{R}\psi_{R}\psi^{\dagger}_{L}\psi_{L}\,\Bigr{\}}\,.$$
(2.8)
A precise relation between the anisotropy parameter $\Delta$ and
the four-fermion coupling constant ${\tt g}$ depends on the choice of the
regularization procedure for the continuous Hamiltonian (2.8) and is not essential for
our purposes. What is important is that the quantum field theory
model (2.8)
(which is known as the Thirring model [15, 16]) is
conformally invariant
and equivalent to the Gaussian CFT (see
the preprint collection[17] for a historical
review of bosonization).
2.2 Exponential fields in the Gaussian model
The Gaussian model is defined in
terms of one scalar field $\varphi$,
which satisfies
the D’Alembert equation111
The value of the spin-wave velocity $v$ is determined from the slope of the
dispersion relation (2.6) at the Fermi points and
from the identification of
$x$ with $l\,\varepsilon$.
$$(\partial_{t}^{2}-v^{2}\partial_{x}^{2})\,\varphi=0\,,\qquad\text{with}\ \ v=1\,,$$
(2.9)
and the boundary condition
$$\partial_{t}\varphi(x,t)|_{x\to\pm\infty}=\partial_{x}\varphi(x,t)|_{x\to\pm%
\infty}=0\ .$$
(2.10)
Assuming that the equal-time
canonical commutation relations are imposed on the field,
$$[\,\varphi(x),\partial_{t}\varphi(x^{\prime})\,]=8\pi i\ \delta(x-x^{\prime})%
\,,\ $$
one can write the Hamiltonian
of the model in the form
$${\boldsymbol{\mathrm{H}}}_{\mathrm{Gauss}}=\int\frac{dx}{2\pi}\Big{\{}\mathbf{%
T}_{R}+\mathbf{T}_{L}\Big{\}}+\text{const}\,,$$
(2.11)
where
$$\mathbf{T}_{R}(x)=\frac{1}{16}(\partial_{x}\varphi-\partial_{t}\varphi)^{2}\,,%
\ \qquad\ \mathbf{T}_{L}(x)=\frac{1}{16}(\partial_{x}\varphi+\partial_{t}%
\varphi)^{2}\,.$$
(2.12)
As is usual in CFT [18],
it is convenient to set a class of conformal
primary fields
among all scaling fields. In the case of the Gaussian model,
these conformal primaries include
right and left currents,
$$(\partial_{x}-\partial_{t})\varphi\,,\ \qquad\ (\partial_{x}+\partial_{t})%
\varphi\,,$$
(2.13)
along with exponential fields222Here we are not considering
the orbifold (Ashkin-Teller) sector of the Gaussian model. [10].
With the
boundary condition (2.10), the latter can be defined
as
$$\mathcal{O}_{s,n}(x,t)=\Lambda^{d_{s,n}}\lim_{\varepsilon\to+0}\,\exp\bigg{\{}%
\frac{in}{4\sqrt{\eta}}\int_{-\infty}^{x}dx^{\prime}\,\partial_{t}\varphi(x^{%
\prime},t)\bigg{\}}\,\exp\bigg{\{}\frac{is\sqrt{\eta}}{2}\ \varphi(x+%
\varepsilon,t)\bigg{\}}\,,$$
(2.14)
where $s$, $n$ are integers and
$$d_{s,n}=\frac{s^{2}\eta}{2}+\frac{n^{2}}{8\eta}\ .$$
(2.15)
The regularization
parameter $\Lambda$, which has the dimension $[\,\text{length}\,]^{-1}$,
is introduced
in the definition (2.14) in order to provide a multiplicative renormalization of the fields.
Notice that
$\mathcal{O}_{s,n}$ obey the simple Hermiticity relation
$$\mathcal{O}_{s,n}^{\dagger}=\mathcal{O}_{-s,-n}\ .$$
(2.16)
To completely define these exponential fields,
we should also specify some condition which fixes their
multiplicative normalization.
By a proper choice of $\Lambda$ in (2.14),
one can impose the following form of the causal Green’s functions
in the Euclidean domain $x^{2}-t^{2}>0$:
$$\langle\,T\,\mathcal{O}_{s,n}(x,t)\,\mathcal{O}^{\dagger}_{s,n}(0,0)\,\rangle=%
\Big{(}\frac{t-x}{t+x}\Big{)}^{\frac{sn}{2}}\ (x^{2}-t^{2})^{-d_{s,n}}\,.$$
(2.17)
We will later refer to Eq. (2.17) as the
“CFT normalization condition”.
2.3 Global symmetries
To draw a precise link between the Gaussian CFT and the XXZ spin chain,
it is important
to examine and identify their global symmetries.
Let us first consider the Gaussian model.
The Hamiltonian (2.11) is
manifestly invariant under the $U(1)$ rotations,
$${\mathbb{U}}_{\alpha}\,\varphi\,{\mathbb{U}}^{-1}_{\alpha}=\varphi+\frac{2%
\alpha}{\sqrt{\eta}}\ ,\qquad\quad{\mathbb{U}}_{\alpha}\,\mathcal{O}_{s,n}\,{%
\mathbb{U}}^{-1}_{\alpha}=e^{is\alpha}\,\mathcal{O}_{s,n}\ ,$$
(2.18)
where the operator ${\mathbb{U}}_{\alpha}$ can be written in the form
$${\mathbb{U}}_{\alpha}=\exp\bigg{\{}\frac{i\alpha}{4\pi\sqrt{\eta}}\ \int_{-%
\infty}^{\infty}dx\,\partial_{t}\varphi\,\bigg{\}}\ .$$
(2.19)
The CFT model is also invariant under
the parity transformation
${\mathbb{P}}\,:\,\varphi(x,t)\to\varphi(-x,t),$
the time reversal ${\mathbb{T}}\,:\,\varphi(x,t)\to\varphi(x,-t),$
and the reflection ${\mathbb{C}}\,:\,\varphi\to-\varphi$.
Using the definition (2.14), one can show
that these transformations act on the
exponential fields as
$$\displaystyle{\mathbb{P}}\,\mathcal{O}_{s,n}(x,t)\,{\mathbb{P}}=e^{-\frac{isn%
\pi}{2}}\ {\mathbb{U}}_{\pi n}\ \mathcal{O}_{s,-n}(-x,t)\,,$$
(2.20)
$$\displaystyle{\mathbb{T}}\,\mathcal{O}_{s,n}(x,t)\,{\mathbb{T}}=\mathcal{O}_{-%
s,n}(x,-t)\,,$$
(2.21)
$$\displaystyle{\mathbb{C}}\,\mathcal{O}_{s,n}\,{\mathbb{C}}=\mathcal{O}_{-s,-n}%
(x,t)\,.$$
(2.22)
Note that the reflection ${\mathbb{C}}$ can be naturally considered
as a charge conjugation in the theory. As usual,
${\mathbb{C}}$ and ${\mathbb{P}}$ are intrinsic automorphisms
of the operator algebra, contrary to the anti-unitary
transformation ${\mathbb{T}}$ which acts on $c$-numbers as follows,
$${\mathbb{T}}\,(\text{$c$-number})\,{\mathbb{T}}=(\text{$c$-number})^{*}\ .$$
Let us now identify
the above transformations
with the global symmetries of the spin chain.
First of all, it is natural to define
the action of the $U(1)$ rotation
on the lattice as follows,
$${\mathbb{U}}_{\alpha}\,\sigma^{\pm}_{l}\,{\mathbb{U}}^{-1}_{\alpha}=e^{\pm i%
\alpha}\ \sigma^{\pm}_{l}\,,\qquad\quad{\mathbb{U}}_{\alpha}\,\sigma^{z}_{l}\,%
{\mathbb{U}}^{-1}_{\alpha}=\sigma^{z}_{l}\ ,$$
(2.23)
where
$${\mathbb{U}}_{\alpha}=e^{i\alpha S_{z}}\,\ \qquad\qquad\ \ \text{with}\quad S_%
{z}=\frac{1}{2}\,\sum_{j}\sigma^{z}_{j}\ .$$
(2.24)
Such an identification has an important consequence.
Indeed, as the two rotations of angle $2\pi$ and $-2\pi$ are
indistinguishable
trivial transformations in the lattice theory, we should set
$${\mathbb{U}}_{2\pi}={\mathbb{U}}_{-2\pi}=\nu\,{\mathbb{I}}\,\qquad{\rm with}%
\quad\nu=\pm 1$$
(2.25)
in the corresponding Gaussian model.
The condition (2.25), along with (2.18),
implies that $\varphi$ has to be
treated as a compactified field:
$$\varphi\equiv\varphi+\frac{4\pi}{\sqrt{\eta}}\ {\mathbb{Z}}\ .$$
According to (2.24), the sign factor $\nu=+1$ in Eq. (2.25)
occurs for states with an integer eigenvalue of the
operator $S_{z}$. The corresponding
linear subspace of the whole Hilbert space
can be constructed from the thermodynamic limit
of finite chains with an even number of sites,
and will therefore be referred to
as the “even sector”. Similarly, the condition $\nu=-1$
defines another linear subspace which will be called the “odd sector”.
The actions of charge conjugation and time reversal
can be naturally identified,
in the lattice theory, with the following transformations,
$$\displaystyle{\mathbb{C}}\,\sigma^{\pm}_{l}\,{\mathbb{C}}=\sigma^{\mp}_{l}\,,$$
$$\displaystyle{\mathbb{C}}\,\sigma^{z}_{l}\,{\mathbb{C}}=-\sigma^{z}_{l}\,,$$
(2.26)
$$\displaystyle{\mathbb{T}}\,\sigma^{\pm}_{l}\,{\mathbb{T}}=\sigma^{\mp}_{l}\,,$$
$$\displaystyle{\mathbb{T}}\,\sigma^{z}_{l}\,{\mathbb{T}}=-\sigma^{z}_{l}\ .$$
(2.27)
Let us recall here
that the time reversal is an anti-unitary transformation, so
that ${\mathbb{C}}$ and ${\mathbb{T}}$ correspond to different
symmetries of the XXZ chain, even thought
they act identically on the spin operators.
As for the parity transformation, its action in the lattice model depends
on the choice of the sector specified by the sign
factor $\nu$ in (2.25). Indeed,
$\nu=+1$ implies that the considered
infinite chain is defined as the thermodynamic limit of
finite lattices with an even number of sites.
Such finite lattices clearly do not possess
any invariant site with respect to the parity transformation.
Therefore,
$${\mathbb{P}}\,\sigma^{a}_{l}\,{\mathbb{P}}=\sigma^{a}_{1-l}\ \qquad\ {\rm for}%
\qquad\nu=+1\,,$$
(2.28)
whereas the “naive” action of ${\mathbb{P}}$
is valid only in the odd sector:
$${\mathbb{P}}\,\sigma^{a}_{l}\,{\mathbb{P}}=\sigma^{a}_{-l}\ \qquad\ \ {\rm for%
}\qquad\nu=-1\,.$$
(2.29)
2.4 Selection rules
The global symmetries and the knowledge of their action
on lattice and continuous operators
provide selection rules for the set of
scaling fields which can occur in expansions (1.1) and
(1.2).
Here we examine what are these
selection rules for the conformal primary
fields in the expansions of the spin operators.
First, since the whole operator content of
the Gaussian model is given by
the primary fields described above and by their conformal descendants,
it is easy to see that $\mathcal{O}_{\pm 1,0}$ are the local fields
with the lowest scaling dimension which
may occur in the expansions of $\sigma_{0}^{\pm}$:
$$\sigma_{0}^{\pm}\sim C_{0}\ \varepsilon^{d_{1,0}}\ \mathcal{O}_{\pm 1,0}(0)+%
\ldots\ .$$
(2.30)
Furthermore, assuming that the conjugation in the lattice theory is defined
in such a way that $\sigma_{l}^{a}\ (a=x,y,z)$ are Hermitian operators,
we conclude
from (2.16) that the constant $C_{0}$ in (2.30) is real.
Let us now consider the expansion of the lattice operator $\sigma_{0}^{z}$.
Due to the $U(1)$ invariance, it can contain
only the primary fields $(\partial_{x}\pm\partial_{t})\varphi$ and
$\mathcal{O}_{0,n}$, $n\in{\mathbb{Z}}$, along with their conformal descendants.
Using definition (2.14), it is
easy to show that the fields $\mathcal{O}_{0,n}$ with odd
$n$ are not mutually local
with respect to $\mathcal{O}_{\pm 1,0}$. As the latter are the leading terms
in the expansions (2.30) of $\sigma_{0}^{\pm}$,
it would contradict
the mutual locality of the lattice operators $\sigma_{j}^{\pm}$ and
$\sigma_{l}^{z}$ if there were any $\mathcal{O}_{0,2m+1}$, $m\in{\mathbb{Z}}$,
in the series for $\sigma_{0}^{z}$.
From the ${\mathbb{C}},\ {\mathbb{T}}$ invariances and
the Hermiticity of $\sigma_{0}^{z}$, one can moreover
predict that the primary fields are allowed to appear
only as linear combinations of
$\partial_{t}\varphi$ and
$i\,(\mathcal{O}_{0,2m}-\mathcal{O}_{0,-2m})$ with real coefficients.
Finally, let us consider the parity transformation.
According to equations (2.20) and (2.25),
$${\mathbb{P}}\,\mathcal{O}_{s,2m}(x,t)\,{\mathbb{P}}=\nu^{m}\ \mathcal{O}_{s,-2%
m}(-x,t)\,,$$
(2.31)
and thus one has to examine even and odd sectors separately.
In the odd sector ($\nu=-1$),
the parity (2.29) prohibits the
presence of the fields $\mathcal{O}_{0,4m}-\mathcal{O}_{0,-4m}$ with $m\in{\mathbb{Z}}$.
Therefore the expansion has to be of the form,
$$\displaystyle\sigma_{0}^{z}\sim$$
$$\displaystyle\,\varepsilon\,C_{0}^{z}\,\partial_{t}\varphi(0)$$
$$\displaystyle+\frac{1}{2i}\,\sum\limits_{m=1}^{+\infty}\,C^{z}_{m}\ %
\varepsilon^{d_{0,4m-2}}\ \big{(}\mathcal{O}_{0,4m-2}-\mathcal{O}_{0,-4m+2}%
\big{)}(0)+\,\text{descendants}\,.$$
(2.32)
But, since (2.4) is a local operator expansion,
it cannot depend on the
choice of the sector in the Hilbert space.
Thus, (2.4) should be valid in the even sector as well.
Owing to this and to Eqs. (2.28), (2.31), we can determine
the action of the lattice translation,
$$\mathbb{K}\,\sigma_{l}^{a}\,\mathbb{K}^{-1}=\sigma_{l+1}^{a}\,,$$
(2.33)
on the primary fields $\mathcal{O}_{0,4m-2}$:
$$\mathbb{K}\,\mathcal{O}_{0,4m-2}(x)\,\mathbb{K}^{-1}=-\mathcal{O}_{0,4m-2}(x+%
\varepsilon)\,.$$
(2.34)
With this relation, the expansion (2.4) can
be written in
the more general form,
$$\displaystyle\sigma_{l}^{z}\sim$$
$$\displaystyle\,\frac{\varepsilon}{2\pi\sqrt{\eta}}\,\partial_{t}\varphi(x)$$
$$\displaystyle+\frac{(-1)^{l}}{2i}\,\sum\limits_{m=1}^{+\infty}\,C^{z}_{m}\ %
\varepsilon^{d_{0,4m-2}}\,\big{(}\mathcal{O}_{0,4m-2}-\mathcal{O}_{0,-4m+2}%
\big{)}(x)+\,\text{descendants}\,,$$
(2.35)
where $x=l\,\varepsilon$. Notice that the exact value of the coefficient
$C_{0}^{z}$ in (2.4) is determined
from the comparison of Eqs. (2.19)
and (2.24).
Using the same line of arguments, one can extend the
expansion (2.30),
$$\sigma_{l}^{\pm}\sim\,\frac{1}{2}\,\sum\limits_{m=0}^{+\infty}(-1)^{lm}\,C_{m}%
\ \varepsilon^{d_{1,2m}}\ \big{(}\mathcal{O}_{\pm 1,2m}+\mathcal{O}_{\pm 1,-2m%
}\big{)}(x)+\ \text{descendants}\ ,$$
(2.36)
and determine the action of the lattice translation
$\mathbb{K}$ on $\mathcal{O}_{\pm 1,2m}$:
$$\mathbb{K}\,\mathcal{O}_{\pm 1,2m}(x)\,\mathbb{K}^{-1}=(-1)^{m}\ \mathcal{O}_{%
\pm 1,2m}(x+\varepsilon)\ .$$
(2.37)
Since all exponential fields $\mathcal{O}_{\pm s,2m}(x)$ with
integers $s$ and $m$
can be obtained by means of operator product expansions
of the fields $\mathcal{O}_{\pm 1,2m}(x)$
and $\mathcal{O}_{0,4m-2}(x)$, we deduce
from (2.34) and (2.37)
that
$$\mathbb{K}\,\mathcal{O}_{s,2m}(x)\,\mathbb{K}^{-1}=(-1)^{m}\ \mathcal{O}_{s,2m%
}(x+\varepsilon)\ .$$
(2.38)
Notice that, in the process of bosonization,
the fermionic fields $\psi_{R}$ and $\psi_{L}$ (2.7)
are identified with the
exponential fields $\mathcal{O}_{1,-1}$ and $\mathcal{O}_{1,1}$ respectively.
Hence equation (2.38) is indeed
consistent with (2.7).
It is not difficult to extend our symmetry analysis
to derive the selection rules for
expansion (1.2) of any local lattice operator.
This procedure can, in particular, be applied to the case of the
Hamiltonian density. As a result,
the following form
of the low energy effective Hamiltonian for the XXZ spin chain is suggested:
$${\boldsymbol{\mathrm{H}}}_{\mathrm{XXZ}}\sim{\boldsymbol{\mathrm{H}}}_{\text{%
Gauss}}+\int\frac{dx}{4\pi}\bigg{\{}\,\sum_{m=1}^{\infty}\lambda_{m}\,%
\varepsilon^{d_{0,4m}-2}\,\big{(}\mathcal{O}_{0,4m}+\mathcal{O}_{0,-4m}\big{)}%
(x)+{\rm descendants}\,\bigg{\}}\,.$$
(2.39)
As was mentioned in Introduction, it is necessary
to choose the normalization of the scaling fields to give a precise
meaning to the couplings in (2.39) as well as to
the real constants $C_{m},\ C_{m}^{z}$ in expansions
(2.4) and (2.36). In this paper we
adapt the CFT normalization (2.17).
With this normalization condition, the first coupling constant $\lambda_{1}$
in (2.39) has been obtained in [12],
together with
the leading contributions of descendant fields
(see also Ref.[19] for a qualitative
analysis of the descendent field
contributions):
$$\displaystyle{\boldsymbol{\mathrm{H}}}_{\mathrm{XXZ}}\sim{\boldsymbol{\mathrm{%
H}}}_{\text{Gauss}}+\lambda_{1}\ \varepsilon^{2/\eta-2}\,\int\frac{dx}{4\pi}\,%
\big{(}\mathcal{O}_{0,4}+\mathcal{O}_{0,-4}\big{)}(x)\\
\displaystyle-\varepsilon^{2}\int\frac{dx}{2\pi}\,\Big{\{}\,\lambda_{+}\ %
\mathbf{T}_{R}\mathbf{T}_{L}(x)+\lambda_{-}\ \big{(}\mathbf{T}^{2}_{R}+\mathbf%
{T}^{2}_{L}\big{)}(x)\,\Big{\}}+\dots\,,$$
(2.40)
where
$$\displaystyle\lambda_{1}=-\frac{4\,\Gamma(1/\eta)}{\Gamma\big{(}1-1/\eta\big{)%
}}\bigg{[}\frac{\Gamma\big{(}1+\frac{\eta}{2-2\eta}\big{)}}{2\sqrt{\pi}\Gamma%
\big{(}1+\frac{1}{2-2\eta}\big{)}}\bigg{]}^{2/\eta-2},$$
(2.41)
$$\displaystyle\lambda_{+}=\frac{1}{2\pi}\tan\Big{(}\frac{\pi}{2-2\eta}\Big{)}\,,$$
(2.42)
$$\displaystyle\lambda_{-}=\frac{\eta}{12\pi}\frac{\Gamma\big{(}\frac{3}{2-2\eta%
}\big{)}\,\Gamma^{3}\big{(}\frac{\eta}{2-2\eta}\big{)}}{\Gamma\big{(}\frac{3%
\eta}{2-2\eta}\big{)}\,\Gamma^{3}\big{(}\frac{1}{2-2\eta}\big{)}}\ .$$
(2.43)
2.5 Vacuum spin-spin correlation functions
At this stage, without the knowledge of the constants
occurring in expansions
(2.4) and (2.36), it is already possible
to predict333It has been done for the first time in
Ref.[9].
the exact values of the exponents for the vacuum spin-spin
correlation functions. They follow immediately
from the scaling dimensions of the fields occurring
in (2.4), (2.36) and (2.40).
It may be worth recalling at this point, that the vacuum sector
of the infinite XXZ chain is infinitely degenerate.
In general, the boundary conditions imposed on the finite critical
chain do not preserve all the global symmetries discussed above,
and they may
be spontaneously broken at the thermodynamic limit.
Here we consider only
translational invariant vacuums with unbroken parity:
$${\mathbb{P}}\,|\,\mathrm{vac}\,\rangle={\mathbb{K}}\,|\,\mathrm{vac}\,\rangle=%
|\,\mathrm{vac}\,\rangle\ .$$
(2.44)
To fulfill these requirements, we shall treat the infinite XXZ chain as
the thermodynamic
limit of finite chains subject to periodic boundary
conditions.
Then, different vacuum
states can be distinguished by means of the
operator $S_{z}$ (2.24):
$$S_{z}\,|\,s\,\rangle=s\,|\,s\,\rangle\,,\qquad 2s\in{\mathbb{Z}}\ .$$
(2.45)
Also, by a proper choice of phase factors,
one can always set up the conditions
$${\mathbb{C}}\,|\,s\,\rangle=|-s\,\rangle\ \qquad\ \text{and}\ \qquad\ \ {%
\mathbb{T}}\,|\,s\,\rangle=|-s\,\rangle\ .$$
(2.46)
In the continuous limit, the vacuum $|\,s\,\rangle$
flows toward the conformal primary state with right
and left conformal dimensions equal to $d_{s,0}/2$,
where $d_{s,n}$ is given by (2.15)
[20, 21, 22].
This implies in particular that, for the
XXZ chain with a finite number of sites $N\gg 1$,
the difference of vacuum energies corresponding to the
states $|\,s\,\rangle$ and $|\,0\,\rangle$ is $2\pi d_{s,0}/N+o(N^{-1})$.
From our previous analysis,
one can now predict the following asymptotic
expansions for the time-ordered correlation functions:
$$\displaystyle\langle\,T\,\sigma^{x}_{l+j}(t)\,\sigma_{j}^{x}(0)\,\rangle\sim%
\frac{A}{(l_{+}l_{-})^{\frac{\eta}{2}}}\,\bigg{\{}\,1-\frac{B}{(l_{+}l_{-})^{%
\frac{2}{\eta}-2}}+O\big{(}l^{-2}\log l,l^{8-8/\eta}\big{)}\,\bigg{\}}$$
$$\displaystyle\qquad-\frac{(-1)^{l}\ {\tilde{A}}}{(l_{+}l_{-})^{\frac{\eta}{2}+%
\frac{1}{2\eta}}}\,\bigg{\{}\,\frac{1}{2}\,\Big{(}\,\frac{l_{+}}{l_{-}}+\frac{%
l_{-}}{l_{+}}\,\Big{)}+\frac{{\tilde{B}}}{(l_{+}l_{-})^{\frac{1}{\eta}-1}}+O%
\big{(}l^{-2}\log l,l^{4-4/\eta}\big{)}\,\bigg{\}}+\ldots\ ,$$
(2.47)
$$\displaystyle\langle\,T\,\sigma^{z}_{l+j}(t)\,\sigma_{j}^{z}(0)\,\rangle\sim-%
\frac{1}{\pi^{2}\eta\ l_{+}l_{-}}\,\bigg{\{}\,\frac{1}{2}\,\Big{(}\,\frac{l_{+%
}}{l_{-}}+\frac{l_{-}}{l_{+}}\,\Big{)}$$
$$\displaystyle\qquad+\frac{{\tilde{B}}_{z}}{(l_{+}l_{-})^{\frac{2}{\eta}-2}}\,%
\bigg{(}\,1+\frac{2-\eta}{4(1-\eta)}\,\Big{(}\,\frac{l_{+}}{l_{-}}+\frac{l_{-}%
}{l_{+}}\,\Big{)}\,\bigg{)}+O\big{(}l^{-2}\log l,l^{8-8/\eta}\big{)}\,\bigg{\}}$$
$$\displaystyle\qquad+\frac{(-1)^{l}\ A_{z}}{(l_{+}l_{-})^{\frac{1}{2\eta}}}\,%
\bigg{\{}\,1-\frac{B_{z}}{(l_{+}l_{-})^{\frac{1}{\eta}-1}}+O\big{(}l^{-2}\log l%
,l^{4-4/\eta}\big{)}\,\bigg{\}}+\ldots\ ,$$
(2.48)
where
$$l_{\pm}=l\pm\frac{t}{\varepsilon}\gg 1\ .$$
(2.49)
The coefficients in the asymptotic expansions
(2.47) and (2.48)
do not really depend on the choice of the vacuum
state $|\,s\,\rangle$, thus any finite $s\in{\mathbb{Z}}/2$ can be chosen
for the averaging
$\langle\,\ldots\,\rangle\equiv\frac{\langle\,s\,|\,\ldots\,|\,s\,\rangle}{%
\langle\,s\,|\,s\,\rangle}$.
In Eqs. (2.47), (2.48), the correlation amplitudes
$A$, ${\tilde{A}}$ and $A_{z}$ are simply related to
the first constants
occurring in the expansions (2.4) and (2.36):
$$A=2\,(C_{0})^{2}\,,\qquad{\tilde{A}}=(C_{1})^{2}\,,\qquad A_{z}=\frac{1}{2}\ (%
C_{1}^{z})^{2}\ .$$
(2.50)
They will be computed in the next section.
At the same time,
the constants $B,\,{\tilde{B}},\,B_{z}$ and ${\tilde{B}}_{z}$ appearing in
(2.47) and (2.48)
can be determined by
methods of conformal perturbation theory [23]
based on the effective Hamiltonian (2.40).
In particular, the constant $B$
was obtained in Ref. [12] from
second order perturbative calculations,
$$B=\frac{\lambda_{1}^{2}}{16}\ \bigg{\{}\,\frac{2\pi^{2}}{\sin^{2}(2\pi/\eta)}-%
\frac{\eta^{2}}{(1-\eta)(2-\eta)}-\psi^{\prime}(1/\eta)-\psi^{\prime}(3/2-1/%
\eta)\,\bigg{\}},$$
(2.51)
where $\psi^{\prime}(z)=\partial_{z}^{2}\log\Gamma(z)$ and $\lambda_{1}$ is given by
(2.41).
The constant ${\tilde{B}}_{z}$ can be computed
similarly:
$${\tilde{B}}_{z}=\frac{\lambda_{1}^{2}}{4}\ \frac{\eta}{(\eta-2)^{2}}\ .$$
(2.52)
On the contrary, the determination
of ${\tilde{B}}$ and $B_{z}$ do not require
calculations beyond the first order. They read explicitly,
$$\displaystyle B_{z}=-\lambda_{1}\ 2^{\frac{4}{\eta}-5}\ \frac{\Gamma(\frac{1}{%
\eta}-\frac{1}{2})\,\Gamma(1-\frac{1}{\eta})}{\Gamma(\frac{3}{2}-\frac{1}{\eta%
})\,\Gamma(\frac{1}{\eta})}\ ,$$
(2.53)
$$\displaystyle{\tilde{B}}=(1-\eta)^{2}\ B_{z}\ .$$
(2.54)
3 Calculation of correlation amplitudes
The purpose of this section is to explain how the local
operators $\sigma_{l}^{\pm}$ and $\sigma_{l}^{z}$ can be quantitatively related
to the scaling fields (2.13)
and (2.14), that is, how one can compute
explicit analytic expressions for the constants $C_{m}$ and $C_{m}^{z}$
occurring in (2.4) and (2.36) for the fixed CFT normalization
(2.17).
We concentrate here on the
first terms of the expansions (2.4), (2.36), which provide the
main asymptotic behavior of the spin-spin correlation functions
(2.47) and (2.48):
$$\displaystyle\sigma^{\pm}_{l}\sim C_{0}\,\varepsilon^{\frac{\eta}{2}}\ %
\mathcal{O}_{\pm 1,0}(x)+(-1)^{l}\ \frac{C_{1}}{2}\,\varepsilon^{\frac{\eta}{2%
}+\frac{1}{2\eta}}\ \big{(}\mathcal{O}_{\pm 1,2}+\mathcal{O}_{\pm 1,-2}\big{)}%
(x)+\dots\ ,$$
(3.1)
$$\displaystyle\sigma^{z}_{l}\sim\frac{\varepsilon}{2\pi\sqrt{\eta}}\ \partial_{%
t}\varphi(x)+(-1)^{l}\ \frac{C^{z}_{1}}{2i}\,\varepsilon^{\frac{1}{2\eta}}\,%
\big{(}\mathcal{O}_{0,2}-\mathcal{O}_{0,-2}\big{)}(x)+\dots\ .$$
(3.2)
Note that in (3.2) each of the
two terms is either leading or sub-leading according to the value of
$\eta$, i.e. of the anisotropy parameter $\Delta$, whereas in
(3.1) the second term is always subleading. Nevertheless, this
term gives rise to noticeable numerical
corrections to the leading asymptotic
behavior of correlation functions and has already been studied
numerically (see Section 4).
3.1 Even sector of the infinite XYZ chain
To find the quantitative relation between the spin operators
and the local scaling fields,
it is useful to move slightly away from
criticality and, instead of (2.1), to consider the XYZ chain,
$${\boldsymbol{\mathrm{H}}}_{\mathrm{XYZ}}=-\frac{1}{2}\sum_{l=-\infty}^{\infty}%
\Bigl{\{}J_{x}\,\sigma_{l}^{x}\sigma_{l+1}^{x}+J_{y}\,\sigma_{l}^{y}\sigma_{l+%
1}^{y}+J_{z}\,\sigma_{l}^{z}\sigma_{l+1}^{z}\Bigr{\}}\ .$$
(3.3)
Without loss of generality we assume here that
$J_{x}>J_{y}\geq|J_{z}|.$
The XYZ deformation has a remarkable feature:
it preserves the integrability of the original theory
[8, 24].
Nowadays, the structure of the
Hilbert space of the infinite XYZ chain is well understood.
We recall here some basic facts that will be useful for our analysis.
In the case of the XYZ spin chain, the global
symmetry group discussed in the previous section is
explicitly broken to the subgroup generated
by the lattice translation ${\mathbb{K}}$,
the ${\mathbb{C}}$, ${\mathbb{P}}$, ${\mathbb{T}}$ transformations
and the rotation (2.23) with
$\alpha=\pi$: ${\mathbb{U}}_{\pi}$.
In this section, we concentrate on the even sector of the
XYZ spin chain
(defined as the thermodynamic limit of finite chains with
an even number of sites),
which implies the condition
$${\mathbb{U}}^{2}_{\pi}=1\ .$$
(3.4)
Let us denote by ${\cal H}_{s}$ the eigenspace of the operator
$S_{z}$ corresponding to a given eigenvalue $s\in{\mathbb{Z}}$.
Then, the Hamiltonian (3.3) acts as,
$${\boldsymbol{\mathrm{H}}}_{\mathrm{XYZ}}\,:\,{\cal H}_{s}\to{\cal H}_{s-2}%
\oplus{\cal H}_{s}\oplus{\cal H}_{s+2}\ ,$$
and therefore the infinite degeneracy
in the vacuum sector of the XXZ chain is reduced to the two states
$$|\,s\,\rangle_{\mathrm{XYZ}}\in\oplus_{k=-\infty}^{\infty}\,{\cal H}_{s+2k}%
\qquad(\,s=0,1\,)$$
satisfying the condition444
The corresponding energies $E^{(s)}_{N}$ for a chain
with a finite number of sites $N\gg 1$ are
asymptotically degenerate in the sense that
$E^{(1)}_{N}-E^{(0)}_{N}=O\big{(}e^{-\text{const}\,N}\big{)}$
(see e.g. Ref.[8]).
$${\mathbb{U}_{\pi}}\,|\,s\,\rangle_{\mathrm{XYZ}}=e^{i\pi s}\ |\,s\,\rangle_{%
\mathrm{XYZ}}\ .$$
(3.5)
This requirement, along with the conventional normalization
of vacuum states,
$\langle\,{\mathrm{vac}}\,|\,{\mathrm{vac}}\,\rangle=1$, defines
$|\,s\,\rangle_{\mathrm{XYZ}}$ up to an overall complex phase factor.
Since the time reversal
transformation acts on states as the complex conjugation,
one can eliminate (up to sign) the ambiguity of such a phase
by imposing the condition
$${\mathbb{T}}\,|\,s\,\rangle_{\mathrm{XYZ}}=|\,s\,\rangle_{\mathrm{XYZ}}\ .$$
To gain physical intuition about the vacuum sector,
it is useful to consider a limiting case
where the Hamiltonian (3.3) simplifies drastically.
For $J_{y}=J_{z}=0$, the vacuum sector of the finite periodic XYZ chain
contains two pure ferromagnetic states
$|\,\mathrm{vac}\,\rangle^{(j)}$ ($j=0,1$).
With a proper choice of the overall phases
of these states, one may always set up the
conditions
$${\mathbb{U}}_{\pi}\,|\,\mathrm{vac}\,\rangle^{(j)}=|\,\mathrm{vac}\,\rangle^{(%
1-j)}\,,\qquad\quad{\mathbb{T}}\,|\,\mathrm{vac}\,\rangle^{(j)}=|\,\mathrm{vac%
}\,\rangle^{(j)}\ .$$
(3.6)
Moreover, since the charge conjugation matrix (2.26)
can be identify with $\prod_{l}\sigma_{l}^{x}$, one has
$${\mathbb{C}}\,|\,\mathrm{vac}\,\rangle^{(j)}=|\,\mathrm{vac}\,\rangle^{(j)}\ .$$
(3.7)
When the couplings $J_{y}$ and $J_{z}$ are non-vanishing,
the pure ferromagnetic states are no longer stationary
states, but it is still possible to introduce two vacuums
$|\,\mathrm{vac}\,\rangle^{(j)}$ satisfying
(3.6) and (3.7)
through the relation
$$|\,s\,\rangle_{\mathrm{XYZ}}=\frac{1}{\sqrt{2}}\ \big{\{}\,|\,\mathrm{vac}\,%
\rangle^{(0)}+(-1)^{s}\,|\,\mathrm{vac}\,\rangle^{(1)}\,\big{\}}\qquad\,(s=0,1%
)\ .$$
(3.8)
The Hilbert space of the XYZ chain contains two linear
subspaces ${\cal V}^{(j)}$ ($j=0,1$) associated with the vacuums
$|\,\mathrm{vac}\,\rangle^{(j)}$.
In the spectrum of the model,
there exist kink-like “massive” excitations,
${\bf B}_{+}$ and ${\bf B}_{-}$, such that the corresponding
Zamolodchikov-Faddeev operators intertwine these
subspaces ${\cal V}^{(j)}$:
$${\bf B}_{\pm}\,:\ {\cal V}^{(j)}\to{\cal V}^{(1-j)}\ .$$
One can therefore generate two sets of asymptotic states in the form
$${\bf B}_{\sigma_{2m}}(k_{2m})\ldots{\bf B}_{\sigma_{1}}(k_{1})\,|\,\mathrm{vac%
}\,\rangle^{(j)}\in{\cal V}^{(j)}\,\qquad\ (m=0,1,2\ldots)\ ,$$
(3.9)
where the Zamolodchikov-Faddeev operators
${\bf B}_{\sigma}(k)\ (\sigma=\pm)$ depend on
a quasi-momentum $k$:
$${\mathbb{K}}\,{\bf B}_{\pm}(k)\,{\mathbb{K}}^{-1}=e^{ik}\ {\bf B}_{\pm}(k)\,,%
\qquad\quad\big{[}\,{\boldsymbol{\mathrm{H}}}_{\mathrm{XYZ}}\,,\,{\bf B}_{\pm}%
(k)\,\big{]}={\cal E}(k)\ {\bf B}_{\pm}(k)\ .$$
(3.10)
The dispersion relation ${\cal E}={\cal E}(k)$ of the fundamental
excitations was calculated in work [13]. It is recalled
in Appendix A (see Eqs. (A.8), (A.9)).
The operators ${\bf B}_{\pm}$
satisfy also the conditions:
$${\mathbb{C}}\,{\bf B}_{\pm}^{(1-j,j)}\,{\mathbb{C}}=\mp\,(-1)^{j}\ {\bf B}^{(1%
-j,j)}_{\pm}\,,\qquad\quad{\mathbb{U}}_{\pi}\,{\bf B}^{(1-j,j)}_{\pm}\,{%
\mathbb{U}}_{\pi}=\pm\,(-1)^{j}\ {\bf B}^{(j,1-j)}_{\mp}\ .$$
(3.11)
In Eqs. (3.11), ${\bf B}^{(1-j,j)}_{\pm}$ denotes
the restriction of the operator ${\bf B}_{\pm}$ when it acts
on the subspace ${\cal V}^{(j)}$.
Any local lattice operator ${\cal O}_{\rm latt}$
(i.e. any operator which can be written as
a local combination of $\sigma_{l}^{a}$)
leaves the subspaces ${\cal V}^{(j)}$ invariant:
$${\cal O}_{\rm latt}\,:\,{\cal V}^{(j)}\to{\cal V}^{(j)}\ .$$
Furthermore,
the algebra of local operators ${\cal A}_{\rm loc}$ acts
invariantly on the
component of the subspace ${\cal V}^{(j)}$ generated
by the states (3.9).
For $J_{z}\leq 0$, the kinks ${\bf B}_{\pm}$ do not produce bound
states, and the two sets of
asymptotic states (3.9) obeying the condition
$$-\pi/2\leq k_{1}<k_{2}\ldots<k_{2m}<\pi/2$$
form complete in-bases in these unitary equivalent
spaces of representation of ${\cal A}_{\rm loc}$.
At first glance, to construct explicitly
the representations of ${\cal A}_{\rm loc}$,
one should put at one’s disposal the whole collection of
in-basis matrix elements for an arbitrary local operator
${\cal O}_{\text{latt}}\in{\cal A}_{\rm loc}$.
As a matter of fact, using the
so-called crossing symmetry (see e.g.[2, 25]),
one can express all
possible matrix elements in terms of those of the form
$${}^{(j)}\langle\,\mathrm{vac}\,|\,{\cal O}_{\text{latt}}\,{\bf B}_{\sigma_{2m}%
}(k_{2m})\ldots{\bf B}_{\sigma_{1}}(k_{1})\,|\,\mathrm{vac}\,\rangle^{(j)}%
\equiv{}^{(j)}\big{\langle}\,{\cal O}_{\text{latt}}\,|\,{\bf B}_{\sigma_{1}}(k%
_{1})\ldots{\bf B}_{\sigma_{2m}}(k_{2m})\,\big{\rangle}_{\rm in}\,.$$
(3.12)
Such matrix elements are known as form factors.
Currently, there exists a formal
procedure which allows one
to express form factors of local operators in
terms of multiple integrals [26, 27].
Unfortunately,
it is difficult to apply in practice,
even in the case of the local spin operators
$\sigma^{a}_{l}$ themselves and for
form factors involving only a small number of excitations.
For our purposes, we merely need the explicit form of
Vacuum Expectation Values (VEV) and of two-particle form factors.
From the relations (2.44), (3.6) and (3.7),
it follows immediately
that the VEVs of $\sigma^{y}_{l}$ and $\sigma^{z}_{l}$
vanish, whereas
$${}^{(j)}\big{\langle}\,\sigma^{x}_{l}\,\big{\rangle}=(-1)^{j}\ F\ ,$$
(3.13)
where $F$ depends on the two ratios
$J_{y}/J_{x}$ and $J_{z}/J_{x}$.
This VEV was found in work [28]
(see Appendix A, Eq. (A.7)).
As for the two-particle form factors,
the ${\mathbb{Z}}_{2}$-symmetry generated
by ${\mathbb{U}}_{\pi}$ (3.11) enables one to predict
their general form:
$$\displaystyle{}^{(j)}\big{\langle}\,\sigma^{x}_{0}\,|\,{\bf B}_{\pm}(k_{1}){%
\bf B}_{\mp}(k_{2})\big{\rangle}_{\rm in}=(-1)^{j}\ F^{x}_{1}(k_{1},k_{2})\pm F%
^{x}_{2}(k_{1},k_{2})\,,$$
(3.14)
$$\displaystyle{}^{(j)}\big{\langle}\,\sigma^{y}_{0}\,|\,{\bf B}_{\pm}(k_{1}){%
\bf B}_{\pm}(k_{2})\big{\rangle}_{\rm in}=F^{y}_{1}(k_{1},k_{2})\pm(-1)^{j}\ F%
^{y}_{2}(k_{1},k_{2})\,,$$
(3.15)
$$\displaystyle{}^{(j)}\big{\langle}\,\sigma^{z}_{0}\,|\,{\bf B}_{\pm}(k_{1}){%
\bf B}_{\pm}(k_{2})\big{\rangle}_{\rm in}=(-1)^{j}\ F^{z}_{1}(k_{1},k_{2})\pm F%
^{z}_{2}(k_{1},k_{2})\,,$$
(3.16)
where $F^{a}_{1,2}$ are some functions of the quasi-momentums $k_{1}$ and
$k_{2}$.
The invariance with respect to charge conjugation
${\mathbb{C}}$ (3.11) dictates that all other
two-particle form factors vanish.
Form factor (3.14) was calculated in
Ref.[27].
It is possible to extend the result of that work and calculate
form factors (3.15) and (3.16) as well.
We collect the explicit expressions of $F$ and $F^{a}_{1,2}$
in Appendix A.
To conclude this subsection,
let us note that, in the current treatment
of the infinite XYZ model as the thermodynamic
limit of finite periodical chains with an even
number of sites, we cannot construct excited states
containing an odd number of the elementary excitations
$${\bf B}_{\sigma_{2m+1}}(k_{2m+1})\ldots{\bf B}_{\sigma_{1}}(k_{1})\,|\,{\rm vac%
}\rangle^{(1-j)}\in{\cal V}^{(j)}\,\ \ \ \ \quad\ \ (m=0,1\ldots)\ .$$
(3.17)
Such states are deduced from finite chains
with boundary conditions breaking the translation
invariance. In particular, the linear subspace of ${\cal V}^{(j)}$
spanned by the states (3.17) can be constructed from the
thermodynamic limit
of chains with an odd number
of sites $N$ and
subject to the so called twist boundary condition555This
fact follows from the result of work [22]. :
$$\sigma_{-\frac{N-1}{2}}^{\pm}=-\sigma_{\frac{N+1}{2}}^{\pm}\,,\qquad\ \ \sigma%
_{-\frac{N-1}{2}}^{z}=\sigma_{\frac{N+1}{2}}^{z}\ .$$
The in-asymptotic states (3.9), (3.17)
form
complete bases in ${\cal V}^{(j)}$ for $J_{z}\leq 0$.
3.2 XYZ spin chain in the scaling limit
For $J_{x}=J_{y}$, the gap in the spectrum of the XYZ chain
vanishes, and its correlation length [29]
$$R_{c}\simeq\frac{1}{4}\ \bigg{[}\frac{8\,(J^{2}_{x}-J_{z}^{2})}{J_{x}(J_{x}-J_%
{y})}\bigg{]}^{\frac{1}{2-2\eta}}\,\quad\qquad\Big{(}\eta=\frac{1}{\pi}\,%
\arccos(J_{z}/J_{x})\,\Big{)}$$
(3.18)
becomes infinite. In the
limit $J_{x}\to J_{y}$, the correlation functions at large lattice
separation ($\sim R_{c}$)
assume a certain scaling form
which can be described by quantum field theory.
If $(J_{x}-J_{y})/J_{x}\ll 1$, it is natural to treat the XYZ
model as the perturbation
of the XXZ chain by the lattice operator
$$\sigma^{x}_{l}\sigma^{x}_{l+1}-\sigma^{y}_{l}\sigma^{y}_{l+1}\ .$$
The leading term in
the expansion (1.2) for this operator
is given by the relevant field ${\cal O}_{2,0}+{\cal O}_{-2,0}$
of scaling dimension $2\eta$.
Therefore, the scaling limit of the XYZ chain is described by
the sine-Gordon quantum field theory [29],
$${\bf H}_{\mathrm{sg}}={\bf H}_{\mathrm{Gauss}}-\mu\,\int dx\,\big{(}{\cal O}_{%
2,0}+{\cal O}_{-2,0}\big{)}(x)\ .$$
(3.19)
Up to a numerical factor, which was obtained in Ref. [30],
the coupling constant $\mu$ coincides with the quantity $M^{2-2\eta}$,
where the combination
$$M=(\varepsilon R_{c})^{-1}$$
(3.20)
can be naturally identified with the soliton mass in the sine-Gordon
model (3.19).
The sine-Gordon theory
admits a class of local soliton-creating operators characterized
by two integers $s,n\in{\mathbb{Z}}$, where
$n$ gives the topological charge and $sn/2$ represents
the Lorentz spin of the field.
These operators can also be expressed in a form similar to (2.14),
in which $\varphi$ denotes the sine-Gordon field
instead of the Gaussian field
obeying the simple D’Alembert equation (2.9)
(see e.g. Ref. [31] for details).
They moreover coincide with the Gaussian fields (2.14)
in the conformal limit $\mu\rightarrow 0$. Hence, with some abuse
of notation, we will denote
such soliton-creating operators in the sine-Gordon model by the
same symbol $\mathcal{O}_{s,n}$666In Ref. [31],
the soliton-creating operators
were denoted as $\mathcal{O}_{a}^{n}$, where
$a=s\sqrt{\eta}/2$. The parameter $\beta$ in [31]
coincides with $\sqrt{\eta}$.
Note that there, the quantity $2a/\beta$ was
not assumed to be integer..
To proceed further, one needs to draw a link
between local lattice operators in the XYZ chain
and local fields in the sine-Gordon model.
Let us note at this point that local expansions of the type (1.2)
are based on dimensional analysis and do not necessarily imply
the criticality of the original lattice system.
Similar expansions are expected to be applicable
to describe the near-critical behavior of lattice systems.
Usually, the term with
the smallest scaling dimension in (1.2)
governs the universal scaling behavior of
lattice correlators, whereas
the next terms produce non-universal corrections.
In particular,
relations (3.1), (3.2)
obtained for the XXZ spin chain
can be used to study the leading scaling behavior and
first non-universal
corrections of the XYZ correlation functions and form factors.
In the XYZ case, the continuous fields which appear in
(3.1), (3.2)
should be understood
as operators in the sine-Gordon model rather
than their conformal limits.
The numerical constants $C_{0}$, $C_{1}$ and $C_{1}^{z}$
remain, of course, the same as for the critical XXZ chain.
Let us now discuss the relation between the Hilbert spaces of
the XYZ and sine-Gordon models. In general, the theory (3.19) admits a discrete symmetry $\varphi\to\varphi+2\pi j/\sqrt{\eta}$
$(j\in{\mathbb{Z}})$, which is generated
by the operator ${\mathbb{U}}_{\pi}$ defined similarly to
(2.19).
For $0<\eta\leq 1$, the above symmetry
is spontaneously broken, so that
the theory has an infinite number of ground
states $|\,0_{j}\,\rangle$ $(j\in{\mathbb{Z}})$
characterized by the corresponding VEVs of the
field $\varphi$:
$$\frac{\langle\,0_{j}\,|\,\varphi(x)\,|\,0_{j}\,\rangle}{\langle\,0_{j}\,|\,0_{%
j}\,\rangle}=\frac{2\pi j}{\sqrt{\eta}}\ .$$
(3.21)
The sine-Gordon model which governs the scaling behavior of the even
sector of the XYZ chain is subject to the additional constraint
${\mathbb{U}}_{\pi}^{2}=1$.
This equation implies in particular
that the field $\varphi$ is compactified,
$\varphi\equiv\varphi+4\pi/\sqrt{\eta}$, and that,
unlike the uncompactified case,
there exist only two non-equivalent vacuum states
$|\,0_{j}\,\rangle$ with
$j=0,1$. These states are naturally identified with the scaling
limit of the two XYZ vacuums $|\,{\rm vac}\,\rangle^{(j)}$.
To describe the scaling limit of XYZ excited states,
one has to relate
the Zamolodchikov-Faddeev operators of the lattice and continuous
theories. Let us recall here that the sine-Gordon model
admits a global continuous $U(1)$ symmetry generated by
the operator
$${\mathbb{V}}_{\alpha}=e^{i\alpha Q}\,,\quad\ {\rm where}\ \quad Q=\frac{\sqrt{%
\eta}}{2\pi}\,\int_{-\infty}^{\infty}dx\,\partial_{x}\varphi\ ,$$
(3.22)
which acts on the exponential fields as follows,
$${\mathbb{V}}_{\alpha}\,{\cal O}_{s,n}\,{\mathbb{V}}^{-1}_{\alpha}=e^{in\alpha}%
\ {\cal O}_{s,n}\ .$$
Notice that the Gaussian CFT
also possesses such a global symmetry,
contrary to the XXZ and XYZ lattice models.
Nevertheless, the form of the expansions (2.4), (2.36)
and (2.39) suggests that
the lattice models are invariant with respect to the
transformation ${\mathbb{V}}_{\pi}$ which
acts trivially on
all local lattice fields.
Such symmetry manifests
itself in the existence of two subspaces ${\cal V}^{(j)}$
($j=0,1$) which can be treated as eigenspaces
of the operator ${\mathbb{V}}_{\pi}$:
$${\mathbb{V}}_{\pi}\,{\cal V}^{(j)}=(-1)^{j}\ {\cal V}^{(j)}\ .$$
The fundamental sine-Gordon kink-like excitations,
the soliton ${\bf A}_{-}$ and the antisoliton ${\bf A}_{+}$,
carry, respectively, negative and positive units
of the topological charge $Q$ (3.22):
$${\mathbb{V}}_{\alpha}\,{\bf A}_{\pm}(\theta)\,{\mathbb{V}}^{-1}_{\alpha}=e^{%
\pm i\alpha}\ {\bf A}_{\pm}(\theta)\ ,$$
(3.23)
where the argument $\theta$ denotes kink rapidity.
The relation with the Zamolodchikov-Faddeev operators of the XYZ
model was established in Ref.[32]. In our notations
it can be summarized as follows:
the operators ${\bf A}_{\pm}(k)$ defined
in the lattice model as
$${\bf A}^{(1,0)}_{\pm}=\frac{i}{\sqrt{2}}\,\big{(}\,\pm{\bf B}^{(1,0)}_{+}+{\bf
B%
}^{(1,0)}_{-}\,\big{)}\,,\qquad\ {\bf A}^{(0,1)}_{\pm}=\frac{i}{\sqrt{2}}\,%
\big{(}-{\bf B}^{(0,1)}_{+}\pm{\bf B}^{(0,1)}_{-}\,\big{)}\,,$$
(3.24)
turn out, in the scaling limit, to be the Zamolodchikov-Faddeev
operators ${\bf A}_{\pm}(\theta)$ (3.23) of the sine-Gordon model.
Here, as well as in Eq. (3.11), we denote the restriction of the
operators ${\bf B}_{\pm}$ and ${\bf A}_{\pm}$ acting
on the subspace ${\cal V}^{(j)}$ as ${\bf B}^{(1-j,j)}_{\pm}$ and
${\bf A}^{(1-j,j)}_{\pm}$.
Again, with some abuse of notation, we use the same symbol
${\bf A}_{\pm}$ for the XYZ operators and for their scaling limits.
Notice that the quasi-momentum $k$ of
the low-lying fundamental excitation becomes the
usual particle momentum in the scaling limit:
$$\lim_{\varepsilon\to 0}\,\frac{k}{\varepsilon}=M\,\sinh(\theta)\quad\text{and}%
\quad\lim_{\varepsilon\to 0}\,\frac{{\cal E}(k)}{\varepsilon}=M\,\cosh(\theta)\,,$$
where $M$ is the soliton mass (3.20).
3.3 Scaling behavior of the form factors
We are now in a position to calculate the
constants appearing in
expansions (3.1), (3.2).
To illustrate the procedure,
let us consider first the scaling behavior of the lattice
VEV (3.13). From relation
(3.1) we deduce that
$$F\sim\,2\,C_{0}\,\varepsilon^{\frac{\eta}{2}}\,\big{\langle}\,{\cal O}_{1,0}\,%
\big{\rangle}+\ldots\ .$$
(3.25)
Here $\langle\,\ldots\,\rangle$ means an averaging with respect to the
vacuum (3.21) with $j=0$. To write the equation (3.25),
we also use the fact that the VEV
of the fields ${\cal O}_{1,0}$ and ${\cal O}_{-1,0}$ are equal
by virtue of charge conjugation symmetry.
The VEV of the operator
${\cal O}_{s,0}$ was found in [33]. We denote it
as
$$\big{\langle}\,{\cal O}_{s,0}\,\big{\rangle}=\sqrt{{Z}_{s,0}}\ .$$
On the other hand,
the function $F$ is given by the Baxter-Kelland formula
[28] (see Eq.(A.7)).
Its leading scaling behavior reads
$$F\sim\frac{1}{1-\eta}\,(4R_{c})^{-{\frac{\eta}{2}}}+\ldots\,.$$
(3.26)
Comparing Eqs. (3.25) and (3.26),
and using the relation
$R^{-1}_{c}=M\varepsilon$, one can deduce the constant $C_{0}$:
$$C_{0}=\frac{1}{2(1-\eta)\sqrt{{{Z}}_{1,0}}}\,\Big{(}\frac{M}{4}\Big{)}^{\frac{%
\eta}{2}}\,.$$
(3.27)
A similar strategy can be applied to calculate the constants $C_{1}$ and
$C_{1}^{z}$ in (3.1), (3.2).
The relations (3.24) allows one to express
the leading scaling behavior of the
functions that appear in Eqs. (3.14)-(3.16) through the form factors of the
sine-Gordon fields:
$$\displaystyle F_{1}^{x}\pm i\,F_{1}^{y}\sim 2\,C_{0}\,\varepsilon^{\frac{\eta}%
{2}}\,\big{\langle}\,{\cal O}_{\pm 1,0}(0)\,|\,{\bf A}_{+}(\theta_{1}){\bf A}_%
{-}(\theta_{2})\,\big{\rangle}_{\rm in}+\ldots\,,$$
(3.28)
$$\displaystyle F_{2}^{x}\pm i\,F_{2}^{y}\sim-C_{1}\,\varepsilon^{\frac{\eta}{2}%
+\frac{1}{2\eta}}\,\big{\langle}\,{\cal O}_{\pm 1,2}(0)\,|\,{\bf A}_{-}(\theta%
_{1}){\bf A}_{-}(\theta_{2})\,\big{\rangle}_{\rm in}+\ldots\,,$$
(3.29)
and
$$\displaystyle F_{1}^{z}\sim\frac{\varepsilon}{2\pi\,\sqrt{\eta}}\ \big{\langle%
}\,\partial_{t}\varphi(0)\,|\,{\bf A}_{+}(\theta_{1}){\bf A}_{-}(\theta_{2})\,%
\big{\rangle}_{\rm in}+\ldots\,,$$
(3.30)
$$\displaystyle F_{2}^{z}\sim\frac{i}{2}\ C_{1}^{z}\ \varepsilon^{\frac{1}{2\eta%
}}\ \big{\langle}\,{\cal O}_{0,2}(0)\,|\,{\bf A}_{-}(\theta_{1}){\bf A}_{-}(%
\theta_{2})\,\big{\rangle}_{\rm in}+\ldots\ .$$
(3.31)
We use here an abbreviated notation similar to (3.12),
except that the index specifying the vacuum states is omitted
since it is always assumed to be $j=0$.
The two-particle form factors of the topologically neutral operators
${\cal O}_{\pm 1,0}$ and $\partial_{t}\varphi$ have been known for
a long time (see e.g.[25]). They read explicitly,
$$\displaystyle\big{\langle}\,{\cal O}_{\pm 1,0}(0)\,|\,{\bf A}_{+}(\theta_{1}){%
\bf A}_{-}(\theta_{2})\,\big{\rangle}_{\rm in}=\sqrt{{Z}_{1,0}}\ \frac{G(%
\theta_{1}-\theta_{2})}{\xi\ G(-i\pi)}\ \frac{2i\ e^{\mp(\theta_{1}-\theta_{2}%
+i\pi)/(2\xi)}}{\sinh\big{(}(\theta_{1}-\theta_{2}+i\pi)/\xi\big{)}}\,,$$
(3.32)
$$\displaystyle\big{\langle}\,\partial_{t}\varphi(0)\,|\,{\bf A}_{+}(\theta_{1})%
{\bf A}_{-}(\theta_{2})\,\big{\rangle}_{\rm in}=\frac{G(\theta_{1}-\theta_{2})%
}{\sqrt{\eta}\ G(-i\pi)}\ \frac{i\pi\,M\ \big{(}e^{\frac{\theta_{1}+\theta_{2}%
}{2}}+e^{-\frac{\theta_{1}+\theta_{2}}{2}}\big{)}}{\cosh\big{(}(\theta_{1}-%
\theta_{2}+i\pi)/(2\xi)\big{)}}\,,$$
(3.33)
where $\xi=\frac{\eta}{1-\eta}$ and
the function $G(\theta)$ is a so-called minimal form factor.
The form factors of
the topologically charged operators $\mathcal{O}_{s,n}$ have been proposed in
[31]. In particular, for $n=2$ one has:
$$\big{\langle}\,{\mathcal{O}}_{s,2}(0)\,|\,{\bf A}_{-}(\theta_{1}){\bf A}_{-}(%
\theta_{2})\,\big{\rangle}_{\text{in}}=\sqrt{{Z}_{s,2}}\ e^{\frac{i{\pi s}}{{2%
}}}\,e^{\frac{s\theta_{1}}{2}+\frac{s\theta_{2}}{2}}\,G(\theta_{1}-\theta_{2})\,.$$
(3.34)
The explicit formulae for the minimal form factor $G(\theta)$ and
the field-strength renormalization ${Z}_{s,n}$
are recalled in Appendix B.
We are now able to compare (3.28)-(3.31)
with expansions of the exact lattice form factors
(3.13)-(3.16)
given in Appendix A, and to relate
the values of the constants $C_{0}$, $C_{1}$ and
$C_{1}^{z}$ to the constants ${Z}_{s,n}$:
$$\displaystyle C_{1}=\frac{4}{\eta\,G(-i\pi)\,\sqrt{{{Z}}_{1,2}}}\,\Big{(}\frac%
{M}{4}\Big{)}^{\frac{\eta}{2}+\frac{1}{2\eta}}\,,$$
(3.35)
$$\displaystyle C_{1}^{z}=\frac{8}{\eta\,G(-i\pi)\,\sqrt{{{Z}}_{0,2}}}\,\Big{(}%
\frac{M}{4}\Big{)}^{\frac{1}{2\eta}}\,.$$
(3.36)
4 Correlation amplitudes. Comparison with numerical results
With the explicit expression (B.3) for the normalization constants
${{Z}}_{1,0}$, ${{Z}}_{1,2}$ and ${{Z}}_{0,2}$,
the relations (3.27), (3.35) and (3.36)
lead to the following formulae
for the correlation amplitudes $A$,
${\tilde{A}}$ and $A_{z}$ (2.50):
$${A}=\frac{1}{2(1-\eta)^{2}}\,\bigg{[}\frac{\Gamma(\frac{\eta}{2-2\eta})}{2%
\sqrt{\pi}\Gamma(\frac{1}{2-2\eta})}\bigg{]}^{\eta}\exp\bigg{\{}-\int_{0}^{%
\infty}\frac{dt}{t}\Big{(}\frac{\sinh(\eta t)}{\sinh(t)\cosh((1-\eta)t)}-{\eta%
}\,e^{-2t}\,\Big{)}\bigg{\}}\,,$$
(4.1)
$$\displaystyle\!\!\!\!\!{\tilde{A}}=\frac{2}{\eta(1-\eta)}\,\bigg{[}\frac{%
\Gamma(\frac{\eta}{2-2\eta})}{2\sqrt{\pi}\Gamma(\frac{1}{2-2\eta})}\bigg{]}^{%
\eta+\frac{1}{\eta}}$$
$$\displaystyle\ \times\exp\bigg{\{}-\int_{0}^{\infty}\frac{dt}{t}\Big{(}\frac{%
\cosh(2\eta t)e^{-2t}-1}{2\sinh(\eta t)\sinh(t)\cosh((1-\eta)t)}+\frac{1}{%
\sinh(\eta t)}-\frac{\eta^{2}+1}{\eta}\,e^{-2t}\,\Big{)}\bigg{\}}\,,$$
(4.2)
and
$$A_{z}=\frac{8}{\pi^{2}}\,\bigg{[}\frac{\Gamma(\frac{\eta}{2-2\eta})}{2\sqrt{%
\pi}\Gamma(\frac{1}{2-2\eta})}\bigg{]}^{\frac{1}{\eta}}\exp\bigg{\{}\int_{0}^{%
\infty}\frac{dt}{t}\Big{(}\frac{\sinh((2\eta-1)t)}{\sinh(\eta t)\cosh((1-\eta)%
t)}-\frac{2\eta-1}{\eta}\,e^{-2t}\,\Big{)}\bigg{\}}\,.$$
(4.3)
Note that these amplitudes obey the simple relation:
$$\frac{{\tilde{A}}}{AA_{z}}=\frac{\pi}{4}\,\frac{\Gamma^{2}\big{(}1+\frac{\eta}%
{2-2\eta}\big{)}}{\Gamma^{2}\big{(}\frac{3}{2}+\frac{\eta}{2-2\eta}\big{)}}\,.$$
(4.4)
The correlation amplitude $A$ was already obtained in [33].
Our computations also confirm the conjecture
from [34] concerning the amplitude $A_{z}$.
In [35], numerical values of the spin-spin equal-time
correlation functions
have been obtained for an open chain of 200 sites by the density-matrix
renormalization-group technique [36].
In Table 1, we compare,
for different values of the anisotropy parameter
$\Delta$, the numerical values that follow
from (4.2), (4.3) with those estimated in [35] from the
fitting of the numerical data. The corresponding plots are represented
in Figure 1.
5 Spin-spin correlation functions in the isotropic limit
5.1 Marginal perturbations of the Wess-Zumino-Witten model
As long as the parameter $\eta$ is not too close to unity,
the first terms of the
asymptotic expansions (2.47), (2.48)
provide a good
approximation to the spin-spin correlation functions even for
moderate space separations $l$. However, these expansions cannot
be directly applied in the isotropic limit $\eta\to 1$.
Indeed, in this limit, the operator ${\cal O}_{0,4}+{\cal O}_{0,-4}$ in the effective Hamiltonian (2.40)
becomes marginal and induces
logarithmic corrections to
the leading power-law asymptotic behavior.
The suitable way to explore the $\eta\to 1$ limit is
based on the low-energy
effective theory defined as a perturbation of the
Gaussian model with
$\eta=1$.
As is well known
(see e.g.[17]), the Gaussian model coincides
in this case with the SU(2) level one Wess-Zumino-Witten
(WZW) theory.
In particular, the WZW holomorphic currents
are identified with the following primary operators
of the Gaussian CFT,
$$\displaystyle{\boldsymbol{\mathnormal{J}}}^{z}_{R}=\frac{1}{4}\,(\partial_{t}-%
\partial_{x})\varphi\,,$$
$$\displaystyle{\boldsymbol{\mathnormal{J}}}^{\pm}_{R}={\cal O}_{\pm 1,\mp 2}\,,$$
(5.1)
$$\displaystyle{\boldsymbol{\mathnormal{J}}}^{z}_{L}=\frac{1}{4}\,(\partial_{t}+%
\partial_{x})\varphi\,,$$
$$\displaystyle{\boldsymbol{\mathnormal{J}}}^{\pm}_{L}={\cal O}_{\pm 1,\pm 2}\,,$$
(5.2)
whereas the matrix of the fundamental WZW field is bosonized as
$$\begin{pmatrix}{\cal O}_{0,2}&i\,{\cal O}_{-1,0}\\
i\,{\cal O}_{1,0}&{\cal O}_{0,-2}\end{pmatrix}\,.$$
The low-energy effective Hamiltonian
can be expressed as a marginal current-current perturbation of
the WZW Hamiltonian [37],
$${\boldsymbol{\mathrm{H}}}_{\text{XXZ}}={\boldsymbol{\mathrm{H}}}_{\text{WZW}}+%
\int\frac{dx}{2\pi}\,\Big{\{}\,g_{\parallel}\,{\boldsymbol{\mathnormal{J}}}^{z%
}_{\!R}{\boldsymbol{\mathnormal{J}}}^{z}_{\!L}+\frac{g_{\perp}}{2}\,\big{(}\,{%
\boldsymbol{\mathnormal{J}}}_{\!R}^{+}{\boldsymbol{\mathnormal{J}}}_{\!L}^{-}+%
{\boldsymbol{\mathnormal{J}}}_{\!R}^{-}{\boldsymbol{\mathnormal{J}}}_{\!L}^{+}%
\,\big{)}+\cdots\,\Big{\}}\,.$$
(5.3)
In this expression, the coupling constants $g_{\parallel}$ and $g_{\perp}$
should be understood as “running” ones, i.e. depending on
the renormalization scale $r$ which has the dimension of length.
The corresponding Renormalization Group (RG)
flow is known as the Kosterlitz-Thouless flow.
For our purpose, we need only to consider the domain
$$|g_{\perp}|\leqslant g_{\parallel}\,,$$
(5.4)
in which all RG
trajectories flow toward the line $g_{\perp}=0$ of the infrared-stable
fixed points associated with the Gaussian CFT.
These trajectories are characterized by the limiting values of
the running coupling $g_{\parallel}$,
$$\epsilon=\frac{1}{2}\,\lim_{r\to+\infty}g_{\parallel}(r)\,,$$
(5.5)
and the parameter $\epsilon$
is simply related with the parameter $\eta$ of the Gaussian model,
$$\epsilon=1-\eta\,.$$
(5.6)
The RG flow of the running coupling constants is defined by
a system of ordinary differential equations,
$$r\,\frac{dg_{\parallel}}{dr}=-\frac{g_{\perp}^{2}}{f_{\parallel}(g_{\parallel}%
,g_{\perp})}\ ,\qquad\qquad r\,\frac{dg_{\perp}}{dr}=-\frac{g_{\parallel}\,g_{%
\perp}}{f_{\perp}(g_{\parallel},g_{\perp})}\ .$$
(5.7)
Perturbatively, the functions $f_{\parallel,\perp}(g_{\parallel},g_{\perp})=1+O(g)$
admit loop expansions as power series in $g_{\parallel}$ and
$g_{\perp}$, and their precise form depends on
the choice of a renormalization scheme.
We use here the scheme introduced by
Al.B. Zamolodchikov[30, 38],
who showed that, under a suitable diffeomorphism in $g_{\parallel}$
and $g_{\perp}$, the functions $f_{\parallel}$ and $f_{\perp}$
can be taken equal to each other and to the quantity777The
analysis of Refs.[30, 38] concerns the
ultraviolet-stable domain of the running coupling constants,
which in the current notations is defined as
$-g_{\parallel}>|g_{\perp}|$. Due to the perturbative nature of
the RG flow equations, the $\beta$-function
for the infrared-stable domain (5.4) can
be obtained through the
formal substitution $g_{\parallel}\to-g_{\parallel}$ in the original
Zamolodchikov equations.
$$f_{\parallel}=f_{\perp}=1-\frac{g_{\parallel}}{2}\,.$$
(5.8)
With this particular choice of the $\beta$-function,
it is possible to integrate the RG flow equations exactly.
To do this, let us first note that the system
of differential equations (5.7), (5.8) admits a first
integral, the numerical value of which
is determined by means of the condition (5.5):
$$g_{\parallel}^{2}-g_{\perp}^{2}=(2\epsilon)^{2}\,.$$
(5.9)
Then the equations (5.7) are solved as
$$g_{\parallel}=2\,\epsilon\ \frac{1+q}{1-q}\,,\qquad\qquad g_{\perp}=4\,%
\epsilon\ \frac{\sqrt{q}}{1-q}\,,$$
(5.10)
where $q=q(r)$ is the solution of
$$q^{\frac{1}{2\epsilon}-\frac{1}{2}}\ (1-q)=\epsilon\,\Big{(}\frac{r_{0}}{r}%
\Big{)}^{2}\,.$$
(5.11)
As well as $\epsilon$,
the dimensional parameter $r_{0}$ is a RG invariant.
It is of the same order as the lattice spacing $\varepsilon$,
and is supposed to have a regular loop expansion of the form
$$\frac{\varepsilon}{r_{0}}=\exp\big{(}c_{0}+c_{1}\epsilon+c_{2}\epsilon^{2}+%
\ldots\,\big{)}\,.$$
(5.12)
It should be noted that the even coefficients $c_{0}$, $c_{2},\ \ldots$
in (5.12) are
essentially ambiguous and can be chosen as one wants. A variation
of these coefficients corresponds to a smooth redefinition of
the coupling constants which does not affect the $\beta$-function.
On the contrary, the odd constants $c_{2k+1}$ are unambiguous and
precisely specified, once the
form of the RG equations is fixed. It is possible to
show [30, 38] that the odd constants vanish in
Zamolodchikov’s scheme: $c_{2k+1}=0\ (k=0,1\ldots).$
Therefore, once the coefficients $c_{2k}$ in (5.12) are chosen,
the renormalization scheme is
completely specified.
As already mentioned, the perturbation by the marginal operators
produces logarithmic corrections
to the scale-invariant correlation functions.
Hence, the conformal normalization condition imposed on the
field ${\cal O}_{s,n}$ turns out to be singular for $\epsilon=0$,
and we should define renormalized fields
${\cal O}^{(\text{ren})}_{s,n}$, which
are rescaled version of the “bare” exponential
operators:
$${\cal O}^{(\text{ren})}_{s,n}(x,t;r)={\cal Z}^{-\frac{1}{2}}_{s,n}(r)\ {\cal O%
}_{s,n}(x,t)\,.$$
(5.13)
Notice that, in writing (5.13), we assume that there
is no resonance mixing of the operator ${\cal O}_{s,n}$ with other
fields, so that it is renormalized as
a singlet. In particular, one can easily check that this is indeed the case
when
$$|n|<2+2\,|s|\,.$$
(5.14)
The renormalized fields (5.13) are no longer
singular at $\epsilon=0$, but depend on the auxiliary RG scale $r$.
To specify them completely, we have to impose some non-singular
normalization condition.
The conventional condition, which is usually
imposed on Green’s function for a space-like interval
$t^{2}-x^{2}<0$, is
$$\big{\langle}\,T\,{\cal O}^{(\text{ren})}_{s,n}(x,t;r)\,{\cal O}^{(\text{ren})%
\dagger}_{s,n}(0,0;r)\,\big{\rangle}\big{|}_{\sqrt{x^{2}-t^{2}}=r}=\Big{(}%
\frac{t-x}{t+x}\Big{)}^{\frac{sn}{2}}\,.$$
(5.15)
Eqs. (5.13) and (5.15) imply that
the correlators of the “bare” exponential fields should take the
form:
$$\big{\langle}\,T\,{\cal O}_{s,n}(x,t)\,{\cal O}^{\dagger}_{s,n}(0,0)\,\big{%
\rangle}=\Big{(}\frac{t-x}{t+x}\Big{)}^{\frac{sn}{2}}\,{\cal Z}_{s,n}\big{(}%
\sqrt{x^{2}-t^{2}}\,\big{)}\,.$$
(5.16)
Then, using (3.1), (3.2),
one can express the spin-spin correlation functions
through the renormalization factors ${\cal Z}_{s,n}$.
For example, for the time-ordered correlation function of $\sigma^{x}$,
one has:
$$\displaystyle\langle\,T\,\sigma_{l+j}^{x}(t)\,\sigma_{j}^{x}(0)\,\rangle\sim A%
\,\varepsilon^{2d_{1,0}}\,{\cal Z}_{1,0}\big{(}\varepsilon\sqrt{l_{+}l_{-}}%
\big{)}\\
\displaystyle-(-1)^{l}\,{\tilde{A}}\,\varepsilon^{2d_{1,2}}\,{\cal Z}_{1,2}(%
\varepsilon\sqrt{l_{+}l_{-}})\,\bigg{\{}\,\frac{1}{2}\,\Big{(}\,\frac{l_{+}}{l%
_{-}}+\frac{l_{-}}{l_{+}}\,\Big{)}-\mathcal{R}(\varepsilon\sqrt{l_{+}l_{-}})\,%
\bigg{\}}+\ldots\,.$$
(5.17)
Here we use notations (2.49),
and the function $\mathcal{R}$ is related to the following
causal Green’s function as,
$$\big{\langle}\,T\,{\cal O}_{1,2}(x,t)\,{\cal O}_{-1,2}(0,0)\,\big{\rangle}={%
\cal Z}_{1,2}(r)\,\mathcal{R}(r)\Big{|}_{r=\sqrt{x^{2}-t^{2}}}\,.$$
(5.18)
The first terms of the perturbative expansion for
the scalar factor ${\cal Z}_{s,n}(r)$ in (5.13)
can be deduced from the results of work [39]:
$$\displaystyle{\cal Z}_{s,n}(r)={\bar{Z}}_{s,n}\,\Big{(}\frac{\varepsilon}{r}%
\Big{)}^{\frac{n^{2}}{4}+s^{2}(1+\epsilon^{2})}\ \big{(}g_{\perp}^{2}\big{)}^{%
\frac{n^{2}}{16}-\frac{s^{2}}{4}(1-\epsilon^{2})}\\
\displaystyle\times e^{u_{1}g_{\parallel}+u_{2}g_{\parallel}^{3}}\ \Big{(}1+g_%
{\perp}^{2}\,(v_{1}-v_{2}\,g_{\parallel})+O(g^{4})\,\Big{)}\,,$$
(5.19)
where
$${\bar{Z}}_{s,n}=\varepsilon^{-2d_{s,n}}\ \Big{(}\,2^{1-\epsilon}\,\sqrt{%
\epsilon}\ e^{-c_{0}\epsilon-c_{2}\epsilon^{3}+\ldots}\Big{)}^{2s^{2}-2d_{s,n}%
}\ e^{-2\,\epsilon\,u_{1}-(2\epsilon)^{3}\,u_{2}+\ldots}\,,$$
and $d_{s,n}$ is given by (2.15) with $\eta=1-\epsilon$.
The coefficients $u_{1}$, $u_{2}$, $v_{1}$ and $v_{2}$ in
these equations are listed in Appendix C. Notice that
to derive (5.19) one should assume that the field
$\mathcal{O}_{s,n}$ is mutually local with respect to the
density of the effective Hamiltonian (2.40). This
assumption implies that $s\in{\mathbb{Z}}$, but does not impose any
restriction on
$n$ in addition to (5.14).
As follows from the Callan-Symanzik equation, the
function $\mathcal{R}$ in (5.18) admits
a perturbative expansion in terms
of the running coupling constants. Explicitly, one can obtain
$$\mathcal{R}=-\frac{g_{\perp}}{4}\ \bigg{\{}\,g_{\parallel}+\Big{(}\,c-\frac{1}%
{2}\,\Big{)}\,g^{2}_{\parallel}+c\,g^{2}_{\perp}+O(g^{3})\,\bigg{\}}\ .$$
(5.20)
The constant $c$ appearing in Eq. (5.20) is
related to $c_{0}$ from (5.12) as
$$c_{0}=c+\gamma_{E}+\frac{1}{2}\,\ln(2\pi)\,,$$
(5.21)
where $\gamma_{E}=0.5772\ldots$ is the Euler constant.
Combining relation (5.17) with (5.19) and (5.20),
one can deduce the RG improved expansion of the $\sigma^{x}$ lattice
correlator
which is applicable for $\epsilon\ll 1$.
We can similarly derive an expansion
for the correlation function of $\sigma^{z}$.
The relation
$$\frac{1}{2\pi\sqrt{\eta}}\,\partial_{t}\varphi=\frac{2}{i\pi}\,\partial_{x}%
\partial_{n}{\cal O}_{0,n}\Big{|}_{n=0}\,,$$
(5.22)
which follows from the definition (2.14),
is useful to perform this computation.
Note also that the operators
$${\cal O_{+}}=\frac{{\cal O}_{0,2}+{\cal O}_{0,-2}}{\sqrt{2}}\,,\qquad\qquad{%
\cal O_{-}}=\frac{{\cal O}_{0,2}-{\cal O}_{0,-2}}{\sqrt{2}\,i}\,,$$
renormalize as singlets:
$${\cal O}^{(\text{ren})}_{\pm}(x,t;r)={\cal Z}^{-\frac{1}{2}}_{\pm}(r)\ {\cal O%
}_{\pm}(x,t)\,.$$
(5.23)
Indeed, since
$${\mathbb{C}}\,{\cal O_{\pm}}\,{\mathbb{C}}=\pm{\cal O}_{\pm}\,,$$
invariance with respect to the charge conjugation
prevents
resonance mixing of ${\cal O}_{+}$ and ${\cal O}_{-}$.
Now, using equations (5.22), (5.23)
and (3.2), one obtains
$$\displaystyle\langle\,T\,\sigma_{l+j}^{z}(t)\,\sigma_{j}^{z}(0)\,\rangle\sim-%
\frac{2}{\pi^{2}}\,(\partial_{l_{+}}+\partial_{l_{-}})^{2}\partial_{n}^{2}\,{%
\cal Z}_{0,n}\big{(}\varepsilon\sqrt{l_{+}l_{-}}\big{)}\Big{|}_{n=0}\\
\displaystyle+(-1)^{l}\,A_{z}\,\varepsilon^{2d_{0,2}}\,{\cal Z}_{-}(%
\varepsilon\sqrt{l_{+}l_{-}})+\ldots\,.$$
(5.24)
We collect in Appendix D the RG improved expansions
of the different-time two-point correlation functions
(5.17) and (5.24).
5.2 Equal-time correlation functions for the XXX spin chain.
Comparison with numerical results
Using expansions (D.1), (D.2), it
is easy to perform the isotropic limit.
Setting $\epsilon=0$ and
$g_{\perp}=g_{\parallel}=g$, one obtains the following
large $l$ expansion
for the equal-time spin-spin correlation functions888The
coefficient $\sqrt{2/\pi^{3}}$ in Eq. (5.25) was originally
obtained in Ref. [40].:
$$\displaystyle\langle\,\sigma^{x}_{l+j}\,\sigma_{j}^{x}\,\rangle$$
$$\displaystyle=(-1)^{l}\ \langle\,\sigma^{z}_{l+j}\,\sigma_{j}^{z}\,\rangle$$
$$\displaystyle\sim\sqrt{\frac{2}{\pi^{3}}}\,\frac{1}{l\,\sqrt{g}}\,\bigg{\{}\,1%
+\Big{(}\,\frac{3}{8}-\frac{c}{2}\,\Big{)}\,g+\Big{(}\,\frac{5}{128}-\frac{c}{%
16}-\frac{c^{2}}{8}\,\Big{)}\,g^{2}$$
$$\displaystyle\ \qquad\ +\Big{(}\,\frac{21}{1024}+\frac{7c}{256}-\frac{7c^{2}}{%
64}-\frac{c^{3}}{16}+\frac{13\,\zeta(3)}{32}\,\Big{)}\,g^{3}+O(g^{4})\,\bigg{\}}$$
$$\displaystyle\ \qquad\ -\frac{(-1)^{l}}{\pi^{2}\,l^{2}}\,\bigg{\{}\,1+\frac{g}%
{2}+\Big{(}c+\frac{3}{4}\,\Big{)}\,\frac{g^{2}}{2}+\frac{c(c+2)}{2}\,g^{3}+O(g%
^{4})\,\bigg{\}}+\ldots\ .$$
(5.25)
Here $g=g(l)$ is a solution of the equation
$$\sqrt{g}\,e^{\frac{1}{g}}=2\sqrt{2\pi}\,e^{\gamma_{E}+c}\,l\,,$$
(5.26)
which corresponds to the limit $\epsilon\to 0$ of
Eqs. (5.10) and (5.11).
Let us stress that, if the perturbation series in (5.25) can be summed,
then the correlation function should not
depend on the auxiliary parameter $c$ (5.12), (5.21).
This, however, is not true
if we truncate the perturbative series at some finite order.
Thus, when fitting the numerical
data with (5.25), we may treat $c$ as
an optimization parameter, allowing us to minimize the
remainder of the series, or at least to develop
some feeling concerning the effects of this remainder.
The correlation function (5.25) has been computed numerically in
[43] for $1\leqslant l\leqslant 30$.
The authors used the density-matrix algorithm [36]
to study the
large-distance decay of the correlation function for XXX spin chains
with $14\leq N\leq 70$ sites. To extract the values of the
correlation function in the infinite chain case, they adopted
the phenomenological scaling relation of Kaplan et al.[41]
(see also [42]).
The relative error of the interpolation procedure was estimated
to be of order $1\%$
for the largest $l$ values.
In Table 2, we compare those numerical data to the
results obtained from (5.25) in the cases $c=-1$ and $c=-2$.
The corresponding plots (numerical data against RG result for $c=-1$)
are given in Figure 2.
It appears that the numerical data are
consistent with our prediction within the given errors.
5.3 Erratum of [12]
The spin-spin correlation function
in the limit $\epsilon\to 0$
was previously studied in Section 5 of Ref. [12].
The analysis performed in that paper, along
with the numerical results obtained in [43] (see
Fig. 2 of [12]), strongly suggested the existence of
an additional staggered term of the
form
$$\propto\frac{(-1)^{l}+1}{l^{\eta+1}}$$
in the large distance asymptotic expansion of
the correlation function $\langle\,\sigma_{l+j}^{x}\,\sigma_{j}^{x}\,\rangle$.
It was argued in [12] that
such a term occurs because of
the presence of a correction
of
the type
$(-1)^{l}\ \partial_{x}{\cal O}_{\pm 1,0}$
in expansions (3.1).
However, the RG computation from [12] appears to be
erroneous. Indeed, Eqs. (5.6) and (5.7) from [12] have
to be replaced respectively by our equations
(D.1) and (5.25).
Therefore, contrary to what was claimed in [12],
the numerical data are consistent (within the numerical errors)
with (3.1).
6 Conclusion and further remarks
The purpose of this article is the quantitative study of the long-distance
behavior of spin-spin correlation functions for the
XXZ Heisenberg chain in the critical regime.
Our main result here is the determination of analytical expressions
for the correlation amplitudes
involved in the corresponding asymptotic expansions.
To obtain these values, we considered quantum field theory
which describes the scaling limit of the lattice model,
and compared, in this limit,
the respective normalizations
of the lattice operators and of the corresponding local fields.
This comparison was achieved by considering known exact
matrix elements
(form factors).
We would like to conclude the article with the following remarks.
$\bullet$ The method used in this work can be
applied to higher order terms in expansion
(2.4) or (2.36).
For example, we were able to
compute (up to sign factors)
all constants $C_{m}$ from (2.36) for odd integers $m=2p+1$:
$$\displaystyle(C_{2p+1})^{2}=\frac{2}{\eta(1-\eta)}\,\bigg{[}\frac{\Gamma(\frac%
{\eta}{2-2\eta})}{2\sqrt{\pi}\,\Gamma(\frac{1}{2-2\eta})}\bigg{]}^{\eta+\frac{%
(2p+1)^{2}}{\eta}}\prod_{j=1}^{p}\bigg{\{}\sin^{2}\Big{(}\frac{2\pi j}{\eta}%
\Big{)}\,\cot^{2}\Big{(}\frac{\pi(2j-1)}{2-2\eta}\Big{)}\bigg{\}}\\
\displaystyle\qquad\qquad\times\exp\bigg{\{}-\int_{0}^{\infty}\frac{dt}{t}\,%
\Big{(}\frac{\cosh(2\eta t)e^{-2(2p+1)t}-1}{2\sinh(\eta t)\sinh(t)\cosh((1-%
\eta)t)}\\
\displaystyle+\frac{2p+1}{\sinh(\eta t)}-\Big{(}\eta+\frac{(2p+1)^{2}}{\eta}\,%
\Big{)}\,e^{-2t}\,\Big{)}\,\bigg{\}}\ .$$
(6.1)
$\bullet$
One can also study expansions of other local
lattice operators in terms of the scaling fields.
For example, we have calculated the constant $C_{0}^{(s)}$ in
the leading term of the expansion of the lattice operators
$$\sigma^{\pm}_{l}\sigma^{\pm}_{l+1}\dots\sigma^{\pm}_{l+s-1}\sim C^{(s)}_{0}\,%
\varepsilon^{\frac{s^{2}\eta}{2}}\,\mathcal{O}_{\pm s,0}+\ldots\,,$$
(6.2)
for which
we obtained $(p=0,1\ldots)$:
$$\displaystyle C^{(2p)}_{0}=\bigg{[}\frac{\Gamma(\frac{\eta}{2-2\eta})}{2\sqrt{%
\pi}\,\Gamma(\frac{1}{2-2\eta})}\bigg{]}^{2\eta p^{2}}\frac{1}{\pi^{p}(1-\eta)%
^{2p^{2}-p}}\prod_{j=1}^{p}\frac{\Gamma^{2}(\frac{1}{2}+\frac{\eta(2j-1)}{2-2%
\eta})\,\Gamma^{2}(\eta(2j-1))}{\Gamma^{2}(\frac{\eta(2j-1)}{2-2\eta})}\ ,$$
(6.3)
$$\displaystyle C^{(2p+1)}_{0}=\bigg{[}\frac{\Gamma(\frac{\eta}{2-2\eta})}{2%
\sqrt{\pi}\,\Gamma(\frac{1}{2-2\eta})}\bigg{]}^{2\eta(p+\frac{1}{2})^{2}}\frac%
{1}{2\pi^{p}(1-\eta)^{2p^{2}+p+1}}\prod_{j=1}^{p}\frac{\Gamma^{2}(\frac{1}{2}+%
\frac{\eta j}{1-\eta})\,\Gamma^{2}(2\eta j)}{\Gamma^{2}(\frac{\eta j}{1-\eta})}$$
$$\displaystyle \times\exp\bigg{\{}-\int_{0}^{\infty%
}\frac{dt}{t}\ \Big{(}\,\frac{\sinh(\eta t)}{2\sinh(t)\cosh((1-\eta)t)}-\frac{%
\eta}{2}\ e^{-2t}\,\Big{)}\,\bigg{\}}\ .$$
(6.4)
$\bullet$
Eventually, one can wonder if it is possible to confirm
our predictions
from existing integral representations of lattice correlators.
Up to now, although
explicit expressions for the equal-time
spin-spin correlation functions at finite lattice
distances are known [44, 4],
their long-distance behavior
was studied only for the so-called “free fermion point”,
$\Delta=0$. In this case the XXZ spin chain
can be mapped onto
two non-interacting critical Ising models and the long-distance
asymptotics are readily derived from results of
works [45, 46, 47].
$\bullet$ The approach of this work
is actually quite general for lattice solvable models at criticality:
from a knowledge of particular form factors
of a lattice theory and of its quantum field theory counterpart at the
scaling limit, it is possible to predict the amplitudes
which govern the large distance behavior of lattice correlation
functions. It would indeed be interesting to obtain effective results for
other critical exactly solvable lattice models.
Acknowledgments
We would like to thank A. Furusaki for kindly providing us
with the numerical
data of work [35] and A.B. Zamolodchikov for
interesting discussions.
S.L. acknowledge helpful discussions with V.V. Bazhanov.
We are also grateful to Daniela Kusmierek for her careful reading of the
manuscript.
This research is supported in part by DOE grant
$\#$DE-FG02-96ER40959. V.T. is also supported in part by CNRS.
Appendix A Two-particle form factors in the XYZ model
In this appendix, we collect explicit expressions of
two-particle form factors
of local spin operators in the XYZ model.
Following Baxter [24, 8], we use the parameterization
of the coupling constants $J_{x}>J_{y}>|J_{z}|$ of the
Hamiltonian (3.3) in terms of $0<\eta<1$ and of the
elliptic nome $0<p<1$:
$$\displaystyle J_{x}=\frac{1-\eta}{\pi\varepsilon}\ \biggl{(}\frac{\vartheta_{4%
}(\eta)\vartheta^{\prime}_{1}(0)}{\vartheta_{4}(0)\vartheta_{1}(\eta)}+\frac{%
\vartheta_{1}(\eta)\vartheta^{\prime}_{1}(0)}{\vartheta_{4}(0)\vartheta_{4}(%
\eta)}\biggr{)}\,,$$
(A.1)
$$\displaystyle J_{y}=\frac{1-\eta}{\pi\varepsilon}\ \biggl{(}\frac{\vartheta_{4%
}(\eta)\vartheta^{\prime}_{1}(0)}{\vartheta_{4}(0)\vartheta_{1}(\eta)}-\frac{%
\vartheta_{1}(\eta)\vartheta^{\prime}_{1}(0)}{\vartheta_{4}(0)\vartheta_{4}(%
\eta)}\biggr{)}\,,$$
(A.2)
$$\displaystyle J_{z}=\frac{1-\eta}{\pi\varepsilon}\ \biggl{(}\frac{\vartheta^{%
\prime}_{1}(\eta)}{\vartheta_{1}(\eta)}-\frac{\vartheta^{\prime}_{4}(\eta)}{%
\vartheta_{4}(\eta)}\biggr{)}\,.$$
(A.3)
Here $\vartheta_{i}(u)\equiv\vartheta_{i}(u,p)$
denote the elliptic theta-functions
$$\displaystyle\vartheta_{1}(u,p)=2p^{1/4}\ \sin(\pi u)\ \prod_{n=1}^{\infty}(1-%
p^{2n})(1-2p^{2n}\cos(2\pi u)+p^{4n})\,,$$
(A.4)
$$\displaystyle\vartheta_{4}(u,p)=\prod_{n=1}^{\infty}(1-p^{2n})(1-2p^{2n-1}\cos%
(2\pi u)+p^{2(2n-1)})\,,$$
(A.5)
and the prime in Eqs. (A.1)-(A.3) means a derivative: $\vartheta^{\prime}_{1}=\partial_{u}\vartheta_{1}$. We shall also
use the other conventional theta-functions
$$\vartheta_{2}(u)=\vartheta_{1}(u+1/2)\,,\qquad\qquad\ \vartheta_{3}(u)=%
\vartheta_{4}(u+1/2)\,,$$
and the notation,
$$\xi=\frac{\eta}{1-\eta}\ .$$
(A.6)
With this parameterization,
the VEV of $\sigma^{x}$ (3.13) is given
by the Baxter-Kelland formula [28]:
$$F=(1+\xi)\ p^{\frac{\xi}{8}}\ \prod\limits_{n=1}^{\infty}\biggl{(}\,\frac{1-p^%
{n(1+\xi)}}{1-p^{(n-\frac{1}{2})(1+\xi)}}\ \frac{1-p^{n-\frac{1}{2}}}{1-p^{n}}%
\,\biggr{)}^{2}\ .$$
(A.7)
In order to describe the two-particle form factors,
one needs to know the explicit form
of the dispersion relation (3.10).
For this purpose, it is convenient to parameterize
the quasi-momentum $k$ by means of the so-called rapidity variable
$\theta$:
$$e^{ik(\theta)}=\frac{\vartheta_{4}\big{(}\,\frac{\theta}{2i\pi}-\frac{1}{4}\,,%
\,p^{\frac{1+\xi}{4}}\,\big{)}}{\vartheta_{4}\big{(}\,\frac{\theta}{2i\pi}+%
\frac{1}{4}\,,\,p^{\frac{1+\xi}{4}}\,\big{)}}\,.$$
(A.8)
As a function of $\theta$, the excitation energy explicitly reads
[13],
$${\cal E}(\theta)=\frac{\partial k(\theta)}{\partial\theta}\,.$$
(A.9)
Equations (A.8) and (A.9) define
the dispersion relation ${\cal E}={\cal E}(k)$ in
parametric form.
The two-particle form factors of the spin operators can be computed
by means of the $q$-vertex operator approach,
for which progress has been made recently in the XYZ case
[26, 27]. In [27], the two-particle form factors of the
$\sigma^{x}$ operator were obtained999Note that the regime considered in [27] is the so-called principal
one ($-J_{z}>J_{x}\geq|J_{y}|$).
To apply the results obtained there to the
case with
$J_{x}>J_{y}\geq|J_{z}|$, one has to replace $\sigma_{l}^{x}$, $\sigma_{l}^{y}$ and
$\sigma_{l}^{z}$ from [27]
respectively by $\sigma_{l}^{y}$, $(-1)^{l}\sigma_{l}^{z}$ and $(-1)^{l}\sigma_{l}^{x}$, which
corresponds to a similarity transformation of the Hamiltonian
(3.3).
:
$$\displaystyle F_{1}^{x}=\frac{F_{0}\ {\bar{G}}(\theta_{1}-\theta_{2}\,,\,p)\ %
\vartheta_{4}\big{(}\,0\,,\,p^{\frac{1}{2}}\,\big{)}}{\vartheta_{4}\big{(}\,%
\frac{\theta_{1}}{2i\pi}-\frac{1}{4}\,,\,p^{\frac{1+\xi}{4}}\,\big{)}\ %
\vartheta_{4}\big{(}\,\frac{\theta_{2}}{2i\pi}-\frac{1}{4}\,,\,p^{\frac{1+\xi}%
{4}}\,\big{)}}\ \ \frac{\vartheta_{4}\big{(}\,\frac{\theta_{1}+\theta_{2}}{2i%
\pi}\,,\,p^{\frac{1+\xi}{2}}\,\big{)}}{\vartheta_{1}\big{(}\,\frac{\theta_{1}-%
\theta_{2}+i\pi}{2i\pi\xi}\,,\,p^{\frac{1+\xi}{2\xi}}\,\big{)}}\,,$$
(A.10)
$$\displaystyle F_{2}^{x}=\frac{F_{0}\ {\bar{G}}(\theta_{1}-\theta_{2}\,,\,p)\ %
\vartheta_{4}\big{(}\,0\,,\,p^{\frac{1}{2}}\,\big{)}}{\vartheta_{4}\big{(}\,%
\frac{\theta_{1}}{2i\pi}-\frac{1}{4}\,,\,p^{\frac{1+\xi}{4}}\,\big{)}\ %
\vartheta_{4}\big{(}\,\frac{\theta_{2}}{2i\pi}-\frac{1}{4}\,,\,p^{\frac{1+\xi}%
{4}}\,\big{)}}\ \ \frac{\vartheta_{1}\big{(}\,\frac{\theta_{1}+\theta_{2}}{2i%
\pi}\,,\,p^{\frac{1+\xi}{2}}\,\big{)}}{\vartheta_{4}\big{(}\,\frac{\theta_{1}-%
\theta_{2}+i\pi}{2i\pi\xi}\,,\,p^{\frac{1+\xi}{2\xi}}\,\big{)}}\,.$$
(A.11)
For $-2\pi<\Im m(\theta)<0$ the
meromorphic function ${\bar{G}}$ reads
$$\bar{G}(\theta,\,p)=e^{\frac{\delta(1+\xi)}{8\pi\xi}\,(\theta+i\pi)^{2}}\,\exp%
\bigg{\{}\,\sum_{n=1}^{\infty}\,\frac{1}{n}\ \frac{\sin^{2}(\delta n(\theta+i%
\pi)/2)\,\sinh(\pi\delta(\xi+1)n/2)}{\sinh(\pi\delta n)\,\sinh(\pi\delta\xi n/%
2)\,\cosh(\pi\delta n/2)}\,\bigg{\}}\,,$$
(A.12)
and it is defined through an analytic continuation
outside this domain.
The parameter $\delta$ in (A.12) is
related to the elliptic nome as
$p=e^{-\frac{4\pi}{\delta(\xi+1)}},$ and
the constant $F_{0}$ in (A.10), (A.11) is given by
$$F_{0}=\frac{1+\xi}{\pi\xi}\ \ \frac{\theta^{\prime}_{1}\big{(}\,0\,,\,p^{\frac%
{1+\xi}{2}}\,\big{)}\ \theta^{\prime}_{1}\big{(}\,0\,,\,p^{\frac{1+\xi}{2\xi}}%
\,\big{)}}{\theta^{\prime}_{1}\big{(}\,0\,,\,p^{\frac{1}{2}}\,\big{)}}\,.$$
Notice that, in writing the form factors, we always
assume the conventional normalization of vacuum states,
$\langle\,{\rm vac}\,|\,{\rm vac}\,\rangle=1$, and of
in-asymptotic states:
$${}_{\rm in}\big{\langle}\,{\bf B}_{\sigma^{\prime}_{n}}(k^{\prime}_{n})\ldots{%
\bf B}_{\sigma^{\prime}_{1}}(k^{\prime}_{1})\,|\,{\bf B}_{\sigma_{1}}(k_{1})%
\ldots{\bf B}_{\sigma_{n}}(k_{n})\,\big{\rangle}_{\rm in}=(2\pi)^{n}\ \prod_{j%
=1}^{n}\delta_{\sigma_{j}\sigma^{\prime}_{j}}\ \delta(\theta_{j}-\theta^{%
\prime}_{j})\,,$$
(A.13)
where $k_{j}=k(\theta_{j})$ and $k^{\prime}_{j}=k(\theta^{\prime}_{j})$.
Using the method proposed in [27], one can also compute the
two-particle form factors of the other spin fields, $\sigma^{y}$ and
$\sigma^{z}$:
$$\displaystyle F_{1}^{y}=-\ \frac{F_{0}\ {\bar{G}}(\theta_{1}-\theta_{2}\,,\,p)%
\ \vartheta_{3}\big{(}\,0\,,\,p^{\frac{1}{2}}\,\big{)}}{\vartheta_{4}\big{(}\,%
\frac{\theta_{1}}{2i\pi}-\frac{1}{4}\,,\,p^{\frac{1+\xi}{4}}\,\big{)}\ %
\vartheta_{4}\big{(}\,\frac{\theta_{2}}{2i\pi}-\frac{1}{4}\,,\,p^{\frac{1+\xi}%
{4}}\,\big{)}}\ \ \frac{\vartheta_{3}\big{(}\,\frac{\theta_{1}+\theta_{2}}{2i%
\pi}\,,\,p^{\frac{1+\xi}{2}}\,\big{)}}{\vartheta_{2}\big{(}\,\frac{\theta_{1}-%
\theta_{2}+i\pi}{2i\pi\xi}\,,\,p^{\frac{1+\xi}{2\xi}}\,\big{)}}\,,$$
(A.14)
$$\displaystyle F_{2}^{y}=-\ \frac{F_{0}\ {\bar{G}}(\theta_{1}-\theta_{2}\,,\,p)%
\ \vartheta_{3}\big{(}\,0\,,\,p^{\frac{1}{2}}\,\big{)}}{\vartheta_{4}\big{(}\,%
\frac{\theta_{1}}{2i\pi}-\frac{1}{4}\,,\,p^{\frac{1+\xi}{4}}\,\big{)}\ %
\vartheta_{4}\big{(}\,\frac{\theta_{2}}{2i\pi}-\frac{1}{4}\,,\,p^{\frac{1+\xi}%
{4}}\,\big{)}}\ \ \frac{\vartheta_{2}\big{(}\,\frac{\theta_{1}+\theta_{2}}{2i%
\pi}\,,\,p^{\frac{1+\xi}{2}}\,\big{)}}{\vartheta_{3}\big{(}\,\frac{\theta_{1}-%
\theta_{2}+i\pi}{2i\pi\xi}\,,\,p^{\frac{1+\xi}{2\xi}}\,\big{)}}\,.$$
(A.15)
and
$$\displaystyle F_{1}^{z}=i\ \ \frac{F_{0}\ {\bar{G}}(\theta_{1}-\theta_{2}\,,\,%
p)\ \vartheta_{2}\big{(}\,0\,,\,p^{\frac{1}{2}}\,\big{)}}{\vartheta_{4}\big{(}%
\,\frac{\theta_{1}}{2i\pi}-\frac{1}{4}\,,\,p^{\frac{1+\xi}{4}}\,\big{)}\ %
\vartheta_{4}\big{(}\,\frac{\theta_{2}}{2i\pi}-\frac{1}{4}\,,\,p^{\frac{1+\xi}%
{4}}\,\big{)}}\ \ \frac{\vartheta_{2}\big{(}\,\frac{\theta_{1}+\theta_{2}}{2i%
\pi}\,,\,p^{\frac{1+\xi}{2}}\,\big{)}}{\vartheta_{2}\big{(}\,\frac{\theta_{1}-%
\theta_{2}+i\pi}{2i\pi\xi}\,,\,p^{\frac{1+\xi}{2\xi}}\,\big{)}}\,,$$
(A.16)
$$\displaystyle F^{z}_{2}=i\ \ \frac{F_{0}\ {\bar{G}}(\theta_{1}-\theta_{2}\,,\,%
p)\ \vartheta_{2}\big{(}\,0\,,\,p^{\frac{1}{2}}\,\big{)}}{\vartheta_{4}\big{(}%
\,\frac{\theta_{1}}{2i\pi}-\frac{1}{4}\,,\,p^{\frac{1+\xi}{4}}\,\big{)}\ %
\vartheta_{4}\big{(}\,\frac{\theta_{2}}{2i\pi}-\frac{1}{4}\,,\,p^{\frac{1+\xi}%
{4}}\,\big{)}}\ \ \frac{\vartheta_{3}\big{(}\,\frac{\theta_{1}+\theta_{2}}{2i%
\pi}\,,\,p^{\frac{1+\xi}{2}}\,\big{)}}{\vartheta_{3}\big{(}\,\frac{\theta_{1}-%
\theta_{2}+i\pi}{2i\pi\xi}\,,\,p^{\frac{1+\xi}{2\xi}}\,\big{)}}\,.$$
(A.17)
To compare the lattice and the sine-Gordon two-particle form factors,
one should take the
limit $p\rightarrow 0$. Notice that
$$p\simeq(4\,R_{c})^{-\frac{4}{\xi+1}}\,,$$
where the correlation length is defined as in (3.18).
The following relation between the function
${\bar{G}}$ (A.12) and
the sine-Gordon minimal form factor (B.2):
$$\lim_{p\to 0}{\bar{G}}(\theta\,,\,p)=\frac{G(\theta)}{G(-i\pi)}\,.$$
is useful to proceed with this limit.
Appendix B Form factors of topologically charged operators in the
sine-Gordon model
In this appendix, we recall the expressions obtained in [31]
concerning the form factors of
the topologically charged (or soliton-creating)
operators $\mathcal{O}_{s,n}$ in the sine-Gordon model.
The simplest non-vanishing form factor of
$\mathcal{O}_{s,n}$ is given by the
formula:
$$\big{\langle}\,{\mathcal{O}}_{s,n}(0)\,|\,{\bf A}_{-}(\theta_{1})\cdots{\bf A}%
_{-}(\theta_{n})\big{\rangle}_{\rm in}=\sqrt{{Z}_{s,n}}\ e^{\frac{i{\pi ns}}{4%
}}\,\prod_{m=1}^{n}\,e^{\frac{s\theta_{m}}{2}}\,\prod_{m<j}\,G(\theta_{m}-%
\theta_{j})\,.$$
(B.1)
Here, the minimal form factor $G$ has a form:
$$G(\theta)=i\,{\mathcal{C}}_{1}\,\sinh(\theta/2)\ \exp\bigg{\{}\int_{0}^{\infty%
}\frac{dt}{t}\ \frac{\sinh^{2}(t(1-i\theta/\pi))\,\sinh((\xi-1)t)}{\sinh(2t)\,%
\cosh(t)\,\sinh(\xi t)}\,\bigg{\}}\ .$$
(B.2)
The explicit expression of the normalization constant
${Z}_{s,n}$, which has been conjectured in [31],
is the following:
$$\displaystyle{Z}_{s,n}=\Big{(}\frac{{\mathcal{C}}_{2}}{2\,\mathcal{C}_{1}^{2}}%
\Big{)}^{\frac{n}{2}}\ \Big{(}\frac{\xi\,{\mathcal{C}}_{2}}{16}\Big{)}^{-\frac%
{n^{2}}{4}}\ \bigg{[}\,\frac{\sqrt{\pi}M\Gamma\big{(}\frac{3}{2}+\frac{\xi}{2}%
\big{)}}{\Gamma\big{(}\frac{\xi}{2}\big{)}}\,\bigg{]}^{2d_{s,n}}$$
$$\displaystyle\ \times\exp\bigg{\{}\int_{0}^{\infty}\frac{dt}{t}\,\Big{[}\,%
\frac{\cosh(2\xi st)\,e^{-(1+\xi)nt}-1}{2\,\sinh(\xi t)\sinh((1+\xi)t)\cosh(t)%
}+\frac{n}{2\,\sinh(t\xi)}-2\,d_{s,n}\,e^{-2t}\Big{]}\,\bigg{\}}\,.$$
(B.3)
In the previous formulae we use the notations,
$$\displaystyle{\mathcal{C}}_{1}\equiv G(-i\pi)=\exp\bigg{\{}\,-\int_{0}^{\infty%
}\frac{dt}{t}\frac{\sinh^{2}(t/2)\,\sinh((\xi-1)t)}{\sinh(2t)\,\cosh(t)\,\sinh%
(\xi t)}\,\bigg{\}}\,,$$
(B.4)
$$\displaystyle{\mathcal{C}}_{2}=\exp\bigg{\{}\,4\,\int_{0}^{\infty}\frac{dt}{t}%
\frac{\sinh^{2}(t/2)\,\sinh((\xi-1)t)}{\sinh(2t)\,\sinh(\xi t)}\,\bigg{\}}\,,$$
(B.5)
and $\xi$ is given by (A.6).
Appendix C Numerical coefficients for equation (5.19)
We collect in this appendix the explicit expressions of the coefficients
$u_{1}$, $u_{2}$, $v_{1}$ and $v_{2}$ which occur in
the expansion (5.19).
$$\displaystyle u_{1}=$$
$$\displaystyle\frac{n^{2}-4s^{2}}{16}\,\Big{(}\,T_{s}\Big{(}\frac{n}{2}\Big{)}-%
\frac{3}{2}\,\Big{)}+\frac{s(s-1)}{4}\,,$$
(C.1)
$$\displaystyle u_{2}=$$
$$\displaystyle\frac{(n^{2}-4s^{2})(n^{2}+4s^{2}-8)}{3072}\ T_{s}^{\prime\prime}%
\Big{(}\frac{n}{2}\Big{)}+\frac{n(n^{2}-4)}{192}\,T_{s}^{\prime}\Big{(}\frac{n%
}{2}\Big{)}$$
$$\displaystyle\quad+\frac{3n^{2}-4}{192}\,T_{s}\Big{(}\frac{n}{2}\Big{)}-\frac{%
s(s+2)}{192}-\frac{11n^{2}}{768}+\frac{c}{24}+\frac{c_{2}\,(n^{2}-4s^{2})}{32}\,,$$
where
$$\displaystyle T_{s}(z)=\psi(z+s)+\psi(-z+s)+2\gamma_{E}+2c\,,$$
$$\displaystyle T_{s}^{\prime}(z)=\partial_{z}\,T_{s}(z)\,,\qquad T_{s}^{\prime%
\prime}(z)=\partial_{z}^{2}\,T_{s}(z)\,,$$
and $\psi(z)=\partial_{z}\log\Gamma(z)$.
The constants $c$ and $c_{2}$ are the same as in Eqs. (5.12),
(5.21).
The coefficients $v_{1}$ and (using the expressions $u_{1,2}$ from
(C.1)) $v_{2}$ are
$$\displaystyle v_{1}=$$
$$\displaystyle\frac{n(n^{2}-4s^{2})}{128}\,T_{s}^{\prime}\Big{(}\frac{n}{2}\Big%
{)}+\frac{n^{2}-4s^{2}}{64}\,T_{s}^{2}\Big{(}\frac{n}{2}\Big{)}$$
(C.2)
$$\displaystyle\quad-\frac{3n^{2}+4s(2-5s)}{64}\,T_{s}\Big{(}\frac{n}{2}\Big{)}+%
\frac{7n^{2}+4s(10-17s)}{128}+\frac{u_{1}}{2}\,,$$
$$\displaystyle v_{2}=$$
$$\displaystyle\frac{(n^{2}-4s^{2})(8-3n^{2})}{3072}\,T_{s}^{\prime\prime}\Big{(%
}\frac{n}{2}\Big{)}$$
$$\displaystyle\quad-\bigg{(}\,\frac{n(n^{2}-4s^{2})}{128}\,T_{s}\Big{(}\frac{n}%
{2}\Big{)}+\frac{n(n^{2}+4s^{2}-8s-8)}{256}\,\bigg{)}\,T^{\prime}_{s}\Big{(}%
\frac{n}{2}\Big{)}$$
$$\displaystyle\quad-\frac{n^{2}-4s^{2}}{192}\,T^{3}_{s}\Big{(}\frac{n}{2}\Big{)%
}-\frac{n^{2}+4s(s-2)}{128}\,T^{2}_{s}\Big{(}\frac{n}{2}\Big{)}-\frac{n^{2}-8(%
s^{2}-s+1)}{128}\ T_{s}\Big{(}\frac{n}{2}\Big{)}$$
$$\displaystyle\quad-\frac{n^{2}-4s^{2}}{128}\,\big{(}2\,c_{2}-14\,\zeta(3)-3%
\big{)}-\frac{s(s-4)}{64}+\frac{u_{1}}{8}+\frac{v_{1}}{2}+\frac{3u_{2}}{2}-%
\frac{c}{8}\ .$$
Appendix D Spin-spin correlation
functions for $\epsilon\ll 1$
In this appendix,
we give RG improved expansions for the different-time
spin-spin correlation functions which were
discussed in Section 5:
$$\displaystyle\langle\,\,T\,\sigma_{l+j}^{x}(t)\,\sigma_{j}^{x}(0)\,\,\rangle%
\sim\sqrt{\frac{2}{\pi^{3}}}\ \frac{e^{-(c+\gamma_{E}+\frac{1}{2}\log(8\pi)+%
\frac{1}{4})\epsilon^{2}}}{(\sqrt{l_{+}l_{-}})^{1+\epsilon^{2}}\ (g^{2}_{\perp%
})^{\frac{1-\epsilon^{2}}{4}}}\\
\displaystyle\quad\,\times e^{u_{1}^{x}g_{\parallel}+u_{2}^{x}g_{\parallel}^{3%
}}\,\Big{\{}\,1+g_{\perp}^{2}(v_{1}^{x}-v_{2}^{x}g_{\parallel})+O(g^{4})\,\Big%
{\}}-\frac{(-1)^{l}}{\pi^{2}}\,\frac{(g^{2}_{\perp})^{\frac{\epsilon^{2}}{4}}%
\ e^{-(\gamma_{E}+\frac{1}{2}\log(8\pi))\epsilon^{2}}}{(\sqrt{l_{+}l_{-}})^{2+%
\epsilon^{2}}}\\
\displaystyle\quad\,\times\exp\bigg{\{}\,\frac{g_{\parallel}}{2}+\frac{c}{2}\,%
g_{\perp}^{2}+\Big{(}\frac{1}{96}+\frac{c}{8}\Big{)}\,g_{\parallel}^{3}-\Big{(%
}\frac{1}{32}-\frac{c}{8}-\frac{c^{2}}{2}\Big{)}\,g_{\parallel}g_{\perp}^{2}+O%
(g^{4})\bigg{\}}\\
\displaystyle\times\bigg{[}\,\frac{1}{2}\ \Big{(}\frac{l_{+}}{l_{-}}+\frac{l_{%
-}}{l_{+}}\Big{)}+\frac{{g_{\perp}}}{4}\,\Big{(}g_{\parallel}+\Big{(}c-\frac{1%
}{2}\Big{)}\,g^{2}_{\parallel}+c\,g_{\perp}^{2}+O(g^{3})\Big{)}\,\bigg{]}+\cdots$$
(D.1)
and
$$\displaystyle\langle\,\,T\,\sigma_{l+j}^{z}(t)\,\sigma_{j}^{z}(0)\,\,\rangle%
\sim\sqrt{\frac{8}{\pi^{3}}}\ \ \frac{(-1)^{l}\ \sqrt{g_{\perp}}\ e^{(\frac{1}%
{4}+c)\epsilon^{2}}}{\sqrt{l_{+}l_{-}}\ (g_{\parallel}+g_{\perp})}\\
\displaystyle\quad\,\times e^{u_{1}^{z}g_{\parallel}+u_{2}^{z}g_{\parallel}^{3%
}+g_{\perp}^{2}(v_{1}^{z}-v_{2}^{z}g_{\parallel})}\ \Big{(}1-g_{\perp}\,(w_{1}%
-w_{2}\,g_{\parallel}+w_{3}\,g_{\parallel}^{2}+w_{4}\,g_{\perp}^{2})+O(g^{4})%
\,\Big{)}\\
\displaystyle\quad\,-\frac{1}{\pi^{2}}\ \frac{1}{l_{+}l_{-}\,(1-\frac{g_{%
\parallel}}{2})}\ \bigg{[}\,\frac{1}{2}\ \Big{(}\frac{l_{+}}{l_{-}}+\frac{l_{-%
}}{l_{+}}\Big{)}\,\Big{(}\,1+\Big{(}c-\frac{1}{4}\Big{)}\frac{g_{\perp}^{2}}{2%
}\\
\displaystyle+\Big{(}2c^{2}+c-\frac{1}{4}\Big{)}\,\frac{g_{\perp}^{2}g_{%
\parallel}}{4}+O(g^{4})\,\Big{)}+\frac{g_{\perp}^{2}}{4}\Big{(}\,1+\Big{(}\,2c%
-\frac{1}{2}\,\Big{)}\,g_{\parallel}+O(g^{2})\,\Big{)}\,\bigg{]}+\dots\,.$$
(D.2)
In these expressions, the constants are given by
$$\displaystyle u_{1}^{x}=\frac{3}{8}-\frac{c}{2}\,,$$
$$\displaystyle u_{2}^{x}=-\frac{1}{64}-\frac{\zeta(3)}{48}-\frac{c_{2}}{8}\,,$$
$$\displaystyle v_{1}^{x}=-\frac{1}{32}+\frac{c}{8}-\frac{c^{2}}{4}\,,$$
$$\displaystyle v_{2}^{x}=-\frac{5}{128}-\frac{41}{96}\,\zeta(3)+\frac{c^{3}}{6}%
-\frac{c_{2}}{8}\,,$$
and
$$\displaystyle w_{1}=c\,,$$
$$\displaystyle w_{3}=-\frac{1}{16}-\frac{7}{12}\,\zeta(3)+\frac{c}{8}+\frac{c^{%
2}}{2}+\frac{c^{3}}{6}+\frac{c_{2}}{4}\,,$$
$$\displaystyle w_{2}=\frac{1}{8}-\frac{c(1+c)}{2}\,,$$
$$\displaystyle w_{4}=-\frac{1}{32}-\frac{13}{24}\,\zeta(3)-\frac{c}{8}+\frac{c^%
{2}}{4}-\frac{c^{3}}{6}-\frac{c_{2}}{4}\,,$$
$$\displaystyle u_{1}^{z}=\frac{3}{8}+\frac{c}{2}\,,$$
$$\displaystyle u_{2}^{z}=\frac{1}{64}+\frac{\zeta(3)}{48}+\frac{c}{8}+\frac{c_{%
2}}{8}\,,$$
$$\displaystyle v_{1}^{z}=-\frac{5}{32}+\frac{5c}{8}+\frac{3c^{2}}{4}\,,$$
$$\displaystyle v_{2}^{z}=\frac{11}{128}+\frac{71}{96}\,\zeta(3)+\frac{c}{4}-%
\frac{5\,c^{2}}{4}-\frac{2}{3}c^{3}+\frac{c_{2}}{8}\,.$$
The running couplings $g_{\parallel},\,g_{\perp}$ in
(D.1), (D.2) are defined by equations
(5.10) and (5.11) where
$$\frac{r}{r_{0}}=\sqrt{2\pi\,l_{+}l_{-}}\ e^{\gamma_{E}+c+c_{2}\,\epsilon^{2}+%
\ldots}\,\qquad\qquad\ {\rm and}\qquad\qquad l_{\pm}=l\pm\frac{t}{\varepsilon}\,.$$
Setting $l_{+}=l_{-}=l$ (equal time) and $g_{\perp}=g_{\parallel}=g,\ \epsilon=0$
(isotropic limit) in (D.1), (D.2),
one obtains (5.25).
References
[1]
V.E. Korepin, N.M. Bogoliubov and A.G. Izergin,
Quantum inverse scattering method and correlation functions
(Cambridge Univ. Press, 1993)
[2]
M. Jimbo and T. Miwa,
Algebraic analysis of solvable lattice models
(American Mathematical Society, 1995)
[3]
N. Kitanine, J.M. Maillet and V. Terras,
Nucl. Phys. B567 (2000) 554
[4]
N. Kitanine, J.M. Maillet, N. Slavnov and V. Terras,
Spin-spin correlation functions of the
XXZ-1/2 Heisenberg chain in a magnetic field,
LPENSL-TH-01/02
(hep-th/0201045)
[5]
A.M. Polyakov,
Sov. Phys. JETP, 39 (1970) 10
[6]
A.Z. Patashinskii and V.N. Pokrovskii,
Fluctuation theory of phase transition
(Pergamon Press, Oxford, 1979)
[7]
L.P. Kadanoff,
Statistical Physics. Statics, Dynamics and
Renormalization (World Scientific,
Singapore, 2000)
[8]
R.J. Baxter,
Exactly solved models in statistical mechanics (Academic Press,
London - New York, 1982)
[9]
A. Luther and I. Peschel,
Phys. Rev. B12 (1975) 3908
[10]
L.P. Kadanoff and A.C. Brown,
Ann. Phys. (N.Y.)
121 (1979) 318
[11]
I. Affleck,
In: Les Houches,
Session XLIX, Champs, Cordes et Phénomènes
critiques, eds. E. Brezin and J. Zinn-Justin (Elsevier
Science, 1989)
[12]
S. Lukyanov,
Nucl. Phys. B522 (1998) 533
[13]
J.D. Johnson, S. Krinsky and B.M. McCoy,
Phys. Rev. A8 (1973) 2526
[14]
L.D. Faddeev and L.A. Takhtajan,
Phys. Lett. A85 (1981) 375
[15]
W. Thirring,
Ann. Phys. (NY) 3 (1958) 91
[16]
B. Klaiber,
In: Cargèse Lectures in Theoretical Physics,
1964, ed. M. Levy (Gordon and Breach, New York, 1966)
[17]
Bosonization,
ed. M. Stone (World Scientific, Singapore, 1996)
[18]
A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov,
Nucl. Phys. B241 (1984) 333
[19]
J. Cardy,
Nucl. Phys. B270 (FS16) (1986) 186
[20]
F. Woynarovich, H.P. Eckle,
J. Phys. A: Math. and Gen. 20 (1987) L97
[21]
F.C. Alcaraz, M.N. Barber and M.T. Batchelor,
Phys. Rev. Lett. 58 (1987) 771
[22]
F.C. Alcaraz, M. Baake, U. Grimm and V. Rittenberg,
J. Phys. A21 (1988) L117
[23]
Al.B. Zamolodchikov,
Nucl. Phys. B348 (1991) 619
[24]
R. Baxter.
Ann. of Phys. (NY) 70 (1972), 193, 323
[25]
F.A. Smirnov.
Form-factors in completely
integrable models of
quantum field theory (World Scientific, Singapore, 1992)
[26]
M. Lashkevich and Y. Pugai,
Nucl. Phys. B516 (1998) 623
[27]
M. Lashkevich,
Nucl. Phys. B621 (2002) 587
[28]
R.J. Baxter and S.B. Kelland,
J. Phys. C7 (1974) L403
[29]
A. Luther,
Phys. Rev. B14 (1976) 2153
[30]
A.B. Zamolodchikov,
Int. J. Mod. Phys. A10 (1995) 1125
[31]
S. Lukyanov and A. Zamolodchikov,
Nucl. Phys. B607 (2001) 437
[32]
M. Jimbo, H. Konno and T. Miwa,
In: Ascona 1996, Deformation theory and symplectic geometry, 117
(hep-th/9610079)
[33]
S. Lukyanov and A. Zamolodchikov,
Nucl. Phys. B493 (1997) 571
[34]
S. Lukyanov,
Phys. Rev. B59 (1999) 11163
[35]
T. Hikihara and A. Furusaki,
Phys. Rev. B58 (1998) R583
[36]
S.R. White,
Phys. Rev. Lett. 69 (1992) 2863;
Phys. Rev. B 48 (1993) 10345
[37]
I. Affleck,
Phys. Rev. Lett. 55 (1985) 1355
[38]
Al.B. Zamolodchikov,
unpublished
[39]
B. Doyon and S. Lukyanov,
Fermion Schwinger’s function for the SU(2)-Thirring model,
RUNHETC-2002-7 (hep-th/0203135)
[40]
I. Affleck,
J. Phys. A31 (1998) 4573
[41]
T.A. Kaplan, P. Horsch, and J. Borisowicz,
Phys. Rev. B35 (1987) 1877
[42]
V. Barsykin and I. Affleck,
Finite-size scaling for
the S=1/2 Heisenberg antiferromagnetic chain
(cond-mat/9810075)
[43]
K.A. Hallberg, P. Horsch and G. Martinez,
Phys. Rev. B59 (1995) R179.
[44]
M. Jimbo and T. Miwa,
Jour. Phys. A: Math. and Gen., 29
(1996) 2923
[45]
E. Lieb, T. Schultz and D. Mattis,
Ann. Phys. 16 (1961) 407
[46]
B.M. McCoy, T.T. Wu, C.A. Tracy and E. Borouch,
Phys.Rev. B13 (1976) 316
[47]
H. Au-Yang and J.H.H. Perk,
Phys. Lett. A104 (1984) 131 |
\NewEnviron
shortproof
Proof.
\BODY
\NewEnviron
longproof
Characterizing Realizability in Abstract Argumentation††thanks: This research has been supported by DFG (project BR 1817/7-1) and FWF (projects I1102 and P25518).
Thomas Linsbichler
TU Wien
Austria
&Jörg Pührer
Hannes Strass
Leipzig University
Germany
Abstract
Realizability for knowledge representation formalisms studies the following question:
Given a semantics and a set of interpretations, is there a knowledge base whose semantics coincides exactly with the given interpretation set?
We introduce a general framework for analyzing realizability in abstract dialectical frameworks (ADFs) and various of its subclasses.
In particular, the framework applies to Dung argumentation frameworks, SETAFs by Nielsen and Parsons, and bipolar ADFs.
We present a uniform characterization method for the admissible, complete, preferred and model/stable semantics.
We employ this method to devise an algorithm that decides realizability for the mentioned formalisms and semantics;
moreover the algorithm allows for constructing a desired knowledge base whenever one exists.
The algorithm is built in a modular way and thus easily extensible to new formalisms and semantics.
We have also implemented our approach in answer set programming, and used the implementation to obtain several novel results on the relative expressiveness of the abovementioned formalisms.
1 Introduction
The abstract argumentation frameworks (AFs) introduced by Dung (1995) have garnered increasing attention in the recent past.
In his seminal paper, Dung showed how an abstract notion of argument (seen as an atomic entity) and the notion of individual attacks between arguments together could reconstruct several established KR formalisms in argumentative terms.
Despite the generality of those and many more results in the field that was sparked by that paper, researchers also noticed that the restriction to individual attacks is often overly limiting, and devised extensions and generalizations of Dung’s frameworks:
directions included generalizing individual attacks to collective attacks (Nielsen and Parsons, 2006), leading to so-called SETAFs;
others started offering a support relation between arguments (Cayrol and
Lagasquie-Schiex, 2005),
preferences among arguments (Amgoud and
Cayrol, 2002; Modgil, 2009),
or attacks on attacks into arbitrary depth (Baroni et al., 2011).
This is only the tip of an iceberg, for a more comprehensive overview we refer to the work of Brewka, Polberg, and
Woltran (2014).
One of the most recent and most comprehensive generalizations of AFs has been presented by Brewka and Woltran (2010) (and later continued by Brewka et al., 2013) in the form of abstract dialectical frameworks (ADFs).
These ADFs offer any type of link between arguments:
individual attacks (as in AFs), collective attacks (as in SETAFs), and individual and collective support, to name only a few.
This generality is achieved through so-called acceptance conditions associated to each statement.
Roughly, the meaning of relationships between arguments is not fixed in ADFs, but is specified by the user for each argument in the form of Boolean functions (acceptance functions) on the argument’s parents.
However, this generality comes with a price:
Strass and Wallner (2015) found that the complexity of the associated reasoning problems of ADFs is in general higher than in AFs (one level up in the polynomial hierarchy).
Fortunately, the subclass of bipolar ADFs (defined by Brewka and Woltran, 2010) is as complex as AFs (for all considered semantics) while still offering a wide range of modeling capacities (Strass and Wallner, 2015).
However, there has only been little concerted effort so far to exactly analyze and compare the expressiveness of the abovementioned languages.
This paper is about exactly analyzing means of expression for argumentation formalisms.
Instead of motivating expressiveness in natural language and showing examples that some formalisms seem to be able to express but others do not, we tackle the problem in a formal way. We use a precise mathematical definition of expressiveness:
a set of interpretations is realizable by a formalism under a semantics if and only if there exists a knowledge base of the formalism whose semantics is exactly the given set of interpretations.
Studying realizability in AFs has been started by Dunne et al. (2013, 2015), who analyzed realizability for extension-based semantics, that is, interpretations represented by sets where arguments are either accepted (in the extension set) or not accepted (not in the extension set).
While their initial work disregarded arguments that are never accepted, there have been continuations where the existence of such “invisible” arguments is ruled out (Baumann et al., 2014; Linsbichler, Spanring, and
Woltran, 2015).
Dyrkolbotn (2014) began to analyze realizability for labeling-based semantics of AFs, that is, three-valued semantics where arguments can be accepted (mapped to true), rejected (mapped to false) or neither (mapped to unknown).
Strass (2015) started to analyze the relative expressiveness of two-valued semantics for ADFs (relative with respect to related formalisms).
Most recently, Pührer (2015) presented precise characterizations of realizability for ADFs under several three-valued semantics, namely admissible, grounded, complete, and preferred.
The term “precise characterizations” means that he gave necessary and sufficient conditions for an interpretation set to be ADF-realizable under a semantics.
The present paper continues this line of work by lifting it to a much more general setting.
We combine the works of Dunne et al. (2015), Pührer (2015), and Strass (2015) into a unifying framework, and at the same time extend them to formalisms and semantics not considered in the respective papers: we treat several formalisms, namely AFs, SETAFs, and (B)ADFs, while the previous works all used different approaches and techniques.
This is possible because all of these formalisms can be seen as subclasses of ADFs that are obtained by suitably restricting the acceptance conditions.
Another important feature of our framework is that we uniformly use three-valued interpretations as the underlying model theory.
In particular, this means that arguments cannot be “invisible” any more since the underlying vocabulary of arguments is always implicit in each interpretation.
Technically, we always assume a fixed underlying vocabulary and consider our results parametric in that vocabulary.
In contrast, for example, Dyrkolbotn (2014) presents a construction for realizability that introduces new arguments into the realizing knowledge base;
we do not allow that.
While sometimes the introduction of new arguments can make sense, for example if new information becomes available about a domain or a debate,
it is not sensible in general, as these new arguments would be purely technical with an unclear dialectical meaning.
Moreover, it would lead to a different notion of realizability, where most of the realizability problems would be significantly easier, if not trivial.
The paper proceeds as follows.
We begin with recalling and introducing the basis and basics of our work – the formalisms we analyze and the methodology with which we analyze them.
Next we introduce our general framework for realizability;
the major novelty is our consistent use of so-called characterization functions, firstly introduced by Pührer (2015), which we adapt to further semantics.
The main workhorse of our approach will be a parametric propagate-and-guess algorithm for deciding whether a given interpretation set is realizable in a formalism under a semantics.
We then analyze the relative expressiveness of the considered formalisms, presenting several new results that we obtained using an implementation of our framework.
We conclude with a discussion.
2 Preliminaries
We make use of standard mathematical concepts like functions and partially ordered sets.
For a function $f:X\to Y$ we denote the update of $f$ with a pair $(x,y)\in X\times Y$ by $f|^{x}_{y}:X\to Y$ with $z\mapsto y$ if $z=x$, and $z\mapsto f(z)$ otherwise.
For a function $f:X\to Y$ and $y\in Y$, its preimage is $f^{-1}(y)=\left\{x\in X\ \middle|\ f(x)=y\right\}$.
A partially ordered set is a pair $(S,\sqsubseteq)$ with $\sqsubseteq$ a partial order on $S$. A partially ordered set $(S,\sqsubseteq)$ is a complete lattice if and only if every $S^{\prime}\subseteq S$ has both a greatest lower bound (glb) $\bigsqcap S^{\prime}\in S$ and a least upper bound (lub) $\bigsqcup S^{\prime}\in S$.
A partially ordered set $(S,\sqsubseteq)$ is a complete meet-semilattice iff every non-empty subset $S^{\prime}\subseteq S$ has a greatest lower bound $\bigsqcap S^{\prime}\in S$ (the meet) and every ascending chain $C\subseteq S$ has a least upper bound $\bigsqcup C\in S$.
Three-Valued Interpretations
Let $A$ be a fixed finite set of statements.
An interpretation is a mapping $v:A\to\left\{\mathbf{t},\mathbf{f},\mathbf{u}\right\}$ that assigns one of the truth values true ($\mathbf{t}$), false ($\mathbf{f}$) or unknown ($\mathbf{u}$) to each statement.
An interpretation is two-valued if $v(A)\subseteq\left\{\mathbf{t},\mathbf{f}\right\}$, that is, the truth value $\mathbf{u}$ is not assigned.
Two-valued interpretations $v$ can be extended to assign truth values $v(\varphi)\in\left\{\mathbf{t},\mathbf{f}\right\}$ to propositional formulas $\varphi$ as usual.
The three truth values are partially ordered according to their information content:
we have $\mathbf{u}<_{i}\mathbf{t}$ and $\mathbf{u}<_{i}\mathbf{f}$ and no other pair in $<_{i}$, which intuitively means that the classical truth values contain more information than the truth value unknown.
As usual, we denote by $\leq_{i}$ the partial order associated to the strict partial order $<_{i}$.
The pair $(\left\{\mathbf{t},\mathbf{f},\mathbf{u}\right\},\leq_{i})$ forms a complete meet-semilattice with the information meet operation $\sqcap_{i}$.
This meet can intuitively be interpreted as consensus and assigns
$\mathbf{t}\sqcap_{i}\mathbf{t}=\mathbf{t}$, $\mathbf{f}\sqcap_{i}\mathbf{f}=\mathbf{f}$, and returns $\mathbf{u}$ otherwise.
The information ordering $\leq_{i}$ extends in a straightforward way to interpretations $v_{1},v_{2}$ over $A$ in that
$v_{1}\leq_{i}v_{2}$ iff $v_{1}(a)\leq_{i}v_{2}(a)$ for all $a\in A$.
We say for two interpretations $v_{1},v_{2}$ that $v_{2}$ extends $v_{1}$ iff $v_{1}\leq_{i}v_{2}$.
The set $\mathcal{V}$ of all interpretations over $A$ forms a complete meet-semilattice with respect to the information ordering $\leq_{i}$.
The consensus meet operation $\sqcap_{i}$ of this semilattice is given by $(v_{1}\sqcap_{i}v_{2})(a)=v_{1}(a)\sqcap_{i}v_{2}(a)$ for all $a\in A$.
The least element of $(\mathcal{V},\leq_{i})$ is the valuation $v_{\mathbf{u}}:A\to\left\{\mathbf{u}\right\}$ mapping all statements to unknown – the least informative interpretation.
By $\mathcal{V}_{2}$ we denote the set of two-valued interpretations;
they are the $\leq_{i}$-maximal elements of the meet-semilattice $(\mathcal{V},\leq_{i})$.
We denote by $[{v}]_{2}$ the set of all two-valued interpretations that extend $v$.
The elements of $[{v}]_{2}$ form an $\leq_{i}$-antichain with greatest lower bound $v=\bigsqcap_{i}[{v}]_{2}$.
Abstract Argumentation Formalisms
An abstract dialectical framework (ADF) is a tuple $D=(A,L,C)$ where
$A$ is a set of statements (representing positions one can take or not take in a debate),
$L\subseteq A\times A$ is a set of links (representing dependencies between the positions),
$C=\left\{C_{a}\right\}_{a\in A}$ is a collection of functions
$C_{a}:2^{\mathit{par}(a)}\to\left\{\mathbf{t},\mathbf{f}\right\}$, one for each statement $a\in A$.
The function $C_{a}$ is the acceptance condition of $a$ and expresses whether $a$ can be accepted, given the acceptance status of its parents $\mathit{par}(a)=\left\{b\in S\ \middle|\ (b,a)\in L\right\}$.
We usually represent each $C_{a}$ by a propositional formula $\varphi_{a}$ over $\mathit{par}(a)$.
To specify an acceptance condition, then, we take $C_{a}(M\cap\mathit{par}(a))=\mathbf{t}$ to hold iff $M$ is a model for $\varphi_{a}$.
Brewka and Woltran (2010) introduced a useful subclass of ADFs:
an ADF $D=(A,L,C)$ is bipolar iff all links in $L$ are supporting or attacking (or both).
A link $(b,a)\in L$ is supporting in $D$ iff for all $M\subseteq\mathit{par}(a)$, we have that $C_{a}(M)=\mathbf{t}$ implies $C_{a}(M\cup\left\{b\right\})=\mathbf{t}$.
Symmetrically, a link $(b,a)\in L$ is attacking in $D$ iff for all $M\subseteq\mathit{par}(a)$, we have that $C_{a}(M\cup\left\{b\right\})=\mathbf{t}$ implies $C_{a}(M)=\mathbf{t}$.
If a link $(b,a)$ is both supporting and attacking then $b$ has no actual influence on $a$.
(But the link does not violate bipolarity.)
We write BADFs as $D=(A,L^{+}\cup L^{-},C)$ and mean that $L^{+}$ contains all supporting links and $L^{-}$ all attacking links.
The semantics of ADFs can be defined using an operator $\Gamma_{D}$ over three-valued interpretations (Brewka and Woltran, 2010; Brewka et al., 2013).
For an ADF $D$ and a three-valued interpretation $v$, the interpretation $\Gamma_{D}({v})$ is given by
$$\displaystyle a\mapsto{\textstyle\bigsqcap_{i}}\left\{w(\varphi_{a})\ \middle|\ w\in[{v}]_{2}\right\}$$
That is, for each statement $a$, the operator returns the consensus truth value for its acceptance formula $\varphi_{a}$, where the consensus takes into account all possible two-valued interpretations $w$ that extend the input valuation $v$.
If this $v$ is two-valued, we get $[{v}]_{2}=\left\{v\right\}$ and thus $\Gamma_{D}({v})(a)=v(\varphi_{a})$.
The standard semantics of ADFs are now defined as follows.
For ADF $D$, an interpretation $v:A\to\left\{\mathbf{t},\mathbf{f},\mathbf{u}\right\}$ is
•
admissible iff $v\leq_{i}\Gamma_{D}({v})$;
•
complete iff $\Gamma_{D}({v})=v$;
•
preferred iff it is $\leq_{i}$-maximal admissible;
•
a two-valued model iff it is two-valued and $\Gamma_{D}({v})=v$.
We denote the sets of interpretations that are
admissible, complete, preferred, and two-valued models by $\mathit{adm}(D)$, $\mathit{com}(D)$, $\mathit{prf}(D)$ and $\mathit{mod}(D)$, respectively.
These definitions are proper generalizations of Dung’s notions for AFs: For an AF $(A,R)$,
where $R\subseteq A\times A$ is the attack relation,
the ADF associated to $(A,R)$ is
$D_{(A,R)}=(A,R,C)$ with $C=\{\varphi_{a}\}_{a\in A}$ and
$\varphi_{a}=\bigwedge_{b:(b,a)\in R}\neg b$ for all $a\in A$.
AFs inherit their semantics from the definitions for ADFs (Brewka et al., 2013, Theorems 2 and 4).
In particular, an interpretation is stable for an AF $(A,R)$ if and only if it is a two-valued model of $D_{(A,R)}$.
A SETAF is a pair $S=(A,X)$ where
$X\subseteq(2^{A}\setminus\{\emptyset\})\times A$
is the (set) attack relation.
We define three-valued counterparts of
the semantics introduced by Nielsen and Parsons (2006),
following the same conventions as in
three-valued semantics of AFs (Caminada and Gabbay, 2009) and
argumentation formalisms in general.
Given a statement $a\in A$ and an interpretation $v$
we say that
$a$ is acceptable wrt. $v$
if $\forall(B,a)\in X\exists a^{\prime}\in B:v(a^{\prime})=\mathbf{f}$ and
$a$ is unacceptable wrt. $v$
if $\exists(B,a)\in X\forall a^{\prime}\in B:v(a^{\prime})=\mathbf{t}$.
For an interpretation $v:A\to\left\{\mathbf{t},\mathbf{f},\mathbf{u}\right\}$ it holds that
•
$v\in\mathit{adm}(S)$ iff for all $a\in A$,
$a$ is acceptable wrt. $v$ if $v(a)=\mathbf{t}$ and
$a$ is unacceptable wrt. $v$ if $v(a)=\mathbf{f}$;
•
$v\in\mathit{com}(S)$ iff for all $a\in A$,
$a$ is acceptable wrt. $v$ iff $v(a)=\mathbf{t}$ and
$a$ is unacceptable wrt. $v$ iff $v(a)=\mathbf{f}$;
•
$v\in\mathit{prf}(S)$ iff $v$ is $\leq_{i}$-maximal admissible; and
•
$v\in\mathit{mod}(S)$ iff $v\in\mathit{adm}(F)$ and $\nexists a\in A:v(a)=\mathbf{u}$.
For a SETAF $S=(A,X)$
the corresponding ADF $D_{S}$
has acceptance formula $\varphi_{a}=\bigwedge_{(B,a)\in X}\bigvee_{a^{\prime}\in B}\neg a^{\prime}$ for each statement $a\in A$. (Polberg, 2016)
Proposition 1.
For any SETAF $S=(A,X)$ it holds that $\sigma(S)=\sigma(D_{S})$,
where $\sigma\in\{\mathit{adm},\mathit{com},\mathit{prf},\mathit{mod}\}$.
{longproof}
Given interpretation $v$ and statement $a$, it holds that
$\Gamma_{D_{S}}({v})(a)=\mathbf{t}$ iff $\forall w\in[{v}]_{2}:w(a)=\mathbf{t}$ iff
$\forall(B,a)\in X$ $\exists a^{\prime}\in B:v(a^{\prime})=\mathbf{f}$ iff $a$ is acceptable wrt. $v$ and
$\Gamma_{D_{S}}({v})(a)=\mathbf{f}$ iff $\forall w\in[{v}]_{2}:w(a)=\mathbf{f}$ iff
$\exists(B,a)\in X$ $\forall a^{\prime}\in B:v(a^{\prime})=\mathbf{t}$ iff $a$ is unacceptable wrt. $v$.
Hence $\sigma(S)=\sigma(D_{S})$ for $\sigma\in\{\mathit{adm},\mathit{com},\mathit{prf},\mathit{mod}\}$. $\Box$
Realizability
A set $V\subseteq\mathcal{V}$ of interpretations is realizable in a formalism $\mathcal{F}$ under a semantics $\sigma$
if and only if there exists a knowledge base $\mathsf{kb}\in\mathcal{F}$ having exactly $\sigma(\mathsf{kb})=V$.
Pührer (2015) characterized realizability for ADFs under various three-valued semantics.
We will reuse the central notions for capturing the complete semantics in this work.
Definition 1 (Pührer 2015).
Let $V$ be a set of interpretations.
A function $f:\mathcal{V}_{2}\to\mathcal{V}_{2}$ is a $\mathit{com}$-characterization of $V$ iff: for each $v\in\mathcal{V}$ we have $v\in V$ iff for each $a\in A$:
•
$v(a)\neq\mathbf{u}$ implies $f(v_{2})(a)=v(a)$ for all $v_{2}\in[{v}]_{2}$ and
•
$v(a)=\mathbf{u}$ implies $f(v^{\prime}_{2})(a)=\mathbf{t}$ and $f(v^{\prime\prime}_{2})(a)=\mathbf{f}$ for some $v^{\prime}_{2},v^{\prime\prime}_{2}\in[{v}]_{2}$.
$\blacktriangle$
From a function of this kind we can build a corresponding ADF by the following construction.
For a function $f:\mathcal{V}_{2}\to\mathcal{V}_{2}$, we define $D_{f}$ as the ADF where the acceptance formula for each statement $a$ is given by
$$\displaystyle\varphi^{f}_{a}=\mathop{\bigvee_{w\in\mathcal{V}_{2},}}_{f(w)(a)=\mathbf{t}}\phi_{w}\quad\text{with}\quad\phi_{w}=\bigwedge_{w(a^{\prime})=\mathbf{t}}a^{\prime}\land\bigwedge_{w(a^{\prime})=\mathbf{f}}\neg a^{\prime}$$
Observe that we have $v(\phi_{w})=\mathbf{t}$ iff $v=w$ by definition.
Intuitively, the acceptance condition $\varphi_{a}^{f}$ is constructed such that $v$ is a model of $\varphi_{a}^{f}$ if and only if we find $f(v)(a)=\mathbf{t}$.
Proposition 2 (Pührer 2015).
Let $V\subseteq\mathcal{V}$ be a set of interpretations.
(1) For each ADF $D$ with $\mathit{com}(D)=V$, there is a $\mathit{com}$-characterization $f_{D}$ for $V$;
(2) for each $\mathit{com}$-characterization $f:\mathcal{V}_{2}\to\mathcal{V}_{2}$ for $V$ we have $\mathit{com}(D_{f})=V$.
The result shows that $V$ can be realized under complete semantics if and only if there is a $\mathit{com}$-characterization for $V$.
3 A General Framework for Realizability
The main underlying idea of our framework is that all abstract argumentation formalisms introduced in the previous section can be viewed as subclasses of abstract dialectical frameworks.
This is clear for ADFs themselves and for BADFs by definition;
for AFs and SETAFs it is fairly easy to see.
However, knowing that these formalisms can be recast as ADFs is not everything.
To employ this knowledge for realizability, we must be able to precisely characterize the corresponding subclasses in terms of restricting the ADFs’ acceptance functions.
Alas, this is also possible and paves the way for the framework we present in this section.
Most importantly, we will make use of the fact that different formalisms and different semantics can be characterized modularly, that is, independently of each other.
Towards a uniform account of realizability for ADFs under different semantics,
we start with a new characterization of realizability for ADFs under admissible semantics
that is based on a notion similar in spirit to $\mathit{com}$-characterizations.
Definition 2.
Let $V$ be a set of interpretations.
A function $f:\mathcal{V}_{2}\to\mathcal{V}_{2}$ is an $\mathit{adm}$-characterization of $V$ iff:
for each $v\in\mathcal{V}$ we have $v\in V$ iff for every $a\in A$:
•
$v(a)\neq\mathbf{u}$ implies $f(v_{2})(a)=v(a)$ for all $v_{2}\in[{v}]_{2}$.
$\blacktriangle$
Note that the only difference to Definition 1 is dropping the second condition related to statements with truth value $\mathbf{u}$.
Proposition 3.
Let $V\subseteq\mathcal{V}$ be a set of interpretations.
(1) For each ADF $D$ such that $\mathit{adm}(D)=V$, there is an $\mathit{adm}$-characterization $f_{D}$ for $V$;
(2) for each adm-characterization $f:\mathcal{V}_{2}\to\mathcal{V}_{2}$ for $V$ we have $\mathit{adm}(D_{f})=V$.
{longproof}
(1) We define the function $f_{D}:\mathcal{V}_{2}\to\mathcal{V}_{2}$ as
$f_{D}(v_{2})(a)=v_{2}(\varphi_{a})$
for every $v_{2}\in\mathcal{V}_{2}$ and $a\in A$ where $\varphi_{a}$ is the acceptance formula of $a$ in $D$.
We will show that $f_{D}$ is an $\mathit{adm}$-characterization for $V=\mathit{adm}(D)$.
Let $v$ be an interpretation. Consider the case $v\in\mathit{adm}(D)$
and $v(a)\neq u$ for some $a\in A$ and some $v_{2}\in[{v}]_{2}$.
From $v\leq_{i}\Gamma_{D}({v})$ we get $v_{2}(\varphi_{a})=v(a)$.
By definition of $f_{D}$ is follows that $f_{D}(v_{2})(a)=v(a)$.
Now assume $v\not\in\mathit{adm}(D)$ and consequently $v\not\leq_{i}\Gamma_{D}({v})$.
There must be some $a\in A$ such that $v(a)\neq\mathbf{u}$ and $v(a)\neq\Gamma_{D}({v})(a)$.
Hence, there is some $v_{2}\in[{v}]_{2}$ with $v_{2}(\varphi_{a})\neq v(a)$
and $f_{D}(v_{2})(a)\neq v(a)$ by definition of $f_{D}$.
Thus, $f_{D}$ is an $\mathit{adm}$-characterization
(2)
Observe that for every two-valued interpretation $v_{2}$
and every $a\in A$ we have
$f(v_{2})(a)=v_{2}(\varphi^{f}_{a})$.
$(\subseteq)$:
Let $v\in\mathit{adm}(D_{f})$ be an interpretation and $a\in A$ a statement
such that $v(a)\neq\mathbf{u}$.
Let $v_{2}$ be a two-valued interpretation with $v_{2}\in[{v}]_{2}$.
Since $v\leq_{i}\Gamma_{D_{f}}({v})$ we have $v(a)=v_{2}(\varphi^{f}_{a})$.
Therefore, by our observation it must also hold that $f(v_{2})(a)=v(a)$.
Thus, by Definition 2, $v\in V$.
$(\supseteq)$:
Consider an interpretation $v$ such that $v\not\in\mathit{adm}(D_{f})$.
We show that $v\not\in V$.
From $v\not\in\mathit{adm}(D_{f})$ we get $v\not\leq_{i}\Gamma_{D_{f}}({v})$.
There must be some $a\in A$ such that $v(a)\neq\mathbf{u}$ and $v(a)\neq\Gamma_{D_{f}}({v})(a)$.
Hence, there is some $v_{2}\in[{v}]_{2}$ with $v_{2}(\varphi^{f}_{a})\neq v(a)$
and consequently $f(v_{2})(a)\neq v(a)$. Thus, by Definition 2 we have $v\not\in V$.
$\Box$
When listing sets of interpretations in examples, for the sake of readability we represent three-valued interpretations by sequences of truth values, tacitly assuming that the underlying vocabulary is given and has an associated total ordering.
For example, for the vocabulary $A=\left\{a,b,c\right\}$ we represent the interpretation $\left\{a\mapsto\mathbf{t},b\mapsto\mathbf{f},c\mapsto\mathbf{u}\right\}$ by the sequence $\mathbf{t}\mathbf{f}\mathbf{u}$.
Example 1.
Consider the sets $V_{1}=\{\mathbf{u}\mathbf{u}\mathbf{u},\mathbf{t}\mathbf{f}\mathbf{f},\mathbf{f}\mathbf{t}\mathbf{u}\}$ and $V_{2}=\{\mathbf{t}\mathbf{f}\mathbf{f},\mathbf{f}\mathbf{t}\mathbf{u}\}$ of interpretations over $A=\left\{a,b,c\right\}$.
The mapping
$f=\{\mathbf{t}\mathbf{t}\mathbf{t}\mapsto\mathbf{f}\mathbf{t}\mathbf{t},\mathbf{t}\mathbf{t}\mathbf{f}\mapsto\mathbf{t}\mathbf{f}\mathbf{t},\mathbf{t}\mathbf{f}\mathbf{t}\mapsto\mathbf{t}\mathbf{t}\mathbf{t},\mathbf{t}\mathbf{f}\mathbf{f}\mapsto\mathbf{t}\mathbf{f}\mathbf{f},\mathbf{f}\mathbf{t}\mathbf{t}\mapsto\mathbf{f}\mathbf{t}\mathbf{f},\mathbf{f}\mathbf{t}\mathbf{f}\mapsto\mathbf{f}\mathbf{t}\mathbf{t},\mathbf{f}\mathbf{f}\mathbf{t}\mapsto\mathbf{t}\mathbf{t}\mathbf{f},\mathbf{f}\mathbf{f}\mathbf{f}\mapsto\mathbf{f}\mathbf{t}\mathbf{f}\}$ is an $\mathit{adm}$-characterization for $V_{1}$.
Thus, the ADF $D_{f}$ has $V_{1}$ as its admissible interpretations.
Indeed, the realizing ADF has the following acceptance conditions:
$$\begin{array}[]{lll}\varphi^{f}_{a}&\equiv&(a\land b\land\neg c)\lor(a\land\neg b)\lor(\neg a\land\neg b\land c)\\
\varphi^{f}_{b}&\equiv&(a\land c)\lor(\neg a\land b)\lor(\neg a\land\neg b\land\neg c)\\
\varphi^{f}_{c}&\equiv&(a\land b)\lor(\neg a\land b\land\neg c)\lor(\neg b\land c)\end{array}$$
For $V_{2}$ no $\mathit{adm}$-characterization exists because $\mathbf{u}\mathbf{u}\mathbf{u}\not\in V_{2}$ but the implication of Definition 2 trivially holds for $a$, $b$, and $c$.
$\blacksquare$
We have seen that the construction $D_{f}$ for realizing under complete semantics can also be used for realizing a set $V$ of interpretations under admissible semantics.
The only difference is that we here
require $f$ to be an $\mathit{adm}$-characterization instead of a $\mathit{com}$-characterization for $V$.
Note that admissible semantics can be characterized by properties that are easier to check than existence of an $\mathit{adm}$-characterization (see the work of Pührer, 2015).
However, using the same type of characterizations for different semantics allows for a unified approach for checking realizability and constructing a realizing ADF in case one exists.
For realizing under the model semantics, we can likewise present an adjusted version of $\mathit{com}$-characterizations.
Definition 3.
Let $V\subseteq\mathcal{V}$ be a set of interpretations.
A function $f:\mathcal{V}_{2}\to\mathcal{V}_{2}$ is a $\mathit{mod}$-characterization of $V$ if and only if:
(1) $f$ is defined on $V$ (that is, $V\subseteq\mathcal{V}_{2}$) and
(2) for each $v\in\mathcal{V}_{2}$, we have $v\in V$ iff $f(v)=v$.
$\blacktriangle$
As we can show, there is a one-to-one correspondence between $\mathit{mod}$-characterizations and ADF realizations.
Proposition 4.
Let $V\subseteq\mathcal{V}$ be a set of interpretations.
(1) For each ADF $D$ such that $\mathit{mod}(D)=V$, there is a $\mathit{mod}$-characterization $f_{D}$ for $V$;
(2) vice versa, for each $\mathit{mod}$-characterization $f:\mathcal{V}_{2}\to\mathcal{V}_{2}$ for $V$ we find $\mathit{mod}(D_{f})=V$.{longproof}
(1) Let $D$ be an ADF with $\mathit{mod}(D)=V$.
It immediately follows that $V\subseteq\mathcal{V}_{2}$.
To define $f_{D}$ we can use the construction in the proof of Proposition 3.
It follows directly that for any $v\in\mathcal{V}_{2}$, we find $f_{D}(v)=v$ iff $v\in V$.
Thus $f_{D}$ is a $\mathit{mod}$-characterization for $V$.
(2) Let $V\subseteq\mathcal{V}_{2}$ and $f:\mathcal{V}_{2}\to\mathcal{V}_{2}$ be a $\mathit{mod}$-characterization of $V$.
For any $v\in\mathcal{V}_{2}$ we have:
$$\displaystyle\mathrel{\phantom{\iff}}v\in V\iff v=f(v)$$
$$\displaystyle\iff\forall a\in A:\left(v(a)=f(v)(a)\right)$$
$$\displaystyle\iff\forall a\in A:\left(v(a)=\mathbf{t}\leftrightarrow f(v)(a)=\mathbf{t}\right)$$
$$\displaystyle\iff\forall a\in A:(v(a)=\mathbf{t}\ \leftrightarrow(\exists w\in\mathcal{V}_{2}:f(w)(a)=\mathbf{t}$$
$$\displaystyle\hskip 182.09746pt\land v=w))$$
$$\displaystyle\iff\forall a\in A:(v(a)=\mathbf{t}\ \leftrightarrow(\exists w\in\mathcal{V}_{2}:f(w)(a)=\mathbf{t}$$
$$\displaystyle\hskip 165.02606pt\land v(\phi_{w})=\mathbf{t}))$$
$$\displaystyle\iff\forall a\in A:\left(v(a)=\mathbf{t}\ \leftrightarrow v\!\left(\mathop{\bigvee_{w\in\mathcal{V}_{2},}}_{f(w)(a)=\mathbf{t}}\phi_{w}\right)=\mathbf{t}\right)$$
$$\displaystyle\iff\forall a\in A:v(a)=v\!\left(\mathop{\bigvee_{w\in\mathcal{V}_{2},}}_{f(w)(a)=\mathbf{t}}\phi_{w}\right)$$
$$\displaystyle\iff\forall a\in A:v(a)=v(\varphi_{a}^{f})\iff v\in\mathit{mod}(D_{f})\hskip 19.91692pt\Box$$
A related result was given by Strass (2015, Proposition 10).
The characterization we presented here fits into the general framework of this paper and is directly usable for our realizability algorithm.
Wrapping up, the next result summarizes how ADF realizability can be captured by different types of characterizations for the semantics we considered so far.
Theorem 5.
Let $V\subseteq\mathcal{V}$ be a set of interpretations and consider $\sigma\in\{adm,com,mod\}$.
There is an ADF $D$ such that $\sigma(D)=V$ if and only if there is a $\sigma$-characterization for $V$.
The preferred semantics of an ADF $D$ is closely related to its admissible semantics as,
by definition, the preferred interpretations of $D$ are its $\leq_{i}$-maximal admissible interpretations.
As a consequence we can also describe preferred realizability in terms of $\mathit{adm}$-characterizations.
We use the lattice-theoretic standard notation $\max_{\leq_{i}}V$ to select the $\leq_{i}$-maximal
elements of a given set $V$ of interpretations.
Corollary 6.
Let $V\subseteq\mathcal{V}$ be a set of interpretations.
There is an ADF $D$ with $\mathit{prf}(D)=V$ iff there is an $\mathit{adm}$-characterization for some $V^{\prime}\subseteq\mathcal{V}$ with $V\subseteq V^{\prime}$ and $\max_{\leq_{i}}V^{\prime}=V$.
Finally, we give a result on the complexity of deciding realizability for the mentioned formalisms and semantics.
Proposition 7.
Let $\mathcal{F}\in\left\{\textrm{AF},\textrm{SETAF},\textrm{BADF},\textrm{ADF}\right\}$ be a formalism and
$\sigma\in\left\{\mathit{adm},\mathit{com},\mathit{prf},\mathit{mod}\right\}$ be a semantics.
The decision problem
“Given a vocabulary $A$ and a set $V\subseteq\mathcal{V}$ of interpretations over $A$, is there a $\mathsf{kb}\in\mathcal{F}$ such that $\sigma(\mathsf{kb})=V$?”
can be decided in nondeterministic time that is polynomial in the size of $V$.111We assume here that the representation of any $V$ over $A$ has size $\Theta(3^{\left\lvert A\right\rvert})$. There might be specific $V$ with smaller representations, but we cannot assume any better for the general case.
Proof.
For all considered $\mathcal{F}$ and $\sigma$, computing all $\sigma$-interpretations of a given witness $\mathsf{kb}\in\mathcal{F}$ can be done in time that is linear in the size of $V$.
Comparing the result to $V$ can also be done in linear time.
$\Box$
3.1 Deciding Realizability: Algorithm 1
Our main algorithm for deciding realizability is a propagate-and-guess algorithm in the spirit of the DPLL algorithm for deciding propositional satisfiability (Gomes et al., 2008).
It is generic with respect to (1) the formalism $\mathcal{F}$ and (2) the semantics $\sigma$ for which should be realized.
To this end, the propagation part of the algorithm is kept exchangeable and will vary depending on formalism and semantics.
Roughly, in the propagation step the algorithm uses the desired set $V$ of interpretations to derive certain necessary properties of the realizing knowledge base (line 2).
This is the essential part of the algorithm:
the derivation rules (propagators) used there are based on characterizations of realizability with respect to formalism and semantics.
Once propagation of properties has reached a fixed point (line 7), the algorithm checks whether the derived information is sufficient to construct a knowledge base.
If so, the knowledge base can be constructed and returned (line 9).
Otherwise (no more information can be obtained through propagation and there is not enough information to construct a knowledge base yet), the algorithm guesses another assignment for the characterization (line 11) and calls itself recursively.
The main data structure that Algorithm 1 operates on is a set of triples $(v,a,\mathbf{x})$ consisting of a two-valued interpretation $v\in\mathcal{V}_{2}$, an atom $a\in A$ and a truth value $\mathbf{x}\in\left\{\mathbf{t},\mathbf{f}\right\}$.
This data structure is intended to represent the $\sigma$-characterizations introduced in Definitions 1, 2 and 3.
There, a $\sigma$-characterization is a function $f:\mathcal{V}_{2}\to\mathcal{V}_{2}$ from two-valued interpretations to two-valued interpretations.
However, as the algorithm builds the $\sigma$-characterization step by step and there might not even be a $\sigma$-characterization in the end (because $V$ is not realizable), we use a set $F$ of triples $(v,a,\mathbf{x})$ to be able to represent both partial and incoherent states of affairs.
The $\sigma$-characterization candidate induced by $F$ is partial if we have that for some $v$ and $a$, neither $(v,a,\mathbf{t})\in F$ nor $(v,a,\mathbf{f})\in F$;
likewise, the candidate is incoherent if for some $v$ and $a$, both $(v,a,\mathbf{t})\in F$ and $(v,a,\mathbf{f})\in F$.
If $F$ is neither partial nor incoherent, it gives rise to a unique $\sigma$-characterization that can be used to construct the knowledge base realizing the desired set of interpretations.
The correspondence to the characterization-function is then such that $f(v)(a)=\mathbf{x}$ iff $(v,a,\mathbf{x})\in F$.
In our presentation of the algorithm we focused on its main features, therefore the guessing step (line 11) is completely “blind”.
It is possible to use common CSP techniques, such as shaving (removing guessing possibilities that directly lead to inconsistency).
Finally, we remark that the algorithm can be extended to enumerate all possible realizations of a given interpretation set – by keeping all choice points in the guessing step and thus exhaustively exploring the whole search space.
In the case where the constructed relation $F$ becomes functional at some point, the algorithm returns a realizing knowledge base $\mathit{kb}^{\mathcal{F}}_{\sigma}(F)$.
For ADFs, this just means that we denote by $f$ the $\sigma$-characterization represented by $F$ and set $\mathit{kb}^{\textrm{ADF}}_{\sigma}(F)=D^{f}$.
For the remaining formalisms we will introduce the respective constructions in later subsections.
The algorithm is parametric in two dimensions, namely with respect to the formalism $\mathcal{F}$ and with respect to the semantics $\sigma$.
These two aspects come into the algorithm via so-called propagators.
A propagator is a formalism-specific or semantics-specific set of derivation rules.
Given a set $V$ of desired interpretations and a partial $\sigma$-characterization $F$, a propagator $p$ derives new triples $(v,a,\mathbf{x})$ that must necessarily be part of any total $\sigma$-characterization $f$ for $V$ such that $f$ extends $F$.
In the following, we present semantics propagators for admissible, complete and two-valued model (in (SET)AF terms stable) semantics, and formalism propagators for BADFs, AFs, and SETAFs.
3.2 Semantics Propagators
These propagators (cf. Figure 1)
are directly derived from the properties
of $\sigma$-characterizations presented in
Definitions 1, 2 and 3.
While the definitions provide exact conditions to check
whether a given function is a $\sigma$-characterization,
the propagators allow us to derive definite values of partial characterizations that are necessary to fulfill the conditions for being a $\sigma$-characterization.
For admissible semantics, the condition for a function
$f$ to be an $\mathit{adm}$-characterization of a desired set of interpretations $V$
(cf. Definition 2)
can be split into a condition for desired interpretations $v\in V$
and two conditions for undesired interpretations $v\notin V$.
Propagator $p^{\in}_{adm}$ derives new triples
by considering interpretations $v\in V$.
Here, for all two-valued interpretations $v_{2}$ that extend $v$,
the value $f(v_{2})$ has to be in accordance with $v$ on $v$’s Boolean part,
that is, the algorithm adds $(v_{2},a,v(a))$ whenever $v(a)\neq\mathbf{u}$.
On the other hand, $p^{\notin}_{adm}$ derives new triples
for $v\notin V$ in order to ensure that there is a
two-valued interpretation $v_{2}$ extending $v$
where $f(v_{2})$ differs from $v$ on a Boolean value of $v$.
Note that while $p^{\in}_{adm}$ immediately allows us to derive information about $F$
for each desired interpretation $v\in V$, propagator $p^{\notin}_{adm}$ is much weaker in the sense that it only derives a triple of $F$ if there is no other way to meet the conditions for an undesired interpretation.
Special treatment is required for the interpretation $v_{\mathbf{u}}$ that maps all statements to $\mathbf{u}$ and is admissible for every ADF.
This is not captured by $p^{\in}_{adm}$ and $p^{\notin}_{adm}$ as these deal only with interpretations that have Boolean mappings.
Thus, propagator $p^{\lightning}_{\mathit{adm}}$ serves to check whether $v_{\mathbf{u}}\in V$.
If this is not the case, the propagator immediately makes the relation $F$ incoherent and the algorithm correctly answers “no”.
For complete semantics and interpretations $v\in V$, propagator $p^{\in,\mathbf{t}\mathbf{f}}_{\mathit{com}}$ derives triples just like in the admissible case.
Propagator $p^{\in,\mathbf{u}}_{\mathit{com}}$ deals with statements $a\in A$ having $v(a)=\mathbf{u}$ for which there have to be at least two $v_{2},v_{2}^{\prime}\in[{v}]_{2}$ having $f(v_{2})(a)=\mathbf{t}$ and $f(v_{2}^{\prime})(a)=\mathbf{f}$.
Hence $p^{\in,\mathbf{u}}_{\mathit{com}}$ derives triple $(v_{2},a,\neg\mathbf{x})$ if for all other $v_{2}^{\prime}\in[{v}]_{2}$ we find a triple $(v_{2}^{\prime},a,\mathbf{x})$.
For interpretations $v\notin V$ it must hold that there is some $a\in A$ such that
(i) $v(a)\neq\mathbf{u}$ and $f(v_{2})(a)\neq v(a)$ for some $v_{2}\in[{v}]_{2}$
or
(ii) $v(a)=\mathbf{u}$ but for all $v_{2}\in[{v}]_{2}$, $f(v_{2})$ assigns the same Boolean truth value $\mathbf{x}$ to $a$.
Now if neither (i) nor (ii) can be fulfilled by any statement $b\in A\setminus\{a\}$ due to the current contents of $F$,
propagators $p^{\not\in,\mathbf{t}\mathbf{f}}_{\mathit{com}}$ and $p^{\not\in,\mathbf{u}}_{\mathit{com}}$ derive triple
$(v_{2},a,\neg v(a))$ for $v(a)\neq\mathbf{u}$ if needed for $a$ to fulfill (i) and
$(v_{2},a,\neg\mathbf{x})$ for $v(a)=\mathbf{u}$ if needed for $a$ to fulfill (ii), respectively.
Example 2.
Consider the set $V_{3}=\{\mathbf{u}\mathbf{u}\mathbf{u},\mathbf{f}\mathbf{u}\mathbf{u},\mathbf{u}\mathbf{u}\mathbf{f},\mathbf{f}\mathbf{t}\mathbf{f}\}$.
First, we consider a run of $\mathit{realize}(\textrm{ADF},\mathit{adm},V_{3},\emptyset)$.
In the first iteration, propagator $p^{\in}_{\mathit{adm}}$ ensures that $F_{\Delta}$ in line 2
contains $(\mathbf{f}\mathbf{f}\mathbf{f},a,\mathbf{f})$, $(\mathbf{f}\mathbf{t}\mathbf{f},a,\mathbf{f})$, $(\mathbf{f}\mathbf{t}\mathbf{f},c,\mathbf{f})$, and $(\mathbf{f}\mathbf{f}\mathbf{f},c,\mathbf{f})$.
Based on the latter three tuples and $\mathbf{f}\mathbf{u}\mathbf{f}\notin V_{3}$, propagator $p^{\notin}_{\mathit{adm}}$ derives $(\mathbf{f}\mathbf{f}\mathbf{f},a,\mathbf{t})$ in the second iteration
which together with $(\mathbf{f}\mathbf{f}\mathbf{f},a,\mathbf{f})$ causes the algorithm to return “no”. Consequently, $V_{3}$ is not $\mathit{adm}$-realizable.
A run of $\mathit{realize}(\textrm{ADF},\mathit{com},V_{3},\emptyset)$ on the other hand returns $\mathit{com}$-characterization $f$ for $V_{3}$
that maps $\mathbf{t}\mathbf{t}\mathbf{f}$ to $\mathbf{t}\mathbf{f}\mathbf{f}$, $\mathbf{f}\mathbf{t}\mathbf{t}$ to $\mathbf{f}\mathbf{f}\mathbf{t}$, $\mathbf{f}\mathbf{t}\mathbf{f}$ and $\mathbf{f}\mathbf{f}\mathbf{f}$ to $\mathbf{f}\mathbf{t}\mathbf{f}$
and all other $v_{2}\in\mathcal{V}_{2}$ to $\mathbf{f}\mathbf{f}\mathbf{f}$.
Hence, ADF $D_{f}$, given by the acceptance conditions
$$\begin{array}[]{ll}\varphi^{f}_{a}=a\land b\land\neg c,\qquad\varphi^{f}_{c}=\neg a\land b\land c,\\
\varphi^{f}_{b}=(\neg a\land b\neg\land\neg c)\lor(\neg a\land\neg b\land\neg c)\\
\end{array}$$
has $V_{3}$ as its complete semantics.
$\blacksquare$
Finally, for two-valued model semantics,
propagator $p^{\in}_{\mathit{mod}}$ derives new triples by looking at interpretations $v\in V$.
For those, we must find $f(v)=v$ in each $\mathit{mod}$-characterization $f$ by definition.
Thus the algorithm adds $(v,a,v(a))$ for each $a\in A$ to the partial characterization $F$.
Propagator $p^{\notin}_{\mathit{mod}}$ looks at interpretations $v\in\mathcal{V}_{2}\setminus V$, for which it must hold that $f(v)\neq v$.
Thus there must be a statement $a\in A$ with $v(a)\neq f(v)(a)$, which is exactly what this propagator derives whenever it is clear that there is only one statement candidate left.
This, in turn, is the case whenever all $b\in A$ with the opposite truth value $\neg v(a)$ and all $c\in A$ with $c\neq a$ cannot coherently become the necessary witness any more.
The propagator $p^{\lightning}_{\mathit{mod}}$ checks whether $V\subseteq\mathcal{V}_{2}$, that is, the desired set of interpretations consists entirely of two-valued interpretations.
In that case this propagator makes the relation $F$ incoherent, following a similar strategy as $p^{\lightning}_{\mathit{adm}}$.
Preferred Semantics
Realizing a given set of interpretations $V$ under preferred semantics requires special treatment.
We do not have a $\sigma$-characterization function for $\sigma=\mathit{prf}$ at hand
to directly check realizability of $V$ but
have to find some $V^{\prime}\subseteq\{v\in\mathcal{V}\mid\exists v^{\prime}\in V:v<_{i}v^{\prime}\}$
such that $V\cup V^{\prime}$ is realizable under admissible semantics (cf. Corollary 6).
Algorithm 2 implements this idea by
guessing such a $V^{\prime}$ (line 7)
and then using Algorithm 1 to try to realize $V\cup V^{\prime}$
under admissible semantics (line 11).
If $\mathit{realize}$ returns a knowledge base $\mathsf{kb}$ realizing $V\cup V^{\prime}$ under $\mathit{adm}$
we can directly use $\mathsf{kb}$ as solution of $\mathit{realizePrf}$
since it holds that $\mathit{prf}(\mathsf{kb})=V$,
given that $V$ is an $\leq_{i}$-antichain (line 2).
3.3 Formalism Propagators
When constructing an ADF realizing a given set $V$ of interpretations under a semantics $\sigma$,
the function $\mathit{kb}^{\textrm{ADF}}_{\sigma}(F)$ makes use of the $\sigma$-characterization given by $F$ in the following way:
$v$ is a model of the acceptance condition $\varphi_{a}$ if and only if we find $(v,a,\mathbf{t})\in F$.
Now as bipolar ADFs, SETAFs and AFs are all subclasses of ADFs by
restricting the acceptance conditions of statements,
these restrictions also carry over to the $\sigma$-characterizations.
The propagators defined below use structural knowledge on the form of acceptance conditions
of the respective formalisms to reduce the search space or to induce incoherence of $F$ whenever $V$ is not realizable.
Bipolar ADFs
For bipolar ADFs, we use the fact that each of their links must have at least one polarity, that is, must be supporting or attacking.
Therefore, if a link is not supporting, it must be attacking, and vice versa.
For canonical realization, we obtain the polarities of links,
i.e. the sets $L^{+}$ and $L^{-}$,
as defined in Figure 2.
AFs
To explain the AF propagators, we first need some more definitions.
On the two classical truth values, we define the truth ordering $\mathbf{f}<_{t}\mathbf{t}$, whence the operations $\sqcup_{t}$ and $\sqcap_{t}$ with $\mathbf{f}\sqcup_{t}\mathbf{t}=\mathbf{t}$ and $\mathbf{f}\sqcap_{t}\mathbf{t}=\mathbf{f}$ result.
These operations can be lifted pointwise to two-valued interpretations as usual, that is,
$(v_{1}\sqcup_{t}v_{2})(a)=v_{1}(a)\sqcup_{t}v_{2}(a)$ and
$(v_{1}\sqcap_{t}v_{2})(a)=v_{1}(a)\sqcap_{t}v_{2}(a)$.
Again, the reflexive version of $<_{t}$ is denoted by $\leq_{t}$.
The pair $(\mathcal{V}_{2},\leq_{t})$ of two-valued interpretations ordered by the truth ordering forms a complete lattice with glb $\sqcap_{t}$ and lub $\sqcup_{t}$.
This complete lattice has the least element $v_{\mathbf{f}}:A\to\left\{\mathbf{f}\right\}$, the interpretation mapping all statements to false, and the greatest element $v_{\mathbf{t}}:A\to\left\{\mathbf{t}\right\}$ mapping all statements to true, respectively.
Acceptance conditions of AF-based ADFs have the form of conjunctions of negative literals.
In the complete lattice $(\mathcal{V}_{2},\leq_{t})$, the model sets of AF acceptance conditions correspond to the lattice-theoretic concept of an ideal, a subset of $\mathcal{V}_{2}$ that is downward-closed with respect to $\leq_{t}$ and upward-closed with respect to $\sqcup_{t}$.
The propagator directly implements these closure properties:
application of $p^{\textrm{AF}}$ ensures that when a $\sigma$-characterization $F$
that is neither incoherent nor partial
is found in line 8 of Algorithm 1,
then there is, for each $a\in A$, an interpretation $v_{a}$ such that
$(v_{a},a,\mathbf{t})\in F$ and
$v\leq_{t}v_{a}$ for each $(v,a,\mathbf{t})\in F$. Hence $v_{a}$ is crucial for the acceptance condition,
or in AF terms the attacks,
of $a$ and we can define
$\mathit{kb}^{\textrm{AF}}_{\sigma}(F)=(A,\{(b,a)\mid a,b\in A,v_{a}(b)=\mathbf{f}\})$.
SETAFs
The propagator for SETAFs, $p^{\textrm{SETAF}}$, is a weaker version of that of AFs, since we cannot presume upward-closure with respect to $\sqcup_{t}$.
In SETAF-based ADFs the acceptance formula is in conjunctive normal form
containing only negative literals.
By a transformation preserving logical equivalence we obtain
an acceptance condition in disjunctive normal form, again with only negative literals;
in other words, a disjunction of AF acceptance formulas.
Thus, the model set of a SETAF acceptance condition is not necessarily an ideal, but a union of ideals.
For the canonical realization we can make use of the fact
that, for each $a\in A$, the set
$V^{\mathbf{t}}_{a}=\{v\in\mathcal{V}_{2}\mid(v,a,\mathbf{t})\in F\}$
is downward-closed with respect to $\leq_{t}$,
hence the set of models of
$\bigvee_{v\in\max_{\leq_{t}}V^{\mathbf{t}}}\bigwedge_{v(b)=\mathbf{f}}\neg b$
is exactly $V^{\mathbf{t}}_{a}$.
The clauses of its corresponding CNF-formula exactly coincide with
the sets of arguments attacking $a$ in $\mathit{kb}^{\textrm{SETAF}}_{\sigma}(F)$.
3.4 Correctness
For a lack of space, we could not include a formal proof of soundness and completeness of Algorithm 1, but rather present arguments for termination and correctness.
Termination
With each recursive call, the set $F$ can never decrease in size, as the only changes to $F$ are adding the results of propagation in line 3 and adding the guesses in line 11.
Also within the until-loop, the set $F$ can never decrease in size;
furthermore there is only an overall finite number of triples that can be added to $F$.
Thus at some point we must have $F_{\Delta}=\emptyset$ and leave the until-loop.
Since $F$ always increases in size, at some point it must either become functional or incoherent, whence the algorithm terminates.
Soundness
If the algorithm returns a realizing knowledge base $\mathit{kb}^{\mathcal{F}}_{\sigma}(F)$, then according to the condition in line 8 the relation $F$ induced a total function $f:\mathcal{V}_{2}\to\mathcal{V}_{2}$.
In particular, because the until-loop must have been run through at least once, there was at least one propagation step (line 2).
Since the propagators are defined such that they enforce everything that must hold in a $\sigma$-characterization, we conclude that the induced function $f$ indeed is a $\sigma$-characterization for $V$.
By construction, we consequently find that $\sigma(\mathit{kb}^{\mathcal{F}}_{\sigma}(F))=V$.
Completeness
If the algorithm answers “no”, then the execution reached line 5.
Thus, for the constructed set $F$, there must have been an interpretation $v\in\mathcal{V}_{2}$ and a statement $a\in A$ such that $\left\{(v,a,\mathbf{t}),(v,a,\mathbf{f})\right\}\subseteq F$, that is, $F$ is incoherent.
Since $F$ is initially empty, the only way it could get incoherent is in the propagation step in line 2.
(The guessing step cannot create incoherence, since exactly one truth value is guessed for $v$ and $a$.)
However, the propagators are defined such that they infer only assignments (triples) that are necessary for the given $F$.
Consequently, the given interpretation set $V$ is such that either there is no realization within the ADF fragment corresponding to formalism $\mathcal{F}$ (that is, the formalism propagator derived the incoherence) or there is no $\sigma$-characterization for $V$ with respect to general ADFs (that is, the semantics propagator derived the incoherence).
In any case, $V$ is not $\sigma$-realizable for $\mathcal{F}$.
4 Implementation
As Algorithm 1 is based on propagation, guessing, and checking it is perfectly suited for an implementation using answer set programming (ASP) (Niemelä, 1999; Marek and Truszczyński, 1999)
as this allows for exploiting conflict learning strategies and heuristics of modern ASP solvers.
Thus, we developed ASP encodings in the Gringo language (Gebser et al., 2012) for our approach.
Similar as the algorithm, our declarative encodings are modular, consisting of a main part responsible for constructing set $F$ and separate encodings for the individual propagators.
If one wants, e.g., to compute an AF realization under admissible semantics for a set $V$ of interpretations, an input program encoding $V$ is joined with the main encoding, the propagator encoding for admissible semantics as well as the propagator encoding for AFs.
Every answer set of such a program encodes a respective characterization function.
Our ASP encoding for preferred semantics is based on the admissible encoding and guesses further interpretations following the essential idea of Algorithm 2.
For constructing a knowledge base with the desired semantics, we also provide two ASP encodings that transform the output to an ADF in the syntax of the DIAMOND tool (Ellmauthaler and
Strass, 2014), respectively an AF in ASPARTIX syntax (Egly, Gaggl, and Woltran, 2010; Gaggl et al., 2015).
Both argumentation tools are based on ASP themselves.
The encodings for all the semantics and formalisms we covered in the paper can be downloaded from http://www.dbai.tuwien.ac.at/research/project/adf/unreal/.
A selection of them is depicted in Figure 3 on the next page.
5 Expressiveness Results
In this section we briefly present some results that we have obtained using our implementation.
We first introduce some necessary notation to describe the relative expressiveness of knowledge representation formalisms (Gogic et al., 1995; Strass, 2015).
For formalisms $\mathcal{F}_{1}$ and $\mathcal{F}_{2}$ with semantics $\sigma_{1}$ and $\sigma_{2}$, we say that $\mathcal{F}_{2}$ under $\sigma_{2}$ is at least as expressive as $\mathcal{F}_{1}$ under $\sigma_{1}$ and write $\mathcal{F}_{1}^{\sigma_{1}}\leq_{e}\mathcal{F}_{2}^{\sigma_{2}}$ if and only if $\Sigma_{\mathcal{F}_{1}}^{\sigma_{1}}\subseteq\Sigma_{\mathcal{F}_{2}}^{\sigma_{2}}$, where $\Sigma_{\mathcal{F}}^{\sigma}=\left\{\sigma(\mathsf{kb})\ \middle|\ \mathsf{kb}\in\mathcal{F}\right\}$ is the signature of $\mathcal{F}$ under $\sigma$.
As usual, we define $\mathcal{F}_{1}<_{e}\mathcal{F}_{2}$ iff $F_{1}\leq_{e}\mathcal{F}_{2}$ and $F_{2}\not\leq_{e}\mathcal{F}_{1}$.
We now start by considering the signatures of AFs, SETAFs and (B)ADFs for the unary vocabulary $\left\{a\right\}$:
$$\displaystyle\Sigma_{\textrm{AF}}^{\mathit{adm}}=\Sigma_{\textrm{SETAF}}^{\mathit{adm}}$$
$$\displaystyle=\left\{\left\{\mathbf{u}\right\},\left\{\mathbf{u},\mathbf{t}\right\}\right\}$$
$$\displaystyle\Sigma_{\textrm{AF}}^{\mathit{com}}=\Sigma_{\textrm{SETAF}}^{\mathit{com}}$$
$$\displaystyle=\left\{\left\{\mathbf{u}\right\},\left\{\mathbf{t}\right\}\right\}$$
$$\displaystyle\Sigma_{\textrm{AF}}^{\mathit{prf}}=\Sigma_{\textrm{SETAF}}^{\mathit{prf}}$$
$$\displaystyle=\left\{\left\{\mathbf{u}\right\},\left\{\mathbf{t}\right\}\right\}$$
$$\displaystyle\Sigma_{\textrm{AF}}^{\mathit{mod}}=\Sigma_{\textrm{SETAF}}^{\mathit{mod}}$$
$$\displaystyle=\left\{\emptyset,\left\{\mathbf{t}\right\}\right\}$$
$$\displaystyle\Sigma_{\textrm{ADF}}^{\mathit{adm}}=\Sigma_{\textrm{BADF}}^{\mathit{adm}}$$
$$\displaystyle=\Sigma_{\textrm{AF}}^{\mathit{adm}}\cup\left\{\left\{\mathbf{u},\mathbf{f}\right\},\left\{\mathbf{u},\mathbf{t},\mathbf{f}\right\}\right\}$$
$$\displaystyle\Sigma_{\textrm{ADF}}^{\mathit{com}}=\Sigma_{\textrm{BADF}}^{\mathit{com}}$$
$$\displaystyle=\Sigma_{\textrm{AF}}^{\mathit{com}}\cup\left\{\left\{\mathbf{f}\right\},\left\{\mathbf{u},\mathbf{t},\mathbf{f}\right\}\right\}$$
$$\displaystyle\Sigma_{\textrm{ADF}}^{\mathit{prf}}=\Sigma_{\textrm{BADF}}^{\mathit{prf}}$$
$$\displaystyle=\Sigma_{\textrm{AF}}^{\mathit{prf}}\cup\left\{\left\{\mathbf{f}\right\},\left\{\mathbf{t},\mathbf{f}\right\}\right\}$$
$$\displaystyle\Sigma_{\textrm{ADF}}^{\mathit{mod}}=\Sigma_{\textrm{BADF}}^{\mathit{mod}}$$
$$\displaystyle=\Sigma_{\textrm{AF}}^{\mathit{mod}}\cup\left\{\left\{\mathbf{f}\right\},\left\{\mathbf{t},\mathbf{f}\right\}\right\}$$
The following result shows that the expressiveness of the formalisms under consideration
is in line with the amount of restrictions they impose on acceptance formulas.
Theorem 8.
For any $\sigma\in\left\{\mathit{adm},\mathit{com},\mathit{prf},\mathit{mod}\right\}$:
1.
$\textrm{AF}^{\sigma}<_{e}\textrm{SETAF}^{\sigma}$.
2.
$\textrm{SETAF}^{\sigma}<_{e}\textrm{BADF}^{\sigma}$.
3.
$\textrm{BADF}^{\sigma}<_{e}\textrm{ADF}^{\sigma}$.
{longproof}
(1)
$\textrm{AF}^{\sigma}\leq_{e}\textrm{SETAF}^{\sigma}$ is clear (by modeling individual attacks via singletons).
For $\textrm{SETAF}^{\sigma}\not\leq_{e}\textrm{AF}^{\sigma}$ the witnessing model sets over vocabulary $A=\left\{a,b,c\right\}$ are
$\left\{\mathbf{u}\mathbf{u}\mathbf{u},\mathbf{t}\mathbf{t}\mathbf{f},\mathbf{t}\mathbf{f}\mathbf{t},\mathbf{f}\mathbf{t}\mathbf{t}\right\}\in\Sigma_{\textrm{SETAF}}^{\sigma}\setminus\Sigma_{\textrm{AF}}^{\sigma}$ and
$\left\{\mathbf{t}\mathbf{t}\mathbf{f},\mathbf{t}\mathbf{f}\mathbf{t},\mathbf{f}\mathbf{t}\mathbf{t}\right\}\in\Sigma_{\textrm{SETAF}}^{\tau}\setminus\Sigma_{\textrm{AF}}^{\tau}$
with $\sigma\in\{\mathit{adm},\mathit{com}\}$ and $\tau\in\{\mathit{prf},\mathit{mod}\}$.
By each pair of arguments of $A$ being $\mathbf{t}$ in at least one model,
a realizing AF cannot feature any attack,
immediately giving rise to the model $\mathbf{t}\mathbf{t}\mathbf{t}$.
The respective realizing SETAF is given by the attack relation
$R=\left\{(\{a,b\},c),(\{a,c\},b),(\{b,c\},a)\right\}$.
(2)
It is clear that $\textrm{SETAF}^{\sigma}\leq_{e}\textrm{BADF}^{\sigma}$ holds (all parents are always attacking).
For $\textrm{BADF}^{\sigma}\not\leq_{e}\textrm{SETAF}^{\sigma}$ the respective counterexamples can be read off the signatures above:
for $\sigma\in\left\{\mathit{adm},\mathit{com}\right\}$ we find
$\left\{\mathbf{u},\mathbf{t},\mathbf{f}\right\}\in\Sigma_{\textrm{BADF}}^{\sigma}\setminus\Sigma_{\textrm{SETAF}}^{\sigma}$
and for $\tau\in\left\{\mathit{prf},\mathit{mod}\right\}$ we find
$\left\{\mathbf{t},\mathbf{f}\right\}\in\Sigma_{\textrm{BADF}}^{\tau}\setminus\Sigma_{\textrm{SETAF}}^{\tau}$.
(3)
For $\sigma=\mathit{mod}$ the result is known (Strass, 2015, Theorem 14);
for the remaining semantics the model sets witnessing $\textrm{ADF}^{\sigma}\not\leq_{e}\textrm{BADF}^{\sigma}$ over vocabulary $A=\left\{a,b\right\}$ are
$$\displaystyle\left\{\mathbf{u}\mathbf{u},\mathbf{t}\mathbf{u},\mathbf{t}\mathbf{t},\mathbf{t}\mathbf{f},\mathbf{f}\mathbf{u}\right\}$$
$$\displaystyle\in\Sigma_{\textrm{ADF}}^{\mathit{adm}}\setminus\Sigma_{\textrm{BADF}}^{\mathit{adm}}$$
$$\displaystyle\left\{\mathbf{u}\mathbf{u},\mathbf{t}\mathbf{u},\mathbf{t}\mathbf{t},\mathbf{t}\mathbf{f},\mathbf{f}\mathbf{u}\right\}$$
$$\displaystyle\in\Sigma_{\textrm{ADF}}^{\mathit{com}}\setminus\Sigma_{\textrm{BADF}}^{\mathit{com}}$$
$$\displaystyle\left\{\mathbf{t}\mathbf{t},\mathbf{t}\mathbf{f},\mathbf{f}\mathbf{u}\right\}$$
$$\displaystyle\in\Sigma_{\textrm{ADF}}^{\mathit{prf}}\setminus\Sigma_{\textrm{BADF}}^{\mathit{prf}}$$
A witnessing ADF is given by $\varphi_{a}=a$ and $\varphi_{b}=a\leftrightarrow b$.
$\Box$
Theorem 8 is concerned with the relative expressiveness of the formalisms under consideration, given a certain semantics.
Considering different semantics we find that for all formalisms the signatures become incomparable:
Proposition 9.
$\mathcal{F}_{1}^{\sigma_{1}}\not\leq_{e}\mathcal{F}_{2}^{\sigma_{2}}$ and
$\mathcal{F}_{2}^{\sigma_{2}}\not\leq_{e}\mathcal{F}_{1}^{\sigma_{1}}$
for all formalisms $\mathcal{F}_{1},\mathcal{F}_{2}\in\{\textrm{AF},\textrm{SETAF},\textrm{BADF},\textrm{ADF}\}$ and all semantics
$\sigma_{1},\sigma_{2}\in\{\mathit{adm},\mathit{com},\mathit{prf},\mathit{mod}\}$ with
$\sigma_{1}\neq\sigma_{2}$.
{longproof}
First, the result for $\mathit{adm}$ and $\mathit{com}$
follows by
$\{\mathbf{u},\mathbf{t}\}\in\Sigma^{\mathit{adm}}_{\textrm{AF}}$, but $\{\mathbf{u},\mathbf{t}\}\notin\Sigma^{\mathit{com}}_{\textrm{ADF}}$
and
$\{\mathbf{t}\}\in\Sigma^{\mathit{com}}_{\textrm{AF}}$, but $\{\mathbf{t}\}\notin\Sigma^{\mathit{adm}}_{\textrm{ADF}}$.
Moreover,
taking into account that the set of preferred interpretations (resp. two-valued models)
always forms a $\leq_{i}$-antichain while
the set of admissible (resp. complete) interpretations never does,
the result follows
for $\sigma_{1}\in\{\mathit{adm},\mathit{com}\}$ and $\sigma_{2}\in\{\mathit{prf},\mathit{mod}\}$.
Finally, since a $\mathsf{kb}\in\mathcal{F}$ may not have any two-valued models
and a preferred interpretation is not necessarily two-valued,
the result for $\mathit{prf}$ and $\mathit{mod}$ follows.
$\Box$
Disregarding the possibility of realizing the empty set of interpretations
under the two-valued model semantics,
we obtain the following relation for ADFs.
Proposition 10.
$(\Sigma_{\textrm{ADF}}^{\mathit{mod}}\setminus\{\emptyset\})\subseteq\Sigma_{\textrm{ADF}}^{\mathit{prf}}$.
{longproof}
Consider some $V\in\Sigma_{\textrm{ADF}}^{\mathit{mod}}$ with $V\neq\emptyset$.
Clearly $V\subseteq\mathcal{V}_{2}$ and by Proposition 4 there is a $\mathit{mod}$-characterization $f:\mathcal{V}_{2}\to\mathcal{V}_{2}$ for $V$, that is, $f(v)=v$ iff $v\in V$.
Define $f^{\prime}:\mathcal{V}_{2}\to\mathcal{V}_{2}$ such that $f^{\prime}(v)=f(v)=v$ for all $v\in V$
and $f^{\prime}(v)(a)=\neg v(a)$ for all $v\in\mathcal{V}\setminus V$ and $a\in A$.
Now it holds that $f^{\prime}$ is an $\mathit{adm}$-characterization of
$V^{\prime}=\{v\in\mathcal{V}\mid\forall v_{2}\in[{v}]_{2}:v_{2}\in V\}\cup\{v_{\mathbf{u}}\}$.
Since $\max_{\leq_{i}}V^{\prime}=V$ we get that the ADF $D$ with acceptance formula
$\varphi^{f^{\prime}}_{a}$ for each $a\in A$ has $\mathit{prf}(D)=V$ whence $V\in\Sigma_{\textrm{ADF}}^{\mathit{prf}}$.
$\Box$
In contrast, this relation does not hold for AFs,
which was shown for extension-based semantics
by Linsbichler, Spanring, and
Woltran (2015) (Theorem 5) and immediately follows for the three-valued case.
6 Discussion
We presented a framework for realizability in which AFs, SETAFs, BADFs and general ADFs can be treated in a uniform way.
The centerpiece of our approach is an algorithm for deciding realizability of a given interpretation-set in a formalism under a semantics.
The algorithm makes use of so-called propagators, by which it can be adapted to the different formalisms and semantics.
We also presented an implementation of our framework in answer set programming and several novel expressiveness results that we obtained using our implementation.
In related work, Polberg (2016) studies a wide range of abstract argumentation formalisms, in particular their relationship with ADFs.
This can be the basis for including further formalisms into our realizability framework:
all that remains to do is figuring out suitable ADF fragments and developing propagators for them, just like we did exemplarily for Nielsen and Parsons’s SETAFs.
For further future work, we could also streamline existing propagators such that they do not only derive absolutely necessary assignments, but also logically weaker conclusions, such as disjunctions of (non-)assignments.
References
Amgoud and
Cayrol (2002)
Amgoud, L., and Cayrol, C.
2002.
A reasoning model based on the production of acceptable arguments.
Ann. Math. Artif. Intell. 34(1–3):197–215.
Baroni et al. (2011)
Baroni, P.; Cerutti, F.; Giacomin, M.; and Guida, G.
2011.
AFRA: Argumentation framework with recursive attacks.
Int. J. Approx. Reasoning 52(1):19–37.
Baumann et al. (2014)
Baumann, R.; Dvořák, W.; Linsbichler, T.; Strass, H.; and Woltran, S.
2014.
Compact argumentation frameworks.
In Proc. ECAI, volume 263 of FAIA, 69–74.
Brewka and Woltran (2010)
Brewka, G., and Woltran, S.
2010.
Abstract Dialectical Frameworks.
In Proc. KR, 102–111.
Brewka et al. (2013)
Brewka, G.; Ellmauthaler, S.; Strass, H.; Wallner, J. P.; and Woltran, S.
2013.
Abstract Dialectical Frameworks Revisited.
In Proc. IJCAI, 803–809.
Brewka, Polberg, and
Woltran (2014)
Brewka, G.; Polberg, S.; and Woltran, S.
2014.
Generalizations of Dung frameworks and their role in formal
argumentation.
IEEE Intelligent Systems 29(1):30–38.
Caminada and Gabbay (2009)
Caminada, M., and Gabbay, D.
2009.
A logical account of formal argumentation.
Studia Logica 93(2-3):109–145.
Cayrol and
Lagasquie-Schiex (2005)
Cayrol, C., and Lagasquie-Schiex, M.
2005.
On the acceptability of arguments in bipolar argumentation
frameworks.
In Proc. ECSQARU, volume 3571 of LNCS, 378–389.
Dung (1995)
Dung, P. M.
1995.
On the acceptability of arguments and its fundamental role in
nonmonotonic reasoning, logic programming and n-person games.
Artif. Intell. 77(2):321–357.
Dunne et al. (2013)
Dunne, P. E.; Dvořák, W.; Linsbichler, T.; and Woltran, S.
2013.
Characteristics of multiple viewpoints in abstract argumentation.
In Proc. DKB, 16–30.
Dunne et al. (2015)
Dunne, P. E.; Dvořák, W.; Linsbichler, T.; and Woltran, S.
2015.
Characteristics of multiple viewpoints in abstract argumentation.
Artif. Intell. 228:153–178.
Dyrkolbotn (2014)
Dyrkolbotn, S. K.
2014.
How to Argue for Anything: Enforcing Arbitrary Sets of Labellings
using AFs.
In Proc. KR, 626–629.
Egly, Gaggl, and Woltran (2010)
Egly, U.; Gaggl, S. A.; and Woltran, S.
2010.
Answer-set programming encodings for argumentation frameworks.
Argument & Computation 1(2):147–177.
Ellmauthaler and
Strass (2014)
Ellmauthaler, S., and Strass, H.
2014.
The DIAMOND system for computing with abstract dialectical
frameworks.
In Proc. COMMA, volume 266 of FAIA, 233–240.
Gaggl et al. (2015)
Gaggl, S. A.; Manthey, N.; Ronca, A.; Wallner, J. P.; and Woltran, S.
2015.
Improved answer-set programming encodings for abstract argumentation.
TPLP 15(4-5):434–448.
Gebser et al. (2012)
Gebser, M.; Kaminski, R.; Kaufmann, B.; and Schaub, T.
2012.
Answer Set Solving in Practice.
Synthesis Lectures on Artificial Intelligence and Machine Learning.
Morgan and Claypool Publishers.
Gogic et al. (1995)
Gogic, G.; Kautz, H.; Papadimitriou, C.; and Selman, B.
1995.
The comparative linguistics of knowledge representation.
In Proc. IJCAI, 862–869.
Gomes et al. (2008)
Gomes, C. P.; Kautz, H. A.; Sabharwal, A.; and Selman, B.
2008.
Satisfiability Solvers.
In Handbook of Knowledge Representation, volume 3 of Foundations of AI. Elsevier.
89–134.
Linsbichler, Spanring, and
Woltran (2015)
Linsbichler, T.; Spanring, C.; and Woltran, S.
2015.
The hidden power of abstract argumentation semantics.
In Proc. TAFA, volume 9524 of LNCS, 146–162.
Marek and Truszczyński (1999)
Marek, V. W., and Truszczyński, M.
1999.
Stable models and an alternative logic programming paradigm.
In In The Logic Programming Paradigm: a 25-Year Perspective.
Springer.
375–398.
Modgil (2009)
Modgil, S.
2009.
Reasoning about preferences in argumentation frameworks.
Artif. Intell. 173(9–10):901–934.
Nielsen and Parsons (2006)
Nielsen, S. H., and Parsons, S.
2006.
A generalization of Dung’s abstract framework for argumentation:
Arguing with sets of attacking arguments.
In Proc. ArgMAS, volume 4766 of LNCS, 54–73.
Niemelä (1999)
Niemelä, I.
1999.
Logic programs with stable model semantics as a constraint
programming paradigm.
Ann. Math. Artif. Intell. 25(3-4):241–273.
Polberg (2016)
Polberg, S.
2016.
Developing and Extending the Abstract Dialectical Framework.
Ph.D. Dissertation, TU Wien, Austria.
Upcoming.
Pührer (2015)
Pührer, J.
2015.
Realizability of Three-Valued Semantics for Abstract Dialectical
Frameworks.
In Proc. IJCAI, 3171–3177.
Strass and Wallner (2015)
Strass, H., and Wallner, J. P.
2015.
Analyzing the Computational Complexity of Abstract Dialectical
Frameworks via Approximation Fixpoint Theory.
Artif. Intell. 226:34–74.
Strass (2015)
Strass, H.
2015.
Expressiveness of Two-Valued Semantics for Abstract Dialectical
Frameworks.
J. Artif. Intell. Res. (JAIR) 54:193–231. |
Strangeness Production Incorporating Chiral Symmetry
††thanks: To appear in the proceedings of the 4th International Symposium on
Strangeness in Quark Matter, Padova, Italy, July 20–24.
P. Rehberg
SUBATECH
Laboratoire de Physique Subatomique et des Technologies
Associées
UMR Universté de Nantes, IN2P3/CNRS, Ecole des Mines de Nantes
4 Rue Alfred Kastler, F-44070 Nantes Cedex 3, France
Abstract
Chiral symmetry is known to be decisive for an understanding of the
low energy sector of strong interactions. It is thus important for a
model of relativistic heavy ion collisions to incorporate the dynamical
breaking and restoration of chiral symmetry. Thus we study an expansion
scenario for a quark-meson plasma using the Nambu–Jona-Lasinio (NJL)
model in its three flavor version. The equations of motion for light and
strange quarks as well as for pions, kaons and etas are solved using a
QMD type algorithm, which is based on a parametrization of the Wigner
function. The scattering processes incorporated into this calculation are
of the types $qq\leftrightarrow qq$, $q\bar{q}\leftrightarrow q\bar{q}$, $q\bar{q}\leftrightarrow MM$
and $M\leftrightarrow q\bar{q}$.
††preprint:
In relativistic heavy ion collisions, a very hot and dense system of
strongly interacting matter is created. At these high temperatures
and densities, the state of matter changes drastically. The two main
effects which are predicted from quantum chromodynamics (QCD) are (i)
deconfinement and (ii) chiral symmetry restauration. Lattice simulations
indicate, that these two effects are not independent of each other, but
occur at the same time [1]. Although experiments searching for
this new state of matter have been performed at the Cern SPS and will be
performed at RHIC and LHC, the theoretical description still leaves many
open questions. Some of these are (i) the question, if a local thermal
and/or chemical equilibrium will be established, (ii) the influence of the
in-medium changes of hadronic properties and (iii) the description of the
hadronization. The treatment of these problems is by no means simple,
since a satisfactory, simple model for the description of the phase
transition to a quark-gluon plasma (QGP) does not yet exist. Since the
QGP, if created, is a transient state, a realistic model for its evolution
should be able to handle nonequilibrium effects. A nonequilibrium theory
of nonabelian gauge theories such as QCD, however, cannot be provided
using present days knowledge, since these theories even in thermal
equilibrium are not yet sufficiently understood. Lattice gauge theories,
on the other hand, have been successfully applied in order to extract
the thermal behaviour of strongly interacting matter. They work,
however, only in equilibrium situations. Another practical approach
which is frequently used is hydrodynamics. These models, however,
suffer from the fact that they work in the limit of infinitely
many collisions per time interval, which might not be true in the
first stages and surely is not true in the late stages of the collision.
The ansatz presented here thus the modelization of the expansion of a
hot system using an effective Lagrangian. The model interaction we use
is the three flavor Nambu–Jona-Lasinio (NJL) model [2],
defined by
$$\displaystyle{\cal L}$$
$$\displaystyle=$$
$$\displaystyle\sum_{f=u,d,s}\bar{\psi}_{f}(i\!\not\!\partial-m_{0f})\psi_{f}+G%
\sum_{a=0}^{8}\left[\left(\bar{\psi}\lambda_{a}\psi\right)^{2}+\left(\bar{\psi%
}i\gamma_{5}\lambda_{a}\psi\right)^{2}\right]$$
$$\displaystyle+$$
$$\displaystyle K\left[\det\bar{\psi}\left(1+\gamma_{5}\right)\psi+\det\bar{\psi%
}\left(1-\gamma_{5}\right)\psi\right]\quad.$$
In this model, the exchange interaction has been contracted to give a
pointlike interaction in flavor space. This approximation works as long
as the effective mass of the gluon is high. Recent QCD calculations
support this assumption even at high temperatures [3].
The most important feature of the Lagrangian (Strangeness Production Incorporating Chiral Symmetry
††thanks: To appear in the proceedings of the 4th International Symposium on
Strangeness in Quark Matter, Padova, Italy, July 20–24.) is that it
preserves the chiral symmetry of QCD, i. e. in the limit
$m_{0f}\to 0$ it is invariant under transformations of the form
$$\psi\to\exp\left(-i\theta_{a}\lambda_{a}\gamma_{5}\right)\psi\quad.$$
(2)
It is a well established fact, that this symmetry is the essential
ingredient for the description of the low lying hadronic states
which are produced copiously in a heavy ion collision [4].
The equilibrium properties of the NJL model in thermal
equilibrium have been studied in great detail elsewhere
[2, 5, 6, 7, 8], so that only those
properties will be mentioned here, which are important for the
understanding of the present article. For simplicity, first the case
$m_{0f}=0$ will be detailled. At temperature $T=0$, chiral symmetry is
spontaneously broken, which leads to a finite effective quark mass. As a
consequence of the Goldstone theorem, $N_{f}^{2}-1$ massless modes appear as
quark-antiquark bound states. These states are identified with the $\pi$,
$K$ and $\eta$ mesons. At a certain finite temperature, however, chiral
symmetry gets restored and one has again massless quarks, whereas the
mesons become massive. In this phase, they do no longer exist as bound
states but rather as resonances. If one introduces a finite current quark
mass, $m_{0f}\neq 0$, this picture gets slightly changed in that chiral
symmetry is no longer an exact, but rather an approximate symmetry. Thus
mesons have a small but finite mass at $T=0$, as they have in nature,
whereas the quark mass does not go to zero at large temperatures, but
stays finite and for $T\to\infty$ goes to the current quark mass.
The quark mass spectrum for this case is shown in Fig. 1.
The light quarks $u$ and $d$ are taken to be degenerate and will
be denoted by the generic index $q$ in the following. The current
quark masses used in Fig. 1 are $m_{0q}=5.5$ MeV and
$m_{0s}=140$ MeV. Due to chiral symmetry breaking, one has at $T=0$ an
effective mass of $m_{q}=368$ MeV and $m_{s}=550$ MeV. In the temperature
region $T\approx 200$ MeV, these masses drop and the effective mass
of the light quarks becomes low. For the strange quarks, however,
the situation is different. Due to the high current quark mass, the
effective mass stays relatively high. At $T=350$ MeV, which is far
beyond any temperature to be realistically expected in present days heavy
ion experiments, one still has $m_{s}=300$ MeV, which is about twice
as high as the current quark mass. The pseudoscalar mesonic mass spectrum
using the same parameters is shown in Fig. 2. At $T=0$, the
mesons have their physical masses of $m_{\pi}=135$ MeV, $m_{K}=498$ MeV
and $m_{\eta}=515$ MeV. With rising temperatures, these values stay more
or less constant, until to a certain temperature, where their masses
become equal to the masses of their constituents, i. e. $m_{\pi}=2m_{q}$,
$m_{K}=m_{q}+m_{s}$ or $m_{\eta}=2m_{q}$, respectively. This happens when the
$m_{\pi}$ and $m_{\eta}$ lines of Fig. 2 cross the $2m_{q}$ line, or
the $m_{K}$ line crosses the $m_{q}+m_{s}$ line, respectively, as is indicated
by the arrows. At these temperatures, the mesons become unstable via a
Mott transition [6]. At higher temperatures, they are unstable
with respect to the decay $M\to q\bar{q}$ and obtain a finite width.
The qualitative form of the mass spectrum of Fig. 2 has been
confirmed by lattice calculations [1].
A finite temperature study of this model will surely not lead to an event
generator, since the model still contains free quark states, which are
not observed in nature. It will, however, be able to study qualitative
features of the plasma expansion, such as expansion time scales, the
approach to equilibrium and thus the applicability of hydrodynamics,
the production mechanism of hadrons etc. While the simulation program
presently is limited to zero baryochemical potential, future extensions
will remove this limitation and also allow for the study of strangeness
destillation and DCC formation.
The treatment of the NJL model in nonequilibrium starts from the
observation, that both quark and meson degrees of freedom can
be simultaneously described by an equation of the
Boltzmann type [9],
$$\left(\partial_{t}+\vec{\partial}_{p}E\vec{\partial}_{x}-\vec{\partial}_{x}E%
\vec{\partial}_{p}\right)n(t,\vec{x},\vec{p})=I_{\rm coll}\left[n(t,\vec{x},%
\vec{p})\right]\quad,$$
(3)
where $I_{\rm coll}[n]$ is a collision integral. In principle, one
is thus able to describe a transition from a pure quark regime to a
hadronic regime, where quarks are converted to hadrons via collision
processes. The solution method we choose for Eq. (3) is an
algorithm of the QMD type [10], i. e. we parametrize the particle
distribution functions by
$$n(t,\vec{x},\vec{p})=\sum_{i}\exp\left(-\frac{(\vec{x}-\vec{x}_{i}(t))^{2}}{2w%
^{2}}\right)\exp\left(-\frac{w^{2}}{2}(\vec{p}-\vec{p}_{i}(t))^{2}\right)\quad.$$
(4)
The center points of the distribution move on the characteristics of
Eq. (3), i. e.
$$\dot{\vec{x}}_{i}(t)=\vec{p}_{i}(t)/E\hskip 56.905512pt\dot{\vec{p}}_{i}(t)=-%
\vec{\partial}_{x}E+\mbox{collision contributions}\quad.$$
(5)
Equation (5) has to be solved together with the gap equation
$$\displaystyle m_{i}$$
$$\displaystyle=$$
$$\displaystyle m_{0i}-\frac{GN_{c}}{\pi^{2}}m_{i}A_{i}+\frac{KN_{c}^{2}}{8\pi^{%
4}}m_{j}A_{j}m_{k}A_{k},\qquad i\neq j\neq k\neq i$$
(6a)
$$\displaystyle A_{i}$$
$$\displaystyle=$$
$$\displaystyle-8\pi^{2}\int\frac{d^{3}p}{(2\pi)^{3}}\frac{1}{E}\left(1-\frac{n_%
{i}+n_{\bar{i}}}{2N_{c}}\right)\quad,$$
(6b)
where the indices $i$, $j$ and $k$ run over all three quark flavors. For
the computation of mesonic properties we take a shortcut by defining
an effective temperature via $m_{q}(\vec{x},t)=m_{q}^{\rm eq}(T_{\rm eff}(\vec{x},t))$, where $m_{q}^{\rm eq}(T)$ is the equilibrium form of the
temperature dependence of $m_{q}$ as shown in Fig. 1. The mesonic
properties can then be obtained using the equilibrium expressions, which
are functions of $T_{\rm eff}$, $m_{q}$ and $m_{s}$. The collision processes
entering Eq. (5) are (i) quark quark scattering $qq\leftrightarrow qq$,
$q\bar{q}\leftrightarrow q\bar{q}$ and $\bar{q}\bar{q}\leftrightarrow\bar{q}\bar{q}$
[7], (ii) hadronization $q\bar{q}\leftrightarrow MM$ [8],
and (iii) meson decay $M\to q\bar{q}$ [9]. The latter process is
only possible in the early phases of the expansion, when the effective
temperature is higher than the Mott temperature.
The initial conditions chosen presently do not contain strange quarks,
whereas light quarks are distributed thermally within a sphere of a given
radius. This kind of initial conditions has the immediate consequence,
that the strange quark mass is even higher than in thermal
equilibrium. The reason for this can be seen by writing Eq. (6)
explicitly for strange quarks. One recognizes that the second term on
the right hand side, which is proportional to $G$, only couples to the
strange quark condensate. Since initially no strange quarks are present,
this term does not receive medium corrections and thus makes the effective
quark mass higher than in thermal equilibrium. Medium corrections do
arise from the third term of Eq. (6), which is proportional
to the product of the up and down quark condensate. This contribution
is, however, not sufficient to produce a large mass drop.
Quantitatively, this can be seen from Fig. 3, which shows
the quark masses and the effective temperatures as a function of $r$
at various times during the initial phase of the expansion. At $t=0$,
the light quark mass in the center of the system is low, according to
the high particle density here. At larger radii, the mass goes up due
to the gaussian form of the particle distribution. Note that the quark
masses are only known at those points where a particle is present, thus
the curves stop at the edge of the fireball. The effective temperature
in the centre amounts to approximately 250 MeV. At this temperature in
equilibrium, one would expect a strange quark mass of 380 MeV according
to Fig. 1. In reality, however, one has a strange quark mass
of approximately 450 MeV in the centre due to nonequilibrium effects. At
later times, the system expands and thus the quark mass curve becomes
flatter, while its width grows. Accordingly, the effective temperature
drops. In the final state, the quark masses will tend towards the vacuum
value, which means that the mean field part of the interaction
ceases [11].
Mesons are created during the evolution by collisions of quarks and
antiquarks. Most of all mesons created are pions, which are most
frequently produced by the collision of two light quarks. Kaons,
on the other hand, are produced most easily by the collision of
a light with a strange quark or two strange quarks, while the
hadronization cross section for the creation of a kaon pair from a light
quark antiquark pair is rather low [8]. Eta mesons
are most frequently created collisions of two light quarks, forming
a pion and an eta. The time dependence of the multiplicities is
shown in Fig. 4. It can be seen here, that the production of
mesons starts immediately after the beginning of the expansion.
The production rate is maximal at $t=0$, when the particle density
is high, giving thus rise to a large number of collisions. At later
times the density drops and thus the number of collisions per
time decreases. This in turn leads to a flattening of the multiplicity
curves.
Figure 5 shows the particle density, averaged over all solid angles,
at time $t=10$ fm$/c$. It is clearly visible that the meson density
is maximal at the same places where also the quark density is high.
This confirms the picture gained from Fig. 4, i. e. that
mesons are created within the bulk of the fireball. More insight
can be gained from Fig. 6, which shows the distribution of
the effective temperatures at the creation points of the mesons
for pions, kaons and etas respectively. All three distributions
agree more or less with each other, up to the effects of lower
statistics for the kaon and the eta. It can be seen that mesons are
created most likely at temperatures directly below the Mott temperature,
and within a temperature range of 50 MeV. This plot should be compared
to the mean hadronization times for light and strange quarks in
equilibrium, as have been calculated in [8]. It has
been shown there, that these hadronization times have a minimum
in the temperature range 150 MeV$<T<$200 MeV, which agrees nicely
with Fig. 6.
Figure 7 shows the density along the coordinate axes for
$t=0$ and $t=30$ fm$/c$. It can be seen here, that, although one starts
with a system, which is to a good approximation spherically symmetric,
one ends up with a final state which shows large fluctuations of the
density with respect to the direction. This behaviour contradicts
hydrodynamics, where an initially symmetric system stays symmetric.
To conclude, it hat been demonstrated how a chirally symmetric
quark-meson plasma containing strangeness behaves out of equilibrium
and how mesons are produced. This investigation has been performed
at zero baryochemical potential. Future investigations will remove
this limitation and thus be able to study the mechanism of strangeness
destillation. Also the study of DCC formation is planned.
Acknowledgments. Fruitful discussions with L. Bot and J. Aichelin
are greatfully acknowledged.
References
[1]
E. Laermann, Nucl. Phys. A 610, 1c (1996);
Nucl. Phys. B (Proc. Suppl.) 60A, 180 (1998).
[2]
S. P. Klevansky, Rev. Mod. Phys. 64, 649 (1992);
T. Hatsuda and T. Kunihiro, Phys. Rep. 247, 221 (1994).
[3]
K. Kajantie, M. Laine, J. Peisa, A. Rajantie, K. Rummukainen
and M. Shaposhnikov, Phys. Rev. Lett. 79, 3130 (1997);
Nucl. Phys. (Proc. Suppl.) 63, 418 (1998).
[4]
J. F. Donoghue, E. Golowich and B. R. Holstein, Dynamics
of the Standard Model (Cambridge University Press, 1992).
[5]
J. Hüfner, S. P. Klevansky, P. Zhuang and H. Voss,
Ann. Phys. (NY) 234, 225 (1993);
P. Zhuang, J. Hüfner and S. P. Klevansky, Nucl. Phys. A
576, 525 (1994).
[6]
J. Hüfner, S. P. Klevansky and P. Rehberg, Nucl. Phys. A 606, 260 (1996).
[7]
P. Rehberg, S. P. Klevansky and J. Hüfner, Nucl. Phys. A 608, 356 (1996).
[8]
P. Rehberg, S. P. Klevansky and J. Hüfner, Phys. Rev. C 53, 410 (1996).
[9]
P. Rehberg, Phys. Rev. C 57, 3299 (1998).
[10]
J. Aichelin, Phys. Rep. 202, 233 (1991).
[11]
P. Rehberg and J. Hüfner, Nucl. Phys. A 635, 511 (1998). |
Updating neutrino magnetic moment constraints
B. C. Cañas ${}^{1}$
bcorduz@fis.cinvestav.mx
O. G. Miranda ${}^{1}$
omr@fis.cinvestav.mx
A. Parada ${}^{2}$
alexander.parada00@usc.edu.co
M. Tórtola ${}^{3}$
mariam@ific.uv.es
J. W. F. Valle ${}^{3}$
valle@ific.uv.es, URL:
http://astroparticles.es/
${}^{1}$ Departamento de Física, Centro de
Investigación y de Estudios Avanzados del IPN
Apdo. Postal
14-740 07000 Mexico, DF, Mexico
${}^{2}$ Universidad Santiago de Cali, Campus
Pampalinda, Calle 5 No. 6200, 760001, Santiago de Cali, Colombia
${}^{3}$ AHEP Group, Institut de Física Corpuscular –
C.S.I.C./Universitat de València, Parc Cientific de Paterna.
C/Catedratico José Beltrán, 2 E-46980 Paterna (València) - SPAIN
Abstract
In this paper we provide an updated analysis of the neutrino
magnetic moments (NMMs), discussing both the constraints on the
magnitudes of the three transition moments $\Lambda_{i}$ as well as
the role of the CP violating phases present both in the mixing
matrix and in the NMM matrix.
The scattering of solar neutrinos off electrons in Borexino provides
the most stringent restrictions, due to its robust statistics and
the low energies observed, below 1 MeV. Our new limit on the
effective neutrino magnetic moment which follows from the most
recent Borexino data is $3.1\times 10^{-11}\mu_{B}$ at 90% C.L.
This corresponds to the individual transition magnetic moment
constraints: $|\Lambda_{1}|\leq 5.6\times 10^{-11}\mu_{B}$,
$|\Lambda_{2}|\leq 4.0\times 10^{-11}\mu_{B}$, and $|\Lambda_{3}|\leq 3.1\times 10^{-11}\mu_{B}$ (90% C.L.), irrespective of any complex
phase.
Indeed, the incoherent admixture of neutrino mass eigenstates
present in the solar flux makes Borexino insensitive to the Majorana
phases present in the NMM matrix.
For this reason we also provide a global analysis including the
case of reactor and accelerator neutrino sources, and presenting the
resulting constraints for different values of the relevant CP
phases.
Improved reactor and accelerator neutrino experiments will be needed
in order to underpin the full profile of the neutrino
electromagnetic properties.
pacs: 13.15.+g,14.60.St,12.60.-i,13.40.Em
††preprint: IFIC/15-XX
I Introduction
Neutrino physics has now reached the precision age characterizing a
mature science.
Underpinning the origin of neutrino mass remains an open challenge,
whose investigation could help us find our way towards the ultimate
theory of everything Valle and Romao (2015).
Indeed, the search for new phenomenological signatures associated to
massive neutrinos may yield valuable clues towards the structure of
the electroweak theory beyond the Standard Model (SM).
Although the field is very active, most of the experimental efforts
are devoted to explore the neutrino mass pattern through the study of
oscillations Forero et al. (2014); Maltoni et al. (2004). However it is also
of great importance to investigate the implications of dimension-6
non-standard
interactions Miranda et al. (2006); Barranco et al. (2006, 2008) as
well as electromagnetic properties of the
neutrinos Cisneros (1971); Schechter and Valle (1981); Pal and Wolfenstein (1982); Kayser (1982); Nieves (1982); Akhmedov (1988); Lim and Marciano (1988). Here
we focus on the latter, which has also been a lively subject of
phenomenological research in the last few
years Beacom and Vogel (1999); Grimus et al. (2003); Barranco et al. (2002); Giunti and Studenikin (2014); Giunti et al. (2015); Barranco et al. (2012).
Indeed, different experiments have set constraints coming mainly from
reactor neutrino studies Li et al. (2002); Deniz et al. (2010) as well as from
solar neutrino data Beacom and Vogel (1999); Grimus et al. (2003).
Future tests from experiments measuring coherent neutrino-nucleus
scattering are expected to improve the current bounds on neutrino
electromagnetic
properties Wong et al. (2006); Wong (2010); Bolozdynya et al. (2012); Kosmas et al. (2015a, b).
Most of the constraints reported by the experiments refer to the case
of a Dirac neutrino magnetic moment, despite the fact that Majorana
neutrinos are better motivated from the theoretical point of
view Schechter and Valle (1980). However the Majorana case has been
considered in Refs. Barranco et al. (2002); Grimus et al. (2003) where a more
complete analysis was performed.
In this article we perform a combined analysis of reactor, accelerator
and solar neutrino data, in order to obtain constraints on the
Majorana neutrino transition magnetic moments. We include the most
recent results from the TEXONO reactor experiment Deniz et al. (2010),
as well as the recent results from the Borexino experiment Bellini et al. (2011).
Data from the reactor experiments Krasnoyarsk Vidyakin et al. (1992),
Rovno Derbin et al. (1993) and MUNU Daraktchieva et al. (2005) as well
as the accelerator experiments LAMPF Allen et al. (1993) and
LSND Auerbach et al. (2001) are also included.
Moreover, in our analysis we take into account the updated values of
the neutrino mixing parameters as determined in global oscillation
fits Forero et al. (2014), including the value of $\theta_{13}$
implied by Daya-Bay An et al. (2012, 2014) and RENO reactor
data Ahn et al. (2012), as well as accelerator
data Abe et al. (2013). Moreover, we pay attention to the role of
the, yet unknown, leptonic CP violating phases.
II The neutrino magnetic moment
In this section we will establish the notation used in the description
of neutrino magnetic moments. This will be very important in order to
understand the constraints and the differences between Dirac and
Majorana cases.
For the general Majorana case we have the effective
Hamiltonian Schechter and Valle (1981)
$$\textit{H}^{M}_{em}=-\frac{1}{4}\nu^{T}_{L}C^{-1}~{}\lambda~{}\sigma^{\alpha%
\beta}\nu_{L}F_{\alpha\beta}+h.c.,$$
(1)
where $\lambda=\mu-id$ is an antisymmetric complex matrix
$\lambda_{\alpha\beta}=-\lambda_{\beta\alpha}$, so that $\mu^{T}=-\mu$
and $d^{T}=-d$ are imaginary. Hence, three complex or six real
parameters are needed to describe the Majorana neutrino case.
On the other hand, for the particular case 111A Dirac neutrino
is equivalent to two Majorana neutrinos of same mass and opposite
CP Schechter and Valle (1980). Indeed, in two-component form, the three
Dirac neutrinos are described by a 6$\times 6$ transition moment
matrix. of Dirac neutrino magnetic moments, the corresponding
Hamiltonian is given by Grimus and Schwetz (2000)
$$\textit{H}^{D}_{em}=\frac{1}{2}\bar{\nu}_{R}~{}\lambda~{}\sigma^{\alpha\beta}%
\nu_{L}F_{\alpha\beta}+h.c.,$$
(2)
with $\lambda=\mu-id$ being an arbitrary complex matrix.
Hermiticity now implies that $\mu$ and $d$ obey $\mu=\mu^{{\dagger}}$ and
$d=d^{{\dagger}}$.
We should stress that experimental measurements usually constrain some
process-dependent effective parameter combination. Even in the case of
laboratory neutrino experiments, where the initial neutrino flux is
fixed to have a well determined given flavor, there is no sensitivity
to the final neutrino state and therefore several possibilities must
be envisaged.
For the case of solar neutrino experiments, one needs to take into
account that the original electron neutrino flux experiences
oscillations on its way to the Earth.
Therefore, most of the neutrino magnetic moment constraints discussed
in the literature correspond to restrictions upon some
process-dependent effective parameter. The latter is expressed in
terms of the fundamental parameters describing the transition magnetic
moments and their phases, as well as the neutrino mixing parameters.
From now on we are concerned with the case of three “genuine” active
Majorana neutrinos. As already mentioned, the Dirac case, with three
active plus three sterile neutrinos, would be a particular case of the
six-dimensional Majorana neutrino picture, in which the standard Dirac
magnetic moment is viewed as a transition moment connecting an
“active” with a “sterile” neutrino.
Before we express our results in terms of a general phenomenological
notation, we can illustrate the general features of the neutrino
magnetic moment for the simplest model, namely we consider the case of
Majorana neutrino masses in the standard $\mathrm{SU(2)_{L}\otimes U(1)_{Y}}$ gauge
theory Pal and Wolfenstein (1982), in which case the charged current
contribution gives
$$\mu_{ij}=\frac{3eG_{F}}{16\pi^{2}\sqrt{2}}({m_{\nu}}_{i}+{m_{\nu}}_{j})\sum^{%
\tau}_{\alpha=e}i\,{\mathcal{I}m}\left[U^{*}_{\alpha i}U_{\alpha j}\left(\frac%
{{m_{l}}_{\alpha}}{M_{\rm W}}\right)^{2}\right].$$
(3)
Notice that, in this example, if the masses of the charged leptons
were degenerate, then the off-diagonal transition magnetic moments
would be zero, due to the assumed unitarity of the $U$
matrix. However, in reality, this is not the case and the transition
magnetic moments are nonzero. Moreover, the phases in $\mu_{ij}$ will
be the same as present in the lepton mixing matrix $U$ and, therefore,
could in principle be reconstructed. However, due to the
proportionality with the neutrino mass, the magnetic moments expected
just from the $\mathrm{SU(2)_{L}\otimes U(1)_{Y}}$ gauge sector are too small to be phenomenologically
relevant.
Although enhanced Majorana transition moments are possible in extended
theories, this discussion is beyond the scope of this paper. However,
we quote, as an illustrative example, the case of an extended model
with a charged scalar singlet $\eta^{+}$ suggested in
Ref. Fukugita and Yanagida (1987). In this case the neutrino transition
magnetic moment would be dominated by a charged Higgs boson
contribution, and has been estimated as
$$\mu_{ij}=e\sum_{k}\frac{f_{ki}g^{\dagger}_{kj}+g_{ik}f^{\dagger}_{kj}}{32\pi^{%
2}}\frac{{m_{l}}_{k}}{{m^{2}}_{\eta}}\left(\ln\frac{m^{2}_{\eta}}{{m_{l}}^{2}_%
{k}}-1\right).$$
(4)
Indeed, in principle this scalar contribution could be higher than the
one discussed in Eq. (3). Note that in the case of
Higgs-dominated NMM one could, in principle, introduce new CP phases
in addition to those characterizing the lepton mixing matrix.
The above discussion could be translated into a more phenomenological
approach in which the Dirac NMM is described by an arbitrary complex
matrix $\lambda=\mu+id$ ($\tilde{\lambda}$) in the flavor (or
mass) basis, while for the Majorana case the matrix $\lambda$ takes
the form
$$\lambda=\left(\begin{array}[]{ccc}0&\Lambda_{\tau}&-\Lambda_{\mu}\\
-\Lambda_{\tau}&0&\Lambda_{e}\\
\Lambda_{\mu}&-\Lambda_{e}&0\end{array}\right),\qquad\tilde{\lambda}=\left(%
\begin{array}[]{ccc}0&\Lambda_{3}&-\Lambda_{2}\\
-\Lambda_{3}&0&\Lambda_{1}\\
\Lambda_{2}&-\Lambda_{1}&0\end{array}\right),$$
(5)
where we have used the notation $\lambda_{\alpha\beta}=\varepsilon_{\alpha\beta\gamma}\Lambda_{\gamma}$, where we assume the
transition magnetic moments $\Lambda_{\alpha}$ and $\Lambda_{i}$ to be
complex parameters:
$\Lambda_{\alpha}=|\Lambda_{\alpha}|e^{{i\mskip 1.0mu }\zeta_{\alpha}}$,
$\Lambda_{i}=|\Lambda_{i}|e^{{i\mskip 1.0mu }\zeta_{i}}$. We now turn to the issue
of extracting information on these parameters from experiment.
II.1 The effective neutrino magnetic moment
For the particular case of neutrino scattering off electrons, the
differential cross section for the magnetic moment contribution will
be given by
$$\left(\frac{d\sigma}{dT}\right)_{{em}}=\frac{\pi\alpha^{2}}{m^{2}_{e}\mu^{2}_{%
B}}\left(\frac{1}{T}-\frac{1}{E_{\nu}}\right){\mu_{\nu}}^{2},$$
(6)
where $\mu_{\nu}$ is an effective magnetic moment accounting for the
NMM contribution to the scattering process.
The effective magnetic moment $\mu_{\nu}$ is defined in terms of the
components of the NMM matrix in Eq. (5). In the
flavor basis this can be written as
Grimus and Schwetz (2000); Grimus et al. (2003)
$$({\mu_{\nu}^{F}})^{2}=a^{{\dagger}}_{-}\lambda^{{\dagger}}\lambda a_{-}+a^{{%
\dagger}}_{+}\lambda\lambda^{{\dagger}}a_{+}$$
(7)
where $a_{-}$ and $a_{+}$ denote the negative and positive helicity
neutrino amplitudes, respectively. One finds
$$\displaystyle({\mu_{\nu}^{F}})^{2}$$
$$\displaystyle=$$
$$\displaystyle|a^{1}_{-}\Lambda_{\mu}-a^{2}_{-}\Lambda_{e}|^{2}+|a^{1}_{-}%
\Lambda_{\tau}-a^{3}_{-}\Lambda_{e}|^{2}+|a^{2}_{-}\Lambda_{\tau}-a^{3}_{-}%
\Lambda_{\mu}|^{2}+$$
(8)
$$\displaystyle|a^{1}_{+}\Lambda_{\mu}-a^{2}_{+}\Lambda_{e}|^{2}+|a^{1}_{+}%
\Lambda_{\tau}-a^{3}_{+}\Lambda_{e}|^{2}+|a^{2}_{+}\Lambda_{\tau}-a^{3}_{+}%
\Lambda_{\mu}|^{2}.$$
In order to write the expression for the effective neutrino magnetic
moment in the mass basis we will need the transformations
$$\tilde{a}_{-}=U^{{\dagger}}a_{-},\qquad\tilde{a}_{+}=U^{T}a_{+},\qquad\tilde{%
\lambda}=U^{T}\lambda U.$$
(9)
leading to the expression
$$({\mu_{\nu}^{M}})^{2}=\tilde{a}^{{\dagger}}_{-}\tilde{\lambda}^{{\dagger}}%
\tilde{\lambda}\tilde{a}_{-}+\tilde{a}^{{\dagger}}_{+}\tilde{\lambda}\tilde{%
\lambda}^{{\dagger}}\tilde{a}_{+}.$$
(10)
so that
$$\displaystyle({\mu_{\nu}^{M}})^{2}$$
$$\displaystyle=$$
$$\displaystyle|\tilde{a}^{1}_{-}\Lambda_{2}-\tilde{a}^{2}_{-}\Lambda_{1}|^{2}+|%
\tilde{a}^{1}_{-}\Lambda_{3}-\tilde{a}^{3}_{-}\Lambda_{1}|^{2}+|\tilde{a}^{2}_%
{-}\Lambda_{3}-\tilde{a}^{3}_{-}\Lambda_{2}|^{2}+$$
(11)
$$\displaystyle|\tilde{a}^{1}_{+}\Lambda_{2}-\tilde{a}^{2}_{+}\Lambda_{1}|^{2}+|%
\tilde{a}^{1}_{+}\Lambda_{3}-\tilde{a}^{3}_{+}\Lambda_{1}|^{2}+|\tilde{a}^{2}_%
{+}\Lambda_{3}-\tilde{a}^{3}_{+}\Lambda_{2}|^{2},$$
where $\tilde{a}^{i}_{\pm}$ denotes the $i$-th component of the
$\tilde{a}_{\pm}$ vector.
Before starting the calculations of the effective Majorana magnetic
moment parameter combination corresponding to the different
experimental setups we would like to comment on the counting of
relevant complex phases.
First we write the three complex phases in the transition
magnetic moment matrix as $\zeta_{1}$, $\zeta_{2}$ and $\zeta_{3}$.
From the leptonic mixing matrix we have another 3 CP-violating phases:
the Dirac phase characterizing neutrino oscillations, $\delta$, and
the two Majorana phases involved in lepton number violating
processes Schechter and Valle (1980).
As noticed in Ref. Grimus and Schwetz (2000), three of these six complex
phases are irrelevant, as they can be reabsorbed in different ways. In
what follows we give our results in terms of the Dirac CP phase
$\delta$ and the relative difference between the transition magnetic
moment phases, $\xi_{1}=\zeta_{3}-\zeta_{2}$, $\xi_{2}=\zeta_{3}-\zeta_{1}$, $\xi_{3}=\zeta_{2}-\zeta_{1}$, of which only two are
independent.
II.1.1 Effective neutrino magnetic moment at reactor
experiments.
We now consider the effective neutrino magnetic moment parameter
relevant for the case of reactor neutrinos. In this case we have an
initial electron antineutrino flux, so that the only non–zero entry
in the flavor basis will be $a^{1}_{{+}}=1$. Therefore, from
Eq. (8) we get the following expression for the
effective Majorana transition magnetic moment strength parameter
describing reactor neutrino experiments:
$$(\mu^{F}_{R})^{2}=|\Lambda_{\mu}|^{2}+|\Lambda_{\tau}|^{2}.$$
(12)
which in the mass basis leads to the expression
$$\displaystyle(\mu^{M}_{R})^{2}$$
$$\displaystyle=$$
$$\displaystyle{|{\bf\Lambda}|^{2}}-s^{2}_{12}c^{2}_{13}|\Lambda_{2}|^{2}-c^{2}_%
{12}c^{2}_{13}|\Lambda_{1}|^{2}-s^{2}_{13}|\Lambda_{3}|^{2}$$
$$\displaystyle-$$
$$\displaystyle 2s_{12}c_{12}c^{2}_{13}|\Lambda_{1}||\Lambda_{2}|\cos\delta_{12}%
-2c_{12}c_{13}s_{13}|\Lambda_{1}||\Lambda_{3}|\cos\delta_{13}$$
$$\displaystyle-$$
$$\displaystyle 2s_{12}c_{13}s_{13}|\Lambda_{2}||\Lambda_{3}|\cos\delta_{23}$$
where $c_{ij}=\cos\theta_{ij}$, $s_{ij}=\sin\theta_{ij}$ and
$\delta_{12}=\xi_{3}$, $\delta_{23}=\xi_{2}-\delta$, and $\delta_{13}=\delta_{12}-\delta_{23}$. As already noted, $\delta$ is the Dirac
phase of the leptonic mixing matrix and $\xi_{3}=\zeta_{2}-\zeta_{1}$, $\xi_{2}=\zeta_{3}-\zeta_{1}$, are the
relative phases introduced by the presence of the magnetic moment.
This expression takes into account that $\theta_{13}$ is different
from zero, and hence generalizes the previous result given
in Grimus et al. (2003).
It in important to notice that the effective magnetic moment in
Eq. (II.1.1) implies a degeneracy between the leptonic phase
$\delta$ and those present in the neutrino transition magnetic
moments, $\xi_{2}$ and $\xi_{3}$. As a result, it will not be possible to
disentangle these phases without further
independent experimental information.
In order to illustrate the dependence on the different relative phases
$\delta_{ij}$ we show in Fig. 1 the value of the
effective Majorana transition magnetic moment for three particular
cases, in which the magnitude of one transition magnetic moment
$|\Lambda_{i}|$ is assumed to vanish. This implies that the magnetic
moment would depend only on one effective phase $\delta_{ij}$.
Comparing the three curves in Fig. 1, one sees a
strong dependence on the phase $\delta_{12}$ (see solid black line)
while, due to the smallness of $\sin\theta_{13}$, the value of the
phases $\delta_{13}$ and $\delta_{23}$ has little impact on the
magnitude of the effective magnetic moment $\mu_{R}^{M}$.
II.1.2 Effective neutrino magnetic moment at accelerator
experiments.
Another relevant measurement for neutrino magnetic moment comes from
accelerator–produced neutrinos arising from pion
decays Allen et al. (1993); Auerbach et al. (2001). In this case, pion decay
produces a muon neutrino, while the subsequent muon decay generates an
electron neutrino plus a muon antineutrino. We can write the
effective magnetic moment strength parameter in the flavor basis,
considering for the moment the same proportion of $\nu_{e}$, $\nu_{\mu}$
and $\bar{\nu}_{\mu}$ ($a^{1}_{-}=1$, $a^{2}_{-}=1$, $a^{2}_{+}=1$):
$$(\mu^{F}_{A})^{2}=|\mathbf{\Lambda}|^{2}+|\Lambda_{e}|^{2}+2\,|\Lambda_{\tau}|%
^{2}-2\,|\Lambda_{\mu}||\Lambda_{e}|\cos\eta,$$
(14)
where $|\mathbf{\Lambda}|^{2}=|\Lambda_{e}|^{2}+|\Lambda_{\mu}|^{2}+|\Lambda_{\tau}|^%
{2}$ and $\eta=\zeta_{e}-\zeta_{\mu}$ is the relative phase between the transition
magnetic moments $\Lambda_{e}$ and $\Lambda_{\mu}$.
The corresponding expression for the effective neutrino magnetic
moment strength parameter in the mass basis, for $\theta_{13}=0$ will
be given by
$$\displaystyle(\mu^{M}_{A})^{2}$$
$$\displaystyle=$$
$$\displaystyle|\Lambda_{1}|^{2}[2-(c^{2}_{23}-s^{2}_{23})s^{2}_{12}+2s_{12}c_{1%
2}c_{23}]$$
$$\displaystyle+$$
$$\displaystyle|\Lambda_{2}|^{2}[2-(c^{2}_{23}-s^{2}_{23})c^{2}_{12}-2s_{12}c_{1%
2}c_{23}]+|\Lambda_{3}|^{2}[1+2c^{2}_{23}]$$
$$\displaystyle+$$
$$\displaystyle 2|\Lambda_{1}||\Lambda_{2}|\cos\xi_{3}[s_{12}c_{12}(c^{2}_{23}-s%
^{2}_{23})-(c^{2}_{12}-s^{2}_{12})c_{23}]$$
$$\displaystyle+$$
$$\displaystyle 2|\Lambda_{1}||\Lambda_{3}|\cos\xi_{2}[-c_{12}s_{23}+2s_{12}s_{2%
3}c_{23}]$$
$$\displaystyle+$$
$$\displaystyle 2|\Lambda_{2}||\Lambda_{3}|\cos(\xi_{3}-\xi_{2})[-s_{12}s_{23}-2%
c_{12}s_{23}c_{23}]$$
As expected, the Dirac CP phase $\delta$ present in oscillations does
not enter in this expression, and therefore only the two Majorana
phases from the NMM matrix $\xi_{2}$ and $\xi_{3}$ are present.
Note however that in our numerical analysis we have used the full
expression with $\theta_{13}\not=0$, as
$$\displaystyle(\mu^{M}_{A})^{2}$$
$$\displaystyle=$$
$$\displaystyle|\Lambda_{1}|^{2}\left[\sin 2\theta_{12}c_{13}c_{23}+c^{2}_{12}(2%
c^{2}_{23}+\sin 2\theta_{13}s_{23}\cos\delta)\right.$$
$$\displaystyle+$$
$$\displaystyle\left.c^{2}_{13}(s^{2}_{12}+2s^{2}_{23})+s_{13}(s_{13}+2s^{2}_{12%
}s_{13}s^{2}_{23}-\sin 2\theta_{12}\sin 2\theta_{23}\cos\delta)\right]$$
$$\displaystyle+$$
$$\displaystyle\frac{1}{4}|\Lambda_{2}|^{2}\left[8-\cos 2\theta_{23}(1+3\cos 2%
\theta_{12}+2\cos 2\theta_{13}s^{2}_{12})+4s^{2}_{12}\sin 2\theta_{13}s_{23}%
\cos\delta\right.$$
$$\displaystyle+$$
$$\displaystyle\left.4\sin 2\theta_{12}(-c_{13}c_{23}+s_{13}\sin 2\theta_{23}%
\cos\delta)\right]+|\Lambda_{3}|^{2}\left(2+c^{2}_{13}\cos 2\theta_{23}-\sin 2%
\theta_{13}s_{23}\cos\delta\right)$$
$$\displaystyle+$$
$$\displaystyle 2|\Lambda_{1}||\Lambda_{2}|\left\{\cos\xi_{3}\left[-c^{2}_{12}c_%
{13}c_{23}+c_{23}(s^{2}_{12}c_{13}+\sin 2\theta_{12}c_{23})\right.\right.$$
$$\displaystyle+$$
$$\displaystyle\left.s_{12}c_{12}(-1+\cos 2\theta_{23}s^{2}_{13}+\sin 2\theta_{1%
3}s_{23}\cos\delta)\right]$$
$$\displaystyle+$$
$$\displaystyle\left.s_{13}\sin 2\theta_{23}(\cos 2\theta_{12}\cos\delta\cos\xi_%
{3}+\sin\delta\sin\xi_{3})\right\}$$
$$\displaystyle+$$
$$\displaystyle|\Lambda_{1}||\Lambda_{3}|\left\{2\cos(\xi_{2}-\delta)(-c_{12}c_{%
13}\cos 2\theta_{23}+s_{12}c_{23})s_{13}\right.$$
$$\displaystyle+$$
$$\displaystyle 2\left.\left[c_{13}\cos\xi_{2}(-c_{12}c_{13}+2s_{12}c_{23})+c_{1%
2}s^{2}_{13}\cos(\xi_{2}-2\delta)\right]s_{23}\right\}$$
$$\displaystyle-$$
$$\displaystyle 2|\Lambda_{2}||\Lambda_{3}|\left\{\frac{1}{2}s_{12}\cos(\xi_{1}-%
\delta)(\cos 2\theta_{23}\sin 2\theta_{13}+2\cos 2\theta_{13}s_{23}\cos\delta)\right.$$
$$\displaystyle+$$
$$\displaystyle\left.c_{12}\left[c_{23}s_{13}\cos(\xi_{1}-\delta)+c_{13}\sin 2%
\theta_{23}\cos\xi_{1}\right]+s_{12}s_{23}\sin\delta\sin(\delta-\xi_{1})%
\vphantom{\frac{1}{2}}\right\}$$
Notice that we have used here the phase $\xi_{1}=\xi_{2}-\xi_{3}$. Although this is not an independent phase, it is hepful
to simplify the previous formula. Therefore, the final expression is
given in terms of the three independent phases $\delta$, $\xi_{2}$ and
$\xi_{3}$. One can check that in the limit $\theta_{13}$ = 0, the
expression in Eq. (II.1.2) is recovered.
II.1.3 Effective neutrino magnetic moment in Borexino.
Here we calculate the effective magnetic moment strength parameter
relevant for experiments measuring solar neutrinos through their
scattering with electrons, like Borexino 222The same result
will apply for the Super-Kamiokande experiment, not included here
due to its smaller sensitivity to the neutrino magnetic
moment Grimus et al. (2003)..
In this case, the electron neutrinos originally produced in the solar
interior undergo flavor oscillation and they arrive to the Earth
detector as an incoherent sum of mass eigenstates. Using the
well-justified approximation where Grimus et al. (2003)
$$P^{3\nu}_{e3}=\sin^{2}\theta_{13},\quad P^{3\nu}_{e1}=\cos^{2}\theta_{13}P^{2%
\nu}_{e1},\quad P^{3\nu}_{e2}=\cos^{2}\theta_{13}P^{2\nu}_{e2},\quad$$
(17)
with $P^{2\nu}_{ej}$ ($j=1,2$) being the effective two-neutrino
oscillation probabilities for solar neutrinos, we arrive to the
effective neutrino magnetic moment strength parameter in the mass
basis,
$$(\mu^{M}_{\rm{sol}})^{2}=|\mathbf{\Lambda}|^{2}-c^{2}_{13}|\Lambda_{2}|^{2}+(c%
^{2}_{13}-1)|\Lambda_{3}|^{2}+c^{2}_{13}P^{2\nu}_{e1}(|\Lambda_{2}|^{2}-|%
\Lambda_{1}|^{2}).$$
(18)
where the unitarity condition, $P^{2\nu}_{e1}+P^{2\nu}_{e2}=1$, has
also been assumed. The calculation of this expression in the flavor
basis is more complicated due to presence of the neutrino transition
probabilities and therefore we do not include it here.
As we can see from Eq. (18), the expression of the
effective magnetic moment for solar neutrinos is independent of any
phase, as has already been noticed Grimus et al. (2003). Here we take
into account the non-zero value of $\theta_{13}$ for the first time in
this kind of analysis. Taking advantage of the previous equation we
obtain constraints on the individual neutrino transition magnetic
moments.
After obtaining the neutrino magnetic moment expressions for the case
of $\theta_{13}\neq 0$, we now turn our attention to the relevant
experiments for our analysis.
III Neutrino data analysis
Having evaluated the effective neutrino magnetic moment strength
parameter for reactor, accelerator and solar neutrino experiments, we
are ready to perform a combined analysis of the experimental data in
order to get constraints on the three different transition magnetic
moments $\Lambda_{i}$. In order to perform this analysis we make some
assumptions on the phases $\delta$, $\xi_{2}$ and $\xi_{3}$. In the next
section we will describe the data used in the fit and show the
results. We now briefly describe the statistical analysis performed
in this article.
III.1 Reactor antineutrinos
We start by describing the reactor antineutrino experiments. They use
the antineutrino flux coming from a nuclear reactor, in combination
with a detector sensitive to the electron antineutrino scattering off
electrons. The total number of events (in the $i$-th bin) in these
experiments is given by
$$N_{R}^{i}=\kappa\int dE_{\nu}\int dT\int^{T_{i+1}}_{T_{i}}dT^{\prime}\lambda(E%
_{\nu})\frac{d\sigma}{dT}(E_{\nu},T,{\mu})R(T,T^{\prime}),$$
(19)
where the integrals run over the detected electron recoil energy $T^{\prime}$,
the real recoil energy $T$, and the neutrino energy $E_{\nu}$.
$T_{i}$ and $T_{i+1}$ are the minimum and maximum energy of the $i$-th
bin, respectively.
The parameter $\kappa$ stands for the product of the total number of
targets times the total antineutrino flux times the total exposure
time of the experimental run and $\lambda(E_{\nu})$ is the antineutrino
energy spectrum coming from the nuclear
reactor Mueller et al. (2011); Kopeikin et al. (1997). Some of the experiments
under consideration reported their resolution function $R(T,T^{\prime})$,
given by
$$R(T,T^{\prime})=\frac{1}{\sqrt{4\pi}\sigma}\exp{\left(\frac{-(T-T^{\prime})^{2%
}}{2\sigma^{2}}\right)}.$$
(20)
where $\sigma$ stands for the error in the kinetic energy
determination. When the information on this resolution function is
not available, we have assumed perfect energy resolution and taken it
as a delta function: $R(T,T^{\prime})=\delta(T-T^{\prime})$.
Finally, the standard differential cross section for the process of
$\bar{\nu}_{e}$-electron scattering is given by
$$\frac{d\sigma}{dT}=\frac{2G^{2}_{F}m_{e}}{\pi}\left[g^{2}_{R}+g^{2}_{L}(1-%
\frac{T}{E_{\nu}})^{2}-g_{L}g_{R}m_{e}\frac{T}{E_{\nu}^{2}}\right],$$
(21)
where $m_{e}$ is the electron mass and $G_{F}$ is the Fermi constant. For
this process, at tree level, the coupling constants $g_{L,R}$ are
given by $g_{L}=1/2+\sin^{2}\theta_{\rm W}$ and $g_{R}=\sin^{2}\theta_{\rm W}$.
The assumed non-zero neutrino magnetic moment yields a new
contribution to the cross section, given by
$$\left(\frac{d\sigma}{dT}\right)_{{em}}=\frac{\pi\alpha^{2}}{m^{2}_{e}}\left(%
\frac{1}{T}-\frac{1}{E_{\nu}}\right){\mu_{R}}^{2}\,,$$
(22)
where $\mu_{R}=\mu^{F,M}_{R}$ is the reactor effective neutrino
magnetic moment, either in the mass or flavor basis, as already
discussed in Eqs. (12) and (II.1.1).
This gives rise to an additional neutrino signal at reactor
experiments.
Finally, we perform our statistical analysis using the following
$\chi^{2}$ function:
$$\chi^{2}=\sum^{N_{bin}}_{i=1}\left(\frac{O^{i}_{R}-N_{R}^{i}(\mu_{R})}{\Delta_%
{i}}\right)^{2},$$
(23)
where $O_{R}^{i}$ and $N_{R}^{i}$ are the observed number of events and the
predicted number of events in the presence of an effective magnetic
moment $\mu_{R}$ at the $i$-th bin, respectively. Here $\Delta_{i}$ is the
statistical error at each bin.
In our analysis, we have used the experimental results
reported by Krasnoyarsk Vidyakin et al. (1992),
Rovno Derbin et al. (1993), MUNU Daraktchieva et al. (2005), and
TEXONO Deniz et al. (2010) reactor experiments.
As a first step we have calibrated our numerical analysis by
reproducing the constraints on the effective neutrino magnetic moment
reported by each experiment. To do this we performed an analysis as
similar as possible to the original references, using the antineutrino
spectrum description available at the time of the corresponding
experiment as well as the antineutrino electron cross section.
Afterwards, we have recalculated our limits on the NMM by introducing
the new antineutrino reactor spectrum.
Our results on reactor neutrino experiments are summarized in
the upper part of Table 1.
Although it is not listed in Table 1, we have also
analyzed the case of the GEMMA Beda et al. (2012) experiment. In this
case there is no detection of the SM signal and therefore, the
statistical analysis is a bit different from what we have described
above. It is important to notice that this experiment gives a
stronger constraint compared with other reactor experiments
($\mu_{\bar{\nu}_{e}}\leq 2.9\times 10^{-11}\mu_{B}$). However, the
different statistical treatment employed to analyze GEMMA’s data makes
it difficult to establish a direct comparison with the remaining
reactor results.
III.2 Accelerator data
For the case of accelerator neutrinos we have considered the
experimental data reported by the LAMPF Allen et al. (1993) and
LSND Auerbach et al. (2001) collaborations. The expected number of
events for electron and muon neutrinos is calculated as
$$N_{A}=\int dE_{\nu}\int^{T_{f}}_{T_{i}}dT^{\prime}\lambda(E_{\nu})\frac{d%
\sigma}{dT}(E_{\nu},T^{\prime},\mu),$$
(24)
where $A$ refers to the type of event ($\nu_{e}$, $\nu_{\mu}$ or
$\bar{\nu}_{\mu}$), $E_{\nu}$ corresponds to the neutrino energy, $T^{\prime}$
is the electron recoil energy, and $\lambda(E_{\nu})$ is the neutrino
energy spectrum from the accelerator
experiments Allen et al. (1993); Auerbach et al. (2001).
The statistical analysis is similar to the one for reactor
antineutrinos described in the previous subsection.
As a first step we try to reproduce the individual limits on the
magnetic moment of electron and muon neutrinos reported by the
experimental collaborations. To do this we have used the $\chi^{2}$
function given by Eq. (23), comparing the expected event
number reported by the experiments with the calculated number of
events. The limits on the muon and electron neutrino magnetic moments
are derived taking into account the following relations for the
effective neutrino magnetic moment (see Refs. Allen et al. (1993)
and Auerbach et al. (2001) for details): $\mu^{2}_{\nu_{e}}+\alpha\mu^{2}_{\nu_{\mu}}<\mu^{2}_{eff}$, where $\alpha$ stands for
the rate between muon and electron neutrinos in the detector. This
ratio is expected to be equal to two as first approximation, since
each pion decay produces a muon antineutrino plus a muon neutrino plus
an electron neutrino. The values reported by the experimental
collaborations are $\alpha=2.1$ for LAMPF Allen et al. (1993) and
$\alpha=2.4$ for LSND Auerbach et al. (2001).
The limits on the effective neutrino magnetic moment derived from
LAMPF and LSND data are reported in the lower part of
Table 1. For the more complete analysis including
the complex phases in the neutrino magnetic moment matrix we take
$\alpha=2$, as included in Eqs. (14)-(II.1.2).
III.3 Borexino data
The Borexino experiment has successfully measured a large part of the
neutrino flux spectrum coming from the
Sun Arpesella
et al. (2008a); Bellini et al. (2010, 2014a, 2014b)
and has set limits on the effective neutrino magnetic moment by using
their observations of the Beryllium solar neutrino
line Back et al. (2003); Arpesella
et al. (2008b). In this paper we will
consider the more recent measurements of the Beryllium solar flux
reported in Ref. Bellini et al. (2011) in order to obtain a stronger
constraint.
For reactor and accelerator experiments, our statistical analysis
followed the covariant approach. In the case of the Borexino, however,
we have adopted the pull approach Fogli et al. (2002).
Focusing on the Beryllium neutrino flux, the expected number of events
at the $i$-th bin, $N_{i}^{th}$, will be given by
$$N_{i}^{th}=\kappa\int\frac{d\sigma}{dT_{e}}(E_{\nu},T_{e})R(T_{e},T^{\prime}_{%
e})dT_{e}dT^{\prime}_{e}+N_{i}^{bg},$$
(25)
where $N_{i}^{bg}$ represents the number of expected background events
at the considered energy bin. Here $\kappa$ stands for the product of
the number of target electrons, the detection time window (740.7 days
in this case), and the total Beryllium neutrino flux. $T_{e}$ is the
real electron kinetic energy and $T^{\prime}_{e}$ is the reconstructed one.
The energy resolution function $R(T_{e},T^{\prime}_{e})$ of the experiment is
given by
$$R(T_{e},T^{\prime}_{e})=\frac{1}{\sqrt{2\pi}\sigma^{2}}exp\left(\frac{(T_{e}-T%
^{\prime}_{e})^{2}}{2\sigma^{2}}\right)$$
(26)
with $\sigma/T_{e}=0.06\sqrt{T_{e}/\rm{MeV}}$ Gonzalez-Garcia
et al. (2010).
Finally the differential cross section is given by
$$\frac{d\sigma_{\alpha}}{dT_{e}}(E_{\nu},T_{e})=\overline{P}_{ee}\frac{d\sigma_%
{e}}{dT_{e}}(E_{\nu},T_{e})+(1-\overline{P}_{ee})\frac{d\sigma_{\mu-\tau}}{dT_%
{e}}(E_{\nu},T_{e}),$$
(27)
where the average survival electron-neutrino probability for the Beryllium line,
$\overline{P}_{ee}$, determines the flavour composition of the neutrino flux detected in the experiment.
According to the most recent analysis of solar neutrino data in
Ref. Forero et al. (2014) (excluding Borexino data to avoid any
correlation with the present analysis) this value is set to
$\overline{P_{ee}^{\rm{th}}}=0.54\pm 0.03$.
In order to explore the sensitivity of the Borexino experiment to the
neutrino magnetic moments, we include the new contribution to the
differential cross section in Eq. (27):
$$\left(\frac{d\sigma}{dT}\right)_{{em}}=\frac{\pi\alpha^{2}}{m^{2}_{e}\mu^{2}_{%
B}}\left(\frac{1}{T}-\frac{1}{E_{\nu}}\right){\mu_{\mathrm{sol}}}^{2},$$
(28)
where $\mu_{\mathrm{sol}}$ is the effective magnetic moment strength
parameter relevant for the Borexino solar neutrino experiment derived
in Eq. (18) in the mass basis.
This yields a new contribution to the expected number of events,
which will determine the sensitivity to the presence of a neutrino
magnetic moment.
With the expected event number, we have fitted our predictions to
the experimental data in the statistical analysis. There we have
considered the Borexino systematic errors associated to the fiducial
mass ratio uncertainty ($\pi_{vol}=6\%$), the energy scale
uncertainty ($\pi_{scl}^{b}=1\%$) and the energy resolution
uncertainty ($\pi_{res}=10\%$). We have also included in the fit
the electron-neutrino survival probability $\overline{P}_{ee}$ as a
free parameter (using the value of $\overline{P_{ee}^{\rm{th}}}$
given above as a prior) with the corresponding penalty in the
$\chi^{2}$ function.
The constraint we have obtained for the effective neutrino magnetic
moment using the latest Borexino data is given in
Table 2. For comparison, we have also included
in the table the previous bound, derived by the Borexino
Collaboration in Ref. Arpesella
et al. (2008b). Note that our updated
limit is comparable to the strongest bound reported by the GEMMA
experiment and previously discussed in Sect. III.1.
Using the expression of the effective neutrino magnetic moment in
Borexino given by Eq. (18), we can also obtain limits
on the individual elements of the transition magnetic moment matrix
$\Lambda_{i}$. In this case, the calculations involve the neutrino
oscillation probability $P^{2\nu}_{e1}$, which, as before, is
considered in our $\chi^{2}$ analysis as a free parameter with an
associated penalty term. As a prior, we have considered again the
value of the probability predicted by the analysis of all other
solar neutrino data except Borexino, given by
$P^{2\nu}_{e1}\big{|}_{\rm th}=0.61\pm 0.06$ Forero et al. (2014).
Our results are summarized in the last row of Table 3.
IV Limits on the neutrino magnetic moment
In the previous section we have derived bounds on the effective
neutrino magnetic moment parameter combinations relevant in reactor,
accelerator and solar neutrino experiments. Our results are summarized
in Tables 1 and 2. The most
remarkable result is the limit obtained with the latest Borexino data:
$\mu_{\mathrm{sol}}\leq 3.1\times 10^{-11}\mu_{B}$ , which is
comparable to the constraint reported by the GEMMA Beda et al. (2012)
collaboration using reactor antineutrinos, $\mu_{\mathrm{R}}\leq 2.9\times 10^{-11}\mu_{B}$ 333Both limits correspond to 90% C.L..
One can go one step further and make a combined analysis using all the
data studied so far.
This combined study can not be done in terms of the effective magnetic
moments, since they are different for each type of experiment, but we
need to use a more general formalism, as the one we have discussed in
section II.
We choose to work in the mass basis and hence we consider the NMM
parameters $\Lambda_{1}$, $\Lambda_{2}$ and $\Lambda_{3}$.
As a first step in our analysis, we take all elements as real, setting
the complex phases to zero, and we also take one nonzero transition
magnetic moment element $\Lambda_{i}$ at a time. The results from this
analysis are shown in Table 3, where one sees that
the Borexino constraint is considerably stronger than the others,
except for GEMMA, as we already commented 444Due to the
complexity of the statistical analysis in GEMMA, here we have only
translated their reported bound Beda et al. (2012) into $\Lambda_{i}$
by using Eq. (II.1.1), instead of including GEMMA data
explicitly in the global analysis..
We have also considered a more complete analysis taking into account
the role of the phases in the reactor and accelerator data. Notice
that the effective magnetic moment for the Borexino experiment is
independent of all the complex phases (see Eq. (18))
since solar neutrinos arrive to the Earth as an incoherent sum of mass
eigenstates and therefore, no interference terms appear in the
calculation.
For the case of reactor neutrinos, we have performed a statistical
analysis of TEXONO data Deniz et al. (2010) for different choices of
the complex phases of $\Lambda_{i}$, $\zeta_{i}$, and taking all
transition magnetic moment amplitudes as nonzero. The results of this
analysis are shown in Fig. 2. There we present
the 90% C.L. allowed regions for the transition magnetic moments in
the mass basis in the form of two-dimensional projections in the
planes ($|\Lambda_{i}|$, $|\Lambda_{j}|$). In all cases the regions have
been obtained marginalizing over the undisplayed parameter
$|\Lambda_{k}|$. Concerning the complex phases, in the two cases
considered we have fixed the mixing matrix CP phase $\delta$ to its
currently preferred value Forero et al. (2014): $\delta=3\pi/2$. For
the complex phases in the transition magnetic moments we have
considered two cases. The magenta (outer) region in
Fig. 2 corresponds to the case with all phases
equal to zero: $\xi_{2}=\xi_{3}=0$ while the orange (inner) displayed
region has been obtained for $\xi_{2}=0$ and $\xi_{3}=\pi/2$.
One can see in this plot the role of the CP phases, since the
resulting restrictions on the transition magnetic moments
$|\Lambda_{1}|$ and $|\Lambda_{2}|$ depend on the chosen phase
combinations.
Note, however, that in the two cases analyzed the bound on
$|\Lambda_{3}|$ is practically unchanged, showing that in this
particular case the complex phases are not very relevant. As discussed
in Fig. 1, this is due to the fact that the
terms involving simultaneously $|\Lambda_{3}|$ and any complex phase in
the expression of the effective magnetic moment in
Eq. (II.1.1) are proportional the small quantity
$\sin\theta_{13}$ and therefore they are subdominant with respect to
the real terms in $\mu_{R}^{M}$.
Finally, we have performed a combined analysis of all the reactor and
accelerator data discussed in this paper, for a particular choice of
phases ($\delta=3\pi/2$ and $\xi_{i}=0$) and compared it with the
corresponding $\chi^{2}$ analysis of Borexino data. The results, shown
in Fig. 3, illustrate how Borexino is more
sensitive in constraining the magnitude of the transition neutrino
magnetic moments. Since the Borexino effective magnetic moment depends
only on the square magnitudes of these transition magnetic moments,
its constraints are in practice the same as those in the
one-parameter-at-a-time analysis. In this sense, a detailed analysis
of GEMMA data, not performed here, is not expected to give a result as
robust as the one obtained with Borexino data. However, one should
notice that future reactor and accelerator experiments are the only
ones that could give information on individual transition magnetic
moments as well as on the Majorana phases discussed here, an
information inaccessible at Borexino.
V Conclusions
In this work we have analyzed the current status of the constraints on
neutrino magnetic moments. We have presented a detailed discussion of
the constraints on the absolute value of the transition magnetic
moments, as well as the role of the CP phases, stressing the
complementarity of different experiments.
Thanks to the low energies observed, below 1 MeV, and its robust
statistics, the Borexino solar experiment plays a very important role
in constraining the electromagnetic neutrino properties.
Indeed, it provides stringent constraints on the absolute magnitude of
the the transition magnetic moments, which we obtain as
$$\displaystyle|\Lambda_{1}|$$
$$\displaystyle\leq 5.6\times 10^{-11}\mu_{B}\,,$$
$$\displaystyle|\Lambda_{2}|$$
$$\displaystyle\leq 4.0\times 10^{-11}\mu_{B}\,,$$
(29)
$$\displaystyle|\Lambda_{3}|$$
$$\displaystyle\leq 3.1\times 10^{-11}\mu_{B}\,,$$
However, the incoherent nature of the solar neutrino flux makes
Borexino insensitive to the Majorana phases which characterize the
transition moments matrix.
Although less sensitive to the absolute value of the transition
magnetic moment strengths, reactor and accelerator experiments provide
the only chance to obtain a hint of the complex CP phases. We
illustrate this fact by presenting the constraints resulting from our
global analysis for different values of the relevant CP phases.
Although less stringent than astrophysical limits say, from globular
clusters Raffelt (1990) or searches for anti-neutrinos from the
sun Miranda
et al. (2004a, b), laboratory limits are model
independent and should be further pursued.
Indeed, as we have illustrated, improved reactor and accelerator
neutrino experiments will be crucial towards obtaining the detailed
structure of the neutrino electromagnetic properties.
Acknowledgements
Work supported by MINECO grants FPA2014-58183-P, Multidark CSD2009-
00064 and SEV-2014-0398 (MINECO); by EPLANET, by the CONACyT grant
166639 (Mexico) and the PROMETEOII/2014/084 grant from Generalitat
Valenciana.
M. Tórtola is also supported by a Ramon y Cajal contract of the
Spanish MINECO.
The work of A.Parada was partially supported by Universidad Santiago
de Cali under grant 935-621114-031.
References
Valle and Romao (2015)
J. W. Valle and
J. C. Romao,
Neutrinos in high energy and astroparticle physics
(Wiley-VCH, Berlin,
2015), 1st ed.
Forero et al. (2014)
D. Forero,
M. Tortola, and
J. Valle,
Phys.Rev. D90,
093006 (2014), eprint 1405.7540.
Maltoni et al. (2004)
M. Maltoni,
T. Schwetz,
M. Tortola, and
J. W. F. Valle,
New J.Phys. 6,
122 (2004), this review gives
a comprehensive set of references, eprint hep-ph/0405172.
Miranda et al. (2006)
O. G. Miranda,
M. A. Tortola,
and J. W. F.
Valle, JHEP 10,
008 (2006).
Barranco et al. (2006)
J. Barranco
et al., Phys. Rev.
D73, 113001
(2006), eprint hep-ph/0512195.
Barranco et al. (2008)
J. Barranco
et al., Phys. Rev.
D77, 093014
(2008), eprint 0711.0698.
Cisneros (1971)
A. Cisneros,
Astrophys. Space Sci. 10,
87 (1971).
Schechter and Valle (1981)
J. Schechter and
J. W. F. Valle,
Phys. Rev. D24,
1883 (1981), err. D25, 283
(1982).
Pal and Wolfenstein (1982)
P. B. Pal and
L. Wolfenstein,
Phys. Rev. D25,
766 (1982).
Kayser (1982)
B. Kayser,
Phys.Rev. D26,
1662 (1982).
Nieves (1982)
J. F. Nieves,
Phys. Rev. D26,
3152 (1982).
Akhmedov (1988)
E. K. Akhmedov,
Phys. Lett. B213,
64 (1988).
Lim and Marciano (1988)
C.-S. Lim and
W. J. Marciano,
Phys. Rev. D37,
1368 (1988).
Beacom and Vogel (1999)
J. F. Beacom and
P. Vogel,
Phys. Rev. Lett. 83,
5222 (1999), eprint hep-ph/9907383.
Grimus et al. (2003)
W. Grimus,
M. Maltoni,
T. Schwetz,
M. A. Tortola,
and J. W. F.
Valle, Nucl. Phys.
B648, 376 (2003),
eprint hep-ph/0208132.
Barranco et al. (2002)
J. Barranco
et al., Phys. Rev.
D66, 093009
(2002), eprint hep-ph/0207326 v3 KamLAND-updated version.
Giunti and Studenikin (2014)
C. Giunti and
A. Studenikin
(2014), eprint 1403.6344.
Giunti et al. (2015)
C. Giunti,
K. A. Kouzakov,
Y.-F. Li,
A. V. Lokhov,
A. I. Studenikin,
et al. (2015), eprint 1506.05387.
Barranco et al. (2012)
J. Barranco,
O. Miranda,
C. Moura, and
A. Parada,
Phys.Lett. B718,
26 (2012), eprint 1205.4285.
Li et al. (2002)
H. B. Li et al.
(TEXONO collaboration) (2002),
eprint hep-ex/0212003.
Deniz et al. (2010)
M. Deniz et al.
(TEXONO collaboration), Phys.
Rev. D81, 072001
(2010), eprint 0911.1597.
Wong et al. (2006)
H. T. Wong,
H.-B. Li,
J. Li,
Q. Yue, and
Z.-Y. Zhou,
J. Phys. Conf. Ser. 39,
266 (2006), [,344(2005)],
eprint hep-ex/0511001.
Wong (2010)
H. T. Wong,
Nucl. Phys. A844,
229C (2010).
Bolozdynya et al. (2012)
A. Bolozdynya
et al. (2012), eprint 1211.5199,
URL http://inspirehep.net/record/1203677/files/arXiv:1211.5199.pdf.
Kosmas et al. (2015a)
T. S. Kosmas,
O. G. Miranda,
D. K. Papoulias,
M. Tortola, and
J. W. F. Valle,
Phys. Rev. D92,
013011 (2015a),
eprint 1505.03202.
Kosmas et al. (2015b)
T. S. Kosmas,
O. G. Miranda,
D. K. Papoulias,
M. Tortola, and
J. W. F. Valle
(2015b), eprint 1506.08377.
Schechter and Valle (1980)
J. Schechter and
J. W. F. Valle,
Phys. Rev. D22,
2227 (1980).
Bellini et al. (2011)
G. Bellini et al.,
Phys. Rev. Lett. 107,
141302 (2011), eprint 1104.1816.
Vidyakin et al. (1992)
G. Vidyakin,
V. Vyrodov,
I. Gurevich,
Y. Kozlov,
V. Martemyanov,
et al., JETP Lett.
55, 206 (1992).
Derbin et al. (1993)
A. I. Derbin
et al., JETP Lett.
57, 768 (1993).
Daraktchieva et al. (2005)
Z. Daraktchieva
et al. (MUNU),
Phys.Lett. B615,
153 (2005), eprint hep-ex/0502037.
Allen et al. (1993)
R. C. Allen
et al., Phys. Rev.
D47, 11 (1993).
Auerbach et al. (2001)
L. B. Auerbach
et al. (LSND), Phys.
Rev. D63, 112001
(2001), eprint hep-ex/0101039.
An et al. (2012)
F. An et al.
(DAYA-BAY Collaboration),
Phys.Rev.Lett. 108,
171803 (2012), eprint 1203.1669.
An et al. (2014)
F. An et al.
(Daya Bay), Phys.Rev.Lett.
112, 061801
(2014), eprint 1310.6732.
Ahn et al. (2012)
J. Ahn et al.
(RENO collaboration),
Phys.Rev.Lett. 108,
191802 (2012), eprint 1204.0626.
Abe et al. (2013)
K. Abe et al.
(T2K Collaboration) (2013),
eprint 1311.4750.
Grimus and Schwetz (2000)
W. Grimus and
T. Schwetz,
Nucl. Phys. B587,
45 (2000), eprint hep-ph/0006028.
Fukugita and Yanagida (1987)
M. Fukugita and
T. Yanagida,
Phys. Rev. Lett. 58,
1807 (1987).
Mueller et al. (2011)
T. Mueller,
D. Lhuillier,
M. Fallot,
A. Letourneau,
S. Cormon,
et al., Phys.Rev.
C83, 054615
(2011), eprint 1101.2663.
Kopeikin et al. (1997)
V. Kopeikin,
L. Mikaelyan,
and V. Sinev,
Phys.Atom.Nucl. 60,
172 (1997).
Beda et al. (2012)
A. Beda,
V. Brudanin,
V. Egorov,
D. Medvedev,
V. Pogosov,
et al., Adv.High Energy Phys.
2012, 350150
(2012).
Arpesella
et al. (2008a)
C. Arpesella
et al. (Borexino),
Phys. Lett. B658,
101 (2008a),
eprint 0708.2251.
Bellini et al. (2010)
G. Bellini et al.
(Borexino), Phys. Rev.
D82, 033006
(2010), eprint 0808.2868.
Bellini et al. (2014a)
G. Bellini et al.
(Borexino), Phys. Rev.
D89, 112007
(2014a), eprint 1308.0443.
Bellini et al. (2014b)
G. Bellini et al.
(BOREXINO), Nature
512, 383
(2014b).
Back et al. (2003)
H. O. Back et al.,
Phys. Lett. B563,
35 (2003).
Arpesella
et al. (2008b)
C. Arpesella
et al. (Borexino),
Phys. Rev. Lett. 101,
091302 (2008b),
eprint 0805.3843.
Fogli et al. (2002)
G. L. Fogli,
E. Lisi,
A. Marrone,
D. Montanino,
and A. Palazzo,
Phys. Rev. D66,
053010 (2002), eprint hep-ph/0206162.
Gonzalez-Garcia
et al. (2010)
M. C. Gonzalez-Garcia,
M. Maltoni, and
J. Salvado,
JHEP 05, 072
(2010), eprint 0910.4584.
Raffelt (1990)
G. G. Raffelt,
Phys. Rev. Lett. 64,
2856 (1990).
Miranda
et al. (2004a)
O. G. Miranda,
T. I. Rashba,
A. I. Rez, and
J. W. F. Valle,
Phys. Rev. D70,
113002 (2004a),
eprint hep-ph/0406066.
Miranda
et al. (2004b)
O. G. Miranda,
T. I. Rashba,
A. I. Rez, and
J. W. F. Valle,
Phys. Rev. Lett. 93,
051304 (2004b),
eprint hep-ph/0311014. |
Securing Microservices and Microservice Architectures: A Systematic Mapping Study
Abdelhakim Hannousse
Department of Computer Science
Universté 8 Mai 1945, Guelma
BP 401, Guelma 24000, Algeria
hannousse.abdelhakim@univ-guelma.dz
& Salima Yahiouche
Department of Computer Science
LRS laboratory, Badji Mokhtar University
BP 12, Annaba 23000, Algeria
yahiouche.salima@univ-annaba.dz
Abstract
Microservice architectures are becoming trending alternatives to existing software development paradigms notably for developing complex and distributed applications.
Microservices emerged as an architectural design pattern aiming to address the scalability and ease the maintenance of online services. However, security breaches also increased threatening the availability, integrity and confidentiality of microservice-based systems. A growing body of literature is found addressing security threats and security mechanisms to individual microservices and microservice architectures.
In this paper, we conduct a systematic mapping study in order to categorize threats on microservice architectures and security proposals along with their applicability levels, platforms and validation techniques. The aim of this study is to provide a helpful guide to developers about already recognized threats on microservices and how they can be detected, mitigated or prevented; we also aim to identify potential research gaps on securing microservice architectures.
The systematic search yielded 1067 studies of which 47 are selected as primary studies. The results of the mapping revealed an unbalanced research focus on external attacks, authentication and authorization techniques compared with internal attacks and mitigation techniques. Additionally, we found that microservice layer is the most addressed layer in the architecture. We also found that performance analysis and case studies are the most used validation technique of security proposals.
Securing Microservices and Microservice Architectures: A Systematic Mapping Study
A Preprint
Abdelhakim Hannousse
Department of Computer Science
Universté 8 Mai 1945, Guelma
BP 401, Guelma 24000, Algeria
hannousse.abdelhakim@univ-guelma.dz
Salima Yahiouche
Department of Computer Science
LRS laboratory, Badji Mokhtar University
BP 12, Annaba 23000, Algeria
yahiouche.salima@univ-annaba.dz
Keywords microservices $\cdot$
microservice architectures $\cdot$
security $\cdot$
systematic mapping
1 Introduction
Nowadays, systems are becoming more complex, larger and more expensive due to the rapid growth of requirements and adoption of new technologies. Moreover, due to competitors, many companies need to make changes to their systems as fast as possible and without affecting their systems availability. This requires appropriate designs, architectural styles and development processes. Software engineering provides different paradigms to partially meet those needs by decomposing software systems into fine-grained software units for better modularity, maintainability and reusability, and hence reduce time-to-market.
Recently, microservice architectures (MSA) [1] has emerged as a new architectural style that allows building software systems by composing lightweight services that perform very cohesive business functions. Microservices are the mainstay of MSA. A mircoservice is a fine-grained software unit that can be created, initialized, duplicated, and destroyed independently from other microservices of the same system. Moreover, micorservices can be deployed across heterogeneous execution platforms over the network. Using microservices enables high scalability and flexibility of large scale and distributed software systems.
Although the advantages brought by adopting microservice architectures in developing complex systems, MSA as a novel technology comes with many flaws [2] and security is one of the serious challenges that need to be tackled [1]. In fact, security is a longstanding problem in networking systems, but with microsevices, security becomes more challenging. This is due to the large number of entry points and overload on communication traffic emerged by decomposing systems into smaller, independent and distributed software units. Moreover, trusts cannot simply be established between individual microservices in the network that often come from different and unknown providers.
Due to the massive attacks reported on companies adopting MSA such as Netflix and Amazon111https://threatpost.com, dealing with security breaches became an urgent need. Several works in the literature have noticed the need to investigate security of MSA [1, 3, 4]. However, security threats are diverse and are continually increasing. Security proposals are also increasing and varies from securing individual microservices into complete architectures and infrastructures.
In this article, we conduct a systematic mapping study to uncover the main threats menacing the security of microservice-based systems. We systematically identify existing studies addressing threats and proposing security solutions to MSA. We apply a thorough protocol to extract, classify, and organize reported threats with the security solutions proposed to mitigate and prevent them. The contributions of the study can be summarized as:
1.
identify the most relevant threats concerning microservices and microservice architectures
2.
point out the set of security mechanisms used to detect, mitigate and prevent those threats
3.
determine the set of techniques and tools used to examine and validate proposed solutions
The reminder of this paper is structured as follows: section 2 gives a succinct background on used techniques and approaches in this paper, specifically, microservice architectures and systematic mapping; section 3 overviews and discusses related works; section 4 details our research methodology; section 5 presents and discusses the mapping results; section 6 discusses threats to validity related to the study and section 7 concludes the paper.
2 Background
2.1 Micorservice Architectures
Microservices is a trending architectural style that aim to design complex systems as collections of fine grained and loosely coupled software artifacts called microservices; each microservice implements a small part or even a single function of the business logic of applications [3]. Their efficient loose coupling enables their development using different programming languages, use different database technologies, and be tested in isolation with respect to the rest of underlying systems. Microsevices may communicate with each other directly using an HTTP resource API or indirectly by means of message brokers (see Figure 1). Microservices can either be deployed in virtual machines or lightweight containers. The use of containers for deploying micorservices is preferred due to their simplicity, lower cost, and their fast initialization and execution.
Regarding software quality attributes, adopting microservices increases reusability and interoperability, enables scalability and enhances maintainability of complex software systems. Within adequate distributed platforms and technologies, microservices can easily be deployed, replicated, replaced, and destroyed independently without affecting systems availability. Moreover, implementing a single business capability per microservices allows their use in different applications and application domains. The main characteristics that differentiate microservices architectural style from monolithic and its ancestor service-oriented architectures is the smaller size, scalability and independence of each unit constituting a system.
Microservices are getting more attention and becoming adopted in industry. Currently, microservices are used by widely recognized companies such as Coca Cola, Amazon, eBay and NetFlix. Specifically, microservices is becoming more popular in software and IT service companies [5].
Although the advantages brought by adopting microservice architectures in developing complex systems, security is one of the serious challenges that need to be tackled. Thus, there is an urgent need to identify and check current trends in overcoming security challenges in microservice architectures which is the aim of the present paper.
2.2 Systematic mapping
A systematic mapping is a kind of evidence-based software engineering (EBSE) [6]. The aim of a systematic mapping is to provide an overview of a research area by building a classification scheme and structuring evidences on a research field. Peterson et al. [7, 8] has proposed an overall process for the elaboration of systematic mapping.
The process is composed of three main steps: planning, conduction and report. By planning, one can start by justifying the need and scope of the mapping, formulate the set of research questions, develop and validate a protocol specifying all the decisions relevant to the conduction of the mapping. The protocol includes the identification of search terms, search strategy, literature sources that need to be used to retrieve relevant papers, how and in which base found papers are selected and included in the mapping, what data need to be extracted from selected papers and how extracted data are synthesized and classified. By conduction, the initially validated protocol from the planning step is executed; thus, the identified sources are used to retrieve papers, found papers are examined for relevance; useful data are then extracted from admitted papers and extracted data are synthesized and classified. By reporting, extracted data from the included papers are visualized and results are interpreted, research questions are answered and the mapping is validated and documented. Figure 2 depicts the overall process for conducting systematic mapping studies as proposed in [8]. The quality assessment step is depicted in Figure 2 within a dashed line box since it is optional as stated in [6, 8].
3 Related Work
We found in the literature several secondary studies (systematic reviews or mappings) dedicated to investigate the state-of-the-art of MSA in general. Surprisingly, few works are found focusing on security aspects in MSA. All found studies, except the work of Vale et al. [9], are either platform or technology dependent investigations.
Vale et al. [9] conducted a similar investigation to our study. They conducted a systematic mapping to reveal adopted security mechanisms for microservice-based systems. The study examined 26 papers published from November 2018 to March 2019. Vale et al. [9] focused only on security mechanisms and categorized the 18 identified mechanisms according to their focus, and classified validation techniques according to their nature. The study revealed that (1) authentication and authorization are the most frequently adopted mechanisms for securing microservices, (2) case studies and experiments are the most validation techniques used for security proposals, (3) absence of patterns for microservices-based systems security. Our study is broader and improves Vale et al. [9] work in several ways. In this study we include published papers since 2011. Moreover, besides security mechanisms, we also focus on identifying security threats and the applicability of proposed solutions regarding their execution platforms and architectural layers.
Yu et al. [10] presented a survey on security in microservice-based fog applications. The survey included papers published between 2010 and 2017. The focus of Yu et al. [10] was domain specific; they focused on determining security challenges and potential solutions of adopting micorservices in fog computing. The security issues identified by the study concerns containers, data, and network vulnerabilities. They also proposed a solution for inter-service communication in fog applications.
Monteiro et al. [11] identified a set of elements related to microservices implementation in cloud computing. They reported the same security aspects discussed by Yu et al. [10]. They concluded that availability and trustworthiness are the two major security requirements in MSA.
Nkomo et al. [12] conducted a systematic review on practices that can be incorporated into the development process of microservice-based systems. The focus of Nkomo et al. [12] was to propose general guidelines where security can be tackled earlier when developing microservice-based applications putting much emphasis on microservices composition. They ended up with five security-focused development activities: (1) identify security requirements of microservice composition, (2) adopt secure programming best practices, (3) validate security requirements and secure programming best practices, (4) secure configuration of runtime infrastructure, and (5) continuously monitor the behavior of microservices. Considering security as a primary concern throughout the life-cycle of microservice-based systems is mandatory; however, experiences show that all security threats cannot be identified earlier especially with the continuous evolving technologies.
Sultan et al. [13] presented a survey on the security of containers; they identified main threats due to images, registries, orchestration, containers themselves, side channels and host OS risks. They distinguished two kind of solutions to containers security: software-based and hardware-based solutions without further investigation of proposed solutions. Belair et al. [14] complements the work of Sultan et al. [13] by proposing a taxonomy for containers’ security proposals. Belair et al. [14] focused on security solutions at the infrastructure level putting much emphasis on data transmission through virtualization. Three categories were identified: configuration-based, code-based and rule-based defense. They reported the fact that Linux security model (LSM), the powerful defense framework targeting Linux, cannot be easily adapted to containers to improve security.
Compared with the works of Yu et al. [10] and Monteiro et al. [11], our study is domain and platform independent and includes more recent endeavors with broader focus on proposed solutions to security threats. Compared with the works in [13] and [14], we focus on our study on security issues concerning MSA in general and not only containers.
4 Research methodology
In this section we present the details of the protocol adopted for conducting this mapping study. Following the guidelines of Peterson et al. [7], a systematic mapping study should include the following primary steps: a definition of research questions, search for relevant papers, screening of found papers, propose or use an existing classification scheme, data extraction and studies mapping. In the sequel, we describe the details of each step.
4.1 Research Questions
The aim of this study is to identify the set of security vulnerabilities and how to be tackled in microservice-based systems. Thus, we formulate our research questions in light of the aims of our study and following the guidelines of Kuhrmann et al. [15]. This study is conducted with five main questions in mind:
RQ1. What are the most addressed security threats, risks, and vulnerabilities of microservices and microservice architectures and how they can be classified? This research question distinguishes the list of mostly treated vulnerabilities from those needing further investigations.
RQ2. What are existing approaches and techniques used for securing mircoservices and microservice architectures and how they can be classified? This research question provides an overview of existing approaches and techniques used for securing microservice-based systems.
RQ3. At what level of architecture the proposed techniques and approaches are applicable for securing microservcies? This research question indicates where security is applied highlighting the less focused levels of microservice architectures .
RQ4. What domains or platforms are the focus of existing solutions for securing microservices and microservice architectures? This research question shows whether the focus of the proposed solutions is platform specific or platform independent.
RQ5. What kind of evidence is given regarding the evaluation and validation of proposed approaches and techniques for securing microservices and microservice architectures? This research question evaluates the maturity of existing security techniques highlighting the set of empirical strategies used to validate proposed solutions.
4.2 Search process
Search string used in this study is designed to be generic and simple. It is constructed based on search terms concerned with population and intervention as suggested by Petticrew and Roberts in [16]. Population refers to the application area which is microservices and microservice architectures where intervention is security, vulnerabilities and attacks. Accordingly, final adopted search string is :
("microservice" OR "micro-service" OR "micro service")
AND
("architecture" OR "design" OR "system" OR "structure")
AND
("security" OR "vulnerability" OR "attack")
For retrieving relevant studies, we followed the guidelines of Kuhrmann et al. [15]. Thus, we adopted the use of the following online academic libraries:
•
IEEE Xplorer (https://ieeexplore.ieee.org)
•
ACM Digital Library (https://dl.acm.org)
•
SpringerLink (https://link.springer.com)
•
ScienceDirect (https://www.sciencedirect.com/)
•
Wiley Online Library (https://onlinelibrary.wiley.com)
To avoid missing relevant studies, we complement our automatic search by conducting recursive backward and forward snowballing on selected studies as suggested by Wohllin [17, 18]. By backward snowballing, we check the relevance of references in approved papers. By forward snowballing, we check the relevance of papers citing approved papers. The snowballing is recursively applied to each newly approved paper. Google Scholar is used as a sole source for forward snowballing.
4.3 Study selection process
The set of retrieved papers by automatic search followed two screening stages. In the first stage, titles and abstracts were read to measure relevance. In the second stage, full texts of papers were examined to check if they meet our inclusion criteria. The list of all the papers are screened separately by the two authors; decisions are exchanged and conflicts are discussed and solved. Found papers from snowballing are also screened separately by the two authors before deciding whether to be included or excluded.
4.4 Inclusion and exclusion criteria
The number of retrieved papers by online academic libraries is reduced by specifying a strict number of inclusion and exclusion criteria. In this study, only peer-reviewed papers from journals and conferences are included. The automatic search is conducted to cover all published papers since 2011 including early publications. The starting year 2011 is adopted since there was no consensus on the term microservice architectures prior 2011 [3]. Only English written papers addressing security aspects or security solutions to microservices or microservice architectures are included. The full list of adopted inclusion and exclusion criteria are presented in Table 1 and Table 2 respectively.
4.5 Data extraction process
Following the guidelines of Peterson et al. [7] a data extraction form is designed as illustrated in Table 3. Each paper is described in terms of its metadata such as year of publication, source and type. In addition, a set of required information for our analysis are extracted. These include the list of security threats or attacks addressed by the study, proposed solutions, application level of proposed solutions, validation method and application platforms.
4.6 Data synthesis
We noticed a lack of a consensus on detailed taxonomies for security threats and security mechanisms; this prevents mapping all the selected studies to appropriate and distinct categories answering research questions RQ1 and RQ2. Moreover, due to the diversity of applications used in selected studies, their targeted platforms and used verification and validation techniques, it was necessary to properly categorize those studies answering RQ3, RQ4 and RQ5.
For mapping properly all the selected studies to proper categories for each research question, we used our experiences and existing taxonomies [11, 1, 19] in identifying categories and their relationships. We also used grounded theory [20] as a complementary approach to generate missing categories from extracted data items. Specifically, we used open coding and selective coding to identify categories and their relationships with existing categories from D6, D9 and D10-11. In this study, grounded theory is used in an iterative process, where categories and subcategories are changed in each iteration until reaching a stability state.
5 Results of the mapping
In this section we describe and detail the results of the mapping study answering the five research questions outlined in section 4.1.
5.1 Overview of selected studies
The search process is conducted in December 2019 and yielded 47 distinct papers published since 2011. The designed query is applied to the set of selected libraries. Table 4 shows the number of returned papers by each library.
The set of 1067 retrieved papers by the different search engines are gathered and duplicate papers are removed. This reduces the number to 1065. By screening titles and abstracts of remaining papers, 1015 papers are excluded for their irrelevance. After checking the inclusion and exclusion criteria, only 38 papers are approved. By conducting recursive backward and forward snowballing, 9 more papers are added. Two snowballing cycles are performed before reaching a steady state. In the first round, 7 new papers are included; in the second round, 2 more papers are added. Figure 3 depicts the overall selection process.
Figure 4 shows the distribution of selected studies according to their publication year and source. We notice that, although the earlier emergence of MSA in 2011, the interest into securing microservices and microservie architectures is considered few years later and start getting more attentions since 2015. Figure 4 also shows that the maximum number of publications come from IEEE Xplorer and none from Wiley meets our inclusion criteria.
Table 5 shows the complete list of selected studies with their year and type of publication and how they are found.
Following the guidelines of Kuhrmann et al. [15], we also experienced the use of Word Clouds to analyze the appropriateness of our result set of primary studies. Figure 5 illustrates the most frequent words used in selected papers based on their titles and abstracts. The Figure shows that the most used words are microservice, security, application, architecture, system and service. Attacks, vulnerabilities and risks are rarely used in titles and abstracts.
5.2 MSA security threats (RQ1)
Microservice architecture as an emerging development paradigm in software engineering brings new security threats and vulnerabilities. These threats may come from insiders (i.e. internal attacks) or from outsiders (i.e. external attacks). For proper securing microservice-based systems, all threats, regardless of their origin, need to be detected and prevented using either available mitigation techniques or through proposing innovative solutions. In this study, we identify the focus of existing endeavors with respect to the source of threats (internal, external or both). Figure 6 depicts the distribution of identified and selected studies regarding the addressed source of threats. Figure 6 shows that 70% of primary studies focus on external attacks, only 11% focus on internal attacks and 19% focus on both source of threats. This clearly indicates an unbalanced research focus towards external attacks.
Due to the plethora of taxonomies of security threats and lack of a consensus among their categorization, we adopted, a classification based on targets of attacks. Accordingly, threats in MSA can be classified into:
•
User-based attacks: attacks where users are involved directly (i.e. malicious user actions) or indirectly (i.e. inadvertent insider actions).
•
Data attacks: threats targeting sensitive data that can be disclosed and manipulated by attackers.
•
Infrastructure attacks: attacks targeting MSA architectural elements and platforms such as monitors, discovery service, message broker, load balancer, etc.
•
Software attacks: threats involving code transformation or injection for malicious purposes.
Table 6 shows the set of MSA security threats addressed by primary studies grouped by category. The results revealed that unauthorized access, sensitive data exposure and compromising individual microservices are the most treated and addressed threats by contemporary studies. In addition, infrastructure attacks are the most diverse and less addressed attacks in selected studies.
Although IBM X-Force [66] reported that 60% of all attacks were carried out by insiders, the study shows that only 11% of primary studies focus on internal attacks. This is probably due to the fact that external threats are easier to be handled compared with internal threats. External threats are common in networking systems and can usually be identified and prevented by means of strong firewalls and intrusion detection systems; internal threats often requires considerable policy changes and continuous monitoring of internal traffics. This is owing to privileges awarded and sensitive data exposed to insiders.
The diversity of attacks is due to the adoption of Zero Trust model [67] that suggests to afford no default trust to users, devices, applications, or packets; instead every action and entity need to be authenticated and authorized appropriately. Moreover, infrastructure attacks are less addressed due to their complexity since most attacks require low level solutions especially those related to hardware, nodes and operating systems. Attacks from other categories often require high level or software-based solutions that can easily be integrated into existing platforms and technologies. This justifies why software, user-based, and data attacks earned more attention than infrastructure attacks.
Thus, we advocate for research studies that investigate threats caused by insiders in microservice-based applications. In addition, we suggest to investigate all OWASP identified vulnerabilities with their effects when adopting microservice architectures.
5.3 Micorservice security mechanisms (RQ2)
Due to the diversity of proposed solutions, we classify MSA security mechanisms addressed in primary studies regarding the nature of their proposals as follows:
•
General protection measures: use of general security techniques to mitigate common known threats in MSA, or a set of general guidelines on choosing appropriate languages and technologies.
•
Framework-based solutions: architectural frameworks for MSA incorporating specific modules to handle some security aspects and mechanisms such as authorization, continuous monitoring, diagnosis, etc.
•
Technique-based solutions: newly designed or adopted techniques from other domains to mitigate or prevent some security threats in MSA.
•
Tool-based solutions: newly developed tools or algorithms implementing security mechanisms.
•
Methodologies: set of decision rules for the selection of appropriate security policies throughout the development lifecycle of microservice-based systems.
Our investigation (see Figure 7) shows that 54% of the studies propose new techniques for securing MSA and 23% propose framework-based solutions. Few studies propose general protection measures, tools or methodologies.
Proposed solutions for securing microservices and microservice architectures can be classified into proposals for enforcing authentication and/or authorization policies, auditing, mitigation and prevention:
•
Authentication: techniques used to verify the identity of users requiring access to MSA resources and data.
•
Authorization: techniques used to check users’ permissions for accessing specific MSA resources or data.
•
Auditing: techniques applied at runtime for discovering security gaps and may: (1) subsequently initiate appropriate measures or (2) simply report security breaches to relevant supervisory authority.
•
Mitigation: techniques that limit the damage of attacks when they appear. Mitigation techniques can be integrated into existing mircoservice-based systems.
•
Prevention: techniques that try to stop attacks from happening in the first place. Prevention techniques need to be considered when developing new mircoservice-based systems.
Table 7 shows the list of the proposed solutions mapped into our classification with the proportion rate for each proposal with respect to the total set of primary studies. The results show that much emphasis is put on proposing prevention techniques (44.68%), enforcing authentication and/or authorization (40.42%222This value is calculated by collecting all the papers from authorization and/or authentication categories and removing duplicates; the obtained number is divided by the total number of papers.), and auditing (34.04%), where less attention is being paid to mitigation (12.77%).
The much emphasis put on authentication and authorization techniques is defensible. In fact, authentication and authorization are basic security mechanisms to any secure systems. They form a front defense line in the protection of the different microservice architecture elements (i.e. individual microservices, API gateway, containers, microservice registry, etc.). However, studies considering authentication and authorization are less innovative since they propose combination of existing techniques and standards. For example, the authors of P8 proposed a combination of OAuth 2.0, JWT, Open ID and SSO used by a special authentication and authorization orchestrator.
Audition proposals are showing the integration of artificial intelligence techniques such as machine learning and self-learning algorithms (P2, P10, P22-23, P36). Those techniques are based on the runtime analysis of user and/or microservice behaviors and (semi-)automatically take predefined actions in reaction to suspicious behaviors.
Most, if not all, proposals for mitigation are Moving Target Defense-based solutions (MTD) [68]. The idea behind MTD is to continuously perform transformation of system components and configurations preventing attackers from acquiring knowledge about target systems to be used to initiate harmful attacks. This includes, periodically update or restart microservices, IP shuffling, etc.
Prevention proposals are the most diverse and innovative techniques. They varies from using physical computing devices such as HMS, powerful techniques and technologies such as encryption and Blockchain into adopting software and intelligent techniques such as using secure programming languages and deception through live cloning of containers to deceive attackers.
Due to the lower rate of mitigation techniques and their applicability to existing microservice-based systems, we advocate more research studies on mitigation techniques.
5.3.1 Micorservice security application levels (RQ3)
The adoption of MSA as an architectural design of distributed applications introduces security vulnerabilities in different architectural layers. Thus, security measures need to be taken in every layer of MSA. In this study, we distinguish the following layers:
•
Microservice: Individual microservices are the mainstays of MSA, those micorservices can be blocked or compromized through injection of malicious code. Thus, security measures to adopt only trusted micorservices and protect them from internal and external attacks need to be taken.
•
Architecture: Connections among microservices can be broken. In addition, unauthorized access to sensitive data can be gained due to the insecure configuration options of microservice-based systems or the static and insecure locations chosen for microservices handling sensitive data. Several security measures can be taken at this level to secure the overall architecture of microservice-based systems.
•
API: Finely tuned attacks on APIs can bypass traditional security measures provided by API gateways. Hence, assets can be accessed and controlled by malicious users. Appropriate security measures should be taken at API gateways to avoid such vulnerabilities.
•
Communication: Data exchanged between microservices through event-buses can be intercepted and altered by malicious insiders. Thus, securing communication channels between microservices is mandatory for securing microservice-based systems.
•
Deployment: Containers holding micorservices can also be sources of vulnerabilities. Containers can be compromised through gaining an unauthorized access or deriving vulnerabilities from using images from untrusted sources. Thus, appropriate security measures should also be taken at this level.
•
Soft-Infrastructure: Infrastructure vulnerabilities are lower level vulnerabilities that can affect practically every software entity running on the network including monitors, registries, message brokers, load balancers and other orchestrators. Thus, introducing techniques at this level to guarantee the security of the diverse software network entities and the safety of their configuration is of higher importance.
•
Hard-Infrastructure: Hardware components are also vulnerable to attacks. Attackers may use bugs and backdoors intentionally or unintentionally introduced at manufacturing [69] to initiate attacks. These vulnerabilities need to be tackled by introducing appropriate error and backdoor detection mechanisms.
Table 8 shows the distribution of primary studies per our described application layers. The study revealed that much emphasis are put to secure individual microservices, soft-infrastructure and API gateways where less attention is being paid to deployment and hard-infrastructure layers.
The higher rate of securing individual microservices is not surprising since microservices form the mainstay of MSA. The less interest in hard-infrastructure solutions is also defensible due to their complexity and cost-intensive compared with soft-infrastructure based solutions. However, communication should have much attention than that has been revealed. Communication level protection is of a high importance regarding the huge number and nature of transmitted data in the communication channels.
5.3.2 MSA security mechanisms target platforms (RQ4)
The identified papers in this study are classified regarding the target platforms and applications for their proposed solutions. Table 9 shows that 29.79% of papers proposed MSA security solutions that work for different platforms; the same proportion is found also for solutions to deal with securing microservices in the cloud platform. However, few studies proposed platform specific solutions such as 5G platform, IoT, Web applications, kebernetes patforms and Springer framework.
Cloud-focused and platform independent solutions are found used in equal measure within a higher rate equal to 29.79%. The interest to cloud computing is understandable due to different facilities provided to companies by adopting MSA for developing their applications. Adopting MSA for developing applications in the cloud allow companies to integrate existing legacy systems, to grow with demands and to use up-to-date and intuitive interfaces. Solutions provided for IoT applications are also getting more attentions due to the specificity and the growing needs to those applications in the market.
5.3.3 Micorservice security V&V methods (RQ5)
For validating the proposed solutions, we distinguished the use of several verification and validation approaches:
•
Validation by simulation: this includes: (1) use of simulated lab environments or testbeds for testing proposed designs, (2) simulation of attacks with check bypassing of proposed security measures.
•
Testing: this includes: (1) use case-based testing, (2) reconfigure-testing cycles, and (3) use of appropriate datasets for training and/or testing.
•
Performance analysis: this is performed by measuring overheads, latency, throughput, memory storage, CPU usage, response time.
•
Qualitative analysis: compare or verify and validate a set of qualitative requirements.
•
Quantitative comparison: comparing the proposed solution with similar proposals using quantitative metrics such as performance metrics.
•
Adhoc metrics: proposing specific metrics for the evaluation of proposals. For example, P18 proposed a diversification index as a security measure to validate the proposal.
•
Case study based validation: use case studies to validate the feasibility of the proposed solution.
•
Proof of concept (POC): develop a prototype to demonstrate the feasibility of the proposed solution.
•
Tool-based testing: use automatic vulnerability scanners such as OWASP and WSVB-benches.
•
Formal verification: use model checkers or theorem provers to check the validity of specified properties.
•
Complexity measuring: estimate temporal complexity of proposed algorithms implementing solutions.
Table 10 shows that performance analysis, case study based validation and use case based testing are the most adopted techniques for verification and validation. Formal verification and complexity measures are the least used methods for validation. This is due to the nature of proposed solutions. Formal verification can only be adopted when formal specification of systems and their properties are described, where complexity measures are adopted for algorithmic based solutions. Validation by simulation and qualitative analysis are found used in equal measure, with medium rate equal to 12.77%. Most used simulation methods used simulated lab environments or testbeds. Only P3 used simulated attacks to evaluate the proposed technique.
6 Threats to validity
In this section we discuss the threats to the validity and how we mitigated their effects on the obtained results.
An internal validity threat to our study concerns the identification of primary studies from the large set of papers found in the literature. For these sake, we adopted the guidelines of Kuhrmann et al. [15] for the selection of search engines. To void bias to search engines, we completed our search by snowballing technique [18] over the already identified papers. The use of several iterations of the snowballing technique allowed the identification of nine more relevant papers in which five were not indexed by the selected search engines. For ensuring the inclusion of high quality papers, we adopted a set of strict inclusion and exclusion criteria that accept only peer-reviewed journal and conference papers for their completeness and sufficient results. However, since only papers explicitly referring to microservices or microservice architectures were included, some papers focusing on securing the deployment layer, specifically Docker containers [70] were omitted.
A conclusion validity threat to our study concerns the adoption of taxonomies for security threats and mechanisms. In fact, several taxonomies are investigated [11, 1, 19], however, none of those taxonomies enable the proper classification of all the identified studies. Thus, we used open and selective coding from grounded theory [20] and we adopted a classification based on deeper analysis of the focus and the proposed solutions of identified papers. Some of the categories of our classifications are already used in existing taxonomies some they are either used as they are or adapted to fulfill the context of our study.
7 Conclusion
In this study, we conducted a systematic mapping on securing microservices focusing on threats, nature, applicability platforms, and validation techniques of security proposals. The study examined 47 papers published since 2011. The results revealed that unauthorized access, sensitive data exposure and compromising individual microservices are the most treated and addressed threats by contemporary studies. The results also revealed that prevention and auditing based solutions are the most proposed security mechanisms. Additionally, we found that microservice layer is the most attacked layer in the MSA architecture, followed by the software infrastructure level and then the API. Our study shows that 29.79% of papers proposed MSA security solutions that work for different platforms, the same proportion is noticed for cloud-based solutions. Finally, we found that most verification and validation methods were based on performance analysis, and case studies.
We noticed that most addressed threats are well-known for other architectural styles and few are concerned only with MSA. Specifically, compromising individual microservices that can radically lead to a chain defection in MSA. Moreover, continuous monitoring became very popular among MSA designers to prevent possibly future threats. Encryption remain the most used technique facing sensitive data exposure. Regarding noticed unbalanced research focus on external attacks and prevention techniques, we advocate more studies studying internal attacks and proposing mitigation techniques. Moreover, more studies are suggested for treating communication layer vulnerabilities.
References
[1]
T. Yarygina and A. H. Bagge.
Overcoming security challenges in microservice architectures.
In 2018 IEEE Symposium on Service-Oriented System Engineering
(SOSE), pages 11–20, March 2018.
[2]
Saša Baškarada, Vivian Nguyen, and Andy Koronios.
Architecting microservices: Practical opportunities and challenges.
Journal of Computer Information Systems, pages 1–9, 2018.
[3]
Nicola Dragoni, Saverio Giallorenzo, Alberto Lluch Lafuente, Manuel Mazzara,
Fabrizio Montesi, Ruslan Mustafin, and Larisa Safina.
Microservices: Yesterday, Today, and Tomorrow, chapter 12,
pages 195–216.
Springer International Publishing, Cham, 2017.
[4]
Nuha Alshuqayran, Nour Ali, and Roger Evans.
A systematic mapping study in microservice architecture.
In 2016 IEEE 9th International Conference on Service-Oriented
Computing and Applications (SOCA), pages 44–51. IEEE, 2016.
[5]
J. Bogner, J. Fritzsch, S. Wagner, and A. Zimmermann.
Microservices in industry: Insights into technologies,
characteristics, and software quality.
In 2019 IEEE International Conference on Software Architecture
Companion (ICSA-C), pages 187–195, 2019.
[6]
Barbara Ann Kitchenham, David Budgen, and Pearl Brereton.
Evidence-Based Software Engineering and Systematic Reviews.
Chapman & Hall/CRC, 2015.
[7]
Kai Petersen, Robert Feldt, Shahid Mujtaba, and Michael Mattsson.
Systematic mapping studies in software engineering.
In Proceedings of the 12th International Conference on
Evaluation and Assessment in Software Engineering, EASE’08, pages 68–77,
Swindon, UK, 2008.
[8]
Kai Petersen, Sairam Vakkalanka, and Ludwik Kuzniarz.
Guidelines for conducting systematic mapping studies in software
engineering: An update.
Information and Software Technology, 64:1–18, 2015.
[9]
Anelis Pereira Vale, Gastón Márquez, Hernán Astudillo, and
Eduardo B Fernandez.
Security mechanisms used in microservices-based systems: A systematic
mapping.
In XLV Latin American Computing Conference, pages 1–10, 2019.
[10]
Dongjin Yu, Yike Jin, Yuqun Zhang, and Xi Zheng.
A survey on security issues in services communication of
microservices-enabled fog applications.
Concurrency and Computation: Practice and Experience,
31(22):e4436, 2019.
e4436 cpe.4436.
[11]
Luciano de Aguiar Monteiro, Washington Henrique Carvalho Almeida,
Raphael Rodrigues Hazin, Anderson Cavalcanti de Lima, Sahra Karolina Gomes
e Silva, and Felipe Silva Ferraz.
A survey on microservice security–trends in architecture, privacy
and standardization on cloud computing environments.
International Journal on Advances in Security,
11(3-4):201–213, 2018.
[12]
Peter Nkomo and Marijke Coetzee.
Software development activities for secure microservices.
In Sanjay Misra, Osvaldo Gervasi, Beniamino Murgante, Elena Stankova,
Vladimir Korkhov, Carmelo Torre, Ana Maria A.C. Rocha, David Taniar,
Bernady O. Apduhan, and Eufemia Tarantino, editors, Poceedings of
International Conference on Computational Science and Its Applications, ICCSA
2019, pages 573–585, Cham, 2019. Springer International Publishing.
[13]
S. Sultan, I. Ahmad, and T. Dimitriou.
Container security: Issues, challenges, and the road ahead.
IEEE Access, 7:52976–52996, 2019.
[14]
Maxime Bélair, Sylvie Laniepce, and Jean-Marc Menaud.
Leveraging kernel security mechanisms to improve container security:
A survey.
In Proceedings of the 14th International Conference on
Availability, Reliability and Security, ARES ’19, pages 76:1–76:6, New
York, NY, USA, 2019. ACM.
[15]
Marco Kuhrmann, Daniel Méndez Fernández, and Maya Daneva.
On the pragmatic design of literature studies in software
engineering: an experience-based guideline.
Empirical Software Engineering, 22(6):2852–2891, 2017.
[16]
Mark Petticrew and Helen Roberts.
Systematic Reviews in the Social Sciences: A Practical Guide.
John Wiley & Sons, Ltd, 2006.
[17]
Claes Wohlin.
Guidelines for snowballing in systematic literature studies and a
replication in software engineering.
In Proceedings of the 18th International Conference on
Evaluation and Assessment in Software Engineering, EASE ’14, pages 1–10,
New York, NY, USA, 2014. Association for Computing Machinery.
[18]
Claes Wohlin.
Second-generation systematic literature studies using snowballing.
In Proceedings of the 20th International Conference on
Evaluation and Assessment in Software Engineering, EASE ’16, pages 1–6, New
York, NY, USA, 2016. Association for Computing Machinery.
[19]
OWASP.
Owasp top 10: The ten most critical web application security risks.
Technical report, OWASP Foundation, 2017.
[20]
Anselm L. Strauss and Juliet M. Corbin.
Basics of qualitative research: techniques and procedures for
developing grounded theory.
Sage Publications, Thousand Oaks, Calif, 1998.
[21]
Mohsen Ahmadvand, Alexander Pretschner, Keith Ball, and Daniel Eyring.
Integrity protection against insiders in microservice-based
infrastructures: From threats to a security framework.
In Manuel Mazzara, Iulian Ober, and Gwen Salaün, editors, Proceedings of Federation of International Conferences on Software
Technologies: Applications and Foundations, pages 573–588, Cham, 2018.
Springer International Publishing.
[22]
Nico Surantha and Felix Ivan.
Secure kubernetes networking design based on zero trust model: A case
study of financial service enterprise in indonesia.
In Leonard Barolli, Fatos Xhafa, and Omar K. Hussain, editors, Proceedings of International Conference on Innovative Mobile and Internet
Services in Ubiquitous Computing, pages 348–361, Cham, 2020. Springer
International Publishing.
[23]
Stefan Brenner, Tobias Hundt, Giovanni Mazzeo, and Rüdiger Kapitza.
Secure cloud micro services using intel sgx.
In Lydia Y. Chen and Hans P. Reiser, editors, Proceedings of
IFIP International Conference on Distributed Applications and Interoperable
Systems, pages 177–191, Cham, 2017. Springer International Publishing.
[24]
Christian Otterstad and Tetiana Yarygina.
Low-level exploitation mitigation by diverse microservices.
In Flavio De Paoli, Stefan Schulte, and Einar Broch Johnsen, editors,
Proceedings of the 6th IFIP WG 2.14 European Conference on
Service-Oriented and Cloud Computing, pages 49–56, Cham, 2017. Springer
International Publishing.
[25]
Tetiana Yarygina and Christian Otterstad.
A game of microservices: Automated intrusion response.
In Silvia Bonomi and Etienne Rivière, editors, Proceedings
of IFIP International Conference on Distributed Applications and
Interoperable Systems, pages 169–177, Cham, 2018. Springer International
Publishing.
[26]
Antonio Nehme, Vitor Jesus, Khaled Mahbub, and Ali Abdallah.
Fine-grained access control for microservices.
In Nur Zincir-Heywood, Guillaume Bonfante, Mourad Debbabi, and
Joaquin Garcia-Alfaro, editors, Proceedings of the International
Symposium on Foundations and Practice of Security, pages 285–300, Cham,
2019. Springer International Publishing.
[27]
A. Bánáti, E. Kail, K. Karóczkai, and M. Kozlovszky.
Authentication and authorization orchestrator for microservice-based
software architectures.
In 2018 41st International Convention on Information and
Communication Technology, Electronics and Microelectronics (MIPRO), pages
1180–1184, May 2018.
[28]
D. Nagothu, R. Xu, S. Y. Nikouei, and Y. Chen.
A microservice-enabled architecture for smart surveillance using
blockchain technology.
In 2018 IEEE International Smart Cities Conference (ISC2),
pages 1–4, Sep. 2018.
[29]
M. Pahl, F. Aubet, and S. Liebald.
Graph-based iot microservice security.
In NOMS 2018 - 2018 IEEE/IFIP Network Operations and Management
Symposium, pages 1–3, April 2018.
[30]
Tran Quang Thanh, S. Covaci, T. Magedanz, P. Gouvas, and
A. Zafeiropoulos.
Embedding security and privacy into the development and operation of
cloud applications and services.
In 2016 17th International Telecommunications Network Strategy
and Planning Symposium (Networks), pages 31–36, Sep. 2016.
[31]
Y. Sun, S. Nanda, and T. Jaeger.
Security-as-a-service for microservices-based cloud applications.
In 2015 IEEE 7th International Conference on Cloud Computing
Technology and Science (CloudCom), pages 50–57, Nov 2015.
[32]
A. Buzachis and M. Villari.
Basic principles of osmotic computing: Secure and dependable
microelements (mels) orchestration leveraging blockchain facilities.
In 2018 IEEE/ACM International Conference on Utility and Cloud
Computing Companion (UCC Companion), pages 47–52, Dec 2018.
[33]
V. M. George and Q. H. Mahmoud.
Claimsware: A claims-based middleware for securing iot services.
In 2017 IEEE 41st Annual Computer Software and Applications
Conference (COMPSAC), volume 1, pages 649–654, July 2017.
[34]
A. Ranjbar, M. Komu, P. Salmela, and T. Aura.
Synaptic: Secure and persistent connectivity for containers.
In 2017 17th IEEE/ACM International Symposium on Cluster, Cloud
and Grid Computing (CCGRID), pages 262–267, May 2017.
[35]
M. Ahmadvand and A. Ibrahim.
Requirements reconciliation for scalable and secure microservice
(de)composition.
In 2016 IEEE 24th International Requirements Engineering
Conference Workshops (REW), pages 68–73, Sep. 2016.
[36]
K. A. Torkura, M. I. H. Sukmana, and A. V. D. M. Kayem.
A cyber risk based moving target defense mechanism for microservice
architectures.
In 2018 IEEE Intl Conf on Parallel Distributed Processing with
Applications, Ubiquitous Computing Communications, Big Data Cloud Computing,
Social Computing Networking, Sustainable Computing Communications
(ISPA/IUCC/BDCloud/SocialCom/SustainCom), pages 932–939, Dec 2018.
[37]
H. Jin, Z. Li, D. Zou, and B. Yuan.
Dseom: A framework for dynamic security evaluation and optimization
of mtd in container-based cloud.
IEEE Transactions on Dependable and Secure Computing, pages
1–12, 2019.
[38]
C. Gerking and D. Schubert.
Component-based refinement and verification of information-flow
security policies for cyber-physical microservice architectures.
In 2019 IEEE International Conference on Software Architecture
(ICSA), pages 61–70, March 2019.
[39]
A. Osman, P. Bruckner, H. Salah, F. H. P. Fitzek, T. Strufe, and
M. Fischer.
Sandnet: Towards high quality of deception in container-based
microservice architectures.
In ICC 2019 - 2019 IEEE International Conference on
Communications (ICC), pages 1–7, May 2019.
[40]
M. Pahl and F. Aubet.
All eyes on you: Distributed multi-dimensional iot microservice
anomaly detection.
In 2018 14th International Conference on Network and Service
Management (CNSM), pages 72–80, Nov 2018.
[41]
R. Ravichandiran, H. Bannazadeh, and A. Leon-Garcia.
Anomaly detection using resource behaviour analysis for autoscaling
systems.
In 2018 4th IEEE Conference on Network Softwarization and
Workshops (NetSoft), pages 192–196, June 2018.
[42]
Z. Wen, T. Lin, R. Yang, S. Ji, r. Ranjan, A. Romanovsky, C. Lin,
and J. Xu.
Ga-par: Dependable microservice orchestration framework for
geo-distributed clouds.
IEEE Transactions on Parallel and Distributed Systems, pages
1–16, 2019.
[43]
D. Lu, D. Huang, A. Walenstein, and D. Medhi.
A secure microservice framework for iot.
In 2017 IEEE Symposium on Service-Oriented System Engineering
(SOSE), pages 9–18, April 2017.
[44]
M. Pahl and L. Donini.
Securing iot microservices with certificates.
In NOMS 2018 - 2018 IEEE/IFIP Network Operations and Management
Symposium, pages 1–5, April 2018.
[45]
A. Nehme, V. Jesus, K. Mahbub, and A. Abdallah.
Securing microservices.
IT Professional, 21(1):42–49, Jan 2019.
[46]
C. Fetzer.
Building critical applications using microservices.
IEEE Security Privacy, 14(6):86–89, Nov 2016.
[47]
Quy Nguyen and Oras F. Baker.
Applying spring security framework and oauth2 to protect microservice
architecture API.
JSW, 14(6):257–264, 2019.
[48]
Xiuyu He and Xudong Yang.
Authentication and authorization of end user in microservice
architecture.
Journal of Physics: Conference Series, 910:012060, oct 2017.
[49]
Oras Baker and Quy Nguyen.
A novel approach to secure microservice architecture from owasp
vulnerabilities.
In Proceedings of the 10th Annual CITRENZ Conference (2019),
pages 54–58, Nelson, NZ, October 2019. ITx New Zealand’s Conference of IT.
[50]
Jyothi Salibindla.
Microservices api security.
International Journal of Engineering Research & Technology,
7(1):277–281, 2018.
[51]
Kai Jander, Lars Braubach, and Alexander Pokahr.
Practical defense-in-depth solution for microservice systems.
Journal of Ubiquitous Systems and Pervasive Networks, 11(1):17
– 25, 2019.
[52]
Kennedy A Torkura, Muhammad IH Sukmana, Feng Cheng, and Christoph Meinel.
Cavas: Neutralizing application and container security
vulnerabilities in the cloud native era.
In International Conference on Security and Privacy in
Communication Systems, pages 471–490. Springer, 2018.
[53]
Jiyu Chen, Heqing Huang, and Hao Chen.
Informer: Irregular traffic detection for containerized microservices
rpc in the real world.
In Proceedings of the 4th ACM/IEEE Symposium on Edge Computing,
SEC ’19, pages 389–394, New York, NY, USA, 2019. ACM.
[54]
Kennedy A. Torkura, Muhammad I.H. Sukmana, and Christoph Meinel.
Integrating continuous security assessments in microservices and
cloud native applications.
In Proceedings of the10th International Conference on Utility
and Cloud Computing, UCC ’17, pages 171–180, New York, NY, USA, 2017. ACM.
[55]
Sven Akkermans, Bruno Crispo, Wouter Joosen, and Danny Hughes.
Polyglot cerberos: Resource security, interoperability and
multi-tenancy for iot services on a multilingual platform.
In Proceedings of the 15th EAI International Conference on
Mobile and Ubiquitous Systems: Computing, Networking and Services,
MobiQuitous ’18, pages 59–68, New York, NY, USA, 2018. ACM.
[56]
Jethro G. Beekman and Donald E. Porter.
Challenges for scaling applications across enclaves.
In Proceedings of the 2Nd Workshop on System Software for
Trusted Execution, SysTEX’17, pages 8:1–8:2, New York, NY, USA, 2017. ACM.
[57]
Daniel Guija and Muhammad Shuaib Siddiqui.
Identity and access control for micro-services based 5g nfv
platforms.
In Proceedings of the 13th International Conference on
Availability, Reliability and Security, ARES 2018, pages 46:1–46:10, New
York, NY, USA, 2018. ACM.
[58]
Xing Li, Yan Chen, and Zhiqiang Lin.
Towards automated inter-service authorization for microservice
applications.
In Proceedings of the ACM SIGCOMM 2019 Conference Posters and
Demos, SIGCOMM Posters and Demos ’19, pages 3–5, New York, NY, USA, 2019.
ACM.
[59]
Gastón Márquez and Hernán Astudillo.
Identifying availability tactics to support security architectural
design of microservice-based systems.
In Proceedings of the 13th European Conference on Software
Architecture - Volume 2, ECSA ’19, pages 123–129, New York, NY, USA, 2019.
ACM.
[60]
Amjad Ibrahim, Stevica Bozhinoski, and Alexander Pretschner.
Attack graph generation for microservice architecture.
In Proceedings of the 34th ACM/SIGAPP Symposium on Applied
Computing, SAC ’19, pages 1235–1242, New York, NY, USA, 2019. ACM.
[61]
Dimitri Michel Stallenberg and Annibale Panichella.
Jcomix: A search-based tool to detect xml injection vulnerabilities
in web applications.
In Proceedings of the 2019 27th ACM Joint Meeting on European
Software Engineering Conference and Symposium on the Foundations of Software
Engineering, ESEC/FSE 2019, pages 1090–1094, New York, NY, USA, 2019. ACM.
[62]
Michel Krämer, Sven Frese, and Arjan Kuijper.
Implementing secure applications in smart city clouds using
microservices.
Future Generation Computer Systems, 99:308 – 320, 2019.
[63]
Kai Jander, Lars Braubach, and Alexander Pokahr.
Defense-in-depth and role authentication for microservice systems.
Procedia Computer Science, 130:456 – 463, 2018.
The 9th International Conference on Ambient Systems, Networks and
Technologies (ANT 2018) / The 8th International Conference on Sustainable
Energy Information Technology (SEIT-2018) / Affiliated Workshops.
[64]
Sarra Abidi, Mehrez Essafi, Chirine Ghedira Guegan, Myriam Fakhri, Hamad Witti,
and Henda Hjjami Ben Ghezala.
A web service security governance approach based on dedicated
micro-services.
Procedia Computer Science, 159:372 – 386, 2019.
Knowledge-Based and Intelligent Information & Engineering Systems:
Proceedings of the 23rd International Conference KES2019.
[65]
Marwa Elsayed and Mohammad Zulkernine.
Offering security diagnosis as a service for cloud saas applications.
Journal of Information Security and Applications, 44:32 – 48,
2019.
[66]
IBM.
An integrated approach to insider threat protection, 2016.
[67]
John Kindervag, Stephanie Balaouras, and Kelley Mak.
Build security into your network’s dna: The zero trust network
architecture.
Technical report, Forrester Research, November 2012.
[68]
Rui Zhuang, Scott A. DeLoach, and Xinming Ou.
Towards a theory of moving target defense.
In Proceedings of the First ACM Workshop on Moving Target
Defense, MTD ’14, pages 31–40, New York, NY, USA, 2014. Association for
Computing Machinery.
[69]
Vasilios Mavroudis, Andrea Cerulli, Petr Svenda, Dan Cvrcek, Dusan Klinec, and
George Danezis.
A touch of evil: High-assurance cryptographic hardware from untrusted
components.
In Proceedings of the 2017 ACM SIGSAC Conference on Computer and
Communications Security, CCS ’17, pages 1583–1600, New York, NY, USA, 2017.
Association for Computing Machinery.
[70]
Dirk Merkel.
Docker: lightweight linux containers for consistent development and
deployment.
Linux journal, 2014(239):2, 2014. |
Tensor products and $q$-characters of HL-modules
and monoidal categorifications
Matheus Brito and Vyjayanthi Chari
Departamento de Matemática, UFPR, Curitiba - PR - Brazil, 81530-015.
mbrito@ufpr.br
Department of Mathematics
University of California, Riverside
900 University Ave., Riverside, CA 92521
chari@math.ucr.edu
Abstract.
We study certain monoidal subcategories (introduced by David Hernandez and Bernard Leclerc) of finite–dimensional representations of a quantum affine algebra of type $A$. We
classify the set of prime representations in these subcategories and give necessary and sufficient conditions for a tensor product of two prime representations to be irreducible. In the case of a reducible tensor product we
describe the prime decomposition of the simple factors. As a consequence we prove that these subcategories are monoidal categorifications of a cluster algebra of type A with coefficients.
MB was partially
supported by CNPq, grant 205281/2014-1
VC was partially supported by DMS 1719357
Introduction
The study of the category $\cal F$ finite–dimensional representations of a quantum affine algebra goes back nearly thirty years and continues to be of significant interest. The irreducible objects in this category are indexed by elements of a free abelian monoid (denoted $\cal P^{+}$) with generators ${\mbox{\boldmath$\omega$}}_{i,a}$ where $i$ varies over the index set for the simple roots and $a$ varies over non–zero elements of the field of rational functions in a variable $q$. The category is not semisimple and there are many interesting indecomposable objects in it. In recent years, there has been new insight in the study of $\cal F$ coming from connections with cluster algebras through the work of [26], [28], [35] and also from KLR algebras through the work of [30, 31].
The category $\cal F$ is a monoidal tensor category and an interesting feature is that a tensor product of generic simple objects is simple. An obviously related notion is that of a prime simple object; this is one which cannot be written in a non-trivial way as a tensor product of objects of $\cal F$.
An open and very difficult question is the following: classify prime simple objects in $\cal F$ and describe the factorization of an arbitrary simple object as a tensor product of primes.
The answer to this question for $\mathfrak{sl}_{2}$ was given in [8] where it was also proved that the factorization was unique. In higher rank the question along with that of uniqueness remains unanswered. However, in [24] and[25] an important result was established which greatly simplifies the problem by reducing it to following: give a necessary and sufficient condition for the tensor product of a pair of prime simple objects to be simple.
In this paper we focus on this question for
certain subcategories of $\cal F$ associated with quantum affine $\mathfrak{sl}_{n+1}$.
These subcategories were introduced by David Hernandez and Bernard Leclerc ([26], [28]) and the definition has its roots in the theory of cluster algebras. The remarkable insight was that prime representations were analogous to cluster variables and the irreducibility of a tensor product of prime objects was analogous to the idea of two elements belonging to the same cluster. The role of the quiver in the theory of cluster algebras is played by the height function;
a height function (of type $A_{n}$) is a function $\xi:[1,n]\to\mathbb{Z}$ satisfying the condition $|\xi(i)-\xi(i+1)|=1$ for $1\leq i\leq n-1$. Define $\cal P^{+}_{\xi}$ to be the submonoid of $\cal P^{+}$ generated by elements ${\mbox{\boldmath$\omega$}}_{i,q^{\xi(i)\pm 1}}$ and
let $\cal F_{\xi}$ be the full subcategory of $\cal F$ consisting of objects whose Jordan–Holder constituents are indexed by elements of $\cal P^{+}_{\xi}$. It was proved in [28] that $\cal F_{\xi}$ is a monoidal tensor category and we let $\cal K_{0}(\cal F_{\xi})$ be the Grothendieck ring of $\cal F_{\xi}$. In the case when $\xi$ is the bipartite height function, i.e, $\xi(i-1)=\xi(i+1)$ for $2\leq i\leq n-2$ or the monotonic function $\xi(i)=i$ they showed that $\cal K_{0}(\cal F_{\xi})$ is isomorphic to a cluster algebra with coefficients of type $A$.
In this paper we prove the result for all height functions of type $A$ by representation theoretic methods. We define a subset $\mathbf{P}\mathbf{r}_{\xi}$ of $\cal P^{+}_{\xi}$ such that the corresponding irreducible representations (which we call HL-modules) are prime.
Working entirely in $\cal F_{\xi}$ we
show that the HL-modules are precisely all the prime objects in this category. To do this, we establish necessary and sufficient conditions for a tensor product of HL-modules to be irreducible. In the case when the tensor product is reducible we describe the Jordan-Holder constituents and their factorization as a tensor product of HL-modules.
The connection with cluster algebras is then made as follows. We define a the quiver $Q_{\xi}$ associated with $\xi$; since we are working in the general case the quiver we use is a mutation of the quivers in [26] and [28]. This mutation allows us to map a non–frozen variable in the initial seed of the cluster algebra to the class of the irreducible module corresponding to either ${\mbox{\boldmath$\omega$}}_{i,\xi(i)+1}$ or ${\mbox{\boldmath$\omega$}}_{i,\xi(i)-1}$. The first mutation at any element of the initial seed is easily described; however is not necessarily of the form ${\mbox{\boldmath$\omega$}}_{i,\xi(i)\pm 1}$. Our tensor product formulae now allow us to prove the existence of an algebra
isomorphism between the cluster algebra with $n$ frozen variables and $\cal K_{0}(\cal F_{\xi})$. The isomorphism maps a cluster variable to an HL-module and we identify this module explicitly. We also show that the isomorphism maps cluster monomials to simple tensor products of HL-modules.
As a consequence of this result we give an alternate proof for the product of a pair of cluster variables to be a cluster monomial; equivalently we give an alternate proof of the criterion for a pair of roots to be compatible.
In Proposition 2.5 we give a closed formula for a cluster variable in terms of the original seed. In terms of representation theory this can be interpreted as giving a $q$-character formula for the prime representations in $\cal F_{\xi}$. It is useful to remark here that other explicit formulae for cluster variables can be found in the literature see for instance, [1], [5], [13], [14]. Not all these papers deal with frozen variables and even those that do impose conditions on the frozen variables which are not satisfied by the quivers considered in this paper. The role of the frozen variable in the connection with representation theory is important and motivates our formulae.
The paper is organized as follows. In Section 1 we recall the definition of the height function $\xi$ and introduce the associated quiver $Q_{\xi}$. We then
state and prove our main result modulo the key Propositions 1.5, 1.6 and 1.7. In Section 2 we prove Proposition 1.5 which gives a recursive formula for a cluster variable. This is done by a simple analysis of the quiver obtained by mutating at successive nodes. The answer we obtain is in a form which is well adapted to the representation theory of quantum affine algebras and can be viewed as an analog of Pieri’s rule in classical representation theory.
We then solve the recursion to give a closed formula for the cluster variable in terms of the initial cluster which includes the frozen variables. In Sections 3, 4 and 5 we provide sufficient and necessary conditions, for the tensor product of two HL-modules to be irreducible. We also analyze the Jordan-Holder series of a reducible tensor product of HL-modules. The proof of Propositions 1.6 and 1.7 can be found in Section 4.
Acknowledgements. MB is grateful to the Department of Mathematics, UCR, for
their hospitality during a visit when part of this research was carried out. He also thanks David Hernandez for supporting his visit to Paris 7 and many helpful discussions. VC thanks David Hernandez, Bernard Leclerc and Salvatore Stella for helpful conversations.
1. The main results
Throughout the paper we denote by $\mathbb{C}$, $\mathbb{Z}$, $\mathbb{Z}_{+}$ and $\mathbb{N}$ the set of complex numbers, integers, non–negative and positive integers respectively. For $i,j\in\mathbb{Z}_{+}$ with $i\leq j$ we let $[i,j]=\{i,i+1,\cdots,j\}$.
Given a commutative ring $A$ we denote by $A[q]$ (resp. $A(q)$) the ring (resp. quotient field) of polynomials in an indeterminate $q$ with coefficients in $A$.
1.1. The cluster algebra $\cal A(\mathbf{x},Q_{\xi})$
Let $\xi:[1,n]\to\mathbb{Z}$ be a height function; namely a function which satisfies the conditions
$$\ |\xi(i)-\xi(i-1)|=1,\ \ 2\leq i\leq n.$$
It will be convenient to extend $\xi$ to $[0,n+1]$ by setting $\xi(0)=\xi(2)$ and $\xi(n-1)=\xi(n+1)$.
Remark.
Although trivial, it is useful to note that $\{\xi(i+1),\xi(i-1)\}\subset\{\xi(i)+1,\xi(i)-1\}$ and that the inclusion can be strict.
For $i\in[1,n-2]$, let $i_{\diamond}\in[i,n]$ be minimal such that $\xi(i_{\diamond})=\xi(i_{\diamond}+2)$ and set $(n-1)_{\diamond}=(n-1)$ and $n_{\diamond}=n$. Let
$Q_{\xi}$ be a quiver with $2n$ vertices labeled $\{1,\cdots,n,1^{\prime},\cdots,n^{\prime}\}$ and with the set of edges given as follows:
•
there are no edges between the primed vertices; in other words the vertices $\{1^{\prime},\cdots,n^{\prime}\}$ are frozen,
•
if $1\leq j\leq n-1$ and $\xi(j)=\xi(j+1)+1$, the edges at $j$ are:
$$\xymatrix{&&(j-1)&j\ar[l]\ar[rrd]_{1-\delta_{j,j_{\diamond}}}\ar@/^{1}pc/[rr]^%
{\delta_{j,j_{\diamond}}}&&(j+1)\ar@/^{1}pc/[ll]_{1-\delta_{j,j_{\diamond}}}\\
&&&j^{\prime}\ar[u]&&(j+1)^{\prime}&}\\
$$
and the reverse orientations if $\xi(j)=\xi(j+1)-1$, where $\delta_{j,j_{\diamond}}$ is the Kronecker delta function and we adopt the convention that a labeled edge exists iff the label is one,
•
at the vertex $n$ we have edges $(n-1)\to n\to n^{\prime}$ if $\xi(n-1)=\xi(n)+1$ and the reverse orientation otherwise.
Clearly $j$ is a sink or source of $Q_{\xi}$ (where we ignore the frozen vertices) iff $j=1$ or $j=j_{\diamond}$. Given $2\leq j\leq n$ let $2_{\bullet}=1$ and for $j>2$ let $j_{\bullet}$ be the maximal sink or source of $Q_{\xi}$ satisfying $j_{\bullet}<j$.
Fix a set $\mathbf{x}=\{x_{1},\cdots,x_{n},f_{1},\cdots,f_{n}\}$ of algebraically independent variables and let
$\cal A(\mathbf{x},Q_{\xi})$
be the cluster algebra (with coefficients) with initial seed $(\mathbf{x},Q_{\xi})$.
The definition of a cluster algebra is recalled briefly in Section 2.1; for the rest of this section we shall freely use the language of cluster algebras. Since the principal unfrozen part of $Q_{\xi}$ is a quiver of type $A_{n}$, the set of non-frozen cluster variables in $\cal A(\mathbf{x},Q_{\xi})$ are indexed by the set $\Phi_{\geq-1}$
of almost positive roots of a root system of type $A_{n}$. In other words if we let $\{\alpha_{i}:1\leq i\leq n\}$, be a set of simple roots for $A_{n}$ and set $\alpha_{i,j}=\alpha_{i}+\cdots+\alpha_{j}$, $1\leq i\leq j\leq n$,
then
$$\Phi_{\geq-1}=\{-\alpha_{i},\ \alpha_{i,j}:1\leq i\leq j\leq n\},$$
and the cluster variables are denoted
$$\{x_{i}:=x[-\alpha_{i}],\ x[\alpha_{i,j}],\ f_{i}:\ 1\leq i\leq j\leq n\}.$$
1.2. The category $\cal F_{\xi}$
Let $\widehat{\mathbf{U}}_{q}$ be the quantum loop algebra over $\mathbb{C}(q)$ associated to $\mathfrak{sl}_{n+1}$ and let $\cal F$ be the monoidal tensor category whose objects are finite–dimensional representations of $\widehat{\mathbf{U}}_{q}$.
Given a height function $\xi:[1,n]\to\mathbb{Z}$ we take
$\cal P^{+}_{\xi}$ to be the free abelian monoid with generators $\{{\mbox{\boldmath$\omega$}}_{i,\xi(i)\pm 1}:i\in[1,n]\}.$
It is known that $\cal P^{+}_{\xi}$ is the index set for a (sub)-family of isomorphism classes of irreducible objects of $\cal F$. We define $\cal F_{\xi}$ to be the full subcategory of $\cal F$ consisting of objects all of whose Jordan–Holder constituents are indexed by elements of $\cal P^{+}_{\xi}$. It was proved in [28] that $\cal F_{\xi}$ is a monodial category and we let $\cal K_{0}(\cal F_{\xi})$ be the corresponding Grothendieck ring. For ${\mbox{\boldmath$\omega$}}\in\cal P^{+}_{\xi}$ let $[{\mbox{\boldmath$\omega$}}]\in\cal K_{0}(\cal F_{\xi})$ be the isomorphism class of the corresponding object in $\cal F_{\xi}$.
Remark.
It is important to keep in mind that the assignment ${\mbox{\boldmath$\omega$}}\to[{\mbox{\boldmath$\omega$}}]$ is not a morphism of monoids $\cal P^{+}_{\xi}\to\cal K_{0}(\cal F_{\xi})$, i.e., $[{\mbox{\boldmath$\omega$}}][{\mbox{\boldmath$\omega$}}^{\prime}]$ is not always equal to $[{\mbox{\boldmath$\omega$}}{\mbox{\boldmath$\omega$}}^{\prime}]$. One of the goals of this paper is to determine a necessary and sufficient condition for equality to hold.
For $i\in[1,n]$ set
$$\mathbf{f}_{i}={\mbox{\boldmath$\omega$}}_{i,\xi(i)+1}{\mbox{\boldmath$\omega$%
}}_{i,\xi(i)-1}.$$
If $1\leq i<j\leq n$,
let $i_{2}<\cdots<i_{k-1}$ be an ordered enumeration of the subset $\{p:i<p<j,\ \xi(p-1)=\xi(p+1)\}$, $i_{1}=i$, $i_{k}=j$ and
define an element ${\mbox{\boldmath$\omega$}}(i,j)\in\cal P^{+}_{\xi}$ by:
$$\displaystyle{\mbox{\boldmath$\omega$}}_{\xi}(i,j)={\mbox{\boldmath$\omega$}}_%
{i_{1},a_{1}}\cdots{\mbox{\boldmath$\omega$}}_{i_{k},a_{k}},$$
where $a_{1}=\xi(i)\pm 1$ if $\xi(i+1)=\xi(i)\mp 1$ and $a_{m}=\xi(i_{m})\pm 1\ \ {\rm{if}}\ \ \xi(i_{m})=\xi(i_{m}-1)\pm 1$, for $m\geq 2$.
Set
$$\mathbf{P}\mathbf{r}_{\xi}=\{{\mbox{\boldmath$\omega$}}_{i,\xi(i)\pm 1},\ {%
\mbox{\boldmath$\omega$}}(i,j):1\leq i\leq j\leq n,\ i\neq j\}.$$
Clearly the set $\mathbf{P}\mathbf{r}_{\xi}$ has the same cardinality as the set of unfrozen cluster variables in $\cal A(\mathbf{x},Q_{\xi})$.
Recall that an object of $\cal F$ is said to be prime if it cannot be written in a nontrivial way as a tensor product of objects of $\cal F$. The following is a special case of the main result of [7].
Lemma.
The irreducible object of $\cal F_{\xi}$ associated to an element ${\mbox{\boldmath$\omega$}}\in\mathbf{P}\mathbf{r}_{\xi}\cup\{\mathbf{f}_{i}:1%
\leq i\leq n\}$ is prime. ∎
1.3. Main Theorem
Recall that by definition $n=n_{\diamond}$ and $(n-1)=(n-1)_{\diamond}$ which in particular implies that $n_{\bullet}=n-1$. For $k\geq 2$, set
$$\bar{k}=(k+1)(1-\delta_{k,k_{\diamond}})+(k_{\bullet}+1)\delta_{k,k_{\diamond}}.$$
(1.1)
Theorem 1.
Let $\xi:[1,n]\to\mathbb{Z}$ be a height function. The assignment
$$\iota(x_{i})=[{\mbox{\boldmath$\omega$}}_{i,\xi(i+1)}],\ \ \ \iota(f_{i})=[%
\mathbf{f}_{i}],$$
extends to an isomorphism of
rings
$\iota:\cal A(\mathbf{x},Q_{\xi})\to\cal K_{0}(\cal F_{\xi})$ such that
for $1\leq i\leq k\leq n$,
$$\displaystyle\iota(x[\alpha_{i,i_{\diamond}}])$$
$$\displaystyle=[{\mbox{\boldmath$\omega$}}_{i,\xi(i+1)\pm 2}],\ \ \ \ \xi(i)=%
\xi(i+1)\pm 1$$
$$\displaystyle\iota(x[\alpha_{i,k}])$$
$$\displaystyle=[{\mbox{\boldmath$\omega$}}(i,\bar{k})],\ \ k\neq i_{\diamond},$$
$$\displaystyle\iota(f_{p}x[\alpha])$$
$$\displaystyle=[\mathbf{f}_{p}{\mbox{\boldmath$\omega$}}]\ \ p\in[1,n],\ \ %
\alpha\in\Phi_{\geq-1},\ \ \ [{\mbox{\boldmath$\omega$}}]=\iota(x[\alpha]).$$
(1.4)
In particular $\iota$ maps cluster variable to a prime object of $\cal F_{\xi}$. Moreover,
if $\beta_{1},\beta_{2}\in\Phi_{\geq-1}$ are such that $x[\beta_{1}]x[\beta_{2}]$ is a cluster monomial then
$$[{\mbox{\boldmath$\omega$}}_{1}{\mbox{\boldmath$\omega$}}_{2}]=[{\mbox{%
\boldmath$\omega$}}_{1}][{\mbox{\boldmath$\omega$}}_{2}],\ \ [{\mbox{\boldmath%
$\omega$}}_{s}]=\iota(x[\beta_{s}]),\ \ s=1,2.$$
Corollary.
The homomorphism $\iota$ sends a cluster monomial to the equivalence class of an irreducible object of $\cal F_{\xi}$. In particular, $\cal F_{\xi}$ is a monoidal categorification of $\cal A(\mathbf{x},\xi)$.
Proof of Corollary. Let $x[\beta_{1}]\cdots x[\beta_{r}]$ be a cluster monomial for some $\beta_{1},\cdots,\beta_{r}\in\Phi_{\geq-1}$ and set $[{\mbox{\boldmath$\omega$}}_{i}]=\iota(x[\beta_{i}])$ for $1\leq i\leq r$. Then the pairs $x[\beta_{j}]x[\beta_{p}]$, $1\leq j\neq p\leq r$ are cluster monomials and hence using the Theorem 1 we have $\iota(x[\beta_{j}]x[\beta_{p}])=[{\mbox{\boldmath$\omega$}}_{j}{\mbox{%
\boldmath$\omega$}}_{p}]$ for $1\leq j\neq p\leq r$.
It follows from the main result of [22] (see section 3 of this paper for the statement) that $\iota(x[\beta_{1}]\cdots x[\beta_{r}])=[{\mbox{\boldmath$\omega$}}_{1}\cdots{%
\mbox{\boldmath$\omega$}}_{r}]$ and the corollary is established. ∎
Remark.
Suppose that $\xi$ satisfies $\xi(i-1)=\xi(i+1)$ for all $1\leq i\leq n$ or that $\xi(j)=\xi(i)+(j-i)$ for all $1\leq i\leq j\leq n$. In these two cases the existence of $\iota$ was established in [26],[28] by very different methods. As was noted in [28] the categories $\cal F_{\xi}$ are not necessarily equivalent for different height functions.
1.4.
In Theorem 3 of this paper we give conditions for the equality $[{\mbox{\boldmath$\pi$}}{\mbox{\boldmath$\pi$}}^{\prime}]=[{\mbox{\boldmath$%
\pi$}}][{\mbox{\boldmath$\pi$}}^{\prime}]$ when ${\mbox{\boldmath$\pi$}},{\mbox{\boldmath$\pi$}}^{\prime}\in\mathbf{P}\mathbf{r%
}_{\xi}$ to hold in $\cal K_{0}(\cal F_{\xi})$. The translation to the language of cluster algebra gives the conditions for describing when two roots are compatible. Thus our theorem gives a proof of the following assertion (compare with the description in Section 10.2.3 of [28] where a similar description in the case of the bipartite height function).
Assume that $i\leq j$, $k\leq\ell$ and $i\leq k$.
If $j\neq i_{\diamond}$ the roots $\alpha_{i,j}$, $\alpha_{k,\ell}$ are compatible iff:
$\bullet$ $k=i\ \ {\rm or}\ k>j+1$,
$\bullet$ $j=j_{\diamond}$ and $j_{\bullet}+1\leq k\leq j,$
$\bullet$ $\ell\neq k_{\diamond}$ and either $\bar{j}=\bar{\ell}$ or
$$\displaystyle i<k<\bar{j}<\bar{\ell},\ \ {\rm{and}}\ \#\{k\leq m<\bar{j}-1:m=m%
_{\diamond}\}\in 2\mathbb{Z}_{+}+1,\ {\rm{or}}$$
$$\displaystyle\ i<k<\bar{\ell}<\bar{j},\ \ {\rm and}\ \#\{k\leq m<\bar{\ell}-1:%
m=m_{\diamond}\}\in 2\mathbb{Z}_{+}.$$
The roots $\alpha_{i,i_{\diamond}}$ and $\alpha_{k,\ell}$ with $i\leq k$ are compatible iff :
$$\displaystyle\ \ k_{\bullet}\neq k-1,\ \ {\rm{or}}\ \ (k-1)_{\bullet}\geq i\ {%
\rm{or}}\ \ \ell\neq k_{\diamond}\ \ {\rm{and}}\ \ i=k.$$
The roots $-\alpha_{i}$ and $\alpha_{k,\ell}$ are in the same cluster iff either $k>i$ or $\ell<i$.
In Theorem 4 we write down the Jordan-Holder series for a reducible tensor product of objects. This amounts to writing down all the non-trivial exchange relations for cluster variables including the frozen variables and is not hard to do using the analysis above.
1.5.
The proof of the theorem involves three principal steps. For $1\leq j\leq n$, set
$$d_{j}=\delta_{j,j_{\diamond}}=\delta_{\xi(j),\xi(j+2)}.$$
The first step is the following proposition which gives a recursive formula for the cluster variables.
We adopt the convention that $\alpha_{i,m}=\alpha_{m},\ m\leq i$.
Proposition.
For $1\leq i<j\leq n$ we have
$$x_{i}x[\alpha_{i}]=f_{i}x_{i+1}^{1-d_{i}}+f_{i+1}^{1-d_{i}}x_{i-1}x_{i+1}^{d_{%
i}},$$
$$\displaystyle x_{j}x[\alpha_{i,j}]$$
$$\displaystyle=f_{j}^{d_{j-1}}x[\alpha_{i,j-1}]x_{j+1}^{1-d_{j}}+$$
$$\displaystyle+f_{j+1}^{1-d_{j}}x_{j+1}^{d_{j}}\left((\delta_{i_{\bullet},j_{%
\bullet}}+\delta_{i,j_{\bullet}})f_{i}^{\delta_{i,j_{\bullet}}}x_{i-1}^{1-%
\delta_{i,j_{\bullet}}}+(1-\delta_{i_{\bullet},j_{\bullet}}-\delta_{i,j_{%
\bullet}})f_{j_{\bullet}}^{d_{j_{\bullet}-1}}x[\alpha_{i,j_{\bullet}-1}]\right).$$
The
proof of this proposition is in Section 2 where we also give a closed formula for $x[\alpha_{i,j}]$ as a Laurent polynomial in the variables $\{x_{1},\cdots,x_{n},f_{1},\cdots,f_{n}\}$.
1.6.
The second step in the proof of the theorem is the following. We adopt the convention that we take ${\mbox{\boldmath$\omega$}}_{i,\xi(i+1)+2}$ if $\xi(i)=\xi(i+1)+1$ and we take ${\mbox{\boldmath$\omega$}}_{i,\xi(i+1)-2}$ if $\xi(i)=\xi(i+1)-1$.
Proposition.
The following equalities hold in $\cal K_{0}(\cal F_{\xi})$ for $1\leq i\leq j\leq n$.
(i)
We have
$$[{\mbox{\boldmath$\omega$}}_{i,\xi(i+1)}][{\mbox{\boldmath$\omega$}}(i,i+1)]^{%
1-d_{i}}[{\mbox{\boldmath$\omega$}}_{i,\xi(i+1)\pm 2}]^{d_{i}}=[\mathbf{f}_{i}%
]\ [{\mbox{\boldmath$\omega$}}_{i+1,\xi(i+2)}]^{1-d_{i}}+[\mathbf{f}_{i+1}]^{1%
-d_{i}}[{\mbox{\boldmath$\omega$}}_{i+1,\xi(i+2)}]^{d_{i}}\ [{\mbox{\boldmath$%
\omega$}}_{i-1,\xi(i)}]\ .$$
(ii)
If $j_{\bullet}\leq i<j$ then
$$\displaystyle[{\mbox{\boldmath$\omega$}}_{j,\xi(j+1)}][{\mbox{\boldmath$\omega%
$}}(i,\bar{j})]^{1-\delta_{j,i_{\diamond}}}[{\mbox{\boldmath$\omega$}}_{i,\xi(%
i+1)\pm 2}]^{\delta_{j,i_{\diamond}}}$$
$$\displaystyle=[\mathbf{f}_{j}]^{d_{j-1}}\ [{\mbox{\boldmath$\omega$}}(i,j)]^{1%
-d_{j-1}}[{\mbox{\boldmath$\omega$}}_{i,\xi(i+1)\pm 2}]^{d_{j-1}}[{\mbox{%
\boldmath$\omega$}}_{j+1,\xi(j+2)}]^{1-d_{j}}$$
$$\displaystyle+[\mathbf{f}_{j+1}]^{1-d_{j}}[\mathbf{f}_{i}]^{\delta_{i,j_{%
\bullet}}}[{\mbox{\boldmath$\omega$}}_{j+1,\xi(j+2)}]^{d_{j}}[{\mbox{\boldmath%
$\omega$}}_{i-1,\xi(i)}]^{1-\delta_{i,j_{\bullet}}}$$
(iii)
If $i<j_{\bullet}$ choose $z\in\{\xi(i)+1,\xi(i)-1\}$ so that ${\mbox{\boldmath$\omega$}}_{i,z}^{-1}{\mbox{\boldmath$\omega$}}(i,j_{\bullet})%
\in\cal P^{+}_{\xi}$ and set $k=(j_{\bullet})_{\bullet}$. Then,
$$\displaystyle[{\mbox{\boldmath$\omega$}}_{j,\xi(j+1)}][{\mbox{\boldmath$\omega%
$}}(i,\bar{j})]=[\mathbf{f}_{j}]^{d_{j-1}}\ [{\mbox{\boldmath$\omega$}}_{j+1,%
\xi(j+2)}]^{1-d_{j}}\ [{\mbox{\boldmath$\omega$}}(i,\overline{j-1})]^{1-\delta%
_{(j-1)_{\bullet},i_{\bullet}}}\ [{\mbox{\boldmath$\omega$}}_{i,z}]^{\delta_{(%
j-1)_{\bullet},i_{\bullet}}}$$
$$\displaystyle+[\mathbf{f}_{j+1}]^{1-d_{j}}\ [\mathbf{f}_{j_{\bullet}}]^{d_{j_{%
\bullet}-1}}[{\mbox{\boldmath$\omega$}}_{j+1,\xi(j+2)}]^{d_{j}}\ [{\mbox{%
\boldmath$\omega$}}(i,\overline{j_{\bullet}-1})]^{1-\delta_{i_{\bullet},k_{%
\bullet}}d_{j_{\bullet}-1}}\ [{\mbox{\boldmath$\omega$}}_{i,z}]^{\delta_{i_{%
\bullet},k_{\bullet}}d_{j_{\bullet}-1}}\ .$$
The proof of this proposition can be found in Section 4.
1.7.
Proposition 1.5 and Proposition 1.6 are enough to establish the existence of $\iota$ and to identify the image of a cluster variable.
The third step needed to establish the theorem
is to show that $\iota$ maps a cluster monomial to the isomorphism class of an irreducible representation. To do this we will need the following result.
Proposition.
Let ${\mbox{\boldmath$\omega$}},{\mbox{\boldmath$\omega$}}^{\prime}\in\mathbf{P}%
\mathbf{r}_{\xi}$. Then either $[{\mbox{\boldmath$\omega$}}][{\mbox{\boldmath$\omega$}}^{\prime}]=[{\mbox{%
\boldmath$\omega$}}{\mbox{\boldmath$\omega$}}^{\prime}]$ or $[{\mbox{\boldmath$\omega$}}][{\mbox{\boldmath$\omega$}}^{\prime}]=[{\mbox{%
\boldmath$\omega$}}_{1}]+[{\mbox{\boldmath$\omega$}}_{2}]$ where $[{\mbox{\boldmath$\omega$}}_{1}]$ and $[{\mbox{\boldmath$\omega$}}_{2}]$ are the images under $\iota$ of cluster monomials.
A much more precise statement can be found in Theorem 3 and Theorem 4 in Sections 3 and 4.
In the rest of this section we assume Proposition 1.5, Proposition 1.6, Proposition 1.7 and prove Theorem 1.
1.8. Existence of $\iota$
Recall [4] that an element of $\cal A(\mathbf{x},Q_{\xi})$ is said to be a standard monomial if it is a monomial in the elements $\{x_{i},x[\alpha_{i}]:i\in[1,n]\}$ and does not involve any product of the form $x_{i}x[\alpha_{i}]$, $i\in[1,n]$. It was
proved in [4] that standard
monomials are a $\mathbb{Z}[f_{i}:i\in I]$–basis of $\cal A(\mathbf{x},\xi)$.
On the other hand consider the quotient of the polynomial ring (with integer coefficients) in variables $X_{i},X[\alpha_{i}],F_{i}$, $i\in[1,n]$ subject to the first relation in Proposition 1.5. It is not hard to show that this ring is the $\mathbb{Z}[F_{i}:i\in I]$ span of
monomials in $X_{i},X[\alpha_{i}]$, $i\in[1,n]$
which do not involve products of $X_{i}X[\alpha_{i}]$ for any $i\in[1,n]$.
It follows that
$\cal A(\mathbf{x},Q_{\xi})$ is is isomorphic to this quotient
(compare with [26, Lemma 4.4]).
Using Proposition 1.6 (i) we have
$$\displaystyle[{\mbox{\boldmath$\omega$}}_{i,\ \xi(i+1)}][{\mbox{\boldmath$%
\omega$}}(i,i+1)]^{1-d_{i}}[{\mbox{\boldmath$\omega$}}_{i,\xi(i+1)\pm 2}]^{d_{%
i}}=[\mathbf{f}_{i}][{\mbox{\boldmath$\omega$}}_{i+1,\xi(i+2)}]^{1-d_{i}}+[%
\mathbf{f}_{i+1}]^{1-d_{i}}[{\mbox{\boldmath$\omega$}}_{i+1,\xi(i+2)}]^{d_{i}}%
[{\mbox{\boldmath$\omega$}}_{i-1,\xi(i)}]\ .$$
It is now immediate that the assignment
$$\displaystyle x_{i}\to[{\mbox{\boldmath$\omega$}}_{i,\xi(i+1)}],\ \ \ f_{i}\to%
\mathbf{f}_{i},\ \ \ x[\alpha_{i}]\to[{\mbox{\boldmath$\omega$}}_{i,\xi(i+1)%
\pm 2}]^{\delta_{i,i_{\diamond}}}[{\mbox{\boldmath$\omega$}}(i,i+1)]^{1-\delta%
_{i,i_{\diamond}}}$$
defines a homomorphism
of rings
$\iota:\cal A(\mathbf{x},Q_{\xi})\to\cal K_{0}(\cal F_{\xi})$.
1.9. The elements $\iota(x[\alpha])$, $\alpha\in\Phi_{\geq-1}$
The formulae given in (1) and (1) can be rewritten as follows:
$$\displaystyle\iota(x[\alpha_{i,j}])=[{\mbox{\boldmath$\omega$}}(i,j+1)]^{1-d_{%
j}}\ [{\mbox{\boldmath$\omega$}}_{i,\xi(i+1)\pm 2}]^{\delta_{i_{\bullet},j_{%
\bullet}}d_{j}}\ [{\mbox{\boldmath$\omega$}}(i,j_{\bullet}+1)]^{d_{j}(1-\delta%
_{i_{\bullet},j_{\bullet}})},\ \ j\geq i.$$
(1.5)
We shall prove this reformulation by induction on $j-i$.
Observe that induction begins when $j=i$ by definition.
For the inductive step apply $\iota$ to both sides of the second equation in Proposition 1.5. We will show that the right hand side of this equation is the same as the right hand side of the equation in Proposition 1.6(ii), (iii). Hence the left hand sides must match up. The inductive step is immediate once we observe that $\cal K_{0}(\cal F_{\xi})$ has no zero divisors.
To prove that the right hand sides are the same, suppose first that $j_{\bullet}\leq i$ (in particular $j_{\bullet}=i_{\bullet}$ or $j_{\bullet}=i$). Applying $\iota$ to both sides of the second equation in Proposition 1.5 gives
$$\displaystyle[{\mbox{\boldmath$\omega$}}_{j,\xi(j+1)}]\iota(x[\alpha_{i,j}])$$
$$\displaystyle=\mathbf{f}_{j}^{d_{j-1}}\iota(x[\alpha_{i,j-1}])[{\mbox{%
\boldmath$\omega$}}_{j+1,\xi(j+2)}]^{1-d_{j}}$$
$$\displaystyle+\ \mathbf{f}_{j+1}^{1-d_{j}}\mathbf{f}_{i}^{\delta_{i,j_{\bullet%
}}}[{\mbox{\boldmath$\omega$}}_{j+1,\xi(j+2)}]^{d_{j}}[{\mbox{\boldmath$\omega%
$}}_{i-1,\xi(i)}]^{1-\delta_{i,j_{\bullet}}}.$$
The second term on the right hand side of the preceding equation is equal to the the second term on the right hand side of the equation in Proposition 1.6(ii). To see that the first terms match up we use the inductive hypothesis for $\iota(x[\alpha_{i,j-1}])$ and see that it suffices to prove that,
$$\displaystyle[{\mbox{\boldmath$\omega$}}_{i,\xi(i+1)\pm 2}]^{d_{j-1}}=\left([{%
\mbox{\boldmath$\omega$}}_{i,\xi(i+1)\pm 2}]^{\delta_{i_{\bullet},(j-1)_{%
\bullet}}}[{\mbox{\boldmath$\omega$}}(i,(j-1)_{\bullet}+1)]^{1-\delta_{i_{%
\bullet},(j-1)_{\bullet}}}\right)^{d_{j-1}}.$$
If $d_{j-1}=0$, then the preceding equality is obviously true. Since
$$d_{j-1}=1\implies(j-1)=(j-1)_{\diamond}=j_{\bullet}=i\implies i_{\bullet}=(j-1%
)_{\bullet}$$
and the equality follows.
If $i<j_{\bullet}$, then the result follows if we prove that,
$$\displaystyle\iota(x[\alpha_{i,j-1}])=[{\mbox{\boldmath$\omega$}}(i,j)]^{1-d_{%
j-1}}\left([{\mbox{\boldmath$\omega$}}_{i,z}]^{\delta_{k,i_{\bullet}}}[{\mbox{%
\boldmath$\omega$}}(i,k+1)]^{1-\delta_{k,i_{\bullet}}}\right)^{d_{j-1}},$$
$$\displaystyle\iota(x[\alpha_{i,j_{\bullet}-1}])=[{\mbox{\boldmath$\omega$}}(i,%
j_{\bullet})]^{1-d_{j_{\bullet}-1}}\left([{\mbox{\boldmath$\omega$}}_{i,z}]^{%
\delta_{i_{\bullet},k_{\bullet}}}[{\mbox{\boldmath$\omega$}}(i,k_{\bullet}+1)]%
^{1-\delta_{i_{\bullet},k_{\bullet}}}\right)^{d_{j_{\bullet}-1}}.$$
where we recall that $k=(j_{\bullet})_{\bullet}$.
If $d_{j-1}=0$ the first equality follows from the definition and the inductive hypothesis and if $d_{j-1}=1$ then $(j-1)=j_{\bullet}$ and so $(j-1)_{\bullet}=k$. The first equality again follows from the inductive hypothesis. The second equality is deduced in the same way from the inductive hypothesis.
1.10.
We prove now that $\iota$ is an isomorphism. Let $\{\omega_{1},\cdots,\omega_{n}\}$ which are dual to the simple roots of $A_{n}$ and $P^{+}$ be their $\mathbb{Z}_{+}$–span. It is convenient to set $\omega_{0}=\omega_{n+1}=0$. Let $\leq$ be the usual partial order on $P^{+}$ given by $\mu\leq\lambda$ iff $\lambda-\mu$ is in the $\mathbb{Z}_{+}$–span of $\{\alpha_{1},\cdots,\alpha_{n}\}$.
Define a morphism of monoids $\operatorname{wt}:\cal P^{+}_{\xi}\to P^{+}$ by setting $\operatorname{wt}{\mbox{\boldmath$\omega$}}_{i,a}=\omega_{i}$. Since $\cal F_{\xi}$ is a tensor category it is well–known that the following holds in $\cal K_{0}(\cal F_{\xi})$; for ${\mbox{\boldmath$\omega$}}={\mbox{\boldmath$\omega$}}_{i_{1},a_{1}}\cdots{%
\mbox{\boldmath$\omega$}}_{i_{k},a_{k}}\in\cal P^{+}_{\xi}$:
$$[{\mbox{\boldmath$\omega$}}_{i_{1},a_{1}}]\cdots[{\mbox{\boldmath$\omega$}}_{i%
_{k},a_{k}}]=[{\mbox{\boldmath$\omega$}}]+\sum_{\stackrel{{\scriptstyle{\mbox{%
\boldmath\scriptsize$\pi$}}\in\cal P^{+}_{\xi}}}{{\operatorname{wt}{\mbox{%
\boldmath\scriptsize$\pi$}}<\operatorname{wt}{\mbox{\boldmath\scriptsize$%
\omega$}}}}}r({\mbox{\boldmath$\omega$}},{\mbox{\boldmath$\pi$}})[{\mbox{%
\boldmath$\pi$}}],\ \ {\rm{for\ some}}\ \ r({\mbox{\boldmath$\omega$}},{\mbox{%
\boldmath$\pi$}})\in\mathbb{Z}_{+}.$$
(1.6)
A straightforward induction on $\operatorname{wt}{\mbox{\boldmath$\omega$}}$ shows that
$\cal K_{0}(\cal F_{\xi})$ is generated as a ring by the elements $[{\mbox{\boldmath$\omega$}}_{i,\xi(i)\pm 1}]$.
By Section 1.9 we see that $\iota(\{x[-\alpha_{i}],x[\alpha_{i,i_{\diamond}}]\})=\{[{\mbox{\boldmath$%
\omega$}}_{i,\xi(i)+1}],[{\mbox{\boldmath$\omega$}}_{i,\xi(i)-1}]\}$
and hence it follows that
$\iota$ is surjective.
We prove that $\iota$ is injective. Set
$$\operatorname{wt}_{\ell}x_{i}={\mbox{\boldmath$\omega$}}_{i,\xi(i+1)},\ \ %
\operatorname{wt}_{\ell}f_{i}=\mathbf{f}_{i},\ \operatorname{wt}_{\ell}x[%
\alpha_{i}]={\mbox{\boldmath$\pi$}},\ \ {\rm such\ that}\ \ \iota(x[\alpha_{i}%
])=[{\mbox{\boldmath$\pi$}}].$$
Extend $\operatorname{wt}_{\ell}$ in the obvious way to the basis of $\cal A(\mathbf{x},\xi)$; if $\mathbf{m}=x_{1}^{p_{1}}\cdots x_{n}^{p_{n}}x[\alpha_{1}]^{m_{1}}\cdots x[%
\alpha_{n}]^{m_{n}}$ is a standard monomial in $\cal A(\mathbf{x},\xi)$ and $\mathbf{f}=f_{1}^{r_{1}}\cdots f_{n}^{r_{n}}\in\mathbb{Z}[f_{1}^{\pm 1},\cdots%
,f_{n}^{\pm 1}]$ then
$$\displaystyle\operatorname{wt}_{\ell}\mathbf{f}\mathbf{m}=\prod_{i=1}^{n}%
\mathbf{f}_{i}^{r_{i}}{\mbox{\boldmath$\omega$}}_{i,\ \xi(i+1)}^{p_{i}}\left({%
\mbox{\boldmath$\omega$}}_{i,\ \xi(i+1)+2}^{\delta_{\xi(i),\xi(i+1)+1}}{\mbox{%
\boldmath$\omega$}}_{i,\xi(i+1)-2}^{\delta_{\xi(i),\xi(i+1)-1}}{\mbox{%
\boldmath$\omega$}}_{i+1,\ \xi(i+2)}^{1-d_{i}}\right)^{m_{i}}.$$
Lemma.
Let $\mathbf{m},\mathbf{m}^{\prime}$ be standard monomials in $\cal A(\mathbf{x},Q_{\xi})$ and $\mathbf{f},\mathbf{f}^{\prime}$ be monomials in $\{f_{i}:i\in[1,n]\}$.
Then
$$\operatorname{wt}_{\ell}\mathbf{f}\mathbf{m}=\operatorname{wt}_{\ell}\mathbf{f%
}^{\prime}\mathbf{m}^{\prime}\iff\mathbf{f}=\mathbf{f}^{\prime}\ \ {\rm{and}}%
\ \ \mathbf{m}=\mathbf{m}^{\prime}.$$
Proof.
Write
$$\mathbf{m}=x_{1}^{p_{1}}\cdots x_{n}^{p_{n}}x[\alpha_{1}]^{m_{1}}\cdots x[%
\alpha_{n}]^{m_{n}},\ \ \mathbf{f}=f_{1}^{r_{1}}\cdots f_{n}^{r_{n}},$$
and let $\mathbf{m}^{\prime},\mathbf{f}^{\prime}$ be defined similarly with $p_{i}$ replaced by $p_{i}^{\prime}$ etc. If $p_{1}>0$ then $m_{1}=0$ and using the fact that $\cal P^{+}_{\xi}$ is a free abelian monoid
we have
$$\mathbf{f}_{1}^{r_{1}}{\mbox{\boldmath$\omega$}}_{1,\xi(2)}^{p_{1}}=\mathbf{f}%
_{1}^{r_{1}^{\prime}}{\mbox{\boldmath$\omega$}}_{1,\ \xi(2)}^{p_{1}^{\prime}}{%
\mbox{\boldmath$\omega$}}_{1,\ \xi(2)+2}^{m_{1}^{\prime}\delta_{\xi(1),\xi(2)+%
1}}{\mbox{\boldmath$\omega$}}_{1,\ \xi(2)-2}^{m_{1}^{\prime}\delta_{\xi(1),\xi%
(2)-1}}.$$
Since $\mathbf{f}_{1}={\mbox{\boldmath$\omega$}}_{1,\xi(1)+1}{\mbox{\boldmath$\omega$%
}}_{i,\xi(1)-1}$, we get
$$r_{1}+p_{1}=r_{1}^{\prime}+p_{1}^{\prime},\ \ r_{1}=m_{1}^{\prime}+r_{1}^{%
\prime}.$$
If $m_{1}^{\prime}\neq 0$ then $p_{1}^{\prime}=0$ and we have $r_{1}>r_{1}^{\prime}$ and
$r_{1}^{\prime}>r_{1}$ which is absurd. Hence $m_{1}^{\prime}=0$ and so $r_{1}^{\prime}=r_{1}$ and $p_{1}^{\prime}=p_{1}$. Writing $\mathbf{m}=x_{1}^{p_{1}}\mathbf{m}_{1}$ and $\mathbf{m}^{\prime}=x_{1}^{p_{1}}\mathbf{m}_{1}^{\prime}$
we see that $\mathbf{m}_{1}$ and $\mathbf{m}_{1}^{\prime}$ are both standard monomials and
$$\operatorname{wt}_{\ell}f_{2}^{r_{2}}\cdots f_{n}^{r_{n}}\mathbf{m}_{1}=%
\operatorname{wt}_{\ell}f_{2}^{r_{2}^{\prime}}\cdots f_{n}^{r_{n}^{\prime}}%
\mathbf{m}_{1}^{\prime}.$$
An obvious iteration of the preceding argument proves the Lemma.
∎
Suppose that
$$\iota\left(\sum_{r,s}c_{r,s}\mathbf{f}(s)\mathbf{m}_{r}\right)=0,$$
where
$\mathbf{m}_{r}$ varies over
standard monomials in $\cal A(\mathbf{x},Q_{\xi})$,
and $\mathbf{f}(s)$ varies over
monomials in $f_{i}$, $i\in[1,n]$ and $c_{r,s}\in\mathbb{Z}$ with only finitely many being non–zero.
Assume for a contradiction that $c_{r,s}\neq 0$ for some $r,s$ and let $\lambda$ be a maximal element (with respect to the partial order on $P^{+}$) of the set
$\{\operatorname{wt}(\operatorname{wt}_{\ell}\mathbf{f}(s)\mathbf{m}_{r}):c_{r,%
s}\neq 0\}.$
Using (1.6) we get
$$0=\sum_{\operatorname{wt}(\operatorname{wt}_{\ell}\mathbf{f}(s)\mathbf{m}_{r})%
=\lambda}c_{r,s}[\operatorname{wt}_{\ell}\mathbf{f}(s)\mathbf{m}_{r}]+\sum_{%
\operatorname{wt}{\mbox{\boldmath\scriptsize$\omega$}}\ngtr\lambda}n_{\mbox{%
\boldmath\scriptsize$\omega$}}[{\mbox{\boldmath$\omega$}}],\ \ \ \ n_{\mbox{%
\boldmath\scriptsize$\omega$}}\in\mathbb{Z}.$$
Since the elements $[{\mbox{\boldmath$\omega$}}]$, ${\mbox{\boldmath$\omega$}}\in\cal P^{+}_{\xi}$ are linearly independent elements of $\cal K_{0}(\cal F_{\xi})$ we get
$$\sum_{\operatorname{wt}(\operatorname{wt}_{\ell}\mathbf{f}(s)\mathbf{m}_{r})=%
\lambda}c_{r,s}[\operatorname{wt}_{\ell}\mathbf{f}(s)\mathbf{m}_{r}]=0.$$
By Lemma 1.10 the elements $[\operatorname{wt}_{\ell}(\mathbf{f}(s)\mathbf{m}_{r})]$ are all distinct and hence also linearly independent. This forces $c_{r,s}=0$ contradicting our assumption and proves that $\iota$ is injective.
1.11. The elements $\iota(x[\beta_{1}]x[\beta_{2}])$
We now prove the final assertion of the theorem. Write $[{\mbox{\boldmath$\omega$}}_{s}]=\iota([x[\beta_{s}])$, $s=1,2$ and let ${\mbox{\boldmath$\omega$}}={\mbox{\boldmath$\omega$}}_{1}{\mbox{\boldmath$%
\omega$}}_{2}$. Assuming that $[{\mbox{\boldmath$\omega$}}]\neq[{\mbox{\boldmath$\omega$}}_{1}][{\mbox{%
\boldmath$\omega$}}_{2}]$ we shall prove that $x[\alpha]x[\beta]$ is not a cluster monomial.
By Proposition 1.7 we can write $[{\mbox{\boldmath$\omega$}}_{1}][{\mbox{\boldmath$\omega$}}_{2}]$ as the non-trivial sum of elements which are imaged under $\iota$ of cluster monomials. Since cluster monomials are linearly independent and $\iota$ is an isomorphism we see that $x[\beta_{1}]x[\beta_{2}]$ is not a cluster monomial and the proof of the main theorem is complete.
2. Proof of Proposition 1.5 and a $q$-character formula.
In this section we prove Proposition 1.5 which is a recursive formula for a cluster variable. We also solve this recursions and give a closed formula for the cluster variable in terms of the initial cluster and the frozen variables. In view of Section 1.9 this formula can also be viewed as giving the $q$–character of $[{\mbox{\boldmath$\omega$}}]$, ${\mbox{\boldmath$\omega$}}\in\mathbf{P}\mathbf{r}_{\xi}$ in terms of the local Weyl modules and Kirillov–Reshetikhin modules.
2.1.
We briefly recall the definition (see [16]) of a cluster algebra. Let $Q$ be a quiver with $(n+m)$-vertices labeled $\{1,\cdots,n,1^{\prime},\cdots,m^{\prime}\}$ and assume that the set of edges has no loops or $2$-cycles. A mutation of $Q$ at a vertex $i$ is the quiver obtained by performing the following three operations.
•
reverse all edges at $i$,
•
given edges $j\to i\to k$ add a new edge $j\to k$,
•
remove any two cycles that may have been created.
We shall assume that mutation is never allowed at the vertices labeled $\{1^{\prime},\cdots,m^{\prime}\}$; these are called the frozen vertices.
Suppose that $\mathbf{x}=\{x_{1},\cdots,x_{n},f_{1},\cdots,f_{m}\}$ is an algebraically independent set and let $\mathbb{Q}(\mathbf{x})$ be the field of rational functions in these variables. The set $\mathbf{x}$ is called the initial cluster and $(\mathbf{x},Q)$ is called the initial seed.
Corresponding to a mutation of $Q$ at a vertex $i$ define a new cluster $\mathbf{x}^{\prime}=\{x_{1}^{\prime},\cdots,x_{n}^{\prime},f_{1},\cdots,f_{m}\}$ by
$$x_{j}^{\prime}=x_{j},\ \ j\neq i,\ \ x_{i}^{\prime}x_{i}=\prod_{\begin{%
subarray}{c}\exists\ j\to i\\
{\rm in}\ Q\end{subarray}}f_{j}\prod_{\begin{subarray}{c}\exists\ j\to i\\
{\rm in}\ Q\end{subarray}}x_{j}+\prod_{\begin{subarray}{c}\exists\ i\to k\\
{\rm in}\ Q\end{subarray}}f_{k}\prod_{\begin{subarray}{c}\exists\ i\to k\\
{\rm in}\ Q\end{subarray}}x_{k}.$$
The new cluster again consists of algebraically independent elements and we have a new seed $(\mathbf{x}^{\prime},Q^{\prime})$ where $Q^{\prime}$ is the mutation of $Q$ at $i$. Iterating this process defines a collection of new clusters and new seeds. An element of a given cluster is called a cluster variable. A cluster monomial is a product of cluster variables all belonging to the same cluster.
The associated cluster algebra is the $\mathbb{Z}$ subring (of the field of rational function $\mathbb{Q}(\mathbf{x})$) generated by all the cluster variables.
2.2. The quiver $Q_{\xi}[i,j]$
Given $1\leq i\leq n-1$ set $Q_{\xi}=Q_{\xi}[i,j]$ if $j<i$ and let $Q_{\xi}[i,i]$ be obtained by mutating $Q_{\xi}$ at $i$. Assume that we have defined $Q_{\xi}[i,j-1]$ for $j>i$ let $Q_{\xi}[i,j]$ be the quiver defined by mutating $Q_{\xi}[i,j-1]$ at $j$.
Proposition 1.5 is a simple inspection when $j=i$ and if $j>i$ then it is a consequence of the discussion in Section 2.1, the following Lemma and an induction on $j-i$.
Lemma.
Suppose that $j>i$ and that we have an arrow $(j-1)\to j$ in $Q_{\xi}$. In $Q_{\xi}[i,j-1]$ we have the following edges at the vertex $j$:
$$\xymatrix{\max\{i-1,j_{\bullet}-1\}\ar@/^{2}pc/[rrr]^{a_{j}}&&(j-1)&j\ar[l]\ar%
[d]_{d_{j-1}}\ar@/^{1}pc/[rr]^{1-d_{j}}&&(j+1)\ar@/^{1}pc/[ll]_{d_{j}}\\
&\max\{i,j_{\bullet}\}^{\prime}\ar[urr]_{b_{j}}&&j^{\prime}&&(j+1)^{\prime}\ar%
[llu]^{1-d_{j}}&},$$
where $a_{j}=1-\delta_{i,j_{\bullet}}$ and $b_{j}=\min\{1,(1-\delta_{j_{\bullet},i_{\bullet}})d_{j_{\bullet}-1}+\delta_{j_%
{\bullet},i}\}$.
Proof.
We proceed by induction on $j-i$. To see that induction begins when $j=i+1$ notice that
$$\displaystyle d_{i}=1\implies i=(i+1)_{\bullet}\implies a_{i+1}=0,\ \ b_{i+1}=1,$$
$$\displaystyle d_{i}=0\implies i_{\bullet}=(i+1)_{\bullet}\implies\ a_{i+1}=1,%
\ \ b_{i+1}=0.$$
On the other hand in $Q_{\xi}[i,i]$ which is the
mutation of $Q_{\xi}$ at $i$ an inspection show that the edges at $i+1$ are given as follows:
$$\xymatrix{i&(i+1)\ar[l]\ar[d]\ar@/^{1}pc/[rr]^{1-d_{i+1}}&&(i+2)\ar@/^{1}pc/[%
ll]_{d_{i+1}}\\
i^{\prime}\ar[ur]&(i+1)^{\prime}&&(i+2)^{\prime}\ar[ull]^{1-d_{i+1}}},\ \ \ d_%
{i}=1,$$
$$\xymatrix{(i-1)\ar@/^{2}pc/[rr]&i&(i+1)\ar[l]\ar@/^{1}pc/[rr]^{1-d_{i+1}}&&(i+%
2)\ar@/^{1}pc/[ll]_{d_{i+1}}\\
&&&&(i+2)^{\prime}\ar[ull]^{1-d_{i+1}}},\ \ d_{i}=0,$$
and it follows that induction begins.
For the inductive step we assume that the result holds for the edges at $j<n$ in $Q_{\xi}[i,j-1]$ for and prove that it holds for the node $j+1$ in $Q_{\xi}[i,j]$.
Case 1. If $d_{j}=1$ then $j$ is a sink of $Q_{\xi}$ by assumption and so we have an edge $(j+1)\to j$ in $Q_{\xi}$. Hence by the inductive hypothesis the edges at $j$ and $(j+1)$ in $Q_{\xi}[i,j-1]$ are
$$\xymatrix{\max\{i-1,j_{\bullet}-1\}\ar@/^{2}pc/[rrr]^{a_{j}}&&(j-1)&j\ar[l]\ar%
[d]_{d_{j-1}}&(j+1)\ar[l]\ar[rrd]_{1-d_{j+1}}\ar@/^{1}pc/[rr]^{d_{j+1}}&&(j+2)%
\ar@/^{1}pc/[ll]_{1-d_{j+1}}\\
&\max\{i,j_{\bullet}\}^{\prime}\ar[urr]^{b_{j}}&&j^{\prime}&(j+1)^{\prime}\ar[%
u]&&(j+2)^{\prime}.}$$
Mutating at $j$ we see that the edges at $(j+1)$ are
$$\xymatrix{(j-1)\ar[r]&j\ar[r]&(j+1)\ar[ld]_{d_{j-1}}\ar@/_{2}pc/[ll]\ar[drr]_{%
1-d_{j+1}}\ar@/^{1}pc/[rr]^{d_{j+1}}&&(j+2)\ar@/^{1}pc/[ll]_{1-d_{j+1}}\\
&j^{\prime}&(j+1)^{\prime}\ar[u]&&(j+2)^{\prime}}.$$
The inductive
step follows since $d_{j}=1\implies(j+1)_{\bullet}=j$ and so
$$\max\{i-1,(j+1)_{\bullet}-1\}=j-1,\ \ \max\{i,(j+1)_{\bullet}\}^{\prime}=j^{%
\prime},\ \ \ \ a_{j+1}=1=d_{j},\ \ b_{j+1}=d_{j-1}.$$
Case 2.
If $d_{j}=0$ or equivalently $j_{\diamond}\neq j$ then in $Q_{\xi}$ we have an edge $j\to j+1$. By the induction hypothesis, the edges at $j$ and $(j+1)$ in $Q_{\xi}[i,j-1]$ are
$$\xymatrix{\max\{i-1,j_{\bullet}-1\}\ar@/^{2}pc/[rrr]^{a_{j}}&&(j-1)&j\ar[d]_{d%
_{j-1}}\ar[l]\ar[r]&(j+1)\ar[d]\ar@/^{1}pc/[rr]^{1-d_{j+1}}&&(j+2)\ar@/^{1}pc/%
[ll]_{d_{j+1}}\\
&\max\{i,j_{\bullet}\}^{\prime}\ar[urr]^{b_{j}}&&j^{\prime}&(j+1)^{\prime}\ar[%
ul]&&(j+2)^{\prime}\ar[ull]^{1-d_{j+1}}}.\\
\\
$$
Mutating at $j$ we obtain
$$\xymatrix{\max\{i-1,j_{\bullet}-1\}\ar@/^{2}pc/[rrr]^{a_{j}}&&j&(j+1)\ar[l]\ar%
@/^{1}pc/[rr]^{1-d_{j+1}}&&(j+2)\ar@/^{1}pc/[ll]_{d_{j+1}}\\
&\max\{i,j_{\bullet}\}^{\prime}\ar[urr]^{b_{j}}&&&&(j+2)^{\prime}\ar[ull]^{1-d%
_{j+1}}}\\
\\
$$
The inductive step follows from the fact that $d_{j}=0\implies(j+1)_{\bullet}=j_{\bullet}<j$ and so
$$\max\{i-1,(j+1)_{\bullet}-1\}=\max\{i-1,j_{\bullet}-1\},\ \ \max\{i,(j+1)_{%
\bullet}\}^{\prime}=\max\{i,j_{\bullet}\}^{\prime},$$
and
$$a_{j+1}=a_{j},\ \ b_{j+1}=b_{j},\ \ d_{j}=0.$$
The proof of the Lemma is complete.
∎
2.3. The set $\Gamma_{i,j}$
We continue to set $d_{m}=\delta_{m,m_{\diamond}}$ for $1\leq m\leq n$.
For $i,j\in[1,n]$ define sets $\Gamma_{i,j}$ as follows: $\Gamma_{i,j}=\{0\}$ if $j<i$ and if $i\leq j$ then $\Gamma_{i,j}$ is the subset of $\mathbb{Z}_{+}^{j-i+2}$ consisting of elements
${\mbox{\boldmath$\epsilon$}}=(\epsilon_{i},\cdots,\epsilon_{j+1})$ satisfying the following conditions: for $r,m\in[i,j]$ with $r\leq m$ and $\sigma_{r,m}({\mbox{\boldmath$\epsilon$}})=\epsilon_{r}+\cdots+\epsilon_{m}$, we have
$$\displaystyle\epsilon_{j+1}=1+(d_{j}-1)\sigma_{\max\{i,j_{\bullet}+1\},j}({%
\mbox{\boldmath$\epsilon$}})$$
(2.1)
$$\displaystyle\sigma_{\max\{i,j_{\bullet}+1\},j}({\mbox{\boldmath$\epsilon$}})%
\leq 1,$$
(2.2)
$$\displaystyle\sigma_{i,i_{\diamond}}({\mbox{\boldmath$\epsilon$}})\leq 1\leq%
\sigma_{i,i_{\diamond}+1}({\mbox{\boldmath$\epsilon$}})\ \ {\rm{if}}\ \ i_{%
\diamond}\leq j,$$
(2.3)
$$\displaystyle\sigma_{m+1,(m+1)_{\diamond}}({\mbox{\boldmath$\epsilon$}})\leq 1%
\leq\sigma_{m+1,(m+1)_{\diamond}+1}({\mbox{\boldmath$\epsilon$}}),\ \ {\rm{if}%
}\ \ i_{\diamond}\leq m=m_{\diamond}<j_{\bullet}$$
(2.4)
Clearly, $\epsilon_{m}\in\{0,1\}$ for $i\leq m\leq j+1$. For $i\leq j$ let
$$\Gamma_{i,j}^{1}=\{{\mbox{\boldmath$\epsilon$}}\in\Gamma_{i,j}:\sigma_{\max\{i%
,j_{\bullet}+1\},j}({\mbox{\boldmath$\epsilon$}})=1\},\ \ \Gamma_{i,j}^{0}=\{{%
\mbox{\boldmath$\epsilon$}}\in\Gamma_{i,j}:\sigma_{\max\{i,j_{\bullet}+1\},j}(%
{\mbox{\boldmath$\epsilon$}})=0\}.$$
The condition in (2.2) shows that
$$\Gamma_{i,j}=\Gamma_{i,j}^{1}\sqcup\Gamma_{i,j}^{0}.$$
We shall use the following freely:
$$d_{m-1}=0\iff(m-1)_{\bullet}=m_{\bullet},\ \ d_{m-1}=1\iff m_{\bullet}=m-1.$$
(2.5)
Lemma.
For $j>i$ the assignments
$$\displaystyle(\epsilon_{i},\cdots,\epsilon_{j})\to(\epsilon_{i},\cdots,%
\epsilon_{j},d_{j}),$$
$$\displaystyle(\epsilon_{i},\cdots,\epsilon_{j_{\bullet}})\to(\epsilon_{i}+%
\delta_{i,j_{\bullet}},\cdots,\epsilon_{j_{\bullet}},0,\cdots,0,1),$$
define bijections $\iota_{j-1}:\Gamma_{i,j-1}\to\Gamma_{i,j}^{1}$ and $\iota_{j_{\bullet}-1}:\Gamma_{i,j_{\bullet}-1}\to\Gamma_{i,j}^{0}$ respectively.
Proof.
For the first assertion of the Lemma we must prove that
$$\tilde{\mbox{\boldmath$\epsilon$}}=(\epsilon_{i},\cdots,\epsilon_{j})\in\Gamma%
_{i,j-1}\iff{\mbox{\boldmath$\epsilon$}}=(\epsilon_{i},\cdots,\epsilon_{j},d_{%
j})\in\Gamma_{i,j}^{1}.$$
Clearly we have $\sigma_{m,r}({\mbox{\boldmath$\epsilon$}})=\sigma_{m,r}(\tilde{\mbox{\boldmath%
$\epsilon$}})$ for all $i\leq m\leq r\leq j.$
Using (2.5) we see that
$$(*)\ \ \epsilon_{j}=1+(d_{j-1}-1)\sigma_{\max\{i,(j-1)_{\bullet}+1\},j-1}(%
\tilde{\mbox{\boldmath$\epsilon$}})\iff\sigma_{\max\{i,j_{\bullet}+1\},j}({%
\mbox{\boldmath$\epsilon$}})=1$$
It follows that $\tilde{\mbox{\boldmath$\epsilon$}}$ satisfies (2.1) if ${\mbox{\boldmath$\epsilon$}}\in\Gamma_{i,j}^{1}$. It also proves that $\epsilon$ satisfies (2.1) and (2.2)
if $\tilde{\mbox{\boldmath$\epsilon$}}\in\Gamma_{i,j-1}$. To see that $\tilde{\mbox{\boldmath$\epsilon$}}$ satisfies (2.2) if ${\mbox{\boldmath$\epsilon$}}\in\Gamma_{i,j}^{1}$ we note that this is clear if $(j-1)_{\bullet}=j_{\bullet}$ and if $j_{\bullet}=j-1$ it follows from the fact that $\epsilon$ satisfies (2.4) with $m=(j-1)_{\bullet}$.
It is obvious
that $\tilde{\mbox{\boldmath$\epsilon$}}$ satisfies (2.3) (resp. (2.4)) if ${\mbox{\boldmath$\epsilon$}}\in\Gamma_{i,j}^{1}$; it is also obvious that $\epsilon$ satisfies these inequalities if $\tilde{\mbox{\boldmath$\epsilon$}}\in\Gamma_{i,j-1}$ as long as $i_{\diamond}\leq j-1$ (resp. $i_{\diamond}\leq m<(j-1)_{\bullet})$).
If $i_{\diamond}=j$ then $d_{j}=1$ and $j_{\bullet}=i_{\bullet}<i$. Using (*) and the fact that we have already proved that $\epsilon$ satisfies (2.1) we get
$$\sigma_{\max\{i,j_{\bullet}+1\},j}({\mbox{\boldmath$\epsilon$}})=1\leq\max%
\sigma_{\max\{i,j_{\bullet}+1\},j+1}({\mbox{\boldmath$\epsilon$}})=2,$$
proving that (2.3) holds for $\epsilon$.
If $(j-1)_{\bullet}\leq m=m_{\diamond}<j_{\bullet}$ then we must have $m=(j-1)_{\bullet}$, and $j_{\bullet}=j-1=(m+1)_{\diamond}$. It follows that $d_{j-1}=1$, $\epsilon_{j}=1$ and so we have
$$\sigma_{(j-1)_{\bullet}+1,j-1}(\tilde{\mbox{\boldmath$\epsilon$}})=\sigma_{(j-%
1)_{\bullet}+1,j-1}({\mbox{\boldmath$\epsilon$}})\leq 1=\epsilon_{j}\leq\sigma%
_{(j-1)_{\bullet}+1,j}({\mbox{\boldmath$\epsilon$}}),$$
proving that $\epsilon$ satisfies (2.4).
The proof of the first assertion is complete.
We prove the second assertion of the Lemma; note that if ${\mbox{\boldmath$\epsilon$}}\in\Gamma_{i,j}^{0}$ then we must have $\epsilon_{m}=0$ for $j_{\bullet}+1\leq m\leq j$ and hence by (2.1) we also have $\epsilon_{j+1}=1$. Since
$$j_{\bullet}\leq i\implies\Gamma_{i,j_{\bullet}-1}=\{0\}\ \ {\rm{and}}\ \ %
\Gamma_{i,j}^{0}=\{(\delta_{i,j_{\bullet}},\cdots,0,1)\}$$
the result is trivially true in this case. We assume from now on that $j_{\bullet}>i$ (in particular $j_{\bullet}\geq i_{\diamond}$) and
let
$$\tilde{\mbox{\boldmath$\epsilon$}}=(\epsilon_{i},\cdots,\epsilon_{j_{\bullet}}%
)\ \qquad{\mbox{\boldmath$\epsilon$}}=(\epsilon_{i},\cdots,\epsilon_{j_{%
\bullet}},0,\cdots,0,1).$$
Suppose that $\tilde{\mbox{\boldmath$\epsilon$}}\in\Gamma_{i,j_{\bullet}-1}$. It is obvious that $\epsilon$ satisfies (2.1) and (2.2) and (2.3) and for $i_{\diamond}\leq m<(j_{\bullet}-1)_{\bullet}$ that $\epsilon$ satisfies (2.4). If $(j_{\bullet}-1)_{\bullet}\leq m=m_{\diamond}\leq j_{\bullet}-1$ then either $m=(j_{\bullet}-1)_{\bullet}$ or $m=j_{\bullet}-1$. In the first case the first inequality in (2.4) for $\epsilon$ is just (2.2) for $\tilde{\mbox{\boldmath$\epsilon$}}$ while the second inequality follows from (2.1) for $\tilde{\mbox{\boldmath$\epsilon$}}$.
If $m=m_{\diamond}=j_{\bullet}-1$, then (2.1) forces $\epsilon_{j_{\bullet}}=1$ and hence we have $\epsilon_{j_{\bullet}}\leq 1\leq\epsilon_{j_{\bullet}}+\epsilon_{j_{\bullet}+1}$. This proves that (2.4) holds for $\epsilon$ and so ${\mbox{\boldmath$\epsilon$}}\in\Gamma_{i,j}^{0}$.
Next we assume that ${\mbox{\boldmath$\epsilon$}}\in\Gamma_{i,j}^{0}$ and prove that $\tilde{\mbox{\boldmath$\epsilon$}}\in\Gamma_{i,j_{\bullet}-1}$.
To prove that (2.1) holds for $\tilde{\mbox{\boldmath$\epsilon$}}$ it suffices to observe that
if $j_{\bullet}=i_{\diamond}$ (resp. $(j_{\bullet})_{\bullet}\geq i_{\diamond}$) then (2.3) (resp. (2.4)) for $\epsilon$ gives
$$\sigma_{\max\{i,(j_{\bullet})_{\bullet}+1\},j_{\bullet}}(\tilde{\mbox{%
\boldmath$\epsilon$}})=\sigma_{\max\{i,(j_{\bullet})_{\bullet}+1\},j_{\bullet}%
}({\mbox{\boldmath$\epsilon$}})=1.$$
If $d_{j_{\bullet}-1}=1$ then $(j_{\bullet})_{\bullet}=j_{\bullet}-1$ and so the preceding equality is $\epsilon_{j_{\bullet}}=1$ as needed. If $d_{j_{\bullet}-1}=0$ then $(j_{\bullet}-1)_{\bullet}=(j_{\bullet})_{\bullet}$ and again the preceding equality is a reformulation of (2.1) for $\tilde{\mbox{\boldmath$\epsilon$}}$.
The fact that $\tilde{\mbox{\boldmath$\epsilon$}}$ satisfies
(2.2) follows by using (2.3) for $\epsilon$ if $(j_{\bullet})_{\bullet}<i_{\diamond}$ and using (2.4) for $\epsilon$ otherwise. It is clear that (2.3) and (2.4) hold for $\tilde{\mbox{\boldmath$\epsilon$}}$ since they are the same as the corresponding ones for $\epsilon$ and the proof of the Lemma is complete.
∎
2.4. The sets $\Gamma_{i,j}^{\prime}$
For $i\leq j$ define a map
$$p_{ij}:\Gamma_{i,j}\to\mathbb{Z}^{(j-i+2)},\ \ p_{i,j}(\epsilon_{i},\cdots,%
\epsilon_{j+1})=(\epsilon_{i}^{\prime},\cdots,\epsilon_{j+1}^{\prime}),$$
as follows:
•
$\epsilon_{j+1}^{\prime}=(1-d_{j})\sigma_{\max\{i,j_{\bullet}+1\},j}({\mbox{%
\boldmath$\epsilon$}})+d_{j}(1-\sigma_{\max\{i,j_{\bullet}+1\},j}({\mbox{%
\boldmath$\epsilon$}})),$
•
if $i_{\bullet}=m_{\bullet}$ or $\sigma_{\max\{i,(m_{\bullet})_{\bullet}+1\},m_{\bullet}}=1$
then,
$$\epsilon_{m}^{\prime}=\begin{cases}(d_{m}-1)\epsilon_{m+1}-d_{m},\ \ \ \ %
\sigma_{\max\{i,m_{\bullet}+1\},m}({\mbox{\boldmath$\epsilon$}})=0,\\
d_{m}-\left(\epsilon_{m}+\epsilon_{m+1}\right),\ \ \ \sigma_{\max\{i,m_{%
\bullet}+1\},m}({\mbox{\boldmath$\epsilon$}})=1,\end{cases}$$
•
if $m_{\bullet}\geq i$ and $\sigma_{\max\{i,(m_{\bullet})_{\bullet}+1,m_{\bullet}\}}({\mbox{\boldmath$%
\epsilon$}})=0$ then
$\epsilon_{m}^{\prime}=d_{m}(1-\epsilon_{m+1})$.
It is easily seen that $\epsilon_{m}^{\prime}\in\{-1,0,1\}$ for $i\leq m\leq j$. For $i\leq j$ let $\Gamma_{i,j}^{\prime}$ be the image of $p_{ij}$ and set $\Gamma_{i,j}^{\prime}=\{0\}$ if $i>j$.
Lemma.
Let $1\leq i\leq j\leq n$.
(i)
If $\tilde{\mbox{\boldmath$\epsilon$}}=(\epsilon_{i},\cdots,\epsilon_{j})\in\Gamma%
_{i,j-1}$ then
$$p_{ij-1}(\tilde{\mbox{\boldmath$\epsilon$}})=(\epsilon_{i}^{\prime},\cdots,%
\epsilon_{j}^{\prime})\implies p_{ij{}}(\iota_{j-1}(\tilde{\mbox{\boldmath$%
\epsilon$}}))=(\epsilon_{i}^{\prime},\cdots,\epsilon_{j-1}^{\prime},-1+%
\epsilon_{j}^{\prime},1-d_{j}).$$
(ii)
If $\tilde{\mbox{\boldmath$\epsilon$}}=(\epsilon_{i},\cdots,\epsilon_{j_{\bullet}}%
)\in\Gamma_{i,j_{\bullet}-1}$ then
$$p_{ij_{\bullet}-1}(\tilde{\mbox{\boldmath$\epsilon$}})=(\epsilon_{i}^{\prime},%
\cdots,\epsilon_{j_{\bullet}}^{\prime})\implies p_{ij{}}(\iota_{j_{\bullet}-1}%
(\tilde{\mbox{\boldmath$\epsilon$}}))=(\epsilon_{i}^{\prime},\cdots,\epsilon_{%
j_{\bullet}}^{\prime},0\cdots,0,-1,d_{j}).$$
Proof.
Let ${\mbox{\boldmath$\epsilon$}}=(\epsilon_{i},\cdots,\epsilon_{j},d_{j})=\iota_{i%
,j-1}(\tilde{\mbox{\boldmath$\epsilon$}})$ and let $p_{ij}({\mbox{\boldmath$\epsilon$}})=(\epsilon_{i}^{\prime\prime},\cdots,%
\epsilon_{j+1}^{\prime\prime})$. Since
$$\sigma_{m,r}(\tilde{\mbox{\boldmath$\epsilon$}})=\sigma_{m,r}({\mbox{\boldmath%
$\epsilon$}}),\ \ m\leq r\leq j$$
it is clear from the definition that $\epsilon_{m}^{\prime}=\epsilon_{m}^{\prime\prime}$ if $m\leq j-1$.
By Lemma 2.3 we have ${\mbox{\boldmath$\epsilon$}}\in\Gamma_{i,j}^{1}$ and hence $\sigma_{\max\{i,j_{\bullet}+1\},j}({\mbox{\boldmath$\epsilon$}})=1$. It is immediate from the definition of $p_{ij}$ that $\epsilon_{j+1}^{\prime\prime}=1-d_{j}$. We now prove that $\epsilon_{j}^{\prime\prime}=-1+\epsilon_{j}^{\prime}$; using the definition of $\epsilon_{j}^{\prime}$ this is equivalent to proving
$$\displaystyle\epsilon_{j}^{\prime\prime}$$
$$\displaystyle=-1+(1-d_{j-1})\sigma_{\max\{i,(j-1)_{\bullet}+1\},j-1}(\tilde{%
\mbox{\boldmath$\epsilon$}})+d_{j-1}(1-\sigma_{\max\{i,(j-1)_{\bullet}+1,\},j-%
1}(\tilde{\mbox{\boldmath$\epsilon$}})),$$
(2.6)
$$\displaystyle=-1+(1-d_{j-1})\sigma_{\max\{i,j_{\bullet}+1\},j-1}({\mbox{%
\boldmath$\epsilon$}})+d_{j-1}(1-\sigma_{\max\{i,(j_{\bullet})_{\bullet}+1\},j%
_{\bullet}}({\mbox{\boldmath$\epsilon$}})).$$
If $j_{\bullet}\geq i$ and $\sigma_{\max\{i,(j_{\bullet})_{\bullet}+1\},j_{\bullet}}({\mbox{\boldmath$%
\epsilon$}})=0$ then by (2.4) we have $\epsilon_{j_{\bullet}+1}=1$ and so $\sigma_{j_{\bullet}+1,j-1}({\mbox{\boldmath$\epsilon$}})=1$. This means that the right hand side of (2.6) is zero. Since by definition $\epsilon_{j}^{\prime\prime}=d_{j}(1-d_{j})=0$ the result is proved in this case.
If $i_{\bullet}=j_{\bullet}$ then $d_{j-1}=0$ and since $\sigma_{\max\{i,j_{\bullet}+1\},j}({\mbox{\boldmath$\epsilon$}})=1$ it follows that the right hand side of (2.6) is $-\epsilon_{j}^{\prime}$ which is precisely the value of $\epsilon_{j}^{\prime\prime}$ in this case.
Suppose that $\sigma_{\max\{i,(j_{\bullet})_{\bullet}+1\},j_{\bullet}}({\mbox{\boldmath$%
\epsilon$}})=1$ and that $j_{\bullet}\geq i$. This means that the second term on the right hand side of (2.6) is zero. Since $\sigma_{\max\{i,j_{\bullet}+1\},j}({\mbox{\boldmath$\epsilon$}})=1$ by definition we have $\epsilon_{j}^{\prime\prime}=-\epsilon_{j}$. Recalling that $\epsilon_{j}=1+(d_{j-1}-1)\sigma_{\max\{i,(j-1)_{\bullet}+1\},j-1}$ we see that the right hand side of (2.6) is also $-\epsilon_{j}^{\prime}$.
The proof of part (i) is now complete.
We prove part (ii). Let
$${\mbox{\boldmath$\epsilon$}}=\iota_{i,j_{\bullet}-1}(\tilde{\mbox{\boldmath$%
\epsilon$}})\in\Gamma_{i,j}^{0},\ \ p_{ij}({\mbox{\boldmath$\epsilon$}})=(%
\epsilon_{i}^{\prime\prime},\cdots,\epsilon_{j+1}^{\prime\prime}).$$
Since $\sigma_{m,r}(\tilde{\mbox{\boldmath$\epsilon$}})=\sigma_{m,r}({\mbox{\boldmath%
$\epsilon$}})$ for all $m\leq r\leq j_{\bullet}-1$ it is clear from the definition that $\epsilon_{m}^{\prime}=\epsilon_{m}^{\prime\prime}$ if $m\leq j_{\bullet}-1$. Since $d_{m}=0$ if $j_{\bullet}+1\leq m\leq j-1$ a simple inspection also shows that
$$\epsilon_{m}^{\prime\prime}=0,\ \ j_{\bullet}+1\leq m\leq j-1,\ \ \epsilon_{j+%
1}^{\prime\prime}=d_{j}.$$
It remains to prove that $\epsilon_{j_{\bullet}}^{\prime\prime}=\epsilon_{j_{\bullet}}^{\prime}$ and that $\epsilon_{j}^{\prime\prime}=-1$.
If $j_{\bullet}=i_{\bullet}$ then $\tilde{\mbox{\boldmath$\epsilon$}}=\{0\}$, $\Gamma_{i,j_{\bullet}-1}=\{0\}$ and ${\mbox{\boldmath$\epsilon$}}=(0,\cdots,0,1)$. By definition $p_{i,j}({\mbox{\boldmath$\epsilon$}})=(0,\cdots,0,-1,d_{j})$ and we are done in this case. If $i=j_{\bullet}$ then $\Gamma_{i,j_{\bullet}-1}=\{0\}$ and ${\mbox{\boldmath$\epsilon$}}=(1,0,\cdots,1)$. and one checks easily that $\epsilon_{i}^{\prime}=0$. On the other hand, by definition $\epsilon_{i}^{\prime\prime}=d_{i}-(\delta_{i,j_{\bullet}}+\epsilon_{i+1})=0$. The fact that $\epsilon_{j}^{\prime\prime}=-1$ is a straightforward checking from the definition.
Suppose that $j_{\bullet}>i$. Since
$\epsilon_{j_{\bullet}+1}=0$ we have $\sigma_{\max\{i,j_{\bullet}+1\},j}({\mbox{\boldmath$\epsilon$}})=0$ and by using (2.4) that $\sigma_{\max\{i,(j_{\bullet})_{\bullet}+1\},j_{\bullet}}({\mbox{\boldmath$%
\epsilon$}})=1$. Since $\epsilon_{j+1}=1$ it follows by definition that $\epsilon_{j}^{\prime\prime}=-1$ as needed.
Finally to show $\epsilon_{j_{\bullet}}^{\prime\prime}=\epsilon_{j_{\bullet}}^{\prime}$, we write $m=j_{\bullet}$ and see that we must prove
$$\displaystyle\epsilon_{m}^{\prime\prime}$$
$$\displaystyle=(1-d_{m-1})\sigma_{\max\{i,(m-1)_{\bullet}+1\},m-1}(\tilde{\mbox%
{\boldmath$\epsilon$}})+d_{m-1}(1-\sigma_{\max\{i,(m-1)_{\bullet}+1\},m-1}(%
\tilde{\mbox{\boldmath$\epsilon$}}))$$
(2.7)
$$\displaystyle=(1-d_{m-1})\sigma_{\max\{i,m_{\bullet}+1\},m-1}({\mbox{\boldmath%
$\epsilon$}})+d_{m-1}(1-\sigma_{\max\{i,(m_{\bullet})_{\bullet}+1\},m_{\bullet%
}}({\mbox{\boldmath$\epsilon$}})),$$
$$\displaystyle=(1-d_{m-1})(1-\epsilon_{m})+d_{m-1}(1-\sigma_{\max\{i,(m_{%
\bullet})_{\bullet}+1\},m_{\bullet}}({\mbox{\boldmath$\epsilon$}})).$$
If $\sigma_{\max\{i,(m_{\bullet})_{\bullet}+1\},m_{\bullet}}({\mbox{\boldmath$%
\epsilon$}}))=1$ then $\epsilon_{m}^{\prime\prime}=1-\epsilon_{m}$ by definition. By (2.4) we have $\epsilon_{m}=1$ if $d_{m-1}=1$ and hence $(1-\epsilon_{m})=(1-d_{m-1})(1-\epsilon_{m})$ and (2.7) is proved.
If $\sigma_{\max\{i,(m_{\bullet})_{\bullet}+1\},m_{\bullet}}({\mbox{\boldmath$%
\epsilon$}})=0$ then by definition $\epsilon_{m}^{\prime\prime}=1$. Hence we must prove that
$$1=(1-d_{m-1})(1-\epsilon_{m})+d_{m-1}.$$
If $d_{m-1}=1$ this is clear from the preceding computation. If $d_{m-1}=0$ then $m_{\bullet}+1<m$ and (2.4) forces $\epsilon_{m_{\bullet}+1}=1$; in particular it follows that $\epsilon_{m}=0$ and (2.7) and is completely proved.
∎
2.5.
Proposition.
For $1\leq i\leq j\leq n$ we have
$$x[\alpha_{i,j}]=\sum_{{\mbox{\boldmath\scriptsize$\epsilon$}}\in\Gamma_{i,j}}f%
_{i,j}^{\mbox{\boldmath\scriptsize$\epsilon$}}m_{i,j}^{\mbox{\boldmath%
\scriptsize$\epsilon$}},$$
where
$$m_{i,j}^{{\mbox{\boldmath\scriptsize$\epsilon$}}}=x_{i-1}^{1-\epsilon_{i}}x_{i%
}^{\epsilon_{i}^{\prime}}\cdots x_{j}^{\epsilon_{j}^{\prime}}x_{j+1}^{\epsilon%
_{j+1}^{\prime}},\ \ f_{i,j}^{\mbox{\boldmath\scriptsize$\epsilon$}}=f_{i}^{%
\epsilon_{i}}\cdots f_{j}^{\epsilon_{j}}f_{j+1}^{(1-d_{j})\epsilon_{j+1}},$$
with ${\mbox{\boldmath$\epsilon$}}=(\epsilon_{i},\cdots,\epsilon_{j+1})$ and $p_{i,j}({\mbox{\boldmath$\epsilon$}})=(\epsilon_{i}^{\prime},\cdots,\epsilon_{%
j+1}^{\prime})$.
Proof.
The proof of the proposition proceeds by an induction on $j-i$. To see that induction begins recall from Proposition 1.5 that
$$x[\alpha_{i}]=\delta_{i,i_{\diamond}}\left(f_{i}x_{i}^{-1}+x_{i-1}x_{i+1}x_{i}%
^{-1}\right)+(1-\delta_{i,i_{\diamond}})\left(x_{i}^{-1}\left(f_{i}x_{i+1}+x_{%
i-1}f_{i+1}\right)\right).$$
Since
$$i=i_{\diamond}\implies\Gamma_{i,i}=\{(1,1),(0,1)\},\ \ \Gamma_{i,i}^{\prime}=%
\{(-1,0),(-1,1)\}$$
and
$$i\neq i_{\diamond}\implies\Gamma_{i,i}=\{(0,1),(1,0)\},\ \ \Gamma_{i,i}^{%
\prime}=\{(-1,0),(-1,1)\}$$
we see that induction begins.
For the inductive step Proposition 1.5 asserts that
$$\displaystyle x_{j}x[\alpha_{i,j}]$$
$$\displaystyle=f_{j}^{d_{j-1}}x[\alpha_{i,j-1}]x_{j+1}^{1-d_{j}}+$$
$$\displaystyle f_{j+1}^{1-d_{j}}x_{j+1}^{d_{j}}\left((\delta_{i_{\bullet},j_{%
\bullet}}+\delta_{i,j_{\bullet}})f_{i}^{\delta_{i,j_{\bullet}}}x_{i-1}^{1-%
\delta_{i,j_{\bullet}}}+(1-\delta_{i_{\bullet},j_{\bullet}}-\delta_{i,j_{%
\bullet}})f_{j_{\bullet}}^{d_{j_{\bullet}-1}}x[\alpha_{i,j_{\bullet}-1}]\right).$$
Let $\tilde{\mbox{\boldmath$\epsilon$}}=(\epsilon_{i},\cdots,\epsilon_{j})\in\Gamma%
_{i,j-1}$. By Lemmas 2.3 and 2.4 we have
$$m_{i,j}^{{\mbox{\boldmath\scriptsize$\epsilon$}}}f_{i,j}^{{\mbox{\boldmath%
\scriptsize$\epsilon$}}}=m_{i,j-1}^{\tilde{\mbox{\boldmath\scriptsize$\epsilon%
$}}}f_{i,j}^{\tilde{\mbox{\boldmath\scriptsize$\epsilon$}}}f_{j}^{d_{j-1}}x_{j%
+1}^{1-d_{j}}x_{j}^{-1},\quad{\mbox{\boldmath$\epsilon$}}=\iota_{i,j-1}(\tilde%
{\mbox{\boldmath$\epsilon$}})$$
once we notice that $(1-d_{j-1})\epsilon_{j}+d_{j-1}=\epsilon_{j}$. Hence using the inductive hypothesis we get
$$f_{j}^{d_{j-1}}x[\alpha_{i,j-1}]x_{j+1}^{1-d_{j}}x_{j}^{-1}=\sum_{{\mbox{%
\boldmath$\epsilon$}}\in\Gamma_{i,j}^{1}}f_{i,j}^{\mbox{\boldmath\scriptsize$%
\epsilon$}}m_{i,j}^{\mbox{\boldmath\scriptsize$\epsilon$}}.$$
Similarly let $\tilde{\mbox{\boldmath$\epsilon$}}=(\epsilon_{i},\cdots,\epsilon_{\max\{i,j_{%
\bullet}+1\}})\in\Gamma_{i,j_{\bullet}-1}$ and ${\mbox{\boldmath$\epsilon$}}=\iota_{i,j_{\bullet}-1}(\tilde{\mbox{\boldmath$%
\epsilon$}})$. Then $\epsilon_{j+1}=1-d_{j}$ by definition and $1-\epsilon_{i}=1-\delta_{i,j_{\bullet}}$ if $j_{\bullet}\leq i$ and we get
$$m_{i,j}^{{\mbox{\boldmath\scriptsize$\epsilon$}}}f_{i,j}^{{\mbox{\boldmath%
\scriptsize$\epsilon$}}}=\begin{cases}f_{j+1}^{1-d_{j}}f_{i}^{\delta_{i,j_{%
\bullet}}}x_{i-1}^{1-\delta_{i,j_{\bullet}}}x_{j}^{-1}x_{j+1}^{d_{j}},\quad%
\quad j_{\bullet}\leq i\\
f_{i,j_{\bullet}-1}^{\tilde{\mbox{\boldmath\scriptsize$\epsilon$}}}f_{j+1}^{1-%
d_{j}}f_{j_{\bullet}}^{d_{\bullet}-1}m_{i,j_{\bullet}-1}^{\tilde{\mbox{%
\boldmath\scriptsize$\epsilon$}}}x_{j}^{-1}x_{j+1}^{d_{j}},\quad\quad i<j_{%
\bullet}.\end{cases}$$
The inductive step follows from the inductive hypothesis and the fact that $\Gamma_{i,j}=\Gamma_{i,j}^{0}\sqcup\Gamma_{i,j}^{1}$.
∎
3. Irreducible tensor products.
In
this section we give a necessary condition (see Section 3.6) for the equality $[{\mbox{\boldmath$\pi$}}_{1}][{\mbox{\boldmath$\pi$}}_{2}]=[{\mbox{\boldmath$%
\pi$}}_{1}{\mbox{\boldmath$\pi$}}_{2}]$ to hold when ${\mbox{\boldmath$\pi$}}_{1},{\mbox{\boldmath$\pi$}}_{2}\in\mathbf{P}\mathbf{r}%
_{\xi}$. We shall see in later sections that the conditions are sufficient as well.
We shall often need to work in the monoidal category $\cal F_{\xi}$ rather than its Grothendieck ring; by abuse of notation we shall use the symbol $[{\mbox{\boldmath$\omega$}}]$ to also denote an irreducible module in $\cal F_{\xi}$ with label $\omega$. To emphasize that we are working in the category we shall write $[{\mbox{\boldmath$\omega$}}]\otimes[{\mbox{\boldmath$\omega$}}^{\prime}]$ for the tensor product of the corresponding objects.
3.1.
We collect some well–known facts on the category $\cal F_{\xi}$. An object of $\cal F_{\xi}$ is said to be $\ell$-highest weight with highest weight $\omega$ if it has $[{\mbox{\boldmath$\omega$}}]$ as its unique irreducible quotient. Clearly any quotient of an $\ell$–highest weight module is also $\ell$–highest weight with the same irreducible quotient. Given ${\mbox{\boldmath$\omega$}}_{1},{\mbox{\boldmath$\omega$}}_{2}\in\cal P_{\xi}^{+}$ the module $[{\mbox{\boldmath$\omega$}}_{1}{\mbox{\boldmath$\omega$}}_{2}]$ occurs in the Jordan–Holder series of $[{\mbox{\boldmath$\omega$}}_{1}]\otimes[{\mbox{\boldmath$\omega$}}_{2}]$ with multiplicity one. In particular if $[{\mbox{\boldmath$\omega$}}_{1}]\otimes[{\mbox{\boldmath$\omega$}}_{2}]$ is an $\ell$–highest weight module then $[{\mbox{\boldmath$\omega$}}_{1}{\mbox{\boldmath$\omega$}}_{2}]$ is its unique irreducible quotient and hence $[{\mbox{\boldmath$\omega$}}_{1}]\otimes[{\mbox{\boldmath$\omega$}}_{2}]$ is irreducible iff $[{\mbox{\boldmath$\omega$}}_{1}]\otimes[{\mbox{\boldmath$\omega$}}_{2}]\cong[{%
\mbox{\boldmath$\omega$}}_{2}]\otimes[{\mbox{\boldmath$\omega$}}_{1}]\cong[{%
\mbox{\boldmath$\omega$}}_{1}{\mbox{\boldmath$\omega$}}_{2}]$.
The following results from [24], [25] play an important role in this section.
Theorem 2.
Let ${\mbox{\boldmath$\omega$}}_{s}\in\cal P^{+}_{\xi}$ for $1\leq s\leq r$. Then $[{\mbox{\boldmath$\omega$}}_{1}]\otimes\cdots\otimes[{\mbox{\boldmath$\omega$}%
}_{s}]$ is $\ell$–highest weight if every pair $[{\mbox{\boldmath$\omega$}}_{s}]\otimes[{\mbox{\boldmath$\omega$}}_{p}]$ with $1\leq s<p\leq r$
is $\ell$–highest weight. Moreover if $[{\mbox{\boldmath$\omega$}}_{s}]\otimes[{\mbox{\boldmath$\omega$}}_{p}]\cong[{%
\mbox{\boldmath$\omega$}}_{s}{\mbox{\boldmath$\omega$}}_{p}]$ for all $1\leq s<p\leq r$ then
$$[{\mbox{\boldmath$\omega$}}_{1}]\otimes\cdots\otimes[{\mbox{\boldmath$\omega$}%
}_{r}]\cong[{\mbox{\boldmath$\omega$}}_{1}\cdots{\mbox{\boldmath$\omega$}}_{r}].$$
∎
3.2.
Given ${\mbox{\boldmath$\omega$}}_{i,a},{\mbox{\boldmath$\omega$}}_{j,b}\in\cal P_{%
\xi}^{+}$ it was shown in [6] that the module $[{\mbox{\boldmath$\omega$}}_{i,a}]\otimes[{\mbox{\boldmath$\omega$}}_{j,b}]$ is $\ell$–highest weight (resp. irreducible) if
$$(b-a)\notin\{2p+2-i-j:\max\{i,j\}<p+1\leq\min\{n+1,i+j\},$$
$${\rm{(resp.}}\ \ \pm(b-a)\notin\{2p+2-i-j:\max\{i,j\}<p+1\leq\min\{n+1,i+j\}).$$
The next proposition is a simple calculation using the preceding criterion and the fact that $|\xi(j)-\xi(i)|\leq|j-i|$ for all $i,j\in[1,n]$.
Proposition.
Let ${\mbox{\boldmath$\omega$}}_{i,a},{\mbox{\boldmath$\omega$}}_{j,b}\in\cal P^{+}%
_{\xi}$. Then $[{\mbox{\boldmath$\omega$}}_{i,a}]\otimes[{\mbox{\boldmath$\omega$}}_{j,b}]$ (resp. $[{\mbox{\boldmath$\omega$}}_{j,b}]\otimes[{\mbox{\boldmath$\omega$}}_{i,a}]$) is $\ell$–highest weight if $a=\xi(i)+1$ (resp. $a=\xi(i)-1$). Moreover,
$${\mbox{\boldmath$\omega$}}_{i,a}{\mbox{\boldmath$\omega$}}_{j,b}\notin\mathbf{%
P}\mathbf{r}_{\xi}\cup\{\mathbf{f}_{i}\}\implies[{\mbox{\boldmath$\omega$}}_{i%
,a}]\otimes[{\mbox{\boldmath$\omega$}}_{j,b}]\cong[{\mbox{\boldmath$\omega$}}_%
{i,a}{\mbox{\boldmath$\omega$}}_{j,b}].$$
∎
3.3.
Let $\xi^{*}$ be the height function defined by $\xi^{*}(i)=\xi(n+1-i)$. The assignment
$${\mbox{\boldmath$\omega$}}_{i,\xi(i)\pm 1}\to{\mbox{\boldmath$\omega$}}_{n+1-i%
,\xi^{*}(n+1-i)\pm 1}$$
extends to an isomorphism $\cal P^{+}_{\xi}\cong\cal P^{+}_{\xi^{*}}$, and if ${\mbox{\boldmath$\omega$}}={\mbox{\boldmath$\omega$}}_{i_{1},a_{1}}\cdots{%
\mbox{\boldmath$\omega$}}_{i_{k},a_{k}}\in\cal P_{\xi}^{+}$ we set
$${\mbox{\boldmath$\omega$}}^{*}={\mbox{\boldmath$\omega$}}_{n+1-i_{1},a_{1}}%
\cdots{\mbox{\boldmath$\omega$}}_{n+1-i_{k},a_{k}}\in\cal P^{+}_{\xi^{*}}.$$
It was proved in [9] that if $[{\mbox{\boldmath$\omega$}}_{1}]\otimes[{\mbox{\boldmath$\omega$}}_{2}]$ and $[{\mbox{\boldmath$\omega$}}_{2}^{*}]\otimes[{\mbox{\boldmath$\omega$}}_{1}^{*}]$ are both $\ell$–highest weight then they are both irreducible with the converse being trivially true.
Say that an ordered triple of elements $({\mbox{\boldmath$\omega$}}_{1},{\mbox{\boldmath$\omega$}}_{2},{\mbox{%
\boldmath$\omega$}}_{3})$ from $\cal P^{+}_{\xi}$ is $\xi$-admissible if:
•
$[{\mbox{\boldmath$\omega$}}_{s}]\otimes[{\mbox{\boldmath$\omega$}}_{3}]$ is irreducible for $s=1,2$,
•
either $[{\mbox{\boldmath$\omega$}}_{1}]\otimes[{\mbox{\boldmath$\omega$}}_{2}]$
or $[{\mbox{\boldmath$\omega$}}_{2}]\otimes[{\mbox{\boldmath$\omega$}}_{1}]$
is $\ell$–highest weight
Lemma.
If $({\mbox{\boldmath$\omega$}}_{1},{\mbox{\boldmath$\omega$}}_{2},{\mbox{%
\boldmath$\omega$}}_{3})$
is $\xi$-admissible and $({\mbox{\boldmath$\omega$}}_{1}^{*},{\mbox{\boldmath$\omega$}}_{2}^{*},{\mbox{%
\boldmath$\omega$}}_{3}^{*})$ is $\xi^{*}$-admissible then $[{\mbox{\boldmath$\omega$}}_{1}{\mbox{\boldmath$\omega$}}_{2}]\otimes[{\mbox{%
\boldmath$\omega$}}_{3}]\cong[{\mbox{\boldmath$\omega$}}_{1}{\mbox{\boldmath$%
\omega$}}_{2}{\mbox{\boldmath$\omega$}}_{3}].$
Proof.
Suppose that $({\mbox{\boldmath$\omega$}}_{1},{\mbox{\boldmath$\omega$}}_{2},{\mbox{%
\boldmath$\omega$}}_{3})$
is $\xi$-admissible and that $[{\mbox{\boldmath$\omega$}}_{1}]\otimes[{\mbox{\boldmath$\omega$}}_{2}]$ is $\ell$–highest weight. Then Theorem 2 shows that the modules $[{\mbox{\boldmath$\omega$}}_{1}]\otimes[{\mbox{\boldmath$\omega$}}_{2}]\otimes%
[{\mbox{\boldmath$\omega$}}_{3}]$ and $[{\mbox{\boldmath$\omega$}}_{3}]\otimes[{\mbox{\boldmath$\omega$}}_{1}]\otimes%
[{\mbox{\boldmath$\omega$}}_{2}]$ are $\ell$–highest weight. Hence the corresponding quotients $[{\mbox{\boldmath$\omega$}}_{1}{\mbox{\boldmath$\omega$}}_{2}]\otimes[{\mbox{%
\boldmath$\omega$}}_{3}]$ and $[{\mbox{\boldmath$\omega$}}_{3}]\otimes[{\mbox{\boldmath$\omega$}}_{1}{\mbox{%
\boldmath$\omega$}}_{2}]$ are $\ell$–highest weight. Working with $\xi^{*}$ we see similarly that $[{\mbox{\boldmath$\omega$}}_{1}^{*}{\mbox{\boldmath$\omega$}}_{2}^{*}]\otimes[%
{\mbox{\boldmath$\omega$}}_{3}^{*}]$ and $[{\mbox{\boldmath$\omega$}}_{3}^{*}]\otimes[{\mbox{\boldmath$\omega$}}_{1}^{*}%
{\mbox{\boldmath$\omega$}}_{2}^{*}]$ are $\ell$–highest weight. The irreducibility of the four quotient modules follows from the discussion preceding the statement of the Lemma.
∎
3.4.
Recall the map $\operatorname{wt}:\cal P^{+}_{\xi}\to P^{+}$ given by extending $\operatorname{wt}{\mbox{\boldmath$\omega$}}_{i,a}=\omega_{i}$ to a morphism of monoids; for ${\mbox{\boldmath$\pi$}}\in\cal P_{\xi}^{+}$ with $\operatorname{wt}{\mbox{\boldmath$\pi$}}=\sum_{i=1}^{n}r_{i}\omega_{i}$ set
$$\displaystyle\operatorname{ht}{\mbox{\boldmath$\pi$}}=\sum_{i=1}^{n}r_{i},\ \ %
\min{\mbox{\boldmath$\pi$}}=\min\{i\in I:r_{i}\neq 0\},\ \ \max{\mbox{%
\boldmath$\pi$}}=\max\{i\in I:r_{i}\neq 0\}.$$
If ${\mbox{\boldmath$\pi$}}\in\mathbf{P}\mathbf{r}_{\xi}$ and $b\in\{\xi(j)+1,\xi(j)-1\}$, $j\in[1,n]$ are such that ${\mbox{\boldmath$\omega$}}_{j,b}^{-1}{\mbox{\boldmath$\pi$}}\in\mathbf{P}%
\mathbf{r}_{\xi}$ then $j\in\{\min{\mbox{\boldmath$\pi$}},\max{\mbox{\boldmath$\pi$}}\}$.
Proposition.
Let $\xi$ be an arbitrary height function and let ${\mbox{\boldmath$\omega$}},{\mbox{\boldmath$\omega$}}^{\prime}\in\mathbf{P}%
\mathbf{r}_{\xi}$.
(i)
Suppose that ${\mbox{\boldmath$\omega$}}{\mbox{\boldmath$\omega$}}^{\prime}\in\mathbf{P}%
\mathbf{r}_{\xi}$ and write ${\mbox{\boldmath$\omega$}}={\mbox{\boldmath$\omega$}}_{1}{\mbox{\boldmath$%
\omega$}}_{i,a}$ with ${\mbox{\boldmath$\omega$}}_{1}\in\mathbf{P}\mathbf{r}_{\xi}$ and ${\mbox{\boldmath$\omega$}}_{i,a}{\mbox{\boldmath$\omega$}}^{\prime}\in\mathbf{%
P}\mathbf{r}_{\xi}$.
If $a=\xi(i)+1$ then $[{\mbox{\boldmath$\omega$}}]\otimes[{\mbox{\boldmath$\omega$}}^{\prime}]$ is $\ell$–highest weight and otherwise $[{\mbox{\boldmath$\omega$}}^{\prime}]\otimes[{\mbox{\boldmath$\omega$}}]$ is $\ell$–highest weight.
(ii)
If $i=\max{\mbox{\boldmath$\omega$}}<j=\min{\mbox{\boldmath$\omega$}}^{\prime}$ and $|\xi(i)-\xi(j)|\neq j-i$, then the module $[{\mbox{\boldmath$\omega$}}]\otimes[{\mbox{\boldmath$\omega$}}^{\prime}]$ is
irreducible.
Proof.
The proof of both parts is by an induction on $p=\operatorname{ht}{\mbox{\boldmath$\omega$}}+\operatorname{ht}{\mbox{%
\boldmath$\omega$}}^{\prime}$ with Proposition 3.2 showing that induction begins when $p=2$. For the inductive step assume that we have proved both parts for $p^{\prime}<p$ and also assume without loss of generality that $\operatorname{ht}{\mbox{\boldmath$\omega$}}^{\prime}\geq 2$.
For part (i) write
$${\mbox{\boldmath$\omega$}}^{\prime}={\mbox{\boldmath$\omega$}}_{j,b}{\mbox{%
\boldmath$\omega$}}^{\prime\prime},\ \ {\rm{with}}\ \ {\mbox{\boldmath$\omega$%
}}^{\prime\prime}\in\mathbf{P}\mathbf{r}_{\xi},\ \ {\mbox{\boldmath$\omega$}}{%
\mbox{\boldmath$\omega$}}_{j,b}\in\mathbf{P}\mathbf{r}_{\xi}$$
and observe that
if $k\in\{\min{\mbox{\boldmath$\omega$}}^{\prime\prime},\max{\mbox{\boldmath$%
\omega$}}^{\prime\prime}\}$ then $|\xi(i)-\xi(k)|\neq|k-i|$. The inductive hypothesis applies to the pairs $({\mbox{\boldmath$\omega$}}_{j,b},{\mbox{\boldmath$\omega$}}^{\prime\prime})$ and to $({\mbox{\boldmath$\omega$}},{\mbox{\boldmath$\omega$}}_{j,b})$; hence either both of $[{\mbox{\boldmath$\omega$}}_{j,b}]\otimes[{\mbox{\boldmath$\omega$}}^{\prime%
\prime}]$ and $[{\mbox{\boldmath$\omega$}}_{j,b}]\otimes[{\mbox{\boldmath$\omega$}}]$ or both of $[{\mbox{\boldmath$\omega$}}^{\prime\prime}]\otimes[{\mbox{\boldmath$\omega$}}_%
{j,b}]$ and $[{\mbox{\boldmath$\omega$}}]\otimes[{\mbox{\boldmath$\omega$}}_{j,b}]$ are $\ell$–highest weight.
The inductive hypothesis from part (ii) applies to the pair $({\mbox{\boldmath$\omega$}},{\mbox{\boldmath$\omega$}}^{\prime\prime})$ and so $[{\mbox{\boldmath$\omega$}}]\otimes[{\mbox{\boldmath$\omega$}}^{\prime\prime}]$ is irreducible. It follows from Theorem 2 that the module $[{\mbox{\boldmath$\omega$}}_{j,b}]\otimes[{\mbox{\boldmath$\omega$}}^{\prime%
\prime}]\otimes[{\mbox{\boldmath$\omega$}}]$ is $\ell$–highest weight (or $[{\mbox{\boldmath$\omega$}}]\otimes[{\mbox{\boldmath$\omega$}}^{\prime\prime}]%
\otimes[{\mbox{\boldmath$\omega$}}_{j,b}]$ is $\ell$–highest weight). Hence the quotient $[{\mbox{\boldmath$\omega$}}^{\prime}]\otimes[{\mbox{\boldmath$\omega$}}]$ (or $[{\mbox{\boldmath$\omega$}}]\otimes[{\mbox{\boldmath$\omega$}}^{\prime}])$ is $\ell$–highest weight and the inductive step for (i) is proved.
For part (ii) we continue to write ${\mbox{\boldmath$\omega$}}^{\prime}={\mbox{\boldmath$\omega$}}_{j,b}{\mbox{%
\boldmath$\omega$}}^{\prime\prime}$ and observe that the inductive hypothesis applies to the pairs $({\mbox{\boldmath$\omega$}},{\mbox{\boldmath$\omega$}}_{j,b})$ and $({\mbox{\boldmath$\omega$}},{\mbox{\boldmath$\omega$}}^{\prime\prime})$ and gives that $[{\mbox{\boldmath$\omega$}}]\otimes[{\mbox{\boldmath$\omega$}}_{j,b}]$ and $[{\mbox{\boldmath$\omega$}}]\otimes[{\mbox{\boldmath$\omega$}}^{\prime\prime}]$ are irreducible. Since the inductive step has been proved for part (i) it applies to the pair $({\mbox{\boldmath$\omega$}}_{j,b},{\mbox{\boldmath$\omega$}}^{\prime\prime})$ and so we see that $({\mbox{\boldmath$\omega$}}^{\prime\prime},{\mbox{\boldmath$\omega$}}_{j,b},{%
\mbox{\boldmath$\omega$}})$ is $\xi$-admissible. The conditions of the proposition obviously hold for $\xi$ iff they hold for $\xi^{*}$; hence it follows from Lemma 3.3 that $[{\mbox{\boldmath$\omega$}}]\otimes[{\mbox{\boldmath$\omega$}}^{\prime}]$ is irreducible.
∎
3.5.
The next proposition is essential to prove our main result.
Proposition.
Suppose that $1\leq j_{1}\leq j_{2}\leq j_{3}\leq j_{4}\leq n$ are such that $|\xi(j_{s})-\xi(j_{s+1})|=j_{s+1}-j_{s}$ for all $1\leq s\leq 3$.
(i)
Let $j_{1}<j_{2}$ and ${\mbox{\boldmath$\omega$}}_{j_{3},a}\in\mathbf{P}\mathbf{r}_{\xi}$. The module
$[{\mbox{\boldmath$\omega$}}(j_{1},j_{2})]\otimes[{\mbox{\boldmath$\omega$}}_{j%
_{3},a}]$ is irreducible if
$${\mbox{\boldmath$\omega$}}(j_{1},j_{2}){\mbox{\boldmath$\omega$}}_{j_{3},a}%
\notin\{{\mbox{\boldmath$\omega$}}(j_{1},j_{3}),\ \ \mathbf{f}_{3}{\mbox{%
\boldmath$\omega$}}_{j_{1},\xi(j_{1})\pm 1}\}.$$
An analogous statement holds if $j_{2}<j_{3}$ and ${\mbox{\boldmath$\omega$}}_{j_{1},a}\in\mathbf{P}\mathbf{r}_{\xi}$.
(ii)
Let $j_{1}<j_{2}<j_{3}$. The following modules are irreducible :
$\bullet$ $[{\mbox{\boldmath$\omega$}}(j_{1},j_{2})]\otimes[{\mbox{\boldmath$\omega$}}(j_%
{1},j_{2})]$,
$\bullet$ $[{\mbox{\boldmath$\omega$}}(j_{1},j_{2})]\otimes[{\mbox{\boldmath$\omega$}}(j_%
{1},j_{3})]$ if $\xi(j_{2}-1)\neq\xi(j_{2}+1)$,
$\bullet$ $[{\mbox{\boldmath$\omega$}}(j_{1},j_{2})]\otimes[{\mbox{\boldmath$\omega$}}(j_%
{2},j_{3})]$ if
$\xi(j_{2}-1)=\xi(j_{2}+1)$.
(iii)
Assume that $j_{1}<j_{2}<j_{3}<j_{4}$. Then the following are irreducible:
$\bullet$
$[{\mbox{\boldmath$\omega$}}(j_{1},j_{4})]\otimes[{\mbox{\boldmath$\omega$}}(j_%
{2},j_{3})]$ if $|\xi(j_{4})-\xi(j_{1})|=j_{4}-j_{1}$,
$\bullet$ $[{\mbox{\boldmath$\omega$}}(j_{1},j_{2})]\otimes[{\mbox{\boldmath$\omega$}}(j_%
{3},j_{4})]$ if ${\mbox{\boldmath$\omega$}}(j_{1},j_{2}){\mbox{\boldmath$\omega$}}(j_{3},j_{4})%
\neq{\mbox{\boldmath$\omega$}}(j_{1},j_{4}).$
Proof.
Part (i) was proved in [28] if $|\xi(j_{3})-\xi(j_{1})|=j_{3}-j_{1}$. If $|\xi(j_{3})-\xi(j_{1})|\neq j_{3}-j_{1}$writing ${\mbox{\boldmath$\omega$}}(j_{1},j_{2})={\mbox{\boldmath$\omega$}}_{j_{1},a_{1%
}}{\mbox{\boldmath$\omega$}}_{j_{2},a_{2}}$, our assumptions force,
$$j_{2}\neq j_{3},\ \ \xi(j_{2}-1)=\xi(j_{2}+1),\ \ {\mbox{\boldmath$\omega$}}_{%
j_{2},a_{2}}{\mbox{\boldmath$\omega$}}_{j_{3},a}\neq{\mbox{\boldmath$\omega$}}%
(j_{2},j_{3}).$$
Proposition 3.2 now shows that $({\mbox{\boldmath$\omega$}}_{j_{1},a_{1}},{\mbox{\boldmath$\omega$}}_{j_{2},a_%
{2}},{\mbox{\boldmath$\omega$}}_{j_{3},a})$ is a $\xi$–admissible triple. It also proves that
$({\mbox{\boldmath$\omega$}}_{n+1-j_{1},a_{1}},{\mbox{\boldmath$\omega$}}_{n+1-%
j_{2},a_{2}},{\mbox{\boldmath$\omega$}}_{n+1-j_{3},a})$ is $\xi^{*}$-admissible and the hence Lemma 3.3 gives the result.
The proof of the analogous statement for ${\mbox{\boldmath$\omega$}}_{j_{1},a}$ is entirely similar.
The first two assertions in part (ii) were proved in [28]. Suppose that $j_{2}<j_{3}$ and $\xi(j_{2}-1)=\xi(j_{2}+1)$ and write
$${\mbox{\boldmath$\omega$}}(j_{1},j_{2})={\mbox{\boldmath$\omega$}}_{j_{1},a_{1%
}}{\mbox{\boldmath$\omega$}}_{j_{2},a_{2}},\ \ {\mbox{\boldmath$\omega$}}(j_{2%
},j_{3})={\mbox{\boldmath$\omega$}}_{j_{2},a_{2}}{\mbox{\boldmath$\omega$}}_{j%
_{3},a_{3}}.$$
Then
$$a_{1}=\xi(j_{1})\pm 1\iff a_{2}=\xi(j_{2})\mp 1,\ \ a_{3}=\xi(j_{3})\pm 1.$$
Assuming that $a_{1}=\xi(j_{1})+1$ we use Proposition 3.4 and Theorem 2 and part (i) of this proposition to see that $[{\mbox{\boldmath$\omega$}}_{j_{3},a_{3}}]\otimes[{\mbox{\boldmath$\omega$}}_{%
j_{2},a_{2}}]\otimes[{\mbox{\boldmath$\omega$}}(j_{1},j_{2})]$ is $\ell$–highest weight and hence so is the quotient $[{\mbox{\boldmath$\omega$}}(j_{2},j_{3})]\otimes[{\mbox{\boldmath$\omega$}}(j_%
{1},j_{2})]$. Similarly working with $[{\mbox{\boldmath$\omega$}}_{j_{1},a_{1}}]\otimes[{\mbox{\boldmath$\omega$}}_{%
j_{2},a_{2}}]\otimes[{\mbox{\boldmath$\omega$}}(j_{2},j_{3})]$ we see that $[{\mbox{\boldmath$\omega$}}(j_{1},j_{2})]\otimes[{\mbox{\boldmath$\omega$}}(j_%
{2},j_{3})]$ is $\ell$–highest weight. Repeating the argument with $\xi^{*}$ proves the irreducibility and proves the third assertion of part (ii).
The first assertion in (iii) was proved in [28].
If $|\xi(j_{4})-\xi(j_{1})|\neq j_{4}-j_{1}$ then either $\xi(j_{2}-1)=\xi(j_{2}+1)\ \ {\rm{or}}\ \ \xi(j_{3}-1)=\xi(j_{3}+1)$. Write
$${\mbox{\boldmath$\omega$}}(j_{1},j_{2})={\mbox{\boldmath$\omega$}}_{j_{1},a_{1%
}}{\mbox{\boldmath$\omega$}}_{j_{2},a_{2}},\qquad{\mbox{\boldmath$\omega$}}(j_%
{3},j_{4})={\mbox{\boldmath$\omega$}}_{j_{3},a_{3}}{\mbox{\boldmath$\omega$}}_%
{j_{4},a_{4}},$$
and observe that since $j_{2}<j_{3}$ and ${\mbox{\boldmath$\omega$}}(j_{1},j_{2}){\mbox{\boldmath$\omega$}}(j_{3},j_{4})%
\neq{\mbox{\boldmath$\omega$}}(j_{1},j_{4})$ we get
$$\displaystyle(a_{1},a_{2})=(\xi(j_{1})\mp 1,\xi(j_{2})\pm 1)\iff(a_{3},a_{4})=%
(\xi(j_{3})\pm 1,\xi(j_{4})\mp 1).$$
If $\xi(j_{3}-1)=\xi(j_{3}+1)$ then $|\xi(j_{4})-\xi(j_{2})|\neq(j_{4}-j_{2})$. Using parts (i) and (ii) of the current proposition and Proposition 3.4 we see that $({\mbox{\boldmath$\omega$}}_{j_{3},a_{3}},{\mbox{\boldmath$\omega$}}_{j_{4},a_%
{4}},{\mbox{\boldmath$\omega$}}(j_{1},j_{2}))$ is $\xi$-admissible. If $\xi(j_{2}-1)=\xi(j_{2}+1)$ then an identical argument shows that
$({\mbox{\boldmath$\omega$}}_{j_{1},a_{1}},{\mbox{\boldmath$\omega$}}_{j_{2},a_%
{2}},{\mbox{\boldmath$\omega$}}(j_{3},j_{4}))$ is $\xi$-admissible.
Since the analogous equalities hold for $\xi^{*}$ Lemma 3.3 now proves the irreduciblity of $[{\mbox{\boldmath$\omega$}}(j_{1},j_{2})]\otimes[{\mbox{\boldmath$\omega$}}(j_%
{3},j_{4})]$.
∎
3.6.
In the rest of the section we shall prove the following theorem.
Theorem 3.
(a)
For all ${\mbox{\boldmath$\pi$}}\in\mathbf{P}\mathbf{r}_{\xi}$ and $k\in[1,n]$ we have $[{\mbox{\boldmath$\pi$}}][\mathbf{f}_{k}]=[{\mbox{\boldmath$\pi$}}\mathbf{f}_{%
k}]$.
(b)
Let ${\mbox{\boldmath$\pi$}}_{1},{\mbox{\boldmath$\pi$}}_{2}\in\mathbf{P}\mathbf{r}%
_{\xi}$. The equality $[{\mbox{\boldmath$\pi$}}_{1}{\mbox{\boldmath$\pi$}}_{2}]=[{\mbox{\boldmath$\pi%
$}}_{1}][{\mbox{\boldmath$\pi$}}_{2}]$ holds in $\cal K_{0}(\cal F_{\xi})$ if
one of the following conditions is satisfied:
(i)
${\mbox{\boldmath$\pi$}}_{1}{\mbox{\boldmath$\pi$}}_{2}\notin\mathbf{P}\mathbf{%
r}_{\xi}$ and $\max{\mbox{\boldmath$\pi$}}_{s}<\min{\mbox{\boldmath$\pi$}}_{m}$ for some $s,m\in[1,2]$.
(ii)
there exists $1\leq i\leq n$ and $a\in\{\xi(i)+1,\xi(i)-1\}$ such that
${\mbox{\boldmath$\omega$}}_{i,a}^{-1}{\mbox{\boldmath$\pi$}}_{s}\in\mathbf{P}%
\mathbf{r}_{\xi}$ for $s=1,2$,
(iii)
there exists $s,m\in\{1,2\}$ such that either
(a)
$\min{\mbox{\boldmath$\pi$}}_{s}<i=\min{\mbox{\boldmath$\pi$}}_{m}<j=\max{\mbox%
{\boldmath$\pi$}}_{s}<\max{\mbox{\boldmath$\pi$}}_{m}$ and $\operatorname{ht}{\mbox{\boldmath$\omega$}}(i,j)$ is odd, or
(b) $\min{\mbox{\boldmath$\pi$}}_{s}<i=\min{\mbox{\boldmath$\pi$}}_{m}<j=\max{\mbox%
{\boldmath$\pi$}}_{m}<\max{\mbox{\boldmath$\pi$}}_{s}$ and $\operatorname{ht}{\mbox{\boldmath$\omega$}}(i,j)$ is even.
3.7. Proof of Theorem 3
Notice that the hypothesis of the theorem hold for the pair $({\mbox{\boldmath$\pi$}}_{1},{\mbox{\boldmath$\pi$}}_{2})$ if and only if they hold for the pair $({\mbox{\boldmath$\pi$}}_{1}^{*},{\mbox{\boldmath$\pi$}}_{2}^{*})$ of elements in $\mathbf{P}\mathbf{r}_{\xi^{*}}$. In particular if we show that the conditions imply that we can write ${\mbox{\boldmath$\pi$}}_{1}={\mbox{\boldmath$\omega$}}_{1}{\mbox{\boldmath$%
\omega$}}_{2}$ so that $({\mbox{\boldmath$\omega$}}_{1},{\mbox{\boldmath$\omega$}}_{2},{\mbox{%
\boldmath$\pi$}}_{2})$ is $\xi$–admissible then the triple $({\mbox{\boldmath$\omega$}}_{1}^{*},{\mbox{\boldmath$\omega$}}_{2}^{*},{\mbox{%
\boldmath$\pi$}}_{2}^{*})$ is $\xi^{*}$–admissible. Lemma 3.3 then proves that $[{\mbox{\boldmath$\pi$}}_{1}]\otimes[{\mbox{\boldmath$\pi$}}_{2}]$ is irreducible. A similar comment applies to the pair $(\mathbf{f}_{p},{\mbox{\boldmath$\pi$}})$. This observation will be frequently used without further mention in the proof of the theorem.
We proceed by induction on $\operatorname{ht}{\mbox{\boldmath$\pi$}}$. If ${\mbox{\boldmath$\pi$}}={\mbox{\boldmath$\omega$}}_{i,a}$ and $|\xi(i)-\xi(k)|=|k-i|$ the result was proved in [28]. If $|\xi(i)-\xi(k)|\neq|k-i|$ then Proposition 3.4 shows that the triple $({\mbox{\boldmath$\omega$}}_{k,\xi(k)+1},{\mbox{\boldmath$\omega$}}_{k,\xi(k)-%
1},{\mbox{\boldmath$\omega$}}_{i,a})$ is $\xi$–admissible proving that $[{\mbox{\boldmath$\omega$}}_{i,a}]\otimes[\mathbf{f}_{k}]$ is irreducible.
If $\operatorname{ht}{\mbox{\boldmath$\pi$}}>1$ write ${\mbox{\boldmath$\pi$}}={\mbox{\boldmath$\omega$}}{\mbox{\boldmath$\omega$}}_{%
i,a}$ with $i=\max{\mbox{\boldmath$\pi$}}$ and ${\mbox{\boldmath$\omega$}}\in\mathbf{P}\mathbf{r}_{\xi}$. The inductive hypothesis and Proposition 3.4 show that the triples $({\mbox{\boldmath$\omega$}},{\mbox{\boldmath$\omega$}}_{i,a},\mathbf{f}_{k})$ is $\xi$-admissible and the inductive step is proved.
All three assertions in part (b) are proved by an induction on $p=\operatorname{ht}{\mbox{\boldmath$\pi$}}_{1}+\operatorname{ht}{\mbox{%
\boldmath$\pi$}}_{2}$. Proposition 3.5 shows that induction begins when $p\leq 3$. It also shows that the result hold when $\operatorname{ht}{\mbox{\boldmath$\pi$}}_{1}=\operatorname{ht}{\mbox{\boldmath%
$\pi$}}_{2}=2$. Hence for the inductive step we assume that the results hold for all $3\leq p^{\prime}<p$ and that either $\operatorname{ht}{\mbox{\boldmath$\pi$}}_{1}>2$ or $\operatorname{ht}{\mbox{\boldmath$\pi$}}_{2}>2$.
To prove the inductive step for (i) assume without loss of generality that $\operatorname{ht}{\mbox{\boldmath$\pi$}}_{1}>2$ and write ${\mbox{\boldmath$\pi$}}_{1}={\mbox{\boldmath$\omega$}}_{1}{\mbox{\boldmath$%
\omega$}}_{j_{1},a_{1}}{\mbox{\boldmath$\omega$}}_{j_{2},a_{2}}$ with ${\mbox{\boldmath$\omega$}}_{1}\in\mathbf{P}\mathbf{r}_{\xi}$ such that one of the following holds:
$$\displaystyle\max{\mbox{\boldmath$\pi$}}_{1}<\min{\mbox{\boldmath$\pi$}}_{2},%
\ \ \max{\mbox{\boldmath$\omega$}}_{1}<j_{1}<j_{2}=\max{\mbox{\boldmath$\pi$}}%
_{1}\ \ {\rm{and}}\ \ \xi(j_{1}-1)=\xi(j_{1}+1),$$
$$\displaystyle\max{\mbox{\boldmath$\pi$}}_{2}<\min{\mbox{\boldmath$\pi$}}_{1},%
\ \ \min{\mbox{\boldmath$\pi$}}_{1}=j_{1}<j_{2}<\min{\mbox{\boldmath$\omega$}}%
_{1}\ \ {\rm{and}}\ \ \xi(j_{2}-1)=\xi(j_{2}+1).$$
It follows that ${\mbox{\boldmath$\omega$}}_{1}{\mbox{\boldmath$\pi$}}_{2}\notin\mathbf{P}%
\mathbf{r}_{\xi}$ and since ${\mbox{\boldmath$\pi$}}_{1}{\mbox{\boldmath$\pi$}}_{2}\notin\mathbf{P}\mathbf{%
r}_{\xi}$ we also have ${\mbox{\boldmath$\omega$}}_{j_{1},a_{1}}{\mbox{\boldmath$\omega$}}_{j_{2},a_{2%
}}{\mbox{\boldmath$\pi$}}_{2}\notin\mathbf{P}\mathbf{r}_{\xi}$. Hence $[{\mbox{\boldmath$\omega$}}_{j_{1},a_{1}}{\mbox{\boldmath$\omega$}}_{j_{2},a_{%
2}}]\otimes[{\mbox{\boldmath$\pi$}}_{2}]$ and $[{\mbox{\boldmath$\omega$}}_{1}]\otimes[{\mbox{\boldmath$\pi$}}_{2}]$ are irreducible by the inductive hypothesis. Proposition 3.4 shows that either $[{\mbox{\boldmath$\omega$}}_{1}]\otimes[{\mbox{\boldmath$\omega$}}_{j_{1},a_{1%
}}{\mbox{\boldmath$\omega$}}_{j_{2},a_{2}}]$ or $[{\mbox{\boldmath$\omega$}}_{j_{1},a_{1}}{\mbox{\boldmath$\omega$}}_{j_{2},a_{%
2}}]\otimes[{\mbox{\boldmath$\omega$}}_{1}]$ is $\ell$–highest weight proving that the triple $({\mbox{\boldmath$\omega$}}_{1},{\mbox{\boldmath$\omega$}}_{j_{1},a_{1}}{\mbox%
{\boldmath$\omega$}}_{j_{2},a_{2}},{\mbox{\boldmath$\pi$}}_{2})$ is $\xi$–admissible. The proof of the inductive step for (i) is complete.
To prove the inductive step for (ii) notice that
the conditions on ${\mbox{\boldmath$\pi$}}_{1}$ and ${\mbox{\boldmath$\pi$}}_{2}$ imply that one of the following hold: $\max{\mbox{\boldmath$\pi$}}_{1}=\min{\mbox{\boldmath$\pi$}}_{2}=i$ and $\xi(i-1)=\xi(i+1)$ or $\min{\mbox{\boldmath$\pi$}}_{1}=\min{\mbox{\boldmath$\pi$}}_{2}=i$ or $\max{\mbox{\boldmath$\pi$}}_{1}=\max{\mbox{\boldmath$\pi$}}_{2}=i$.
Assume first that $\max{\mbox{\boldmath$\pi$}}_{1}=\min{\mbox{\boldmath$\pi$}}_{2}=i$. If $\operatorname{ht}{\mbox{\boldmath$\pi$}}_{1}\geq 3$ write ${\mbox{\boldmath$\pi$}}_{1}={\mbox{\boldmath$\omega$}}_{k,c}{\mbox{\boldmath$%
\omega$}}_{1}$ with $\max{\mbox{\boldmath$\omega$}}_{1}=i$; otherwise $\operatorname{ht}{\mbox{\boldmath$\pi$}}_{2}\geq 3$ write ${\mbox{\boldmath$\pi$}}_{2}={\mbox{\boldmath$\omega$}}_{k,c}{\mbox{\boldmath$%
\omega$}}_{2}$ and $\min{\mbox{\boldmath$\omega$}}_{2}=i$. In the first case, Proposition 3.4 and the inductive hypothesis show that the triple $({\mbox{\boldmath$\omega$}}_{k,c},{\mbox{\boldmath$\omega$}}_{1},{\mbox{%
\boldmath$\pi$}}_{2})$ is $\xi$–admissible while in the second case we get that $({\mbox{\boldmath$\omega$}}_{k,c},{\mbox{\boldmath$\omega$}}_{2},{\mbox{%
\boldmath$\pi$}}_{1})$ is $\xi$-admissible. In either case the irreducibility of $[{\mbox{\boldmath$\pi$}}_{1}]\otimes[{\mbox{\boldmath$\pi$}}_{2}]$ follows from Lemma 3.3. If $\min{\mbox{\boldmath$\pi$}}_{1}=\min{\mbox{\boldmath$\pi$}}_{2}$ assume without loss of generality that $\operatorname{ht}{\mbox{\boldmath$\pi$}}_{1}\leq\operatorname{ht}{\mbox{%
\boldmath$\pi$}}_{2}$ and let $k=\max{\mbox{\boldmath$\pi$}}_{1}$. Write ${\mbox{\boldmath$\pi$}}_{2}={\mbox{\boldmath$\omega$}}{\mbox{\boldmath$\omega$%
}}^{\prime}$ with ${\mbox{\boldmath$\omega$}},{\mbox{\boldmath$\omega$}}^{\prime}\in\mathbf{P}%
\mathbf{r}_{\xi}$ satisfying: $\min{\mbox{\boldmath$\omega$}}=\min{\mbox{\boldmath$\pi$}}_{1}$, $\max{\mbox{\boldmath$\omega$}}<k,$ $\min{\mbox{\boldmath$\omega$}}^{\prime}\geq k$ and $\min{\mbox{\boldmath$\omega$}}^{\prime}=k$ if $\xi(k-1)=\xi(k+1)$.
The inductive hypothesis applies to $({\mbox{\boldmath$\pi$}}_{1},{\mbox{\boldmath$\omega$}})$, it also applies to $({\mbox{\boldmath$\pi$}}_{1},{\mbox{\boldmath$\omega$}}^{\prime})$ if $\xi(k-1)=\xi(k+1)$ and otherwise ${\mbox{\boldmath$\pi$}}_{1}{\mbox{\boldmath$\omega$}}^{\prime}\notin\mathbf{P}%
\mathbf{r}_{\xi}$ and part (b)(i) applies and shows that the corresponding tensor products are irreducible.
Since Proposition 3.4 applies to $({\mbox{\boldmath$\omega$}},{\mbox{\boldmath$\omega$}}^{\prime})$ we have now shown that $({\mbox{\boldmath$\omega$}},{\mbox{\boldmath$\omega$}}^{\prime},{\mbox{%
\boldmath$\pi$}}_{1})$ is $\xi$–admissible and the inductive step is proved in this case
Finally, we prove the inductive step for (iii). This amounts to proving the following: if
$1\leq i_{1}<i_{2}<i_{3}<i_{4}\leq n$ then the tensor product $[{\mbox{\boldmath$\omega$}}(i_{1},i_{3})]\otimes[{\mbox{\boldmath$\omega$}}(i_%
{2},i_{4})]$ is irreducible if $\operatorname{ht}{\mbox{\boldmath$\omega$}}(i_{2},i_{3})$ is odd and $[{\mbox{\boldmath$\omega$}}(i_{1},i_{4})]\otimes[{\mbox{\boldmath$\omega$}}(i_%
{2},i_{3})]$ is irreducible if $\operatorname{ht}{\mbox{\boldmath$\omega$}}(i_{2},i_{3})$ is even. It is simple to see that $p=\operatorname{ht}{\mbox{\boldmath$\omega$}}(i_{1},i_{3})+\operatorname{ht}{%
\mbox{\boldmath$\omega$}}(i_{2},i_{4})=\operatorname{ht}{\mbox{\boldmath$%
\omega$}}(i_{1},i_{4})+\operatorname{ht}{\mbox{\boldmath$\omega$}}(i_{2},i_{3})$.
Since Proposition 3.5 shows that the result holds when $p=4$ it means that it holds whwn $\operatorname{ht}{\mbox{\boldmath$\omega$}}(i_{1},i_{4})=2$. Hence for the inductive step we may assume $\operatorname{ht}{\mbox{\boldmath$\omega$}}(i_{1},i_{4})\geq 3$.
Suppose that $\operatorname{ht}{\mbox{\boldmath$\omega$}}(i_{2},i_{3})=2$. Using Proposition 3.4, the inductive hypothesis and parts (b)(i),(ii) of this theorem we see that one of the following holds:
$\bullet$ there exists $i_{1}<m\leq i_{2}$ and $i_{4}>p\geq i_{3}$ such that ${\mbox{\boldmath$\omega$}}(i_{1},i_{4})={\mbox{\boldmath$\omega$}}(i_{1},m){%
\mbox{\boldmath$\omega$}}(p,i_{4})$ and $({\mbox{\boldmath$\omega$}}(i_{1},m),{\mbox{\boldmath$\omega$}}(p,i_{4}),{%
\mbox{\boldmath$\omega$}}(i_{2},i_{3}))$ is $\xi$–admissible,
$\bullet$ there exists $b\in\{\xi(i_{4})+1,\xi(i_{4})-1\}$ with ${\mbox{\boldmath$\omega$}}(i_{1},i_{4})={\mbox{\boldmath$\omega$}}(i_{1},i_{2}%
){\mbox{\boldmath$\omega$}}_{i_{4},b}$ and $({\mbox{\boldmath$\omega$}}(i_{1},i_{2}),{\mbox{\boldmath$\omega$}}_{i_{4},b},%
{\mbox{\boldmath$\omega$}}(i_{2},i_{3}))$ is $\xi$–admissible,
$\bullet$ there exists $a\in\{\xi(i_{1})+1,\xi(i_{1})-1\}$ with
${\mbox{\boldmath$\omega$}}(i_{1},i_{4})={\mbox{\boldmath$\omega$}}_{i_{1},a}{%
\mbox{\boldmath$\omega$}}(i_{3},i_{4})$
and $({\mbox{\boldmath$\omega$}}(i_{3},i_{4}),{\mbox{\boldmath$\omega$}}_{i_{1},a},%
{\mbox{\boldmath$\omega$}}(i_{2},i_{3}))$ is $\xi$–admissible.
In all cases the irreducibility of $[{\mbox{\boldmath$\omega$}}(i_{1},i_{4})]\otimes[{\mbox{\boldmath$\omega$}}(i_%
{2},i_{3})]$ is proved.
Suppose that $\operatorname{ht}{\mbox{\boldmath$\omega$}}(i_{2},i_{3})\geq 3$ and let $i_{2}<p<i_{3}$ be minimal such that $|\xi(p)-\xi(i_{2})|=p-i_{2}$ with $\xi(p-1)=\xi(p+1)$. Similarly, let $i_{2}<m<i_{3}$ be maximal so that $|\xi(i_{3})-\xi(m)|=i_{3}-m$ and $\xi(m-1)=\xi(m+1)$. Then Proposition 3.4, parts (b)(i) and (b)(ii) and the inductive hypothesis show that one of the following hold:
$\bullet$
if $\xi(i_{2}-1)=\xi(i_{2}+1)$, then
$$\displaystyle\operatorname{ht}{\mbox{\boldmath$\omega$}}(i_{2},i_{3})\ {\rm{%
odd}}\ \implies({\mbox{\boldmath$\omega$}}(i_{1},i_{2}),{\mbox{\boldmath$%
\omega$}}(p,i_{3}),{\mbox{\boldmath$\omega$}}(i_{2},i_{4}))\ \ {\rm{is}}\ \xi-%
{\rm{admissible}},$$
$$\displaystyle\operatorname{ht}{\mbox{\boldmath$\omega$}}(i_{2},i_{3})\ {\rm{%
even}}\ \implies({\mbox{\boldmath$\omega$}}(i_{1},i_{2}),{\mbox{\boldmath$%
\omega$}}(p,i_{4}),{\mbox{\boldmath$\omega$}}(i_{2},i_{3}))\ \ {\rm{is}}\ \ %
\xi-{\rm{admissible}},$$
$\bullet$
if $\xi(i_{3}-1)=\xi(i_{3}+1)$, then
$$\displaystyle\operatorname{ht}{\mbox{\boldmath$\omega$}}(i_{2},i_{3})\ {\rm{%
odd}}\ \implies({\mbox{\boldmath$\omega$}}(i_{2},m),{\mbox{\boldmath$\omega$}}%
(i_{3},i_{4}),{\mbox{\boldmath$\omega$}}(i_{1},i_{3}))\ \ {\rm{is}}\ \ \xi-{%
\rm{admissible}}$$
$$\displaystyle\operatorname{ht}{\mbox{\boldmath$\omega$}}(i_{2},i_{3})\ {\rm{%
even}}\ \implies{\mbox{\boldmath$\omega$}}(i_{1},m),{\mbox{\boldmath$\omega$}}%
(i_{3},i_{4}),{\mbox{\boldmath$\omega$}}(i_{2},i_{3}))\ \ {\rm{is}}\ \xi-{\rm{%
admissible}},$$
$\bullet$
$\xi(i_{j}+1)\neq\xi(i_{j}-1)$ for $j=2,3$.
If $p=m$ there exists $b\in\mathbb{C}(q)^{\times}$ such that ${\mbox{\boldmath$\omega$}}_{i_{4},b}^{-1}{\mbox{\boldmath$\omega$}}(i_{2},i_{4%
})\in\mathbf{P}\mathbf{r}_{\xi}$ and the triple
$({\mbox{\boldmath$\omega$}}(i_{2},p),{\mbox{\boldmath$\omega$}}_{i_{4},b},{%
\mbox{\boldmath$\omega$}}(i_{1},i_{3}))$ is $\xi$-admissible.
If $p\neq m$ let $p<p^{\prime}\leq m$ be minimum such that $\xi(p^{\prime}-1)=\xi(p^{\prime}+1)$. Then
$({\mbox{\boldmath$\omega$}}(i_{2},p),{\mbox{\boldmath$\omega$}}(p^{\prime},i_{%
4}),{\mbox{\boldmath$\omega$}}(i_{1},i_{3}))$ is $\xi$-admissible if $\operatorname{ht}{\mbox{\boldmath$\omega$}}(i_{2},i_{3})$ is odd and otherwise $({\mbox{\boldmath$\omega$}}(i_{2},p),{\mbox{\boldmath$\omega$}}(p^{\prime},i_{%
3}),{\mbox{\boldmath$\omega$}}(i_{1},i_{4}))$ is $\xi$–admissible.
In all cases the inductive step follows and the proof of the theorem is complete.
4. Identities in $\cal K_{0}(\cal F_{\xi})$
In this section we establish Proposition 1.6 and Proposition 1.7.
4.1.
We will need the converse of Theorem 3(b). The most elementary case is the following well–known. Namely, let $i\leq j$ satisfy $\xi(i)-\xi(j)=\pm(j-i)$; then the following equality holds in $\cal K_{0}(\cal F_{\xi})$:
$$[{\mbox{\boldmath$\omega$}}_{i,\xi(i)\pm 1}][{\mbox{\boldmath$\omega$}}_{j,\xi%
(j)\mp 1}]=[{\mbox{\boldmath$\omega$}}_{i,\xi(i)\pm 1}{\mbox{\boldmath$\omega$%
}}_{j,\xi(j)\mp 1}]+[{\mbox{\boldmath$\omega$}}_{i-1,\xi(i)}][{\mbox{\boldmath%
$\omega$}}_{j+1,\xi(j)}].$$
(4.1)
Given ${\mbox{\boldmath$\pi$}}={\mbox{\boldmath$\omega$}}{\mbox{\boldmath$\omega$}}_{%
i,a}\in\mathbf{P}\mathbf{r}_{\xi}$ with ${\mbox{\boldmath$\omega$}}\in\mathbf{P}\mathbf{r}_{\xi}$, set
$${\mbox{\boldmath$\pi$}}^{\prime}={\mbox{\boldmath$\omega$}}{\mbox{\boldmath$%
\omega$}}_{i-1,\xi(i)},\ \ i=\max{\mbox{\boldmath$\pi$}},\ \ ^{\prime}{\mbox{%
\boldmath$\pi$}}={\mbox{\boldmath$\omega$}}_{i+1,\xi(i)}{\mbox{\boldmath$%
\omega$}},\ \ i=\min{\mbox{\boldmath$\pi$}}.$$
In the remaining cases the converse is
most conveniently stated as follows.
Theorem 4.
(i)
Suppose that ${\mbox{\boldmath$\pi$}}_{1}{\mbox{\boldmath$\pi$}}_{2}\in\mathbf{P}\mathbf{r}_%
{\xi}$ and $\max{\mbox{\boldmath$\pi$}}_{1}<\min{\mbox{\boldmath$\pi$}}_{2}$. Then
$$[{\mbox{\boldmath$\pi$}}_{1}][{\mbox{\boldmath$\pi$}}_{2}]=[{\mbox{\boldmath$%
\pi$}}_{1}{\mbox{\boldmath$\pi$}}_{2}]+[{\mbox{\boldmath$\pi$}}_{1}^{\prime}][%
\,^{\prime}{\mbox{\boldmath$\pi$}}_{2}].$$
(4.2)
(ii)
Suppose that ${\mbox{\boldmath$\omega$}}(m,p)\in\mathbf{P}\mathbf{r}_{\xi}$
and for $m<i<p$, write
$${\mbox{\boldmath$\omega$}}(m,i)={\mbox{\boldmath$\omega$}}_{1}{\mbox{\boldmath%
$\omega$}}_{i,a},\ \ {\mbox{\boldmath$\omega$}}(i,p)={\mbox{\boldmath$\omega$}%
}_{i,b}{\mbox{\boldmath$\omega$}}_{2}.$$
If $a\neq b$ then
$$\displaystyle(*)\ \ [{\mbox{\boldmath$\omega$}}(m,p)][{\mbox{\boldmath$\omega$%
}}_{i,a}]=[{\mbox{\boldmath$\omega$}}(m,i)][{\mbox{\boldmath$\omega$}}_{2}]+[{%
\mbox{\boldmath$\omega$}}_{1}^{\prime}][^{\prime}{\mbox{\boldmath$\omega$}}(i,%
p)],$$
$$\displaystyle(**)\ \ [{\mbox{\boldmath$\omega$}}(m,p)][{\mbox{\boldmath$\omega%
$}}_{i,b}]=[{\mbox{\boldmath$\omega$}}_{1}][{\mbox{\boldmath$\omega$}}(i,p)]+[%
{\mbox{\boldmath$\omega$}}(m,i)^{\prime}][^{\prime}{\mbox{\boldmath$\omega$}}_%
{2}].$$
If $a=b$ then
$$\displaystyle(\dagger)\ \ [{\mbox{\boldmath$\omega$}}(m,p)][{\mbox{\boldmath$%
\omega$}}_{i,a^{\prime}}]=[{\mbox{\boldmath$\omega$}}_{1}][\mathbf{f}_{i}][{%
\mbox{\boldmath$\omega$}}_{2}]+[{\mbox{\boldmath$\omega$}}(m,i)^{\prime}][^{%
\prime}{\mbox{\boldmath$\omega$}}(i,p)],\quad\{a,a^{\prime}\}=\{\xi(i)+1,\xi(i%
)-1\},$$
$$\displaystyle(\dagger\dagger)\ \ [{\mbox{\boldmath$\omega$}}(m,p)][{\mbox{%
\boldmath$\omega$}}_{i,a}]=[{\mbox{\boldmath$\omega$}}(m,i)][{\mbox{\boldmath$%
\omega$}}(i,p)]+[{\mbox{\boldmath$\omega$}}_{1}^{\prime}][\mathbf{f}_{i}][^{%
\prime}{\mbox{\boldmath$\omega$}}_{2}].$$
Finally if ${\mbox{\boldmath$\pi$}}_{1}={\mbox{\boldmath$\omega$}}_{1}{\mbox{\boldmath$%
\omega$}}_{i,a}$ and ${\mbox{\boldmath$\pi$}}_{2}={\mbox{\boldmath$\omega$}}_{i,b}{\mbox{\boldmath$%
\omega$}}_{2}$ are in $\mathbf{P}\mathbf{r}_{\xi}$ with $\max{\mbox{\boldmath$\pi$}}_{1}=\min{\mbox{\boldmath$\pi$}}_{2}$ and $a\neq b$ then
$$[{\mbox{\boldmath$\pi$}}_{1}][{\mbox{\boldmath$\pi$}}_{2}]=[{\mbox{\boldmath$%
\omega$}}_{1}][\mathbf{f}_{i}][{\mbox{\boldmath$\omega$}}_{2}]+[{\mbox{%
\boldmath$\pi$}}_{1}^{\prime}][^{\prime}{\mbox{\boldmath$\pi$}}_{2}].$$
(4.3)
(iii)
Assume that $i_{1}<i_{2}<i_{3}<i_{4}$ and write
$${\mbox{\boldmath$\omega$}}(i_{1},i_{2})={\mbox{\boldmath$\omega$}}_{1}{\mbox{%
\boldmath$\omega$}}_{i_{2},a},\ \ {\mbox{\boldmath$\omega$}}(i_{2},i_{3})={%
\mbox{\boldmath$\omega$}}_{i_{2},b}{\mbox{\boldmath$\omega$}}{\mbox{\boldmath$%
\omega$}}_{i_{3},c},\ \ {\mbox{\boldmath$\omega$}}(i_{3},i_{4})={\mbox{%
\boldmath$\omega$}}_{i_{3},d}{\mbox{\boldmath$\omega$}}_{2}.$$
Then$(-1)^{\operatorname{ht}({\mbox{\boldmath\scriptsize$\omega$}}(i_{2},i_{3}))}%
\left([{\mbox{\boldmath$\omega$}}(i_{1},i_{3})][{\mbox{\boldmath$\omega$}}(i_{%
2},i_{4})]-[{\mbox{\boldmath$\omega$}}(i_{1},i_{4})][{\mbox{\boldmath$\omega$}%
}(i_{2},i_{3})]\right)$ is equal to
$$[{\mbox{\boldmath$\omega$}}_{1}^{\prime}]^{\delta_{a,b}}[{\mbox{\boldmath$%
\omega$}}(i_{1},i_{2})^{\prime})]^{1-\delta_{a,b}}\left(\prod_{s=i_{2}}^{i_{3}%
}[\mathbf{f}_{s}]^{\delta_{\xi(s-1),\xi(s+1)}}\right)[^{\prime}{\mbox{%
\boldmath$\omega$}}_{2}]^{\delta_{c,d}}[^{\prime}{\mbox{\boldmath$\omega$}}(i_%
{3},i_{4}))]^{1-\delta_{c,d}}.$$
From now on we freely use (often without mention) the results of Theorem 3. We deduce Proposition 1.6 and Proposition 1.7 before proving Theorem 4.
4.2. Proof of Proposition 1.6
The proposition is obviously a special case of equation (4.1) and Theorem 4(i),(ii). However the translation from the formulation in this section to the one in Section 1 which is adapted to cluster algebras needs some clarification which we provide for the readers convenience. We recall that $d_{j}=\delta_{\xi(j),\xi(j+2)}=\delta_{j,j_{\diamond}}$.
For part (i) of Proposition 1.6 we take
$${\mbox{\boldmath$\pi$}}_{1}={\mbox{\boldmath$\omega$}}_{i,\xi(i+1)},\qquad{%
\mbox{\boldmath$\pi$}}_{2}={\mbox{\boldmath$\omega$}}(i,i+1)^{1-d_{i}}{\mbox{%
\boldmath$\omega$}}_{i,\xi(i+1)\pm 2}^{d_{i}}={\mbox{\boldmath$\omega$}}_{i,%
\xi(i+1)\pm 2}{\mbox{\boldmath$\omega$}}_{i+1,\xi(i+2)}^{1-d_{i}},$$
where the second formula for ${\mbox{\boldmath$\pi$}}_{2}$ uses the fact that $\xi(i+2)=\xi(i+1)\mp 1=\xi(i)\mp 2$ if $d_{i}=0$. Theorem 3 gives
$$[{\mbox{\boldmath$\pi$}}_{1}{\mbox{\boldmath$\pi$}}_{2}]=[\mathbf{f}_{i}][{%
\mbox{\boldmath$\omega$}}_{i+1,\xi(i+2)}]^{1-d_{i}},\ \ \ \ [{\mbox{\boldmath$%
\omega$}}_{i+1,\xi(i)}{\mbox{\boldmath$\omega$}}_{i+1,\xi(i+2)}^{1-d_{i}}]=[%
\mathbf{f}_{i+1}]^{1-d_{i}}[{\mbox{\boldmath$\omega$}}_{i+1,\xi(i)}]^{d_{i}},$$
Using either (4.1) or (4.3) we get
$$[{\mbox{\boldmath$\pi$}}_{1}][{\mbox{\boldmath$\pi$}}_{2}]=[\mathbf{f}_{i}][{%
\mbox{\boldmath$\omega$}}_{i+1,\xi(i+2)}]^{1-d_{i}}+[{\mbox{\boldmath$\omega$}%
}_{i-1,\xi(i)}][\mathbf{f}_{i+1}]^{1-d_{i}}[{\mbox{\boldmath$\omega$}}_{i+1,%
\xi(i)}]^{d_{i}}$$
as needed.
For Proposition 1.6(ii) using the definition of $\bar{j}$ we can rewrite its left hand side as
$$[{\mbox{\boldmath$\omega$}}(i,\bar{j})]^{1-\delta_{j,i_{\diamond}}}[{\mbox{%
\boldmath$\omega$}}_{i,\xi(i+1)\pm 2}]^{\delta_{j,i_{\diamond}}}=[{\mbox{%
\boldmath$\omega$}}(i,j+1)]^{1-d_{j}}\left([{\mbox{\boldmath$\omega$}}_{i,\xi(%
i+1)\pm 2}]^{1-\delta_{i,j_{\bullet}}}[{\mbox{\boldmath$\omega$}}(i,j_{\bullet%
}+1)]^{\delta_{i,j_{\bullet}}}\right)^{d_{j}}.$$
It is easiest to verify the four cases given by $d_{j}\in\{0,1\}$ and $\delta_{j_{\bullet},i}\in\{0,1\}$ separately.
If $d_{j}=1$ the left hand side of Proposition 1.6(ii) is $[{\mbox{\boldmath$\pi$}}_{1}][{\mbox{\boldmath$\pi$}}_{2}]$ where
$${\mbox{\boldmath$\pi$}}_{1}={\mbox{\boldmath$\omega$}}_{j,\xi(j+1)},\ \ {\mbox%
{\boldmath$\pi$}}_{2}={\mbox{\boldmath$\omega$}}_{i,\xi(i+1)\pm 2}^{1-\delta_{%
i,j_{\bullet}}}{\mbox{\boldmath$\omega$}}(i,i+1)^{\delta_{i,j_{\bullet}}}={%
\mbox{\boldmath$\omega$}}_{i,\xi(i+1)\pm 2}{\mbox{\boldmath$\omega$}}_{i+1,\xi%
(i+2)\mp 2}^{\delta_{i,j_{\bullet}}}.$$
and the right hand side is
$$[\mathbf{f}_{j}]^{d_{j-1}}\ [{\mbox{\boldmath$\omega$}}(i,j)]^{1-d_{j-1}}[{%
\mbox{\boldmath$\omega$}}_{i,\xi(i+1)\pm 2}]^{d_{j-1}}+[\mathbf{f}_{i}]^{%
\delta_{i,j_{\bullet}}}[{\mbox{\boldmath$\omega$}}_{j+1,\xi(j+2)}][{\mbox{%
\boldmath$\omega$}}_{i-1,\xi(i)}]^{1-\delta_{i,j_{\bullet}}}.$$
Since $\delta_{j_{\bullet},i}=0\implies j_{\bullet}<i\leq j-1\implies d_{j-1}=0,$ and ${\mbox{\boldmath$\omega$}}(i,j)={\mbox{\boldmath$\omega$}}_{i,\xi(i+1)\pm 2}{%
\mbox{\boldmath$\omega$}}_{j,\xi(j+1)}$ we see that the right hand side of Proposition 1.6(ii) is precisely the right hand side of (4.1) and we are done. Otherwise
$$\delta_{j_{\bullet},i}=1\ \ {\rm and\ either}\ \ j_{\bullet}=j-1\ \ {\rm{or}}%
\ \ j_{\bullet}<j-1.$$
In the first case $d_{j-1}=1$ and $i+1=j$ and so the result follows from (4.3); in the second case we have $i+1<j$ and $d_{j-1}=0$. Since $i=j_{\bullet}=i_{\diamond}$ we also have $\xi(i)=\xi(i+2)$ which implies that ${\mbox{\boldmath$\omega$}}(i,i+1){\mbox{\boldmath$\omega$}}_{j,\xi(j+1)}={%
\mbox{\boldmath$\omega$}}(i,j)$, The result follows from Theorem 4(i).
If $d_{j}=0$ then the left hand side of Proposition 1.6 is $[{\mbox{\boldmath$\pi$}}_{1}][{\mbox{\boldmath$\pi$}}_{2}]$ where
$${\mbox{\boldmath$\pi$}}_{1}={\mbox{\boldmath$\omega$}}_{j,\xi(j)+1},\ \ \ {%
\mbox{\boldmath$\pi$}}_{2}={\mbox{\boldmath$\omega$}}(i,j+1)={\mbox{\boldmath$%
\omega$}}_{i,\xi(i+1)\pm 2}{\mbox{\boldmath$\omega$}}_{j,\xi(j+1)\mp 2}^{d_{j-%
1}}{\mbox{\boldmath$\omega$}}_{j+1,\xi(j+2)}$$
and the right hand side of Proposition 1.6 is
$$[\mathbf{f}_{j}]^{d_{j-1}}\ [{\mbox{\boldmath$\omega$}}(i,j)]^{1-d_{j-1}}[{%
\mbox{\boldmath$\omega$}}_{i,\xi(i+1)\pm 2}]^{d_{j-1}}[{\mbox{\boldmath$\omega%
$}}_{j+1,\xi(j+2)}]+[\mathbf{f}_{j+1}][\mathbf{f}_{i}]^{\delta_{i,j_{\bullet}}%
}[{\mbox{\boldmath$\omega$}}_{i-1,\xi(i)}]^{1-\delta_{i,j_{\bullet}}}.$$
If $d_{j-1}=1$ then $i=j-1=j_{\bullet}$
and the result follows from ($\dagger$) in Theorem 4(ii).
If $d_{j-1}=0$, then the result follows from equation ($*$) in Theorem 4(ii).
The proof of part (iii) is a similar detailed analyses. Note that $[{\mbox{\boldmath$\omega$}}(i,\bar{j})]=[{\mbox{\boldmath$\omega$}}(i,j_{%
\bullet}+1)]^{d_{j}}[{\mbox{\boldmath$\omega$}}(i,j+1)]^{1-d_{j}}$.
If $d_{j}=1$, we take
$${\mbox{\boldmath$\pi$}}_{1}={\mbox{\boldmath$\omega$}}_{j,\xi(j+1)},\ \ {\mbox%
{\boldmath$\pi$}}_{2}={\mbox{\boldmath$\omega$}}(i,j_{\bullet}+1)$$
and use Theorem 4(i) if $j_{\bullet}+1<j$ and (4.3) if $j_{\bullet}+1=j$. If $d_{j}=0$ we take
$${\mbox{\boldmath$\pi$}}_{1}={\mbox{\boldmath$\omega$}}_{j,\xi(j+1)},\ \ \ {%
\mbox{\boldmath$\pi$}}={\mbox{\boldmath$\omega$}}(i,j+1).$$
Note that ${\mbox{\boldmath$\omega$}}(i,j_{\bullet}+1){\mbox{\boldmath$\omega$}}_{j,\xi(j%
+1)}={\mbox{\boldmath$\omega$}}(i,j)\in\mathbf{P}\mathbf{r}$ if $j_{\bullet}+1<j$ and that if we write $k=(j_{\bullet})_{\bullet}$ then
$${\mbox{\boldmath$\pi$}}_{2}{\mbox{\boldmath$\omega$}}_{j,\xi(j+1)}=\begin{%
cases}({\mbox{\boldmath$\omega$}}_{i,\xi(i+1)\pm 2}^{\delta_{k,i_{\bullet}}}{%
\mbox{\boldmath$\omega$}}(i,k+1)^{1-\delta_{k,i_{\bullet}}})\mathbf{f}_{j}{%
\mbox{\boldmath$\omega$}}_{j+1,\xi(j+2)},\ \ d_{j-1}=1,\\
\\
{\mbox{\boldmath$\omega$}}(i,j_{\bullet}+1){\mbox{\boldmath$\omega$}}_{j+1,\xi%
(j+2)},\ \ d_{j-1}=0.\end{cases}$$
An application of Theorem 4 as in the other cases completes the proof.
4.3. Proof of Proposition 1.7
. Let ${\mbox{\boldmath$\omega$}},{\mbox{\boldmath$\omega$}}^{\prime}\in\mathbf{P}%
\mathbf{r}$ be such that $[{\mbox{\boldmath$\omega$}}{\mbox{\boldmath$\omega$}}^{\prime}]=[{\mbox{%
\boldmath$\omega$}}][{\mbox{\boldmath$\omega$}}^{\prime}]$. By Section 1.9 we can choose $\alpha,\beta\in\Phi_{\geq-1}$ such that
$$[{\mbox{\boldmath$\omega$}}]=\iota(x[\alpha]),\ \ [{\mbox{\boldmath$\omega$}}^%
{\prime}]=\iota(x[\beta]).$$
We claim that $x[\alpha]x[\beta]$ is a cluster monomial. If not, we can write
$$x[\alpha]x[\beta]=m_{1}x[\gamma]x[\eta]+m_{2}x[\gamma^{\prime}]x[\eta^{\prime}],$$
where $m_{1},m_{2}\in\mathbb{Z}_{\geq 0}[f_{i}:i\in I]$ where $\gamma,\gamma^{\prime},\eta),\eta^{\prime})$ are in $\Phi_{\geq-1}$.
Applying $\iota$ to both sides of the equation we get
$$[{\mbox{\boldmath$\omega$}}{\mbox{\boldmath$\omega$}}^{\prime}]=[{\mbox{%
\boldmath$\omega$}}][{\mbox{\boldmath$\omega$}}^{\prime}]=\iota(x[\alpha]x[%
\beta])=\iota(m_{1})\iota(x[\gamma])\iota(x[\eta])+\iota(m_{2})\iota(x[\gamma^%
{\prime}])\iota(x[\eta^{\prime}]).$$
Since $\iota(m_{1}),\iota(m_{2})\in\mathbb{Z}_{\geq 0}[\mathbf{f}_{i}:i\in I]$ this means that we can write $[{\mbox{\boldmath$\omega$}}{\mbox{\boldmath$\omega$}}^{\prime}]$ as a nontrivial linear combination of elements $\{[{\mbox{\boldmath$\pi$}}]:{\mbox{\boldmath$\pi$}}\in\cal P^{+}_{\xi}\}$ which is absurd.
Suppose now that ${\mbox{\boldmath$\pi$}},{\mbox{\boldmath$\pi$}}^{\prime}\in\mathbf{P}\mathbf{r%
}_{\xi}$ are such that $[{\mbox{\boldmath$\pi$}}][{\mbox{\boldmath$\pi$}}^{\prime}]\neq[{\mbox{%
\boldmath$\pi$}}{\mbox{\boldmath$\pi$}}^{\prime}]$. Theorem 3 amd Theorem 4 imply that $[{\mbox{\boldmath$\pi$}}{\mbox{\boldmath$\pi$}}^{\prime}]$ and $[{\mbox{\boldmath$\pi$}}][{\mbox{\boldmath$\pi$}}^{\prime}]-[{\mbox{\boldmath$%
\pi$}}{\mbox{\boldmath$\pi$}}^{\prime}]$ are in the $\mathbb{Z}_{\geq 0}[\mathbf{f}_{i}:i\in I]$- span of elements of the form $[{\mbox{\boldmath$\omega$}}]=[{\mbox{\boldmath$\omega$}}_{1}]\cdots[{\mbox{%
\boldmath$\omega$}}_{p}]$ with ${\mbox{\boldmath$\omega$}}_{s}\in\mathbf{P}\mathbf{r}_{\xi}$, $1\leq s\leq p$. By the previous discussion it follows that such products are the images of cluster monomials. Hence the inverse image under $\iota$ of $[{\mbox{\boldmath$\pi$}}][{\mbox{\boldmath$\pi$}}^{\prime}]$ is a positive linear combination of certain cluster monomials; in particular the inverse image is not a cluster monomial and the proof is complete.
4.4.
In the rest of the paper we prove Theorem 4. The crucial step is the following proposition whose proof we postpone to the next section.
Proposition.
Let ${\mbox{\boldmath$\omega$}}_{i,a},{\mbox{\boldmath$\pi$}}$ be elements of $\mathbf{P}\mathbf{r}_{\xi}$ with $i<\min{\mbox{\boldmath$\pi$}}=j$ or $i>\max{\mbox{\boldmath$\pi$}}=k$ and ${\mbox{\boldmath$\omega$}}_{i,a}{\mbox{\boldmath$\pi$}}\in\mathbf{P}\mathbf{r}%
_{\xi}$. Let $b,c\in\mathbb{C}(q)$ be such that ${\mbox{\boldmath$\omega$}}_{j,b}^{-1}{\mbox{\boldmath$\pi$}}$ and ${\mbox{\boldmath$\omega$}}_{k,c}^{-1}{\mbox{\boldmath$\pi$}}$ are elements of $\mathbf{P}\mathbf{r}_{\xi}$. We have
$$[{\mbox{\boldmath$\omega$}}_{i,a}][{\mbox{\boldmath$\pi$}}]-[{\mbox{\boldmath$%
\omega$}}_{i,a}{\mbox{\boldmath$\pi$}}]=[{\mbox{\boldmath$\omega$}}_{i-1,\xi(i%
)}][^{\prime}{\mbox{\boldmath$\pi$}}],\ \ i<j,$$
$$[{\mbox{\boldmath$\omega$}}_{i,a}][{\mbox{\boldmath$\pi$}}]-[{\mbox{\boldmath$%
\omega$}}_{i,a}{\mbox{\boldmath$\pi$}}]=[{\mbox{\boldmath$\omega$}}_{i+1,\xi(i%
)}][{\mbox{\boldmath$\pi$}}^{\prime}],\ \ i>k.$$
4.5. Proof of Theorem 4(i)
We need the following consequence of Proposition 4.4.
Lemma.
Let ${\mbox{\boldmath$\omega$}}_{i,a},{\mbox{\boldmath$\omega$}}_{i,b}{\mbox{%
\boldmath$\pi$}}\in\mathbf{P}\mathbf{r}_{\xi}$ and assume that $a\neq b$ and $\min{\mbox{\boldmath$\pi$}}>i$ (resp. $\max{\mbox{\boldmath$\pi$}}<i$). Then
$$[{\mbox{\boldmath$\omega$}}_{i,a}][{\mbox{\boldmath$\omega$}}_{i,b}{\mbox{%
\boldmath$\pi$}}]=[\mathbf{f}_{i}][{\mbox{\boldmath$\pi$}}]+[{\mbox{\boldmath$%
\omega$}}_{i-1,\xi(i)}][{\mbox{\boldmath$\omega$}}_{i+1,\xi(i)}{\mbox{%
\boldmath$\pi$}}],$$
$$({\rm{resp.}}\ \ [{\mbox{\boldmath$\omega$}}_{i,a}][{\mbox{\boldmath$\omega$}}%
_{i,b}{\mbox{\boldmath$\pi$}}]=[\mathbf{f}_{i}][{\mbox{\boldmath$\pi$}}]+[{%
\mbox{\boldmath$\omega$}}_{i+1,\xi(i)}][{\mbox{\boldmath$\omega$}}_{i-1,\xi(i)%
}{\mbox{\boldmath$\pi$}}].)$$
Proof.
Proceed by induction on $\operatorname{ht}{\mbox{\boldmath$\omega$}}_{i,b}{\mbox{\boldmath$\pi$}}$. If $\operatorname{ht}{\mbox{\boldmath$\omega$}}_{i,b}{\mbox{\boldmath$\pi$}}=1$ then the result is well–known (see for instance [20]). Assume that we have proved the result if $\operatorname{ht}{\mbox{\boldmath$\omega$}}_{i,b}{\mbox{\boldmath$\pi$}}<r$. Write ${\mbox{\boldmath$\pi$}}={\mbox{\boldmath$\omega$}}_{m,c}{\mbox{\boldmath$%
\omega$}}$ with $m=\min{\mbox{\boldmath$\pi$}}$
and note that
$$a=\xi(i)\mp 1\iff b=\xi(i)\pm 1\iff\xi(i+1)\pm 1=\xi(i)\ {\rm{and}}\ \ c=\xi(m%
)\mp 1.$$
It follows that the pair $({\mbox{\boldmath$\omega$}}_{i+1,\xi(i)},{\mbox{\boldmath$\omega$}}_{m,c}{%
\mbox{\boldmath$\omega$}})$ satisfies the conditions of Proposition 4.4 if $i+1\neq m$ and the inductive hypothesis of this Lemma if $i+1=m$ and so we have,
$$(*)\ \ [{\mbox{\boldmath$\omega$}}_{i+1,\xi(i)}][{\mbox{\boldmath$\omega$}}_{m%
,c}{\mbox{\boldmath$\omega$}}]=[{\mbox{\boldmath$\omega$}}_{i+1,\xi(i)}{\mbox{%
\boldmath$\omega$}}_{m,c}{\mbox{\boldmath$\omega$}}]+[{\mbox{\boldmath$\omega$%
}}_{i,a}][{\mbox{\boldmath$\omega$}}_{m+1,\xi(m)}{\mbox{\boldmath$\omega$}}].$$
The inductive hypothesis and Proposition 4.4 also give
$$\displaystyle[{\mbox{\boldmath$\omega$}}_{i,a}][{\mbox{\boldmath$\omega$}}_{i,%
b}][{\mbox{\boldmath$\pi$}}]$$
$$\displaystyle=[{\mbox{\boldmath$\omega$}}_{i,a}]\left([{\mbox{\boldmath$\omega%
$}}_{i,b}{\mbox{\boldmath$\pi$}}]+[{\mbox{\boldmath$\omega$}}_{i-1,\xi(i)}][{%
\mbox{\boldmath$\omega$}}{\mbox{\boldmath$\omega$}}_{m+1,\xi(m)}]\right)$$
$$\displaystyle=\left([\mathbf{f}_{i}]+[{\mbox{\boldmath$\omega$}}_{i-1,\xi(i)}]%
[{\mbox{\boldmath$\omega$}}_{i+1,\xi(i)}]\right)[{\mbox{\boldmath$\pi$}}]$$
$$\displaystyle=[\mathbf{f}_{i}][{\mbox{\boldmath$\pi$}}]+[{\mbox{\boldmath$%
\omega$}}_{i-1,\xi(i)}]\left([{\mbox{\boldmath$\omega$}}_{i+1,\xi(i)}{\mbox{%
\boldmath$\omega$}}_{m,c}{\mbox{\boldmath$\omega$}}]+[{\mbox{\boldmath$\omega$%
}}_{i,a}][{\mbox{\boldmath$\omega$}}_{m+1,\xi(m)}{\mbox{\boldmath$\omega$}}]%
\right).$$
Equating the first and third terms on the right hand side and using (*) gives
$$[\mathbf{f}_{i}][{\mbox{\boldmath$\pi$}}]+[{\mbox{\boldmath$\omega$}}_{i-1,\xi%
(i)}][{\mbox{\boldmath$\omega$}}_{i+1,\xi(i)}{\mbox{\boldmath$\omega$}}_{m,c}{%
\mbox{\boldmath$\omega$}}]=[{\mbox{\boldmath$\omega$}}_{i,a}][{\mbox{\boldmath%
$\omega$}}_{i,b}{\mbox{\boldmath$\pi$}}],$$
which establishes the inductive step.
∎
The proof of Theorem 4(i) proceeds by an induction on $\operatorname{ht}{\mbox{\boldmath$\pi$}}_{1}$ with Proposition 4.4 showing that induction begins when $\operatorname{ht}{\mbox{\boldmath$\pi$}}_{1}=1$.
For the inductive step, recall that ${\mbox{\boldmath$\pi$}}_{1}={\mbox{\boldmath$\omega$}}_{i,a}{\mbox{\boldmath$%
\omega$}}_{1}$ and ${\mbox{\boldmath$\pi$}}_{2}={\mbox{\boldmath$\omega$}}_{j,b}{\mbox{\boldmath$%
\omega$}}_{2}$ with $\max{\mbox{\boldmath$\pi$}}_{1}=i<j=\min{\mbox{\boldmath$\pi$}}_{2}$.
Since ${\mbox{\boldmath$\pi$}}_{1}{\mbox{\boldmath$\pi$}}_{2}\in\mathbf{P}\mathbf{r}_%
{\xi}$ we see that Proposition 4.4 applies to the pairs $({\mbox{\boldmath$\omega$}}_{1},{\mbox{\boldmath$\omega$}}_{i,a})$, $({\mbox{\boldmath$\omega$}}_{i,a},{\mbox{\boldmath$\pi$}}_{2})$ and also to the pairs $({\mbox{\boldmath$\omega$}}_{i+1,\xi(i)},{\mbox{\boldmath$\pi$}}_{2})$ and $({\mbox{\boldmath$\omega$}}_{1},{\mbox{\boldmath$\omega$}}_{i-1,\xi(i)})$ if $i+1\neq j$ and $i-1\neq\min{\mbox{\boldmath$\omega$}}_{1}$. If $i+1=j$ (resp. $i-1=\min{\mbox{\boldmath$\omega$}}_{1}$) then Lemma 4.5 applies to $({\mbox{\boldmath$\omega$}}_{i+1,\xi(i)},{\mbox{\boldmath$\pi$}}_{2})$ (resp.$({\mbox{\boldmath$\omega$}}_{1},{\mbox{\boldmath$\omega$}}_{i-1,\xi(i)})$). Together with the inductive hypothesis which applies to $({\mbox{\boldmath$\omega$}}_{1},{\mbox{\boldmath$\omega$}}_{i,a}{\mbox{%
\boldmath$\pi$}}_{2})$ we get the following series of equalities:
$$\displaystyle[{\mbox{\boldmath$\pi$}}_{1}][{\mbox{\boldmath$\pi$}}_{2}]+[{%
\mbox{\boldmath$\omega$}}_{1}^{\prime}]\left([{\mbox{\boldmath$\omega$}}_{i+1,%
\xi(i)}{\mbox{\boldmath$\pi$}}_{2}]+[{\mbox{\boldmath$\omega$}}_{i,\xi(i+1)}][%
^{\prime}{\mbox{\boldmath$\pi$}}_{2}]\right)$$
$$\displaystyle=\left([{\mbox{\boldmath$\pi$}}_{1}+[{\mbox{\boldmath$\omega$}}_{%
1}^{\prime}][{\mbox{\boldmath$\omega$}}_{i+1,\xi(i)}]\right)[{\mbox{\boldmath$%
\pi$}}_{2}]$$
$$\displaystyle=[{\mbox{\boldmath$\omega$}}_{1}][{\mbox{\boldmath$\omega$}}_{i,a%
}][{\mbox{\boldmath$\pi$}}_{2}]$$
$$\displaystyle=[{\mbox{\boldmath$\omega$}}_{1}]\left([{\mbox{\boldmath$\omega$}%
}_{i,a}{\mbox{\boldmath$\pi$}}_{2}]+[{\mbox{\boldmath$\omega$}}_{i-1,\xi(i)}][%
^{\prime}{\mbox{\boldmath$\pi$}}_{2}]\right)$$
$$\displaystyle=[{\mbox{\boldmath$\pi$}}_{1}{\mbox{\boldmath$\pi$}}_{2}]+[{\mbox%
{\boldmath$\omega$}}_{1}^{\prime}][{\mbox{\boldmath$\omega$}}_{i+1,\xi(i)}{%
\mbox{\boldmath$\pi$}}_{2}]+[{\mbox{\boldmath$\omega$}}_{1}{\mbox{\boldmath$%
\omega$}}_{i-1\xi(i)}][^{\prime}{\mbox{\boldmath$\pi$}}_{2}]+[{\mbox{\boldmath%
$\omega$}}_{1}^{\prime}][{\mbox{\boldmath$\omega$}}_{i,\xi(i-1)}][^{\prime}{%
\mbox{\boldmath$\pi$}}_{2}].$$
Equating the first and the fifth terms gives the inductive step since $\xi(i-1)=\xi(i+1)$ and part (i) is proved.
4.6. Proof of Theorem 4
(ii)
Suppose that $a\neq b$ which means that $\xi(i-1)\neq\xi(i+1)$ and hence ${\mbox{\boldmath$\omega$}}(m,p)={\mbox{\boldmath$\omega$}}_{1}{\mbox{\boldmath%
$\omega$}}_{2}$. We prove equation (**); the proof of (*) being an obvious modification.
Using Theorem 3(b)(i) gives that $[{\mbox{\boldmath$\omega$}}(m,p){\mbox{\boldmath$\omega$}}_{i,b}]=[{\mbox{%
\boldmath$\omega$}}_{1}][{\mbox{\boldmath$\omega$}}(i,p)]$
and we have to prove that
$$[{\mbox{\boldmath$\omega$}}(m,p)][{\mbox{\boldmath$\omega$}}_{i,b}]-[{\mbox{%
\boldmath$\omega$}}_{1}][{\mbox{\boldmath$\omega$}}(i,p)]=[{\mbox{\boldmath$%
\omega$}}(m,i)^{\prime}][^{\prime}{\mbox{\boldmath$\omega$}}_{2}].$$
For this we calculate $[{\mbox{\boldmath$\omega$}}_{1}][{\mbox{\boldmath$\omega$}}_{i,b}][{\mbox{%
\boldmath$\omega$}}_{2}]$ in two ways by using Proposition 4.4 on $({\mbox{\boldmath$\omega$}}_{i,b},{\mbox{\boldmath$\omega$}}_{2})$ and part (i) of the theorem on $({\mbox{\boldmath$\omega$}}_{1},{\mbox{\boldmath$\omega$}}_{2})$. This gives,
$$\displaystyle[{\mbox{\boldmath$\omega$}}_{1}][{\mbox{\boldmath$\omega$}}_{i,b}%
][{\mbox{\boldmath$\omega$}}_{2}]$$
$$\displaystyle=[{\mbox{\boldmath$\omega$}}_{1}][{\mbox{\boldmath$\omega$}}(i,p)%
]+[{\mbox{\boldmath$\omega$}}_{1}][{\mbox{\boldmath$\omega$}}_{i-1,\xi(i)}][^{%
\prime}{\mbox{\boldmath$\omega$}}_{2}]$$
$$\displaystyle=[{\mbox{\boldmath$\omega$}}_{i,b}][{\mbox{\boldmath$\omega$}}(m,%
p)]+[{\mbox{\boldmath$\omega$}}_{i,b}][{\mbox{\boldmath$\omega$}}_{1}^{\prime}%
][^{\prime}{\mbox{\boldmath$\omega$}}_{2}].$$
Equating we see that we must prove that
$$[{\mbox{\boldmath$\omega$}}_{1}][{\mbox{\boldmath$\omega$}}_{i-1,\xi(i)}]-[{%
\mbox{\boldmath$\omega$}}_{1}^{\prime}][{\mbox{\boldmath$\omega$}}_{i,b}]=[{%
\mbox{\boldmath$\omega$}}_{1}{\mbox{\boldmath$\omega$}}_{i-1,\xi(i)}]=[{\mbox{%
\boldmath$\omega$}}(m,i)^{\prime}].$$
(4.4)
This follows since Proposition 4.4 applies if $\min{\mbox{\boldmath$\omega$}}_{1}<i-1$ (and Lemma 4.5 if $\min{\mbox{\boldmath$\omega$}}_{1}=i-1$) to the pair $({\mbox{\boldmath$\omega$}}_{1},{\mbox{\boldmath$\omega$}}_{i-1,\xi(i)})$.
If $a=b$ then $\xi(i-1)=\xi(i+1)$ and hence Theorem 3 shows that $[{\mbox{\boldmath$\omega$}}(m,i)][{\mbox{\boldmath$\omega$}}(i,p)]=[{\mbox{%
\boldmath$\omega$}}(m,p){\mbox{\boldmath$\omega$}}_{i,a}]$ and $[{\mbox{\boldmath$\omega$}}(m,p){\mbox{\boldmath$\omega$}}_{i,a^{\prime}}]=[{%
\mbox{\boldmath$\omega$}}_{1}][\mathbf{f}_{i}][{\mbox{\boldmath$\omega$}}_{2}]$. To prove $(\dagger)$
we use part (i) of the theorem on the pair $({\mbox{\boldmath$\omega$}}_{1},{\mbox{\boldmath$\omega$}}(i,p))$ and Lemma 4.5 on the pair $({\mbox{\boldmath$\omega$}}_{i,a^{\prime}},{\mbox{\boldmath$\omega$}}(i,p))$ to get
$$\displaystyle[{\mbox{\boldmath$\omega$}}_{1}][{\mbox{\boldmath$\omega$}}_{i,a^%
{\prime}}][{\mbox{\boldmath$\omega$}}(i,p)]$$
$$\displaystyle=[{\mbox{\boldmath$\omega$}}_{i,a^{\prime}}][{\mbox{\boldmath$%
\omega$}}(m,p)]+[{\mbox{\boldmath$\omega$}}_{i,a}][{\mbox{\boldmath$\omega$}}_%
{1}^{\prime}][^{\prime}{\mbox{\boldmath$\omega$}}(i,p)],$$
$$\displaystyle=[{\mbox{\boldmath$\omega$}}_{1}][\mathbf{f}_{i}][{\mbox{%
\boldmath$\omega$}}_{2}]+[{\mbox{\boldmath$\omega$}}_{1}][{\mbox{\boldmath$%
\omega$}}_{i-1,\xi(i)}][^{\prime}{\mbox{\boldmath$\omega$}}(i,p)].$$
Equating the right hand sides and using (4.4) gives the result.
The proof of $(\dagger\dagger)$ is similar;
we calculate $[{\mbox{\boldmath$\omega$}}_{i,a}][{\mbox{\boldmath$\omega$}}_{1}][{\mbox{%
\boldmath$\omega$}}(i,p)]$ in two ways by using Proposition 4.4 on $({\mbox{\boldmath$\omega$}}_{i,a},{\mbox{\boldmath$\omega$}}_{1})$ and part (i) of the theorem on $({\mbox{\boldmath$\omega$}}_{1},{\mbox{\boldmath$\omega$}}(i,p))$. This gives
$$\displaystyle[{\mbox{\boldmath$\omega$}}_{i,a}][{\mbox{\boldmath$\omega$}}_{1}%
][{\mbox{\boldmath$\omega$}}(i,p)]$$
$$\displaystyle=[{\mbox{\boldmath$\omega$}}(m,i)][{\mbox{\boldmath$\omega$}}(i,p%
)]+[{\mbox{\boldmath$\omega$}}_{1}^{\prime}][{\mbox{\boldmath$\omega$}}_{i+1,%
\xi(i)}][{\mbox{\boldmath$\omega$}}(i,p)]$$
$$\displaystyle=[{\mbox{\boldmath$\omega$}}_{i,a}][{\mbox{\boldmath$\omega$}}(m,%
p)]+[{\mbox{\boldmath$\omega$}}_{i,a}][{\mbox{\boldmath$\omega$}}_{1}^{\prime}%
][^{\prime}{\mbox{\boldmath$\omega$}}(i,p)]$$
We then observe that ${\mbox{\boldmath$\omega$}}_{i+1,\xi(i)}{\mbox{\boldmath$\omega$}}_{2}\in%
\mathbf{P}\mathbf{r}_{\xi}$ if $\xi(i)\neq\xi(i+2)$ and
${\mbox{\boldmath$\omega$}}_{i+1,\xi(i)}{\mbox{\boldmath$\omega$}}_{2}=\mathbf{%
f}_{i+1}{\mbox{\boldmath$\omega$}}^{\prime}$, with ${\mbox{\boldmath$\omega$}}^{\prime}\in\mathbf{P}\mathbf{r}_{\xi}$, if $\xi(i+1)=\xi(i+3)$. Then we can apply the results proved above of part (ii) of this theorem to the
pair $({\mbox{\boldmath$\omega$}}_{i+1,\xi(i)},{\mbox{\boldmath$\omega$}}(i,p))$, and hence either by $(**)$ or by $(\dagger)$ we get
$$[{\mbox{\boldmath$\omega$}}_{i+1,\xi(i)}][{\mbox{\boldmath$\omega$}}(i,p)]=[{%
\mbox{\boldmath$\omega$}}_{i,a}][{\mbox{\boldmath$\omega$}}_{i+1,\xi(i)}{\mbox%
{\boldmath$\omega$}}_{2}]+[^{\prime}{\mbox{\boldmath$\omega$}}_{2}][\mathbf{f}%
_{i}]$$
Equation $(\dagger\dagger)$ now follows by a substitution, recalling that $[^{\prime}{\mbox{\boldmath$\omega$}}(i,p)]=[{\mbox{\boldmath$\omega$}}_{i+1,%
\xi(i)}{\mbox{\boldmath$\omega$}}_{2}]$, by definition.
Finally we prove (4.3). If $\operatorname{ht}{\mbox{\boldmath$\pi$}}_{1}=1$ or $\operatorname{ht}{\mbox{\boldmath$\pi$}}_{2}=1$ this was proved in Lemma 4.5. Hence we may assume that ${\mbox{\boldmath$\pi$}}_{1}={\mbox{\boldmath$\omega$}}(m,i)$ and ${\mbox{\boldmath$\pi$}}_{2}={\mbox{\boldmath$\omega$}}(i,p)$ for some $m<i<p$.
Since $a\neq b$ we use Theorem 3 to see that $[{\mbox{\boldmath$\omega$}}(m,i){\mbox{\boldmath$\omega$}}(i,p)]=[\mathbf{f}_{%
i}][{\mbox{\boldmath$\omega$}}(m,p)]$ and we prove that
$$[{\mbox{\boldmath$\omega$}}(m,i)][{\mbox{\boldmath$\omega$}}(i,p)]-[\mathbf{f}%
_{i}][{\mbox{\boldmath$\omega$}}(m,p)]=[{\mbox{\boldmath$\omega$}}(m,i)^{%
\prime}][^{\prime}{\mbox{\boldmath$\omega$}}(i,p)].$$
(4.5)
For this we note that ${\mbox{\boldmath$\omega$}}_{1}{\mbox{\boldmath$\omega$}}_{2}={\mbox{\boldmath$%
\omega$}}(m,p)$ and hence, using part (i) of the theorem to the pair $({\mbox{\boldmath$\omega$}}_{1},{\mbox{\boldmath$\omega$}}_{2})$ and Proposition 4.4 or Lemma 4.5 to the pairs $({\mbox{\boldmath$\omega$}}_{1},{\mbox{\boldmath$\omega$}}_{i-1,\xi(i)})$ and $({\mbox{\boldmath$\omega$}}_{i,a},{\mbox{\boldmath$\omega$}}(i,p))$ we get
$$\displaystyle[\mathbf{f}_{i}]\left([{\mbox{\boldmath$\omega$}}(m,p)]+[{\mbox{%
\boldmath$\omega$}}_{1}^{\prime}][^{\prime}{\mbox{\boldmath$\omega$}}_{2}]%
\right)+\left([{\mbox{\boldmath$\omega$}}(m,i)^{\prime}]+[{\mbox{\boldmath$%
\omega$}}_{1}^{\prime}][{\mbox{\boldmath$\omega$}}_{i,b}]\right)[^{\prime}{%
\mbox{\boldmath$\omega$}}(i,p)]$$
$$\displaystyle=[{\mbox{\boldmath$\omega$}}_{1}][\mathbf{f}_{i}][{\mbox{%
\boldmath$\omega$}}_{2}]+[{\mbox{\boldmath$\omega$}}_{1}][{\mbox{\boldmath$%
\omega$}}_{i-1,\xi(i)}][^{\prime}{\mbox{\boldmath$\omega$}}(i,p)]=[{\mbox{%
\boldmath$\omega$}}_{1}][{\mbox{\boldmath$\omega$}}_{i,a}][{\mbox{\boldmath$%
\omega$}}(i,p)]$$
$$\displaystyle=[{\mbox{\boldmath$\omega$}}(m,i)][{\mbox{\boldmath$\omega$}}(i,p%
)]+[{\mbox{\boldmath$\omega$}}_{1}^{\prime}][{\mbox{\boldmath$\omega$}}_{i+1,%
\xi(i)}][{\mbox{\boldmath$\omega$}}(i,p)].$$
Equating the first and last terms we see that (4.5) follows if we prove that
$$[\mathbf{f}_{i}][{\mbox{\boldmath$\omega$}}_{2}^{\prime}]+[{\mbox{\boldmath$%
\omega$}}_{i,b}][^{\prime}{\mbox{\boldmath$\omega$}}(i,p)]=[{\mbox{\boldmath$%
\omega$}}_{i+1,\xi(i)}][{\mbox{\boldmath$\omega$}}(i,p)].$$
But this follows from the cases of part (ii) of this theorem proved above. This completes the proof of part (ii).
4.7. Proof of Theorem 4(iii)
We proceed by induction on $N=\operatorname{ht}{\mbox{\boldmath$\omega$}}(i_{1},i_{3})+\operatorname{ht}{%
\mbox{\boldmath$\omega$}}(i_{2},i_{4})$ with [28] showing that induction begins when $N=4$. Recall that
$${\mbox{\boldmath$\omega$}}(i_{1},i_{2})={\mbox{\boldmath$\omega$}}_{1}{\mbox{%
\boldmath$\omega$}}_{i_{2},a},\ \ {\mbox{\boldmath$\omega$}}(i_{2},i_{3})={%
\mbox{\boldmath$\omega$}}_{i_{2},b}{\mbox{\boldmath$\omega$}}{\mbox{\boldmath$%
\omega$}}_{i_{3},c},\ \ {\mbox{\boldmath$\omega$}}(i_{3},i_{4})={\mbox{%
\boldmath$\omega$}}_{i_{3},d}{\mbox{\boldmath$\omega$}}_{2}.$$
Set
$$[\mathbf{f}_{i_{2},i_{3}}]=\prod_{s=i_{2}}^{i_{3}}[\mathbf{f}_{s}]^{\delta_{%
\xi(s-1),\xi(s+1)}},$$
and note that ${\mbox{\boldmath$\omega$}}(i_{2},i_{4})={\mbox{\boldmath$\omega$}}_{i_{2},b}{%
\mbox{\boldmath$\omega$}}{\mbox{\boldmath$\omega$}}_{i_{3},c}^{\delta_{c,d}}{%
\mbox{\boldmath$\omega$}}_{2}$.
Case 1: $a=b$ or $c=d$ Suppose that $a=b$; the proof is similar when $c=d$. Then ${\mbox{\boldmath$\omega$}}_{1}{\mbox{\boldmath$\omega$}}(i_{2},i_{s})\in%
\mathbf{P}\mathbf{r}_{\xi}$ for $s=3,4$. Hence
(4.2) gives
$$\displaystyle[{\mbox{\boldmath$\omega$}}_{1}][{\mbox{\boldmath$\omega$}}(i_{2}%
,i_{4})][{\mbox{\boldmath$\omega$}}(i_{2},i_{3})]$$
$$\displaystyle=\left([{\mbox{\boldmath$\omega$}}(i_{1},i_{4})]+[{\mbox{%
\boldmath$\omega$}}_{1}^{\prime}][^{\prime}{\mbox{\boldmath$\omega$}}(i_{2},i_%
{4})]\right)[{\mbox{\boldmath$\omega$}}(i_{2},i_{3})]$$
$$\displaystyle=\left([{\mbox{\boldmath$\omega$}}(i_{1},i_{3})]+[{\mbox{%
\boldmath$\omega$}}_{1}^{\prime}][^{\prime}{\mbox{\boldmath$\omega$}}(i_{2},i_%
{3})])\right)[{\mbox{\boldmath$\omega$}}(i_{2},i_{4})].$$
The result follows if we prove that
$$[^{\prime}{\mbox{\boldmath$\omega$}}(i_{2},i_{3})][{\mbox{\boldmath$\omega$}}(%
i_{2},i_{4})]-[^{\prime}{\mbox{\boldmath$\omega$}}(i_{2},i_{4})][{\mbox{%
\boldmath$\omega$}}(i_{2},i_{3})]=(-1)^{\operatorname{ht}{\mbox{\boldmath%
\scriptsize$\omega$}}(i_{2},i_{3})}[\mathbf{f}_{i_{2},i_{3}}][^{\prime}{\mbox{%
\boldmath$\omega$}}_{2}]^{\delta_{c,d}}[^{\prime}{\mbox{\boldmath$\omega$}}(i_%
{3},i_{4}))]^{1-\delta_{c,d}}.$$
Note that we have the following possibilities for the pair $(^{\prime}{\mbox{\boldmath$\omega$}}(i_{2},i_{3}),^{\prime}{\mbox{\boldmath$%
\omega$}}(i_{2},i_{4}))$:
$$({\mbox{\boldmath$\omega$}}(i_{2}+1,i_{3}),{\mbox{\boldmath$\omega$}}(i_{2}+1,%
i_{4})),\ \ (\mathbf{f}_{i_{2}+1}{\mbox{\boldmath$\omega$}}(m,i_{3}),\mathbf{f%
}_{i_{2}+1}({\mbox{\boldmath$\omega$}}(m,i_{4})),$$
$$(\mathbf{f}_{i_{2}+1}{\mbox{\boldmath$\omega$}}_{i_{3},c},\ \mathbf{f}_{i_{2}+%
1}{\mbox{\boldmath$\omega$}}_{i_{3},c}^{\delta_{c,d}}{\mbox{\boldmath$\omega$}%
}_{2}),\ \ (\mathbf{f}_{i_{2}+1},(\mathbf{f}_{i_{2}+1}{\mbox{\boldmath$\omega$%
}}_{2})^{\delta_{c,d}}{\mbox{\boldmath$\omega$}}(i_{2}+1,i_{4})^{1-\delta_{c,d%
}}).$$
In the first case, $\operatorname{ht}{\mbox{\boldmath$\omega$}}(i_{2},i_{3})=\operatorname{ht}{%
\mbox{\boldmath$\omega$}}(i_{2}+1,i_{3})$, $\operatorname{ht}{\mbox{\boldmath$\omega$}}(i_{2},i_{3})<\operatorname{ht}{%
\mbox{\boldmath$\omega$}}(i_{1},i_{3})$ the inductive hypothesis applies to $i_{2}<i_{2}+1<i_{3}<i_{4}$ and gives the result. In the second case the inductive hypothesis applies to $i_{2}<m<i_{3}<i_{4}$ and gives the result. In the third case we use equations $(*)$ and $(\dagger\dagger)$ of Theorem 4(ii) to get the result. In the fourth case we use Theorem 4(i) if $c=d$ and (4.3) if $c\neq d$.
Case 2. Assume that $a\neq b$ and $c\neq d$. Since $N\geq 5$ we may assume without loss of generality that $\operatorname{ht}{\mbox{\boldmath$\omega$}}(i_{1},i_{3})\geq 3$.
If $\operatorname{ht}{\mbox{\boldmath$\omega$}}(i_{1},i_{2})\geq 3$ let $i_{1}<j<i_{2}$ be minimal with $\xi(j-1)=\xi(j+1)$. We choose $z\in\mathbb{C}(q)^{\times}$ so that ${\mbox{\boldmath$\omega$}}(i_{1},i_{2}){\mbox{\boldmath$\omega$}}_{i_{1},z}^{-%
1}\in\mathbf{P}\mathbf{r}_{\xi}$ and calculate $[{\mbox{\boldmath$\omega$}}_{i_{1},z}]\left([{\mbox{\boldmath$\omega$}}(j,i_{4%
})][{\mbox{\boldmath$\omega$}}(i_{2},i_{3})]-[{\mbox{\boldmath$\omega$}}(j,i_{%
3})][{\mbox{\boldmath$\omega$}}(i_{2},i_{4})]\right)$ in two ways to get two expressions for it; the first one by using the inductive hypothesis which shows that it is equal to
$$(-1)^{\operatorname{ht}{\mbox{\boldmath\scriptsize$\omega$}}(i_{2},i_{3})}[{%
\mbox{\boldmath$\omega$}}_{i_{1},z}][{\mbox{\boldmath$\omega$}}(j,i_{2})^{%
\prime}][\mathbf{f}_{i_{2},i_{3}}][^{\prime}{\mbox{\boldmath$\omega$}}(i_{3},i%
_{4})]$$
and the second by using Theorem 4(i) on the pairs $({\mbox{\boldmath$\omega$}}_{i_{1},z},{\mbox{\boldmath$\omega$}}(j,i_{s}))$, $s=3,4$ which gives that it is equal to
$$\displaystyle[{\mbox{\boldmath$\omega$}}(i_{1},i_{4})][{\mbox{\boldmath$\omega%
$}}(i_{2},i_{3})]-[{\mbox{\boldmath$\omega$}}(i_{1},i_{3})][{\mbox{\boldmath$%
\omega$}}(i_{2},i_{4})]+[{\mbox{\boldmath$\omega$}}_{i_{1}-1,\xi(i_{1})}]([^{%
\prime}{\mbox{\boldmath$\omega$}}(j,i_{4})][{\mbox{\boldmath$\omega$}}(i_{2},i%
_{3})]-[^{\prime}{\mbox{\boldmath$\omega$}}(j,i_{3})][{\mbox{\boldmath$\omega$%
}}(i_{2},i_{4})]$$
Hence the inductive step follows if we prove that
$$\displaystyle\left([{\mbox{\boldmath$\omega$}}_{i_{1},z}][{\mbox{\boldmath$%
\omega$}}(j,i_{2})^{\prime}]-[{\mbox{\boldmath$\omega$}}(i_{1},i_{2})^{\prime}%
]\right)[\mathbf{f}_{i_{2},i_{3}}][^{\prime}{\mbox{\boldmath$\omega$}}(i_{3},i%
_{4})]=$$
$$\displaystyle(-1)^{\operatorname{ht}{\mbox{\boldmath\scriptsize$\omega$}}(i_{2%
},i_{3})}[{\mbox{\boldmath$\omega$}}_{i-1,\xi(i)}]\left([^{\prime}{\mbox{%
\boldmath$\omega$}}(j,i_{4})][{\mbox{\boldmath$\omega$}}(i_{2},i_{3})]-[^{%
\prime}{\mbox{\boldmath$\omega$}}(j,i_{3})][{\mbox{\boldmath$\omega$}}(i_{2},i%
_{4})]\right).$$
This is proved by noting that
$${\mbox{\boldmath$\omega$}}(j,i_{2})^{\prime}={\mbox{\boldmath$\omega$}}(j,i_{2%
}-1)^{(1-\delta_{j,i_{2}-1})(1-\delta_{\xi(i_{2}),\xi(i_{2})-2})}({\mbox{%
\boldmath$\omega$}}_{3}^{(1-\delta_{j,i_{2}-1})}\mathbf{f}_{i_{2}-1})^{\delta_%
{\xi(i_{2}),\xi(i_{2})-2}},$$
where ${\mbox{\boldmath$\omega$}}_{1}={\mbox{\boldmath$\omega$}}_{3}{\mbox{\boldmath$%
\omega$}}_{i_{2}-1,\xi(i_{2})\pm 2}$ if $\xi(i_{2})=\xi(i_{2}-2)$ and considering the different cases. In each case, Theorem 4(i) applies to the left hand side while the induction hypothesis or Theorem 4(ii) applies to the right hand side and gives the answer. As an example suppose that $j=i_{2}-1$ and $\xi(i_{2}-2)=\xi(i_{2})$. Then ${\mbox{\boldmath$\omega$}}(j,i_{2})^{\prime}=\mathbf{f}_{j}$ and the minimality of $j$ shows that ${\mbox{\boldmath$\omega$}}(i_{1},i_{2})^{\prime}={\mbox{\boldmath$\omega$}}_{i%
_{1},z}\mathbf{f}_{j}$ and hence the left hand side is zero. On the right hand side since $\xi(i_{2}-1)\neq\xi(i_{2}+1)$ by assumption we get ${}^{\prime}{\mbox{\boldmath$\omega$}}(j,i_{s})={\mbox{\boldmath$\omega$}}(i_{2%
},i_{s})$ and so the right hand side is zero as well. We omit the details in other cases.
Finally suppose that $j>i_{2}$ and let $b^{\prime}\in\mathbb{C}(q)$ be such that $\{b,b^{\prime}\}=\{\xi(i_{2})+1,\xi(i_{2})-1\}$; we have the following series of equalities.
$$\displaystyle\left([{\mbox{\boldmath$\omega$}}(i_{1},i_{4})]+[{\mbox{\boldmath%
$\omega$}}_{i_{1}-1,\xi(i_{1})}][^{\prime}{\mbox{\boldmath$\omega$}}(j,i_{4})]%
\right)[{\mbox{\boldmath$\omega$}}(i_{2},i_{3})]+[{\mbox{\boldmath$\omega$}}(i%
_{1},i_{2})^{\prime}][^{\prime}{\mbox{\boldmath$\omega$}}(j,i_{3})][{\mbox{%
\boldmath$\omega$}}(j,i_{4})]$$
$$\displaystyle=\left([{\mbox{\boldmath$\omega$}}_{i_{1},a}][{\mbox{\boldmath$%
\omega$}}(i_{2},i_{3})]+[{\mbox{\boldmath$\omega$}}(i_{1},i_{2})^{\prime}][^{%
\prime}{\mbox{\boldmath$\omega$}}(j,i_{3})]\right)[{\mbox{\boldmath$\omega$}}(%
j,i_{4})]$$
$$\displaystyle=[{\mbox{\boldmath$\omega$}}_{i_{2},b^{\prime}}][{\mbox{\boldmath%
$\omega$}}(i_{1},i_{3})][{\mbox{\boldmath$\omega$}}(j,i_{4})]$$
$$\displaystyle=\left([{\mbox{\boldmath$\omega$}}(i_{2},i_{4})]+[{\mbox{%
\boldmath$\omega$}}_{i_{2}-1,\xi(i_{2})}][^{\prime}{\mbox{\boldmath$\omega$}}(%
j,i_{4})]\right)[{\mbox{\boldmath$\omega$}}(i_{1},i_{3})]$$
$$\displaystyle=[{\mbox{\boldmath$\omega$}}(i_{2},i_{4})][{\mbox{\boldmath$%
\omega$}}(i_{1},i_{3})]+[^{\prime}{\mbox{\boldmath$\omega$}}(j,i_{4})]\left([{%
\mbox{\boldmath$\omega$}}(i_{1},i_{2})^{\prime}][{\mbox{\boldmath$\omega$}}(j,%
i_{3})]+[{\mbox{\boldmath$\omega$}}_{i_{1}-1,\xi(i_{1})}][{\mbox{\boldmath$%
\omega$}}(i_{2},i_{3})]\right).$$
where the first and third equality follow from applying (4.2) to the pairs $({\mbox{\boldmath$\omega$}}_{i_{1},a},{\mbox{\boldmath$\omega$}}(j,i_{4}))$ and $({\mbox{\boldmath$\omega$}}_{i_{2},b^{\prime}},{\mbox{\boldmath$\omega$}}(j,i_%
{4}))$, respectively, and the second and fourth equality follow busing $(*),(**)$ of Theorem 4(ii) to $({\mbox{\boldmath$\omega$}}_{i_{2},b^{\prime}},{\mbox{\boldmath$\omega$}}(i_{1%
},i_{3}))$ and $({\mbox{\boldmath$\omega$}}_{i_{2}-1,\xi(i_{2})},{\mbox{\boldmath$\omega$}}(i_%
{1},i_{3}))$. The inductive step follows by establishing
$$(-1)^{\operatorname{ht}{\mbox{\boldmath$\omega$}}(i_{2},i_{3})}[\mathbf{f}_{i_%
{2},i_{3}}][^{\prime}{\mbox{\boldmath$\omega$}}(i_{3},i_{4})]=[{\mbox{%
\boldmath$\omega$}}(j,i_{3})][^{\prime}{\mbox{\boldmath$\omega$}}(j,i_{4})]-[^%
{\prime}{\mbox{\boldmath$\omega$}}(j,i_{3})][{\mbox{\boldmath$\omega$}}(j,i_{4%
})].$$
(4.6)
The calculations are similar to the ones done so far and we omit further details.
5. Proof of Proposition 4.4
In this section we prove Proposition 4.4 when $i<j$; the proof in the case $i>k$ is identical. We recall the statement of the proposition for the readers convenience.
Proposition.
Suppose that ${\mbox{\boldmath$\omega$}}_{ia}{\mbox{\boldmath$\omega$}}_{j,b}{\mbox{%
\boldmath$\omega$}}\in\mathbf{P}\mathbf{r}_{\xi}$ with $i<j<\min{\mbox{\boldmath$\omega$}}$ and set ${\mbox{\boldmath$\pi$}}={\mbox{\boldmath$\omega$}}_{j,b}{\mbox{\boldmath$%
\omega$}}$. We have
$$[{\mbox{\boldmath$\omega$}}_{i,a}][{\mbox{\boldmath$\pi$}}]-[{\mbox{\boldmath$%
\omega$}}_{i,a}{\mbox{\boldmath$\pi$}}]=[{\mbox{\boldmath$\omega$}}_{i-1,\xi(i%
)}][{\mbox{\boldmath$\omega$}}_{j+1,\xi(j)}{\mbox{\boldmath$\omega$}}].$$
We make some preliminary remarks about the proof.
Recall from Lemma 1.2 that for all ${\mbox{\boldmath$\omega$}}\in\mathbf{P}\mathbf{r}_{\xi}$ the module $[{\mbox{\boldmath$\omega$}}]$ is prime, i.e., that it cannot be written as a tensor product of non-trivial finite–dimensional representations of $\hat{\mathbf{U}}_{q}$. It follows that the module $[{\mbox{\boldmath$\omega$}}_{i,a}{\mbox{\boldmath$\pi$}}]$ is a proper subquotient of $[{\mbox{\boldmath$\omega$}}_{i,a}]\otimes[{\mbox{\boldmath$\pi$}}]$.
We claim that $[{\mbox{\boldmath$\omega$}}_{i-1,\xi(i)}]\otimes[{\mbox{\boldmath$\omega$}}_{j%
+1,\xi(j)}{\mbox{\boldmath$\omega$}}]$ is irreducible. The condition ${\mbox{\boldmath$\omega$}}_{i,a}{\mbox{\boldmath$\omega$}}_{j,b}{\mbox{%
\boldmath$\omega$}}\in\mathbf{P}\mathbf{r}_{\xi}$ forces $\xi(j-1)=\xi(j+1)$ and hence
$$j+1<\min{\mbox{\boldmath$\omega$}}\implies{\mbox{\boldmath$\omega$}}_{i-1,\xi(%
i)}{\mbox{\boldmath$\omega$}}_{j+1,\xi(j)}{\mbox{\boldmath$\omega$}}\notin%
\mathbf{P}\mathbf{r}_{\xi}$$
and the claim follows from Theorem 3(b)(i). Otherwise, we have
$$j+1=\min{\mbox{\boldmath$\omega$}}\implies{\mbox{\boldmath$\omega$}}_{j+1,\xi(%
j)}{\mbox{\boldmath$\omega$}}=\mathbf{f}_{j+1}{\mbox{\boldmath$\omega$}}^{%
\prime},\ \ {\mbox{\boldmath$\omega$}}^{\prime}\in\mathbf{P}\mathbf{r}_{\xi}%
\cup\{1\},$$
Theorem 3 (a) gives
$$[{\mbox{\boldmath$\omega$}}_{j+1,\xi(j)}{\mbox{\boldmath$\omega$}}^{\prime}]=[%
{\mbox{\boldmath$\omega$}}^{\prime}]\otimes[\mathbf{f}_{j+1}],\ \ [{\mbox{%
\boldmath$\omega$}}_{i-1,\xi(i)}]\otimes[\mathbf{f}_{j+1}]=[{\mbox{\boldmath$%
\omega$}}_{i-1,\xi(i)}\mathbf{f}_{j+1}],$$
while (b)(i),
$$[{\mbox{\boldmath$\omega$}}_{i-1,\xi(i)}]\otimes[{\mbox{\boldmath$\omega$}}^{%
\prime}]=[{\mbox{\boldmath$\omega$}}_{i-1,\xi(i)}{\mbox{\boldmath$\omega$}}^{%
\prime}].$$
An application of Theorem 2 now proves the claim in this case.
In the first part of this section we shall show that $[{\mbox{\boldmath$\omega$}}_{i-1,\xi(i)}]\otimes[{\mbox{\boldmath$\omega$}}_{j%
+1,\xi(j)}{\mbox{\boldmath$\omega$}}]$ is also a subquotient of $[{\mbox{\boldmath$\omega$}}_{i,a}]\otimes[{\mbox{\boldmath$\omega$}}_{j,b}{%
\mbox{\boldmath$\omega$}}]$; in particular
$$\dim[{\mbox{\boldmath$\omega$}}_{i,a}]\dim[{\mbox{\boldmath$\omega$}}_{j,b}{%
\mbox{\boldmath$\omega$}}]\geq\dim[{\mbox{\boldmath$\omega$}}_{i,a}{\mbox{%
\boldmath$\omega$}}_{j,b}{\mbox{\boldmath$\omega$}}]+\dim[{\mbox{\boldmath$%
\omega$}}_{i-1,\xi(i)}]\dim[{\mbox{\boldmath$\omega$}}_{j+1,\xi(j)}{\mbox{%
\boldmath$\omega$}}].$$
The proposition clearly follows if we prove the reverse inequality. This is done by using a presentation of the graded limit of the modules $[{\mbox{\boldmath$\pi$}}]$, ${\mbox{\boldmath$\pi$}}\in\mathbf{P}\mathbf{r}_{\xi}$ given
in [3] along with some additional results in the representation theory of current algebras.
5.1.
The proof of the next result is an elementary application of $q$–character theory for quantum affine algebras.
Lemma.
The module $[{\mbox{\boldmath$\omega$}}_{i-1,\xi(i)}{\mbox{\boldmath$\omega$}}_{j+1,\xi(j)%
}{\mbox{\boldmath$\omega$}}]$ occurs in the Jordan–Holder series of $[{\mbox{\boldmath$\omega$}}_{i,a}]\otimes[{\mbox{\boldmath$\omega$}}_{j,b}{%
\mbox{\boldmath$\omega$}}].$
Proof.
It suffices to show that there exists an $\ell$–highest weight vector with $\ell$–highest weight ${\mbox{\boldmath$\omega$}}_{i-1,\xi(i)}{\mbox{\boldmath$\omega$}}_{j+1,\xi(j)}%
{\mbox{\boldmath$\omega$}}$ in $[{\mbox{\boldmath$\omega$}}_{i,a}]\otimes[{\mbox{\boldmath$\pi$}}]$ (resp. $[{\mbox{\boldmath$\pi$}}]\otimes[{\mbox{\boldmath$\omega$}}_{i,a}]$) if $a=\xi(i)+1$ (resp. $a=\xi(i)-1$).
But this is true by a routine argument using $q$–characters. Namely one observes that the element ${\mbox{\boldmath$\omega$}}_{i-1,\xi(i)}{\mbox{\boldmath$\omega$}}_{j+1,\xi(j)}%
{\mbox{\boldmath$\omega$}}$ is an $\ell$-weight of $[{\mbox{\boldmath$\omega$}}_{i,a}]\otimes[{\mbox{\boldmath$\pi$}}]$ but not of $[{\mbox{\boldmath$\omega$}}_{i,a}{\mbox{\boldmath$\pi$}}]$. It is then elementary
to see that the corresponding eigenvector is necessarily highest weight. We omit the details.
∎
5.2.
We need some standard notation from the theory of simple Lie algebras. Thus, $\mathfrak{h}$ denotes a Cartan subalgebra of $\mathfrak{sl}_{n+1}$, $\{\alpha_{i}:1\leq i\leq n\}$ a set of simple roots for $(\mathfrak{sl}_{n+1},\mathfrak{h})$ and $R^{+}=\{\alpha_{i,j}:=\alpha_{i}+\cdots+\alpha_{j}:1\leq i\leq j\leq n\}$ the corresponding set of positive roots. Fix a Chevalley basis $x_{i,j}^{\pm}$, $1\leq i\leq j\leq n$ and $h_{j}$, $1\leq j\leq n$ for $\mathfrak{sl}_{n+1}$. Set $x_{j,j}^{\pm}=x_{j}^{\pm}$ and $h_{i,j}=h_{i}+\cdots+h_{j}$ for all $1\leq i\leq j\leq n$.
As in the earlier sections $P^{+}$ will be the set of dominant integral weights corresponding to a set $\{\omega_{i}:1\leq i\leq n\}$ of fundamental weights and we set
$$P^{+}(1)=\{\lambda\in P^{+}:\lambda(h_{i})\leq 1,\ 1\leq i\leq n\}.$$
For $\lambda\in P^{+}$ let $V(\lambda)$ be an irreducible finite dimensional $\mathfrak{sl}_{n+1}$ with highest weight $\lambda$.
Let $t$ be an indeterminate and $\mathbb{C}[t]$ the corresponding polynomial ring with complex coefficients.
Denote by $\mathfrak{sl}_{n+1}[t]$ the Lie algebra with underlying vector space $\mathfrak{sl}_{n+1}\otimes\mathbb{C}[t]$ and commutator given by
$$[a\otimes f,b\otimes g]=[a,b]\otimes fg,\ \ a,b\in\mathfrak{sl}_{n+1},\ \ f,g%
\in\mathbb{C}[t].$$
Then $\mathfrak{sl}_{n+1}[t]$ and its universal enveloping algebra admit a natural $\mathbb{Z}_{+}$-grading given by declaring a monomial $(a_{1}\otimes t^{r_{1}})\cdots(a_{p}\otimes t^{r_{p}})$ to have grade $r_{1}+\cdots+r_{p}$, where $a_{s}\in\mathfrak{sl}_{n+1}$ and $r_{s}\in\mathbb{Z}_{+}$ for $1\leq s\leq p$.
5.3.
We shall be interested in the category of $\mathbb{Z}_{+}$–graded modules for $\mathfrak{sl}_{n+1}[t]$. An object of this category is a module $V$ for $\mathfrak{sl}_{n+1}[t]$ which admits a compatible $\mathbb{Z}$–grading, i.e.,
$$V=\bigoplus_{s\in\mathbb{Z}}V[s],\ \ (x\otimes t^{r})V[s]\subset V[r+s],\ \ x%
\in\mathfrak{sl}_{n+1},\ \ r\in\mathbb{Z}_{+}.$$
For any $p\in\mathbb{Z}$ we let $\tau_{p}^{*}V$ be the graded module which given by shifting the grades up by $p$ and leaving the action of $\mathfrak{sl}_{n+1}[t]$ unchanged.
The morphisms between graded modules are $\mathfrak{sl}_{n+1}[t]$- maps of grade zero. A $\mathfrak{sl}_{n+1}$–module $M$ will be regarded as an object (denoted $\operatorname{ev}_{0}^{*}M$) of this category by placing $M$ in degree zero and requiring that
$$(a\otimes t^{r})m=\delta_{r,0}am,\ \ a\in\mathfrak{sl}_{n+1},\ \ m\in M\ \ r%
\in\mathbb{Z}_{+}.$$
For $\lambda\in P^{+}$, the local Weyl module $W_{\operatorname{loc}}(\lambda)$ is the $\mathfrak{sl}_{n+1}[t]$–module generated by an element $w_{\lambda}$ with graded defining relations:
$$(x_{i}^{+}\otimes 1)w_{\lambda}=0,\ \ (h\otimes t^{r})w_{\lambda}=\delta_{r,0}%
\lambda(h)w_{\lambda},\ \ \ (x_{i}^{-}\otimes 1)^{\lambda(h_{i})+1}w_{\lambda}%
=0,$$
(5.1)
where $1\leq i\leq n,$ and $r\in\mathbb{Z}_{+}$. Define a grading on $W_{\operatorname{loc}}(\lambda)$ by requiring $\operatorname{gr}w_{\lambda}=0$.
It is straightforward to see that
$$W(\omega_{i})\cong_{\mathfrak{sl}_{n+1}}V(\omega_{i}),\ \ 1\leq i\leq n.$$
In general $W_{\operatorname{loc}}(\lambda)$ has a unique graded irreducible quotient which is isomorphic to $\operatorname{ev}_{0}^{*}V(\lambda)$. It is obtained by imposing the additional relation $(x^{-}_{\alpha}\otimes t)w_{\lambda}=0$ for all $\alpha\in R^{+}$.
5.4.
Given $\mu\in P^{+}(1)$ set
$$\displaystyle\min\mu=\min\{i:\mu(h_{i})=1\},$$
$$\displaystyle R^{+}(\mu)=\{\alpha_{i,j}\in R^{+}:1\leq i<j\leq n,\ \ \mu(h_{i}%
)=1=\mu(h_{j})\ \ {\rm{and}}\ \ \mu(h_{i,j})=2\}.$$
Given $\lambda=2\lambda_{0}+\lambda_{1}\in P^{+}$ with $\lambda_{0}\in P^{+}$ and $\lambda_{1}\in P^{+}(1)$ and $0\leq i<\min\lambda_{1}$, define
$M(\omega_{i},\lambda)$ to be the graded $\mathfrak{sl}_{n+1}[t]$-module generated by an element $m_{i,\lambda}$ of grade zero satisfying the graded relations in (5.1) and
$$(x^{-}_{p}\otimes t^{(\lambda_{0}+\lambda_{1}+\omega_{i})(h_{p})})m_{i,\lambda%
}=0=(x_{\alpha}^{-}\otimes t^{\lambda_{0}(h_{\alpha})+1})m_{i,\lambda},\ \ 1%
\leq p\leq n,\ \alpha\in R^{+}(\lambda_{1}).$$
(5.2)
Clearly $M(\omega_{i},\lambda)$ is a graded quotient of $W_{\operatorname{loc}}(\lambda)$
and
$$M(0,\omega_{i})\cong_{\mathfrak{sl}_{n+1}[t]}M(\omega_{i},0)\cong_{\mathfrak{%
sl}_{n+1}[t]}W_{\operatorname{loc}}(\omega_{i})\cong_{\mathfrak{s}l_{n+1}}V(%
\omega_{i}).$$
If $\lambda_{1}\neq 0$ and $i_{1}=\min\lambda_{1}$, then $R^{+}(\lambda_{1}+\omega_{i})=R^{+}(\lambda_{1})\cup\{\alpha_{i,i_{1}}\}$ and it is simple to check that the assignment $m_{i,\lambda}\to m_{0,\lambda+\omega_{i}}$ gives rise to the following short exact sequence of $\mathfrak{sl}_{n+1}[t]$–modules
$$0\to\mathbf{U}(\mathfrak{g}[t])(x_{i,i_{1}}^{-}\otimes t^{\lambda_{0}(h_{i,i_{%
1}})+1})m_{i,\lambda}\to M(\omega_{i},\lambda)\to M(0,\lambda+\omega_{i})\to 0.$$
(5.3)
The modules $M(0,\lambda)$, $\lambda\in P^{+}$ are examples of level two Demazure modules; the latter have been studied extensively and are usually denoted as $D(2,\lambda)$ in the literature. We now state a result which relates modules for the quantum affine algebra which are defined over $\mathbb{C}(q)$ and modules for $\mathfrak{sl}_{n+1}[t]$ which are defined over $\mathbb{C}$. Denote by $\dim_{\mathbb{C}(q)}V$ the dimension of a module $V$ for the quantum affine algebra and by $\dim M$ the dimension over $\mathbb{C}$ of a module $M$ for $\mathfrak{sl}_{n+1}[t]$.
Part (i) of the following result was proved in [11, Theorem 1] and parts (ii) and (iii) were proved in [3, Theorem 1].
Theorem 5.
(i)
Let $\mu\in P^{+}(1)$, $\nu_{1},\nu\in P^{+}$ with $\nu-\nu_{1}\in P^{+}$. Then
$$\displaystyle\dim M(0,2\nu)\dim M(0,\mu)$$
$$\displaystyle=\dim M(0,2\nu+\mu)$$
$$\displaystyle=\dim M(0,2\nu_{1})\dim M(0,2(\nu-\nu_{1})+\mu).$$
(ii)
Let $\xi:I\to\mathbb{Z}$ be an arbitrary height function and ${\mbox{\boldmath$\pi$}}\in\mathbf{P}\mathbf{r}_{\xi}$. We have
$$\dim M(0,\operatorname{wt}{\mbox{\boldmath$\pi$}})=\dim_{\mathbb{C}(q)}[{\mbox%
{\boldmath$\pi$}}].$$
(iii)
For all $1\leq p\leq n$ we have $\dim M(0,2\omega_{p})=\dim_{\mathbb{C}(q)}[\mathbf{f}_{p}]$.
∎
Corollary.
Let ${\mbox{\boldmath$\omega$}}_{j,b}{\mbox{\boldmath$\omega$}}\in\mathbf{P}\mathbf%
{r}_{\xi}$ with $j<k=\min{\mbox{\boldmath$\omega$}}$. We have
$$\dim_{\mathbb{C}(q)}[{\mbox{\boldmath$\omega$}}_{j+1,\xi(j)}{\mbox{\boldmath$%
\omega$}}]=\dim M(0,\omega_{j+1}+\operatorname{wt}{\mbox{\boldmath$\omega$}}).$$
Proof.
If $j+1\neq k$ then ${\mbox{\boldmath$\omega$}}_{j+1,\xi(j)}{\mbox{\boldmath$\omega$}}\in\mathbf{P}%
\mathbf{r}_{\xi}$ and the corollary is immediate from Theorem 5(ii).
Suppose that $j+1=k$. If ${\mbox{\boldmath$\omega$}}_{j+1,\xi(j)}{\mbox{\boldmath$\omega$}}=\mathbf{f}_{%
j+1}$ then the assertion of the corollary is just Theorem 5(iii).
Otherwise
$${\mbox{\boldmath$\omega$}}_{j+1,\xi(j)}{\mbox{\boldmath$\omega$}}=\mathbf{f}_{%
j+1}{\mbox{\boldmath$\omega$}}^{\prime},\ \ {\mbox{\boldmath$\omega$}}^{\prime%
}\in\mathbf{P}\mathbf{r}_{\xi},\ \ \operatorname{wt}{\mbox{\boldmath$\omega$}}%
^{\prime}=\operatorname{wt}{\mbox{\boldmath$\omega$}}-\omega_{j+1}.$$
Theorem 3(a) gives
$[{\mbox{\boldmath$\omega$}}_{j+1,\xi(j)}{\mbox{\boldmath$\omega$}}]=[\mathbf{f%
}_{j+1}][{\mbox{\boldmath$\omega$}}^{\prime}]$. Together with
parts (ii) and (iii) of Theorem 5 we get
$$\dim_{\mathbb{C}(q)}[{\mbox{\boldmath$\omega$}}_{j+1,\xi(j)}{\mbox{\boldmath$%
\omega$}}]=\dim_{\mathbb{C}(q)}[\mathbf{f}_{j+1}]\dim_{\mathbb{C}(q)}[{\mbox{%
\boldmath$\omega$}}^{\prime}]=\dim M(0,2\omega_{j+1})\dim M(0,\operatorname{wt%
}{\mbox{\boldmath$\omega$}}-\omega_{j+1}).$$
Now using part (i) of the theorem we see that the right hand side is $\dim M(0,\omega_{j+1}+\operatorname{wt}{\mbox{\boldmath$\omega$}})$ and the corollary is established.
∎
Along with Section 5.1 we have now established the following inequality. Let ${\mbox{\boldmath$\omega$}}_{i,a}{\mbox{\boldmath$\omega$}}_{j,b}{\mbox{%
\boldmath$\omega$}}\in\mathbf{P}\mathbf{r}_{\xi}$ with $i<j<\min{\mbox{\boldmath$\omega$}}$. Then
$$\displaystyle\dim M(\omega_{i},0)\dim M(0,\omega_{j}+\operatorname{wt}{\mbox{%
\boldmath$\omega$}})\geq\dim M(0,\operatorname{wt}{\mbox{\boldmath$\omega$}}+%
\omega_{i}+\omega_{j})\\
\displaystyle+\dim M(0,\omega_{i-1})\dim M(0,\omega_{j+1}+\operatorname{wt}{%
\mbox{\boldmath$\omega$}}).$$
(5.4)
and Proposition 5 follows if we prove that the preceding inequality is actually an equality. This is done in the rest of the section.
5.5.
We deduce a consequence of the preceding discussion.
Lemma.
Let $\lambda_{0}\in P^{+}$, $\lambda_{1}\in P^{+}(1)$, $\lambda=2\lambda_{0}+\lambda_{1}$ and $1\leq i<i_{1}=\min\lambda_{1}$.
Then
$$\displaystyle\dim M(\omega_{i},0)\dim M(0,\lambda)$$
$$\displaystyle\geq\dim M(0,\lambda+\omega_{i})+\dim M(0,\omega_{i-1})\dim M(0,%
\omega_{i_{1}+1}+\lambda-\omega_{i_{1}}).$$
Proof.
By Theorem 5(i) we see that for $\mu\in\{\lambda,\lambda+\omega_{i},\lambda+\omega_{i_{1}+1}-\omega_{i_{1}}\}$ we can write
$$\dim M(0,\mu)=\dim M(0,2\lambda_{0})\dim M(0,\mu-2\lambda_{0}).$$
Hence the Lemma follows if we prove that
$$\displaystyle\dim M(\omega_{i},0)\dim M(0,\lambda_{1})$$
$$\displaystyle\geq\dim M(0,\lambda_{1}+\omega_{i})+\dim M(0,\omega_{i-1})\dim M%
(0,\omega_{i_{1}+1}+\lambda_{1}-\omega_{i_{1}}).$$
Comparing this with (5.4) we see that it suffices to prove that we can find a height function $\xi$ such that there exists an element ${\mbox{\boldmath$\omega$}}_{i,a}{\mbox{\boldmath$\pi$}}\in\mathbf{P}\mathbf{r}%
_{\xi}$ with $\lambda_{1}=\operatorname{wt}{\mbox{\boldmath$\pi$}}$. Writing $\lambda_{1}=\omega_{i_{1}}+\cdots+\omega_{i_{k}}$ take $\xi:I\to\mathbb{Z}_{+}$ such that
$$\displaystyle\xi(m)=m,\ \ 1\leq m\leq i_{1},\ \ \ \ \xi(i_{k}+j)=\xi(i_{k})+(-%
1)^{k}j,\ \ 1\leq j\leq n-i_{k},\ \ {\rm and}$$
$$\displaystyle\xi(i_{j+1})-\xi(i_{j})=(-1)^{j}(i_{j+1}-i_{j}),\ \ 1\leq j\leq k%
-1.$$
If $k=1$ then ${\mbox{\boldmath$\omega$}}_{i,i-1}{\mbox{\boldmath$\omega$}}_{i_{1},i_{1}+1}%
\in\mathbf{P}\mathbf{r}_{\xi}$ and otherwise ${\mbox{\boldmath$\omega$}}_{i,i-1}{\mbox{\boldmath$\omega$}}(i_{1},i_{k})\in%
\mathbf{P}\mathbf{r}_{\xi}$ and the Lemma is proved.
∎
5.6.
Given a module $V$ for $\mathfrak{sl}_{n+1}[t]$ and $z\in\mathbb{C}$ denote by $V^{z}$ the $\mathfrak{sl}_{n+1}[t]$-module with underlying vector space $V$ and action given by,
$$(x\otimes t^{r})w=(x\otimes(t+z)^{r})w,\ \ x\in\mathfrak{sl}_{n+1},\ \ r\in%
\mathbb{Z}_{+},\ w\in V.$$
Suppose that $V_{1},V_{2}$ are cyclic finite–dimensional $\mathfrak{sl}_{n+1}[t]$-modules with cyclic vectors $v_{1}$ and $v_{2}$ respectively. It was proved in [15] that if $z_{1},z_{2}$ are distinct complex numbers, then the tensor product
$V_{1}^{z_{1}}\otimes V_{2}^{z_{2}}$ is a cyclic $\mathfrak{sl}_{n+1}[t]$-module generated by $v_{1}\otimes v_{2}$. Further this module admits a filtration by the non–negative integers: the $r$-th filtered piece of $V_{1}^{z_{1}}\otimes V_{2}^{z_{2}}$ is spanned by elements of the form $(y_{1}\otimes t^{s_{1}})\cdots(y_{m}\otimes t^{s_{m}})(v_{1}\otimes v_{2})$ where $m\geq 0$, $y_{1},\cdots,y_{m}\in\mathfrak{sl}_{n+1}$, $s_{1},\cdots,s_{m}\in\mathbb{Z}_{+}$ and $s_{1}+\cdots+s_{m}\leq r$.
The associated graded space is called a fusion product and is denoted
$V_{1}^{z_{1}}*V_{2}^{z_{2}}$. It admits a canonical $\mathfrak{sl}_{n+1}[t]$-module structure and
is generated by the image of $v_{1}\otimes v_{2}$ and,
$$\dim\left(V_{1}^{z_{1}}*V_{2}^{z_{2}}\right)=\dim V_{1}\dim V_{2}.$$
Proposition.
Let $\lambda_{0}\in P^{+}$, $\lambda_{1}\in P^{+}(1)$, $\lambda=2\lambda_{0}+\lambda_{1}$, $1\leq i<\min\lambda_{1}$ and $z_{1}\neq z_{2}\in\mathbb{C}$. There exists a surjective map of $\mathfrak{sl}_{n+1}[t]$–modules
$$M(\omega_{i},\lambda)\to M^{z_{1}}(0,\lambda)*M^{z_{2}}(0,\omega_{i}),\ \ \ m_%
{i,\lambda}\to m_{0,\lambda}*m_{0,\omega_{i}}.$$
In particular,
$\dim M(\omega_{i},\lambda)\geq\dim M(0,\lambda)\dim M(0,\omega_{i}).$
Proof.
The proposition follows if we prove that the element $\mathbf{m}:=m_{0,\lambda}*m_{0,\omega_{i}}$ (which generates $M^{z_{1}}(0,\lambda)*M^{z_{2}}(0,\omega_{i})$) satisfies the same relations as $m_{i,\lambda}$. We first prove that $\mathbf{m}$ satisfies the three relations in (5.1). The first relation in that equation is true in the tensor product $M^{z_{1}}(0,\lambda)\otimes M^{z_{2}}(0,\omega_{i})$ and hence hold in the fusion product as well. For the second relation, we use the definition of $M^{z_{1}}(0,\lambda)$ and $M^{z_{2}}(0,\omega_{i})$ to see that
$$\displaystyle(h\otimes(t-z_{1})^{r})(m_{0,\lambda}\otimes m_{0,\omega_{i}})=((%
h\otimes t^{r})m_{0,\lambda})\otimes m_{0,\omega_{i}}+m_{0,\lambda}\otimes(h%
\otimes(t+z_{2}-z_{1})^{r})m_{0,\omega_{i}}.$$
where the action on the right hand side is in $M(0,\lambda)\otimes M(0,\omega_{i})$.
If $r=0$ the relation holds in the tensor product and we are done. If $r\geq 1$,
the first term on the right hand side is zero and
the second term is $\omega_{i}(h)(z_{2}-z_{1})^{r}(m_{0,\lambda}\otimes m_{0,\omega_{i}})$. Hence
$$(h\otimes(t-z_{1})^{r})(m_{0,\lambda}\otimes m_{0,\omega_{i}})\in\mathbf{U}(%
\mathfrak{sl}_{n+1}[t])[0],$$
and so
in the associated graded space we get
$$(h\otimes t^{r})\mathbf{m}=(h\otimes(t-z_{1})^{r})\mathbf{m}=0,\ \ r\geq 1.$$
The third relation in (5.1) is immediate from the finite–dimensional representation theory of $\mathfrak{sl}_{n+1}$. Next a straightforward calculation gives,
$$(x_{p}^{-}\otimes(t-z_{1})^{(\lambda_{0}+\lambda_{1})(h_{p})}(t-z_{2})^{\omega%
_{i}(h_{p})})(m_{0,\lambda}\otimes m_{0,\omega_{i}})=0,\ \ 1\leq p\leq n,$$
and
$$(x_{\alpha}^{-}\otimes(t-z_{1})^{\lambda_{0}(h_{\alpha})}(t-z_{2}))(m_{0,%
\lambda}\otimes m_{0,\omega_{i}})=0,\ \ \alpha\in R^{+}(\lambda_{1}).$$
This means that in the fusion product we have
$$(x_{p}^{-}\otimes t^{(\lambda_{0}+\lambda_{1})(h_{p})+\omega_{i}(h_{p})})%
\mathbf{m}=0,\ \ 1\leq p\leq n,$$
and
$$(x_{\alpha}^{-}\otimes t^{\lambda_{0}(h_{\alpha})+1})\mathbf{m}=0,\ \ \alpha%
\in R^{+}(\lambda_{1}),$$
which proves that $\mathbf{m}$ satisfies the relations in (5.2). This completes the proof of the proposition.
∎
5.7.
We deduce some additional relations satisfied by $m_{i,\lambda}$. Note that by the second relation in (5.1) we get for $\alpha\in R^{+}$, $r\in\mathbb{Z}_{+}$,
$$(x_{\alpha}^{\pm}\otimes t^{r})m_{i,\lambda}=0\implies(h_{\alpha}\otimes t^{p}%
)(x_{\alpha}^{\pm}\otimes t^{r})m_{i,\lambda}=0\implies(x_{\alpha}^{\pm}%
\otimes t^{r+p})m_{i,\lambda}=0,\ \ p\in\mathbb{Z}_{+}.$$
Together with the first relation in (5.2) we have by a simple induction on $k-j$ that for all $1\leq j\leq k\leq n$,
$$(x^{-}_{j,k}\otimes t^{r})m_{i,\lambda}=0,\ \ {\rm{if}}\ \ r\geq(\lambda_{0}+%
\lambda_{1}+\omega_{i})(h_{j,k}).$$
(5.5)
Since
$(x_{j}^{-}\otimes t^{\lambda_{0}(h_{j})+1})m_{i,\lambda}=0$, $1\leq j\leq n$, a simple calculation (see [12] for instance) shows that
$$0=(x_{j}^{+}\otimes t)^{2\lambda_{0}(h_{j})}(x_{j}^{-}\otimes 1)^{2\lambda_{0}%
(h_{j})+2}m_{i,\lambda}=(x_{j}^{-}\otimes t^{\lambda_{0}(h_{j})})^{2}m_{i,%
\lambda}.$$
If $\alpha_{j,k}\in R^{+}(\lambda_{1})$ then by using the preceding two relations we get
$$0=(x^{-}_{j+1,k}\otimes t^{\lambda_{0}(h_{j+1,k})+1})(x_{j}^{-}\otimes t^{%
\lambda_{0}(h_{j})})^{2}m_{i,\lambda}=(x^{-}_{j,k}\otimes t^{\lambda_{0}(h_{j,%
k})+1})(x_{j}^{-}\otimes t^{\lambda_{0}(h_{j})})m_{i,\lambda}.$$
(5.6)
Proposition.
Suppose that $\lambda=2\lambda_{0}+\lambda_{1}$ with $\lambda_{1}\in P^{+}(1)$ and let $i<i_{1}=\min\lambda$.
There exists a right exact sequence of $\mathfrak{sl}_{n+1}[t]$–modules
$$M(\omega_{i-1},\lambda-\omega_{i_{1}}+\omega_{i_{1}+1})\to M(\omega_{i},%
\lambda)\to M(0,\lambda+\omega_{i})\to 0.$$
Proof.
Set
$$\displaystyle s=(\lambda_{0}+\lambda_{1})(h_{i,i_{1}}),\ \ \ \min(\lambda_{1}-%
\omega_{i_{1}})=i_{2},$$
$$\displaystyle\lambda_{2}=\lambda-\omega_{i_{1}}+\omega_{i_{1}+1}=2(\lambda_{0}%
+\delta_{i_{1}+1,i_{2}}\omega_{i_{1}+1})+\lambda_{1}-\omega_{i_{1}}+(1-\delta_%
{i_{1}+1,i_{2}})\omega_{i_{1}+1}.$$
In view of the short exact sequence in (5.3) it suffices to prove that the assignment
$$m_{i-1,\lambda_{2}}\to(x_{i,i_{1}}^{-}\otimes t^{s})m_{i,\lambda}$$
extends to a well–defined morphism $M(\omega_{i-1},\lambda_{2})\to M(\omega_{i},\lambda)$ of $\mathfrak{g}[t]$–modules. In other words it is enough to check that the element $m=(x_{i,i_{1}}^{-}\otimes t^{s})m_{i,\lambda}$ satisfies the defining relations of $M(\omega_{i-1},\lambda_{2})$. This is a tedious but straightforward checking. The first thing to check is that $m$ satisfies the defining relations of $W_{\operatorname{loc}}(\lambda_{2}+\omega_{i-1})$. For this, we observe that for $1\leq j\leq n$,
$$\displaystyle(x_{j}^{+}\otimes 1)m$$
$$\displaystyle=[(x_{j}^{+}\otimes 1),(x_{{i,i_{1}}}^{-}\otimes t^{s})]m_{i,%
\lambda}=(A\delta_{j,i}(x^{-}_{{i+1,i_{1}}}\otimes t^{s})+B\delta_{j,i_{1}}(x^%
{-}_{{i,i_{1}-1}}\otimes t^{s}))m_{i,\lambda},$$
for some $A,B\in\mathbb{C}$. It follows from (5.5) that the right hand side is zero once we note that
$$s=(\lambda_{0}+\lambda_{1})(h_{i,i_{1}})\geq\max\left\{(\lambda_{0}+\lambda_{1%
})(h_{i+1,i_{1}}),\ \ (\lambda_{0}+\lambda_{1})(h_{i,i_{1}-1})\right\}.$$
For the second relation in (5.1) we observe
$$(h\otimes t^{r})m=[h\otimes t^{r},x_{i,i_{1}}^{-}\otimes t^{s}]m_{i,\lambda}=-%
(\delta_{r,0}\lambda-\alpha_{i,i_{1}})(h)(x^{-}_{{i,i_{1}}}\otimes t^{s+r})m_{%
i,\lambda}.$$
If $r\geq 1$ then $s+r\geq(\lambda_{0}+\lambda_{1}+\omega_{i})(h_{i,i_{1}})$ and hence the right hand side is zero by (5.5).
The final relation in (5.1) holds by the standard representation theory of $\mathfrak{sl}_{n+1}$. Next we check that $m$ satisfies the relations in (5.2). We first show that
$$\displaystyle(x_{p}^{-}\otimes t^{r_{p}})m=0,\ \ r_{p}=(\lambda_{0}+\lambda_{1%
}-\omega_{i_{1}}+\omega_{i_{1}+1}+\omega_{i-1})(h_{p}),\ \ 1\leq p\leq n.$$
If $p\in\{i,i_{1}\}$ this follows from (5.6).
Assume that $p\notin\{i,i_{1}\}$.
Then
$$(x^{-}_{p}\otimes t^{r_{p}})m=\begin{cases}(x_{{i,i_{1}}}^{-}\otimes t^{s})(x_%
{p}^{-}\otimes t^{r_{p}})m_{i,\lambda},\ \ p\notin\{i-1,i_{1}+1\},\\
(x_{{i-1,i_{1}}}^{-}\otimes t^{s+r_{i-1}})m_{i,\lambda},\ \ p=i-1,\\
(x_{{p,i_{1}+1}}^{-}\otimes t^{s+r_{i_{1}+1}})m_{i,\lambda},\ \ p=i_{1}+1<i_{2%
},\\
m_{i,\lambda},\ \ p=i_{1}+1=i_{2}.\end{cases}$$
If $p\notin\{i-1,i_{1}+1\}$ then $r_{p}=(\lambda_{0}+\lambda_{1}+\omega_{i})(h_{p})$ and $(x^{-}_{p}\otimes t^{r_{p}})m_{i,\lambda}=0$.
If $p=i-1$, then
$$s+r_{i-1}=(\lambda_{0}+\lambda_{1})(h_{i,i_{1}})+(\lambda_{0}+\lambda_{1}+%
\omega_{i-1})(h_{i-1})=(\lambda_{0}+\lambda_{1}+\omega_{i})(h_{i-1,i_{1}})$$
and (5.5) gives $(x_{{i-1,i_{1}}}^{-}\otimes t^{s+r_{i-1}})m_{i,\lambda}=0$.
If $p=i_{1}+1$ and $i_{1}+1\neq i_{2}$ a similar argument shows that $(x_{{i,i_{1}+1}}^{-}\otimes t^{s+r_{i_{1}+1}})m_{i,\lambda}=0$.
If $p=i_{1}+1=i_{2}$, then one checks
$$(x^{-}_{i,i_{1}-1}\otimes t^{s})m_{i,\lambda}=0=(x^{-}_{i_{1},i_{1}+1}\otimes t%
^{r_{i_{1}+1}})m_{i,\lambda}.$$
In all cases the first relation in (5.2) is now established.
The second relation in (5.2) follows if we prove that
$$(x_{\alpha}^{-}\otimes t^{(\lambda_{0}+\delta_{i_{1}+1,i_{2}}\omega_{i_{1}+1})%
(h_{\alpha})+1})(x_{{i,i_{1}}}^{-}\otimes t^{s})m_{i,\lambda}=0,\ \ \alpha\in R%
^{+}(\lambda_{1}-\omega_{i_{1}}+(1-2\delta_{i_{1}+1,i_{2}})\omega_{i_{1}+1}).$$
If $i_{1}+1=i_{2}$, then
$$R^{+}(\lambda_{1}-\omega_{i_{1}}+(1-2\delta_{i_{1}+1,i_{2}})\omega_{i_{1}+1})%
\subset R^{+}(\lambda_{1})-\{\alpha_{i_{1},i_{1}+1}\}$$
and if $i_{1}+1<i_{2}$ then
$$R^{+}(\lambda_{1}-\omega_{i_{1}}+(1-2\delta_{i_{1}+1,i_{2}})\omega_{i_{1}+1})=%
(R^{+}(\lambda_{1})-\{\alpha_{i_{1},i_{2}}\})\cup\{\alpha_{i_{1}+1,i_{2}}\}.$$
If
$\alpha\neq\alpha_{i_{1}+1,i_{2}}$ then $[x_{\alpha}\otimes t^{r},x_{i,i_{1}}\otimes t^{s}]=0$, for each $r\in\mathbb{Z}_{+}$ and hence we get
$$(x_{\alpha}^{-}\otimes t^{\lambda_{0}(h_{\alpha})+1})(x_{i,i_{1}}^{-}\otimes t%
^{s})m_{i,\lambda}=(x_{i,i_{1}}^{-}\otimes t^{s})(x_{\alpha}^{-}\otimes t^{%
\lambda_{0}(h_{\alpha})+1})m_{i,\lambda}=0.$$
If $\alpha=\alpha_{i_{1}+1,i_{2}}$ then $i_{1}+1<i_{2}$ and so by the defining relations of $M(\omega_{i},\lambda)$ we have
$$(x^{-}_{i_{1},i_{2}}\otimes t^{\lambda_{0}(h_{i_{1},i_{2}})+1})m_{i,\lambda}=0%
=(x^{-}_{i,i_{1}-1}\otimes t^{\lambda_{0}(h_{i,i_{1}-1})+1})m_{i,\lambda}=0,$$
and so
$$(x_{i,i_{2}}^{-}\otimes t^{\lambda_{0}(h_{i,i_{2}})+2})m_{i,\lambda}=0.$$
It follows that
$$(x_{i_{1}+1,i_{2}}^{-}\otimes t^{\lambda_{0}(h_{i_{1}+1,i_{2}})+1})(x^{-}_{i,i%
_{1}}\otimes t^{s})m_{i,\lambda}=A(x_{i,i_{2}}^{-}\otimes t^{\lambda_{0}(h_{i,%
i_{2}})+2})m_{i,\lambda}=0,\ \ A\in\mathbb{C},$$
which completes the proof of (5.2) and so also of the Proposition.
∎
5.8.
The proof of Proposition 5 is completed in the course of establishing the following claim: for $\lambda=2\lambda_{0}+\lambda_{1}\in P^{+}$ and $i<\min\lambda_{1}$, we have
$$\displaystyle\dim M(\omega_{i},\lambda)=\dim M(\omega_{i},0)\dim M(0,\lambda).$$
(5.7)
The claim is proved by an induction on $i$. Induction begins at $i=0$ when there is nothing to prove since $M(0,0)\cong\mathbb{C}$. Otherwise using Proposition 5.7 we have
$$\dim M(\omega_{i},\lambda)\leq\dim M(\omega_{i-1},\lambda-\omega_{i_{1}}+%
\omega_{i_{1}+1})+\dim M(0,\lambda+\omega_{i}).$$
The following equality is clear if $i=1$,
and otherwise holds by the inductive hypothesis,
$$\dim M(\omega_{i-1},\lambda-\omega_{i_{1}}+\omega_{i_{1}+1})=\dim M(0,\omega_{%
i-1})\dim M(0,\lambda-\omega_{i_{1}}+\omega_{i_{1}+1}),$$
and hence
$$\dim M(\omega_{i},\lambda)\leq\dim M(0,\omega_{i-1})\dim M(0,\lambda-\omega_{i%
_{1}}+\omega_{i_{1}+1})+\dim M(0,\lambda+\omega_{i}).$$
By Proposition 5.6 we have $\dim M(\omega_{i},0)\dim M(0,\lambda)\leq\dim M(\omega_{i},\lambda)$ and hecne we get
$$\dim M(\omega_{i},0)\dim M(0,\lambda)\leq\dim M(\omega_{i-1})\dim M(0,\lambda-%
\omega_{i_{1}}+\omega_{i_{1}+1})+\dim M(0,\lambda+\omega_{i}).$$
Lemma 5.5 now shows that all the inequalities are actually equalities and the proof of the inductive step is complete. Notice that we have also proved that the inequality in (5.4) is an equality and so the proof of Proposition 5 is also complete.
References
[1]
F. Borges, Pierin, T. C. A new cluster character with coefficients for cluster category, arXiv:1804.10606
[2]
J. Beck and H. Nakajima, Crystal bases and two–sided cells of quantum affine algebras, Duke Math.
J. 123 (2004), no. 2, 335–402.
[3]
M. Brito, V. Chari and A. Moura, Demazure modules of level two and prime representations of quantum affine $\mathfrak{sl}_{n+1}$, J. Inst. Math. Jussieu (2015), 31 pages.
[4]
A. Berenstein, S. Fomin and A. Zelevinsky, Cluster algebras III: upper bounds and double Bruhat cells, Duke Math.
J. 126 (2005), 1â52.
[5]
P. Caldero and F. Chapoton, Cluster algebras as Hall algebras of quiver representations, Commentarii Mathematici Helvetici 81 (2006), 596â616.
[6]
V. Chari, Braid group actions and tensor products, Int. Math. Res. Notices (2002), 357–382.
[7]
V. Chari, A. Moura and C. Young, Prime representations from a
homological perspective, Math. Z. 274 (2013), 613–645.
[8]
V. Chari and A. Pressley, Quantum affine algebras, Comm. Math. Phys. 142 (1991), 261–283.
[9]
V. Chari and A. Pressley, Minimal affinizations of representations of quantum
groups: the nonsimply laced case, Lett. Math. Phys. 35 (1995), 99–114.
[10]
V. Chari and A. Pressley, Minimal affinizations of representations of quantum groups:
the simply–laced case, J. Alg. 184 (1996), no. 1, 1-30.
[11]
V. Chari, P. Shereen, R. Venkatesh and J. Wand, A Steinberg type decomposition theorem for higher level Demazure modules, arXiv:1408.4090.
[12]
V. Chari, R. Venkatesh, Demazure modules, Fusion Products, and $Q$-systems, Commun. Math. Phys. 333 (2015), no. 2, 799-830.
[13]
G. Dupont, Quantized Chebyshev polynomials and cluster characters with coefficients, Journal of Algebraic Combinatorics 31 (2010), 501â532, 10.1007/s10801-009-0198-8.
[14]
C. Fu and B. Keller, On cluster algebras with coefficients and 2-Calabi-Yau categories, Trans. Amer. Math. Soc. 362 (2010), 859â895.
[15]
B. Feigin and S. Loktev, On Generalized Kostka Polynomials and the Quantum Verlinde RuleDifferential topology,infinite–dimensional Lie algebras, and applications, Amer.Math. Soc. Transl. Ser. 2, Vol. 194 (1999), p. 61–79,math.QA/9812093.
[16]
S. Fomin, A. Zelevinsky, Cluster algebras: notes for the CDM-03 conference. Current developments in mathematics (2003), 1–34, Int. Press, Somerville, MA, 2003.
[17]
G. Fourier, P. Littelmann. Weyl modules, Demazure modules, KR-modules, crystals, fusion products, and limit constructions. Adv. Math. 211 (2007), no.2, 566-593.
[18]
E. Frenkel and E. Mukhin, Combinatorics of $q$-character of finite dimensional representations of quantum affine algebras, Commun. Math. Phys. 216 (2001), 23–57.
[19]
E. Frenkel and N. Reshetikhin, The $q$-characters of representations of quantum affine algebras and deformations of $W$-algebras, Contem. Math. 248 (1998), 163–205.
[20]
D. Hernandez, Algebraic approach to $q,t$-characters, Adv. Math. 187 (2004), 1–52.
[21]
D. Hernandez, Monomials of $q$ and $q,t$-characters for non simply–laced quantum affinizations, Math. Z. 250 (2005),
443–473.
[22]
D. Hernandez, The Kirillov–Reshetikhin conjecture and solutions of $T$-systems, J. Reine Angew. Math. 2006 (2006), 63–87
[23]
D. Hernandez, On minimal affinizations of representations of quantum groups, Comm. Math. Phys. 277 (2007), 221–
259.
[24]
D. Hernandez, Simple tensor product, Invent. Math 181 (2010), no. 3, 649–675.
[25]
D. Hernandez, Cyclicity and R-matrices, arXiv:1710.05756.
[26]
D. Hernandez and B. Leclerc, Cluster algebras and quantum affine algebras, Duke Math. J. 154 (2010), 265–341,
DOI 10.1215/00127094-2010-040.
[27]
D. Hernandez and B. Leclerc, Quantum Grothendieck rings and derived Hall algebras, J. Reine Angew. Math. 701 (2015), 77–126.
[28]
D. Hernandez and B. Leclerc, Monoidal categorifications of cluster algebras of type A and D, Symmetries, Integrable Systems and Representations, Springer Proceedings in Mathematics & Statistics 40 (2013), 175–193.
[29]
D. Hernandez and B. Leclerc, A cluster algebra approach to q–characters of Kirillov–Reshetikhin module, J. Eur. Math. Soc. 18 (2016), no. 5, 1113–1159.
[30]
S–J. Kang, M. Kashiwara, M. Kim and S–J. Oh, Monoidal categorification of cluster algebras,
Preprint arXiv:1412.8106
[31]
S–J. Kang, M. Kashiwara, M. Kim and S–J. Oh, Monoidal categorification of cluster algebras
II, Preprint arXiv:1502.06714.
[32]
G. Muller, J. Rajchot, AND B. Zykoski, Lower bound cluster algebras,: presentations, Cohen–Macaulayness and normality, arxiv 1508.02314.pdf.
[33]
H. Nakajima, $t$-analogs of $q$-characters of quantum affine algebras of type $A_{n}$, $D_{n}$, in Combinatorial and geometric
representation theory (Seoul, 2001), 141–160, Contemp. Math., 325, Amer. Math. Soc., Providence, RI (2003).
[34]
H. Nakajima, Quiver Varieties and $t$–Analogs of $q$–Characters of Quantum Affine Algebras,
Ann. of Math. 160, 1057–1097 (2004).
[35]
H. Nakajima, Quiver varieties and cluster algebras, Kyoto J. Math. 51 (2011), 71–126.
[36]
K. Naoi, Weyl Modules, Demazure modules and finite crystals for non-simply laced type, Adv. Math. 229 (2012), no.2, 875–934.
[37]
M. Varagnolo and E. Vasserot, Standard modules of quantum affine algebras, Duke Math. J. 111, no.
3, 509–533 (2002).
[38]
J. Wand, Demazure flags of local Weyl modules, Ph.D. Thesis, UC Riverside, http://escholarship.org/uc/item/8nr0m842 |
Photovoltaic Chiral Magnetic Effect
Katsuhisa Taguchi${}^{1,3}$, Tatsushi Imaeda${}^{1,3}$, Masatoshi Sato${}^{2}$, and Yukio Tanaka${}^{1,3}$
${}^{1}$Department of Applied Physics, Nagoya University, Nagoya, 464-8603, Japan
${}^{2}$Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto, 606-8502, Japan
${}^{3}$CREST, Japan Science and Technology Corporation (JST), Nagoya 464-8603, Japan
Abstract
We theoretically predict a generation of a current in Weyl semimetals
by applying circularly polarized light.
The electric field of the light can drive an
effective magnetic field of order of ten Tesla.
For lower frequency light, a non-equilibrium spin distribution is
formed near the Fermi surface.
Due to the spin-momentum locking,
a giant electric current proportional to the effective magnetic field is
induced.
On the other hand, higher frequency light realizes a quasi-static
Floquet state with no induced electric current.
We discuss relevant materials and estimate order of magnitude of the
induced current.
Introduction—
Recently, Dirac and Weyl semimetals,
which host bulk gapless excitations obeying quasi-relativistic fermion
equations, have attracted much attention in condensed matter physics
rf:Murakami07 ; rf:burkov11 ; rf:Yang14 ; rf:biinse1 ; rf:biinse2 ; rf:nabi1 ; rf:nabi2 ; rf:cd2as3theo ; rf:cd2as3exp ; rf:taas1 ; rf:taas2 ; rf:taas3 ; rf:taas4 ; rf:tominaga14 .
Dirac semimetals
have been theoretically predictedrf:Murakami07 ; rf:burkov11 ; rf:Yang14
and experimentally demonstrated
in (Bi${}_{1-x}$In${}_{x}$)${}_{2}$Se${}_{3}$rf:biinse1 ; rf:biinse2 ,
Na${}_{3}$Birf:nabi1 ; rf:nabi2 and
Cd${}_{3}$As${}_{2}$rf:cd2as3theo ; rf:cd2as3exp .
There are also several experiments
supporting the realization of Weyl semimetals
in TaAsrf:taas1 ; rf:taas2 ; rf:taas3 ; rf:taas4 .
Moreover, Dirac and Weyl semimetals have been theoretically predicted
in a superlattice heterostructure of topological insulator (TI)/normal insulator
(NI)rf:burkov11 , and a Dirac semimetal has been realized in the
GeTe/Sb${}_{2}$Te${}_{3}$ superlatticerf:tominaga14 .
Low energy bulk excitations in Dirac and Weyl semimetals come in pairs
of left and right-handed Weyl fermions,
because of the Nielsen-Ninomiya’s no go theoremrf:Nielsen81 . In the low energy limit, each charge flow of left and right-handed Weyl
fermions preserves classically, but their difference,
the axial current, is not conserved in the quantum theory, due to the
chiral anomaly.
In an analogy of relativistic high energy physics
rf:Vilenkin80 ; rf:Metlitski05 ; rf:Kharzeev08 ; rf:Fukushima08 ; rf:Kharzeev13 ; rf:Kharzeev14 ,
the anomaly related effects have been discussed in condensed matter
physicsrf:Jackiw99 ; rf:Vazifeh13 ; rf:Yamamoto12 ; rf:Zyuzin12 ; rf:Zyuzin12b ; rf:Chen13 ; rf:Burkov15 ; rf:Sumiyoshi15 ; rf:Nomura15 ; rf:Taguchi15 .
The anomaly induced currents are dissipationless and thus they
have potential applications to unique electronics.
Among the anomaly related effects, one of the most interesting phenomena is
the chiral magnetic effect.
In the presence of a time-dependent $\theta$ term in the Dirac-Weyl theory,
a current proportional to an applied static magnetic field has been predicted
theoreticallyrf:Vilenkin80 ; rf:Metlitski05 ; rf:Kharzeev08 ; rf:Fukushima08 ; rf:Kharzeev13 ; rf:Zyuzin12 ; rf:Zyuzin12b ; rf:Vazifeh13 ; rf:Yamamoto12 .
The flow due to the static magnetic field, however, might
be problematic in condensed matter physics.
First, in Weyl semimetals, the time-dependent $\theta$ term is
obtained in the ground state, in the presence of the energy difference of
left and right-handed Weyl points rf:Zyuzin12b .
However, the system stays the ground state under a static magnetic field, so no
actual current should flow eventuallyrf:Vazifeh13 .
Moreover, the detection can be difficult because there is no driving force
to get out the current in such an equilibrium state.
Hence, instead of a static magnetic field, one should consider a
non-equilibrium magnetic field to obtain a net current of the chiral
magnetic effect.
Recent studies using femtosecond laser pulses have established
a method to generate
non-equilibrium magnetic fields by circularly polarized light in ferrimagnetsrf:Stanciu07 ; rf:Vahaplar09 ; rf:Kirilyuk10 .
The light-induced effective magnetic field
$\bm{B}^{\textrm{eff}}$ is given by
$$\displaystyle\bm{B}^{\textrm{eff}}\propto i\bm{\mathcal{E}}\times\bm{\mathcal{E}}^{*}.$$
(1)
with the circularly polarized complex electric field $\bm{\mathcal{E}}$rf:Pitaevskii61 ; rf:Pershan66 .
The generation of the effective magnetic field is due to the conversion of
spin-angular momentum from light to electrons via the spin-orbit
couplingrf:Pershan66 ; rf:Taguchi11 ; rf:Misawa11 .
The direction of $\bm{B}^{\textrm{eff}}$ depends on the chirality of
the circularly polarized light.
Its magnitude is proportional
to the intensity of laser and
can be 20 T for a sufficient strong laser pulserf:Stanciu07 ; rf:Vahaplar09 ; rf:Kirilyuk10 .
In this Letter, we theoretically predict a current $\bm{j}$ induced by the effective magnetic field (Fig.1).
The photovoltaic current is due to a non-equilibrium spin
distribution near the Fermi surface.
For lower frequency light, the conversion of spin-angular momentum
between light and electrons occurs only near the Fermi surface.
Thus, like the relativistic theory, the low energy description using
Weyl fermions gives a good approximation to evaluate the chiral
magnetic effect.
On the basis of the Keldysh Green’s function, we show that a net current
is obtained by applying circularly polarized light.
The current is proportional to the effective magnetic field in an analogous form
of the chiral magnetic effect.
On the other hand, when light has a frequency higher than the energy
scale of the band width, a quasi-static Floquet state is realized,
where the chiral magnetic effect is cancelled due to the occupied band electrons.
In the latter case, Weyl points are shifted in the momentum
space, resulting in the change of the anomalous Hall effect, instead.
Model—We consider
the following Hamiltonian to describe
Weyl/Dirac semimetals in the presence of circularly polarized light
$$\displaystyle H$$
$$\displaystyle=H_{\textrm{Weyl}}+H_{\textrm{em}}+V_{\textrm{imp}}.$$
(2)
The first term is the Hamiltonian of Weyl/Dirac semimetals.
In low energy, it takes the form
$$\displaystyle H_{\textrm{Weyl}}$$
$$\displaystyle=\sum_{\bm{k}}\psi^{\dagger}_{\bm{k}}\mathcal{H}_{\textrm{Weyl}}\psi_{\bm{k}},$$
(3)
$$\displaystyle\mathcal{H}_{\textrm{Weyl}}$$
$$\displaystyle=\hbar v_{\textrm{F}}\sigma^{z}(\bm{k}-\sigma^{z}\bm{b})\cdot\bm{s}-\mu\sigma^{0}s^{0}-\mu_{5}\sigma^{z}s^{0},$$
(4)
where $\psi_{\bm{k}}={}^{t}\!(\psi_{\uparrow,+}\ \psi_{\downarrow,+}\ \psi_{\uparrow,-}\ \psi_{\downarrow,-})$
is the annihilation operator of electron with (pseudo)spin
($\uparrow,\downarrow$) and helicity ($+,-$).
$s^{\mu}$ and $\sigma^{\mu}$ are the Pauli matrices of (pseudo)spin and
helicity, $v_{\textrm{F}}$ is the Fermi velocity, and $\mu$
is the chemical
potential.
The parameters $2\bm{b}$ and $2\mu_{5}$ denote the difference of the position
of left and right-handed Weyl points in the momentum and energy spaces,
respectively.
For Dirac semimetals, $\bm{b}=\bm{0}$ and $\mu_{5}=0$.
The second term in Eq. (3) is the gauge coupling between
Weyl/Dirac semimetals and light
$$\displaystyle H_{\textrm{em}}$$
$$\displaystyle=-\sum_{\bm{k}}\bm{j}\cdot\bm{A}^{\textrm{em}},$$
(5)
where $\bm{j}$ denotes the charge current,
and $\bm{A}^{\textrm{em}}$ is the vector potential of light.
For circlarily polarized light, the electric field $\bm{E}^{\textrm{em}}=-\partial_{t}\bm{A}^{\textrm{em}}$ is
given by
$$\displaystyle\bm{E}^{\textrm{em}}$$
$$\displaystyle=\textrm{Re}\left[\bm{\mathcal{E}}e^{i\Omega t}\right],$$
(6)
where $\bm{\mathcal{E}}$ is a complex vector and
$\Omega$ is the angular frequency of light.
The third term in Eq. (2) expresses the
impurity scattering in Weyl/Dirac semimetalsrf:book1 ; rf:Hosur12 ,
$$\displaystyle V_{\textrm{imp}}$$
$$\displaystyle=\sum_{\bm{k},\bm{q}}\psi^{\dagger}_{\bm{k}+\bm{q}}\sigma^{0}s^{0}u_{\textrm{imp}}(\bm{q})\psi_{\bm{k}}.$$
(7)
The impurity scattering potential $u_{\textrm{imp}}$ is
assumed to be short-ranged
and triggers a finite relaxation time,
which is given within the Born approximation as
$\tau_{{\rm e},\sigma}=\hbar/(\pi\nu_{{\rm e},\sigma}n_{\textrm{c}}u_{\textrm{imp}}^{2})$
with a concentration of nonmagnetic impurities $n_{\textrm{c}}$.
Current induced by circularly polarized light—
We calculate the current induced by light,
using the Keldysh Green’s function techniquerf:book1 .
Below, we assume that $\hbar\Omega$ is much lower than the
band width, so the low energy effective Hamiltonian (3) gives a
good approximation.
For Eq.(3),
the current is defined by
$$\displaystyle\langle\bm{j}\rangle\equiv ev_{\textrm{F}}\langle\psi^{\dagger}(\bm{x},t)\sigma^{z}\bm{s}\psi(\bm{x},t)\rangle,$$
(8)
which is decomposed as
$$\displaystyle\langle\bm{j}\rangle\equiv\langle\bm{j}_{+}\rangle+\langle\bm{j}_{-}\rangle,$$
(9)
with $\langle\bm{j}_{\sigma=\pm}\rangle\equiv\sigma ev_{\textrm{F}}\langle\psi^{\dagger}_{\sigma}\bm{s}\psi_{\sigma}\rangle$.
Here $\psi^{\dagger}_{\sigma}=(\psi^{\dagger}_{\sigma,\uparrow},\psi^{\dagger}_{\sigma,\downarrow})$ is the creation operator of Weyl
fermions with helicity $\sigma=\pm$.
There is no mixing term between $\psi^{\dagger}_{+}$
and $\psi_{-}$ in $H$, and thus
$\langle{\bm{j}}_{+}\rangle$ and $\langle{\bm{j}}_{-}\rangle$ can be
calculated separately.
For a while, we consider the ${\bm{b}}=0$ case.
In terms of the Keldysh Green’s function, the chiral current $\langle\bm{j}_{\sigma}\rangle$ is represented as
$\langle\bm{j}_{\sigma}\rangle=-\sigma i\hbar ev_{\textrm{F}}\textrm{tr}[\bm{s}G^{<}_{\sigma}(\bm{x},t:\bm{x},t)]$
with the $2\times 2$ matrix lesser Green function $G_{\sigma}^{<}(\bm{x},t:\bm{x},t)=-i\hbar\langle\psi^{\dagger}_{\sigma}(\bm{x},t)\psi_{\sigma}(\bm{x},t)\rangle$.
The contribution from $\bm{B}^{\textrm{eff}}\propto i\bm{\mathcal{E}}\times\bm{\mathcal{E}^{*}}$ is given by the diagrams in Fig.2.
It is written as
$$\displaystyle\langle j^{i}_{\sigma}\rangle=-i\hbar ev_{\rm F}[{\cal I}_{\sigma}^{ijk}(\Omega)+{\cal I}_{\sigma}^{ijk}(-\Omega)]{\cal E}^{j}{{\cal E}^{*}}^{k}$$
(10)
with
$$\displaystyle{\cal I}_{\sigma}^{ijk}(\Omega)=\frac{e^{2}v_{\rm F}^{2}}{4\Omega^{2}}\sum_{I=a,b,c,d}{\cal C}_{\sigma}^{(I),ijk}.$$
(11)
Each diagram in Fig.2 gives the following ${\cal C}_{\sigma}^{(I=a,b,c,d),ijk}$
$$\displaystyle{\cal C}_{\sigma}^{(a),ijk}=\sum_{{\bm{k}},\omega}{\rm tr}\left[s^{i}g_{{\bm{k}},\omega,\sigma}s^{j}g_{{\bm{k}},\omega+\Omega,\sigma}s^{k}g_{{\bm{k}},\omega,\sigma}\right]^{<},$$
$$\displaystyle{\cal C}_{\sigma}^{(b),ijk}=\sum_{{\bm{k}},\omega}{\rm tr}\left[s^{i}g_{{\bm{k}},\omega,\sigma}{\cal S}^{j}_{\omega,\omega+\Omega}g_{{\bm{k}},\omega+\Omega,\sigma}s^{k}g_{{\bm{k}},\omega,\sigma}\right]^{<},$$
$$\displaystyle{\cal C}_{\sigma}^{(c),ijk}=\sum_{{\bm{k}},\omega}{\rm tr}\left[s^{i}g_{{\bm{k}},\omega,\sigma}s^{j}g_{{\bm{k}},\omega+\Omega,\sigma}{\cal S}^{k}_{\omega+\Omega,\omega}g_{{\bm{k}},\omega,\sigma}\right]^{<},$$
$$\displaystyle{\cal C}_{\sigma}^{(d),ijk}=\sum_{{\bm{k}},\omega}{\rm tr}\left[{\cal S}^{i}_{\omega,\omega}g_{{\bm{k}},\omega,\sigma}s^{j}g_{{\bm{k}},\omega+\Omega,\sigma}s^{k}g_{{\bm{k}},\omega,\sigma}\right]^{<},$$
(12)
where $g_{{\bm{k}},\omega,\sigma}^{<}$ is given by
$$\displaystyle g_{{\bm{k}},\omega,\sigma}^{<}=f_{\omega}\left[g_{{\bm{k}},\omega,\sigma}^{\rm a}-g_{{\bm{k}},\omega,\sigma}^{\rm r}\right]$$
(13)
with the Fermi distribution function $f_{\omega}$ and the retarded and
advanced Green’s functions
$$\displaystyle g^{\rm r}_{{\bm{k}},\omega,\sigma}=\left[\hbar\omega-\sigma\hbar v_{\rm F}{\bm{k}}\cdot{\bm{s}}+\mu+\sigma\mu_{5}+\frac{i\hbar}{2\tau_{{\rm e},\sigma}}\right]^{-1},$$
$$\displaystyle g^{\rm a}_{{\bm{k}},\omega,\sigma}=\left[g^{\rm r}_{{\bm{k}},\omega,\sigma}\right]^{\dagger}.$$
(14)
${\cal S}^{i}_{\omega,\omega^{\prime}}$ is the vertex correction due to the
nonmagnetic impurity scattering $V_{\rm imp}$rf:suppl-A .
Using Eq.(13), one can rewrite $C_{\sigma}^{(I=a,b,c,d),ijk}$ in terms
of the retarded and advanced Green’s functions.
For $|\mu+\sigma\mu_{5}|\gg\hbar/\tau_{\rm e}$, we find that
$$\displaystyle C_{\sigma}^{(I=a,b,c,d),ijk}\propto\sum_{\omega}(f_{\omega+\Omega}-f_{\omega}).$$
(15)
This means that only fermions near the Fermi surface contribute the
light-induced current, which justifies our Weyl fermion approximation.
We also find that $C_{\sigma}^{(I),ijk}$ contains both of the
retarded and advanced Green’s functions, and it is expressed by their product.
This is a signal of a non-equilibrium processrf:book1 .
After some calculation rf:suppl-A , we obtain
$$\displaystyle\langle{\bm{j}}_{\sigma}\rangle=\sigma\frac{2\nu_{{\rm e},\sigma}e^{3}v_{\rm F}^{3}\tau_{{\rm e},\sigma}^{4}}{3\hbar}\Omega i({\bm{\mathcal{E}}}\times{\bm{\mathcal{E}}^{*}}),$$
(16)
where $\nu_{\rm e,\sigma}=\frac{(\mu+\sigma\mu_{5})^{2}}{2\pi^{2}\hbar^{3}v_{\textrm{F}}^{3}}$ is the density of state of the Weyl cone with
helicity $\sigma$.
From Eq. (16), the total current $\langle\bm{j}\rangle$ is
$$\displaystyle\langle{\bm{j}}\rangle=\frac{2(\nu_{{\rm e},+}\tau^{4}_{{\rm e},+}-\nu_{{\rm e},-}\tau^{4}_{{\rm e},-})e^{3}v_{\rm F}^{3}}{3\hbar}\Omega i({\bm{\mathcal{E}}}\times{\bm{\mathcal{E}}^{*}}),$$
(17)
which is non-zero
when $\nu_{{\rm e},+}\tau^{4}_{{\rm e},+}\neq\nu_{{\rm e},-}\tau^{4}_{{\rm e},-}$, namely when $\mu_{5}\neq 0$.
The obtained current originates from a non-equilibrium distribution of spin:
When one exposes the system to circularly polarized light,
the conversion of spin-angular momentum
between light and electrons occurs due to the spin-orbit interaction.
As a result, there arises a non-equilibrium distribution of spin near
the Fermi surface [Fig.1(a)].
For Weyl fermions, because of the spin-momentum locking, the
non-equilibrium spin distribution gives rise to the current flow
[Fig.1(b)].
Indeed, for Eq.(3),
the current operator is essentially the same as the
spin operator, and thus, from the same calculation, one can show that the circularly
polarized light induces a non-zero spin polarization of electrons
$$\displaystyle\langle\psi^{\dagger}_{\sigma}{\bm{s}}\psi_{\sigma}\rangle=\frac{2\nu_{{\rm e},\sigma}e^{2}v_{\rm F}^{2}\tau^{4}_{{\rm e},\sigma}}{3\hbar}\Omega i({\bm{\mathcal{E}}}\times{\bm{\mathcal{E}}^{*}}),$$
(18)
near the Fermi surface.
Since the circularly polarized light induces the spin-polarization of
electrons, it effectively acts as a Zeeman magnetic field near the Fermi surface
$$\displaystyle\bm{B}^{\textrm{eff}}_{\sigma}$$
$$\displaystyle\equiv\chi_{\sigma}\Omega i\bm{\mathcal{E}}\times\bm{\mathcal{E}}^{*}=\sigma_{\textrm{L}}\chi_{\sigma}\Omega|\bm{\mathcal{E}}|^{2}\hat{\bm{q}},$$
(19)
with $\chi_{\sigma}\equiv\frac{2}{3}\frac{e^{2}v_{\textrm{F}}^{2}\tau^{4}_{{\rm e},\sigma}}{g\mu_{\textrm{B}}\hbar}$.
Here $\hat{\bm{q}}$ is the unit vector of the direction of
light propagation,
$\sigma_{\textrm{L}}=\pm 1$ specifies the chirality (clockwise or
counter-clockwise polarization) of light, $g$ is
the Landé factor, and $\mu_{\rm B}$ is
the Bohr magneton.
It is noted that the light-induced current has a similarity to
the chiral magnetic effect.
In both cases, the current flows in the direction of an applied magnetic
or effective magnetic field, and its magnitude is proportional to the
difference of the chemical potential between left and right-handed fermions.
Indeed, like our case, the spin-polarization and the spin-momentum
locking are essential to obtain the current in the chiral magnetic
effectrf:Kharzeev08 .
Under a static magnetic field, electrons form the Landau levels.
For Weyl fermions, the zeroth Landau level is fully
spin-polarized in the direction of the applied magnetic field, and thus
the ground state of the system is also spin-polarized.
As a result, the current flows
due to the spin-momentum lockingrf:Kharzeev08 . We dub our light-induced current effect as photovoltaic chiral magnetic effect.
Here we would like to mention that there is an important difference
between our photovoltaic chiral magnetic effect and the original one.
In the original case, the chiral magnetic effect is caused by a static
magnetic field, and thus the resultant current is
equilibrium (and dissipationless).
In condensed matter physics, however, an analogous current of Weyl
fermions, even if exists, is completely cancelled by other current in
the conduction band rf:Vazifeh13 .
On the other hand, the photovoltaic chiral magnetic effect is due to the
time-dependent electric field, so the current is non-equilibrium and dissipative.
The current comes only from Weyl fermions near the Fermi surface, so no
cancellation occurs.
The effective magnetic field also generates the axial current, which is
the difference between charge currents with
different helicity: $\langle\bm{j}_{\textrm{axial}}\rangle\equiv\langle\bm{j}_{+}\rangle-\langle\bm{j}_{-}\rangle=ev_{\textrm{F}}[\langle\psi^{\dagger}_{+}{\bm{s}}\psi_{+}\rangle+\langle\psi^{\dagger}_{-}{\bm{s}}\psi_{-}\rangle]$.
As mentioned above, for lower $\Omega$, the system is well described by
Weyl fermions, and thus the axial current can be well-defined as well.
The axial current is nonzero even for Dirac semimetals with ${\bm{b}}=\mu_{5}=0$.
The axial current can be detected as total spin polarization, by using
pump-probe techniquesrf:Kirilyuk10 .
We can easily generalize the above result for $\langle{\bm{j}}\rangle$ in the case with ${\bm{b}}\neq 0$.
Since ${\bm{b}}$ behaves like a static Zeeman field in ${\cal H}_{\rm Weyl}$,
it can shift $\langle\psi^{\dagger}_{\sigma}{\bm{s}}\psi_{\sigma}\rangle$ by
the Pauli paramagnetism.
However, ${\bm{b}}$ cannot drive a net current
since it is static.
Moreover, the circularly polarized light affects only electrons near the
Fermi surface, which structure does not depend on ${\bm{b}}$.
Therefore, we have the same current $\langle{\bm{j}}\rangle$ in Eq.(17)
even when ${\bm{b}}\neq 0$.
We estimate the magnitude of ${\bm{B}}^{\textrm{eff}}_{\sigma}$ and $\langle\bm{j}\rangle$ by using
material parameters for TaAsrf:C-Zhang15 ,
$v_{\textrm{F}}=3\times 10^{5}$ m/s,
$\tau_{e}=4.5\times 10^{-11}$ s, and
$\mu=11.5$ meV.
If the difference of the chemical potential is $\mu_{5}=1$ meV,
$|{\bm{B}}^{\textrm{eff}}_{\sigma}|$ can be estimated as
$|{\bm{B}}^{\textrm{eff}}_{\sigma=\pm}|=(4.3\mp 2.6)\times 10^{-16}(\frac{\Omega}{[\rm s^{-1}]})(\frac{|\bm{\mathcal{E}}|^{2}}{[\rm V^{2}/m^{2}]})$ T.
For $|\bm{\mathcal{E}}|=4$ kV/m and $\Omega=2.2\times 10^{9}$ s${}^{-1}$,
$|{\bm{B}}^{\textrm{eff}}_{\sigma=\pm}|$ can reach up to $15\mp 9$ T.
Then, the induced charge current
becomes $|\langle\bm{j}\rangle|\simeq 2\times 10^{6}$A/m${}^{2}$,
whose current density is much larger than the anomalous Hall current density
due to the chiral anomalyrf:Sekine15 .
This giant current density is caused from the giant magnetic field ${\bm{B}}^{\textrm{eff}}_{\sigma}$.
We would like to point out that
$\langle\bm{j}\rangle$ is distinguished from
the longitudinal rf:Hosur12 and the
transverse charge current
rf:Zyuzin12 ; rf:Chen13 ; rf:Burkov15 ; rf:Sekine15 ,
since $\langle\bm{j}\rangle$ is parallel to the light traveling
direction and it flows in an opposite direction when
the chirality of the light is reversed.
Floquet state—
So far, we have assumed that the frequency $\Omega$ of light is much
lower than a scale of the band width.
Now, we consider the opposite case.
In contrast to the lower $\Omega$ case, in which only electrons near the
Fermi surface are influenced by light, the higher frequency light can
affect the whole electrons in valence bands.
To consider this situation, we adapt
the Floquet method:
Because $H$ in Eq.(2) is periodic in $t$, i.e. $H(t)=H(t+2\pi/\Omega)$,
the wave function of the Schödinger equation
$i\hbar\partial_{t}\psi(t)=H(t)\psi(t)$ has the form of $\psi(t)=\sum_{m}\phi_{m}e^{-i(\varepsilon+m\hbar\Omega)t/\hbar}$, where the summation is taken for all
integers $m$.
Substituting this form into the Schödinger equation, we have
the Floquet equation, $\sum_{n}H_{m,n}\phi_{n}=(\varepsilon+m\hbar\Omega)\phi_{m}$, with
$H_{m,n}=(\Omega/2\pi)\int_{0}^{2\pi/\Omega}dtH(t)e^{i(m-n)\Omega t}+m\hbar\Omega\delta_{m,n}$.
For the Hamiltonian in Eq.(2), the diagonal term of the Floquet
Hamiltonian is given by $H_{m,m}=H_{\rm Weyl}+V_{\rm imp}+m\hbar\Omega$, and
the off-diagonal ones are
$H_{m,m+1}=H_{m+1,m}^{*}=(\Omega/2\pi)\int_{0}^{2\pi/\Omega}dtH_{\rm em}e^{-i\Omega t}=-\frac{iev_{\textrm{F}}|\bm{\mathcal{E}}|}{2\Omega}\sigma^{z}(s^{x}-i\sigma_{\rm{L}}s^{y})$ when light is along $z$ axis.
Other off-diagonal terms are identically zero.
Each solution of the Floquet equation gives a periodic steady state.
For large $\Omega$, the diagonal terms are dominant, so one can treat the
off-diagonal ones as a perturbation.
In the zeroth order, our system is described by
$H_{0,0}=H_{\rm Weyl}+V_{\rm imp}$, then
the first non-zero correction in the perturbation theory appears in the
second order as
$\frac{1}{\hbar\Omega}\left[H_{0,-1},H_{0,1}\right].$
Thus, we obtain the following effective Hamiltonian
$$\displaystyle H_{\rm eff}=H_{\rm Weyl}+V_{\rm imp}+i\sigma^{0}\frac{e^{2}v_{\rm F}^{2}}{\hbar\Omega^{3}}(\bm{\mathcal{E}}\times\bm{\mathcal{E}}^{*})\cdot{\bm{s}},$$
(20)
by which a periodic steady state of our system is described.
From Eq. (20), it is found that the higher
frequency light induces a different effective Zeeman magnetic field,
$$\displaystyle{\bm{B}}^{\rm eff}_{\rm Floquet}\equiv\frac{e^{2}v_{\rm F}^{2}}{g\mu_{\rm B}\hbar\Omega^{3}}i\sigma^{0}(\bm{\mathcal{E}}\times\bm{\mathcal{E}}^{*}).$$
(21)
Here we note that the physical origin is completely different from
that in the lower frequency case.
While ${\bm{B}}^{\rm eff}_{\sigma}$ in Eq.(19) originates from a dissipative
process so it depends on $\tau_{{\rm e},\sigma}$,
${\bm{B}}^{\rm eff}_{\rm Floquet}$ in Eq.(21) is independent of the impurity scattering.
Furthermore, the former magnetic field only affects on electrons near
the Fermi surface, but the latter acts on the whole of the band.
Consequently, the resultant phenomena can be different.
We find that no net current $\langle{\bm{j}}\rangle$ is obtained
by ${\bm{B}}^{\rm eff}_{\rm Floquet}$:
According to Eq.(20), ${\bm{B}}^{\rm eff}_{\rm Floquet}$ just provides a uniform Zeeman splitting (or shift) in the
whole band spectrum of the Weyl semimetal, like a static Zeeman field.
Therefore, in a steady state, electrons fill the band up
to the Fermi energy. In this situation, one can use the same argument in
Ref.rf:Vazifeh13 , and prove that $\langle{\bm{j}}\rangle=0$.
Whereas Weyl fermions may have a nonzero spin $\langle\psi_{\sigma}^{\dagger}{\bm{s}}\psi_{\sigma}\rangle$ due to the Pauli magnetism of ${\bm{B}}^{\rm eff}_{\rm Floquet}$, the current due to the spin-momentum locking is
totally cancelled by the current from the rest of the band.
In other words, no photovoltaic chiral magnetic effect occurs for the
higher frequency light.
It is helpful to regard the frequency $\Omega$ as an energy cutoff
for the chiral magnetic effect.
For lower $\Omega$, the light can excite only Weyl fermions near the Fermi
surface, and thus the quasi relativistic phenomena like the chiral
magnetic effect may occur. As $\Omega$ increases, electrons in a
lower position of the band can participate in the current, then
eventually, when $\Omega$ is large enough to affect the whole spectrum
of the band, the chiral magnetic effect is completely cancelled.
Instead, for higher $\Omega$, one can
expect the light induced anomalous Hall effect.
Substituting Eq.(3) for $H_{\rm Weyl}$ in
Eq.(20), one finds that ${\bm{B}}^{\rm eff}_{\rm Floquet}$ shifts ${\bm{b}}$ by $\delta{\bm{b}}=-(g\mu_{\rm B}/2\hbar v_{\rm F}){\bm{B}}^{\rm eff}_{\rm Floquet}$.
The change of ${\bm{b}}$
induces the change of $\theta$-term in the Weyl semimetals rf:Zyuzin12b , which
results in
$$\displaystyle\langle\delta\rho\rangle=\frac{2\alpha c\epsilon_{0}}{\pi}\delta{\bm{b}}\cdot{\bm{B}},\quad\langle\delta{\bm{j}}\rangle=-\frac{2\alpha c\epsilon_{0}}{\pi}\delta{\bm{b}}\times{\bm{E}},$$
(22)
in the presence of external magnetic and electric fields, ${\bm{B}}$ and
${\bm{E}}$. Here $\alpha$ is the fine structure constant, $c$ is the speed of light, and $\epsilon_{0}$ is the vacuum permittivity.
The light induced charge pump $\langle\delta\rho\rangle$ and anomalous
Hall current $\langle\delta{\bm{j}}\rangle$ have been discussed
recently in Refs.rf:Oka15 ; rf:Chan15 ; rf:oka09 .
Conclusion—
We theoretically predict photovoltaic chiral magnetic effect, which is induced by the effective magnetic field due to circularly polarized light.
In the low light frequency regime,
the effective magnetic field affects only fermions near the Fermi surface.
As a result, the effective magnetic field plays the role to trigger a
finite spin polarization of Weyl fermions and drive the finite charge current in
Eq.(17).
On the other hand, in the high frequency regime, the Floquet quasi
steady state is realized.
The circularly polarized light induces the effective magnetic field in
Eq.(21),
which is completely different from that in the lower frequency regime.
The magnetic field in the high frequency regime behaves like the Zeeman
field and shifts the whole
band structure.
The current of Weyl fermions are completely cancelled by other band
contribution.
Our photovoltaic chiral magnetic effect, which drastically
depends on the light frequency,
realizes the chiral magnetic effect in condensed matter
physics.
Acknowledgments
The authors acknowledge the fruitful discussion with K. T. Law and W. Y. He.
This work was supported by Grants-in-Aid for the Core Research for Evolutional Science and Technology (CREST) of the Japan Science, for Japan Society for the Promotion of Science (JSPS) Fellows,
for Topological Materials Science (TMS) (Grant No.15H05855, No.15H05853),
for Scientific Research B (Grant No.25287085, No.15H03686),
and
for Challenging Exploratory Research (Grant No.15K13498).
References
(1)
S. Murakami,
New J. Phys. 9, 356 (2007).
(2)
B-J. Yang and N. Nagaosa,
Nat. Comm.
5, 4898 (2014).
(3)
A. A. Burkov and Leon Balents,
Phys. Rev. Lett. 107, 127205 (2011).
(4)
M. Brahlek, N. Bansal, N. Koirala, S. Y. Xu, M. Neupane,
C. Liu, M. Z. Hasan, and S. Oh,
Phys. Rev. B. 109, 186403 (2012).
(5)
L. Wu, M. Brahlek, R. Valdes A., A. V. Stier, C. M.
Morris, Y. Lubashevsky, L. S. Bilbro, N. Bansal, S. Oh,
and N. P. Armitage,
Nature Physics 9, 410 (2013).
(6)
Z. Wang, Y. Sun, X. Q. Chen, C. Franchini, G. Xu,
H. Weng, X. Dai, and Z. Fang,
Phys. Rev. B 85, 195320 (2012).
(7)
Z. K. Liu, B. Zhou, Y. Zhang, Z. J. Wang, H. M. Weng,
D. Prabhakaran, S. K. Mo, Z. X. Shen, Z. Fang, X. Dai,
Z. Hussain, and Y. L. Chen,
Science
343, 864 (2014).
(8)
Wang, Zhijun and Weng, Hongming and Wu, Quansheng and Dai, Xi and Fang, Zhon,
Phys. Rev. B. 88, 125427 (2013).
(9)
Z. K. Liu, J. Jiang, B. Zhou, Z. J. Wang, Y. Zhang, H. M. Weng, D. Prabhakaran, S-K. Mo, H. Peng, P. Dudin, T. Kim, M. Hoesch, Z. Fang, X. Dai, Z. X. Shen, D. L. Feng, Z. Hussain and Y. L. Chen,
Nature mat. 13, 677 (2014).
(10)
S. M. Huang, S. Y. Xu, I. Belopolski, C. C. Lee,
G. Chang, B. K. Wang, N. Alidoust, G. Bian, M. Neupane,
A. Bansil, H. Lin, and M. Z. Hasan,,
Nat. Comm. 6, 7373 (2014).
(11)
H. Weng, C. Fang, Z. Fang, A. Bernevig, and X. Dai,
Phys. Rev. X 5, 011029 (2015).
(12)
B. Q. Lv, H. M. Weng, B. B. Fu, X. P. Wang, H. Miao, J. Ma,
P. Richard, X. C. Huang, L. X. Zhao, G. F. Chen, Z. Fang, X. Dai, T. Qian, and H. Ding,
Phys. Rev. X 5, 031013 (2015).
(13)
S. Y. Xu, I. Belopolski, N. Alidoust, M. Neupane,
C. Zhang, R. Sankar, S. M. Huang, C. C. Lee, G. C.,
B. K. Wang, G. Bian, H. Zheng, D. S. Sanchez, F. Chou,
H. Lin, S. Jia, and M. Z. Hasan,
Science 349, 613 (2015).
(14)
J. Tominaga, A. V. Kolobov, P. Fons, T. Nakano, and S. Murakami,
Adv. Mater. Interfaces 1, 1300027 (2014).
(15)
H. B. Nielsen and M. Ninomiya,
Phys. Lett. B 105.2 219 (1981).
(16)
A. Vilenkin,
Phys. Rev. D 22, 3080 (1980).
(17)
D. E. Kharzeev, L. D. McLerran, and H. J. Warringa,
Nucl. Phys. A 803, 227 (2008).
(18)
K. Fukushima, D. E. Kharzeev, and H. J. Warringa,
Phys. Rev. D 78, 074033 (2008).
(19)
M. A. Metlitski and A. R. Zhitnitsky,
Phys. Rev. D 72, 045011 (2005).
(20)
D. Kharzeev, K.Landsteiner, A.Schmitt, H.-U. Yee,
Strongly Interacting Matter in Magnetic Fields
(Springer, New York, 2013), Chap.9.
(21)
D. E. Kharzeev,
Prog. Part. Nucl. Phys. 75, 133 (2014).
(22)
R. Jackiw and V. Alan Kosteleck,
Phys. Rev. Lett. 82, 3572 (1999).
(23)
M. M. Vazifeh and M. Franz, Phys. Rev. Lett. 111, 027201 (2013).
(24)
D. T. Son and N. Yamamoto,
Phys. Rev. Lett. 109.181602 (2012)
(25)
A. A. Zyuzin and A. A. Burkov,
Phys. Rev. B 86, 115133 (2012).
(26)
A. A. Zyuzin, S. Wu, and A. A. Burkov,
Phys. Rev. B 85, 165110 (2012).
(27)
Y. Chen, S. Wu, and A. A. Burkov,
Phys. Rev. B 88, 125105 (2013).
(28)
A. A. Burkov,
J. Phys. Condens. Matter 27, 113201 (2015).
(29)
H. Sumiyoshi and S. Fujimoto,
arXiv1509.03981v2.
(30)
K. Nomura and D. Kurebayashi,
Phys. Rev. Lett. 115, 127201 (2015).
(31)
K. Taguchi and Y. Tanaka,
Phys. Rev. B 91, 054422 (2015).
(32)
A. Kirilyuk, A. V. Kimel, and T. Rasing,
Rev. Mod. Phys. 82, 2731 (2010).
(33)
C. D. Stanciu, F. Hansteen, A. V. Kimel, A. Kirilyuk, A. Tsukamoto, A. Itoh, and T. Rasing,
Phys. Rev. Lett. 99, 047601 (2007).
(34)
K. Vahaplar, A. M. Kalashnikova, A. V. Kimel, D. Hinzke, U. Nowak, R. Chantrell, A. Tsukamoto, A. Itoh, A. Kirilyuk, and Th. Rasing,
Phys. Rev. Lett. 103, 117201(2009).
(35)
L. P. Pitaevskii,
Sov. Phys. JETP 12, 1008 (1961).
(36)
P. S. Pershan, J. P. van der Ziel, and L. D. Malmstrom,
Phys. Rev. 143, 574 (1966).
(37)
K. Taguchi and G. Tatara,
Phys. Rev. B 84, 174433 (2011).
(38)
T. Misawa, T. Yokoyama, and S. Murakami,
Phys. Rev. B. 84.165407 (2011).
(39)
P. Hosur, S. A. Parameswaran, and A. Vishwanath,
Phys. Rev. Lett. 108, 046602 (2012).
(40)
H. Haug and A. P. Jauho, Quantum Kinetics in Transport and Optics of Semiconductors, 2nd ed. (Springer, New York, 2007).
(41)
See the supplemental material for ”Photovoltaic Chiral Magnetic Effect”.
(42)
C. Zhang, Z. Yuan, S.-Y. Xu, Z. Lin, B. Tong, M. Z. Hasan, J. Wang, C. Zhang, S. Jia,
arXiv:1502.00251.
(43)
A. Sekine and K. Nomura, arXiv 1508.04590.
(44)
S. Ebihara, K. Fukushima, and T. Oka, arXiv 1509.03673.
(45)
C. Chan, P. A. Lee, K. S. Burch, J. H. Han, and Y. Ran, arXiv 1509.05400.
(46)
T. Oka and H. Aoki in Phys. Rev. B 79, 081406(R) (2009). |
Decay of highly-correlated spin states in a dipolar-coupled solid
HyungJoon Cho111Currently address: Schlumberger Doll Research, Ridgefield, CT, Paola Cappellaro, David G. Cory and Chandrasekhar Ramanathan222Author to whom correspondence should be addressed. Electronic address:sekhar@mit.edu
Department of Nuclear Science and Engineering, Massachusetts Institute of Technology, Cambridge 02139, USA
(November 26, 2020)
Abstract
We have measured the decay of NMR multiple quantum coherence intensities both under the internal dipolar Hamiltonian as well as when this interaction is effectively averaged to zero, in the cubic calcium fluoride (CaF${}_{2}$) spin system and the pseudo one-dimensional system of fluoroapatite. In calcium fluoride the decay rates depend both on the number of correlated spins in the cluster, as well as on the coherence number. For smaller clusters, the decays depend strongly on coherence number, but this dependence weakens as the size of the cluster increases. The same scaling was observed when the coherence distribution was measured in both the usual Zeeman or $z$ basis and the $x$ basis. The coherence decay in the one dimensional fluoroapatite system did not change significantly as a function of the multiple quantum growth time, in contrast to the calcium fluoride case. While the growth of coherence orders is severely restricted in this case, the number of correlated spins should continue to grow, albeit more slowly. All coherence intensities were observed to decay as Gaussian functions in time. In all cases the standard deviation of the observed decay appeared to scale linearly with coherence number.
I Introduction
While there have been several proposals put forward for scalable quantum computing architectures, experimental realizations have been limited to a handful of qubits at most. Maintaining coherence as the size of the system Hilbert space increases remains extremely challenging. It is essential to understand how decoherence rates in different physical systems scale as a function of system size.
There have been a number of theoretical investigations on the scaling behaviour of decoherence Unruh ; Palma ; Duan ; Reina . These general models have typically been based on the spin-boson model of Leggett and co-workers Leggett . System-specific scaling laws have also been proposed for a few physical implementations (e.g. Dalton ; Ischi ).
Palma et al. Palma showed that for a multi-qubit quantum register, the decay of particular off-diagonal elements of the system density matrix depended on the Hamming distance $f$ between the two states. In the case of independent, uncorrelated noise, the decay was of the form $\exp(-f\Gamma(t))$ while for the case of correlated noise the decay was $\exp(-f^{2}\Gamma(t))$ where $\Gamma(t)$ corresponds to the single qubit decay.
Suter and coworkers recently published an experimental study of the decay of multi-spin states using nuclear magnetic resonance (NMR)Krojanski . They used multiple-quantum (MQ) NMR
experiments to create correlated multi-spin states in a powdered sample of the plastic crystal adamantane, and observed the rate at which these states decay during evolution under the internal dipolar coupling of the spins. They observed that the decay rate increased as a square root of the estimated number of correlated spins in the cluster.
A theoretical analysis of their experimental results has been published recently by Fedorov and Fedichkin Fedorov . Neglecting the flip-flop (XY) terms of the dipolar interaction, they obtained the following expression for the decay of multiple quantum coherence states, in the short time, large spin limit
$$S_{n}(t)=p\exp\left(-\alpha n^{2}t^{2}\right)+\left(1-p\right)\exp\left(-\frac%
{N}{2}\alpha t^{2}\right)$$
(1)
where $p=1/N(\sum_{j}d_{jk})^{2}/\sum_{j}d_{jk}^{2}$ is a correlation parameter, $\alpha$ is proportional to the second moment of the lineshape, $n$ is the coherence number and $N$ is the number of correlated spins. The first term depends strongly on the coherence number and indicates that the spins in the multi-quantum state experience a correlated mean field. The second term does not depend on the coherence number, but only depends on the number of spins in the cluster, and indicates that the fields experienced by the different spins are uncorrelated from each other.
These equations agree well with the measured data at short times, but deviate at longer times. The correlation parameter $p$ appeared to remain constant with increasing spin system size. However since adamantane is a plastic crystal in which the molecules are undergoing rapid rotational motions, the intramolecular dipolar couplings are averaged to zero, and the only residual couplings are motionally averaged intermolecular dipolar couplings. This motional averaging modulates the contributions in Equation 1, making the experimental results obtained peculiar to this class of samples.
In this paper we expand on these preliminary studies. Our test systems are the cubic lattice of 100% abundant ${}^{19}$F spins in a single crystal of calcium fluoride, and the pseudo one dimensional spin chains of fluoroapatite (FAp). These are both rigid crystals, and molecular motions will not affect the results obtained. In addition to characterizing the decay of the system under the dipolar Hamiltonian, we measure the decay rates obtained when we suspend evolution under the dipolar Hamiltonian. We also repeated the experiments, encoding the multi-spin coherences in a different basis, and observed the resulting scaling behaviour.
The standard MQ NMR experiment is shown in Figure 1
Baum ; Munowitz . Assuming that the system is closed, the final density matrix is given by
$$\rho_{f}^{\zeta}(\phi)=U_{DQ}^{\dagger}U_{ev}U_{\zeta}(\phi)U_{DQ}\rho_{i}U_{%
DQ}^{{\dagger}}U_{\zeta}^{\dagger}(\phi)U_{ev}^{\dagger}U_{DQ}$$
(2)
$\phi$ is the phase angle and $\zeta=\{x,z\}$ is the rotation axis for the multiple-quantum encoding. Collective rotations about each axis yield coherence order distributions in the corresponding basis. The propagator $U_{DQ}=\exp\left(-it\mathcal{H}_{DQ}\right)$ represents evolution under the double quantum (DQ) Hamiltonian given by
$$\mathcal{H}_{\mathrm{DQ}}=-\frac{1}{2}\sum_{j<k}D_{jk}\left\{\sigma_{j}^{+}%
\sigma_{k}^{+}+\sigma_{j}^{-}\sigma_{k}^{-}\right\}.\\
$$
(3)
where the dipolar coupling constant $D_{jk}$ between spins $j$ and $k$ is
$$D_{jk}=\frac{\gamma^{2}\hbar^{2}}{{r}_{jk}^{3}}\left(1-3\cos^{2}\theta_{jk}%
\right),\\
$$
(4)
$\gamma$ is the gyromagnetic ratio, $r_{jk}$ is the distance
between spins $j$ and $k$, and $\theta_{jk}$ is the angle between
the external magnetic field and inter-nuclear vector $\vec{r}_{jk}$. This effective DQ Hamiltonian is created by multiple-pulse NMR techniques that toggle the dipolar Hamiltonian to create the appropriate zeroth order Average Hamiltonian.
In Equation 2 above we have assumed that the experimental implementation of $-\mathcal{H}_{DQ}$ is perfect.
The encoding of the coherence orders is performed by the collective rotations $U_{\zeta}(\phi)=\exp\left(-i\phi\sum_{j}\sigma_{\zeta j}\right)$.
The observed signal is given by the overlap
$$\displaystyle S(t)$$
$$\displaystyle=$$
$$\displaystyle Tr\left[\rho_{f}^{\zeta}(\phi)\rho_{i}\right]$$
(5)
$$\displaystyle=$$
$$\displaystyle Tr\left[\rho_{f}^{DQ\zeta}(\phi)\rho_{i}^{DQ}\right]$$
(6)
where $\rho_{f}^{DQ\zeta}(\phi)=U_{ev}U_{\zeta}(\phi)\rho_{iDQ}U_{\zeta}^{\dagger}(%
\phi)U_{ev}^{\dagger}$, and $\rho_{i}^{DQ}=U_{DQ}\rho_{i}U_{DQ}^{\dagger}$, and the evolution $U_{ev}$ is defined below.
Following this evolution the density operator of the spin system in the Zeeman basis contains off-diagonal terms of the form
$$\displaystyle\sigma_{+}^{1}\sigma_{+}^{2}\cdots\sigma_{+}^{n}\sigma-^{n+1}%
\cdots\sigma_{-}^{n+m}\sigma_{z}^{n+m+1}\cdots\sigma_{z}^{n+m+q}$$
$$\displaystyle+\hskip 1.4454pt\sigma_{-}^{1}\sigma_{-}^{2}\cdots\sigma_{-}^{n}%
\sigma_{+}^{n+1}\cdots\sigma_{+}^{n+m}\sigma_{z}^{n+m+1}\cdots\sigma_{z}^{n+m+%
q}.$$
(7)
We are interested in the properties of such coherences in the system. In the experiments here, we create a distribution of states with different coherence orders $M=(m-n)$, and spin numbers $K=n+m+p$.
After this step, we either allow the spin system to evolve under the internal dipolar
interaction or suspend evolution of the spin system by applying a time suspension sequence. In the first case, $U_{ev}=\exp\left(-i\mathcal{H}_{D}t\right)$, where the dipolar
Hamiltonian is
$$\mathcal{H}_{D}=\sum_{j<k}D_{jk}\left\{\sigma_{jz}\sigma_{kz}-\frac{1}{4}(%
\sigma_{j+}\sigma_{k-}+\sigma_{j-}\sigma_{k+})\right\}.$$
(8)
In the second case $U_{ev}=\mathcal{I}$, if the time suspension sequence is perfect and the system is completely isolated. Decays observed during this time could be the result of errors in the control, or couplings to an environment.
The experiments were
performed at room temperature at 2.35 T (94.2 MHz, ${}^{19}$F), using
a Bruker Avance spectrometer and home built probe. The samples used
were a 1 mm${}^{3}$ single crystal of CaF${}_{2}$ with T${}_{1}$ $\sim$ 7s, and a crystal of fluorapatite (FAp) with $T_{1}\sim 200$ ms. The FAp crystal is a mineral crystal specimen from Durango, Mexico.
High power 0.5 $\mu s$ $\pi$/2 pulses were used. The rotation $U_{z}(\phi)=\exp\left(-i\phi\sum_{j}\sigma_{zj}\right)$ was used to encode coherence number in the Zeeman or $z$ basis while the rotation $U_{x}(\phi)=\exp\left(-i\phi\sum_{j}\sigma_{xj}\right)$ was used to encode coherence orders in the $x$-basis Ramanathan ; vanBeek . The phase ($\phi$) was incremented from 0 to 4$\pi$ with $\Delta\phi=\frac{\pi}{32}$ to encode up to 32 quantum coherences for every experiment. A
fixed-time point corresponding to the maximum intensity signal was
sampled for each $\phi$ value, and then was Fourier transformed with
respect to $\phi$ to obtain the coherence order distribution, as seen in Figure 2.
As the evolution time $\tau$ under the DQ Hamiltonian increases (also referred to as the MQ growth time), progressively more spins are correlated into the MQ states. Table 1 shows the size of the spin system for each MQ growth time, as estimated by the method of Baum et al. Baum , and used in Krojanski . A log-log fit indicates that the number of spins is increasing as $N(=n+m+q)\sim t^{2}$.
MQ growth time
System size (N)
43.3 $$\mu$$s
6
86.8 $$\mu$$s
8
130.2 $$\mu$$s
12
173.6 $$\mu$$s
28
217 $$\mu$$s
36
260.4 $$\mu$$s
66
303.8 $$\mu$$s
96
Table 1. Effective size of the spin cluster as a function of the MQ growth time, using the model in Baum .
II Evolution under the secular dipolar Hamiltonian
The MQ states are not stationary under the internal dipolar Hamiltonian of the spins, and evolve as a function of the dipolar interaction time. This uncompensated evolution leads to imperfect refocusing during $-\mathcal{H}_{DQ}$.
Figure 3 shows the intensities of various coherence
orders in the $z$ basis as a function of dipolar evolution time for two different MQ growth times. The signal intensity appears to decay as a Gaussian function in time. We have fit the data to $\exp(-t^{2}/2T_{d}^{2})$ to extract the effective decay times ($T_{d}$ = standard deviation) for each coherence order. The Gaussian shape of the decays indicates that the underlying process is consistent with a time-invariant dispersion of fields (due to the spins) having a normal distribution Abragam .
Figure 4 shows the decay times $T_{d}$ of the different $z$ basis coherence orders as a function of the MQ growth times. Four features are evident in the data: (i) the $T_{d}$ are seen to depend linearly on the coherence number; (ii) for short $\tau$, the $T_{d}$ are seen to depend strongly on the coherence number; (iii) this dependence on coherence number weakens significantly with increasing $\tau$ and (iv) the incremental change in $T_{d}$ with increasing $\tau$ decreases.
We performed a linear fit of $T_{d}$ versus coherence number for each of the MQ growth times as
$$T_{d}=T_{d}(0)-\kappa\cdot n$$
(9)
where $T_{d}(0)$ is the decay of the zero-quantum intercept, and $\kappa$ is the slope. Figure 5 shows the dependence of $\ln(T_{d}(0))$ and $\ln(-\kappa)$ on $\ln(N)$ the size of the spin system from Table 1, as well as the best linear fits obtained. We get
$\ln(T_{d}(0))=3.94-0.57\cdot\ln(N)$ and $\ln(-\kappa)=2.75-1.21\cdot\ln(N)$. Thus $T_{d}(0)\approx A/\sqrt{N}$ and $\kappa\approx-B/N$, where $A=51.4$ and $B=15.6$. We can therefore express the scaling behavior of $T_{d}$ as
$$T_{d}=\frac{A}{\sqrt{N}}-\frac{B\cdot n}{N}\>\>.$$
(10)
We have repeated the experiment encoding the coherence orders in the $x$-basis instead, and obtained identical scaling behaviour. The decays were once again observed to be Gaussian. In Figure 6 we show the results of a multi-dimensional experiment in which we correlate the $z$ and $x$ basis decay times for an MQ growth time $\tau=$ 217.2 $\mu$s. It should be noted that the dipolar Hamiltonian in a strong external magnetic field is anisotropic (see Equation 8), but this is not reflected in the observed decay times.
III Evolution under a time suspension sequence
We then attempted to suppress evolution of the dipolar Hamiltonian using a time-suspension pulse sequence that implements (approximately) the identity operator on the spin system Cory . In the ideal experiment, we should see no decay of the spin coherences due to dipolar couplings within the spin system. The cycle time of the 48-pulse time suspension sequence used here was 132.48 $\mu s$. The change in
intensity of the $z$-basis coherence order was measured as a function of the number of loops of the 48-pulse sequence.
Once again we observed Gaussian decays as a function of time for the coherence intensities, indicating that the underlying noise has a long correlation time. We fit the data to a Gaussian and extracted the $T_{d}$ decay times.
Figure 7 shows the effective $T_{d}$’s of the $z$-basis coherence orders, under the time-suspension sequence. We see that the data is qualitatively identical to that obtained in the case of dipolar evolution, showing the same features discussed above. This correspondence was repeated in the $x$-basis data, as well as in the correlated $z$ and $x$ basis encoding experiments.
Figure 8 shows the ratio of the $z$-basis multiple quantum $T_{d}$’s during evolution under the 48-pulse time suspension sequence to the $T_{d}$’ measured during dipolar evolution. The ratio was measured to be around 70, and appeared to be independent of the size of the spin correlations involved. Given the uniform scaling obtained, we did not repeat the linear fits. However it is clear from the uniform scaling that both $A$ and $B$ are scaled by the same factor of around 70.
IV quasi-1D Fluoroapatite system
We also measured the decay of multiple quantum coherences under the dipolar Hamiltonian in fluoroapatite. Floroapatite is a quasi-one dimensional spin chain, as the distance between spin chains is about three times larger than the distance between adjacent spins in the chain Engelsberg . The one dimensional spin chain with nearest neighbor double quantum Hamiltonian is exactly solvable Feldman1 ; Feldman2 , and it has been shown that starting from a thermal equilibrium state, only zero and double quantum coherences are produced, even as higher order multi-spin states are created. The presence of higher order coherences indicates the importance of next-nearest neighbor and other distant couplings. Consequently it has been observed that higher order coherences grow very slowly in this system Yesinowski .
Consistent with the one dimensional nature of the spin system, the growth of the spin clusters is much slower in this case. More importantly, we see that the character of the decays does not change as a function of the the MQ growth time. A strong dependence on coherence number is observed for both short and long MQ growth times.
However, even at long times, the number of correlated spins is still small in this system.
V Discussion
It is important to ensure that the observed decays are not an artifact of the encoding used to make the measurement. For example, we need to examine if the phases introduced into the system by the encoding $U_{\zeta}(\phi)$ lead to imperfect refocusing in our experiments. We measured the decay time for the particular situation where $U_{\zeta}(\phi)=U_{\zeta}^{\dagger}(\phi)=\mathcal{I}$, as a function of the MQ growth time. The observed scaling for the case of dipolar evolution was $T_{d}\approx 22.7/\sqrt{N}$ and $T_{d}\approx 2276/\sqrt{N}$ for the 48-pulse time suspension sequence. These times are slightly shorter than the corresponding values of $T_{d}(0)$ for the zero quantum coherence. This is reasonable as we are measuring the aggregate decay of all the MQ states here, and suggests that the encoding step is not responsible for the decay rates observed.
The similarity between the dipolar evolution and the time suspension data suggests that the dominant source of noise in the time suspension data are residual dipolar coupling terms that are not effectively averaged out, as these would be expected to scale identically in the two cases. However, it is worth examining an alternative model in which the decay is due to coupling to an external environment. Fedorov et al. have also calculated the effect of a large, bath coupled to the multiple quantum states, and get a similar solution to that obtained under an inter-spin Ising coupling. Thus both theories - residual dipolar errors and the presence of external spins yield identical scaling behaviours and it is not possible a priori to distinguish between theses two models on the basis of the data here. However we can examine the physics of the system under study to understand the origin of the decays.
In principle the environment that the ${}^{19}$F nuclear spins are coupled to could be lattice phonons or other spins—both electron and nuclear—that are present in the system. In order to effectively relax the nuclear spins, the phonons would need to be resonant with the Larmor frequency of the spins (94.2 MHz in these experiments).
The Debye Temperature of calcium fluoride is 510 K, so it can be assumed that the spins are fixed in a rigid lattice in these room temperature measurements, and that phonons do not play a significant role in the relaxation of the spins. In addition to sparse paramagnetic impurities (responsible for $T_{1}$ relaxation), we know that the CaF${}_{2}$ system contains ${}^{43}$Ca which has spin 7/2 and is 0.13$\%$ abundant. There could possibly be other spin defects in the crystal such as protons—albeit at much lower concentrations.
Paramagnetic impurities
We also measured the decay rates under the dipolar evolution in a second crystal with a longer $T_{1}$ relaxation time $\approx$ 110 s, corresponding to a smaller concentration of paramagnetic impurities, and found no difference in the experimentally observed $T_{d}$’s. In Mn-doped CaF${}_{2}$ with a $T_{1}$ of 700 ms, Tse and Lowe estimated that the concentration of impurities ($N_{e}$) was approximately $5.6\times 10^{17}$ cm${}^{-3}$ Tse , yielding an average impurity separation of 15 nm. Since $1/T_{1}\propto N_{e}$, a 7 s $T_{1}$ corresponds to an impurity concentration of $5.6\times 10^{16}$ cm${}^{-3}$ and an average separation of 32.4 nm, while for $T_{1}=110$s, we get a concentration of $3.56\times 10^{15}$ cm${}^{-3}$ and a separation of 81.3 nm.
The magnetic moment of a S=5/2 paramagnetic impurity is about 3500 times larger than that of the fluorine nucleus. The dipolar coupling between such an impurity and a ${}^{19}$F nucleus becomes comparable to the strongest ${}^{19}$F–${}^{19}$F coupling ($a$ = 2.73 Å) at a distance of about 4 nm. This corresponds to an interaction strength of about 15 kHz, and dipolar correlation time of about 66 $\mu$s. This is approximately the size of the frozen core of fluorine spins around each impurity site, in which the “flip-flop” or XY terms of the dipolar Hamiltonian are suppressed. The electron-nuclear interaction further drops to 10 % of the inter-nuclear coupling at a distance of 8.9 nm. Thus, assuming Mn impurities, for a $T_{1}=7$ s crystal, 84 % of the nuclear spins experience a hyperfine field that is less than 1.5 kHz. The correlation time of the hyperfine field seen by the bulk of the nuclear spins is much too long to explain the observed decays.
${}^{43}$Ca spins
The magnetic moment of the calcium spins is about half that of the fluorine spins, and the Ca–F spacing is about 2.36 Å, just a little shorter than the F–F distance. Thus we expect the strongest Ca–F coupling to be 11.6 kHz. However, given the low natural abundance of the Ca spins (0.13 %), very few nuclear spins see a Ca coupling of this strength. Another possibility is that the Ca–Ca dipolar coupling leads to mutual spin flips, that reduces the efficiency of the 48-pulse time suspension sequence. The mean separation between ${}^{43}$Ca spins is 4.1 nm, and the average ${}^{43}$Ca–${}^{43}$Ca coupling is about 2.2 Hz. Since the Ca–F coupling is decoupled on the time-scale of one cycle of the 48-pulse sequence, which is about 44 $\mu$s, the 2 Hz Ca–Ca coupling is too weak to affect the efficiency of the decoupling sequence. Thus we do not expect the ${}^{43}$Ca spins to be the source of the observed decays.
Errors in control
We have assumed to this point that the implementation of all the pulse-sequences has been perfect. The propagators $U_{DQ}$ and $U_{DQ}^{\dagger}$ are idealized propagators for the NMR pulse sequences used, and typically correspond to the zeroth order Average Hamiltonian of the sequence. In reality the presence of higher order terms in the Magnus expansion result in $U_{DQ}^{exp}U_{DQ}^{{\dagger}exp}\neq\mathcal{I}$. More importantly, errors in the implementation of the two sequences—the presence of phase transients during the leading and falling edges of the pulses, as well as errors in the setting of the $\pi/2$ pulse lead to imperfections in the refocusing. This imperfect refocusing is however unlikely to be the source of the observed decay in either experiment. In the dipolar evolution experiment, the strength of the dipolar coupling is much stronger than any residual error terms in the propagator, and it is these couplings that determine the decay rate.
In the time-suspension experiments, the 48-pulse sequence averages the dipolar interaction to zero to second order in the Magnus expansion. Assuming perfect implementation, the leading error terms are likely to be the second-order dipolar-offset term and the second order offset term. Our results indicate that the source of the observed signal decay in the time suspension experiments is the residual errors in the zeroth-order Average Hamiltonian, which are proportional to $\mathcal{H}_{D}$, rather than second-order terms like $\left[\mathcal{H}_{D}(t_{1}),\mathcal{H}_{D}(t_{2})\right]$, or other higher-order terms. The configurational space accessible by a single spin-flip is much smaller than that accessible by two spin-flips, especially at larger spin numbers. This suggests that we should expect significantly different scaling of decay rates for the two types of processes. Given the identical scaling behaviour observed in the two cases, we can conclude that the
the dominant error in the time-suspension sequences has the form of $\mathcal{H}_{D}$ (possibly toggled about some arbitrary axis). The more likely source of error in the control is imperfect implementation of the sequence. Phase transients during the pulses, or small errors in the setting of the $\pi/2$ pulse can accumulate over the hundreds of pulses that are applied in this experiment, and the residual error terms will look like modulated dipolar interactions, and consequently will scale the same way, though with a reduced strength Viola .
Acknowledgements
This work was supported in part by the National Security Agency (NSA) under Army Research Office (ARO) contract numbers DAAD19-03-1-0125 and W911NF-05-1-0469, and by DARPA DSO and the Air Force Office of Scientific Research.
References
(1)
W. Unruh, Phys. Rev. A 51, 992-997 (1995).
(2)
G. M. Palma, K.-A. Suominen, A. K. Ekert, Proc. R. Soc. Lon. A 452, 567–584 (1996).
(3)
L.-M. Duan and G.-C. Guo, Phys. Rev. A 57, 737–741 (1998).
(4)
J. H. Reina, L. Quiroga and N. F. Johnson, Phys. Rev. A 65 Article 032326 (2002).
(5)
A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher and W. Zwerger, Rev. Mod. Phys. 59, 1–85 (1987).
(6)
B. J. Dalton, J. Mod. Optics 50, 951–966 (2003).
(7)
B. Ischi, M. Hilke and M. Dube, Phys. Rev. B 71 Article 195325 (2005).
(8)
H. G. Krojanski and D. Suter, Phys. Rev. Lett. 93, Article 090501 (2004).
(9)
A. Fedorov and L. Fedichkin, J. Phys. C. 18, 3217—3228 (2006).
(10)
J. Baum, M. Munowitz, A.N. Garroway and A. Pines, J. Chem. Phys. 83, 2015 (1985).
(11)
M. Munowitz, A. Pines and M. Mehring, J. Chem. Phys. 86, 3172 (1987).
(12)
A. Abragam, The Principles of Nuclear Magnetism, Oxford University Press, Oxford (1961).
(13)
C. Ramanathan, H. Cho, P. Cappellaro, G.S. Boutis and D. G. Cory, Chem. Phys. Lett. 369, 311–317 (2003).
(14)
J. D. van Beek, M. Carravetta, G. C. Antonioli and M. H. Levitt, J. Chem. Phys. 122, 244510 (2005).
(15)
D.G. Cory, J.B. Miller, and A.N. Garroway, J. Magn. Reson. 90 205 (1990).
(16)
M. Engelsberg, I.J. Lowe, and J.L. Carolan, Phys. Rev. B 7, 924–929 (1973).
(17)
E.B. Fel’dman and S. Lacelle, Chem. Phys. Lett. 253, 27–31 (1996).
(18)
E.B. Fel’dman and S. Lacelle, J. Chem. Phys. 107, 7067–7084 (1997).
(19)
G. Cho and J.P. Yesinowski, J. Phys. Chem. 100, 15716–15725 (1996).
(20)
D. Tse and I.J. Lowe, Phys. Rev. 166, 292–302 (1968).
(21)
L. Viola and E. Knill, Phys. Rev. Lett. 94, 060502 (2005). |
[
[
Abstract
Solar activity, ranging from the background solar wind to energetic coronal mass ejections (CMEs), is the main driver of the conditions in the interplanetary space and in the terrestrial space environment, known as space weather. A better understanding of the Sun-Earth connection carries enormous potential to mitigate negative space weather effects with economic and social benefits. Effective space weather forecasting relies on data and models. In this paper, we discuss some of the most used space weather models, and propose suitable locations for data gathering with space weather purposes.
We report on the application of Representer analysis (RA) and Domain of Influence (DOI) analysis to three models simulating different stages of the Sun-Earth connection: the OpenGGCM and Tsyganenko models, focusing on solar wind - magnetosphere interaction, and the PLUTO model, used to simulate CME propagation in interplanetary space. Our analysis is promising for space weather purposes for several reasons. First, we obtain quantitative information about the most useful locations of observation points, such as solar wind monitors. For example, we find that the absolute values of the DOI are extremely low in the magnetospheric plasma sheet. Since knowledge of that particular sub-system is crucial for space weather, enhanced monitoring of the region would be most beneficial. Second, we are able to better characterize the models. Although the current analysis focuses on spatial rather than temporal correlations, we find that time-independent models are less useful for Data Assimilation activities than time-dependent models. Third, we take the first steps towards the ambitious goal of identifying the most relevant heliospheric parameters for modelling CME propagation in the heliosphere, their arrival time, and their geoeffectiveness at Earth.
\helveticabold
1 Keywords:
solar wind, coronal mass ejections (CMEs), magnetohydrodynamics (MHD), numerical simulations, statistical tools, domain of influence, observations
DOI analysis for space weather]Domain of Influence analysis: implications for Data Assimilation in space weather forecasting
Millas et al.]Dimitrios Millas ${}^{1,4,*}$, Maria Elena Innocenti ${}^{1,*}$, Brecht Laperre ${}^{1}$, Joachim Raeder ${}^{2}$, Stefaan Poedts ${}^{1,3}$ and Giovanni Lapenta ${}^{1}$ \correspondance
\extraAuth
Maria Elena Innocenti
mariaelena.innocenti@kuleuven.be
2 Introduction
Solar activity affects the terrestrial environment with a constantly present but highly variable solar wind and with higher energy, transient events, such as flares and Coronal Mass Ejections (CMEs).
“Space weather” (Bothmer and Daglis, 2007) is the discipline that focuses on the impact of these solar drives on the solar system and in particular on the Earth and its near space environment.
Space weather events can have serious effects on the health of astronauts and on technology, with potentially large economic costs (Eastwood et al., 2017). The importance of space weather forecasting has grown with societal dependence on advanced space technology, on communication and on the electrical grid. For example, the Halloween 2003 solar storms that impacted Earth between 19th of October 2003 and 7th of November 2003 caused an hour long power outage in Sweden (Pulkkinen et al., 2005), forced airline flight reroutes, and affected communication and satellite systems (Plunkett, 2005). The “great geomagnetic storm” of March 13-14, 1989 caused, among other disruptions, a blackout of up to nine hours in most of Quebec Province, due to a massive failure experienced by the power grid Hydro-Quebec Power Company (Allen et al., 1989).
In order to improve space weather forecasting, accurate models of the Sun-Earth connection are needed. Such forecasts are challenging because of the complexity of the processes involved and the large range of spatial and temporal scales. Commonly the heliosphere is divided into sub-systems, where each one is simulated with a different model, such that the models feed into each other (Luhmann et al., 2004; Tóth et al., 2005). These models can be either physics-based or empirical. Empirical models (such as, in the solar domain, Altschuler and Newkirk (1969); Schatten et al. (1969); Schatten (1971); Wang and Sheeley Jr (1992); NikoliÄ (2019)) usually require less computational resources, enabling faster forecasts. They can also serve as a baseline for physics-based models (Siscoe et al., 2004). However, empirical models lack the sophistication of more expensive first-principles based numerical models. Recently, machine learning methods have emerged, that can provide a new approach to space weather forecasting (Camporeale, 2019; Laperre et al., 2020). Most of these methods, while promising, must still undergo extensive validation.
The technique of Data Assimilation (DA) was developed to improve model predictions by properly initializing models from data and by keeping a model on track during its time evolution. (Kalnay, 2003; Bouttier and Courtier, 1999; Evensen, 2009).
DA methods were originally applied to atmosphere and ocean models, which exhibit a large degree of inertia. The latter is also true for the solar wind, but not for the magnetosphere-ionosphere system, which is strongly driven and dissipative.
Therefore, in space weather forecasting, DA aims not only at initializing the models, but also at using information from various observations to bring the evolution of a system as predicted from a model closer to the real system evolution (Kalman, 1960; Bouttier and Courtier, 1999; Bishop et al., 2001; Evensen, 2009; Le Dimet and Talagrand, 1986; Innocenti et al., 2011), making up for model deficiencies in the terms of resolution and incomplete physical description.
The quantity and quality of available data is a critical factor in the effectiveness of Data Assimilation. This is the reason why fields where data can be obtained more easily and continuously have shown early successes in DA implementations. Examples of these fields are meteorology and oceanography, and, in space sciences, ionospheric and radiation belt physics (Bennett, 1992; Ghil and Malanotte-Rizzoli, 1991; Egbert and Bennett, 1996; Kalnay, 2003; Rigler et al., 2004; Schunk et al., 2004; Kondrashov et al., 2007).
Examples of DA applications targeting specifically the interplanetary space environments are Schrijver and DeRosa (2003); Mendoza et al. (2006); Arge et al. (2010); Innocenti et al. (2011); Skandrani et al. (2014); Lang and Owens (2019, 2019); Lang et al. (2020).
Representer analysis (RA) and Domain of Influence analysis (DOI) (Bennett, 1992; Egbert and Bennett, 1996; Echevin et al., 2000; Evensen, 2009; Skandrani et al., 2014), briefly summarized in Section 3, are powerful statistical tools used to estimate the effectiveness of DA techniques when applied to a specific model, without assimilating actual data.
Such analysis can be used in several ways. It allows us
to optimize assimilation strategies, it may uncover model biases
that can then be addressed by further model development, and it may be used to optimize the observation systems that provide operational data for DA. For example, RA/DOI can be used to optimize locations for solar wind monitors, such as locations proposed near the L5 Lagrangian point (Vourlidas, 2015; Lavraud et al., 2016; Pevtsov et al., 2020).
In the present paper, the RA and DOI analysis is applied to three models: the OpenGGCM magnetosphere - ionosphere model (Section 4.1), two of the empirical Tsyganenko magnetosphere magnetic field models (Section 4.2), and a solar wind simulation based on the PLUTO code (Section 4.3). These models simulate critical sub-systems in the Sun-Earth connection with a focus on the terrestrial magnetosphere and Coronal Mass Ejection propagation.
The present paper provides insights into the locations of the terrestrial magnetosphere that should be prioritized (ideally, in absence of orbital constraints) for space weather forecasting and monitoring activity. We compare a time-dependent, physics based model (e.g., OpenGGCM) and time-independent, empirical (e.g., Tsyganenko) models in terms of the expected benefits that DA can provide. We conclude that time-dependent models should be preferentially chosen for DA. We take the first steps towards the goal if understanding the main physical parameters, close to the Sun and in interplanetary space, that control CME propagation and hence their arrival time at Earth.
This manuscript is organized as follows: in Section 3 we introduce the theoretical background on RA and the DOI; Section 4 discusses the application of the method to the different models; in Section 5 we summarize the results and discuss potential improvements and new applications.
3 Representer analysis and Domain of Influence analysis
This Section introduces the mathematical basis of RA and DOI analysis. The reader is referred to Skandrani et al. (2014) and references therein for an in depth derivation.
Let us start from a system described by the state variable vector $\mathbf{x}_{t}\in\;\mathbb{R}^{n}$.$\mathbf{x}_{t}$ is a vector containing the $n$ state variables that describe the system at a time $t$. Let us assume that the evolution of the system can be described as a discrete-time process controlled by an evolution law $\mathbf{A}$. The state of the system then evolves as follows: $\mathbf{x}_{t}=\mathbf{A}(\mathbf{x}_{t-1})+\mathbf{w}_{t-1}$, where $\mathbf{w}\in\;\mathbb{R}^{n}$ is process noise. The process noise is assumed to be Gaussian and with covariance matrix Q.
If a model $\mathbf{M}$ (for example, a simulation model) of the evolution law $\mathbf{A}$ is available, we can obtain, following Kalman (1960); Evensen (2009), a prior estimate $\hat{\mathbf{x}}^{-}_{t}$ of the state variable $\mathbf{x}_{t}$ through the simulation model as
$$\centering\hat{\mathbf{x}}_{t}^{-}=\mathbf{M}(\hat{\mathbf{x}}^{-}_{t-1})+%
\mathbf{w}_{t-1}.\@add@centering$$
(1)
Assume now that we have $m$ observational values or measurements $\mathbf{z}_{t}\in\mathbb{R}^{m}$. These measurements can be mapped to the current state $\mathbf{x_{t}}$ through the so-called observation operator $\mathbf{H}\in\mathbb{R}^{m\times n}$, such that $\mathbf{z}_{t}=\mathbf{H}\mathbf{x}_{t}+\mathbf{\nu}_{t}$.
Here, $\mathbf{\nu}$ is the (assumed Gaussian) measurement noise, with a covariance matrix $\mathbf{R}$.
It can be then shown (Bishop et al., 2001) that a posterior estimate of the state ($\hat{\mathbf{x}}_{t}$) can be obtained from the prior estimate of the state ($\hat{\mathbf{x}}_{t}^{-}$), obtained from Eq. 1, as follows:
$$\centering\hat{\mathbf{x}}_{t}=\hat{\mathbf{x}}_{t}^{-}+\mathbf{K}_{t}\left(%
\mathbf{z}_{t}-\mathbf{H}\hat{\mathbf{x}}_{t}^{-}\right).\@add@centering$$
(2)
Here, the term $\left(\mathbf{z}_{t}-\mathbf{H}\hat{\mathbf{x}}_{t}^{-}\right)$ is called the “innovationâ, and represents the difference between the measurements and their expected values, calculated by applying the observation operator to the prior state estimate. The Kalman gain $\mathbf{K}_{t}$ is obtained by minimizing the posterior error covariance matrix. This is the “correctionâ step of the Kalman filter, where the Kalman gain is calculated and the estimate and error covariance matrix of the posterior state are updated. The “predictionâ (forecast) phase of the filter results in the calculation of the prior state estimate and prior error covariance matrix (used to compute the Kalman gain).
The prior and posterior error covariance matrices are respectively defined as
$$\mathbf{P}_{t}^{-}=\mathbb{E}\left[\left(\mathbf{x}_{t}-\hat{\mathbf{x}}_{t}^{%
-}\right)\left(\mathbf{x}_{t}-\hat{\mathbf{x}}_{t}^{-}\right)^{T}\right],\quad%
\mathbf{P}_{t}=\mathbb{E}\left[\left(\mathbf{x}_{t}-\hat{\mathbf{x}}_{t}\right%
)\left(\mathbf{x}_{t}-\hat{\mathbf{x}}_{t}\right)^{T}\right],$$
(3)
where $\mathbb{E}$ is the expected value, $\mathbf{x}_{t}$ is the “realâ, unknown system state and $\epsilon^{-}=\left(\hat{\mathbf{x}}_{t}^{-}-{\mathbf{x}}_{t}\right)$ and $\epsilon=\left(\hat{\mathbf{x}}_{t}-{\mathbf{x}}_{t}\right)$ are the prior and posterior errors, calculated as the difference between the prior ($\hat{\mathbf{x}}_{t}^{-}$)/posterior ($\hat{\mathbf{x}}_{t}$) state and the real state, $\mathbf{x}_{t}$. Notice that, although these are the definitions of the error covariances, this is not how they are computed in the filter, since the real state is not known.
The formula for the calculation of the posterior state, Eq. 2, can be written as
$$\centering\hat{\mathbf{x}}_{t}=\hat{\mathbf{x}}_{t}^{-}+\mathbf{r}\mathbf{b}\@add@centering$$
(4)
where $\mathbf{r}\in\;\mathbb{R}^{n\times m}$ and $\mathbf{b}\in\;\mathbb{R}^{m}$ are defined as
$$\centering\mathbf{r}=\mathbf{P}^{-}\mathbf{H}^{T},\qquad\mathbf{b}=\left(%
\mathbf{H}\mathbf{P}^{-}\mathbf{H}^{T}+\mathbf{R}\right)^{-1}\left(\mathbf{z}-%
\mathbf{H}\hat{\mathbf{x}}^{-}\right),\@add@centering$$
(5)
with $\mathbf{R}$ the measurement noise covariance matrix. The time index $t$ has been dropped for ease of reading.
We will from now on assume that the system (and in particular, the observation operator $\mathbf{H}$) is linear. Then, each column of the matrix $\mathbf{r}$, denoted as $\mathbf{r}_{j}$ with $j=1,\dots,m$, is the representer associated to a given observation $z_{j}$ (remember that $\mathbf{z}$ is the vector with $m$ observations), and gives a measure of the impact of that observation in âcorrectingâ the prior state estimate. If we further assume that each observation $j$ is located at grid point $k_{j}$, and is associated to a state variable, then each column $\mathbf{r}_{j}$ (now $\mathbf{r}_{k_{j}}$, given the assumption mentioned above) contains the covariances (âcovâ) between the prior errors at the observation point $k_{j}$ and at every other grid node, for all the state variables.
Since the real state is not available for error covariance calculations, an ensemble of simulations can be used to estimate the prior errors instead. An ensemble (Evensen, 2009) can be generated by perturbing one or several of the sources of model errors. In this work, ensembles are generated for each model by perturbing the respective initial / boundary conditions.
Once the ensemble is available, the covariances of the prior errors at a certain simulated time can be approximated as the ensemble covariances (“$\text{cov}^{ens}$”) of the prior errors. These in turn become the ensemble covariances of the simulated state variables, if one assumes that the prior errors are unbiased. The ensemble covariance between the state variable x and y is defined as
$$\text{cov}^{ens}(x,y)=\frac{1}{N}\sum^{N}_{s=1}\left[\left(x^{-}_{s}-\frac{1}{%
N}\sum^{N}x_{s}^{-}\right)\left(y_{s}^{-}-\frac{1}{N}\sum^{N}_{s=1}y_{s}^{-}%
\right)\right]$$
(6)
where $N$ is the number of members in the ensemble, and $x$ and $y$ represent two state variables (notice that, for ease of reading, we indicate here two state variables as x and y, while earlier we indicated as vector $\mathbf{x}$ all the state variables).
In a set of simulations, the prior state variables are the simulation results at a specific time. Being able to calculate the representers associated to observations from the ensemble rather than from assimilating observations simplifies the RA significantly.
Following the discussion of the representer term $\mathbf{r}$ in the calculation of the posterior state, Eq. 4 and Eq. 5, we now examine the term $\mathbf{b}$.
Assuming that the only observation point for observation $j$ is at grid node $k_{j}$ (i.e., the row $j$ of the matrix $\mathbf{H}$ has only the term $k_{j}$ different from zero), the element of $\mathbf{b}$ associated to the observation at grid point $k_{j}$, denoted as $b_{k_{j}}$, becomes, from Eq. 5:
$$b_{k_{j}}=\dfrac{z_{k_{j}}-\hat{x}^{-}_{k_{j}}}{cov\left(\epsilon^{-}_{k_{j}},%
\epsilon^{-}_{k_{j}}\right)+cov\left(\epsilon^{z}_{k_{j}},\epsilon^{z}_{k_{j}}%
\right)},$$
(7)
where we have made use of the simplified form of the matrix $\mathbf{H}$ and where $\epsilon^{z}_{k_{j}}$ is the observation error associated with the observation $z_{k_{j}}$.
If $x_{i}$ is one of the state variables at grid node $i$, the correction to $x_{i}$ brought by the assimilation of the measure $z_{k_{j}}$, following Eq. 4, Eq. 5, Eq. 7 and some straightforward manipulation based on the definitions of covariance, variance, correlation, can be written as (Skandrani et al., 2014)
$$\centering\hat{x_{i}}-\hat{x}_{i}^{-}=\text{corr}^{ens}\left(\hat{x}_{k_{j}}^{%
-},\hat{x}_{i}^{-}\right)F\left(z_{k_{j}}\right).\@add@centering$$
(8)
Here, $F(z_{k_{j}})$ is the modulation factor and $\text{corr}^{ens}\left(\hat{x}_{k_{j}}^{-},\hat{x}_{i}^{-}\right)$ the correlation. The correlation is computed from the ensemble, and is calculated between the state variable at node $k_{j}$ and at node $i$. This correlation reflects how a change at node $k_{j}$, caused e.g. by the assimilation of the measurement $z_{k_{j}}$, will influence the node $i$, and is what we call the DOI. The correlation over the ensemble is defined, using the dummy variables $x$ and $y$ for brevity, as:
$$\centering\text{corr}^{ens}(x,y)=\frac{\text{cov}^{ens}\left(x,y\right)}{\sqrt%
{\text{var}\left(x\right)\text{var}\left(y\right)}}.\@add@centering$$
(9)
The modulation factor $F(z_{k_{j}})$ in Eq. 8 depends, among other things, on the measurement $z_{k_{j}}$ and on the error associated to the measure $z_{k_{j}}$. Hence Data Assimilation has to be performed to calculate this term. The $\text{corr}^{ens}\left(\hat{x}_{k_{j}}^{-},\hat{x}_{i}^{-}\right)$ term reflects how large we can expect the area that will be affected by the assimilation of $z_{k_{j}}$ to be. But to know how large the difference between the posterior and prior state, $\hat{x_{i}}-\hat{x}_{i}^{-}$, will be, we need to know the modulation factor as well.
So now the DOI of the measurement $z_{k_{j}}$ on the state variable at grid point $i$, $x_{i}$, can be defined as
$$\centering DOI(z_{k_{j}},x_{i})=\text{corr}^{ens}\left(\hat{x}_{k_{j}}^{-},%
\hat{x}_{i}^{-}\right).\@add@centering$$
(10)
One can see from its definition that the DOI can be calculated before assimilation by computing the ensemble correlation of the state variable value at the grid point $k_{j}$ with that at grid point $i$.
Dropping the $i$ index, i.e. examining the expected impact of measurement $z_{k_{j}}$ on all the state variables $\mathbf{x}$ at all grid points, we obtain the more general definition of the DOI as
$$\centering DOI(z_{k_{j}})=\text{corr}^{ens}\left(\hat{x}_{k_{j}}^{-},\hat{%
\mathbf{x}}^{-}\right).\@add@centering$$
(11)
We can then draw “DOI maps” that show the correlation between a field at grid point $k_{j}$, the “observation point”, and the other grid points.
Notice that in this derivation we have assumed, for simplicity, that the measurement $z$ and the state variable $x$ refer to the same field, for example, the x component of the magnetic field, or of the velocity. This simplifies the formulation of the numerator of Eq. 7 and improves the readability of the derivation. Skandrani et al. (2014) shows examples where the DOI is calculated between different fields, e.g. magnetic field and velocity.
DOI analysis has the advantage that it can be calculated for all state variables and at any grid point without actual assimilation, i.e. without the need for measurements $\mathbf{z}$. In order to compute the DOI at a time step $t$, we only require evolving the ensemble up to said time step $t$, and then performing the correlation over the ensemble between the state variable value at the observation point $k_{j}$ and at all other grid points.
Because DOI values are derived from a correlation they are bounded between -1 and 1. $|DOI|\sim 1$ indicates that the field at that specific point significantly changes when the same field (or a different field, in the case of cross-correlation) varies at the observation point. $|DOI|\sim 0$ indicates the opposite, i.e., variation at the observation point have little or no effect. Thus, DOI analysis also provides information on how information propagates within the model, and therefore sheds light on the physical processes within the model. We will exploit this property in Section 4.2.
We note that in this study we only focus on spatial correlations, neglecting temporal correlations. In other words, the following analysis (i.e. the calculation of variances, DOI, etc.) is restricted to specific instances in time, rather than examining correlations between fields at difference times as well. The dependence on time will be addressed in a future project.
The RA is applied to “artificial” data, obtained from ensembles of simulations focused on different processes of interest in the Sun-Earth connection: the interaction of the solar wind with the terrestrial magnetosphere (via OpenGGCM and Tsyganenko simulations) and the propagation of a CME-like event in the steady solar wind (PLUTO).
OpenGGCM and the Tsyganenko Geomagnetic Field Models both simulate the interaction of the solar wind with the magnetospheric system. OpenGGCM is a physics-based magnetohydrodynamic (MHD) model, while the Tsyganenko models are semi-empirical best-fit representations for the magnetic field, based on a large number of satellite observations (Tsyganenko, 1995, 2002a; Tsyganenko and Sitnov, 2005).
PLUTO is an MHD-modelling software used to simulate the propagation of a CME in the background solar wind. This software can be used to numerically solve the partial differential equations encountered in plasma physics problems, in conservative form, in different regimes (from hydrodynamics to relativistic MHD). The structure of the software is explained in Mignone et al. (2007, 2012). Full documentation and references can be found in the relevant web page: http://plutocode.ph.unito.it/.
Because the DOI analysis is an ensemble based technique the size of the ensemble and its properties matter. In order to test for sufficient size, we performed the DOI calculation using a limited, random subset of the ensemble, which we gradually expanded. We found that using at 25 runs were sufficient to obtain a consistent ensemble mean, variance, and DOI. We note, however, that this may change for different choices of simulation resolution and parameters used for the generation of the ensemble.
4 Applications
4.1 Magnetospheric applications I: OpenGGCM
The OpenGGCM (Open Geospace General Circulation Model) is a MHD based model that simulates the interaction of the solar wind with the magnetosphere-ionosphere-thermosphere system. OpenGGCM is available at the Community Coordinated Modeling Center at NASA/GSFC for model runs on demand (see: http://ccmc.gsfc.nasa.gov). This model has been developed and continually improved over more than two decades. Besides numerically solving the MHD equations with high spatial resolution in a large volume containing the magnetosphere, the model also includes ionospheric processes and their electrodynamic coupling with the magnetosphere.
The mathematical formulation of the software is described in Raeder (2003).
The latest version of OpenGGCM, used here, is coupled with the Rice Convection Model (RCM), (Toffoletto et al., 2001), which treats the inner magnetosphere drift physics better than MHD and allows for more realistic simulations that involve the ring current (Cramer et al., 2017). The model is both modular and efficiently parallelized using the message passing interface (MPI). It is written in Fortran and C, and uses extensive Perl scripting for pre-processing. The software runs on virtually any massively parallel supercomputer available today.
OpenGGCM uses a stretched Cartesian grid (Raeder, 2003) that is quite flexible. There is a minimal useful resolution, about 150x100x100 cells, that yields the main magnetosphere features but does not resolve mesoscale structures such as FTEs or small plasmoids in the tail plasma sheet. At the other end, we have run simulations with grids as large as $\sim$1000${}^{3}$ (on some 20,000 cores). In terms of computational cost that is almost a 10${}^{4}$ ratio. Here, we used a grid of 325x150x150 cells which is sufficient for the purposes of this study and runs faster than real time on a modest number of cores.
OpenGGCM has been used for numerous studies of magnetospheric phenomena such as storms (Raeder et al., 2001b; Raeder and Lu, 2005; Connor et al., 2016), substorms (Raeder et al., 2001a; Ge et al., 2011; Raeder et al., 2010), magnetic reconnection (Dorelli, 2004; Raeder, 2006; Berchem et al., 1995), field-aligned currents (Moretto et al., 2006; Vennerstrom et al., 2005; Raeder et al., 2017; Anderson et al., 2017), and magnetotail processes (Zhu et al., 2009; Zhou et al., 2012; Shi et al., 2014), to name a few.
The boundary conditions require the specification of the three components of the solar wind velocity and magnetic field, the plasma pressure and the plasma number density at 1 AU, which are obtained from ACE observations (Stone et al., 1998) and applied for the entire duration of the simulation at the sunward boundary.
We generate an ensemble of 50 OpenGGCM simulations by perturbing the $v_{x}$ component of the input solar wind velocity. Changing this particular parameter guarantees a direct and easy way to interpret magnetospheric response.
First, we run a reference simulation using the observed solar wind values at 1 AU starting from May 8${}^{th}$, 2004, 09:00 UTC (denoted as $t_{0}$) until 13:00 UTC on May 8${}^{th}$, 2004. We choose this period because it is relatively quiet: no iCME were registered in the Richardson/ Cane list of near-Earth interplanetary CMEs (Richardson and Cane, 2010) and, as it is common during the declining phase of the solar cycle, geomagnetic activity is driven by Corotating Interaction Regions and High Speed Streams (Tsurutani et al., 2006). The study of outlier events, such as CME arrival at Earth, is left as future work.
To generate the ensemble, the solar wind compression is changed in each of the “perturbed” simulations. The perturbed velocity in the $x$ (Earth-to-Sun) direction is set, for the entire duration of each simulation, to a constant value obtained by multiplying the average observed $v^{avg}_{x}$ by a random number $S$ sampled from a normal distribution with mean $\mu=1$ and standard deviation $\sigma=0.1$:
$$v_{x}=Sv^{avg}_{x},\text{ with }S\in\mathcal{N}(1,(0.1)^{2}).$$
(12)
The time period we use to calculate the average is the duration of the reference simulation, between 9:00 UTC and 13:00 UTC on May 8${}^{th}$, 2004.
Our choices for the generation of the ensemble are determined by the necessity to preserve both the Gaussian characteristic of the model error and the physical significance of the simulations: the solar wind compression in all ensemble members is is not too far from the reference value. With such low standard deviation, the average of the obtained perturbed value plus/minus several sigmas are still within the typical range for the solar wind: the minimum and maximum values of the constant, perturbed input velocities are $|v_{x}|\sim$ 363 km/s and $|v_{x}|\sim$ 583 km/s respectively. The solar wind velocity $v_{x}$ is negative in the Geocentric Solar Ecliptic (GSE) coordinates used here. The $v_{x}$ values that we obtain in this way are not supposed to be representative of the full range of values that $v_{x}$ can assume; they are used to generate an ensemble of simulations “slightly perturbed” with respect to our reference simulation. We note that the real distribution of solar wind velocity is far from a normal distribution, with two distinct peaks and extreme outliers, and would not be appropriate to produce the required ensemble. We refer the reader to Fortin et al. (2014) for optimal procedures on how to choose the variance of the ensemble.
The ensemble analysis requires running 50 simulations to produce the ensemble members, plus the unperturbed reference simulation. Each run takes $\sim$ 12 hours on 52 cores on the supercomputer Marconi-Broadwell (Cineca, Italy), for a total cost of $\sim$ 32000 core hours.
We verify that the prior errors are unbiased (as assumed in the derivation of the method summarized in Section 3) by comparing the reference simulation and the average of the ensemble. We note that the ensemble mean is an appropriate metric to use in this case because the perturbed simulations have not been generated in order to represent all possible solar wind velocity values, but small perturbations around the reference case.
Figure 1 shows this comparison in the xz plane for the x component of the velocity and of the magnetic field at a fixed time, (172 minutes after the beginning of the simulation), for both the reference simulation (panel (a) and (b)) and the average of the ensemble (panel (c) and (d)). The magnetic field lines, depicted in black, are calculated from the reference simulation in panel (a) and (b) and from the ensemble average in panel (c) and (d). The distances are normalized by the Earth radius $R_{E}$.
Visual inspection of panel (a), (b), (c), (d) and of the difference between the ensemble mean and the reference simulation results, depicted in panels (e) and (f) for $v_{x}$ and $B_{x}$ respectively, highlight the areas where the behavior differs most within the ensemble: the bow shock, the plasma sheet, and the neutral line. The former is a plasma discontinuity that moves back and forth in response to the changing solar wind Mach number, and thus gets smeared out in the ensemble. The latter is a region of marginal stability in the magnetosphere that reacts in a non-linear way to solar wind changes.
In order to determine if 50 ensemble members are sufficient for our analysis, we have compared corresponding plots of the difference between the ensemble mean and the reference simulation for $v_{x}$ with decreasing number (40, 30, 20) of ensemble members.
We find that, with decreasingly smaller ensembles, the plasma sheet structure seen in Fig. 1, panel (e), is only minimally affected. However, the differences around the bow shock become more pronounced. The velocity difference increases in the solar wind and magnetosheath as well, and the magnetic field structure at the magnetosphere/ solar wind interface (as shown by the magnetic field lines, which are drawn for the average field in panel (e) and (f) and similar analysis) begin to change significantly with respect to the reference simulation. By comparing the plots with 50, 40, 30 and 20 ensemble members, we conclude that 30 is the minimum number of ensemble members that gives average fields compatible with the reference simulation, with our choice of perturbation to generate the ensemble.
Panels (a) and (b) of Figure 1 show characteristic signatures of magnetic reconnection in the magnetotail, i.e, the X pattern and the formation of dipolarization fronts in the magnetic field lines, and the presence of earthward and tailward jets in $v_{x}$ departing from the X point. We provide a movie showing the dynamic evolution in the supplementary material (ReferenceSimVx.avi). The movie shows the solar wind $b_{z}$ time variation and the subsequent occurrence of several magnetopause/ magnetotail reconnection events. The “formation” of the magnetosphere occurs during the first $\sim 30$ minutes of the simulation and should be disregarded.
The movie MeanVx.avi, also in the supplementary material, shows how the global evolution changes in the ensemble mean: the magnetopause and magnetotail reconnection patterns are still overall visible, but smoothed out by the averaging procedure with respect to the reference simulation, since the different ensemble instances reconnect at different times and the smaller scale features of each single run are averaged away.
In Figure 2 and movies LowerVx.avi and HigherVx.avi we compare the evolution of the members of the ensemble generated with the lower ($|v_{x}|\sim 363$ km/s, panel (a)) and higher ($|v_{x}|\sim$583 km/s, panel (b)) absolute value of the $v_{x}$ velocity component. The movies show that the velocity values and magnetic field line patterns are significantly different from the reference simulation and from the ensemble average, demonstrating that the perturbations are not trivial.
We now discuss the RA and DOI analysis for a set of different observation points, depicted as white stars in the following figures, in the inflowing solar wind (a), in the magnetosheath (b), in the northern lobe (c), and in the plasma sheet (d), for the same plane and time as Figure 1. The coordinates of each of these points in the $xz$-plane are given in Table 1, the $y$ coordinate being $y/R_{E}=0$. Figure 3 and Figure 4 show the DOI maps for $v_{x}$ and $b_{x}$ respectively. Note that the correlations which are displayed are not cross-correlations: the correlation is done between the value of a field at the observation point and the values of the same field at the other grid points.
Figures 3 and 4 show that the correlations are mostly ordered by the principal regions of the magnetosphere such as the lobes, the magnetosheath, and the plasma sheet. For example, Figure 3 shows the results for the $v_{x}$ correlations. The DOI values for the plasma sheet are different from those in the magnetosheath and the lobes in all panels. As expected, the stronger correlations are somewhat localized around the observation point, for example, the strongest correlations in panel (d), where the observation point is in the plasma sheet, are in the plasma sheet itself and its immediate surroundings. However, some other observation points have a much larger DOI, such as the ones in the solar wind and the magnetosheath. This makes physical sense, because $v_{x}$ variations in those regions will propagate through much of the magnetosphere. Figure 4 shows the $B_{x}$ correlations. The northern and southern lobes clearly stand out, with opposite DOI values, and the magnetosheath stands out as well. Because the $B_{x}$ values have opposite signs in the two lobes, the DOI values also have opposite signs. The correlation values depend on the variability of the field value at the observation point, thus the panels that exhibit lower correlations are those with observation point in the plasma sheet, where the intrinsic variance of both $v_{x}$ and $B_{x}$ is higher (see Figure 1) due to the different reconnection patterns in the different members of the ensemble. For example, in Figure 3, panel (d) the velocity value at the observation point in the plasma sheet exhibits little correlation with the $v_{x}$ values outside of the plasma sheet and the neighboring areas. This is a consequence of the jet structure which is caused by internal magnetospheric dynamics rather than the solar wind driver.
The temporal dynamic DOI behavior is similar: the DOI maps of $v_{x}$ and $B_{x}$ with observation point in the plasma sheet exhibit higher temporal variability than those with a observation point in the magnetosheath, as can be seen in the DOI movies DOI_bx_bx_MSheath.avi, DOI_bx_bx_pSheet.avi, DOI_vx_vx_MSheath.avi, DOI_vx_vx_pSheet.avi and in Figure 5. The Figure shows the DOI map for $v_{x}$ (panel (a) and (b)) and $B_{x}$ (panel (c) and (d)) with observation point in the plasma sheet (panel (a) and (c)) and in the magnetosheath (panel (b) and (d)) at $t_{0}$ + 192 minutes. All the previous figures, Fig. 1, 2, 3 and 4, were at $t_{0}$ + 172 minutes.
We note that the plots with observation points in the magnetosheath are not significantly different to earlier plots (see Figures 3, 4), except for the plasma sheet plots, which differ profoundly.
To summarize and interpret the OpenGGCM results, the DOI analysis is well in line with our understanding of the terrestrial magnetosphere. In the $v_{x}$ case, when the observation point is in the solar wind or in the magnetosheath, the $|DOI|$ values are very high in both the solar wind and the magnetosheath region. This is expected, because $v_{x}$ in the solar wind is a correlation with itself (and thus a sanity test for the calculation), whereas the magnetosheath is largely driven by the interaction between solar wind and the bow shock, where the Rankine-Hugoniot conditions predict a positive correlation of the downstream velocity with the upstream velocity. When the observation points are in the solar wind and magnetosheath regions, the $|DOI|$ values in the plasma sheet are expected to be lower due to internal transient dynamics (e.g., reconnection events, bursty bulk flows) in the sheet which may be triggered by local plasma sheet dynamics, rather than solar wind compression. Local dynamics in the sheet is also the reason why, in Figure 3, panel (d), when the observation point is in the plasma sheet, the correlation with the solar wind and magnetosheath regions is close to zero. Even if, in global terms, magnetic reconnection in the plasma sheet were triggered by magnetopause dynamics, in any region of the plasma sheet $v_{x}$ may flow sunward or anti-sunward, depending on the location of the reconnection site, and thus would be uncorrelated with the velocity in the solar wind or in the magnetosheath. The lobe magnetic field is expected to be directly driven by the solar wind dynamic pressure, and thus by $v_{x}$. As the dynamic pressure increases, the lobe flare angle decreases, and vice versa. As the flare angle decreases, the lobe field gets compressed. Figure 4, panels (b) and (c) show that effect, as expected.
Similar consideration broadly apply to the $B_{x}$ DOI results shown in Figure 4. There is, however, a significant difference between panels (a) in 3 and 4. When the observation point is in the solar wind, high correlations are obtained in large parts of the magnetosphere for $v_{x}$, while $B_{x}$ correlations are much lower. This can be attributed to the fact that $B_{x}$ in the solar wind is not a major driver of magnetospheric dynamics, unlike the solar wind speed and solar wind dynamic pressure. The geo-effective component of the interplanetary magnetic field (IMF) is the $B_{z}$ component, which controls reconnection at the magnetopause and thus the dominant energy input into the magnetosphere.
4.2 Magnetospheric applications II: Tsyganenko Model
The Tsyganenko models are a family of empirical, static terrestrial magnetic field models (Tsyganenko, 1987, 1989, 1995, 2002a, 2002b; Tsyganenko et al., 2003; Tsyganenko and Sitnov, 2005). The successive model versions (available at http://geo.phys.spbu.ru/~tsyganenko/modeling.html) reflect increasing knowledge of the magnetospheric systems and are based on an increasing amount of data from all regions in the magnetosphere.
The models are based on a mathematical description of the magnetosphere, which includes contributions from major magnetospheric current sources such as the Chapman-Ferraro current, the ring current, the cross-tail current sheet and large-scale field-aligned currents. Terms are added to account for the magnetopause and for partial penetration of the IMF into the magnetosphere. The most recent versions can also take into account the dipole tilt, the dawn-dusk asymmetry, and allow for open magnetospheric configurations. The parameters of the models are derived from a regression to magnetic field observations, and keyed to magnetic indices and/or solar wind parameters. The model requires the user to specify a date and time for the dipole orientation. The other model parameters, either an index such as the Kp, or solar wind variables, are to be given by the user. In more recent models, Tsyganenko also provides yearly input data files for his models. From these inputs, an approximation of the magnetosphere is created for the specified date and time. Notice that the Tsyganenko models are static, and only provides a snapshot of the magnetosphere. However, since the parameters are time dependent the model can be used in a quasi-dynamic mode.
Several versions of the Tsyganenko model have been tested over the years against observations and physics-based, MHD models (Thomsen et al., 1996; Huang et al., 2006; Woodfield et al., 2007). While the Tsyganenko models do not account for the Earth’s internal magnetic field, methods are provided to add the internal field model as described in the above cited literature.
In order to simulate the evolution of the magnetosphere with the chosen Tsyganenko model, we create snapshots of the magnetosphere at different times. The time May 8th, 2004, 09:00 UTC is taken as $t_{0}$, the same time as the OpenGGCM simulations presented in Section 4.1. The model is “evolved” by using a time series of the required input parameters, which are obtained from the OMNIWeb database (King and Papitashvili, 2005).
We use two versions of the Tsyganenko model, the T96 model (Tsyganenko, 1996) and the TA15 model (Tsyganenko and Andreeva, 2015). We generate the Tsyganenko ensembles in the same way as the OpenGGCM examples, by using a distribution of $v_{x}$ values as described in section 4.1.
Before we analyse the results of the T96 ensemble, we show the magnetospheric configuration computed by the model using the original solar wind data. In Figure 6, the first row of figures shows the results of the “reference” simulation, e.g. the simulation without any perturbed inputs, at time $t_{0}+85$min, for $B_{x}$ (panel (a)) and $B_{z}$ (panel (b)).
Unlike the OpenGGCM, the T96 model cannot model reconnection, although some approximation of reconnection is included in later Tsyganenko models (Tsyganenko, 2002a, b). Also, the day-side magnetospheric structure is only approximated with respect to physics-based models, and bow shock and magnetosheath are not clearly distinguishable. In Figure 6, the second row shows the average of the ensemble at the same time of the reference simulation, for $B_{x}$ (panel (c)) and $B_{z}$ (panel (d)). The results are similar to the reference simulation, as shown by the logarithm of the absolute difference between the reference and ensemble mean, e.g., Figure 6, panel (e) and (f). The only significant difference is located at the magnetopause, which is expected since varying the solar wind velocity changes the standoff distance.
Next we analyse the DOI maps of the T96 model. Figure 7 shows the DOI maps for the $B_{x}$ and $B_{z}$ field components at time $t_{0}+85$ min. Although the T96 model is parameterized by the solar wind velocity, it only models the magnetic field in the magnetosphere. Because of this, we are only able to analyze the DOI maps of the magnetic field components. The observation points are placed in the northern lobe, in proximity to the current sheet, at the dayside magnetosphere, and in the southern lobe. Like in the OpenGGCM case, the DOI maps reflect the general regions of the magnetosphere as reproduced by the T96 model. However, the correlation only takes values of $\pm$1 in the magnetosphere, and zero in the solar wind. The latter is simply a consequence of the fact that the model does not predict the IMF, which is therefore independent of the $v_{x}$ variations of the ensemble.
The former is due to the fact that the model has no intrinsic time dependence. Any variations of the solar wind affect the entire magnetosphere instantly and in proportion to the variation. Thus, after normalization, only the sign matters, i.e., whether a given change at the observation point leads to a positive or negative change at a different point.
Now we focus on the results of the TA15 model. Figure 8 shows the reference simulation and ensemble mean for the $B_{x}$ and $B_{z}$ fields, with superimposed field lines, at time $t_{0}+85$ min, together with the difference between reference and ensemble mean. We observe that the reproduced dayside magnetosphere structure is improved compared to the T96 model, at the expense of unrealistically high magnetic field values in the inflowing solar wind, and correspondingly distorted magnetic field lines. These artificial boundary conditions in the Sunwards boundary are used to obtain an “open” magnetosphere which blends with the inflowing solar wind, without seeming to form a nightside magnetosheath. Notice also the high values at these artificial boundary conditions in the difference plot, indicating that there is a high variability in their values.
From the DOI maps in Figure 9 (with observation points at the same positions as Figure 7), we can confirm that the modelled IMF is used to construct the internal magnetospheric solution. While in the T96 model the solar wind $B_{x}$ and $B_{z}$ values were uncorrelated with the magnetospheric values, here the absolute value (i.e. ignoring the sign) of the correlation is very high: the solar wind input strictly determines the inner magnetospheric solution, making the correlation practically unitary. This could be because of the deterministic analytical formula used to construct the magnetic field, where everything is exactly determined on a global scale.
Note that the correlations reported are spatial and not temporal, therefore no causality is implied. High correlation between the IMF and magnetospheric fields point to the fact that, in an ensemble generated by perturbing the solar wind input, the model is built in such a way that variations in the magnetic field are highly correlated through the system, apparently without highlighting the boundary regions that we were able to spot in the DOI maps for the OpenGGCM and Tsyganenko T96 simulations.
A last remark on the DOI analysis applied to the Tsyganenko models is the following. The analysis helps us understand and visualize how the different models are built, with regards to the relationship between the solar input and the magnetospheric solution. DA analysis then proves useful here as a model investigation tool.
It also highlights that caution should be used when deciding to apply DA techniques to a particular model, depending on the objectives of the investigation. The Tsyganenko models were built to provide time-independent, empirical-based insights into the structure of the magnetosphere at a particular instant in time. They do not aim at representing the state of the pristine solar wind, which is used only to better the magnetospheric solution (hence the somehow unrealistic solar wind patterns identified in Figure 8 and 9). Also, they do not intend to reproduce temporal dynamics in the magnetosphere. These factors result in DOI maps where the absolute value of the correlation is always either 1 or -1. When using DOI techniques with the purposes of identifying useful locations for satellite placement, these are not useful results: we are interested in the value of the DOI, not in the sign. Hence, caution should be used before using empirical, time-independent models for this particular purpose: more significant information will possibly be acquired from their physics-based, time-dependent counterparts. This consideration does not intend to diminish the importance of empirical, time-independent models for other scientific objectives such us, most importantly, quick forecasting.
4.3 Heliospheric application: PLUTO
In this section, we study the propagation of a Coronal Mass Ejection in a solar meridional plane, which is defined by the rotation axis of the Sun and a radial vector in the equatorial plane. In all the runs of the ensemble, the computational domain is $1R_{\odot}\leq r\leq 216R_{\odot}$ and $0\leq\theta\leq\pi$ in spherical coordinates, where $R_{\odot}$ is the solar radius and $\theta$ is the polar angle (or colatitude), corotating with the Sun. Assuming axisymmetry around the solar rotation axis, we may limit our analysis to 2.5D (pseudo 3D) simulations. The grid resolution is uniform in both directions, $384\times 384$ cells, which is sufficient to capture the structure of the background solar wind while keeping the computational cost and output size manageable.
We simulate the background solar wind using a simple adiabatic model with effective polytropic index $\Gamma=1.13$ (Keppens and Goedbloed, 2000). We also assume a time-independent dipole background magnetic field:
$$\displaystyle B_{r}$$
$$\displaystyle=-2B_{o}\cos\theta/r^{3},$$
(13)
$$\displaystyle B_{\theta}$$
$$\displaystyle=-B_{o}\sin\theta/r^{3},$$
(14)
where $B_{r},B_{\theta}$ are the $r,\theta$ components of the magnetic field in spherical coordinates and $B_{o}$ a constant used to scale the field to $B=1.1$G on the solar equator. We impose the density distribution $\rho$ as a function of the latitude $\theta$ at the inner boundary to achieve a âdead zoneâ of low velocity near the equator and a fast solar wind near the poles simultaneously (see Keppens and Goedbloed (2000) and Chané et al. (2008)). The differential rotation of the Sun is also taken into account, following Chané et al. (2005); this is achieved by imposing a varying azimuthal velocity $v_{\phi}=v_{\phi}(\theta)$ at the inflow boundary.
Once the simulation reaches a steady state, roughly after $\sim~{}2.5$ days or $t=10$ in normalized units, the radial velocity at 1 AU is $\sim 300$ km/s near the equator and $\sim 850$ km/s near the poles. This is consistent with the large-scale bimodal solar wind structure that is typically observed during solar minimum (McComas et al., 1998)).
We create two ensembles of 100 simulations each. In the first ensemble, the velocity of the CME in each case is randomly selected from a Gaussian distribution with mean $\mu=900$ km/s and standard deviation $\sigma=25$ km/s. The resulting values are typical of strong CME events. In the second ensemble, the spatial extent of the boundary conditions that launch the CME varies as well, along with the velocity of the CME as described above. The half-width of this region is also randomly selected from a Gaussian distribution with $\mu=10^{\circ}$ and $\sigma=0.5^{\circ}$. All other parameters remain the same in every run.
The values of the CME widths that are used here are comparable to observed events. The choice of parameters in the second ensemble is less constrained by observations and leads to the appearance of very small values of variance. We thus find large areas where the DOI $\sim 1$, since the simulations in the ensemble do not differ significantly. This was confirmed by creating and analyzing a third ensemble, where the width of the CME is chosen from a Gaussian with $\mu=20^{\circ}$ and $\sigma=2.0^{\circ}$.
Figure 10 shows the evolution of the radial velocity average over the whole ensemble. Up to t=11, i.e., before the CME is launched, all runs are identical. The CME is initialized similar to the simplified approach of Keppens and Goedbloed (2000), such that the boundary conditions on the solar surface are modified to represent a change of mass flux. In our case, we modify the boundary conditions at R=1R${}_{\odot}$, in a given region around $\theta=80^{\circ}$. A tracer (a passive scalar only present as an advected quantity within the flow, without effect on the plasma) is also injected with the CME, to facilitate monitoring its propagation. In the middle and right panel of Fig. 10 we show the ejection of the CME and its propagation. The CME front can be clearly distinguished at t=12.
To apply the RA technique, as explained in Section 3, we select a point of interest and perform the analysis based on (a) the plasma density, or (b) the radial velocity. We present results at t=14, when the CME has reached a distance of $\sim 150R_{\odot}$, for two detection points (at $R=90$ and $R=150R_{\odot}$, $\theta=80^{\circ}$). At times earlier than $t=10$ (when the solar wind reaches a steady state and the CME is injected), the DOI is zero, since the observation point is disconnected from the rest of the domain before the CME reaches it.
The propagation of the CME can be monitored in the MHD simulations easily via e.g. a tracer (or the radial velocity). The DOI map, when the tracer is used as a criterion, follows closely the CME propagation pattern observed in the MHD runs. However, this is of limited use, besides testing, as the tracer (in our case) does not represent a real physical quantity.
The DOI map for the first ensemble, where we perturb only the radial velocity of the CME, is shown in Fig. 11. The regions where information from the CME front has not yet arrived have DOI=0, as shown in the radial velocity DOI map (Fig. 11). When only one parameter is modified (first ensemble), the density DOI map shows a very large area of the domain saturated with correlation$\simeq$1. This is probably due to variations in density of the background solar wind induced by the propagation of the CME. The regions with absolute value of the DOI$\simeq 1$ that are located far from the CME propagation front (at small or large angles $\theta$) are the areas of high radial velocity in Fig. 10, where the information on the perturbation introduced when triggering the CME has already propagated. The density and radial velocity of all ensemble members are modified in a similar way, hence the large $|DOI|$ values.
In the second ensemble, where the width of the CME is also modified, the DOI map of the density shows smaller correlation values (compare especially panel (b) and (d) in the two Figures) compared to the previous ensemble, because the differences between the runs of the ensemble are now larger (see Fig. 12). This results in smaller regions where the DOI is close to unity compared to the first case.
The last ensemble, where the CME width and its perturbation are larger compared to the second case, is shown in Fig. 13. The DOI pattern is qualitatively similar to Fig. 12, but due to the larger values in the size of the CME and its perturbation, the regions with high DOI values (meaning the regions affected by CME propagation in at least one member of the ensemble) are slightly larger as well.
Additional analysis, not shown here, was carried out on subsets of the ensembles to ensure the ensemble size is sufficient. We found that in this case convergence was achieved if at least 25 ensemble members were used (as described in Section 3); however, this number may differ in other cases, depending on the specifics of the ensemble.
The DOI analysis applied to the simulations performed with PLUTO are indicative of the versatility of the method. In all PLUTO ensembles, we can monitor the influence of the CME during its propagation and the response of the system via the DOI. Moreover, we can identify certain CME components, such as the leading edge, from the DOI maps. Differences in the response of the system due to the choice of the perturbation or parameters are captured as well. The resolution used here was sufficient to capture the CME injection and propagation within reasonable computational cost; the typical run time for simulating a member of the ensemble was of the order of $\sim 10^{\prime}$ minutes on 28 cores.
However, some limitations of the model must be considered. The axisymmetric assumption simplifies the problem and allows to reduce computation costs, but with the drawback of not accounting for the three-dimensional CME structure. The limited angular resolution imposes a weak constraint on both the perturbed and unperturbed size of the CME that we can simulate. Runs with a higher resolution can remove this constraint at additional computational cost. Simulations in 3D will be part of future work in order to capture the full system, where also differences in the polar direction can be examined. Finally, a more realistic model for the background magnetic field should be used, rather than a simple static dipole. We focused mainly on calculating the DOI at different times and locations, but a similar approach can be used to estimate the arrival time of the CME, as described in Owens et al. (2020).
5 Summary and Conclusions
In this paper, we apply the Representer analysis and the Domain of Influence analysis to two fundamental components of the Sun-Earth connection: the interaction between the solar wind and the terrestrial magnetosphere, simulated with the OpenGGCM MHD code and with the empirical Tsyganenko models, and the propagation of CMEs in the background solar wind, simulated with the MHD PLUTO model.
In each case an ensemble is generated by appropriately perturbing initial/ boundary conditions. Subsequently, the DOI analysis is applied over the ensemble. Localisation methods, which can be used to reduce spurious correlations in the estimated prior covariance matrix (Anderson, 2007; Bishop and Hodyss, 2007; Sakov and Bertino, 2011), are not used at this stage.
Primarily, the DOI analysis is a first step in the application of Data Assimilation techniques to a model, and can be applied before assimilation itself to gain insight on the system and on the model. However, the DOI analysis can also be used to gain physical insight, and to devise optimized observation systems, as discussed below.
Our main results are as follows.
First, we have demonstrated that DOI analysis can provide useful information on the most appropriate locations for future observation points, such as solar wind and magnetospheric monitors. Large absolute values of the DOI, calculated with respect to an observation point, means that observations at that location would provide significant information of that field in the specific, large $|DOI|$ area, but less so in areas with lower $|DOI|$. This can be used in two different ways.
On one hand, DOI analysis can help to find observations points that are connected to large $|DOI|$ areas, in order to increase the amount of information brought in by a single new observation.
On the other hand, the same information can be used for a different objective. Given a particular location, one can ask where observations need to be obtained to improve knowledge of that area. A useful example here is the plasma sheet in the OpenGGCM analysis, Section 4.1. Figure 3 and 4 show that $|DOI|$ values in the plasma sheet are consistently low, notwithstanding the field which is examined ($v_{x}$ or $B_{x}$) and the location of the observation point. $|DOI|$ values in the plasma sheet are low even if the observation point is in the plasma sheet itself: $|DOI|$ values, which are of course 1 at the observation point itself, quickly become smaller even a small distance away. Since the plasma sheet is a location of particular importance for space weather forecasting, or basic research for that matter, single s/c in the plasma sheet are of limited use, and rather a constellation of satellites, such as proposed in Angelopoulos et al. (1998); Raeder and Angelopoulos (1998) would be necessary.
Second, we have used the DOI analysis to improve our knowledge of the models we use, and in particular to investigate whether these models are appropriate for the implementation of Data Assimilation. The DOI analysis for the Tsyganenko models in Section 4.2 powerfully highlights the model evolution from version T96 to version TA15. In version T96, the magnetosphere is a closed system, and solar wind conditions are not correlated (DOI $\sim$ 0 in Figure 7) with the magnetospheric region. In version TA15, the magnetosphere opens up to solar wind driving, and the correlation between the solar wind region and the magnetosphere becomes very high (Figure 9). One should remember that the Tsyganenko models are supposed to be used to investigate the magnetospheric system, and the solar wind configuration is artfully modified as to give the best representation of the magnetosphere under the specified conditions.
One common aspect of the two versions of the Tsyganenko models is that (with the exception of the solar wind region in T96) the DOI values are always either 1 or -1, for all fields and regions examined. These results appear less realistic than the OpenGGCM results obtained in Section 4.1, where DOI values have larger variability. The Tsyganenko models differ with respect to OpenGGCM in two fundamental aspects, in that they (a) empirically reconstruct the magnetospheric magnetic field from an array of observations and (b) that they are not time-dependent. Either of these two aspects can contribute to the unrealistically high correlations we observe. Investigations on other models, and specifically on empirical, time-dependent models, will possibly help disentangle the role of these two aspects. At this stage of the investigation, we advance the hypothesis that time-dependent models may be better suited than time-independent models as background models for Data Assimilation techniques.
Third, with this analysis we have highlighted a possible path for future, targeted improvements of global heliospheric models used, among other things, for simulations of CME propagation in the heliosphere. It has long been known that one of the critical aspects of the simulation of CME arrival time is the estimation of the physical parameters to use as initial conditions in the simulations. While some parameters can be easily estimated from remote sensing, others are more difficult to determine properly and their variability affects the accuracy of the forecast (Falkenberg et al., 2010). In this paper, we have shown that DOI analysis could constitute an important stage of a model analysis effort aimed at clarifying which aspects of a model should be prioritized in order to obtain more accurate simulations of CME propagation.
In this study, as a first step, we show DOI maps obtained from the correlations of a single variable calculated between the variable at the observation point and the same variable in the domain under investigation. As demonstrated in Skandrani et al. (2014), cross-correlations can be used to find the influence of one variable upon another.
The results of a DOI cross-correlation analysis can then be used to determine which quantities and areas in a simulation are most relevant in determining a certain observational quantity (such as the radial velocity of a CME in the case of CME propagation simulations). This analysis can then guide modelers on deciding which aspects of a model could be improved for more realistic results. It could help understanding, for example, if CME propagation in a model is mainly controlled by the background magnetic field configuration or by the properties of the CME itself at launch. In the first case, modeling efforts could be directed into accurate high resolution representation of the magnetic field configuration in the lower corona. In the second case, instead, modeling improvements could be focused on extracting better estimates of CME launch parameters (e.g. CME density, velocity, internal magnetic field configuration with respect to the background wind) from available observations.
The spatial correlations provided by DOI can also be of particular interest in evaluating the effect of actual measurements done at positions different from the traditional L1, such as, for example, missions planned for L5 or missions closer to the Sun.
Future work will extend this study to include temporal and cross correlations between different field components. This will further increase our knowledge of the models used to simulate such critical space weather processes.
The DOI analysis presented here can also be combined with an Observing System Simulation Experiment (OSSE), an approach already used in ionospheric and solar dynamo studies (Hsu et al., 2018; Dikpati, 2017) to help provide a cost-effective approach to the evaluation of the potential impact of new observations. OSSE requires that DA is already implemented and uses independently simulated “data” that are ingested into a different model or a different instance of the same model. The effect of DA can then be investigated, albeit with caveats, since the “data” are not real. DOI analysis would obviate the need to have DA implemented, which can be very costly. Instead, only ensemble runs with an unmodified model are required, and can provide a measure of the usefulness of a model and the available data for a specific situation.
Conflict of Interest Statement
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Author Contributions
MEI, BL and DM created the ensembles using OpenGGCM, Tsyganenko and PLUTO respectively, as they appear in the main body of the article, processed the data and presented the results of the analysis and prepared the manuscript. JR provided OpenGGCM and contributed to the interpretation of the magnetosphere model results. JR, GL and SP provided support during the analysis and the preparation of the article.
Funding
DM aknowledges support from AFRL (Air Force Research Laboratory)/USAF (US Air Force) project (AFRL Award No. FA9550-14-1-0375, 2014-2019) and partial support by UK STFC (Science and Technology Facilities Council) Consolidated Grant ST/S000240/1 (UCL-MSSL, University College London-Mullard Space Science Laboratory, Solar System). M.E.I.’s work is supported by an FWO (Fonds voor Wetenschappelijk Onderzoek â Vlaanderen) postdoctoral fellowship.
BL, MEI, GL, JR acknowledge funding from the European Unionâs Horizon 2020 research and innovation programme under grant agreement No 776262 (AIDA, Artificial Intelligence for Data Analysis, www.aida-space.eu).
JR also acknowledges support through AFOSR grant FA9550-18-1-0483 and from the NASA/THEMIS mission through a subcontract from UC Berkeley.
SP and GL acknowledge funding from the European Unionâs Horizon 2020 research and innovation programme under grant agreement No 870405. These results were also obtained in the framework of the projects C14/19/089 (C1 project Internal Funds KU Leuven), G.0A23.16N (FWO-Vlaanderen), C 90347 (ESA Prodex), and Belspo (Belgian Science Policy) BRAIN project BR/165/A2/CCSOM.
Acknowledgments
The PLUTO simulations were performed using allocated time on the clusters Genius and Breniac. The computational resources and services used in this work were provided by the VSC (Flemish Supercomputer Center), funded by the Research Foundation Flanders (FWO) and the Flemish Government â department EWI. The OpenGGCM simulations were performed on the supercomputer Marconi-Broadwell (Cineca, Italy) under a PRACE allocation.
We acknowledge the NASA National Space Science Data Center, the Space Physics Data Facility, and the ACE Principal Investigator, Edward C. Stone of the California Institute of Technology, for usage of ACE data. We acknowledge use of NASA/GSFC’s Space Physics Data Facility’s OMNIWeb service, and OMNI data.
Supplemental Data
Data Availability Statement
The OMNIWeb database was used to get the solar wind conditions for the Tsyganenko runs. The ACE database was used to get the solar wind conditions for the OpenGGCM runs. The data used in the analysis are available upon reasonable request.
References
Allen et al. (1989)
Allen, J., Sauer, H., Frank, L., and Reiff, P. (1989).
Effects of the march 1989 solar activity.
Eos, Transactions American Geophysical Union 70, 1479–1488
Altschuler and Newkirk (1969)
Altschuler, M. D. and Newkirk, G. (1969).
Magnetic fields and the structure of the solar corona.
Solar Physics 9, 131–149
Anderson et al. (2017)
Anderson, B. J., Korth, H., Welling, D. T., Merkin, V. G., Wiltberger, M. J.,
Raeder, J., et al. (2017).
Comparison of predictive estimates of high-latitude electrodynamics
with observations of global-scale birkeland currents.
Space Weather 15, 352–373.
10.1002/2016sw001529
Anderson (2007)
Anderson, J. L. (2007).
Exploring the need for localization in ensemble data assimilation
using a hierarchical ensemble filter.
Physica D: Nonlinear Phenomena 230, 99–111
Angelopoulos et al. (1998)
Angelopoulos, V., Carlson, C. W., Curtis, D. W., Harvey, P., Lin, R. P., Mozer,
F. S., et al. (1998).
On the necessity and feasability of a equatorial magnetospheric
constellation.
In Science Closure and Enabling Technologies for Constellation
Class Missions, eds. V. Angelopoulos and P. V. Panetta (University of
California, Berkeley, and NASA Goddard Space Flight Center). 14
Arge et al. (2010)
Arge, C. N., Henney, C. J., Koller, J., Compeau, C. R., Young, S., MacKenzie,
D., et al. (2010).
Air force data assimilative photospheric flux transport (adapt)
model.
In AIP Conference Proceedings (AIP), vol. 1216, 343–346
Bennett (1992)
Bennett, A. F. (1992).
Inverse methods in physical oceanography (Cambridge university
press)
Berchem et al. (1995)
Berchem, J., Raeder, J., and Ashour-Abdalla, M. (1995).
Reconnection at the magnetospheric boundary: Results from global
MHD simulations.
In Physics of the Magnetopause, eds. B. U. Sonnerup and
P. Song. vol. 90 of AGU Geophysical Monograph, 205
Bishop and Hodyss (2007)
Bishop, C. H. and Hodyss, D. (2007).
Flow-adaptive moderation of spurious ensemble correlations and its
use in ensemble-based data assimilation.
Quarterly Journal of the Royal Meteorological Society: A
journal of the atmospheric sciences, applied meteorology and physical
oceanography 133, 2029–2044
Bishop et al. (2001)
Bishop, G., Welch, G., et al. (2001).
An introduction to the kalman filter.
Proc of SIGGRAPH, Course 8, 41
Bothmer and Daglis (2007)
Bothmer, V. and Daglis, I. A. (2007).
Space weather: physics and effects (Springer Science &
Business Media)
Bouttier and Courtier (1999)
Bouttier, F. and Courtier, P. (1999).
Data assimilation concepts and methods. meteorological training
course lecture series. ecmwf.
Reading, UK
Camporeale (2019)
Camporeale, E. (2019).
The challenge of machine learning in space weather: Nowcasting and
forecasting.
Space Weather 17, 1166–1207
Chané et al. (2005)
Chané, E., Jacobs, C., van der Holst, B., Poedts, S., and Kimpe,
D. (2005).
On the effect of the initial magnetic polarity and of the background
wind on the evolution of CME shocks.
A&A 432, 331–339.
10.1051/0004-6361:20042005
Chané et al. (2008)
Chané, E., Poedts, S., and van der Holst, B. (2008).
On the combination of ACE data with numerical simulations to
determine the initial characteristics of a CME.
A&A 492, L29–L32.
10.1051/0004-6361:200811022
Connor et al. (2016)
Connor, H. K., Zesta, E., Fedrizzi, M., Shi, Y., Raeder, J., Codrescu, M. V.,
et al. (2016).
Modeling the ionosphere-thermosphere response to a geomagnetic storm
using physics-based magnetospheric energy input: OpenGGCM-CTIM results.
Journal of Space Weather and Space Climate 6, A25.
10.1051/swsc/2016019
Cramer et al. (2017)
Cramer, W. D., Raeder, J., Toffoletto, F., Gilson, M., and Hu, B. (2017).
Plasma sheet injections into the inner magnetosphere: Two-way coupled
openggcm-rcm model results.
Journal of Geophysical Research: Space Physics 122, 5077–5091
Dikpati (2017)
Dikpati, M. (2017).
Ensemble Kalman Filter Data Assimilation in a Solar Dynamo Model.
In AGU Fall Meeting Abstracts. vol. 2017, SM14A–05
Dorelli (2004)
Dorelli, J. C. (2004).
A new look at driven magnetic reconnection at the terrestrial
subsolar magnetopause.
Journal of Geophysical Research 109.
10.1029/2004ja010458
Eastwood et al. (2017)
Eastwood, J. P., Biffis, E., Hapgood, M. A., Green, L., Bisi, M. M., Bentley,
R. D., et al. (2017).
The economic impact of space weather: Where do we stand?
Risk Analysis 37, 206–218.
10.1111/risa.12765
Echevin et al. (2000)
Echevin, V., De Mey, P., and Evensen, G. (2000).
Horizontal and vertical structure of the representer functions for
sea surface measurements in a coastal circulation model.
Journal of physical oceanography 30, 2627–2635
Egbert and Bennett (1996)
[Dataset] Egbert, G. and Bennett, A. (1996).
Data assimilation methods for ocean tides, modern approaches to data
assimilation in ocean modeling, edited by: Malonotte-rizzoli, p
Evensen (2009)
Evensen, G. (2009).
Data assimilation: the ensemble Kalman filter (Springer
Science & Business Media)
Fortin et al. (2014)
Fortin, V., Abaza, M., Anctil, F., and Turcotte, R. (2014).
Why should ensemble spread match the rmse of the ensemble mean?
Journal of Hydrometeorology 15, 1708–1713
Ge et al. (2011)
Ge, Y. S., Raeder, J., Angelopoulos, V., Gilson, M. L., and Runov, A. (2011).
Interaction of dipolarization fronts within multiple bursty bulk
flows in global MHD simulations of a substorm on 27 February 2009.
J. Geophys. Res. 116, A00I23
Ghil and Malanotte-Rizzoli (1991)
Ghil, M. and Malanotte-Rizzoli, P. (1991).
Data assimilation in meteorology and oceanography.
In Advances in geophysics (Elsevier), vol. 33. 141–266
Hsu et al. (2018)
Hsu, C. T., Matsuo, T., and Liu, J. Y. (2018).
Impact of Assimilating the FORMOSAT-3/COSMIC and FORMOSAT-7/COSMIC-2
RO Data on the Midlatitude and Low-Latitude Ionospheric Specification.
Earth and Space Science 5, 875–890.
10.1029/2018EA000447
Huang et al. (2006)
Huang, C.-L., Spence, H. E., Lyon, J. G., Toffoletto, F. R., Singer, H. J., and
Sazykin, S. (2006).
Storm-time configuration of the inner magnetosphere:
Lyon-fedder-mobarry mhd code, tsyganenko model, and goes observations.
Journal of Geophysical Research: Space Physics 111
Innocenti et al. (2011)
Innocenti, M. E., Lapenta, G., Vršnak, B., Crespon, F., Skandrani, C.,
Temmer, M., et al. (2011).
Improved forecasts of solar wind parameters using the kalman filter.
Space Weather 9, 1–15
Kalman (1960)
Kalman, R. E. (1960).
A new approach to linear filtering and prediction problems.
Journal of basic Engineering 82, 35–45
Kalnay (2003)
Kalnay, E. (2003).
Atmospheric modeling, data assimilation and predictability
(Cambridge university press)
Keppens and Goedbloed (2000)
Keppens, R. and Goedbloed, J. P. (2000).
Stellar Winds, Dead Zones, and Coronal Mass Ejections.
ApJ 530, 1036–1048.
10.1086/308395
King and Papitashvili (2005)
King, J. and Papitashvili, N. (2005).
Solar wind spatial scales in and comparisons of hourly wind and ace
plasma and magnetic field data.
Journal of Geophysical Research: Space Physics 110
Kondrashov et al. (2007)
Kondrashov, D., Shprits, Y., Ghil, M., and Thorne, R. (2007).
A kalman filter technique to estimate relativistic electron lifetimes
in the outer radiation belt.
Journal of Geophysical Research: Space Physics 112
Lang et al. (2020)
Lang, M., Owens, M., and Lawless, A. (2020).
Improving solar wind forecasts using data assimilation.
In EGU General Assembly Conference Abstracts. EGU General
Assembly Conference Abstracts, 10909
Lang and Owens (2019)
Lang, M. and Owens, M. J. (2019).
A Variational Approach to Data Assimilation in the Solar Wind.
Space Weather 17, 59–83.
10.1029/2018SW001857
Lang and Owens (2019)
Lang, M. and Owens, M. J. (2019).
A variational approach to data assimilation in the solar wind.
Space Weather 17, 59–83
Laperre et al. (2020)
Laperre, B., Amaya, J., and Lapenta, G. (2020).
Dynamic time warping as a new evaluation for dst forecast with
machine learning.
arXiv preprint arXiv:2006.04667
Lavraud et al. (2016)
Lavraud, B., Liu, Y., Segura, K., He, J., Qin, G., Temmer, M., et al. (2016).
A small mission concept to the sunâearth lagrangian l5 point for
innovative solar, heliospheric and space weather science.
Journal of Atmospheric and Solar-Terrestrial Physics 146, 171
– 185.
https://doi.org/10.1016/j.jastp.2016.06.004
Le Dimet and Talagrand (1986)
Le Dimet, F.-X. and Talagrand, O. (1986).
Variational algorithms for analysis and assimilation of
meteorological observations: theoretical aspects.
Tellus A: Dynamic Meteorology and Oceanography 38, 97–110
Luhmann et al. (2004)
Luhmann, J. G., Solomon, S. C., Linker, J. A., Lyon, J. G., Mikic, Z.,
Odstrcil, D., et al. (2004).
Coupled model simulation of a sun-to-earth space weather event.
Journal of atmospheric and solar-terrestrial physics 66,
1243–1256
McComas et al. (1998)
McComas, D. J., Bame, S. J., Barraclough, B. L., Feldman, W. C., Funsten,
H. O., Gosling, J. T., et al. (1998).
Ulysses’ return to the slow solar wind.
Geophysical Research Letters 25, 1–4.
10.1029/97GL03444
Mendoza et al. (2006)
Mendoza, O. B., De Moor, B., and Bernstein, D. (2006).
Data assimilation for magnetohydrodynamics systems.
Journal of Computational and Applied Mathematics 189, 242–259
Mignone et al. (2007)
Mignone, A., Bodo, G., Massaglia, S., Matsakos, T., Tesileanu, O.,
Zanni, C., et al. (2007).
PLUTO: A Numerical Code for Computational Astrophysics.
ApJS 170, 228–242.
10.1086/513316
Mignone et al. (2012)
Mignone, A., Zanni, C., Tzeferacos, P., van Straalen, B., Colella,
P., and Bodo, G. (2012).
The PLUTO Code for Adaptive Mesh Computations in Astrophysical Fluid
Dynamics.
ApJS 198, 7.
10.1088/0067-0049/198/1/7
Moretto et al. (2006)
Moretto, T., Vennerstrom, S., Olsen, N., Rastaetter, L., and Raeder, J. (2006).
Using global magnetospheric models for simulation and interpretation
of SWARM external field measurements.
Earth, Planets, Space 58, 439–449
NikoliÄ (2019)
NikoliÄ, L. (2019).
On solutions of the pfss model with gong synoptic maps for
2006â2018.
Space Weather 17, 1293–1311.
10.1029/2019SW002205
Owens et al. (2020)
Owens, M., Lang, M., Barnard, L., Riley, P., Ben-Nun, M., Scott,
C. J., et al. (2020).
A Computationally Efficient, Time-Dependent Model of the Solar Wind
for Use as a Surrogate to Three-Dimensional Numerical Magnetohydrodynamic
Simulations.
Sol. Phys. 295, 43.
10.1007/s11207-020-01605-3
Pevtsov et al. (2020)
Pevtsov, A. A., Petrie, G., MacNeice, P., and Virtanen, I. I. (2020).
Effect of additional magnetograph observations from different
lagrangian points in sun-earth system on predicted properties of quasi-steady
solar wind at 1 au.
Space Weather 18, e2020SW002448.
10.1029/2020SW002448.
E2020SW002448 10.1029/2020SW002448
Plunkett (2005)
Plunkett, S. (2005).
The extreme solar storms of October to November 2003.
Tech. rep., NAVAL RESEARCH LAB WASHINGTON DC SPACE SCIENCE DIV
Pulkkinen et al. (2005)
Pulkkinen, A., Lindahl, S., Viljanen, A., and Pirjola, R. (2005).
Geomagnetic storm of 29–31 october 2003: Geomagnetically induced
currents and their relation to problems in the swedish high-voltage power
transmission system.
Space Weather 3
Raeder (2003)
Raeder, J. (2003).
Global Magnetohydrodynamics – A Tutorial.
In Space Plasma Simulation, eds. J. Büchner, C. T.
Dum, and M. Scholer (Berlin Heidelberg New York: Springer Verlag).
10.1007/3-540-36530-3_11
Raeder (2006)
Raeder, J. (2006).
Flux transfer events: 1. generation mechanism for strong southward
IMF.
Annales Geophysicae 24, 381–392.
10.5194/angeo-24-381-2006
Raeder and Angelopoulos (1998)
Raeder, J. and Angelopoulos, V. (1998).
Using global simulations of the magnetosphere for multi-satellite
mission planning and analysis,.
In Science Closure and Enabling Technologies for Constellation
Class Missions, eds. V. Angelopoulos and P. V. Panetta (University of
California, Berkeley, and NASA Goddard Space Flight Center). 78
Raeder et al. (2017)
Raeder, J., Cramer, W. D., Germaschewski, K., and Jensen, J. (2017).
Using OpenGGCM to Compute and Separate Magnetosphere Magnetic
Perturbations Measured on Board Low Earth Orbiting Satellites.
Space Sci. Rev. 206, 601–620.
10.1007/s11214-016-0304-x
Raeder and Lu (2005)
Raeder, J. and Lu, G. (2005).
Polar cap potential saturation during large geomagnetic storms.
Advances in Space Research 36, 1804–1808.
10.1016/j.asr.2004.05.010
Raeder et al. (2001a)
Raeder, J., McPherron, R. L., Frank, L. A., Paterson, W. R., Sigwarth, J. B.,
Lu, G., et al. (2001a).
Global simulation of the geospace environment modeling substorm
challenge event.
J. Geophys. Res. 106, 381
Raeder et al. (2001b)
Raeder, J., Wang, Y. L., Fuller-Rowell, T. J., and Singer, H. J.
(2001b).
Global simulation of space weather effects of the Bastille Day
storm.
Sol. Phys. 204, 325
Raeder et al. (2010)
Raeder, J., Zhu, P., Ge, Y., and Siscoe, G. L. (2010).
OpenGGCM simulation of a substorm: Axial tail instability and
ballooning mode preceding substorm onset.
J. Geophys. Res. 115, A00l16
Richardson and Cane (2010)
Richardson, I. G. and Cane, H. V. (2010).
Near-earth interplanetary coronal mass ejections during solar cycle
23 (1996–2009): Catalog and summary of properties.
Solar Physics 264, 189–237
Rigler et al. (2004)
Rigler, E., Baker, D., Weigel, R., Vassiliadis, D., and Klimas, A. (2004).
Adaptive linear prediction of radiation belt electrons using the
kalman filter.
Space Weather 2, 1–9
Sakov and Bertino (2011)
Sakov, P. and Bertino, L. (2011).
Relation between two common localisation methods for the enkf.
Computational Geosciences 15, 225–237
Schatten (1971)
Schatten, K. H. (1971).
Current sheet magnetic model for the solar corona
Schatten et al. (1969)
Schatten, K. H., Wilcox, J. M., and Ness, N. F. (1969).
A model of interplanetary and coronal magnetic fields.
Solar Physics 6, 442–455
Schrijver and DeRosa (2003)
Schrijver, C. J. and DeRosa, M. L. (2003).
Photospheric and heliospheric magnetic fields.
Solar Physics 212, 165–200
Schunk et al. (2004)
Schunk, R. W., Scherliess, L., Sojka, J. J., Thompson, D. C., Anderson, D. N.,
Codrescu, M., et al. (2004).
Global assimilation of ionospheric measurements (gaim).
Radio Science 39
Shi et al. (2014)
Shi, Q. Q., Hartinger, M., Angelopoulos, V., Tian, A., Fu, S., Zong, Q.-G.,
et al. (2014).
Solar wind pressure pulse-driven magnetospheric vortices and their
global consequences.
Journal of Geophysical Research: Space Physics 119,
4274–4280.
10.1002/2013ja019551
Siscoe et al. (2004)
Siscoe, G., Baker, D., Weigel, R., Hughes, J., and Spence, H. (2004).
Roles of empirical modeling within cism.
Journal of atmospheric and solar-terrestrial physics 66,
1481–1489
Skandrani et al. (2014)
Skandrani, C., Innocenti, M. E., Bettarini, L., Crespon, F., Lamouroux, J., and
Lapenta, G. (2014).
Flip-mhd-based model sensitivity analysis.
Nonlinear processes in geophysics 21, 539–553
Stone et al. (1998)
Stone, E. C., Frandsen, A., Mewaldt, R., Christian, E., Margolies, D., Ormes,
J., et al. (1998).
The advanced composition explorer.
Space Science Reviews 86, 1–22
Thomsen et al. (1996)
Thomsen, M., McComas, D., Reeves, G., and Weiss, L. (1996).
An observational test of the tsyganenko (t89a) model of the
magnetospheric field.
Journal of Geophysical Research: Space Physics 101,
24827–24836
Toffoletto et al. (2001)
Toffoletto, F., Spiro, R., Wolf, R., Birn, J., and Hesse, M. (2001).
Modeling inner magnetospheric electrodynamics.
Washington DC American Geophysical Union Geophysical Monograph
Series 125, 265–272
Tóth et al. (2005)
Tóth, G., Sokolov, I. V., Gombosi, T. I., Chesney, D. R., Clauer, C. R.,
De Zeeuw, D. L., et al. (2005).
Space weather modeling framework: A new tool for the space science
community.
Journal of Geophysical Research: Space Physics 110
Tsurutani et al. (2006)
Tsurutani, B. T., Gonzalez, W. D., Gonzalez, A. L., Guarnieri, F. L.,
Gopalswamy, N., Grande, M., et al. (2006).
Corotating solar wind streams and recurrent geomagnetic activity: A
review.
Journal of Geophysical Research: Space Physics 111
Tsyganenko (1987)
Tsyganenko, N. (1987).
Global quantitative models of the geomagnetic field in the cislunar
magnetosphere for different disturbance levels.
Planetary and space science 35, 1347–1358
Tsyganenko (1989)
Tsyganenko, N. (1989).
A magnetospheric magnetic field model with a warped tail current
sheet.
Planetary and Space Science 37, 5–20
Tsyganenko (2002a)
Tsyganenko, N. (2002a).
A model of the near magnetosphere with a dawn-dusk asymmetry 1.
mathematical structure.
Journal of Geophysical Research: Space Physics 107, SMP–12
Tsyganenko (2002b)
Tsyganenko, N. (2002b).
A model of the near magnetosphere with a dawn-dusk asymmetry 2.
parameterization and fitting to observations.
Journal of Geophysical Research: Space Physics 107, SMP–10
Tsyganenko and Andreeva (2015)
Tsyganenko, N. and Andreeva, V. (2015).
A forecasting model of the magnetosphere driven by an optimal solar
wind coupling function.
Journal of Geophysical Research: Space Physics 120, 8401–8425
Tsyganenko et al. (2003)
Tsyganenko, N., Singer, H., and Kasper, J. (2003).
Storm-time distortion of the inner magnetosphere: How severe can it
get?
Journal of Geophysical Research: Space Physics 108
Tsyganenko and Sitnov (2005)
Tsyganenko, N. and Sitnov, M. (2005).
Modeling the dynamics of the inner magnetosphere during strong
geomagnetic storms.
Journal of Geophysical Research: Space Physics 110
Tsyganenko (1995)
Tsyganenko, N. A. (1995).
Modeling the earth’s magnetospheric magnetic field confined within a
realistic magnetopause.
Journal of Geophysical Research: Space Physics 100, 5599–5612
Tsyganenko (1996)
Tsyganenko, N. A. (1996).
Effects of the solar wind conditions in the global magnetospheric
configurations as deduced from data-based field models.
In International conference on substorms. vol. 389, 181
Vennerstrom et al. (2005)
Vennerstrom, S., Moretto, T., Rastaetter, L., and Raeder, J. (2005).
Field-aligned currents during northward interplanetary field:
Morphology and causes.
J. Geophys. Res. 110, A06205, doi: 10.1029/2004JA010802
Vourlidas (2015)
Vourlidas, A. (2015).
Mission to the sun-earth l5 lagrangian point: An optimal platform for
space weather research.
Space Weather 13, 197–201.
10.1002/2015SW001173
Wang and Sheeley Jr (1992)
Wang, Y.-M. and Sheeley Jr, N. (1992).
On potential field models of the solar corona.
The Astrophysical Journal 392, 310–319
Woodfield et al. (2007)
Woodfield, E., Dunlop, M., Holme, R., Davies, J., and Hapgood, M. (2007).
A comparison of cluster magnetic data with the tsyganenko 2001 model.
Journal of Geophysical Research: Space Physics 112
Zhou et al. (2012)
Zhou, X.-Z., Ge, Y. S., Angelopoulos, V., Runov, A., Liang, J., Xing, X.,
et al. (2012).
Dipolarization fronts and associated auroral activities: 2.
acceleration of ions and their subsequent behavior.
Journal of Geophysical Research: Space Physics 117, 1.
10.1029/2012ja017677
Zhu et al. (2009)
Zhu, P., Raeder, J., Germaschewski, K., and Hegna, C. C. (2009).
Initiation of ballooning instability in the near-Earth plasma sheet
prior to the 23 march 2007 THEMIS substorm expansion onset.
Annales Geophysicae 27, 1129–1138.
10.5194/angeo-27-1129-2009 |
Stationary Measures for Stochastic Differential Equations with Jumps*
Huijie Qiao and Jinqiao Duan
Department of Mathematics,
Southeast University,
Nanjing, Jiangsu 211189, China
hjqiaogean@yahoo.com.cn
Department of Applied Mathematics, Illinois Institute of Technology,
Chicago, IL 60616, USA
duan@iit.edu
(Date:: November 20, 2020)
Abstract.
Three methods for studying stationary measures
of stochastic differential equations with jumps are considered. These stationary measures are given
by Markov measures, solutions of Fokker-Planck equations, and
long time limits for the distributions of system states.
1991 Mathematics Subject Classification:
AMS Subject Classification(2000): 28C10; 60G52, 60H10, 60J35.
Keywords: Stationary measure, Cadlag cocycles, Markov measure,
Fokker-Planck equations.
*This work was partly supported by the NSF of China (No. 11001051 and No. 10971225) and the Fundamental Research Funds for the Central Universities, HUST 2010ZD037.
1. Introduction
Stationary measures for stochastic differential equations (SDEs) and
invariant measures for Markov processes have been studied
extensively, as in [2, 3, 5, 13, 17], among others.
Moreover, there are interesting relationships between stationary
measures and invariant measures, such as the one-to-one
correspondence between the set of invariant Markov measures (c.f.
Definition 3.1 in Section 3) and the set of stationary
measures (c.f. [3, 5]), as well as the correspondence
between the solutions of Fokker-Planck equations and stationary
measures (c.f.[13]). These results play an important role in
the development of the theory for random dynamical systems
associated with SDEs.
SDEs with jumps and random dynamical systems associated with them
are considered by a number of authors (c.f. [1] [2]
[7] [10] [11] [15] [17]). We note that
Albeverio-Rüdiger-Wu [1] discussed stationary measures
for SDEs with jumps, in the context of Lévy type operators and
considered mainly infinitesimal invariant measures. The concept of
infinitesimal invariant measures is weaker than that of the usual
stationary measures. Moreover, Zabczyk [17] studied stationary
measures for linear SDEs with jumps.
We ask, naturally, whether the above correspondences for usual SDEs
also hold for SDEs with jumps. This question will be answered in
Sections 3 and 4 in the present paper.
Now we briefly sketch our method. We begin from SDEs with jumps as
random dynamical systems and examine their Markov measures. Then we
consider SDEs which are driven by Brownian motions and by
$\alpha$-stable processes. By a functional analysis
technique, stationary measures are investigated. When the
coefficients of the SDEs are sufficiently regular, the long time
limits for the distributions of the solutions are shown to be
stationary measures, with the help of Malliavin analysis.
This paper is arranged as follows. In Section 2, we introduce random dynamical systems and
related concepts. Sobolev spaces and $\alpha$-stable processes are also introduced. The content
to obtain stationary measures from Markov measures is in Section 3. In
Section 4, we consider special stochastic differential equations
driven by Brownian motions and $\alpha$-stable processes. In Section 5, we study SDEs with jumps, under
certain regular conditions on the coefficients.
The following convention will be used throughout the paper: $C$ with
or without indices will denote different positive constants
(depending on the indices) whose values may change from one place to
another.
2. Preliminary
2.1. Random dynamical systems and related concepts
Let $D({\mathbb{R}}_{+},{\mathbb{R}}^{d})$ be the set of all functions which are
càdlàg, defined on ${\mathbb{R}}_{+}$ and taking values in ${\mathbb{R}}^{d}$. We take sample space
$\Omega=D({\mathbb{R}}_{+},{\mathbb{R}}^{d})$. This sample space becomes a complete and separable
metric space (c.f.[6]), when endowed with the following Skorohod metric
$d$:
$$\displaystyle d(x,y):=\inf\limits_{\lambda\in\Lambda}\left\{\sup\limits_{s\neq
t%
}\left|\log\frac{\lambda(t)-\lambda(s)}{t-s}\right|+\sum\limits_{m=1}^{\infty}%
\frac{1}{2^{m}}\big{(}1\wedge d_{m}^{\circ}(x^{m},y^{m})\big{)}\right\}$$
for all $x,y\in\Omega$, where $x^{m}(t):=g_{m}(t)x(t)$,
$y^{m}(t):=g_{m}(t)y(t)$ with
$$\displaystyle g_{m}(t):=\left\{\begin{array}[]{c}1,\mbox{if}~{}t\leqslant m,\\
m+1-t,\mbox{if}~{}m<t<m+1,\\
0,\mbox{if}~{}t\geqslant m+1,\end{array}\right.$$
and
$$\displaystyle d_{m}^{\circ}(x,y):=\sup\limits_{0\leqslant t\leqslant m}\left|x%
(t)-y(\lambda(t))\right|.$$
Moreover $\Lambda$
denotes the set of strictly increasing and continuous functions from
${\mathbb{R}}_{+}$ to ${\mathbb{R}}_{+}$, and $a\wedge b:=\min\{a,b\}$.
We identify a càdlàg function $\omega(t)$ with a (canonical) sample $\omega$ in the sample space $\Omega$.
The Borel $\sigma$-algebra in the sample space $\Omega$ under the topology induced by the Skorohod metric
$d$ is denoted as ${\mathcal{F}}$. Note that ${\mathcal{F}}=\sigma(\omega(t),t\in{\mathbb{R}}_{+})$ (c.f.[6]). Let $P$ be the unique probability measure which makes a
canonical process a Lévy process, and denotes $({\mathcal{F}}_{t})_{t\in{\mathbb{R}}_{+}}$ the complete natural
filtration with respect to $P$. Thus $(\Omega,{\mathcal{F}},P;({\mathcal{F}}_{t})_{t\in{\mathbb{R}}_{+}})$ is a complete filtered probability space.
Define for each $t\in{\mathbb{R}}_{+}$,
$$\displaystyle(\theta_{t}\omega)(\cdot)=\omega(t+\cdot)-\omega(t),\quad\omega%
\in\Omega.$$
Then $\theta$ is a one-parameter semigroup on
$\Omega$, $\Omega$ is invariant with respect to $\theta$, i.e.
$$\theta_{t}^{-1}\Omega=\Omega,\quad~{}\mbox{for~{}all}~{}t\in{\mathbb{R}}_{+},$$
and $P$ is $\theta$-invariant, i.e.
$$P(\theta_{t}^{-1}(B))=P(B),\qquad~{}\mbox{for~{} all}~{}B\in{\mathcal{F}},t\in%
{\mathbb{R}}_{+}.$$
Thus $(\Omega,{\mathcal{F}},P,(\theta_{t})_{t\in{\mathbb{R}}_{+}})$ is a metric dynamical
system (DS), as defined in [3]. This metric DS is in fact
ergodic, i.e. all measurable $\theta$-invariant sets have
probability either $0$ or $1$. (c.f.[3])
Definition 2.1.
A measurable random dynamical system on the measurable space $({\mathbb{X}},{\mathcal{B}})$
over a metric DS $(\Omega,{\mathcal{F}},P,(\theta_{t})_{t\in{\mathbb{R}}_{+}})$ with time ${\mathbb{R}}_{+}$ is
a mapping
$$\displaystyle\varphi:{\mathbb{R}}_{+}\times\Omega\times{\mathbb{X}}\mapsto{%
\mathbb{X}},\quad(t,\omega,x)\mapsto\varphi(t,\omega,x),$$
$$\displaystyle\varphi(t,\omega):=\varphi(t,\omega,\cdot):{\mathbb{X}}\mapsto{%
\mathbb{X}},$$
such that
(i) Measurability: $\varphi$ is ${\mathcal{B}}({\mathbb{R}}_{+})\otimes{\mathcal{F}}\otimes{\mathcal{B}}/{%
\mathcal{B}}$-measurable,
where ${\mathcal{B}}({\mathbb{R}}_{+})$ is Borel $\sigma$-algebra of ${\mathbb{R}}_{+}$.
(ii) Càdlàg cocycle property: $\varphi(t,\omega)$ forms a (perfect) càdlàg
cocycle over $\theta$ if it is càdlàg in $t$ and satisfies
$$\displaystyle\varphi(0,\omega)$$
$$\displaystyle=$$
$$\displaystyle id_{{\mathbb{X}}},~{}\mbox{for~{} all}~{}\omega\in\Omega,$$
(1)
$$\displaystyle\varphi(t+s,\omega)$$
$$\displaystyle=$$
$$\displaystyle\varphi(t,\theta_{s}\omega)\circ\varphi(s,\omega),$$
(2)
for all $s,t\in{\mathbb{R}}_{+}$ and $\omega\in\Omega$.
A random dynamical system(RDS) induces a skew product flow of
measurable maps
$$\displaystyle\Theta_{t}:\Omega\times{\mathbb{X}}$$
$$\displaystyle\mapsto$$
$$\displaystyle\Omega\times{\mathbb{X}}$$
$$\displaystyle(\omega,x)$$
$$\displaystyle\mapsto$$
$$\displaystyle\big{(}\theta_{t}\omega,\varphi(t,\omega)x\big{)}.$$
The flow property $\Theta_{t+s}=\Theta_{t}\circ\Theta_{s}$ follows from (2).
Denote by $\mathscr{P}(\Omega\times{\mathbb{X}})$ the probability measures
on $(\Omega\times{\mathbb{X}},{\mathcal{F}}\otimes{\mathcal{B}})$. Moreover, $\Theta_{t}$ acts on $\mu\in\mathscr{P}(\Omega\times{\mathbb{X}})$
by $(\Theta_{t}\mu)(C)=\mu(\Theta_{t}^{-1}C)$, for $C\in{\mathcal{F}}\otimes{\mathcal{B}}$,
$t\in{\mathbb{R}}_{+}$.
Definition 2.2.
A probability measure
$\mu\in\mathscr{P}(\Omega\times{\mathbb{X}})$ is called invariant for the
skew product flow $\Theta_{t}$ if
(i) the marginal of $\mu$ on $\Omega$ is $P$,
(ii) $\Theta_{t}\mu=\mu$ for all $t\in{\mathbb{R}}_{+}$.
If ${\mathbb{X}}$ is a Polish space with its Borel $\sigma$-algebra ${\mathcal{B}}({\mathbb{X}})$, every
measure $\mu\in\mathscr{P}(\Omega\times{\mathbb{X}})$ with marginal $P$ can
be uniquely characterized through its factorization
$$\mu(\mathrm{d}\omega,\mathrm{d}x)=\mu_{\omega}(\mathrm{d}x)P(\mathrm{d}\omega),$$
where $\mu_{\omega}(\mathrm{d}x)$ is a probability kernel, i.e. for any $B\in{\mathcal{B}}({\mathbb{X}})$,
$\mu_{\cdot}(B)$ is ${\mathcal{F}}$-measurable; for $P.a.s.\omega\in\Omega$,
$\mu_{\omega}(\cdot)$ is a probability measure on $({\mathbb{X}},{\mathcal{B}}({\mathbb{X}}))$(c.f.[3, p.23]).
Thus $\mu$ is invariant if and only if
$$\displaystyle{\mathbb{E}}[\varphi(t,\omega)\mu_{\cdot}|\theta_{t}^{-1}{%
\mathcal{F}}](\omega)=\mu_{\theta_{t}\omega},\quad P.a.s,$$
(3)
for all $t\in{\mathbb{R}}_{+}$.
2.2. Sobolev spaces
Let ${\mathcal{C}}_{0}({\mathbb{R}}^{d})$ be the space of continuous functions $f$ on ${\mathbb{R}}^{d}$ satisfying
$\lim\limits_{|x|\rightarrow\infty}f(x)=0$ with norm $\|f\|_{{\mathcal{C}}_{0}({\mathbb{R}}^{d})}=\sup\limits_{x}|f(x)|$.
Let ${\mathcal{C}}^{2}_{0}({\mathbb{R}}^{d})$ be the set of $f\in{\mathcal{C}}_{0}({\mathbb{R}}^{d})$ such that $f$ is $2$
times differentiable and the partial derivatives of $f$ with order $\leqslant 2$ belong
to ${\mathcal{C}}_{0}({\mathbb{R}}^{d})$. Let ${\mathcal{C}}_{c}^{n}({\mathbb{R}}^{d})$ stand for the space of all $n$
times differentiable functions on ${\mathbb{R}}^{d}$ with compact supports. Let $S({\mathbb{R}}^{d})$
be the Schwartz space of all rapidly decreasing real valued ${\mathcal{C}}^{\infty}$
functions on ${\mathbb{R}}^{d}$ and $S^{\prime}({\mathbb{R}}^{d})$ the space of all tempered
distributions on ${\mathbb{R}}^{d}$. Let $\hat{f}$ and $\breve{f}$ denote the
Fourier transform and the Fourier inversion transform of $f\in S({\mathbb{R}}^{d})$,
respectively, i.e.
$$\displaystyle\hat{f}(u)=\frac{1}{(2\pi)^{d/2}}\int_{{\mathbb{R}}^{d}}e^{-i{%
\langle}u,x{\rangle}}f(x)\mathrm{d}x,$$
$$\displaystyle\breve{f}(u)=\frac{1}{(2\pi)^{d/2}}\int_{{\mathbb{R}}^{d}}e^{i{%
\langle}u,x{\rangle}}f(x)\mathrm{d}x,$$
for all $u\in{\mathbb{R}}^{d}$. And Fourier transforms and the Fourier inversion transforms
can be defined on $S^{\prime}({\mathbb{R}}^{d})$ by the same means to the above one.
We introduce the following Sobolev space
$$\displaystyle{\mathbb{H}}^{\lambda,2}({\mathbb{R}}^{d}):=\{f\in S^{\prime}({%
\mathbb{R}}^{d}):\|f\|_{\lambda,2}<\infty\},$$
for any $\lambda\in{\mathbb{R}}$, where
$$\displaystyle\|f\|^{2}_{{\mathbb{H}}^{\lambda,2}({\mathbb{R}}^{d})}:=\int_{{%
\mathbb{R}}^{d}}(1+|u|^{2})^{\lambda}|\hat{f}(u)|^{2}\mathrm{d}u.$$
In particular, ${\mathbb{H}}^{0,2}({\mathbb{R}}^{d})=L^{2}({\mathbb{R}}^{d})$.
2.3. Rotation invariant $\alpha$-stable process
Definition 2.3.
A process $L=(L_{t})_{t\geqslant 0}$ with $L_{0}=0$ a.s. is a
$d$-dimensional Lévy process if
(i) $L$ has independent increments; that is, $L_{t}-L_{s}$ is
independent of $L_{v}-L_{u}$ if $(u,v)\cap(s,t)=\emptyset$;
(ii) $L$ has stationary increments; that is, $L_{t}-L_{s}$ has the same
distribution as $L_{v}-L_{u}$ if $t-s=v-u>0$;
(iii) $L_{t}$ is stochastically continuous;
(iv) $L_{t}$ is right continuous with left limit.
Its characteristic function is given by
$$\displaystyle{\mathbb{E}}\left(\exp\{i{\langle}z,L_{t}{\rangle}\}\right)=\exp%
\{t\Psi(z)\},\quad z\in{\mathbb{R}}^{d}.$$
The function $\Psi:{\mathbb{R}}^{d}\rightarrow\mathcal{C}$ is
called the characteristic exponent of the Lévy process $L$. By
Lévy-Khintchine formula, there exist a nonnegative-definite
$d\times d$ matrix $Q$, a measure $\nu$ on ${\mathbb{R}}^{d}$ satisfying
$$\displaystyle\nu(\{0\})=0~{}\mbox{and}~{}\int_{{\mathbb{R}}^{d}\setminus\{0\}}%
(|u|^{2}\wedge 1)\nu(\mathrm{d}u)<\infty,$$
and $\gamma\in{\mathbb{R}}^{d}$ such that
$$\displaystyle\Psi(z)$$
$$\displaystyle=$$
$$\displaystyle i{\langle}z,\gamma{\rangle}-\frac{1}{2}{\langle}z,Qz{\rangle}$$
(4)
$$\displaystyle+\int_{{\mathbb{R}}^{d}\setminus\{0\}}\big{(}e^{i{\langle}z,u{%
\rangle}}-1-i{\langle}z,u{\rangle}1_{|u|\leqslant 1}\big{)}\nu(\mathrm{d}u).$$
The measure $\nu$ is called the Lévy measure.
Definition 2.4.
For $\alpha\in(0,2)$. A $d$-dimensional rotation invariant $\alpha$-stable
process $L$ is a Lévy process such that its characteristic exponent $\Psi$
is given by
$$\displaystyle\Psi(z)=-C|z|^{\alpha},\quad z\in{\mathbb{R}}^{d}.$$
Thus, for $d$-dimensional rotation invariant $\alpha$-stable process $L$,
Lévy measure $\nu$ is given by
$$\nu(\mathrm{d}u)=\frac{C_{d,\alpha}}{|u|^{d+\alpha}}\mathrm{d}u,$$
and $Q=0$ in (4). Moreover,
$$\displaystyle-C|z|^{\alpha}=\int_{{\mathbb{R}}^{d}\setminus\{0\}}\big{(}e^{i{%
\langle}z,u{\rangle}}-1-i{\langle}z,u{\rangle}1_{|u|\leqslant 1}\big{)}\frac{C%
_{d,\alpha}}{|u|^{d+\alpha}}\mathrm{d}u.$$
Define
$$\displaystyle({\mathcal{L}}_{\alpha}f)(x):=\int_{{\mathbb{R}}^{d}\setminus\{0%
\}}\big{(}f(x+u)-f(x)-{\langle}\partial_{x}f(x),u{\rangle}1_{|u|\leqslant 1}%
\big{)}\frac{C_{d,\alpha}}{|u|^{d+\alpha}}\mathrm{d}u$$
on ${\mathcal{C}}_{0}^{2}({\mathbb{R}}^{d})$. And then for $\xi\in{\mathbb{R}}^{d}$
$$\displaystyle({\mathcal{L}}_{\alpha}e^{i{\langle}\cdot,\xi{\rangle}})(x)=e^{i{%
\langle}x,\xi{\rangle}}\int_{{\mathbb{R}}^{d}\setminus\{0\}}\big{(}e^{i{%
\langle}u,\xi{\rangle}}-1-i{\langle}\xi,u{\rangle}1_{|u|\leqslant 1}\big{)}%
\frac{C_{d,\alpha}}{|u|^{d+\alpha}}\mathrm{d}u.$$
By Courrège’s second theorem (c.f.[2, Theorem 3.5.5, p.183]),
for every $f\in{\mathcal{C}}_{c}^{\infty}({\mathbb{R}}^{d})$
$$\displaystyle({\mathcal{L}}_{\alpha}f)(x)$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{(2\pi)^{d/2}}\int_{{\mathbb{R}}^{d}}e^{i{\langle}z,x{%
\rangle}}\left[e^{-i{\langle}x,z{\rangle}}({\mathcal{L}}_{\alpha}e^{i{\langle}%
\cdot,z{\rangle}})(x)\right]\hat{f}(z)\mathrm{d}z$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{(2\pi)^{d/2}}\int_{{\mathbb{R}}^{d}}e^{i{\langle}z,x{%
\rangle}}\left[\int_{{\mathbb{R}}^{d}\setminus\{0\}}\big{(}e^{i{\langle}u,z{%
\rangle}}-1-i{\langle}z,u{\rangle}1_{|u|\leqslant 1}\big{)}\frac{C_{d,\alpha}}%
{|u|^{d+\alpha}}\mathrm{d}u\right]\hat{f}(z)\mathrm{d}z$$
$$\displaystyle=$$
$$\displaystyle-\frac{C}{(2\pi)^{d/2}}\int_{{\mathbb{R}}^{d}}e^{i{\langle}z,x{%
\rangle}}|z|^{\alpha}\hat{f}(z)\mathrm{d}z$$
$$\displaystyle=$$
$$\displaystyle C\cdot[-(-\Delta)^{\alpha/2}f](x).$$
Moreover, the following result is well-known (c.f.[1]).
Theorem 2.5.
Let ${\mathcal{L}}_{\alpha}$ be as above for $\alpha\in(0,2)$ and ${\mathcal{L}}_{2}=\Delta$,
as defined on ${\mathcal{C}}_{c}^{\infty}({\mathbb{R}}^{d})$ in $L^{2}({\mathbb{R}}^{d})$. Then ${\mathcal{L}}_{\alpha}$,
$0<\alpha\leqslant 2$, has a unique closed extensions to self-adjoint negative
operators on the domain ${\mathbb{H}}^{\alpha,2}({\mathbb{R}}^{d})$.
3. From Markov measures to stationary measures
Let $({\mathbb{U}},{\mathcal{U}},n)$ be a $\sigma$-finite measurable
space. Let $\{W(t)\}_{t\geqslant 0}$ be an $m$-dimensional standard
${\mathcal{F}}_{t}$-adapted Brownian motion, and $\{k_{t},t\geqslant 0\}$ a stationary
${\mathcal{F}}_{t}$-adapted Poisson point process with values in ${\mathbb{U}}$ and with
characteristic measure $n$(cf. [7]). Let $N_{k}((0,t],\mathrm{d}u)$ be the counting measure of $k_{t}$, i.e., for $A\in{\mathcal{U}}$
$$N_{k}((0,t],A):=\#\{0<s\leqslant t:k_{s}\in A\},$$
where $\#$ denotes the cardinality of a set. The compensator measure
of $N_{k}$ is given by
$$\tilde{N}_{k}((0,t],\mathrm{d}u):=N_{k}((0,t],\mathrm{d}u)-tn(\mathrm{d}u).$$
Fix a ${\mathbb{U}}_{0}\in{\mathcal{U}}$ such that $n({\mathbb{U}}-{\mathbb{U}}_{0})<\infty$,
and consider the following SDE with jumps in ${\mathbb{R}}^{d}$:
$$\displaystyle X_{t}(x)$$
$$\displaystyle=$$
$$\displaystyle x+\int^{t}_{0}b(X_{s}(x))\,\mathrm{d}s+\int^{t}_{0}\sigma(X_{s}(%
x))\,\mathrm{d}W_{s}$$
(5)
$$\displaystyle+\int^{t+}_{0}\int_{{\mathbb{U}}_{0}}f(X_{s-}(x),u)\,\tilde{N}_{k%
}(\mathrm{d}s,\mathrm{d}u)$$
$$\displaystyle+\int^{t+}_{0}\int_{{\mathbb{U}}-{\mathbb{U}}_{0}}g(X_{s-}(x),u)%
\,N_{k}(\mathrm{d}s,\mathrm{d}u),\quad t\geqslant 0,$$
where $b:{\mathbb{R}}^{d}\mapsto{\mathbb{R}}^{d}$, $\sigma:{\mathbb{R}}^{d}\mapsto{\mathbb{R}}^{d}\times{\mathbb{R}}^{m}$,
$f,g:{\mathbb{R}}^{d}\times{\mathbb{U}}\mapsto{\mathbb{R}}^{d}$ satisfy the following assumptions:
(H${}_{b}$)
there exists a constant $C_{b}>0$ such that
for $x,y\in{\mathbb{R}}^{d}$
$$|b(x)-b(y)|\leqslant C_{b}|x-y|\cdot\log(|x-y|^{-1}+e);$$
(H${}_{\sigma}$)
there exists a constant $C_{\sigma}>0$ such that
for $x,y\in{\mathbb{R}}^{d}$
$$|\sigma(x)-\sigma(y)|^{2}\leqslant C_{\sigma}|x-y|^{2}\cdot\log(|x-y|^{-1}+e);$$
(H${}_{f}$)
for some $q>(2d)\vee 4$ and any $p\in[2,q]$, there exists a constant $C_{p}>0$ such that
for $x,y\in{\mathbb{R}}^{d}$
$$\int_{{\mathbb{U}}_{0}}|f(x,u)-f(y,u)|^{p}\,n(\mathrm{d}u)\leqslant C_{p}|x-y|%
^{p}\cdot\log(|x-y|^{-1}+e),$$
and
$$\int_{{\mathbb{U}}_{0}}|f(x,u)|^{p}\,n(\mathrm{d}u)\leqslant C_{p}(1+|x|)^{p}.$$
(H${}_{g}$)
for $u\in{\mathbb{U}}-{\mathbb{U}}_{0}$, $x\mapsto g(x,u)\in{\mathcal{C}}({\mathbb{R}}^{d})$, where ${\mathcal{C}}({\mathbb{R}}^{d})$
stands for the total of continuous functions from ${\mathbb{R}}^{d}$ to ${\mathbb{R}}^{d}$.
Here, the second integral of the right hand side in Eq.(5)
is taken in Itô’s sense,
and the definitions of the third and fourth integrals
are referred to [7].
Under (H${}_{b}$), (H${}_{\sigma}$), (H${}_{f}$) and (H${}_{g}$), it
is well known that there exists a unique strong solution to Eq.(5)(cf. [10]).
This solution will be denoted by $X_{t}(x)$. Set
$$\displaystyle\mathscr{F}_{\geqslant t}:=\sigma\{W_{s},N_{p}((0,s],B);s%
\geqslant t,B\in{\mathcal{U}}\},$$
$$\displaystyle\mathscr{F}_{\leqslant t}:=\sigma\{W_{s},N_{p}((0,s],B);s%
\leqslant t,B\in{\mathcal{U}}\},$$
for $t\geqslant 0$. And then
$$\theta_{t}^{-1}\mathscr{F}_{>0}\subset\mathscr{F}_{>t}$$
and
$$\theta_{t}^{-1}\mathscr{F}_{\leqslant s}\subset\mathscr{F}_{\leqslant t+s},%
\quad s\geqslant 0.$$
Moreover, $X_{t}(x)$ is $\mathscr{F}_{\leqslant t}$-measurable
and independent of $\theta_{t}^{-1}\mathscr{F}_{>0}$.
Definition 3.1.
A probability measure $\mu$ on $(\Omega\times{\mathbb{R}}^{d},{\mathcal{F}}\otimes{\mathcal{B}}({\mathbb{R}}^{%
d}))$
is called a Markov measure if $\mu_{\omega}$ satisfies
$$\displaystyle{\mathbb{E}}(\mu_{\cdot}|\mathscr{F}_{<\infty})={\mathbb{E}}(\mu_%
{\cdot}|\mathscr{F}_{=0}),\quad P.a.s..$$
For the Markov process $X_{t}(x)$, the transition probability is defined by
$$p_{t}(x,B):=P\big{(}X_{t}(x)\in B\big{)},\quad t>0,B\in{\mathcal{B}}({\mathbb{%
R}}^{d}).$$
Definition 3.2.
A measure $\bar{\mu}$ on $({\mathbb{R}}^{d},{\mathcal{B}}({\mathbb{R}}^{d}))$ is a stationary measure for $p_{t}$
or Eq.(5) if
$$\displaystyle\int_{{\mathbb{R}}^{d}}p_{t}(x,B)\bar{\mu}(\mathrm{d}x)=\bar{\mu}%
(B),\quad\forall t>0,B\in{\mathcal{B}}({\mathbb{R}}^{d}).$$
Theorem 3.3.
Set $\varphi(t,\omega)x:=X_{t}(x)$, and then for an invariant Markov measure $\mu$
of the skew product flow $\Theta_{t}$, the stationary measure $\bar{\mu}$ for $p_{t}$ is given by
$$\bar{\mu}={\mathbb{E}}(\mu_{\cdot}|\mathscr{F}_{>0}).$$
Proof.
By Definition 3.1
$$\displaystyle{\mathbb{E}}(\mu_{\cdot}|\mathscr{F}_{<\infty})={\mathbb{E}}(\mu_%
{\cdot}|\mathscr{F}_{=0}).$$
Set
$$\displaystyle\bar{\mu}:={\mathbb{E}}(\mu_{\cdot}|\mathscr{F}_{>0}),$$
and then it follows from independence of $\mathscr{F}_{>0}$ and $\mathscr{F}_{=0}$
$$\displaystyle\bar{\mu}$$
$$\displaystyle=$$
$$\displaystyle{\mathbb{E}}\left[{\mathbb{E}}(\mu_{\cdot}|\mathscr{F}_{=0})|%
\mathscr{F}_{>0}\right]$$
$$\displaystyle=$$
$$\displaystyle{\mathbb{E}}\left[{\mathbb{E}}(\mu_{\cdot}|\mathscr{F}_{=0})\right]$$
$$\displaystyle=$$
$$\displaystyle{\mathbb{E}}(\mu_{\cdot}).$$
Therefore $\nu$ is not random.
By (3), it holds that for $t>0$
$$\displaystyle{\mathbb{E}}[\varphi(t,\cdot)\mu_{\cdot}|\theta_{t}^{-1}\mathscr{%
F}_{>0}]$$
$$\displaystyle=$$
$$\displaystyle{\mathbb{E}}[{\mathbb{E}}(\varphi(t,\cdot)\mu_{\cdot}|\theta_{t}^%
{-1}{\mathcal{F}})|\theta_{t}^{-1}\mathscr{F}_{>0}]$$
$$\displaystyle=$$
$$\displaystyle{\mathbb{E}}[\mu_{\theta_{t}\cdot}|\theta_{t}^{-1}\mathscr{F}_{>0}]$$
$$\displaystyle=$$
$$\displaystyle{\mathbb{E}}(\mu_{\cdot}|\mathscr{F}_{>0})(\theta_{t}\omega)$$
$$\displaystyle=$$
$$\displaystyle\bar{\mu}.$$
Besides,
$$\displaystyle{\mathbb{E}}[\varphi(t,\cdot)\mu_{\cdot}|\theta_{t}^{-1}\mathscr{%
F}_{>0}]$$
$$\displaystyle=$$
$$\displaystyle{\mathbb{E}}[{\mathbb{E}}(\varphi(t,\cdot)\mu_{\cdot}|\mathscr{F}%
_{>0})|\theta_{t}^{-1}\mathscr{F}_{>0}]$$
$$\displaystyle=$$
$$\displaystyle{\mathbb{E}}[\varphi(t,\cdot)\bar{\mu}|\theta_{t}^{-1}\mathscr{F}%
_{>0}]$$
$$\displaystyle=$$
$$\displaystyle{\mathbb{E}}[\varphi(t,\cdot)\bar{\mu}].$$
Thus for $B\in{\mathcal{B}}({\mathbb{R}}^{d})$
$$\displaystyle\bar{\mu}(B)$$
$$\displaystyle=$$
$$\displaystyle{\mathbb{E}}[\varphi(t,\cdot)\bar{\mu}(B)]={\mathbb{E}}[\bar{\mu}%
(\varphi(t,\cdot)^{-1}B)]$$
$$\displaystyle=$$
$$\displaystyle\int_{\Omega}P(\mathrm{d}\omega)\int_{{\mathbb{R}}^{d}}1_{B}(%
\varphi(t,\omega)x)\bar{\mu}(\mathrm{d}x)$$
$$\displaystyle=$$
$$\displaystyle\int_{{\mathbb{R}}^{d}}\bar{\mu}(\mathrm{d}x)\int_{\Omega}1_{B}(%
\varphi(t,\omega)x)P(\mathrm{d}\omega)$$
$$\displaystyle=$$
$$\displaystyle\int_{{\mathbb{R}}^{d}}p_{t}(x,B)\bar{\mu}(\mathrm{d}x).$$
By Definition 3.2 ${\mathbb{E}}(\mu_{\cdot}|\mathscr{F}_{>0})$ is a stationary
measure for $p_{t}$.
∎
So, this theorem, together with [10, Theorem 1.1] and
[5, Lemma 5.1], yields
Theorem 3.4.
If $({\mathbb{R}}^{d},{\mathcal{B}}({\mathbb{R}}^{d}))$ is replaced by its one point compactification
$(\hat{{\mathbb{R}}}^{d},{\mathcal{B}}(\hat{{\mathbb{R}}}^{d}))$, stationary measures for Eq.(5)
on $(\hat{{\mathbb{R}}}^{d},{\mathcal{B}}(\hat{{\mathbb{R}}}^{d}))$ exist.
4. From Fokker-Planck equations to stationary measures
Consider the following equation
$$\displaystyle X_{t}(x)=x+\int^{t}_{0}b(X_{s}(x))\,\mathrm{d}s+\int^{t}_{0}%
\sigma(X_{s}(x))\,\mathrm{d}W_{s}+L_{t},$$
where $L_{t}$ is a rotation invariant $\alpha$-stable process independent
of $W_{t}$. Based on the lévy-Itô representation of $L_{t}$, the above
equation can be rewritten as follows:
(i)
for $1\leqslant\alpha<2$,
$$\displaystyle X_{t}(x)$$
$$\displaystyle=$$
$$\displaystyle x+\int^{t}_{0}b(X_{s}(x))\,\mathrm{d}s+\int^{t}_{0}\sigma(X_{s}(%
x))\,\mathrm{d}W_{s}$$
(6)
$$\displaystyle+\int_{0}^{t}\int_{|u|\leqslant\delta}u\tilde{N}_{k}(\mathrm{d}s,%
\mathrm{d}u)+\int_{0}^{t}\int_{|u|>\delta}uN_{k}(\mathrm{d}s,\mathrm{d}u),$$
where $k(t):=L_{t}-L_{t-}$ and $0<\delta<1$ satisfies that for some $q>4d$
$$\displaystyle\frac{q\delta}{(1-\delta)^{q+1}}<1;$$
(ii)
for $0<\alpha<1$
$$\displaystyle X_{t}(x)$$
$$\displaystyle=$$
$$\displaystyle x+\int^{t}_{0}b(X_{s}(x))\,\mathrm{d}s+\int^{t}_{0}\sigma(X_{s}(%
x))\,\mathrm{d}W_{s}$$
(7)
$$\displaystyle+\int_{0}^{t}\int_{{\mathbb{R}}^{d}\setminus\{0\}}uN_{k}(\mathrm{%
d}s,\mathrm{d}u).$$
We study mainly Eq.(6). Eq.(7) can be dealt with similarly.
Under (H${}_{b}$) and (H${}_{\sigma}$), by [10, Theorem 1.3],
for almost all $\omega\in\Omega$, $x\mapsto X_{t}(x,\omega)$ is a homeomorphism
mapping on ${\mathbb{R}}^{d}$, where $X_{t}(x,\omega)$ is the solution of Eq.(6).
Define
$$\displaystyle(p_{t}h)(x):={\mathbb{E}}\big{[}h\big{(}X_{t}(x)\big{)}\big{]}=%
\int_{{\mathbb{R}}^{d}}h(y)p_{t}(x,\mathrm{d}y),$$
for $h\in{\mathcal{C}}_{0}({\mathbb{R}}^{d})$. Thus $p_{t}h\in{\mathcal{C}}_{0}({\mathbb{R}}^{d})$ by dominated convergence
theorem. Let ${\mathcal{M}}_{r}({\mathbb{R}}^{d})$ be the set of all finite regular signed measures
on ${\mathcal{B}}({\mathbb{R}}^{d})$. And then it is adjoint of ${\mathcal{C}}_{0}({\mathbb{R}}^{d})$(c.f. [14, Theorem 5.4.3, p.148].
Lemma 4.1.
The family of operators $\{p_{t},t\geqslant 0\}$ defined above is a strongly continuous
contraction semigroup on ${\mathcal{C}}_{0}({\mathbb{R}}^{d})$.
Proof.
For $t,s\geqslant 0$ and $h\in{\mathcal{C}}_{0}({\mathbb{R}}^{d})$, by C-K equation
$$\displaystyle(p_{t+s}h)(x)$$
$$\displaystyle=$$
$$\displaystyle\int_{{\mathbb{R}}^{d}}h(y)p_{t+s}(x,\mathrm{d}y)=\int_{{\mathbb{%
R}}^{d}}h(y)\int_{{\mathbb{R}}^{d}}p_{s}(z,\mathrm{d}y)p_{t}(x,\mathrm{d}z)$$
$$\displaystyle=$$
$$\displaystyle\int_{{\mathbb{R}}^{d}}p_{t}(x,\mathrm{d}z)\int_{{\mathbb{R}}^{d}%
}h(y)p_{s}(z,\mathrm{d}y)=\int_{{\mathbb{R}}^{d}}p_{t}(x,\mathrm{d}z)(p_{s}h)(z)$$
$$\displaystyle=$$
$$\displaystyle(p_{t}(p_{s}h))(x).$$
So, $p_{t+s}=p_{t}p_{s}$. Obviously $p_{0}=I$.
Next, we prove strong continuity. For $h\in{\mathcal{C}}_{0}({\mathbb{R}}^{d})$, $h$ is uniformly
continuous on ${\mathbb{R}}^{d}$. And for $\forall\varepsilon>0$, there exists an $\eta>0$
such that $|h(x)-h(y)|<\varepsilon$ for $x,y\in{\mathbb{R}}^{d}$, $|x-y|<\eta$. For any
$\lambda\in{\mathcal{M}}_{r}({\mathbb{R}}^{d})$
$$\displaystyle\left|\int_{{\mathbb{R}}^{d}}(p_{t}h)(x)\lambda(\mathrm{d}x)-\int%
_{{\mathbb{R}}^{d}}h(x)\lambda(\mathrm{d}x)\right|$$
$$\displaystyle=$$
$$\displaystyle\left|\int_{{\mathbb{R}}^{d}}\int_{{\mathbb{R}}^{d}}h(y)p_{t}(x,%
\mathrm{d}y)\lambda(\mathrm{d}x)-\int_{{\mathbb{R}}^{d}}\int_{{\mathbb{R}}^{d}%
}h(x)p_{t}(x,\mathrm{d}y)\lambda(\mathrm{d}x)\right|$$
$$\displaystyle\leqslant$$
$$\displaystyle\int_{{\mathbb{R}}^{d}}\int_{{\mathbb{R}}^{d}}|h(y)-h(x)|p_{t}(x,%
\mathrm{d}y)|\lambda|(\mathrm{d}x)$$
$$\displaystyle\leqslant$$
$$\displaystyle\int_{{\mathbb{R}}^{d}}\int_{|x-y|<\eta}|h(y)-h(x)|p_{t}(x,%
\mathrm{d}y)|\lambda|(\mathrm{d}x)$$
$$\displaystyle+\int_{{\mathbb{R}}^{d}}\int_{|x-y|\geqslant\eta}|h(y)-h(x)|p_{t}%
(x,\mathrm{d}y)|\lambda|(\mathrm{d}x)$$
$$\displaystyle\leqslant$$
$$\displaystyle|\lambda|({\mathbb{R}}^{d})\varepsilon+2\|h\|\int_{{\mathbb{R}}^{%
d}}P\{|X_{t}(x)-x|\geqslant\eta\}|\lambda|(\mathrm{d}x),$$
where $|\lambda|$ stands for the variation measure of the signed measure
$\lambda$. For $\int_{{\mathbb{R}}^{d}}P\{|X_{t}(x)-x|\geqslant\eta\}|\lambda|(\mathrm{d}x)$,
by stochastical continuity of $X_{t}$ and dominated convergence theorem,
when $t$ is small enough,
$$\int_{{\mathbb{R}}^{d}}P\{|X_{t}(x)-x|\geqslant\eta\}|\lambda|(\mathrm{d}x)<\varepsilon.$$
So,
$$\displaystyle\lim\limits_{t\downarrow 0}\int_{{\mathbb{R}}^{d}}(p_{t}h)(x)%
\lambda(\mathrm{d}x)=\int_{{\mathbb{R}}^{d}}h(x)\lambda(\mathrm{d}x).$$
That is to say, $p_{t}h$ converges weakly to $h$. By [16, Theorem, p.233],
$p_{t}h$ converges strongly to $h$.
Finally, by [9, Definition 2.1, p.4], $\{p_{t},t\geqslant 0\}$ is a strongly
continuous contraction semigroup on ${\mathcal{C}}_{0}({\mathbb{R}}^{d})$.
∎
Let ${\mathcal{L}}$ be the infinitesimal generator of $\{p_{t},t\geqslant 0\}$.
Lemma 4.2.
For $h\in{\mathcal{C}}^{2}_{c}({\mathbb{R}}^{d})$,
$$\displaystyle({\mathcal{L}}h)(x)={\langle}\partial_{x}h(x),b(x){\rangle}+\frac%
{1}{2}\left(\frac{\partial^{2}}{\partial x_{i}\partial x_{j}}h(x)\right)\sigma%
_{ij}(x)+({\mathcal{L}}_{\alpha}h)(x).$$
Proof.
By the Itô formula, one can obtain for $h\in{\mathcal{C}}^{2}_{c}({\mathbb{R}}^{d})$
$$\displaystyle h(X_{t})-h(x)$$
$$\displaystyle=$$
$$\displaystyle\int_{0}^{t}{\langle}\partial_{y}h(X_{s}),b(X_{s}){\rangle}%
\mathrm{d}s+\int_{0}^{t}{\langle}\partial_{y}h(X_{s}),\sigma(X_{s})\mathrm{d}W%
_{s}{\rangle}$$
$$\displaystyle+\int_{0}^{t}\int_{|u|\leqslant\delta}\left(h(X_{s}+u)-h(X_{s})%
\right)\tilde{N}_{p}(\mathrm{d}s,\mathrm{d}u)$$
$$\displaystyle+\int_{0}^{t}\int_{|u|>\delta}\left(h(X_{s}+u)-h(X_{s})\right)N_{%
p}(\mathrm{d}s,\mathrm{d}u)$$
$$\displaystyle+\frac{1}{2}\int_{0}^{t}\left(\frac{\partial^{2}}{\partial y_{i}%
\partial y_{j}}h(X_{s})\right)\sigma_{ij}(X_{s})\mathrm{d}s$$
$$\displaystyle+\int_{0}^{t}\int_{|u|\leqslant\delta}\big{(}h(X_{s}+u)-h(X_{s})$$
$$\displaystyle\qquad-{\langle}\partial_{y}h(X_{s}),u{\rangle}\big{)}\frac{C_{d,%
\alpha}}{|u|^{d+\alpha}}\mathrm{d}u\mathrm{d}s.$$
Taking expectation on two sides, we get
$$\displaystyle(p_{t}h)(x)-h(x)$$
$$\displaystyle=$$
$$\displaystyle\int_{0}^{t}{\mathbb{E}}\left[{\langle}\partial_{y}h(X_{s}),b(X_{%
s}){\rangle}\right]\mathrm{d}s+\frac{1}{2}\int_{0}^{t}{\mathbb{E}}\left[\left(%
\frac{\partial^{2}}{\partial y_{i}\partial y_{j}}h(X_{s})\right)\sigma_{ij}(X_%
{s})\right]\mathrm{d}s$$
$$\displaystyle+\int_{0}^{t}{\mathbb{E}}\bigg{[}\int_{{\mathbb{R}}^{d}\setminus%
\{0\}}\big{(}h(X_{s}+u)-h(X_{s})$$
$$\displaystyle\qquad-{\langle}\partial_{y}h(X_{s}),u{\rangle}\big{)}\frac{C_{d,%
\alpha}}{|u|^{d+\alpha}}\mathrm{d}u\bigg{]}\mathrm{d}s$$
$$\displaystyle=$$
$$\displaystyle\int_{0}^{t}\int_{{\mathbb{R}}^{d}}{\langle}\partial_{y}h(y),b(y)%
{\rangle}p_{s}(x,\mathrm{d}y)\mathrm{d}s$$
$$\displaystyle+\frac{1}{2}\int_{0}^{t}\int_{{\mathbb{R}}^{d}}\left(\frac{%
\partial^{2}}{\partial y_{i}\partial y_{j}}h(y)\right)\sigma_{ij}(y)p_{s}(x,%
\mathrm{d}y)\mathrm{d}s$$
$$\displaystyle+\int_{0}^{t}\int_{{\mathbb{R}}^{d}}\Big{[}\int_{{\mathbb{R}}^{d}%
\setminus\{0\}}(h(y+u)-h(y)$$
$$\displaystyle\quad-{\langle}\partial_{y}h(y),u{\rangle})\frac{C_{d,\alpha}}{|u%
|^{d+\alpha}}\mathrm{d}u\Big{]}p_{s}(x,\mathrm{d}y)\mathrm{d}s.$$
Thus
$$\displaystyle\lim\limits_{t\downarrow 0}\frac{1}{t}\left((p_{t}h)(x)-h(x)\right)$$
$$\displaystyle=$$
$$\displaystyle{\langle}\partial_{x}h(x),b(x){\rangle}+\frac{1}{2}\left(\frac{%
\partial^{2}}{\partial x_{i}\partial x_{j}}h(x)\right)\sigma_{ij}(x)$$
$$\displaystyle+\int_{{\mathbb{R}}^{d}\setminus\{0\}}\left(h(x+u)-h(x)-{\langle}%
\partial_{x}h(x),u{\rangle}\right)\frac{C_{d,\alpha}}{|u|^{d+\alpha}}\mathrm{d%
}u.$$
By [11, Lemma 31.7, p.209],
$$\displaystyle({\mathcal{L}}h)(x)$$
$$\displaystyle=$$
$$\displaystyle{\langle}\partial_{x}h(x),b(x){\rangle}+\frac{1}{2}\left(\frac{%
\partial^{2}}{\partial x_{i}\partial x_{j}}h(x)\right)\sigma_{ij}(x)$$
$$\displaystyle+\int_{{\mathbb{R}}^{d}\setminus\{0\}}\left(h(x+u)-h(x)-{\langle}%
\partial_{x}h(x),u{\rangle}\right)\frac{C_{d,\alpha}}{|u|^{d+\alpha}}\mathrm{d}u$$
$$\displaystyle=$$
$$\displaystyle{\langle}\partial_{x}h(x),b(x){\rangle}+\frac{1}{2}\left(\frac{%
\partial^{2}}{\partial x_{i}\partial x_{j}}h(x)\right)\sigma_{ij}(x)+({%
\mathcal{L}}_{\alpha}h)(x).$$
∎
To get the main result, we make the following assumptions.
(H${}_{p}$)
For all $t>0$ and $x\in{\mathbb{R}}^{d}$, the transition
probability $p_{t}(x,\mathrm{d}y)$ admits a density $\rho_{t}(x,y)$ and functions
$$\displaystyle(t,y)$$
$$\displaystyle\mapsto$$
$$\displaystyle\frac{\partial}{\partial t}\rho_{t}(x,y),$$
$$\displaystyle(t,y)$$
$$\displaystyle\mapsto$$
$$\displaystyle\frac{\partial}{\partial y_{j}}\big{(}b_{j}(y)\rho_{t}(x,y)\big{)%
},j=1,2,\dots,d,$$
$$\displaystyle(t,y)$$
$$\displaystyle\mapsto$$
$$\displaystyle\frac{\partial^{2}}{\partial y_{i}\partial y_{j}}\big{(}\sigma_{%
ij}(y)\rho_{t}(x,y)\big{)},i,j=1,2,\dots,d,$$
$$\displaystyle(t,y)$$
$$\displaystyle\mapsto$$
$$\displaystyle\big{(}{\mathcal{L}}_{\alpha}\rho_{t}(x,\cdot)\big{)}(y),$$
exist and are continuous on ${\mathbb{R}}_{+}\times{\mathbb{R}}^{d}$.
Under (H${}_{p}$), the distribution of $X_{t}(x)$ has a density
$\rho_{X}(t,y)=\rho_{t}(x,y)$ for $t>0$.
Lemma 4.3.
$\rho_{X}(t,y)$ satisfies the following
Fokker-Planck equation
$$\displaystyle\left\{\begin{array}[]{l}\frac{\partial}{\partial t}u(t,y)=({%
\mathcal{L}}^{*}u(t,\cdot))(y),\\
\lim\limits_{t\downarrow 0}u(t,y)\mathrm{d}y=\delta_{x}(\mathrm{d}y),\end{%
array}\right.$$
(8)
where ${\mathcal{L}}^{*}$ is adjoint of ${\mathcal{L}}$, i.e.
$$({\mathcal{L}}^{*}\phi)(y):=-\frac{\partial}{\partial y_{j}}\big{(}b_{j}(y)%
\phi(y)\big{)}+\frac{1}{2}\frac{\partial^{2}}{\partial y_{i}\partial y_{j}}%
\big{(}\sigma_{ij}(y)\phi(y)\big{)}+({\mathcal{L}}_{\alpha}\phi)(y)$$
for $\phi\in{\mathcal{C}}^{\infty}_{c}({\mathbb{R}}^{d})$.
Proof.
By similar deduction to that of Lemma 4.2, we get for $\psi\in{\mathcal{C}}^{\infty}_{c}({\mathbb{R}}^{d})$
and any $r>0$
$$\displaystyle\int_{{\mathbb{R}}^{d}}\psi(y)\rho_{X}(t+r,y)\mathrm{d}y-\int_{{%
\mathbb{R}}^{d}}\psi(y)\rho_{X}(t,y)\mathrm{d}y$$
$$\displaystyle=$$
$$\displaystyle\int_{t}^{t+r}\int_{{\mathbb{R}}^{d}}{\langle}\partial_{y}\psi(y)%
,b(y){\rangle}\rho_{X}(s,y)\mathrm{d}y\mathrm{d}s$$
$$\displaystyle+\frac{1}{2}\int_{t}^{t+r}\int_{{\mathbb{R}}^{d}}\left(\frac{%
\partial^{2}}{\partial y_{i}\partial y_{j}}\psi(y)\right)\sigma_{ij}(y)\rho_{X%
}(s,y)\mathrm{d}y\mathrm{d}s$$
$$\displaystyle+\int_{t}^{t+r}\int_{{\mathbb{R}}^{d}}({\mathcal{L}}_{\alpha}\psi%
)(y)\rho_{X}(s,y)\mathrm{d}y\mathrm{d}s.$$
Divided by $r$ and Letting $r\downarrow 0$, it follows from dominated
convergence theorem and integration by parts that
$$\displaystyle\int_{{\mathbb{R}}^{d}}\psi(y)\frac{\partial}{\partial t}\rho_{X}%
(t,y)\mathrm{d}y$$
$$\displaystyle=$$
$$\displaystyle\int_{{\mathbb{R}}^{d}}{\langle}\partial_{y}\psi(y),b(y){\rangle}%
\rho_{X}(t,y)\mathrm{d}y$$
$$\displaystyle+\frac{1}{2}\int_{{\mathbb{R}}^{d}}\left(\frac{\partial^{2}}{%
\partial y_{i}\partial y_{j}}\psi(y)\right)\sigma_{ij}(y)\rho_{X}(t,y)\mathrm{%
d}y$$
$$\displaystyle+\int_{{\mathbb{R}}^{d}}({\mathcal{L}}_{\alpha}\psi)(y)\rho_{X}(t%
,y)\mathrm{d}y$$
$$\displaystyle=$$
$$\displaystyle-\int_{{\mathbb{R}}^{d}}\psi(y)\frac{\partial}{\partial y_{j}}%
\big{(}b_{j}(y)\rho_{X}(t,y)\big{)}\mathrm{d}y$$
$$\displaystyle+\frac{1}{2}\int_{{\mathbb{R}}^{d}}\psi(y)\frac{\partial^{2}}{%
\partial y_{i}\partial y_{j}}\big{(}\sigma_{ij}(y)\rho_{X}(t,y)\big{)}\mathrm{%
d}y$$
$$\displaystyle+\int_{{\mathbb{R}}^{d}}\psi(y)({\mathcal{L}}_{\alpha}\rho_{X}(t,%
\cdot))(y)\mathrm{d}y.$$
Thus
$$\displaystyle\frac{\partial}{\partial t}\rho_{X}(t,y)$$
$$\displaystyle=$$
$$\displaystyle-\frac{\partial}{\partial y_{j}}\big{(}b_{j}(y)\rho_{X}(t,y)\big{%
)}+\frac{1}{2}\frac{\partial^{2}}{\partial y_{i}\partial y_{j}}\big{(}\sigma_{%
ij}(y)\rho_{X}(t,y)\big{)}+({\mathcal{L}}_{\alpha}\rho_{X}(t,\cdot))(y)$$
$$\displaystyle=$$
$$\displaystyle({\mathcal{L}}^{*}\rho_{X}(t,\cdot))(y).$$
∎
Note that Schertzer et. al. [12] also derived the
Fokker-Planck equation in this context.
The main result in this section is the following theorem.
Theorem 4.4.
If $\rho(y)\in{\mathbb{H}}^{2,2}({\mathbb{R}}^{d})$ satisfies the following equation
$$\displaystyle{\mathcal{L}}^{*}\rho=0$$
and
$$\displaystyle\rho(y)\geqslant 0,\forall y\in{\mathbb{R}}^{d}\qquad\mbox{and}%
\qquad\int_{{\mathbb{R}}^{d}}\rho(y)\mathrm{d}y=1.$$
Then $\bar{\mu}(\mathrm{d}y):=\rho(y)\mathrm{d}y$ is a stationary measure for $p_{t}$.
Proof.
By Lemma 4.1 and 4.2, $\{p_{t},t\geqslant 0\}$ is a strongly continuous
semigroup on ${\mathcal{C}}_{0}({\mathbb{R}}^{d})$ with the infinitesimal generator ${\mathcal{L}}$. Let $p_{t}^{*}$
be adjoint of $p_{t}$. By Theorem 2.5,
${\mathbb{H}}^{2,2}({\mathbb{R}}^{d})$ is the closure of
${\mathcal{D}}({\mathcal{L}}^{*})$ in ${\mathcal{M}}_{r}({\mathbb{R}}^{d})$. [9, Theorem 10.4, p.41] admits us to get
that the restriction $p_{t}^{+}$ of $p_{t}^{*}$ to ${\mathbb{H}}^{2,2}({\mathbb{R}}^{d})$ is a strongly
continuous semigroup on ${\mathcal{M}}_{r}({\mathbb{R}}^{d})$. Moreover, the infinitesimal
generator ${\mathcal{L}}^{+}$ of $p_{t}^{+}$ is the part of ${\mathcal{L}}^{*}$ in ${\mathbb{H}}^{2,2}({\mathbb{R}}^{d})$,
i.e. ${\mathcal{D}}({\mathcal{L}}^{+})=\{h\in{\mathcal{D}}({\mathcal{L}}^{*})\cap{%
\mathbb{H}}^{2,2}({\mathbb{R}}^{d}),{\mathcal{L}}^{*}h\in{\mathbb{H}}^{2,2}({%
\mathbb{R}}^{d})\}$
and ${\mathcal{L}}^{+}h={\mathcal{L}}^{*}h$ for $h\in{\mathcal{D}}({\mathcal{L}}^{+})$.
Thus, Eq.(8) has a unique solution in ${\mathbb{H}}^{2,2}({\mathbb{R}}^{d})$ by [9, Theorem 1.3, p.102].
Next, since $\rho(y)$ satisfies
$$\displaystyle{\mathcal{L}}^{*}\rho=0$$
and
$$\displaystyle\rho(y)\geqslant 0,\forall y\in{\mathbb{R}}^{d}\qquad\mbox{and}%
\qquad\int_{{\mathbb{R}}^{d}}\rho(y)\mathrm{d}y=1,$$
$\rho(y)$ solves Eq.(8) and is a density function. So,
$$\displaystyle\rho_{X}(t,y)=\rho(y),\quad t>0.$$
For $t>0$ and $B\in{\mathcal{B}}({\mathbb{R}}^{d})$,
$$\displaystyle\int_{{\mathbb{R}}^{d}}p_{t}(x,B)\bar{\mu}(\mathrm{d}x)$$
$$\displaystyle=$$
$$\displaystyle\int_{{\mathbb{R}}^{d}}p_{t}(x,B)\rho(x)\mathrm{d}x=\int_{{%
\mathbb{R}}^{d}}p_{t}(x,B)\rho_{X}(s,x)\mathrm{d}x$$
$$\displaystyle=$$
$$\displaystyle\int_{{\mathbb{R}}^{d}}\int_{B}\rho_{t}(x,y)\rho_{X}(s,x)\mathrm{%
d}y\mathrm{d}x$$
$$\displaystyle=$$
$$\displaystyle\int_{B}\int_{{\mathbb{R}}^{d}}\rho_{t}(x,y)\rho_{X}(s,x)\mathrm{%
d}x\mathrm{d}y$$
$$\displaystyle=$$
$$\displaystyle\int_{B}\rho_{X}(s+t,y)\mathrm{d}y$$
$$\displaystyle=$$
$$\displaystyle\int_{B}\rho(y)\mathrm{d}y=\bar{\mu}(B).$$
By Definition 3.2, $\bar{\mu}(\mathrm{d}y)$ is a stationary measure
for $p_{t}$.
∎
Remark 4.5.
If $b(x)=-x$ and $\sigma(x)=0$, the above theorem is [1, Proposition 3.2(ii)].
Moreover,
$$\hat{\rho}(u)=\exp\{-\frac{1}{\alpha}C|u|^{\alpha}\},\quad u\in{\mathbb{R}}^{d},$$
where $C$ is the same constant as one in Definition 2.4.
5. From probability density functions to stationary measures
In the section we study Eq.(5) with $g=0$. We further make the following assumptions.
(H${}^{1}_{b,\sigma,f}$) $b$ and $\sigma$ are $(4d+6)$-times differentiable with bounded
derivatives of all order between $1$ and $4d+6$. Besides, $f(\cdot,u)$
is $(4d+6)$-times differentiable, and
$$\displaystyle f(0,\cdot)$$
$$\displaystyle\in$$
$$\displaystyle\bigcap_{2\leqslant q<\infty}L^{q}({\mathbb{U}}_{0},n)$$
$$\displaystyle\sup\limits_{x}|\partial^{r}_{x}f(x,\cdot)|$$
$$\displaystyle\in$$
$$\displaystyle\bigcap_{2\leqslant q<\infty}L^{q}({\mathbb{U}}_{0},n),\quad 1%
\leqslant r\leqslant 4d+6,$$
where the space $({\mathbb{U}}_{0},n)$ is equipped with a norm and
$\partial^{r}_{x}f(x,\cdot)$ stands for $r$ order partial derivative of $f(x,\cdot)$
with respect to $x$.
(H${}^{2}_{b,\sigma,f}$) There exist three constants $\varepsilon>0$, $\delta\geqslant 0$ and $C>0$ such that
for all $x,y\in{\mathbb{R}}^{d}$
$$\displaystyle{\langle}y,\sigma(x)\sigma^{T}(x)y{\rangle}\geqslant|y|^{2}\frac{%
\varepsilon}{1+|x|^{\delta}}$$
and
$$\displaystyle|\det\{I+r\partial_{x}f(x,u)\}|\geqslant C$$
for all $r\in[0,1]$.
Under (H${}^{1}_{b,\sigma,f}$) and (H${}^{2}_{b,\sigma,f}$), by
[4, Theorem 2-29, p.15], Eq.(5) has a unique solution
denoted by $X_{t}$. Moreover, the transition probability $p_{t}(x,\mathrm{d}y)$
has a density $\rho_{t}(x,y)$ and $(t,x,y)\mapsto\rho_{t}(x,y)$ is continuous.
Thus, the distribution of $X_{t}$ has a density $\rho_{X}(t,y)=\rho_{t}(x,y)$.
Theorem 5.1.
Suppose that $\lim\limits_{t\rightarrow\infty}\rho_{X}(t,y)=\rho(y)$, where
$\rho(y)$ satisfies
$$\displaystyle\rho(y)\geqslant 0,\forall y\in{\mathbb{R}}^{d}\qquad\mbox{and}%
\qquad\int_{{\mathbb{R}}^{d}}\rho(y)\mathrm{d}y=1.$$
Then $\bar{\mu}(\mathrm{d}y):=\rho(y)\mathrm{d}y$ is a stationary measure for $p_{t}$.
Proof.
For $t>0$ and $B\in{\mathcal{B}}({\mathbb{R}}^{d})$,
$$\displaystyle\int_{{\mathbb{R}}^{d}}p_{t}(x,B)\bar{\mu}(\mathrm{d}x)$$
$$\displaystyle=$$
$$\displaystyle\int_{{\mathbb{R}}^{d}}p_{t}(x,B)\rho(x)\mathrm{d}x=\int_{{%
\mathbb{R}}^{d}}p_{t}(x,B)\lim\limits_{s\rightarrow\infty}\rho_{X}(s,x)\mathrm%
{d}x$$
$$\displaystyle=$$
$$\displaystyle\lim\limits_{s\rightarrow\infty}\int_{{\mathbb{R}}^{d}}p_{t}(x,B)%
\rho_{X}(s,x)\mathrm{d}x=\lim\limits_{s\rightarrow\infty}\int_{{\mathbb{R}}^{d%
}}\int_{B}\rho_{t}(x,y)\rho_{X}(s,x)\mathrm{d}y\mathrm{d}x$$
$$\displaystyle=$$
$$\displaystyle\lim\limits_{s\rightarrow\infty}\int_{B}\int_{{\mathbb{R}}^{d}}%
\rho_{t}(x,y)\rho_{X}(s,x)\mathrm{d}x\mathrm{d}y$$
$$\displaystyle=$$
$$\displaystyle\lim\limits_{s\rightarrow\infty}\int_{B}\rho_{X}(s+t,y)\mathrm{d}y$$
$$\displaystyle=$$
$$\displaystyle\bar{\mu}(B).$$
By Definition 3.2, $\bar{\mu}(\mathrm{d}y)$ is a stationary measure
for $p_{t}$.
∎
Remark 5.2.
By the above theorem, we see that if a limiting distribution exists, it must be a
stationary measure. This theorem also has a corresponding version in the theory of
Markov chains (c.f. [8, p.237]).
References
[1]
S. Albeverio, B. Rüdiger and J. Wu: Invariant measures
and symmetry property of Lévy type operators, Potential Analysis,
13(2000)147-168.
[2]
D. Applebaum: Lévy Processes and Stochastic Calculus,
Second Edition, Cambridge Univ. Press, Cambridge, 2009.
[3]
L. Arnold: Random Dynamical Systems, Springer-Verlag
Berlin Heidelberg New York, 1998.
[4]
K. Bichteler, J. B. Gravereaux and J. Jacod: Malliavin Calculus for
Processes with Jumps, Stochastic Monographs Volume 2, Gordon and
Breach Science Publishers, 1987.
[5]
H. Crauel: Markov measures for random dynamical systems,
Stochastics and Stochastics Reports, 37(1991)153-173.
[6]
S. He, J. Wang and J. Yan: Semimartingale Theory and
Stochastic Calculus, Science Press and CRC Press Inc., 1992.
[7]
N. Ikeda and S. Watanabe: Stochastic Differential Equations
and Diffusion Processes, 2nd ed., North-Holland/Kodanska,
Amsterdam/Tokyo, 1989.
[8]
S. P. Meyn and R.L. Tweedie: Markov Chains and Stochastic
Stability, Cambridge University Press, 2nd Edition, 2008.
[9]
A. Pazy: Semigroups of Linear Operators and Applications to
Partial Differential Equations, Springer-Verlag Berlin Heidelberg
New York, 1983.
[10]
H. Qiao and X. Zhang: Homeomorphism flows for non-Lipschitz
stochastic differential equations with jumps, Stochastic Processes and their
Applications, 118(2008)2254-2268.
[11]
K. Sato: Lévy Processes and Infinitely Divisible
Distributions, Cambridge University Press, 1999.
[12]
D. Schertzer, M. Larcheveque, J. Duan, V. Yanovsky and S.
Lovejoy, Fractional Fokker–Planck equation for nonlinear stochastic
differential equations driven by non-Gaussian Levy stable noises.
J. Math. Phys., 42(2001)200-212.
[13]
C. Soize: The Fokker-Planck Equation for Stochastic Dynamical Systems
and Its Explicit Steady State Solutions, World Scientific, Singapore New Jersey
London Hong Kong, 1994.
[14]
J. Yan: Lectures on Measure Theory (Chinese), Science Press, 2004.
[15]
J. Ying: Invariant measures of symmetric Lévy processes,
Proceedings of the American Mathematical Society, 120(1994)267-273.
[16]
K. Yosida: Functional Analysis, Springer-Verlag Berlin Heidelberg New York,
6th Edition, 1980.
[17]
J. Zabczyk: Stationary distribution for linear equations driven by
general noise, Bulletin of The Polish Academy of Sciences, Mathematics,
31(1983)197-209. |
Solving The Vehicle Routing Problem via Quantum Support Vector Machines
1stNishikanta Mohanty,
Centre for Quantum Software and Information,
University of Technology Sydney, Ultimo, Sydney 2007, NSW, Australia
Nishikanta.M.Mohanty@student.uts.edu.au
2nd Bikash K. Behera,
Bikash’s Quantum (OPC) Pvt. Ltd., Mohanpur 741246, WB, India
bikas.riki@gmail.com
3rd Christopher Ferrie
Centre for Quantum Software and Information,
University of Technology Sydney, Ultimo, Sydney 2007, NSW, Australia
Christopher.Ferrie@uts.edu.au
Abstract
The Vehicle Routing Problem (VRP) is an example of a combinatorial optimization problem that has attracted academic attention due to its potential use in various contexts. VRP aims to arrange vehicle deliveries to several sites in the most efficient and economical manner possible. Quantum machine learning offers a new way to obtain solutions by harnessing the natural speedups of quantum effects, although many solutions and methodologies are modified using classical tools to provide excellent approximations of the VRP. In this paper, we implement and test hybrid quantum machine learning methods for solving VRP of $3$ and $4$-city scenarios, which use $6$ and $12$ qubit circuits, respectively. The method is based on quantum support vector machines (QSVMs) with a variational quantum eigensolver on a fixed or variable ansatz. Different encoding strategies are used in the experiment to transform the VRP formulation into a QSVM and solve it. Multiple optimizers from the IBM Qiskit framework are also evaluated and compared.
Index Terms:
Vehicle Routing Problem, Ising Model, Variational Quantum Eigensolver, Quantum Encoding, Quantum Support Vector Machine, Parameterized Circuit
I Introduction
I-A Quantum Computing
Quantum computing has provided novel approaches for solving computationally complex problems over the last decade by leveraging the inherent speedup(s) of quantum calculations compared to classical computing. Quantum superposition and entanglement are two key factors that give a massive speed up to calculations in the quantum domain compared to classical counterparts [1, 2, 3]. Because of this, addressing Optimization problems by quantum computing is an appealing prospect. Multiple approaches, such as Grover’s algorithm [4], adiabatic computation (AC) [5], and quantum approximate optimization algorithm (QAOA) [6], have been proposed to use quantum effects and, as such, have served as the basis for solving mathematically complex problems using quantum computing. The performance of classical algorithms has generally been found to be subpar when applied to larger dimensional problem spaces [7]. On a multidimensional problem, classical machine learning optimization techniques frequently require a significant amount of CPU and GPU resources and take a long time to compute. The reason for this is because ML techniques are needed to resolve NP-hard optimization problems [8].
I-B Vehicle Routing Problem
The vehicle routing problem is an intriguing optimization problem because of its many uses in routing and fleet management [9], but its computational complexity is NP-hard [10, 11]. Moving automobiles as quickly and cheaply as feasible is always the objective. VRP has inspired a plethora of precise and heuristic approaches [9, 12], all of which struggle to provide fast and trustworthy solutions. The VRP’s bare bones implementation comprises sending a single vehicle to deliver items to many client locations before returning to the depot to restock [13]. By optimizing a collection of routes that are available and all begin and terminate at a single node called the depot, VRP seeks to maximize the reward, which is often the inverse of the total distance traveled or the average service time. It is computationally difficult to find an optimum solution to this issue, even with just a few hundred customer nodes.
Explicitly, in every VRP ($n,k$), there are ($n-1$) stations, $k$ vehicles, and a depot D [14, 9]. The solution is a collection of paths whereby each vehicle takes exactly one journey, and all $k$ vehicles start and conclude at the same location, $D$.
The best route is one that requires $k$ vehicles to drive the fewest total miles. This problem may be thought of as a generalization of the well-known “traveling salesman” problem, whereby a group of $k$ salesmen must service an aggregate of ($n-1$) sites with a single visit to each of those places being guaranteed [9]. In most practical settings, the VRP issue is complicated by other constraints, such as limited vehicle capacity or limited time for coverage. As a consequence, several other approaches, both classical and quantum, have been proposed as potential ways forwards. Currently, available quantum approaches for optimizing a system include the Quantum Approximate Optimization Algorithm (QAOA) [14], the Quadratic Unconstrained Binary Optimization (QUBO) [15], and quantum annealing [16, 17, 18].
I-C Quantum Support Vector Machine(QSVM)
The goal of the support vector machine (SVM) technique is to find the best line (or decision boundary) between two classes in $n$-dimensional space so that new data may be classified quickly. This optimum decision boundary is referred to as a hyperplane. The most extreme vectors and points that help construct the hyperplane are selected using SVM. The SVM method is based on support vectors, which are used to represent these extreme instances. Typically, a hyperplane cannot divide a data point in its original space. In order to find this hyperplane, a nonlinear transformation is applied to the data as a function. A feature map is a function that transforms the features of provided data into the inner product of data points, also known as the kernel [19, 20, 21].
Quantum computing produces implicit calculations in high-dimensional Hilbert spaces using kernel techniques by physically manipulating quantum systems. Feature vectors for SVM in the quantum realm are represented by density operators, which are themselves encodings of quantum states. The kernel of a quantum support vector machine (QSVM) is made up of the fidelities between different feature vectors, as opposed to a classical SVM; the kernel conducts an encoding of classical input into quantum states [19, 22].
I-D Novelty and Contribution
•
In this work, we propose a new method to solve the VRP using a machine-learning approach through the use of QSVM.
•
In this context, we came across recent and older works in QSVM [21, 23, 20] and VQE algorithms [24], which are used to solve optimization problems such as VRP. However, none of them use a hybrid approach to arrive at a solution.
•
Our work implements this new approach of solving VRP in a detailed gate-based simulation of a $3$-city or $4$-city problem on a 6-qubit or 12-qubit system, respectively, using a parameterized circuit that is developed as a solution to VRP.
•
We apply quantum encoding techniques such as amplitude encoding, angle encoding, higher order encoding, IQP Encoding, and quantum algorithms such as QSVM, VQE, and QAOA to construct circuits for VRP and analyze the effects and consolidate our findings.
•
We evaluate our solution using a variety of classical optimizers, as well as fixed and variable Hamiltonians to draw statistical conclusions.
I-E Organization
The paper is organized as follows. Sec. II discusses the fundamental mathematical concepts such as QAOA, the Ising model, quantum support vector machine, Amplitude encoding, Angle encoding, Higher order encoding, IQP encoding, and VQE. Sec. III discusses the formulation and solution of VRP using the concepts discussed in the previous Section. Sub-Sec. III-B covers the basic building blocks of circuits to solve VRP using QSVM. Sec. IV covers the outcomes of the QSVM simulation consisting of two sub-sections. Sub Sec IV-A covers the outcome of simulation results of all the encoding schemes used, Finally in Sub Sec. IV-B, we conclude by comparing the results of QSVM solutions using various optimizers in the Qiskit platform on the VRP circuit and discuss the feasibility of higher qubit solutions as the future directions of research.
II Background
Dealing with methods and processes for resolving combinatorial optimization problems is the foundation of solving routing challenges. The objective function is then created by transforming the mathematical models into a quantum equivalent mathematical model. By maximizing or minimizing the mathematical model iteratively, we arrive at the solution of the objective function. We list the main ideas in this section for our solution approach.
II-A QAOA
A variational approach called the Quantum Approximate Optimization Algorithm (QAOA) was put forth by Farhi et al. in 2014 [5, 6] using adiabatic quantum computation framework as the foundation of this algorithm. It is a hybrid algorithm since it applies both classical and quantum approaches. Simply described, quantum adiabatic computation involves switching from the eigenstate of the driver Hamiltonian to that of the problem Hamiltonian. The problem Hamiltonian can be expressed as,
$$\displaystyle C|z\rangle=\sum^{m}_{\alpha=1}C_{\alpha}|z\rangle.$$
(1)
We are aware that the combinatorial optimization problem is resolved by finding the highest energy eigenstate of C. Similarly, we employ driver Hamiltonian as
$$\displaystyle B=\sum^{n}_{j=1}{\ }{\sigma}^{x}_{j},$$
(2)
where ${\sigma}^{x}_{j}$ represents the ${\sigma}^{x}$ Pauli operator on bit $z_{j}$ and $B$ is the mixing operator. Let’s additionally define $U_{C}\left(\gamma\right){=}e^{{-}i\gamma C}$ and $U_{B}\left(\beta\right){=}e^{{-}i\beta{B}}$ which allow the system to evolve under C for $\gamma$ time and under B for $\beta$ time, respectively. Essentially, QAOA creates a state
$$|\boldsymbol{\beta},\boldsymbol{\gamma}\rangle=e^{-i{\beta}_{p}B}e^{-i{\gamma}_{p}{C}}\cdots e^{-i{\beta}_{2}B}e^{-i{\gamma}_{2}{C}}e^{-i{\beta}_{1}{B}}e^{-i{\gamma}_{1}{C}}|s\rangle,$$
(3)
where $|s\rangle$ denotes the superposition state of all input qubits The expectation value of the cost function $\sum^{m}_{\alpha{=1}}{{\langle}\beta{,}\gamma~{}{|C_{\alpha}}~{}{|}\beta{,}\gamma~{}{\rangle}}$ gives the solution, or an approximate solution to the problem [25].
II-B Ising Model
In statistical mechanics, the Ising model is a well-known mathematical depiction of ferromagnetism[26, 27]. In the model, discrete variables ($+1$ or $-1$) represent the magnetic dipole moments of ”spins” in one of two states. Because the spins are organized in a network, commonly a lattice(when there is periodic repetition in all directions of the local structure), each spin can interact with its neighbors. The spins interact in pairs, with an energy that has one value when the two spins are identical and a second value when they are dissimilar. Nevertheless, heat reverses this tendency, enabling alternate structural phases to arise. The model is a condensed representation of reality that enables the recognition of phase transitions. The following Hamiltonian explains the total spin energy:
$$\displaystyle H_{c}=-\sum_{\left\langle i,j\right\rangle}{\ }J_{ij}{{\sigma}}_{{i}}{{\sigma}}_{{j}}-h\sum{{\sigma}}_{{i}},$$
(4)
where $J_{ij}$ represents the interaction of adjacent spins $i$ and $j$, and $h$ represents an external magnetic field. The ground state at $h=0$ is a ferromagnet if $J$ is positive. If $J$ is negative, the ground-state at $h=0$ is an anti-ferromagnet for a bipartite lattice. As a result, for the sake of simplicity and in the context of this document, we can write the Hamiltonian as
$$\displaystyle H_{c}=-\sum_{\langle i,j\rangle}{\ }J_{ij}{\sigma}_{i}^{z}{\sigma}_{j}^{z}-\sum h_{i}{\sigma}_{i}^{x}.$$
(5)
Here ${\sigma}_{z}$ and ${\sigma}_{x}$ represent Pauli $z$ and $x$ operator. For the sake of simplification, we can assume the following conditions to be ferromagnetic ($J_{ij}>0$) if there is no external impact on the spin: $h=0$. Hence, the Hamiltonian may be rewritten as follows:
$$\displaystyle H_{c}=-\sum_{\langle i,j\rangle}{\ }J_{ij}{\sigma}_{i}^{z}{\sigma}_{j}^{z}=-\sum_{\langle i,j\rangle}{\ }\sigma_{i}^{z}\sigma_{j}^{z}.$$
(6)
II-C Quantum Support Vector Machine
SVM [20, 21] is a supervised algorithm that constructs hyper-plane with $\vec{w}\cdot\vec{x}+b=0$ such that $\vec{w}\cdot\vec{x}+b\geq 1$ for a training point $\vec{x}_{i}$ in the positive class, and $\vec{w}\cdot\vec{x}+b\leq-1$ for a training point $\vec{x}_{i}$ in the negative class. During the training process, the algorithm aims to maximize the gap between the two classes, which is intuitive as we want to separate two classes as far as possible, in order to get a sharper estimate for the classification result of new data samples like $\vec{x_{0}}$. Mathematically we can see the objective of SVM is to find a hyper-plane that maximizes the distance $2/|\vec{w}|$ constraint to $\vec{y_{i}}(\vec{w}\cdot\vec{x_{i}}+b)\geq 1$. The normal vector $\vec{w}$ can be written as $\vec{w}=\sum_{i=1}^{M}\alpha_{i}\vec{x}_{i}$ where $\alpha_{i}$ is the weight of the $i^{th}$ training vector $\vec{x}_{i}$. Thus, obtaining optimal parameters b and $\alpha_{i}$ is the same as finding the optimal hyper-plane. To classify the new vector, is analogous to knowing which side of the hyper-plane it lies, i.e., $y_{i}(\vec{x}_{0})=sign(\vec{w}.\vec{x}+b)$. After having the optimal parameters, classification now becomes a linear operation. From the least-squares approximation of SVM, the optimal parameters can be obtained by solving a linear equation,
$$\displaystyle\vec{F}(b,\alpha_{1},\alpha_{2},\alpha_{3},...,\alpha_{M})^{T}=(0,y_{1},y_{2},y_{3},...y_{M})^{T}.$$
(7)
In a general form of F we adopt the linear kernels $K_{i,j}=\kappa(\vec{x}_{i},\vec{x}_{j})=\vec{x}_{i}.\vec{x}_{j}$. Thus to find the hyper-plane parameters we use matrix inversion of F : $(b,{\vec{\alpha}_{i}}^{T})^{T}={\tilde{F}^{-1}}(0,{\vec{y}_{i}}^{T})^{T}$.
II-C1 Quantum Kernels
The main inspiration of a quantum Support vector machine is to consider quantum feature maps that lead to quantum kernel functions, which are hard to simulate in classical computers. In this case, the quantum computer is only used to estimate a quantum kernel function, which can be later used in kernel-based algorithms. For simplicity assuming the datapoints $x,z\in\mathcal{X}$, the nonlinear feature map of any datapoint is
$$\displaystyle\Phi(\boldsymbol{x})=U(\boldsymbol{x})\left|0^{n}\right\rangle\left\langle 0^{n}\right|U^{\dagger}(\boldsymbol{x}).$$
(8)
The kernel function $\kappa(x,z)$ can be computed as
$$\displaystyle\kappa(x,y)=|\langle\phi(x)\mid\phi(z)\rangle|^{2}.$$
(9)
The state $|\phi(x)\rangle$ can be prepared by using a unitary gate $U(x)$, and thus $|\phi(x)\rangle=U(x)|0\rangle$.Thus the kernel fuction becomes ,
$$\displaystyle\kappa(x,z)=\left|\left\langle 0\left|U^{\dagger}(x)U(z)\right|0\right\rangle\right|^{2}.$$
(10)
From the above we can say that the kernel $\kappa(x,z)$ is simply the probability of getting an all-zero string when the circuit $U^{\dagger}(x)U(z)|0\rangle$ is measured or this kernel is an $\left|0^{n}\right\rangle$ to $\left|0^{n}\right\rangle$ transition probability of a particular unitary quantum circuit on $n$ qubits [19, 28]. This can be implemented using the following kernel estimation circuit (Fig. 1).
II-D Amplitude Encoding(AE)
In the process of amplitude-embedding [29], data is encoded into the amplitudes of a quantum state. A N-dimensional classical datapoint $x$ is represented by the amplitudes of an n-qubit quantum state $\ket{\psi_{x}}$ as
$$\left|\psi_{x}\right\rangle=\sum_{i=1}^{N}x_{i}|i\rangle$$
(11)
where $N=2^{n}$,
$x_{i}$ is the $i$-th element of $x$
and $|i\rangle$ is the $i$-th computational basis state.
In order to encode any data point $x$ into an amplitude-encoded state, we must normalize the same by following
$$\ket{\psi_{x_{norm}}}=\frac{1}{x_{norm}}\sum_{i=1}^{N}x_{i}|i\rangle,$$
(12)
where $x_{norm}$= $\sqrt{\sum_{i=1}^{N}|x_{i}|^{2}}$ .
II-E Angle Encoding(AgE)
While the above-described amplitude encoding expands into a complicated quantum circuit with huge depths, the angle encoding employs N qubits and a quantum circuit with fixed depth, making it favorable to NISQ computers [30, 31]. We define angle encoding as a method of classical information encoding that employs rotation gates(the rotation could be chosen along $x$, $y$ or $z$ axis). In our scenario, the classical information consists of the node and edge weights assigned to the vehicle’s nodes and pathways which are further assigned as parameters to ansatz.
$$|\mathbf{x}\rangle=\bigotimes_{i}^{n}R\left(\mathbf{x}_{i}\right)\left|0^{n}\right\rangle,$$
(13)
where ${x}_{i}$ represents the classical information stored on the angle parameter of rotation operator $R$.
II-F Higher Order Encoding(HO)
Higher order encoding is a variation of angle encoding where we have an entangled layer and an additional sequential operation of rotation angles of two entangled qubits [31]. This can be loosely defined as the following
$$|\mathbf{x}\rangle=\bigotimes_{i=2}^{n}R_{(}x_{i-1}.x_{i})\bigotimes_{i=2}^{n-1}CX_{i,i+1}\bigotimes_{i=1}^{n}R\left(x_{i}\right)\left|0^{n}\right\rangle.$$
(14)
Similar to angle encoding we are free to chose the rotation.
II-G IQP Encoding(IqpE)
IQP-style encoding is a relatively complicated encoding strategy. We encode classical information [32]
$$|{x}\rangle=\left(\mathrm{U}_{\mathrm{Z}}({x})\mathrm{H}^{\otimes n}\right)^{r}\left|0^{n}\right\rangle,$$
(15)
where $r$ is the depth of the circuit, indicating the repeating times of $\mathrm{U}_{\mathrm{Z}}({x})\mathrm{H}^{\otimes n}$. $\mathrm{H}^{\otimes n}$ is a layer of Hadamard gates acting on all qubits. $\mathrm{U}_{\mathrm{Z}}(\mathbf{x})$ is the key step in IQP encoding scheme:
$$\displaystyle\mathrm{U}_{\mathrm{Z}}(\mathrm{x})=\prod_{[i,j]\in S}R_{Z_{t}Z_{j}}\left(x_{i}x_{j}\right)\bigotimes_{k=1}^{n}R_{z}\left(x_{k}\right),$$
(16)
where $S$ is the set containing all pairs of qubits to be entangled using $R_{ZZ}$ gates.
First, we consider a simple two-qubit gate: $R_{Z_{1}Z_{2}}(\theta)$. Its mathematical form $e^{-i\frac{i}{2}Z_{1}\otimes Z_{2}}$ can be seen as a two-qubit rotation gate around $ZZ$, which makes these two qubits entangled.
II-H VQE
Another hybrid quantum classical algorithm is the Variational Quantum Eigensolver (VQE), which is used to estimate the eigenvalue of a large matrix or Hamiltonian $H$ [33]. The basic goal of this method is to find a trial qubit state of a wave function $\ket{\psi(\vec{\theta})}$ that is dependent on a parameter set $\vec{\theta}=\theta_{1},\theta_{2},\cdots$, which is also known as the variational parameters. The expectation of an observable or Hamiltonian $H$ in a state $\ket{\psi(\vec{\theta})}$ can be expressed in quantum theory as,
$$\displaystyle E(\vec{\theta})=\bra{\psi(\vec{\theta})}H\ket{\psi(\vec{\theta})}.$$
(17)
By spectral decomposition
$$\displaystyle H=\lambda_{1}\ket{\psi}_{1}\bra{\psi}_{1}+\lambda_{2}\ket{\psi}_{2}\bra{\psi}_{2}+\ldots+\lambda_{n}\ket{\psi}_{n}\bra{\psi}_{n},$$
where $\lambda_{i}$ and ${\ket{\psi}}_{i}$ are the eigenvalues and eigenstates,
respectively, of matrix $H$. Also, because the eigenstates of $H$ are orthogonal, $\left\langle\psi_{i}\mid\psi_{j}\right\rangle=0$ If $i\neq j$ . The wave function $\ket{\psi(\vec{\theta})}$ can be expressed as a superposition of eigenstates.
$$\displaystyle\ket{\psi(\vec{\theta})}=\alpha_{1}(\vec{\theta})\ket{\psi}_{1}+\alpha_{2}(\vec{\theta})\ket{\psi}_{2}+\ldots+\alpha_{n}(\vec{\theta})\ket{\psi}_{n}.$$
(19)
Thus the expectation becomes
$$\displaystyle E(\vec{\theta})$$
$$\displaystyle=$$
$$\displaystyle|\alpha_{1}(\vec{\theta})|^{2}\lambda_{1}+|\alpha_{2}(\vec{\theta})|^{2}\lambda_{2}+\ldots+|\alpha_{n}(\vec{\theta})|^{2}\lambda_{n}.$$
Clearly, $E(\vec{\theta})\geq\lambda_{\min}$.
So in VQE algorithm, we vary the parameters $\vec{\theta}=\theta_{1},\theta_{2},\ldots$ until $E(\vec{\theta})$ is minimized. This property of VQE is useful when attempting to solve combinatorial optimization problems namely those in which a parameterized circuit is used to set up the trial state of the algorithm, and $E(\vec{\theta})$ is referred to as the cost function, that is also the expected value of the Hamiltonian in this state. The ground state of the desired Hamiltonian may be obtained by iterative minimization of the cost function.The optimization process utilizes a classical optimizer which uses quantum computer to evaluate the cost function and calculate its gradient at each optimization step.
III Methodology
III-A Modelling VRP in QSVM
The vehicle routing problem can be solved by mapping the cost function to an Ising Hamiltonian $H_{c}$ [34]. The answer to the problem is given by minimizing the Ising Hamiltonian $H_{c}$. Consider an arbitrarily connected graph with $n$ vertices and $n-1$ edges. Assuming we need to route a vehicle between two non-adjacent vertices in the graph; Consider a binary decision variable $x_{ij}$ whose value is $1$ if there is an edge between $i$ and $j$ with an edge weight $w_{ij}>0$; otherwise, its value is $0$. Now, the VRP problem requires $n\times(n-1)$ choice variables. We define two sets of nodes for each edge from $i\rightarrow j$: $source\left[i\right]$ and $target[j]$. $source\left[i\right]$ contains the nodes $j$ to which $i$ sends an edge $j\ \epsilon\ source[i]$. The collection $target\left[j\right]$ comprises the nodes $i$ to which the node $i$ delivers the edge $i\ \epsilon\ target[j]$. The VRP is defined as follows[14, 35]:
$$\displaystyle VRP(n,k)=\mathop{min}_{{\left\{x_{ij}\right\}}_{i\to j}\in\{0,1\}}\ \sum_{i\to j}{\ }w_{ij}x_{ij},$$
(21)
where $k$ is the number of vehicles, and $n$ is the total number of locations, we have $n-1$ locations for vehicles to traverse if we consider the starting place to be the $0th$ location or Depot $D$. Noticeably, this is subject to the following restrictions[11]:
$$\displaystyle\sum_{j\in~{}{source}~{}[i]}{\ }x_{ij}$$
$$\displaystyle=$$
$$\displaystyle 1,{\forall}i\in\{1,\cdots,n-1\},$$
$$\displaystyle\sum_{j\in~{}{target}~{}[i]}{\ }x_{ji}$$
$$\displaystyle=$$
$$\displaystyle 1,{\forall}i\in\{1,\cdots,n-1\},$$
$$\displaystyle\sum_{{j}{\in}~{}{source}~{}{[}{0}{]}}{{\ }}{x}_{0j}$$
$$\displaystyle=$$
$$\displaystyle k,$$
$$\displaystyle\sum_{j\in~{}{target}~{}[0]}{\ }x_{j0}$$
$$\displaystyle=$$
$$\displaystyle k$$
$$\displaystyle u_{i}-u_{j}+Qx_{ij}$$
$$\displaystyle\leq$$
$$\displaystyle Q-q_{j},\forall i\sim j,i,j\neq 0,$$
$$\displaystyle q_{i}\leq u_{i}$$
$$\displaystyle\leq$$
$$\displaystyle Q,\forall i,i\neq 0.$$
(22)
The first two restrictions establish the limitation that the delivering vehicle may only visit each node once. After delivering the products, the middle two limitations enforce the requirement that the vehicle must return to the depot.
The last two constraints impose the sub-tour elimination conditions and are bound on $u_{i}$, with $Q>q_{j}>0$, and $u_{i},Q,q_{i}\in\mathbb{R}$. For the VRP equation and restrictions, the Hamiltonian of VRP can be expressed as follows [14].
$$\displaystyle H_{VRP}$$
$$\displaystyle=$$
$$\displaystyle H_{A}+H_{B}+H_{C}+H_{D}+H_{E},$$
$$\displaystyle H_{A}$$
$$\displaystyle=$$
$$\displaystyle~{}\sum_{i~{}\to j}{w_{ij}x_{ij}},$$
$$\displaystyle H_{B}$$
$$\displaystyle=$$
$$\displaystyle A\sum_{i\in 1,\cdots,n-1}{\ }{\left(1-\sum_{j\in~{}{source}~{}[i]}{\ }x_{ij}\right)}^{2},$$
$$\displaystyle H_{C}$$
$$\displaystyle=$$
$$\displaystyle A\sum_{i\in 1,\cdots,n-1}{\ }{\left(1-\sum_{j\in~{}{target}[i]}{\ }x_{ji}\right)}^{2},$$
$$\displaystyle H_{D}$$
$$\displaystyle=$$
$$\displaystyle A{\left(k-\sum_{j\in~{}{source}[0]}{\ }x_{0j}\right)}^{2},$$
$$\displaystyle H_{E}$$
$$\displaystyle=$$
$$\displaystyle A{\left(k-\sum_{j\in~{}{target}[0]}{\ }x_{j0}\right)}^{2}.$$
(23)
$A>0$ represents a constant. In vector form, the collection of all binary decision variables $x_{ij}$ can be written as
$$\displaystyle\overrightarrow{\boldsymbol{{x}}}={\left[x_{(0,1)},x_{(0,2)},\cdots x_{(1,0)},x_{(1,2)},\cdots x_{(n-1,n-2)}\right]}^{\boldsymbol{{T}}}.$$
(24)
Using the preceding vector, we can build two new vectors for each node: $\overrightarrow{z}_{S[i]}$ and $\overrightarrow{z}_{T[i]}$ (in the beginning of the section, we defined two sets for source and target nodes, thus two vectors will represent them).
$$\displaystyle\overrightarrow{z}_{S\left[i\right]}$$
$$\displaystyle=$$
$$\displaystyle\vec{x}\ni x_{ij}=1,\ x_{kj}=0\ ,\ k\neq i\ ,\ \ \forall j,k\ \in\{0,\cdots,n-1\},$$
$$\displaystyle\overrightarrow{z}_{T\left[i\right]}$$
$$\displaystyle=$$
$$\displaystyle\vec{x}\ni x_{ji}=1,\ x_{jk}=0\ ,\ k\neq i\ ,\ \ \forall j,k\ \in\{0,\cdots,n-1\}.$$
$$\displaystyle\sum_{j\in\text{ source }[i]}x_{ij}$$
$$\displaystyle=$$
$$\displaystyle\vec{z}_{S[i]}^{\mathrm{T}}\vec{x},$$
$$\displaystyle\sum_{j\in\text{ target }[i]}x_{ji}$$
$$\displaystyle=$$
$$\displaystyle\vec{z}_{T[i]}^{\mathrm{T}}\vec{x}.$$
(26)
The aforementioned vectors will aid in the development of the QUBO model of VRP [36, 15, 37, 38]. In general, the QUBO model of a connected graph $G=(N,V)$ is specified as follows:
$$\displaystyle f(x)_{QUBO}=\mathop{min}_{x\in\{0,1\}(N\times V)}x^{T}Qx+g^{T}x+c,$$
(27)
where, $Q$ is a quadratic edge weight coefficient, $g$ is a linear node weight coefficient, and $c$ is a constant. In order to find these coefficients in the QUBO formations of $H_{VRP}$ given in Eq. 23 we first put in Eqs. 26 in terms $H_{B}$ and $H_{c}$respectively, then expand and regroup the expression of $H_{VRP}$ according to Eq. 27
$$\displaystyle H$$
$$\displaystyle=$$
$$\displaystyle A\sum_{i=0}^{n-1}\left[z_{S[i]}z_{S[i]}^{T}+z_{T[i]}z_{T[i]}^{T}\right]\vec{x}^{2}$$
$$\displaystyle+$$
$$\displaystyle w^{T}\vec{x}-2A\sum_{i=1}^{n-1}\left[z_{S[i]}^{T}+z_{T[i]}^{T}\right]\vec{x}$$
$$\displaystyle-$$
$$\displaystyle 2Ak\left[z_{S[0]}^{T}+z_{T[0]}^{T}\right]\vec{x}+2A(n-1)+2Ak^{2}.$$
Hence for QUBO formulation of Eq. (23) we get the coefficients $\mathrm{Q}(n(n-1)\times$ $n(n-1)),\mathrm{g}(n(n-1)\times 1)$ and $\mathrm{c}$ : The coefficients for the QUBO formulation of Eq. (23) are therefore as follows:
$$\displaystyle Q$$
$$\displaystyle=$$
$$\displaystyle A\left[\left[z_{T[0]},\ldots,z_{T[n-1]}\right]^{T}\left[z_{T[0]},\ldots,z_{T[n-1]}\right]\right.$$
$$\displaystyle\left.+\left(\mathbb{I}_{n}\otimes\mathbb{J}(n-1,n-1)\right)\right],$$
$$\displaystyle g$$
$$\displaystyle=$$
$$\displaystyle W-2Ak\left(\left(e_{0}\otimes\mathbb{J}_{n-1}\right)+\left[z_{T[0]}\right]^{T}\right),$$
$$\displaystyle+2A\left(\mathbb{J}_{n}\otimes\mathbb{J}_{n-1}\right),$$
$$\displaystyle c$$
$$\displaystyle=$$
$$\displaystyle 2A(n-1)+2Ak^{2}.$$
(29)
$\mathbb{J}$ is the matrix containing all ones, $\mathbb{I}$ is the identity matrix, and $e_{0}={\left[1,0,\cdots..,0\right]}^{T}$ is the identity matrix.The binary decision variable $x_{ij}$ is converted to the spin variable $s_{ij}\in\left\{-1,1\right\}$ using the formula $x_{ij}=(s_{ij}+1)/2$.
From the aforementioned equations, we may expand Eq. (27) to form the Ising Hamiltonian of VRP [15].
$$\displaystyle H_{Ising}=-\sum_{i}{\ }\sum_{i<j}{\ }J_{ij}s_{i}s_{j}-\sum_{i}{\ }h_{i}s_{i}+d.$$
(30)
Following are definitions for the terms $J_{ij},h_{i}$, and $d$:
$$\displaystyle J_{ij}$$
$$\displaystyle=$$
$$\displaystyle\ -\frac{Q_{ij}}{2},\ \forall\ i<j,$$
$$\displaystyle h_{i}$$
$$\displaystyle=$$
$$\displaystyle\frac{g_{i}}{2}+\sum{\frac{Q_{ij}}{4}+\ \sum{\frac{Q_{ji}}{4}\ }\ },$$
$$\displaystyle d$$
$$\displaystyle=$$
$$\displaystyle c+\sum_{i}{\ }\frac{g_{i}}{2}+\sum_{i}{\ }\sum_{j}{\ }\frac{Q_{ij}}{4}.$$
(31)
III-B Analysis And Circuit Building
III-B1 VRP
In this section, we create a gate-based circuit to realize the above formulation using the IBM gate model, which we have implemented using the Qiskit framework [39]. For any arbitrary VRP problem using qubits, we begin with the state of $\ket{+}^{\otimes n(n-1)}$ the ground state of $H_{mixer}$ by applying the Hadamard to all qubits initialized as zero state, and we prepare the following state.
$$\displaystyle\ket{\beta,\gamma}$$
$$\displaystyle=$$
$$\displaystyle e^{-iH_{mixer}\beta_{p}}e^{-iH_{cost}\gamma_{p}}...$$
(32)
$$\displaystyle...e^{-iH_{mixer}\beta_{0}}e^{-iH_{cost}\gamma_{0}}\ket{+}^{n\otimes(n-1)}.$$
The energy E of the state $\ket{\beta,\gamma}$ is calculated by expectation of $H_{cost}$ from Eq. (17). Again From the Ising model, $H_{cost}$ term can be written in terms of Pauli operators as,
$$\displaystyle{H}_{\mathrm{cost}}=-\sum_{i}\sum_{i<j}J_{ij}\sigma_{i}^{z}\sigma_{j}^{z}-\sum_{i}h_{i}\sigma_{i}^{z}-d.$$
(33)
Thus for a single term of state in $\ket{\beta,\gamma}$ as $\beta_{0},\gamma_{0}$, the expression reads,
$e^{-i{H}_{mixer}\beta_{0}}e^{-i{H}_{cost}\gamma_{0}}.$
The first term ${H}_{\mathrm{cost\ }}$ can be expanded to following,
$$\displaystyle{e}^{iJ_{ij}\gamma_{0}\sigma_{i}\sigma_{j}}$$
$$\displaystyle=$$
$$\displaystyle\cos J_{ij}\gamma_{0}I+i\ \sin J_{ij}\gamma_{0}\sigma_{i}\sigma_{j},$$
(34)
$$\displaystyle=$$
$$\displaystyle\left[\begin{matrix}{e}^{i{J}_{ij}{\gamma}_{0}}&0&0&0\\
0&{e}^{-i{J}_{ij}{\gamma}_{0}}&0&0\\
0&0&{e}^{-i{J}_{ij}{\gamma}_{0}}&0\\
0&0&0&{e}^{i{J}_{ij}{\gamma}_{0}}\end{matrix}\right],$$
$$\displaystyle=$$
$$\displaystyle M$$
Applying $CNOT$ gate on before and after the above matrix ‘$M$’ we can swap the diagonal elements,
$$\displaystyle CNOT(M)CNOT=\left[\begin{matrix}{e}^{i{J}_{ij}{\gamma}_{0}}&0&0&0\\
0&{e}^{-i{J}_{ij}{\gamma}_{0}}&0&0\\
0&0&{e}^{i{J}_{ij}{\gamma}_{0}}&0\\
0&0&0&{e}^{-i{J}_{ij}{\gamma}_{0}}\end{matrix}\right].$$
Observing the upper and lower blocks of matrix we can rewrite,
$$\displaystyle\left[\begin{matrix}1&0\\
0&1\end{matrix}\right]\otimes\left[\begin{matrix}{e}^{i{J}_{ij}{\gamma}_{0}}&0\\
0&{e}^{-i{J}_{ij}{\gamma}_{0}}\end{matrix}\right]=I\otimes{e}^{i{J}_{ij}{\gamma}_{0}}\left[\begin{matrix}1&0\\
0&{e}^{-2i{J}_{ij}{\gamma}_{0}}\end{matrix}\right].$$
$\left[\begin{matrix}1&0\\
0&{e}^{-2i{J}_{ij}{\gamma}_{0}}\end{matrix}\right]$ is a phase gate. Looking at the $2$-nd term of ${H}_{\mathrm{cost}}$ we get,
$$\displaystyle{H}_{\mathrm{cost}}$$
$$\displaystyle=$$
$$\displaystyle\sum_{i}h_{i}\sigma_{i}^{z},$$
$$\displaystyle e^{ih_{i}\sigma_{i}}$$
$$\displaystyle=$$
$$\displaystyle{\cos h}_{i}{\gamma_{o}I+i\sin\gamma_{0}\sigma_{i}},$$
(37)
$$\displaystyle=$$
$$\displaystyle\cos h_{i}{\gamma_{o}\left[\begin{matrix}1&0\\
0&1\\
\end{matrix}\right]+i\sin h_{i}\gamma_{0}\left[\begin{matrix}1&0\\
0&-1\\
\end{matrix}\right]},$$
$$\displaystyle=$$
$$\displaystyle\left[\begin{matrix}e^{ih_{i}\gamma_{0}}\ &0\\
0&e^{-ih_{i}\gamma_{0}}\\
\end{matrix}\right].$$
Fig. 3(b) depicts the basic circuit with two qubits along with gate selections for ${H}_{\mathrm{cost}}$. Similarly, $H_{mixer}$ is merely a rotation along the $X$ axis, as depicted by the $U$ gate in Fig. 3(d).
The above sample circuits can be used for the solution of VRP combined with VQE and QAOA approach, However, in this paper, we are focusing on a machine learning solution of VRP by use of QSVM; thus we need to construct a QSVM circuit using various encoding schemes. Simple interpretation and implementation of encoding schemes are described in upcoming subsections.
III-B2 Amplitude Encoding
As we look into AE, a single qubit state is represented by
$$\displaystyle\ket{\psi}(\theta)=\cos(\theta/2)\ket{0}+\sin(\theta/2)\ket{1},$$
(38)
for two qubits
$$\displaystyle|\psi(\theta)\rangle=\alpha|00\rangle+\beta|01\rangle+\gamma|10\rangle+\delta|11\rangle,$$
$$\displaystyle=|0\rangle(\alpha|0\rangle+\beta|1\rangle)+|1\rangle(\gamma|0\rangle+\delta|1\rangle),$$
$$\displaystyle=|0\rangle\sqrt{\left(\alpha^{2}+\beta^{2}\right)}\left(\frac{\alpha|0\rangle+\beta|1\rangle}{\sqrt{\alpha^{2}+\beta^{2}}}\right)$$
$$\displaystyle+|1\rangle\sqrt{\gamma^{2}+\delta^{2}}\frac{\gamma|0\rangle+\delta|1\rangle}{\sqrt{\gamma^{2}+\delta^{2}}}.$$
(39)
Now applying Ctrl U and Anti-CTRL U on the above state we achieve
$$\displaystyle|0\rangle\sqrt{\alpha^{2}+\beta^{2}}|0\rangle+|1\rangle\sqrt{\gamma^{2}+\delta^{2}}|0\rangle$$
$$\displaystyle=\left(\sqrt{\alpha^{2}+\beta^{2}}|0\rangle+\sqrt{\gamma^{2}+\delta^{2}}|1\rangle\right)|0\rangle.$$
(40)
Here
$\theta_{1}=\tan^{-1}\frac{\sqrt{\gamma^{2}+\delta^{2}}}{\sqrt{\alpha^{2}+\beta^{2}}}$ ,
$\theta_{2}=\tan^{-1}\frac{\delta}{\gamma}$ ,
$\theta_{3}=\tan^{-1}\frac{\beta}{\alpha}$
Combining VRP and amplitude encoding circuit eliminates the need of Hadamard gates and $H_{mixer}$ components and we end up with the following skeleton circuits Fig. 3 (a).
III-B3 Angle Encoding
For a 2-qubit scenario, angle encoding translates to the following example. We define the $R_{y}$ gate as follows
$$\displaystyle R_{y}(\theta)$$
$$\displaystyle=$$
$$\displaystyle e^{-iY\theta/2}=\cos\frac{\theta}{2}-i\sin{\theta/2}Y,$$
(43)
$$\displaystyle=$$
$$\displaystyle\left[\begin{array}[]{ll}\cos\theta/2&-\sin\theta/2\\
\sin\theta/2&\cos\theta/2\end{array}\right].$$
$$\displaystyle|00\rangle$$
(44)
$$\displaystyle\xrightarrow[R_{y}(\theta_{2})]{R_{y}(\theta_{1})}\left(\cos\frac{\theta_{1}}{2}|0\rangle+\sin\frac{\theta_{1}}{2}|1\rangle\right)\left(\cos\frac{\theta_{2}}{2}|0\rangle+\sin\frac{\theta_{2}}{2}|1\rangle\right),$$
$$\displaystyle=\cos\frac{\theta_{1}}{2}\cdot\cos\frac{\theta_{2}}{2}|00\rangle+\cos\frac{\theta_{1}}{2}\cdot\sin\frac{\theta_{2}}{2}|01\rangle$$
$$\displaystyle+$$
$$\displaystyle\sin\frac{\theta_{1}}{2}\cdot\cos\frac{\theta_{2}}{2}|10\rangle+\sin\frac{\theta_{1}}{2}\sin\frac{\theta_{2}}{2}|11\rangle$$
$$\displaystyle\xrightarrow[]{CNOT}\cos\frac{\theta_{1}}{2}\cdot\cos\frac{\theta_{2}}{2}|00\rangle+\cos\frac{\theta_{1}}{2}\cdot\sin\frac{\theta_{2}}{2}|01\rangle$$
$$\displaystyle+\sin\frac{\theta_{1}}{2}\cdot\cos\frac{\theta_{2}}{2}|11\rangle+\sin\frac{\theta_{1}}{2}\sin\frac{\theta_{2}}{2}|10\rangle.$$
III-B4 Higher Order Encoding
For a $2$qubit scenario, HO encoding translates to the following
We define the $R_{y}$ gate as follows
$$\displaystyle R_{y}(\theta)$$
$$\displaystyle=$$
$$\displaystyle e^{-iY\theta/2}=\cos\frac{\theta}{2}-i\sin{\theta/2}Y,$$
(47)
$$\displaystyle=$$
$$\displaystyle\left[\begin{array}[]{ll}\cos\theta/2&-\sin\theta/2\\
\sin\theta/2&\cos\theta/2\end{array}\right].$$
$$\displaystyle|00\rangle$$
(48)
$$\displaystyle\xrightarrow[R_{y}(\theta_{2})]{R_{y}(\theta_{1})}\left(\cos\frac{\theta_{1}}{2}|0\rangle+\sin\frac{\theta_{1}}{2}|1\rangle\right)\left(\cos\frac{\theta_{2}}{2}|0\rangle+\sin\frac{\theta_{2}}{2}|1\rangle\right),$$
$$\displaystyle=\cos\frac{\theta_{1}}{2}\cdot\cos\frac{\theta_{2}}{2}|00\rangle+\cos\frac{\theta_{1}}{2}\cdot\sin\frac{\theta_{2}}{2}|01\rangle$$
$$\displaystyle+\sin\frac{\theta_{1}}{2}\cdot\cos\frac{\theta_{2}}{2}|10\rangle+\sin\frac{\theta_{1}}{2}\sin\frac{\theta_{2}}{2}|11\rangle$$
$$\displaystyle\xrightarrow[R_{y}(\theta_{1}.\theta_{2})]{CNOT}\cos\frac{\theta_{1}}{2}\cdot\cos\frac{\theta_{2}}{2}|0\rangle\left(\cos\frac{\theta_{1}\cdot\theta_{2}}{2}|0\rangle+\sin\frac{\theta_{1}\cdot\theta_{2}}{2}|1\rangle\right)$$
$$\displaystyle+\cos\frac{\theta_{1}}{2}\cdot\sin\frac{\theta_{2}}{2}|0\rangle\left(-\sin\frac{\theta_{1}\cdot\theta_{2}}{2}|0\rangle+\cos\frac{\theta_{1}\cdot\theta_{2}}{2}|1\rangle\right)$$
$$\displaystyle+\sin\frac{\theta_{1}}{2}\cos\frac{\theta_{2}}{2}|1\rangle\left(\cos\frac{\theta_{1}\cdot\theta_{2}}{2}|0\rangle+\sin\frac{\theta_{1}\cdot\theta_{2}}{2}|1\rangle\right)$$
$$\displaystyle+\sin\frac{\theta_{1}}{2}\cdot\sin\frac{\theta_{2}}{2}|1\rangle\left(-\sin\frac{\theta_{1}\cdot\theta_{2}}{2}|0\rangle+\cos\frac{\theta_{1}\cdot\theta_{2}}{2}|1\rangle\right).$$
III-B5 IQP Encoding
For a $2$qubit scenario IqpE translates to the following
$$\displaystyle|00\rangle\xrightarrow{H_{1}H_{2}}|++\rangle,$$
(49)
$$\displaystyle=\frac{1}{2}(|00\rangle+|01\rangle+|10\rangle+|11\rangle),$$
$$\displaystyle\xrightarrow[R_{Z}(\theta_{2})]{R_{Z}(\theta_{1})}\frac{1}{2}\left(|00\rangle+e^{i\theta_{2}}|01\rangle+e^{i\theta_{1}}|10\rangle+e^{i\left(\theta_{1}+\theta_{2}\right)}|11\rangle\right)$$
$$\displaystyle\left.\stackrel{{\scriptstyle\text{ CNOT }}}{{\longrightarrow}}\frac{1}{2}(|00\rangle+e^{i\theta_{2}}|01\rangle+e^{i\theta_{1}}\left|11\right\rangle+e^{i\left(\theta_{1}+\theta_{2}\right)}|10\rangle\right)$$
$$\displaystyle\xrightarrow{R_{Z}(\theta_{1}.\theta_{2})}\frac{1}{2}(|00\rangle+e^{i\theta_{2}(1+\theta_{1})}|01\rangle+e^{i\theta_{1}(1+\theta_{2})}|11\rangle$$
$$\displaystyle+e^{i\theta_{1}(1+\theta_{2})}|10\rangle+e^{i(\theta_{1}+\theta_{2})}|11\rangle).$$
IV Results
IV-A VQE Simulation of QSVM and VRP
We build the Hamiltonian with a uniform distribution of weights between $0$ and $1$, and then run it along with the ansatz via IBM’s three available VQE optimizers (COBYLA, L_BFGS_B, and SLSQP). We run the circuit up to two layers and gather data using all of the available optimizers. We run the experiment again with a fixed Hamiltonian and, subsequently, a set of variable Hamiltonians to see whether the QSVM and encoding approach can effectively reach the classical minimum. Our results indicate that COBYLA is the most efficient optimizer, followed by SLSQP and L BFGS B. In the sections that follow, we’ll have a look at the results obtained using various QSVM encoding schemes. We define two terms—Accuracy and Error—in the context of outcomes’ interpretability. An error occurs when the solution deviates from the classical minimum more often than it reaches it, whereas accuracy is defined as the number of times the solution reaches the classical minimum. Percentages based on the distribution of the outcomes are used to evaluate both terms.
$$\displaystyle Acc$$
$$\displaystyle=$$
$$\displaystyle\frac{N}{T},$$
$$\displaystyle Err$$
$$\displaystyle=$$
$$\displaystyle\frac{T-N}{T}.$$
(50)
$T=$ Total number of Simulation runs
$N=$ Total number of times solution reaches classical minimum
IV-A1 Amplitude Encoding
With a large number of gates, the AE circuit has proven to be the most complex of all encoding circuits. We can simulate no more than six qubit computations due to this complexity. Despite its complexity, AE has a high, nearly perfect accuracy rate ($100\%$) and a very low error rate ($0\%$) for 50-iteration fixed Hamiltonian simulations. The trend is present in both the first and second layer. The first layer accuracy for a variable Hamiltonian simulation is $96\%$, and the second layer accuracy is $94\%$ across all optimizers. Figure 4 depicts the results of $50$ iterations of simulating SVM with amplitude encoding on a VRP circuit with fixed and variable Hamiltonian. The decline in accuracy, however, can be attributed to simulation or computational errors, as all the errors are greater than $100$ percent and are therefore considered aberrations. Most likely, the simulation hardware cannot accommodate the VQE procedure.
IV-A2 Angle Encoding
Angle encoding is the second encoding, following amplitude encoding; we have experimented with SVM VRP simulation, which yields high accuracy and low error rates. Observing tables I and II, angle encoding is the second most precise encoding employed in our investigations. For fixed Hamiltonian simulations over $50$ iterations with $6$ qubits angle encoding, the first layer, including all optimizers, achieves $100$ percent accuracy and zero percent error. In the 2nd layer simulation (over $50$ iterations), the accuracy decreases to $98\%$ for COBYLA, $96\%$ for SLSQP, and $86\%$ for L_BGFS_B, which is a greater decrease than the other two. These declines are attributable to optimizer-dependent statistical errors. Similarly, for $12$ qubit simulations of SVM VRP, the accuracy rates are higher in the first layer, which consists of COBYLA at $100\%$, SLSQP at $92\%$, and L_BGFS_B at $88\%$, reiterating that the accuracy is highly dependent on the optimizer. As we move to the second layer of $12$ qubit simulations on Fixed hamiltonian, we observe a decline in precision as the level of optimization rises. In this case, COBYLA winds up with $80\%$, L_BGFS_B with $70\%$, and SLSQP with $84\%$. Here, SLSQP’s accuracy loss is less than that of the other two optimizers. The variable hamiltonian with $12$ qubits demonstrates a comparable trend. On the initial layer, we observe high accuracy with COBYLA at $96\%$, L_BGFS_B at $86\%$, and SLSQP at $90\%$. Moving to the second stratum, the accuracy figures drop significantly, with COBYLA at $76\%$ and L_BGFS_B at $62\%$, while SLSQP maintains excellent accuracy at $86\%$. In every scenario of our investigation, it is evident that over-optimization reduces accuracy rates.
IV-A3 Higher Order Encoding
After Amplitude and Angle Encoding, Higher Order Encoding is the third most prevalent encoding in our SVM VRP simulation experiment. This is also the third most accurate encoding in our experiment. For both $6$ qubit and $12$ qubit simulations, HO encoding yields moderately accurate results; however, as the number of circuit layers is increased, the accuracy of the HO encoding scheme deteriorates, rendering it inappropriate. Figure $5$ depicts the statistics of the HO encoding scheme for fixed and variable hamiltonian simulations of SVM VRP circuits over $50$ iterations for both $6$ qubit and $12$ qubit simulations. COBYLA achieves $78\%$ accuracy for a $6$-qubit HO encoding circuit on a fixed Hamiltonian, while L_BGFS_B achieves $66\%$ accuracy and SLSQP achieves $70\%$ accuracy. As we proceed to the second layer, the accuracy considerably decreases, with COBYLA at $34\%$ and SLSQP, L_BGFS_B at $16\%$, respectively. Similar trends can be observed in variable Hamiltonian simulations of HO encoding with $6$ qubits, with COBYLA at $76\%$, SLSQP at $62\%$, and L_BGFS_B at $58\%$ for the first layer; for the second layer, the accuracy drops to $36\%$, $34\%$, and $36\%$ for COBYLA, L_BGFS_B, and SLSQP, respectively. The $12$ qubit simulation yields superior results than the $6$ qubit simulation and improves COBYLA’s accuracy. For fixed hamiltonian simulations, COBYLA achieves an accuracy of $92\%$, compared to $78\%$ for 6qubit. For variable hamiltonian simulations, COBYLA stores $76\%$ for $6$ qubit in the first layer, and $92\%$ for $12$ qubit in the first layer. The tendencies for L_BGFS_B and SLSQP are ambiguous for both cases (fixed and variable hamiltonian simulations); it is reassuring to conclude that an increase in layer decreases accuracy and that COBYLA outperforms the other two optimizers and ensures stable performance.
IV-A4 IQP Encoding
IQP encoding is the last and least accurate encoding in our experiment to simulate an SVM VRP circuit. The results are plotted in 7 and in tables I and tables II. As we can see from the figures and tables that accuracy is consistently poor for fixed and variable hamiltonian simulations in both $6$ qubit and $12$ qubit circuits. The accuracy further declines as layers increase. Hence this encoding is unsuitable in our experiment of SVM VRP circuits.
IV-B Inferences from Simulation
As we scan through the results of SVM VRP simulations across the encoding schemes we observe some clear and distinct trends regarding the experiment. The tables I , II summarize the results obtained from the plots of all the encoding schemes used in this experiment. We list these trends as our outcomes of this experiment in the below points
•
The approach to solving VRP using machine learning is successful and is capable of accomplishing the same or a superior result than the conventional approach using VQE and QAOA.
•
The use of encoding/decoding schemes can serve the purpose of creating superposition and entanglement and eliminate the additional effort required to construct the mixer hamiltonian when solving the VRP using the standard approach of QAOA and VQE.
•
While the standard approach to solving VRP or any combinatorial optimization problem requires a few layers of circuit depth (2 in most cases), we are able to achieve the same on the first layer itself with this approach, proving that it is more efficient than the standard approach.
•
We also observe a distinct trend that as the number of layers increases, the accuracy decreases, which can be used to determine where to limit the optimization depth.
•
Encoding/decoding schemes reduce the number of optimization layers but increase the circuit’s complexity by introducing more gates. Therefore, when selecting an encoding scheme, we must take into account the complexity of the generated circuit and the number of required gates, as well as the number of classical resources (memory, CPU) it will require. There must be a trade-off between circuit complexity and the desired problem accuracy.
•
Despite the fact that amplitude encoding provided the greatest accuracy, it could not be used to simulate a $12$-qubit VRP scenario due to the large number of gates required. Angel encoding, on the other hand, was found to be much simpler due to a significantly smaller number of gates, as well as providing excellent accuracy ($96\%$ for COBYLA, and $92\%$ for SLSQP and L_BGFS_B in variable hamiltonian simulation) across all the available optimizers. This again demonstrates that the complexity of circuits and the number of gates used are the most important considerations when choosing an encoding/decoding scheme.
•
It can be noticed that AgE performs the best in terms of circuit complexity and accuracy rates due to the formation of a single layer of superposition. In other encodings (HO, IqpE), we observe multi-layered complex superposition structures, which is the reason for fluctuations or error rates. Also in the fact that increasing layers also increases the superposition structures and therefore decreases the accuracy.
•
Using COBYLA as an optimizer, HO encoding yielded intriguing results with reduced accuracy in circuits with fewer qubits ($6$ qubits) and higher accuracy in circuits with more qubits ($12$ qubits) for both fixed and variable hamiltonian simulations. The trend is disregarded by SLSQP and L_BGFS_B. This demonstrates that the algorithm’s performance is extremely dependent on the optimizer; therefore, when evaluating the algorithm’s performance, the most efficient optimizer should be selected by comparing the available optimizers.
•
The IQP encoding scheme performed the worst in this experiment, with the lowest accuracy and highest error rates among all other encodings used for $1$-layer, $2$-layer, fixed, and variable Hamiltonians simulations. Therefore, the IqpE method cannot be used to solve VRP using QSVM.
•
All of the optimizers used in the experiments performed well across AE, AgE, and HO encodings; however, COBYLA outperformed the other two due to its consistently high level of accuracy, but SLSQP is more resistant to accuracy fluctuations caused by an increase in optimization depth or in the presence of multi-layered circuits.
IV-C Experimental Setup, data gathering, and statistics
This experiment is conducted within the ambit of the QISKIT framework. while performing the experiment, we used a quantum instance object, and the ansatz runs inside the quantum instance object. A random seed is added to quantum instance to stabilize VQE results. All the experiments have been run $50+50$ times, one with a fixed Hamiltonian matrix and the other by varying the Hamiltonian matrix. The objective of the experiments is to ensure that the results of experiments are just not dependent on a single Hamiltonian. This is also to ensure that the used circuits achieve classical minimum or near classical minimum regardless of the hamiltonian used. Thus apart from the plots, the tables I, II become the figure of merit. In addition to the many hours of testing and debugging, it is to be noted that the results reported here amounted to $150$ hours of CPU time on a $24$-core AMD workstation using Qiskit’s built-in simulators [39].
V Conclusion
In this paper, we presented a novel technique for solving VRP through the use of a $6$ and $12$-qubit circuit-based quantum support vector machine (QSVM) with a variational quantum eigensolver for both fixed and variable Hamiltonians. In the experiment, multiple encoding strategies were used to convert the VRP formulation into a QSVM and solve it. In addition, we utilized multiple classical optimizers available within the QISKIT framework to measure the output variation and accuracy rates. Consequently, our machine learning-based approach to resolving VRP has proven fruitful thus far. Using a QSVM to implement a gate-based simulation of a $3$-city or $4$-city VRP on a $6$-qubit or $12$-qubit system accomplishes the goal. The method not only resolves VRP, but also outperforms the conventional method of resolving VRP via multiple Optimization phases involving only VQE and QAOA. In addition, selecting appropriate encoding methods establishes the optimal balance between circuit complexity and optimization depth, thereby enabling multiple approaches to solve CO problems using machine learning techniques.
Acknowledgment
The authors are grateful to the IBM Quantum Experience platform and their team for developing the Qiskit platform and providing open access to their simulators for running quantum circuits and performing the experiments reported here [39].
References
[1]
A. Montanaro, “Quantum algorithms: an overview,” npj Quantum
Information, vol. 2, no. 1, p. 15023, Jan. 2016. [Online]. Available:
https://doi.org/10.1038/npjqi.2015.23
[2]
S. Jordan, “https://quantumalgorithmzoo.org/.” [Online]. Available:
https://quantumalgorithmzoo.org/#ONML
[3]
R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum
entanglement,” Rev. Mod. Phys., vol. 81, pp. 865–942, Jun 2009.
[Online]. Available: https://link.aps.org/doi/10.1103/RevModPhys.81.865
[4]
L. K. Grover, A fast quantum mechanical algorithm for database search,
May 1996. [Online]. Available: https://arxiv.org/abs/quant-ph/9605043v3
[5]
E. Farhi, J. Goldstone, S. Gutmann, and M. Sipser, Quantum Computation
by Adiabatic Evolution, Jan. 2000. [Online]. Available:
https://arxiv.org/abs/quant-ph/0001106v1
[6]
E. Farhi, J. Goldstone, and S. Gutmann, A Quantum Approximate
Optimization Algorithm, Nov. 2014. [Online]. Available:
https://arxiv.org/abs/1411.4028v1
[7]
E. National Academies of Sciences, “Chapter: 3 Quantum Algorithms and
Applications,” in Quantum Computing: Progress and Prospects
(2019). [Online]. Available:
https://www.nap.edu/catalog/25196/quantum-computing-progress-and-prospects
[8]
V. Dasari, M. S. Im, and L. Beshaj, Solving machine learning optimization
problems using quantum computers, M. Blowers, R. D. Hall, and V. R. Dasari,
Eds. Online Only, United States: SPIE,
apr 2020. [Online]. Available:
https://www.spiedigitallibrary.org/conference-proceedings-of-spie/11419/2565038/Solving-machine-learning-optimization-problems-using-quantum-computers/10.1117/12.2565038.full
[9]
S. Harwood, C. Gambella, D. Trenev, A. Simonetto, D. Bernal, and D. Greenberg,
“Formulating and Solving Routing Problems on Quantum Computers,”
IEEE Transactions on Quantum Engineering, vol. 2, pp. 1–17, 2021.
[Online]. Available: https://ieeexplore.ieee.org/document/9314905
[10]
S. N. Kumar and R. Panneerselvam, “A survey on the vehicle routing problem and
its variants,” Intelligent Information Management, vol. 4, no. 3,
p. 9, 2012. [Online]. Available:
https://www.scirp.org/html/4-8701154_19355.htm
[11]
N. Mohanty, B. K. Behera, and C. Ferrie, “Analysis of the vehicle routing
problem sovled via hybrid quantum algorithms in presence of noisy channels,”
arXiv:2205.07630, 2021. [Online]. Available:
https://doi.org/10.48550/arXiv.2205.07630
[12]
K. Srinivasan, S. Satyajit, B. K. Behera, and P. K. Panigrahi, “Efficient
quantum algorithm for solving travelling salesman problem: An ibm quantum
experience,” May 2018. [Online]. Available:
https://arxiv.org/abs/1805.10928v1
[13]
S. Feld, C. Roch, T. Gabor, C. Seidel, F. Neukart, I. Galter, W. Mauerer, and
C. Linnhoff-Popien, A Hybrid Solution Method for the
Capacitated Vehicle Routing Problem Using a Quantum Annealer,
2019, vol. 6. [Online]. Available:
https://www.frontiersin.org/article/10.3389/fict.2019.00013
[14]
Utkarsh, B. K. Behera, and P. K. Panigrahi, Solving Vehicle Routing
Problem Using Quantum Approximate Optimization Algorithm, Feb.
2020. [Online]. Available: https://arxiv.org/abs/2002.01351v1
[15]
F. Glover, G. Kochenberger, M. Ma, and Y. Du, “Quantum Bridge Analytics
II: QUBO-Plus, network optimization and combinatorial chaining for
asset exchange,” 4OR, vol. 18, no. 4, pp. 387–417, 2020, publisher:
Springer. [Online]. Available:
https://ideas.repec.org/a/spr/aqjoor/v18y2020i4d10.1007_s10288-020-00464-9.html
[16]
H. Irie, G. Wongpaisarnsin, M. Terabe, A. Miki, and S. Taguchi, “Quantum
Annealing of Vehicle Routing Problem with Time, State and
Capacity,” in Quantum Technology and Optimization Problems,
ser. Lecture Notes in Computer Science, S. Feld and C. Linnhoff-Popien,
Eds. Cham: Springer International
Publishing, 2019, pp. 145–156. [Online]. Available:
https://link.springer.com/chapter/10.1007/978-3-030-14082-3_13
[17]
A. Crispin and A. Syrichas, “Quantum Annealing Algorithm for Vehicle
Scheduling,” in 2013 IEEE International Conference on
Systems, Man, and Cybernetics, Oct. 2013, pp. 3523–3528, iSSN:
1062-922X. [Online]. Available:
https://ieeexplore.ieee.org/document/6722354
[18]
F. E. Office, “Application of Digital Annealer for Faster
Combinatorial Optimization,” FUJITSU Sci. Tech. J., vol. 55,
no. 2, p. 7, 2019. [Online]. Available:
https://www.fujitsu.com/global/documents/about/resources/publications/fstj/archives/vol55-2/paper12.pdf
[19]
V. Havlicek, A. D. Corcoles, K. Temme, A. W. Harrow, A. Kandala, J. M. Chow,
and J. M. Gambetta, “Supervised learning with quantum enhanced feature
spaces,” Nature, vol. 567, pp. 209–212, 2019. [Online]. Available:
https://doi.org/10.1038/s41586-019-0980-2
[20]
P. Rebentrost, M. Mohseni, and S. Lloyd, “Quantum support vector machine for
big data classification,” Physical Review Letters, vol. 113, no. 13,
sep 2014. [Online]. Available:
https://doi.org/10.1103%2Fphysrevlett.113.130503
[21]
A. Kariya and B. K. Behera, “Investigation of quantum support vector machine
for classification in nisq era,” 2021.
[22]
R. Leporini and D. Pastorello, “Support vector machines with quantum state
discrimination,” Quantum Rep., vol. 3, no. 3, pp. 482–499, 2021.
[Online]. Available: https://doi.org/10.3390/quantum3030032
[23]
G. Gentinetta, A. Thomsen, D. Sutter, and S. Woerner, “The complexity of
quantum support vector machines,” 2022.
[24]
M. Cerezo, A. Arrasmith, R. Babbush, S. C. Benjamin, S. Endo, K. Fujii, J. R.
McClean, K. Mitarai, X. Yuan, L. Cincio, and P. J. Coles, “Variational
Quantum Algorithms,” Nat Rev Phys, vol. 3, no. 9, pp. 625–644,
Sep. 2021, arXiv: 2012.09265. [Online]. Available:
http://arxiv.org/abs/2012.09265
[25]
L. Zhou, S.-T. Wang, S. Choi, H. Pichler, and M. D. Lukin, “Quantum
Approximate Optimization Algorithm: Performance, Mechanism, and
Implementation on Near-Term Devices,” Phys. Rev. X, vol. 10,
no. 2, p. 021067, Jun. 2020, publisher: American Physical Society. [Online].
Available: https://link.aps.org/doi/10.1103/PhysRevX.10.021067
[26]
S. P. Singh, The Ising Model: Brief Introduction and Its
Application. IntechOpen, Feb. 2020,
publication Title: Solid State Physics - Metastable, Spintronics Materials
and Mechanics of Deformable Bodies - Recent Progress. [Online]. Available:
https://www.intechopen.com/chapters/71210
[27]
S. G. BRUSH, “History of the lenz-ising model,” Rev. Mod. Phys.,
vol. 39, pp. 883–893, Oct 1967. [Online]. Available:
https://link.aps.org/doi/10.1103/RevModPhys.39.883
[28]
J. R. Glick, T. P. Gujarati, A. D. Corcoles, Y. Kim, A. Kandala, J. M.
Gambetta, and K. Temme, “Covariant quantum kernels for data with group
structure,” 2022.
[29]
I. F. Araujo, D. K. Park, F. Petruccione, and A. J. da Silva, “A
divide-and-conquer algorithm for quantum state preparation,”
Scientific Reports, vol. 11, no. 1, p. 6329, Mar. 2021. [Online].
Available: https://doi.org/10.1038/s41598-021-85474-1
[30]
R. LaRose and B. Coyle, “Robust data encodings for quantum classifiers,”
Phys. Rev. A, vol. 102, p. 032420, Sep 2020. [Online]. Available:
https://link.aps.org/doi/10.1103/PhysRevA.102.032420
[31]
Qiskit, “Lecture 5.1 - Building a Quantum Classifier,” Oct. 2021.
[Online]. Available: https://www.youtube.com/watch?v=-sxlXNz7ZxU
[32]
“Paddle Quantum.” [Online]. Available:
https://qml.baidu.com/tutorials/machine-learning/encoding-classical-data-into-quantum-states.html
[33]
A. Peruzzo, J. McClean, P. Shadbolt, M.-H. Yung, X.-Q. Zhou, P. J. Love,
A. Aspuru-Guzik, and J. L. O’Brien, “A variational eigenvalue solver on a
photonic quantum processor,” Nature Communications, vol. 5, no. 1, p.
4213, Jul. 2014. [Online]. Available:
https://doi.org/10.1038/ncomms5213
[34]
A. Lucas, “Ising formulations of many NP problems,” Frontiers in
Physics, vol. 2, 2014. [Online]. Available:
https://www.frontiersin.org/article/10.3389/fphy.2014.00005
[35]
“Qiskit: Vehicle routing.” [Online]. Available:
https:https://qiskit.org/ecosystem/optimization/tutorials/07_examples_vehicle_routing.html
[36]
P. Date, R. Patton, C. Schuman, and T. Potok, “Efficiently embedding QUBO
problems on adiabatic quantum computers,” Quantum Information
Processing, vol. 18, no. 4, p. 117, Mar. 2019. [Online]. Available:
https://doi.org/10.1007/s11128-019-2236-3
[37]
G. Kochenberger, J.-K. Hao, F. Glover, M. Lewis, Z. Lü, H. Wang, and Y. Wang,
“The unconstrained binary quadratic programming problem: A survey,”
Journal of Combinatorial Optimization, vol. 28, Jul. 2014. [Online].
Available: https://link.springer.com/article/10.1007/s10878-014-9734-0
[38]
G. G. Guerreschi, “Solving Quadratic Unconstrained Binary Optimization
with divide-and-conquer and quantum algorithms,” arXiv:2101.07813
[quant-ph], Jan. 2021, arXiv: 2101.07813. [Online]. Available:
http://arxiv.org/abs/2101.07813
[39]
M. S. A. et al., “Qiskit: An open-source framework for quantum
computing,” 2021. [Online]. Available:
https://github.com/Qiskit/qiskit/tree/0.25.0 |
New Identities for Padovan Numbers
Gamaliel Cerda-Morales
Departamento de Matemática, Facultad de Ciencias Físicas y Matemáticas, Universidad de Concepción, Casilla 160-C, Concepción, Chile.
gamaliel.cerda@usm.cl
http://orcid.org/0000-0003-3164-4434
Abstract.
In [4], the $am+b$ ($0\leq b<a$) subscripted Tribonacci numbers are studied. This work is devoted to study a new generalization of Fibonacci numbers called Padovan numbers. In particular, the $am+b$ subscripted Padovan numbers will be expressed by three $a$ step apart Padovan numbers for any $0\leq b<a$, where $a\in\mathbb{Z}$.
Key words and phrases:
$a$ columns Padovan table, Padovan numbers, third-order recurrence, arithmetic subscripts.
2010 Mathematics Subject Classification: Primary 11B37, 11B39, Secondary 15A36.
1. Introduction
In Fibonacci numbers, there clearly exists the term golden ratio which is defined as the ratio of two consecutive of Fibonacci numbers that converges to $\alpha=\frac{1+\sqrt{5}}{2}$. In a similar manner, the ratio of two consecutive Padovan
numbers converges to
$$\rho=\sqrt[3]{\frac{1}{2}+\frac{1}{6}\sqrt{\frac{23}{3}}}+\sqrt[3]{\frac{1}{2}%
-\frac{1}{6}\sqrt{\frac{23}{3}}}\in\mathbb{R}.$$
There are so many studies in the literature that concern about the generalized Fibonacci numbers such as Tribonacci, third-order Jacobsthal and Padovan numbers (see, for example [1, 2, 4, 5, 6, 8]). Although the study of Padovan numbers started in the beginning of 19 century under different names, the master study was published in 2006 by Shannon et al. in [8]. In the above paper, the authors defined the Padovan $\{P_{n}\}_{n\geq 0}$ sequence as
$$P_{n+3}=P_{n+1}+P_{n},\ P_{0}=P_{1}=P_{2}=1.$$
(1)
A number of properties of the Tribonacci sequence were studied in [4] by Tribonacci table method. Let $a\in\mathbb{N}$ and $m\geq 0$, in this work we shall generalize the identities about $am$ subscripted Padovan numbers $P_{am}$ to any $P_{am+b}$ ($1\leq b\leq a$). One of our main theorem is to express $P_{am+b}$ by $P_{2a+b}$, $P_{a+b}$ and $P_{b}$, which are $a$ step apart terms.
2. Padovan table
For $a\in\mathbb{N}$, when we say $a$ columns Padovan table we mean a rectangle shape having $a$ columns that consists of the all Padovan numbers from $P_{1}$ in order. So,
$$\left|\begin{array}[]{cccc}P_{1}&P_{2}&...&P_{a}\\
P_{a+1}&P_{a+2}&...&P_{2a}\\
P_{2a+1}&P_{2a+2}&...&P_{3a}\\
...&...&...&...\end{array}\right|.$$
We shall investigate a third-order linear recurrence $P_{n}=\rho_{a}P_{n-a}+\sigma_{a}P_{n-2a}+P_{n-3a}$ for Padovan numbers with some $\rho_{a},\sigma_{a}\in\mathbb{Z}$. In the next result, we show some known results for Padovan numbers using the inductive method.
Lemma 1.
Let $P_{n}$ the $n$-th Padovan number, then $P_{n}=2P_{n-2}-P_{n-4}+P_{n-6},$ $P_{n}=3P_{n-3}-2P_{n-6}+P_{n-9}$ and $P_{n}=2P_{n-4}+3P_{n-8}+P_{n-12}.$
Proof.
Observe that $P_{6}=4=2\cdot 2-1+1=2P_{4}-P_{2}+P_{0}$. Then, if we assume $P_{t}=2P_{t-2}-P_{t-4}+P_{t-6}$ for all $t<n$ then
$$\displaystyle\vskip 12.0pt plus 4.0pt minus 4.0ptP_{n}$$
$$\displaystyle=$$
$$\displaystyle P_{n-2}+P_{n-3}$$
$$\displaystyle=$$
$$\displaystyle\left(2P_{n-4}-P_{n-6}+P_{n-8}\right)+\left(2P_{n-5}-P_{n-7}+P_{n%
-9}\right)$$
$$\displaystyle=$$
$$\displaystyle 2\left(P_{n-4}+P_{n-5}\right)-\left(P_{n-6}+P_{n-7}\right)+\left%
(P_{n-8}+P_{n-9}\right)$$
$$\displaystyle=$$
$$\displaystyle 2P_{n-2}-P_{n-4}+P_{n-6}.$$
Similar to this, we notice $P_{9}=9=3\cdot 4-2\cdot 2+P_{0}=3P_{6}-2P_{3}+P_{0}.$ If we assume $P_{t}=3P_{t-3}-2P_{t-6}+P_{t-9}$
for all $t<n$, then the induction hypothesis proves $P_{n}=3P_{n-3}-2P_{n-6}+P_{n-9}.$
Analogously, since $P_{12}=21=2\cdot 7+3\cdot 2+1=2P_{8}+3P_{4}+P_{0},$ the identity $P_{n}=2P_{n-4}+3P_{n-8}+P_{n-12}$ can be proved immediately by induction.
∎
Remark 2.
Note that the identity $P_{4m}=2P_{4(m-1)}+3P_{4(m-2)}+P_{4(m-3)}$ is a
special case of $P_{n}=2P_{n-4}+3P_{n-8}+P_{n-12}$ in above lemma when $n$ is divisible by 4. In general, the identity $P_{am}=\rho_{a}P_{a(m-1)}+\sigma_{a}P_{a(m-2)}+P_{a(m-3)}$ is a special case of $P_{n}=2P_{n-4}+3P_{n-8}+P_{n-12}$ when $n$ is divisible by $a$. We extend Lemma 1 to any
integer $0\leq a\leq 8.$
Theorem 3.
Let $0\leq a\leq 8$. Then, the third-order recurrence $P_{n}=\rho_{a}P_{n-a}+\sigma_{a}P_{n-2a}+P_{n-3a}$ of $P_{n}$ holds with
the following $(\rho_{a},\sigma_{a}).$
$$a$$
$$(\rho_{a},\sigma_{a})$$
$$\begin{array}[]{cccc}P_{n}=&\rho_{a}P_{n-a}&+\sigma_{a}P_{n-2a}&+P_{n-3a}\end{array}$$
$$1$$
$$(0,1)$$
$$\begin{array}[]{cccc}P_{n}=&&P_{n-2}&+P_{n-3}\end{array}$$
$$2$$
$$(2,-1)$$
$$\begin{array}[]{cccc}P_{n}=&2P_{n-2}&-P_{n-4}&+P_{n-6}\end{array}$$
$$3$$
$$(3,-2)$$
$$\begin{array}[]{cccc}P_{n}=&3P_{n-3}&-2P_{n-6}&+P_{n-9}\end{array}$$
$$4$$
$$(2,3)$$
$$\begin{array}[]{cccc}P_{n}=&2P_{n-4}&+3P_{n-8}&+P_{n-12}\end{array}$$
$$5$$
$$(5,-4)$$
$$\begin{array}[]{cccc}P_{n}=&5P_{n-5}&-4P_{n-10}&+P_{n-15}\end{array}$$
$$6$$
$$(5,2)$$
$$\begin{array}[]{cccc}P_{n}=&5P_{n-6}&+2P_{n-12}&+P_{n-18}\end{array}$$
$$7$$
$$(7,1)$$
$$\begin{array}[]{cccc}P_{n}=&7P_{n-7}&+P_{n-14}&+P_{n-21}\end{array}$$
$$8$$
$$(10,-5)$$
$$\begin{array}[]{cccc}P_{n}=&10P_{n-8}&-5P_{n-16}&+P_{n-24}\end{array}$$
Proof.
Clearly $P_{n}=P_{n-2}+P_{n-3}$ shows $(\rho_{1},\sigma_{1})=(0,1).$ And
above Lemma shows $(\rho_{a},\sigma_{a})=(2,-1),$ $(3,-2)$ and $(2,3)$ for
$a=2,3,4$, respectively. For $5\leq a\leq 8$, we shall consider $a$ columns
Padovan tables. Let us begin with $a=5$.
$$\left|\begin{array}[]{ccccc}1&1&2&2&3\\
4&5&7&9&12\\
16&21&28&37&49\\
65&86&114&151&...\end{array}\right|.$$
Then it can be observed that, for instance
$$\left\{\begin{array}[]{c}P_{16}=65=5\cdot 16-4\cdot 4+1=5P_{11}-4P_{6}+P_{1}\\
P_{17}=86=5\cdot 21-4\cdot 5+1=5P_{12}-4P_{7}+P_{2}\\
P_{18}=114=5\cdot 28-4\cdot 7+2=5P_{13}-4P_{8}+P_{3}\end{array}\right..$$
Thus, by assuming $P_{t}=5P_{t-5}-4P_{t-10}+P_{t-15}$ for all $t<n$, the
induction hypothesis gives rise to
$$\displaystyle P_{n}$$
$$\displaystyle=$$
$$\displaystyle P_{n-2}+P_{n-3}$$
$$\displaystyle=$$
$$\displaystyle\left(5P_{n-7}-4P_{n-12}+P_{n-17}\right)+\left(5P_{n-8}-4P_{n-13}%
+P_{n-18}\right)$$
$$\displaystyle=$$
$$\displaystyle 5\left(P_{n-7}+P_{n-8}\right)-4\left(P_{n-12}+P_{n-13}\right)+%
\left(P_{n-17}+P_{n-18}\right)$$
$$\displaystyle=$$
$$\displaystyle 5P_{n-5}-4P_{n-10}+P_{n-15},$$
so $(\rho_{5},\sigma_{5})=(5,-4).$ Moreover from the $6$ columns Padovan
table
$$\left|\begin{array}[]{cccccc}1&1&2&2&3&4\\
5&7&9&12&16&21\\
28&37&49&65&86&114\\
151&200&265&351&465&...\end{array}\right|$$
we can observe that, for instance
$$\left\{\begin{array}[]{c}P_{19}=151=5\cdot 28+2\cdot 5+1=5P_{13}+2P_{7}+P_{1}%
\\
P_{20}=200=5\cdot 37+2\cdot 7+1=5P_{14}+2P_{8}+P_{2}\\
P_{21}=265=5\cdot 49+2\cdot 9+2=5P_{15}+2P_{9}+P_{3}\end{array}\right..$$
By assuming $P_{t}=5P_{t-6}+2P_{t-12}+P_{t-18}$ for all $t<n$, we have
$$\displaystyle P_{n}$$
$$\displaystyle=$$
$$\displaystyle P_{n-2}+P_{n-3}$$
$$\displaystyle=$$
$$\displaystyle\left(5P_{n-8}+P_{n-14}+P_{n-20}\right)+\left(5P_{n-9}+2P_{n-15}+%
P_{n-21}\right)$$
$$\displaystyle=$$
$$\displaystyle 5\left(P_{n-8}+P_{n-9}\right)+2\left(P_{n-14}+P_{n-15}\right)+%
\left(P_{n-20}+P_{n-21}\right)$$
$$\displaystyle=$$
$$\displaystyle 5P_{n-6}-4P_{n-12}+P_{n-18},$$
so $(\rho_{6},\sigma_{6})=(5,2).$ Therefore the observations and
mathematical induction prove that the coefficients $(\rho_{a},\sigma_{a})$
for $a=7,8$ satisfying
$$P_{n}=\rho_{a}P_{n-a}+\sigma_{a}P_{n-2a}+P_{n-3a}$$
are equal to $(7,1),$ $(10,-5),$ respectively.
∎
We note that the subscript $n$ of $P_{n}$ could be negative, for example, in
$6\ $columns Padovan table, $P_{15}=49=5P_{9}+2P_{3}+P_{-3}$. In general,
Definition 4.
A sequence $p_{n}$ is called a Padovan type if it satisfies $p_{n}=p_{n-2}+p_{n-3}$ with any initials $p_{1},$ $p_{2}$ and $p_{3}$.
Theorem 5.
For $0\leq a\leq 8$, let $(\rho_{a},\sigma_{a})$ be the coefficient of the
third order recurrence $P_{n}=\rho_{a}P_{n-a}+\sigma_{a}P_{n-2a}+P_{n-3a}.$
Then,
(1)
For $1\leq b\leq 5,$ $\{\rho_{a}\}$ is a Padovan type sequence $\rho_{b+3}=\rho_{b+1}+\rho_{b}$ with initials $\rho_{1}=0,$ $\rho_{2}=2$ and
$\rho_{3}=3,$ while $\{\sigma_{a}\}$ satisfies $\sigma_{b+3}=\sigma_{b}-\sigma_{b+2}$ with $\sigma_{1}=1,$ $\sigma_{2}=-1$ and $\sigma_{3}=-2.$
(2)
Moreover, $\rho_{a}=3P_{a-2}-P_{a-4}$ and $\sigma_{a}=-\rho_{-a}$
for $0\leq a\leq 8$, where $P_{n}$ is the $n$th Padovan number.
Proof.
By above theorem,
$$\{\rho_{a}\}_{a=1}^{8}=\{0,2,3,2,5,5,7,10\}$$
and
$$\{\sigma_{a}\}_{a=1}^{8}=\{1,-1,-2,3,-4,2,1,-5\}.$$
So it is easy to see
that $\rho_{b+3}=\rho_{b+1}+\rho_{b}$ and $\sigma_{b+3}=\sigma_{b}-\sigma_{b+2}$ for $1\leq b\leq 5$. Moreover, by means of Padovan
numbers $P_{a}$, we notice
$$\rho_{1}=0=3P_{-1}-P_{-3},\text{ }\rho_{2}=2=3P_{0}-P_{-2},\text{ }\rho_{3}=3=%
3P_{1}-P_{-1},$$
and $\rho_{4}=\rho_{2}+\rho_{1}=3P_{2}-P_{0},$ etc. So $\rho_{a}=3P_{a-2}-P_{a-4}$ for $1\leq a\leq 8.$ Now, by considering $P_{n}$ with
negative $n$, the Padovan type sequence $\{\rho_{a}\}$ can be extended to
any $a\in\mathbb{Z}$, as follows.
$$a$$
$$...$$
$$-8$$
$$-7$$
$$-6$$
$$-5$$
$$-4$$
$$-3$$
$$-2$$
$$-1$$
$$0$$
$$1$$
$$2$$
$$3$$
$$...$$
$$\rho_{a}$$
$$...$$
$$5$$
$$-1$$
$$-2$$
$$4$$
$$-3$$
$$2$$
$$1$$
$$-1$$
$$3$$
$$0$$
$$2$$
$$3$$
$$...$$
Then by comparing $\{\rho_{a}\}_{a=-1}^{-8}=\{-1,1,2,-3,4,-2,-1,5\}$ with $\{\sigma_{a}\}_{a=1}^{8}$, we find that $\sigma_{a}=-\rho_{-a}$ for $0\leq a\leq 8$.
∎
3. The third-order linear recurrence of $P_{n}$
We shall generalize the findings in above section for $0\leq a\leq 8$ to any
integer $a$.
Theorem 6.
Let $\rho_{a}=3P_{a-2}-P_{a-4}$ and $\sigma_{a}=-\rho_{-a}$ for any $a\in\mathbb{Z}$. Then, any $n$th Padovan number satisfies $P_{n}=\rho_{a}P_{n-a}+\sigma_{a}P_{n-2a}+P_{n-3a}$ for every $a<n$.
Proof.
It is due to above theorem if $0\leq a\leq 8$. Since $\rho_{a}=3P_{a-2}-P_{a-4}$ for all $a\in\mathbb{Z}$, $\{\rho_{a}\}$ is a Padovan type sequence because
$$\displaystyle\rho_{a+1}+\rho_{a}$$
$$\displaystyle=$$
$$\displaystyle\left(3P_{a-1}-P_{a-3}\right)+\left(3P_{a-2}-P_{a-4}\right)$$
$$\displaystyle=$$
$$\displaystyle 3\left(P_{a-1}+P_{a-2}\right)-\left(P_{a-3}+P_{a-4}\right)$$
$$\displaystyle=$$
$$\displaystyle 3P_{a+1}-P_{a-1}=\rho_{a+3}.$$
Similarly, since $\sigma_{a}=-\rho_{-a}$ for all $a$, $\{\sigma_{a}\}$
satisfies
$$\displaystyle\sigma_{a}-\sigma_{a+2}$$
$$\displaystyle=$$
$$\displaystyle-\rho_{-a}+\rho_{-(a+2)}$$
$$\displaystyle=$$
$$\displaystyle-\left(\rho_{-a-2}+\rho_{-a-3}\right)+\rho_{-(a+2)}$$
$$\displaystyle=$$
$$\displaystyle-\rho_{-(a+3)}=\sigma_{a+3}.$$
We now suppose that the three order recurrence $P_{n}=\rho_{t}P_{n-t}+\sigma_{t}P_{n-2t}+P_{n-3t}$ hold for all $t<a$. Since $P_{n-(a-2)}=P_{n-a}+P_{n-(a+1)},$ $P_{n-2(a-2)}=2P_{n-2(a-1)}-P_{n-2a}+P_{n-2(a+1)}$ and $P_{n-3(a-2)}=3P_{n-3(a-1)}-2P_{n-3a}+P_{n-3(a+1)}.$ Then, by lemma, the
mathematical induction with long calculations proves that
$$\displaystyle\rho_{a+1}P_{n-(a+1)}+\sigma_{a+1}P_{n-2(a+1)}+P_{n-3(a+1)}$$
$$\displaystyle=$$
$$\displaystyle\left(\rho_{a-1}+\rho_{a-2}\right)P_{n-(a+1)}+\left(\sigma_{a-2}-%
\sigma_{a}\right)P_{n-2(a+1)}+P_{n-3(a+1)}$$
$$\displaystyle=$$
$$\displaystyle\left(\rho_{a-1}+\rho_{a-2}\right)\left(P_{n-(a-2)}-P_{n-a}\right)$$
$$\displaystyle+\left(\sigma_{a-2}-\sigma_{a}\right)\left(P_{n-2(a-2)}-2P_{n-2(a%
-1)}+P_{n-2a}\right)$$
$$\displaystyle+\left(P_{n-3(a-2)}-3P_{n-3(a-1)}+2P_{n-3a}\right)$$
$$\displaystyle=$$
$$\displaystyle P_{n}.$$
∎
Above theorem provides a good way to find huge Padovan numbers. For
instance, for $40$th Padovan number $P_{40},$ we may choose any $a$, say $a=10,$ then $\rho_{10}=3P_{8}-P_{6}=17$ and $\sigma_{10}=-\rho_{-10}=-6$,
thus
$$\displaystyle P_{40}$$
$$\displaystyle=$$
$$\displaystyle\rho_{10}P_{40-10}+\sigma_{10}P_{40-20}+P_{40-30}$$
$$\displaystyle=$$
$$\displaystyle 17\cdot 3329-6\cdot 200+12$$
$$\displaystyle=$$
$$\displaystyle 55405.$$
On the other hand, if we take $a=8$ then $\rho_{8}=3P_{6}-P_{4}=10$ and $\sigma_{8}=-\rho_{-8}=-5$, so $P_{40}$ can be obtained by $P_{40}=\rho_{8}P_{32}+\sigma_{10}P_{24}+P_{16}.$
Besides the expression of $\rho_{a}$ by six step apart Padovan numbers,
more identities for $\rho_{a}$ can be developed in terms of three
successive Padovan numbers.
Theorem 7.
Let $a$ be any integer. Then $\rho_{a}=P_{a}-P_{a-1}+2P_{a-2}=2P_{a+1}+P_{a}-3P_{a-1}.$ So, $\rho_{a}=\left[\begin{array}[]{ccc}2&1&-3\end{array}\right]\left[\begin{array}%
[]{c}P_{a+1}\\
P_{a}\\
P_{a-1}\end{array}\right].$
Proof.
Since
$$\displaystyle P_{a-4}$$
$$\displaystyle=$$
$$\displaystyle P_{a-1}-P_{a-3}=P_{a-1}-\left(P_{a}-P_{a-2}\right)$$
$$\displaystyle=$$
$$\displaystyle P_{a-1}-P_{a}+P_{a-2},$$
it follows that
$$\displaystyle\rho_{a}$$
$$\displaystyle=$$
$$\displaystyle 3P_{a-2}-P_{a-4}$$
$$\displaystyle=$$
$$\displaystyle-P_{a-1}+P_{a}+2P_{a-2}$$
$$\displaystyle=$$
$$\displaystyle-P_{a-1}+P_{a}+2\left(P_{a+1}-P_{a-1}\right)$$
$$\displaystyle=$$
$$\displaystyle 2P_{a+1}+P_{a}-3P_{a-1}.$$
Hence we have $\rho_{a}=2P_{a+1}+P_{a}-3P_{a-1}=\left[\begin{array}[]{ccc}2&1&-3\end{array}%
\right]\left[\begin{array}[]{c}P_{a+1}\\
P_{a}\\
P_{a-1}\end{array}\right].$
Therefore it follows immediately
$$\rho_{-a}=2P_{-a+1}+P_{-a}-3P_{-a-1}=\left[\begin{array}[]{ccc}2&1&-3\end{%
array}\right]\left[\begin{array}[]{c}P_{-a+1}\\
P_{-a}\\
P_{-a-1}\end{array}\right].$$
∎
For each $n\in\mathbb{Z}$, we define two sequences
$$Q_{n}=P_{n}+P_{-n}\text{ and }R_{n}=P_{n}-P_{-n}.$$
Then it is easy to have the table
$$n$$
$$...$$
$$1$$
$$2$$
$$3$$
$$4$$
$$5$$
$$6$$
$$7$$
$$8$$
$$9$$
$$10$$
$$11$$
$$12$$
$$...$$
$$P_{n}$$
$$...$$
$$1$$
$$1$$
$$2$$
$$2$$
$$3$$
$$4$$
$$\mathbf{5}$$
$$7$$
$$9$$
$$12$$
$$16$$
$$21$$
$$...$$
$$P_{-n}$$
$$...$$
$$0$$
$$1$$
$$0$$
$$0$$
$$1$$
$$-1$$
$$1$$
$$0$$
$$-1$$
$$2$$
$$-2$$
$$1$$
$$...$$
$$Q_{n}$$
$$...$$
$$1$$
$$2$$
$$\mathbf{2}$$
$$2$$
$$\mathbf{4}$$
$$\mathbf{3}$$
$$6$$
$$7$$
$$8$$
$$14$$
$$14$$
$$22$$
$$...$$
$$R_{n}$$
$$...$$
$$1$$
$$0$$
$$\mathbf{2}$$
$$2$$
$$\mathbf{2}$$
$$\mathbf{5}$$
$$4$$
$$7$$
$$10$$
$$10$$
$$18$$
$$20$$
$$...$$
From the table, we notice $P_{7}=5=3+4-2$ or $P_{7}=Q_{6}+Q_{5}-Q_{3}.$
Theorem 8.
Let $a\in\mathbb{Z}$. Then, the sequences $\{Q_{n}\}$ satisfy the relation $P_{n+1}=Q_{n}+Q_{n-1}-Q_{n-3}.$ Furthermore, $P_{n}=\frac{1}{2}\left(Q_{n}+R_{n}\right)$ and $P_{-n}=\frac{1}{2}\left(Q_{n}-R_{n}\right).$
Proof.
It is easy to see that
$$\displaystyle Q_{n}$$
$$\displaystyle=$$
$$\displaystyle P_{n}+P_{-n}$$
$$\displaystyle=$$
$$\displaystyle\left(P_{n-2}+P_{n-3}\right)+\left(-P_{-(n-1)}+P_{-(n-3)}\right)$$
$$\displaystyle=$$
$$\displaystyle Q_{n-3}+P_{n-2}-P_{-(n-1)}$$
$$\displaystyle=$$
$$\displaystyle Q_{n-3}+\left(P_{n+1}-P_{n-1}\right)-P_{-(n-1)}$$
$$\displaystyle=$$
$$\displaystyle Q_{n-3}-Q_{n-1}+P_{n+1}.$$
Hence $P_{n+1}=Q_{n}+Q_{n-1}-Q_{n-3}$.
∎
Theorem 9.
Let $n=am+b$ with $1\leq b\leq a<n$. Let $(\rho_{a},\sigma_{a})$ be the
coefficient of the third-order recurrence $P_{n}=\rho_{a}P_{n-a}+\sigma_{a}P_{n-2a}+P_{n-3a}.$ Then, $P_{n}$ is a linear combination of any three
consecutive entries of $b$th column in the $a$ columns Padovan table.
Furthermore, $P_{n}$ is expressed by the first three terms $P_{2a+b},$ $P_{a+b}$ and $P_{b}$ of $b$th column.
Proof.
Let $P_{n}=\rho_{a}P_{n-a}+\sigma_{a}P_{n-2a}+P_{n-3a}$ in above theorem.
Then,
$$\displaystyle P_{at+b}$$
$$\displaystyle=$$
$$\displaystyle\rho_{a}P_{a(t-1)+b}+\sigma_{a}P_{a(t-2)+b}+P_{{}_{a(t-3)+b}}$$
$$\displaystyle=$$
$$\displaystyle\rho_{a}\left(\rho_{a}P_{a(t-2)+b}+\sigma_{a}P_{a(t-3)+b}+P_{{}_{%
a(t-4)+b}}\right)$$
$$\displaystyle+\sigma_{a}P_{a(t-2)+b}+P_{{}_{a(t-3)+b}}$$
$$\displaystyle=$$
$$\displaystyle\left(\rho_{a}^{2}+\sigma_{a}\right)P_{a(t-2)+b}+\left(\rho_{a}%
\sigma_{a}+1\right)P_{a(t-3)+b}+\rho_{a}P_{{}_{a(t-4)+b}}$$
Continue this process, then it follows that $P_{n}$ is a linear combination
of $P_{2a+b},$ $P_{a+b}$ and $P_{b}$.
∎
For example, for $P_{38}$ we may take any $a<38$, say $a=7$. Since $(\rho_{7},\sigma_{7})=(7,1)$, $P_{38}$ can be obtained easily by above theorem that
$$\displaystyle P_{38}$$
$$\displaystyle=$$
$$\displaystyle 7P_{31}+P_{24}+P_{17}$$
$$\displaystyle=$$
$$\displaystyle(7^{2}+1)P_{24}+(7+1)P_{17}+7P_{10}$$
$$\displaystyle=$$
$$\displaystyle 50P_{24}+8P_{17}+7P_{10}$$
$$\displaystyle=$$
$$\displaystyle 50\left(7P_{17}+P_{10}+P_{3}\right)+8P_{17}+7P_{10}$$
$$\displaystyle=$$
$$\displaystyle 358P_{17}+57P_{10}+50P_{3}$$
$$\displaystyle=$$
$$\displaystyle 358\cdot 86+57\cdot 12+50\cdot 2$$
$$\displaystyle=$$
$$\displaystyle\allowbreak 31\,572.$$
However, since $P_{n}$ is composed of $P_{n-a},$ $P_{n-2a}$ and $P_{n-3a}$,
it may be better to choose $a\approx\frac{n}{3}.$ Indeed if we take $\frac{56}{3}\approx 18=a$, then $P_{56}=\rho_{18}P_{38}-\rho_{-18}P_{20}+P_{2}$
and the last term $P_{2}$ is known easily.
Remark 10.
Assume the same context $(\rho_{a},\sigma_{a})$ as before. If $n=3a$, then
$P_{n}=\rho_{\frac{n}{3}}P_{\frac{2n}{3}}+\sigma_{\frac{n}{3}}P_{\frac{n}{3}}+1$ since $P_{0}=1.$ In the other hand, if $n=3a+1$ or $n=3a+2$ and $P_{1}=P_{2}=1,$ it follows that
$$P_{n}=\rho_{\left\lfloor\frac{n}{3}\right\rfloor}P_{\left\lfloor\frac{2n}{3}%
\right\rfloor+1}+\sigma_{\left\lfloor\frac{n}{3}\right\rfloor}P_{\left\lfloor%
\frac{n}{3}+\frac{1}{2}\right\rfloor+1}+1$$
For example, if $n=26,$ we have $P_{26}=\rho_{8}P_{18}+\sigma_{8}P_{10}+1=1081.$
4. Partial Sum of Padovan numbers in a row
Lemma 11.
For $m\geq 0$, we have
$$\sum_{k=0}^{m}P_{4k}=\frac{1}{5}\left(P_{4(m+1)}+4P_{4m}+P_{4(m-1)}-1\right).$$
Proof.
If $m=0$, we have $P_{0}=1=\frac{1}{5}\left(P_{4}+4P_{0}+P_{-4}-1\right).$
Now, if $a=4$ and $n=4m$, $P_{4m}=2P_{4(m-1)}+3P_{4(m-2)}+P_{4(m-3)}$. Assume $\sum_{k=0}^{m}P_{4k}=\frac{1}{5}\left(P_{4(m+1)}+4P_{4m}+P_{4(m-1)}-1\right)$ is true. Then it follows that
$$\displaystyle\sum_{k=0}^{m+1}P_{4k}$$
$$\displaystyle=$$
$$\displaystyle\sum_{k=0}^{m}P_{4k}+P_{4(m+1)}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{5}\left(P_{4(m+1)}+4P_{4m}+P_{4(m-1)}-1\right)+P_{4(m+1)}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{5}\left(\left(2P_{4(m+1)}+3P_{4m}+P_{4(m-1)}\right)+P_{4%
m}+4P_{4(m+1)}-1\right)$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{5}\left(P_{4(m+2)}+4P_{4(m+1)}+P_{4m}-1\right).$$
∎
Remark 12.
This theorem is a sum of $4m$ subscripted Padovan numbers. But in our context, it can be explained as a sum of entries of $4$th column in the $4$ columns Padovan table. We now shall study the sum of entries of any $b$th column in the $4$ columns Padovan table.
Consider $n=am+b$ or $P_{am+b}$ ($m\geq 0$ and $1\leq b\leq a$) as an entry
placed at the $(m+1)$th row and $b$th column in the table, and let
$$r_{m}^{(a,b)}=\sum_{k=0}^{m}P_{ak+b}=P_{b}+P_{a+b}+\cdots+P_{am+b}$$
be the partial sum of $m+1$ entries of $b$th column.
Theorem 13.
Consider $r_{m}^{(3,b)}$ with $1\leq b\leq 3$. Then for $m\geq 3$,
$$r_{m}^{(3,b)}=3r_{m-1}^{(3,b)}-2r_{m-2}^{(3,b)}+r_{m-3}^{(3,b)}+1.$$
Proof.
The $3$ columns Padovan table makes the table of $r_{m}^{(3,b)}$ as follows.
$$\left|\begin{tabular}[]{lll}$1$&$1$&$2$\\
$2$&$3$&$4$\\
$5$&$7$&$9$\\
$12$&$16$&$21$\\
$28$&$37$&$\cdots$\end{tabular}\right|\text{ and
\begin{tabular}[]{l|lll}\hline$m$&$r_{m}^{(3,1)}$&$r_{m}^{(3,2)}$&$r_{m}^{(3,3%
)}$\\
\hline$0$&$1$&$1$&$2$\\
$1$&$3$&$4$&$6$\\
$2$&$8$&$11$&$15$\\
$3$&$20$&$27$&$36$\\
$4$&$48$&$64$&$85$\\
\hline\end{tabular}.}$$
When $m=3$, we notice $20=\left(3\cdot 8-2\cdot 3+1\right)+1$, and it
can be written as $r_{3}^{(3,1)}=3r_{2}^{(3,1)}-2r_{1}^{(3,1)}+r_{0}^{(3,1)}+1$. Similarly for $b=2,3$.
Assume that $r_{m}^{(3,1)}=3r_{m-1}^{(3,1)}-2r_{m-2}^{(3,1)}+r_{m-3}^{(3,1)}+1.$ Then by
induction hypothesis yields
$$\displaystyle r_{m+1}^{(3,1)}$$
$$\displaystyle=$$
$$\displaystyle\left(3r_{m-1}^{(3,1)}-2r_{m-2}^{(3,1)}+r_{m-3}^{(3,1)}+1\right)+%
P_{3(m+1)+1}$$
$$\displaystyle=$$
$$\displaystyle 3r_{m-1}^{(3,1)}-2r_{m-2}^{(3,1)}+r_{m-3}^{(3,1)}+1+3P_{3m+1}-2P%
_{3(m-1)+1}+P_{3(m-2)+1}$$
$$\displaystyle=$$
$$\displaystyle 3r_{m}^{(3,1)}-2r_{m-1}^{(3,1)}+r_{m-2}^{(3,1)}+1,$$
this proves the equation if $b=1$. The other cases are similar.
∎
Theorem 14.
Consider $r_{m}^{(4,b)}$ with $1\leq b\leq 4$. Then for $m\geq 3$,
$$r_{m}^{(4,b)}=\left\{\begin{array}[]{ccc}2r_{m-1}^{(4,b)}+3r_{m-2}^{(4,b)}+r_{%
m-3}^{(4,b)}+2&if&b=1,\\
2r_{m-1}^{(4,b)}+3r_{m-2}^{(4,b)}+r_{m-3}^{(4,b)}+4&if&b=2,\\
2r_{m-1}^{(4,b)}+3r_{m-2}^{(4,b)}+r_{m-3}^{(4,b)}+3&if&b=3,\\
2r_{m-1}^{(4,b)}+3r_{m-2}^{(4,b)}+r_{m-3}^{(4,b)}+6&if&b=4.\end{array}\right.$$
Proof.
The $4$ columns Padovan table makes the table of $r_{m}^{(4,b)}$ as follows.
$$\left|\begin{tabular}[]{llll}$1$&$1$&$2$&$2$\\
$3$&$4$&$5$&$7$\\
$9$&$12$&$16$&$21$\\
$28$&$37$&$49$&$65$\\
$86$&$114$&$151$&$\cdots$\end{tabular}\right|\text{ and
\begin{tabular}[]{l|llll}\hline$m$&$r_{m}^{(4,1)}$&$r_{m}^{(4,2)}$&$r_{m}^{(4,%
3)}$&$r_{m}^{(4,4)}$\\
\hline$0$&$1$&$1$&$2$&$2$\\
$1$&$4$&$5$&$7$&$9$\\
$2$&$13$&$17$&$23$&$30$\\
$3$&$41$&$54$&$72$&$95$\\
$4$&$127$&$168$&$223$&$295$\\
\hline\end{tabular}.}$$
When $m=4$, we notice $127=\left(2\cdot 41+3\cdot 13+4\right)+2$, and it
can be written as
$$r_{4}^{(4,1)}=2r_{3}^{(4,1)}+3r_{2}^{(4,1)}+r_{1}^{(4,1)}+2.$$
Similar to this, we observe that
$$\left\{\begin{array}[]{c}2r_{3}^{(4,2)}+3r_{2}^{(4,2)}+r_{1}^{(4,2)}+4\\
2r_{3}^{(4,3)}+3r_{2}^{(4,3)}+r_{1}^{(4,3)}+3\\
2r_{3}^{(4,4)}+3r_{2}^{(4,4)}+r_{1}^{(4,4)}+6\end{array}\right..$$
Assume that $r_{m}^{(4,1)}=2r_{m-1}^{(4,1)}+3r_{m-2}^{(4,1)}+r_{m-3}^{(4,1)}+2.$ Then by
induction hypothesis yields
$$\displaystyle r_{m+1}^{(4,1)}$$
$$\displaystyle=$$
$$\displaystyle\sum_{k=0}^{m}P_{4k+1}+P_{4(m+1)+1}$$
$$\displaystyle=$$
$$\displaystyle 2r_{m-1}^{(4,1)}+3r_{m-2}^{(4,1)}+r_{m-3}^{(4,1)}+2+P_{4(m+1)+1}$$
$$\displaystyle=$$
$$\displaystyle 2r_{m-1}^{(4,1)}+3r_{m-2}^{(4,1)}+r_{m-3}^{(4,1)}+2+2P_{4m+1}+3P%
_{4(m-1)+1}+P_{4(m-2)+1}$$
$$\displaystyle=$$
$$\displaystyle 2r_{m}^{(4,1)}+3r_{m-1}^{(4,1)}+r_{m-2}^{(4,1)}+2,$$
this proves the first equation. Similarly, other relationships are followed.
∎
References
[1]
G. Cerda-Morales, Identities for Third Order Jacobsthal Quaternions, Advances in Applied Clifford Algebras 27(2) (2017), 1043–1053.
[2]
G. Cerda-Morales, On a Generalization of Tribonacci Quaternions, Mediterranean Journal of Mathematics 14:239 (2017), 1–12.
[3]
E. Choi, Modular tribonacci numbers by matrix method, J. Korean Soc. Mathe. Educ., Pure Appl. Math. 20 (2013), no. 3, 207-221.
[4]
E. Choi and J. Jo, Identities involving Tribonacci numbers, Journal of the Chungcheong Mathematical Society, 28(1) (2015), 39-51.
[5]
J. Feng, More identities on the Tribonacci numbers, Ars Comb. 100 (2011), 73-78.
[6]
N. Irmak and M. Alp, Tribonacci numbers with indices in arithmetic progression and their sums, Miskolc Math. Notes 14 (2013), 125-133.
[7]
E. Kilic, Tribonacci sequences with certain indices and their sums, Ars. Comb. 86 (2008), 13-22.
[8]
A. G. Shannon, P.G. Anderson, A.F. Horadam, Properties of Cordonnier, Perrin and Van der Laan numbers, International Journal of Mathematical Education in Science and Technology, 37(7) (2006), 825-831.
[9]
A. G. Shannon and A. F. Horadam, Generating functions for powers of third order recurrence sequences, Duke Mathematical Journal, vol.38 (1971), 791-794. |
Abstract
Standard-Model Higgs bosons are dominantly produced via the gluon-fusion
mechanism $gg\to H$ at the LHC, i.e. in a loop-mediated process with
top loops providing the dominant contribution. For the measured Higgs
boson mass of $\sim 125$ GeV the limit of heavy top quarks provides a
reliable approximation as long as the relative QCD corrections are
scaled with the full mass-dependent LO cross section. In this limit
the Higgs coupling to gluons can be described by an effective
Lagrangian. The same approach can also be applied to the coupling of
more than one Higgs boson to gluons. We will derive the effective
Lagrangian for multi-Higgs couplings to gluons up to N${}^{4}$LO thus
extending previous results for more than one Higgs boson. Moreover we
discuss gluonic Higgs couplings up to NNLO, if several heavy quarks
contribute.
PSI–PR–16–10
Effective Multi–Higgs Couplings to Gluons
[0.3cm]
Michael Spira
[0.3cm]
Paul Scherrer Institut, CH–5232 Villigen PSI, Switzerland
1 Introduction
The discovery of a resonance with 125 GeV mass [1] that is
compatible with the Standard-Model (SM) Higgs boson [2]
marked a milestone in particle physics. The existence of the Higgs
boson is inherently related to the mechanism of spontaneous symmetry
breaking [3] while preserving the full gauge symmetry and the
renormalizability of the SM [4]. The dominant production
process of the Higgs boson at the LHC is the loop-induced gluon-fusion
process mediated by top-quark loops and to a lesser extent bottom- and
charm-quark loops [5]. The QCD corrections are known up to
N${}^{3}$LO in the limit of heavy top quarks [6, 7, 8], while the full quark mass dependence is only known up to NLO
[9, 10]. At NNLO subleading terms in the large top mass
expansion [11] and leading contributions to the top+bottom
interference [12] are known. The limit of heavy top quarks has
also been adopted for threshold-resummed calculations
[13, 14], while the inclusion of finite quark-mass effects
in the resummation has been considered recently [15]. It
has been shown that the limit of heavy top quarks $m_{t}^{2}\gg M_{H}^{2}$
provides a reasonable approximation to the calculation of the
gluon-fusion cross section with full mass dependence as long as the
relative QCD corrections are scaled with the fully massive LO cross
section [9, 13]. In the heavy-top-quark limit the
calculation of the gluon-fusion cross section can be simplified by
starting from an effective Lagrangian describing the Higgs coupling to
gluons after integrating out the top contribution [16]. The same
approach has also been applied to Higgs pair production via gluon
fusion, $gg\to HH$, at NLO [17], NNLO
[18, 19] as well as to threshold resummation up to
NNLL [20]. It has been shown that finite mass effects amount
to about 5% in the single Higgs case and $15\%$ for Higgs boson pairs
[21, 22].
In this letter we will derive the effective Lagrangian for multi-Higgs
couplings to gluons to N${}^{4}$LO for arbitrary numbers of external Higgs
bosons thus extending previous work beyond the single-Higgs case. In
Section 2 we will discuss and present the effective Lagrangian for the
SM Higgs boson up to N${}^{4}$LO, while Section 3 will extend this analysis
to an arbitrary number of heavy quarks contributing to the gluonic Higgs
coupling up to NNLO. In Section 4 we will conclude.
2 Standard-Model Higgs Bosons
The starting point for the derivation of the effective Lagrangian in the
heavy-top-quark limit is the low-energy limit of the top-quark
contributions to the Wilson coefficient of the gluonic field-strength
operator $\hat{G}^{a\mu\nu}\hat{G}^{a}_{\mu\nu}$, where $\hat{G}^{a\mu\nu}$
denotes the ($\overline{\rm MS}$-subtracted) gluonic operator of
colour-SU(3) in the low-energy limit with 6 active flavours111The
same ansatz has also been used in the derivation of the effective $Hgg$
coupling in Refs. [13].,
$${\cal L}_{g}=-\frac{1-\Pi_{t}}{4}\hat{G}^{a\mu\nu}\hat{G}^{a}_{\mu\nu}$$
(1)
The Wilson coefficient $\Pi_{t}$ denotes the gauge-invariant vacuum
polarization function of the gluon that is determined by the top-quark
contribution to the gluon self-energy and the two-point-function parts
of the external vertices attached to the gluons. This boils down to the
inverse top-quark contribution to the strong coupling constant so that
$\Pi_{t}$ is related to the decoupling relation between the strong
coupling constant in an $(N_{F}+1)$- and $N_{F}$-flavour theory ($N_{F}=5$),
$$\alpha_{s}^{(N_{F})}(\mu_{R}^{2})=\zeta_{\alpha_{s}}~{}\alpha_{s}^{(N_{F}+1)}(%
\mu_{R}^{2})\,,\qquad\qquad\zeta_{\alpha_{s}}=1+\sum_{n}D_{n}\left(\frac{%
\alpha_{s}^{(N_{F}+1)}(\mu_{R}^{2})}{\pi}\right)^{n}$$
(2)
with the perturbative coefficients up to fourth order [23, 24, 25] [$L_{t}=\log(\mu_{R}^{2}/\overline{m_{t}}^{2}(\mu_{R}^{2}))$]
$$\displaystyle D_{1}$$
$$\displaystyle=$$
$$\displaystyle-\frac{1}{6}L_{t}\qquad\qquad\qquad\qquad\qquad\qquad D_{2}=\frac%
{11}{72}-\frac{11}{24}L_{t}+\frac{1}{36}L_{t}^{2}$$
(3)
$$\displaystyle D_{3}$$
$$\displaystyle=$$
$$\displaystyle\frac{564731}{124416}-\frac{82043}{27648}\zeta_{3}-\frac{2633}{31%
104}N_{F}-\frac{955-67N_{F}}{576}L_{t}+\frac{53-16N_{F}}{576}L_{t}^{2}-\frac{1%
}{216}L_{t}^{3}$$
$$\displaystyle D_{4}$$
$$\displaystyle=$$
$$\displaystyle\frac{291716893}{6123600}-\frac{121}{4320}\log^{5}2+\frac{3031309%
}{1306368}\log^{4}2+\frac{121}{432}\zeta_{2}\log^{3}2-\frac{3031309}{217728}%
\zeta_{2}\log^{2}2$$
$$\displaystyle+$$
$$\displaystyle\frac{2057}{576}\zeta_{4}\log 2+\frac{1389}{256}\zeta_{5}-\frac{7%
6940219}{2177280}\zeta_{4}-\frac{2362581983}{87091200}\zeta_{3}+\frac{3031309}%
{54432}a_{4}+\frac{121}{36}a_{5}$$
$$\displaystyle-$$
$$\displaystyle\frac{151369}{2177280}X_{0}+N_{F}\left(-\frac{4770941}{2239488}+%
\frac{685}{124416}\log^{4}2-\frac{685}{20736}\zeta_{2}\log^{2}2+\frac{3645913}%
{995328}\zeta_{3}\right.$$
$$\displaystyle\left.-\frac{541549}{165888}\zeta_{4}+\frac{115}{576}\zeta_{5}+%
\frac{685}{5184}a_{4}\right)+N_{F}^{2}\left(-\frac{271883}{4478976}+\frac{167}%
{5184}\zeta_{3}\right)$$
$$\displaystyle-$$
$$\displaystyle\left[\frac{7391699}{746496}+\frac{2529743}{165888}\zeta_{3}+N_{F%
}\left(\frac{110341}{373248}-\frac{110779}{82944}\zeta_{3}\right)-N_{F}^{2}%
\frac{6865}{186624}\right]L_{t}$$
$$\displaystyle+$$
$$\displaystyle\left(\frac{2177}{3456}-N_{F}\frac{1483}{10368}-N_{F}^{2}\frac{77%
}{20736}\right)L_{t}^{2}-\left(\frac{1883}{10368}+N_{F}\frac{127}{5184}-\frac{%
N_{F}^{2}}{324}\right)L_{t}^{3}+\frac{L_{t}^{4}}{1296}$$
where $\overline{m_{t}}^{2}(\mu_{R}^{2}))$ denotes the $\overline{\rm MS}$ top
mass at the renormalization scale $\mu_{R}$. The constants used in this
expression are given by $a_{n}=Li_{n}(1/2)$ and $X_{0}=1.8088795462...$ . The decoupling coefficient contains
one-particle-reducible contributions and the Wilson coefficient
of the Lagrangian Eq. (1) is obtained from the
inverse,
$$\Pi_{t}=1-\frac{1}{\zeta_{\alpha_{s}}}=\sum_{n}C_{n}\left(\frac{\alpha_{s}^{(N%
_{F}+1)}}{\pi}\right)^{n}\nopagebreak$$
(4)
with the perturbative coefficients up to fifth order
$$\displaystyle C_{1}$$
$$\displaystyle=$$
$$\displaystyle-\frac{1}{6}L_{t}\qquad\qquad\qquad\qquad\qquad\qquad C_{2}=\frac%
{11}{72}-\frac{11}{24}L_{t}$$
$$\displaystyle C_{3}$$
$$\displaystyle=$$
$$\displaystyle\frac{564731}{124416}-\frac{82043}{27648}\zeta_{3}-\frac{2633}{31%
104}N_{F}-\frac{2777-201N_{F}}{1728}L_{t}-\frac{35+16N_{F}}{576}L_{t}^{2}$$
$$\displaystyle C_{4}$$
$$\displaystyle=$$
$$\displaystyle\frac{1166295847}{24494400}-\frac{121}{4320}\log^{5}2+\frac{30313%
09}{1306368}\log^{4}2+\frac{121}{432}\zeta_{2}\log^{3}2-\frac{3031309}{217728}%
\zeta_{2}\log^{2}2$$
$$\displaystyle+$$
$$\displaystyle\frac{2057}{576}\zeta_{4}\log 2+\frac{1389}{256}\zeta_{5}-\frac{7%
6940219}{2177280}\zeta_{4}-\frac{2362581983}{87091200}\zeta_{3}+\frac{3031309}%
{54432}a_{4}+\frac{121}{36}a_{5}$$
$$\displaystyle-$$
$$\displaystyle\frac{151369}{2177280}X_{0}+N_{F}\left(-\frac{4770941}{2239488}+%
\frac{685}{124416}\log^{4}2-\frac{685}{20736}\zeta_{2}\log^{2}2+\frac{3645913}%
{995328}\zeta_{3}\right.$$
$$\displaystyle\left.-\frac{541549}{165888}\zeta_{4}+\frac{115}{576}\zeta_{5}+%
\frac{685}{5184}a_{4}\right)+N_{F}^{2}\left(-\frac{271883}{4478976}+\frac{167}%
{5184}\zeta_{3}\right)$$
$$\displaystyle+$$
$$\displaystyle\left[\frac{2875235}{248832}-\frac{897943}{55296}\zeta_{3}-N_{F}%
\left(\frac{40291}{124416}-\frac{110779}{82944}\zeta_{3}\right)+N_{F}^{2}\frac%
{6865}{186624}\right]L_{t}$$
$$\displaystyle-$$
$$\displaystyle\left(\frac{1333}{10368}+N_{F}\frac{1081}{10368}+N_{F}^{2}\frac{7%
7}{20736}\right)L_{t}^{2}-\left(\frac{1697}{10368}+N_{F}\frac{175}{5184}-N_{F}%
^{2}\frac{1}{324}\right)L_{t}^{3}$$
$$\displaystyle C_{5}$$
$$\displaystyle=$$
$$\displaystyle C_{50}+\left(-\frac{685}{10368}N_{F}^{2}a_{4}-\frac{11679301}{43%
5456}N_{F}a_{4}+\frac{93970579}{217728}a_{4}-\frac{121}{72}N_{F}a_{5}+\frac{37%
51}{144}a_{5}\right.$$
(5)
$$\displaystyle\left.+\frac{121}{8640}N_{F}\log^{5}2-\frac{3751}{17280}\log^{5}2%
-\frac{685}{248832}N_{F}^{2}\log^{4}2-\frac{11679301}{10450944}N_{F}\log^{4}2\right.$$
$$\displaystyle\left.+\frac{93970579}{5225472}\log^{4}2-\frac{121}{864}N_{F}%
\zeta_{2}\log^{3}2+\frac{3751}{1728}\zeta_{2}\log^{3}2+\frac{685}{41472}N_{F}^%
{2}\zeta_{2}\log^{2}2\right.$$
$$\displaystyle\left.+\frac{11679301}{1741824}N_{F}\zeta_{2}\log^{2}2-\frac{9397%
0579}{870912}\zeta_{2}\log^{2}2-\frac{2057}{1152}N_{F}\zeta_{4}\log 2+\frac{63%
767}{2304}\zeta_{4}\log 2\right.$$
$$\displaystyle\left.-\frac{211}{10368}N_{F}^{3}\zeta_{3}+\frac{270407}{8957952}%
N_{F}^{3}-\frac{4091305}{1990656}N_{F}^{2}\zeta_{3}+\frac{576757}{331776}N_{F}%
^{2}\zeta_{4}+\frac{115}{2304}N_{F}^{2}\zeta_{5}\right.$$
$$\displaystyle\left.+\frac{48073}{165888}N_{F}^{2}+\frac{151369}{4354560}N_{F}X%
_{0}+\frac{12171659669}{232243200}N_{F}\zeta_{3}-\frac{608462731}{69672960}N_{%
F}\zeta_{4}\right.$$
$$\displaystyle\left.-\frac{313489}{41472}N_{F}\zeta_{5}-\frac{75861299783}{3135%
283200}N_{F}-\frac{4692439}{8709120}X_{0}-\frac{4660543511}{19353600}\zeta_{3}\right.$$
$$\displaystyle\left.-\frac{4674213853}{17418240}\zeta_{4}+\frac{807193}{10368}%
\zeta_{5}+\frac{846138861149}{3135283200}\right)L_{t}+\left(-\frac{481}{62208}%
N_{F}^{3}-\frac{28297}{110592}N_{F}^{2}\zeta_{3}\right.$$
$$\displaystyle\left.+\frac{373637}{746496}N_{F}^{2}+\frac{2985893}{331776}N_{F}%
\zeta_{3}-\frac{47813}{4608}N_{F}-\frac{26296585}{442368}\zeta_{3}+\frac{14393%
9741}{1990656}\right)L_{t}^{2}$$
$$\displaystyle+$$
$$\displaystyle\left(\frac{77}{124416}N_{F}^{3}+\frac{175}{27648}N_{F}^{2}-\frac%
{5855}{124416}N_{F}-\frac{130201}{124416}\right)L_{t}^{3}$$
$$\displaystyle+$$
$$\displaystyle\left(-\frac{1}{2592}N_{F}^{3}+\frac{47}{4608}N_{F}^{2}-\frac{317%
}{6912}N_{F}-\frac{51383}{165888}\right)L_{t}^{4}$$
where the logarithms of the coefficient $C_{5}$ have been reconstructed
from the result of Ref. [25] including the recent five-loop
result of the QCD beta function [26] (partly confirmed by
[27]). The constant $C_{50}$ is irrelevant for our derivation
of the effective Lagrangian for gluonic Higgs couplings. Note that the
highest powers of the logarithmic $L_{t}$ terms disappeared in this
expression as required by the proper RG-evolution of the
one-particle-irreducible part $\Pi_{t}$. Using the low-energy theorem for
a light Higgs boson [16] the effective top-quark contribution to
the Lagrangian of Eq. (1) is related to the couplings of
external Higgs bosons in the heavy-top-quark limit by the
replacement222In the case of an extended Higgs sector with
several scalar Higgs bosons coupling to the top quark the replacement
$\overline{m_{t}}(\mu_{R}^{2})\to\overline{m_{t}}(\mu_{R}^{2})(1+\sum_{i}c_{i}H%
_{i}/v)$ has to be implemented, where $c_{i}$ are the top quark Yukawa
couplings normalized to the SM coupling. This results in the
correspondence $H/v\leftrightarrow\sum_{i}c_{i}H_{i}/v$ for all subsequent
steps. $\overline{m_{t}}(\mu_{R}^{2})\to\overline{m_{t}}(\mu_{R}^{2})(1+H/v)$,
i.e.
$$L_{t}\to\bar{L}_{t}=L_{t}-2\log\left(1+\frac{H}{v}\right)\qquad\mbox{and}%
\qquad\Pi_{t}\to\bar{\Pi}_{t}$$
(6)
where $H$ denotes the physical Higgs field, $v$ the vacuum expectation
value and $\bar{\Pi}_{t}$ the contribution to the Wilson coefficient with
the shifted top-quark mass333Note that diagrammatically for the
single-Higgs case this expression coincides with the replacement
$\frac{1}{\not\!\;p-m_{t}}\to\frac{1}{\not\!\;p-m_{t}}\frac{m_{t}}{v}\frac{1}{%
\not\!\;p-m_{t}}$ of the top-quark propagators inside the
gluonic correlation functions up to 4th order in the gluon fields at the
point where $m_{t}$ is either the unrenormalized or the pure
$\overline{\rm MS}$ mass [9].. Based on this replacement it
is obvious that only the logarithmic $L_{t}$ terms of $\Pi_{t}$ are relevant
for the effective gluonic Higgs couplings. The object $\bar{\Pi}_{t}$ is
expressed in terms of the $(N_{F}+1)$-flavour coupling
$\alpha_{s}^{(N_{F}+1)}$. To derive the low-energy Lagrangian in the
$N_{F}$-flavour theory we have to transform the $(N_{F}+1)$-flavour coupling
into the $N_{F}$-flavour one by means of the relation [23, 24, 25]
$$\displaystyle\alpha_{s}^{(N_{F}+1)}(\mu_{R}^{2})$$
$$\displaystyle=$$
$$\displaystyle\alpha_{s}^{(N_{F})}(\mu_{R}^{2})\left\{1+\frac{\alpha_{s}^{(N_{F%
})}(\mu_{R}^{2})}{\pi}\frac{L_{t}}{6}+\left(\frac{\alpha_{s}^{(N_{F})}(\mu_{R}%
^{2})}{\pi}\right)^{2}\left[-\frac{11}{72}+\frac{11}{24}L_{t}+\frac{L_{t}^{2}}%
{36}\right]\right.$$
(7)
$$\displaystyle+$$
$$\displaystyle\left.\left(\frac{\alpha_{s}^{(N_{F})}(\mu_{R}^{2})}{\pi}\right)^%
{3}\left[-\frac{564731}{124416}+\frac{82043}{27648}\zeta_{3}+\frac{2633}{31104%
}N_{F}\right.\right.$$
$$\displaystyle\left.\left.+\left(\frac{2645}{1728}-\frac{67}{576}N_{F}\right)L_%
{t}+\left(\frac{167}{576}+\frac{N_{F}}{36}\right)L_{t}^{2}+\frac{L_{t}^{3}}{21%
6}\right]+{\cal O}(\alpha_{s}^{4})\right\}$$
derived from inverting Eq. (2). For the proper low-energy
limit the gluonic field-strength operator is expressed in terms of the
one with $N_{F}=5$ active flavours which leads to a global factor
$\zeta_{\alpha_{s}}$ so that the kinetic term of the gluons is properly
normalized in the low-energy limit444Diagrammatically this step
corresponds to adding the external $\overline{\rm MS}$-renormalized
self-energies and two-point-function contributions to the vertices
involving top quarks at vanishing external momentum.. In this way we
arrive at the low-energy Lagrangian in terms of the top $\overline{\rm MS}$ mass. The effective N${}^{4}$LO Lagrangian for (multi-)Higgs couplings
to gluons reads finally
$$\displaystyle{\cal L}_{eff}$$
$$\displaystyle=$$
$$\displaystyle\frac{\alpha_{s}}{12\pi}\left\{(1+\delta)\log\left(1+\frac{H}{v}%
\right)-\frac{\eta}{2}\log^{2}\left(1+\frac{H}{v}\right)\right.$$
(8)
$$\displaystyle \left.+\frac{\rho}{3}\log^{3}\left(1+\frac{H}{v}%
\right)-\frac{\sigma}{4}\log^{4}\left(1+\frac{H}{v}\right)\right\}G^{a\mu\nu}G%
^{a}_{\mu\nu}$$
with the QCD corrections up to N${}^{4}$LO
$$\displaystyle\delta$$
$$\displaystyle=$$
$$\displaystyle\delta_{1}\frac{\alpha_{s}}{\pi}+\delta_{2}\left(\frac{\alpha_{s}%
}{\pi}\right)^{2}+\delta_{3}\left(\frac{\alpha_{s}}{\pi}\right)^{3}+\delta_{4}%
\left(\frac{\alpha_{s}}{\pi}\right)^{4}+{\cal O}(\alpha_{s}^{5})$$
$$\displaystyle\eta$$
$$\displaystyle=$$
$$\displaystyle\eta_{2}\left(\frac{\alpha_{s}}{\pi}\right)^{2}+\eta_{3}\left(%
\frac{\alpha_{s}}{\pi}\right)^{3}+\eta_{4}\left(\frac{\alpha_{s}}{\pi}\right)^%
{4}+{\cal O}(\alpha_{s}^{5})$$
$$\displaystyle\rho$$
$$\displaystyle=$$
$$\displaystyle\rho_{3}\left(\frac{\alpha_{s}}{\pi}\right)^{3}+\rho_{4}\left(%
\frac{\alpha_{s}}{\pi}\right)^{4}+{\cal O}(\alpha_{s}^{5})$$
$$\displaystyle\sigma$$
$$\displaystyle=$$
$$\displaystyle\sigma_{4}\left(\frac{\alpha_{s}}{\pi}\right)^{4}+{\cal O}(\alpha%
_{s}^{5})$$
(9)
The explicit perturbative coefficients are given by
$$\displaystyle\delta_{1}$$
$$\displaystyle=$$
$$\displaystyle\frac{11}{4}\qquad\qquad\qquad\qquad\qquad\qquad\delta_{2}=\frac{%
2777}{288}+\frac{19}{16}L_{t}+N_{F}\left(\frac{L_{t}}{3}-\frac{67}{96}\right)$$
$$\displaystyle\delta_{3}$$
$$\displaystyle=$$
$$\displaystyle\frac{897943}{9216}\zeta_{3}-\frac{2892659}{41472}+\frac{209}{64}%
L_{t}^{2}+\frac{1733}{288}L_{t}$$
$$\displaystyle+$$
$$\displaystyle N_{F}\left(\frac{40291}{20736}-\frac{110779}{13824}\zeta_{3}+%
\frac{23}{32}L_{t}^{2}+\frac{55}{54}L_{t}\right)+N_{F}^{2}\left(-\frac{L_{t}^{%
2}}{18}+\frac{77}{1728}L_{t}-\frac{6865}{31104}\right)$$
$$\displaystyle\delta_{4}$$
$$\displaystyle=$$
$$\displaystyle-\frac{121}{1440}N_{F}\log^{5}2+\frac{3751}{2880}\log^{5}2+\frac{%
685}{41472}N_{F}^{2}\log^{4}2+\frac{11679301}{1741824}N_{F}\log^{4}2$$
(10)
$$\displaystyle-$$
$$\displaystyle\frac{93970579}{870912}\log^{4}2+\frac{121}{144}N_{F}\zeta_{2}%
\log^{3}2-\frac{3751}{288}\zeta_{2}\log^{3}2-\frac{685}{6912}N_{F}^{2}\zeta_{2%
}\log^{2}2$$
$$\displaystyle-$$
$$\displaystyle\frac{11679301}{290304}N_{F}\zeta_{2}\log^{2}2+\frac{93970579}{14%
5152}\zeta_{2}\log^{2}2+\frac{2057}{192}N_{F}\zeta_{4}\log 2-\frac{63767}{384}%
\zeta_{4}\log 2$$
$$\displaystyle+$$
$$\displaystyle\frac{685}{1728}N_{F}^{2}a_{4}+\frac{11679301}{72576}N_{F}a_{4}-%
\frac{93970579}{36288}a_{4}+\frac{121}{12}N_{F}a_{5}-\frac{3751}{24}a_{5}+%
\frac{211}{1728}N_{F}^{3}\zeta_{3}$$
$$\displaystyle-$$
$$\displaystyle\frac{270407}{1492992}N_{F}^{3}+\frac{4091305}{331776}N_{F}^{2}%
\zeta_{3}-\frac{576757}{55296}N_{F}^{2}\zeta_{4}-\frac{115}{384}N_{F}^{2}\zeta%
_{5}-\frac{48073}{27648}N_{F}^{2}$$
$$\displaystyle-$$
$$\displaystyle\frac{151369}{725760}N_{F}X_{0}-\frac{12171659669}{38707200}N_{F}%
\zeta_{3}+\frac{608462731}{11612160}N_{F}\zeta_{4}+\frac{313489}{6912}N_{F}%
\zeta_{5}$$
$$\displaystyle+$$
$$\displaystyle\frac{76094378783}{522547200}N_{F}+\frac{4692439}{1451520}X_{0}+%
\frac{28121193841}{19353600}\zeta_{3}+\frac{4674213853}{2903040}\zeta_{4}-%
\frac{807193}{1728}\zeta_{5}$$
$$\displaystyle-$$
$$\displaystyle\frac{854201072999}{522547200}+\left(\frac{481}{5184}N_{F}^{3}+%
\frac{28297}{9216}N_{F}^{2}\zeta_{3}-\frac{21139}{3456}N_{F}^{2}-\frac{32257}{%
288}N_{F}\zeta_{3}\right.$$
$$\displaystyle\left.+\frac{5160073}{41472}N_{F}+\frac{9364157}{12288}\zeta_{3}-%
\frac{49187545}{55296}\right)L_{t}+\left(-\frac{77}{6912}N_{F}^{3}-\frac{1267}%
{13824}N_{F}^{2}+\frac{4139}{2304}N_{F}\right.$$
$$\displaystyle\left.+\frac{8401}{384}\right)L_{t}^{2}+\left(\frac{1}{108}N_{F}^%
{3}-\frac{157}{576}N_{F}^{2}+\frac{275}{192}N_{F}+\frac{2299}{256}\right)L_{t}%
^{3}$$
and
$$\displaystyle\eta_{2}$$
$$\displaystyle=$$
$$\displaystyle\frac{35}{24}+\frac{2}{3}N_{F}$$
$$\displaystyle\eta_{3}$$
$$\displaystyle=$$
$$\displaystyle\frac{1333}{432}+\frac{589}{48}L_{t}+N_{F}\left(\frac{1081}{432}+%
\frac{191}{72}L_{t}\right)+N_{F}^{2}\left(\frac{77}{864}-\frac{2}{9}L_{t}\right)$$
$$\displaystyle\eta_{4}$$
$$\displaystyle=$$
$$\displaystyle\frac{481}{2592}N_{F}^{3}+N_{F}^{2}\left(\frac{28297}{4608}\zeta_%
{3}-\frac{373637}{31104}\right)+N_{F}\left(\frac{429965}{1728}-\frac{2985893}{%
13824}\zeta_{3}\right)$$
$$\displaystyle+$$
$$\displaystyle\frac{26296585}{18432}\zeta_{3}-\frac{143976701}{82944}+\left(-%
\frac{77}{1728}N_{F}^{3}-\frac{1421}{3456}N_{F}^{2}+\frac{9073}{1728}N_{F}+%
\frac{45059}{576}\right)L_{t}$$
$$\displaystyle+$$
$$\displaystyle\left(\frac{N_{F}^{3}}{18}-\frac{455}{288}N_{F}^{2}+\frac{63}{8}N%
_{F}+\frac{6479}{128}\right)L_{t}^{2}$$
$$\displaystyle\rho_{3}$$
$$\displaystyle=$$
$$\displaystyle\frac{1697}{144}+\frac{175}{72}N_{F}-\frac{2}{9}N_{F}^{2}$$
$$\displaystyle\rho_{4}$$
$$\displaystyle=$$
$$\displaystyle\frac{130201}{1728}+\frac{18259}{192}L_{t}+N_{F}\left(\frac{5855}%
{1728}+\frac{2077}{144}L_{t}\right)-N_{F}^{2}\left(\frac{175}{384}+\frac{439}{%
144}L_{t}\right)$$
$$\displaystyle+$$
$$\displaystyle N_{F}^{3}\left(\frac{L_{t}}{9}-\frac{77}{1728}\right)$$
$$\displaystyle\sigma_{4}$$
$$\displaystyle=$$
$$\displaystyle\frac{51383}{864}+\frac{317}{36}N_{F}-\frac{47}{24}N_{F}^{2}+%
\frac{2}{27}N_{F}^{3}$$
(11)
where $G^{a}_{\mu\nu}$ denotes the gluon field strength tensor and
$\alpha_{s}$ the strong coupling constant with $N_{F}=5$ active flavours.
Note that in accordance with the RG-evolution the coefficients
$\delta_{1},\eta_{2},\rho_{3}$ and $\sigma_{4}$ are free of $L_{t}$ terms.
Numerically we obtain for $N_{F}=5$ light flavours
$$\displaystyle\delta_{1}$$
$$\displaystyle=$$
$$\displaystyle 2.75\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad%
\delta_{2}=6.1528+2.8542L_{t}$$
$$\displaystyle\delta_{3}$$
$$\displaystyle=$$
$$\displaystyle 3.4043+12.2240L_{t}+5.4705L_{t}^{2}$$
$$\displaystyle\delta_{4}$$
$$\displaystyle=$$
$$\displaystyle 36.0373-73.5997L_{t}+27.1760L_{t}^{2}+10.4851L_{t}^{3}$$
$$\displaystyle\eta_{2}$$
$$\displaystyle=$$
$$\displaystyle 4.7917\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\eta%
_{3}=17.8252+19.9792L_{t}$$
$$\displaystyle\eta_{4}$$
$$\displaystyle=$$
$$\displaystyle-167.5239+88.6311L_{t}+57.4401L_{t}^{2}\qquad\qquad\quad\rho_{3}=%
18.3819$$
$$\displaystyle\rho_{4}$$
$$\displaystyle=$$
$$\displaystyle 75.3261+104.8906L_{t}\qquad\qquad\qquad\qquad\qquad\quad\,\,%
\sigma_{4}=63.7998$$
(12)
If the running $\overline{\rm MS}$ top mass is replaced by the top pole
mass $M_{t}$ [28]555Note that the low-energy strong
coupling constant with $N_{F}=5$ active flavours is used in this
relation. [i.e. $L_{t}=\log(\mu_{R}^{2}/M_{t}^{2})$ is used everywhere],
$$\displaystyle\overline{m_{t}}(\mu_{R}^{2})$$
$$\displaystyle=$$
$$\displaystyle M_{t}\left\{1-\left(\frac{4}{3}+\log\frac{\mu_{R}^{2}}{M_{t}^{2}%
}\right)\frac{\alpha_{s}^{(N_{F})}(\mu_{R}^{2})}{\pi}+\left[-\frac{3019}{288}-%
2\zeta_{2}-\frac{2}{3}\zeta_{2}\log 2+\frac{\zeta_{3}}{6}\right.\right.$$
(13)
$$\displaystyle\left.\left.-\frac{461}{72}\log\frac{\mu_{R}^{2}}{M_{t}^{2}}-%
\frac{23}{24}\log^{2}\frac{\mu_{R}^{2}}{M_{t}^{2}}+N_{F}\left(\frac{71}{144}+%
\frac{\zeta_{2}}{3}+\frac{13}{36}\log\frac{\mu_{R}^{2}}{M_{t}^{2}}+\frac{1}{12%
}\log^{2}\frac{\mu_{R}^{2}}{M_{t}^{2}}\right)\right.\right.$$
$$\displaystyle\left.\left.-\frac{4}{3}\sum_{1\leq i\leq N_{F}}\Delta\left(\frac%
{M_{i}}{M_{t}}\right)\right]\left(\frac{\alpha_{s}^{(N_{F})}(\mu_{R}^{2})}{\pi%
}\right)^{2}\right\}+{\cal O}(\alpha_{s}^{3})$$
where the mass-dependent term involving the light flavours can be
approximated by
$$\Delta(x)=\frac{\pi^{2}}{8}~{}x-0.579~{}x^{2}+0.230~{}x^{3}$$
(14)
the QCD corrections are formally different from the $\overline{\rm MS}$
case above only for the coefficients $\delta_{3},\delta_{4}$ and $\eta_{4}$,
$$\displaystyle\delta_{3}$$
$$\displaystyle=$$
$$\displaystyle\frac{897943}{9216}\zeta_{3}-\frac{2761331}{41472}+\frac{209}{64}%
L_{t}^{2}+\frac{2417}{288}L_{t}$$
$$\displaystyle+$$
$$\displaystyle N_{F}\left(\frac{58723}{20736}-\frac{110779}{13824}\zeta_{3}+%
\frac{23}{32}L_{t}^{2}+\frac{91}{54}L_{t}\right)+N_{F}^{2}\left(-\frac{L_{t}^{%
2}}{18}+\frac{77}{1728}L_{t}-\frac{6865}{31104}\right)$$
$$\displaystyle\delta_{4}$$
$$\displaystyle=$$
$$\displaystyle-\frac{121}{1440}N_{F}\log^{5}2+\frac{3751}{2880}\log^{5}2+\frac{%
685}{41472}N_{F}^{2}\log^{4}2+\frac{11679301}{1741824}N_{F}\log^{4}2$$
$$\displaystyle-$$
$$\displaystyle\frac{93970579}{870912}\log^{4}2+\frac{121}{144}N_{F}\zeta_{2}%
\log^{3}2-\frac{3751}{288}\zeta_{2}\log^{3}2-\frac{685}{6912}N_{F}^{2}\zeta_{2%
}\log^{2}2$$
$$\displaystyle-$$
$$\displaystyle\frac{11679301}{290304}N_{F}\zeta_{2}\log^{2}2+\frac{93970579}{14%
5152}\zeta_{2}\log^{2}2+\frac{4}{9}N_{F}\zeta_{2}\log 2+\frac{19}{12}\zeta_{2}\log
2$$
$$\displaystyle+$$
$$\displaystyle\frac{2057}{192}N_{F}\zeta_{4}\log 2-\frac{63767}{384}\zeta_{4}%
\log 2+\frac{685}{1728}N_{F}^{2}a_{4}+\frac{11679301}{72576}N_{F}a_{4}-\frac{9%
3970579}{36288}a_{4}$$
$$\displaystyle+$$
$$\displaystyle\frac{121}{12}N_{F}a_{5}-\frac{3751}{24}a_{5}+\frac{211}{1728}N_{%
F}^{3}\zeta_{3}-\frac{270407}{1492992}N_{F}^{3}-\frac{2}{9}N_{F}^{2}\zeta_{2}+%
\frac{4091305}{331776}N_{F}^{2}\zeta_{3}$$
$$\displaystyle-$$
$$\displaystyle\frac{576757}{55296}N_{F}^{2}\zeta_{4}-\frac{115}{384}N_{F}^{2}%
\zeta_{5}-\frac{161627}{82944}N_{F}^{2}-\frac{151369}{725760}N_{F}X_{0}+\frac{%
13}{24}N_{F}\zeta_{2}+\frac{19}{4}\zeta_{2}$$
$$\displaystyle-$$
$$\displaystyle\frac{12175960469}{38707200}N_{F}\zeta_{3}+\frac{608462731}{11612%
160}N_{F}\zeta_{4}+\frac{313489}{6912}N_{F}\zeta_{5}+\frac{80863176383}{522547%
200}N_{F}$$
$$\displaystyle+$$
$$\displaystyle\frac{4692439}{1451520}X_{0}+\frac{28113533041}{19353600}\zeta_{3%
}+\frac{4674213853}{2903040}\zeta_{4}-\frac{807193}{1728}\zeta_{5}-\frac{83170%
3495799}{522547200}$$
$$\displaystyle+$$
$$\displaystyle\left(\frac{481}{5184}N_{F}^{3}+\frac{28297}{9216}N_{F}^{2}\zeta_%
{3}-\frac{22687}{3456}N_{F}^{2}-\frac{32257}{288}N_{F}\zeta_{3}+\frac{5581849}%
{41472}N_{F}+\frac{9364157}{12288}\zeta_{3}\right.$$
$$\displaystyle\left.-\frac{46543033}{55296}\right)L_{t}+\left(-\frac{77}{6912}N%
_{F}^{3}-\frac{5107}{13824}N_{F}^{2}+\frac{12547}{2304}N_{F}+\frac{14747}{384}%
\right)L_{t}^{2}$$
$$\displaystyle+$$
$$\displaystyle\left(\frac{1}{108}N_{F}^{3}-\frac{157}{576}N_{F}^{2}+\frac{275}{%
192}N_{F}+\frac{2299}{256}\right)L_{t}^{3}+\frac{4}{3}\left(\frac{2}{3}N_{F}+%
\frac{19}{8}\right)\sum_{1\leq i\leq N_{F}}\Delta\left(\frac{M_{i}}{M_{t}}\right)$$
$$\displaystyle\eta_{4}$$
$$\displaystyle=$$
$$\displaystyle\frac{481}{2592}N_{F}^{3}+N_{F}^{2}\left(\frac{28297}{4608}\zeta_%
{3}-\frac{392069}{31104}\right)+N_{F}\left(\frac{442189}{1728}-\frac{2985893}{%
13824}\zeta_{3}\right)$$
(15)
$$\displaystyle+$$
$$\displaystyle\frac{26296585}{18432}\zeta_{3}-\frac{141262589}{82944}+\left(-%
\frac{77}{1728}N_{F}^{3}-\frac{2957}{3456}N_{F}^{2}+\frac{18241}{1728}N_{F}+%
\frac{59195}{576}\right)L_{t}$$
$$\displaystyle+$$
$$\displaystyle\left(\frac{N_{F}^{3}}{18}-\frac{455}{288}N_{F}^{2}+\frac{63}{8}N%
_{F}+\frac{6479}{128}\right)L_{t}^{2}$$
For the on-shell top-quark mass we obtain numerically for $N_{F}=5$ light
flavours
$$\displaystyle\delta_{3}$$
$$\displaystyle=$$
$$\displaystyle 11.0154+17.9323L_{t}+5.4705L_{t}^{2}$$
$$\displaystyle\delta_{4}$$
$$\displaystyle=$$
$$\displaystyle 125.7997+13.8777L_{t}+55.0041L_{t}^{2}+10.4851L_{t}^{3}+7.6111%
\sum_{1\leq i\leq N_{F}}\Delta\left(\frac{M_{i}}{M_{t}}\right)$$
$$\displaystyle\eta_{4}$$
$$\displaystyle=$$
$$\displaystyle-114.2461+128.5894L_{t}+57.4401L_{t}^{2}$$
(16)
The explicit expansion of the Lagrangian of Eq. (8) in powers
of the Higgs field results in
$${\cal L}_{eff}=\frac{\alpha_{s}}{12\pi}\left\{\sum_{n=1}^{\infty}\Delta_{n}%
\frac{(-1)^{n-1}}{n}\left(\frac{H}{v}\right)^{n}\right\}G^{a\mu\nu}G^{a}_{\mu\nu}$$
(17)
with the QCD corrections up to N${}^{4}$LO
$$\displaystyle\Delta_{1}$$
$$\displaystyle=$$
$$\displaystyle 1+\delta_{1}\frac{\alpha_{s}}{\pi}+\delta_{2}\left(\frac{\alpha_%
{s}}{\pi}\right)^{2}+\delta_{3}\left(\frac{\alpha_{s}}{\pi}\right)^{3}+\delta_%
{4}\left(\frac{\alpha_{s}}{\pi}\right)^{4}+{\cal O}(\alpha_{s}^{5})$$
$$\displaystyle\Delta_{2}$$
$$\displaystyle=$$
$$\displaystyle 1+\delta_{1}\frac{\alpha_{s}}{\pi}+(\delta_{2}+\eta_{2})\left(%
\frac{\alpha_{s}}{\pi}\right)^{2}+(\delta_{3}+\eta_{3})\left(\frac{\alpha_{s}}%
{\pi}\right)^{3}+(\delta_{4}+\eta_{4})\left(\frac{\alpha_{s}}{\pi}\right)^{4}+%
{\cal O}(\alpha_{s}^{5})$$
$$\displaystyle\Delta_{3}$$
$$\displaystyle=$$
$$\displaystyle 1+\delta_{1}\frac{\alpha_{s}}{\pi}+\left(\delta_{2}+\frac{3}{2}%
\eta_{2}\right)\left(\frac{\alpha_{s}}{\pi}\right)^{2}+\left(\delta_{3}+\frac{%
3}{2}\eta_{3}+\rho_{3}\right)\left(\frac{\alpha_{s}}{\pi}\right)^{3}$$
$$\displaystyle+$$
$$\displaystyle\left(\delta_{4}+\frac{3}{2}\eta_{4}+\rho_{4}\right)\left(\frac{%
\alpha_{s}}{\pi}\right)^{4}+{\cal O}(\alpha_{s}^{5})$$
$$\displaystyle\Delta_{4}$$
$$\displaystyle=$$
$$\displaystyle 1+\delta_{1}\frac{\alpha_{s}}{\pi}+\left(\delta_{2}+\frac{11}{6}%
\eta_{2}\right)\left(\frac{\alpha_{s}}{\pi}\right)^{2}+\left(\delta_{3}+\frac{%
11}{6}\eta_{3}+2\rho_{3}\right)\left(\frac{\alpha_{s}}{\pi}\right)^{3}$$
$$\displaystyle+$$
$$\displaystyle\left(\delta_{4}+\frac{11}{6}\eta_{4}+2\rho_{4}+\sigma_{4}\right)%
\left(\frac{\alpha_{s}}{\pi}\right)^{4}+{\cal O}(\alpha_{s}^{5})$$
$$\displaystyle\Delta_{5}$$
$$\displaystyle=$$
$$\displaystyle 1+\delta_{1}\frac{\alpha_{s}}{\pi}+\left(\delta_{2}+\frac{25}{12%
}\eta_{2}\right)\left(\frac{\alpha_{s}}{\pi}\right)^{2}+\left(\delta_{3}+\frac%
{25}{12}\eta_{3}+\frac{35}{12}\rho_{3}\right)\left(\frac{\alpha_{s}}{\pi}%
\right)^{3}$$
(18)
$$\displaystyle+$$
$$\displaystyle\left(\delta_{4}+\frac{25}{12}\eta_{4}+\frac{35}{12}\rho_{4}+%
\frac{5}{2}\sigma_{4}\right)\left(\frac{\alpha_{s}}{\pi}\right)^{4}+{\cal O}(%
\alpha_{s}^{5})$$
for up to five external Higgs bosons. It should be noted that the
coefficients $\delta_{1-4}$ of the single-Higgs term $\Delta_{1}$ agree
with previous results up to N${}^{4}$LO [13, 24, 25, 29],
while the coefficient $\eta_{2}$ of the double-Higgs contribution
$\Delta_{2}$ agrees with the explicit diagrammatic calculation of
Ref. [19].
Connecting our approach to derive the effective Lagrangian to the method
of Refs. [24, 25] for the single-Higgs case we can easily derive
their final relation,
$$C_{H}=-\frac{1}{4}\zeta_{\alpha_{s}}~{}g_{t}\partial_{m_{t}}\frac{1}{\zeta_{%
\alpha_{s}}}=\frac{1}{2v}\frac{m_{t}^{2}\partial}{\partial(m_{t}^{2})}\log%
\zeta_{\alpha_{s}}$$
(19)
with $g_{t}=m_{t}/v$, $\partial_{m_{t}}=\partial/\partial m_{t}$ and $C_{H}$
denoting the full coefficient in front of the operator $G^{a\mu\nu}G^{a}_{\mu\nu}H$. This expression agrees with Refs. [24, 25]. For the double-Higgs case we arrive at
$$C_{HH}=\frac{1}{8}\zeta_{\alpha_{s}}~{}g_{t}^{2}\partial_{m_{t}}^{2}\frac{1}{%
\zeta_{\alpha_{s}}}=\frac{1}{4v^{2}}\left\{\left(\frac{m_{t}\partial_{m_{t}}%
\zeta_{\alpha_{s}}}{\zeta_{\alpha_{s}}}\right)^{2}-\frac{m_{t}^{2}\partial_{m_%
{t}}^{2}\zeta_{\alpha_{s}}}{2\zeta_{\alpha_{s}}}\right\}$$
(20)
where $C_{HH}$ denotes the coefficient in front of the operator
$G^{a\mu\nu}G^{a}_{\mu\nu}H^{2}$.
A final comment addresses the removal of one-particle-reducible
contributions in Eq. (4): this corresponds to the removal of
one-particle-reducible diagrams of the type shown in Fig. 1
after attaching external Higgs bosons according to Eq. (6).
We have checked this correspondence explicitly for Higgs boson pair
production in the heavy-top-quark limit at NLO [17].
3 Several Heavy Quarks
Starting from the expression of the effective single-Higgs coupling to
gluons of Ref. [30] with $N_{H}$ heavy quarks contributing we can
reconstruct the corresponding logarithmic parts of the function $\Pi_{Q}$,
$$\displaystyle{\cal L}_{g}$$
$$\displaystyle=$$
$$\displaystyle-\frac{1-\Pi_{Q}}{4}\hat{G}^{a\mu\nu}\hat{G}^{a}_{\mu\nu}$$
$$\displaystyle\Pi_{Q}$$
$$\displaystyle=$$
$$\displaystyle\sum_{n}C_{n}\left(\frac{\alpha_{s}^{(N_{F}+N_{H})}}{\pi}\right)^%
{n}$$
(21)
with the perturbative coefficients up to third order
$$\displaystyle C_{1}$$
$$\displaystyle=$$
$$\displaystyle-\frac{N_{H}}{6}L_{Q}$$
(22)
$$\displaystyle C_{2}$$
$$\displaystyle=$$
$$\displaystyle N_{H}\left[\frac{11}{72}-\frac{11}{24}L_{Q}\right]$$
$$\displaystyle C_{3}$$
$$\displaystyle=$$
$$\displaystyle C_{30}-N_{H}\left(\frac{1877}{1152}-\frac{77}{3456}N_{H}-\frac{6%
7}{576}N_{F}\right)L_{Q}-N_{H}\left(\frac{19}{192}-\frac{11}{288}N_{H}+\frac{N%
_{F}}{36}\right)L_{Q}^{2}$$
where $\hat{G}^{a\mu\nu}$ denotes the gluonic field-strength operator of
colour-SU(3) in the low-energy limit with $N_{F}+N_{H}$ active flavours.
The logarithm is defined as
$$L_{Q}=\frac{1}{N_{H}}\sum_{i=1}^{N_{H}}\log\left(\frac{\mu_{R}^{2}}{M_{i}^{2}}\right)$$
(23)
For the derivation of the effective Lagrangian for the gluonic Higgs
coupling the constant $C_{30}$ is irrelevant. Performing the
replacement666Here we assume SM-type couplings of the heavy
quarks to the Higgs boson as e.g. for a sequential 4th fermion
generation. For the case of different couplings and $N_{S}$ scalar Higgs
bosons this shift has to be replaced by $\log(1+H/v)\to\sum_{i=1}^{N_{H}}\log\left(1+\sum_{j=1}^{N_{S}}c_{ij}H_{j}/v%
\right)/N_{H}$ in all subsequent steps, where the factors $c_{ij}$ denote the
Higgs Yukawa couplings normalized to the SM-Higgs coupling.
$$L_{Q}\to\bar{L}_{Q}=L_{Q}-2\log\left(1+\frac{H}{v}\right)\qquad\mbox{and}%
\qquad\Pi_{Q}\to\bar{\Pi}_{Q}$$
(24)
and decoupling the heavy quarks from the strong coupling constant
$\alpha_{s}$ by
$$\displaystyle\alpha_{s}^{(N_{F}+N_{H})}(\mu_{R}^{2})$$
$$\displaystyle=$$
$$\displaystyle\alpha_{s}^{(N_{F})}(\mu_{R}^{2})\left\{1+\frac{\alpha_{s}^{(N_{F%
})}(\mu_{R}^{2})}{\pi}N_{H}\frac{L_{Q}}{6}\right.$$
(25)
$$\displaystyle+$$
$$\displaystyle\left.\left(\frac{\alpha_{s}^{(N_{F})}(\mu_{R}^{2})}{\pi}\right)^%
{2}N_{H}\left[-\frac{11}{72}+\frac{11}{24}L_{Q}+N_{H}\frac{L_{Q}^{2}}{36}%
\right]\right\}+{\cal O}(\alpha_{s}^{4})$$
and from the gluon-field-strength operator we arrive at the effective
Lagrangian for the gluonic Higgs couplings up to NNLO
$${\cal L}_{eff}=N_{H}\frac{\alpha_{s}}{12\pi}\left\{(1+\delta)\log\left(1+\frac%
{H}{v}\right)-\frac{\eta}{2}\log^{2}\left(1+\frac{H}{v}\right)\right\}G^{a\mu%
\nu}G^{a}_{\mu\nu}$$
(26)
with the QCD corrections up to NNLO
$$\displaystyle\delta$$
$$\displaystyle=$$
$$\displaystyle\delta_{1}\frac{\alpha_{s}}{\pi}+\delta_{2}\left(\frac{\alpha_{s}%
}{\pi}\right)^{2}+{\cal O}(\alpha_{s}^{3})$$
$$\displaystyle\eta$$
$$\displaystyle=$$
$$\displaystyle\eta_{2}\left(\frac{\alpha_{s}}{\pi}\right)^{2}+{\cal O}(\alpha_{%
s}^{3})$$
(27)
The explicit perturbative coefficients read
$$\displaystyle\delta_{1}$$
$$\displaystyle=$$
$$\displaystyle\frac{11}{4}$$
$$\displaystyle\delta_{2}$$
$$\displaystyle=$$
$$\displaystyle\frac{1877}{192}-\frac{77}{576}N_{H}+\frac{19}{16}L_{Q}+N_{F}%
\left(\frac{L_{Q}}{3}-\frac{67}{96}\right)$$
$$\displaystyle\eta_{2}$$
$$\displaystyle=$$
$$\displaystyle\frac{19}{8}-\frac{11}{12}N_{H}+\frac{2}{3}N_{F}$$
(28)
The result for $\delta_{2}$ in the single-Higgs case agrees with the
results of Refs. [30, 31]. The NNLO results for more than
one external Higgs boson are new.
4 Conclusions
In this work we have derived effective (multi-)Higgs couplings to gluons
after integrating out all heavy quarks mediating these couplings. The
effective Lagrangians can be used for the computation of the production
of one or several Higgs bosons in gluon fusion at hadron colliders in
the limit of heavy quarks. In the SM we have extended the effective
Lagrangian for double-Higgs couplings to gluons to N${}^{4}$LO and derived
for the first time the N${}^{4}$LO Lagrangian for more than two SM Higgs
bosons. In the second part we extended the analysis to the case of
several heavy quarks coupling to the Higgs bosons up to NNLO. We
reproduced the existing NNLO results for the single-Higgs case. We have
derived these effective Lagrangians from their connection to the
decoupling relations of the strong coupling constant.
Acknowledgments
We are grateful to M. Mühlleitner and A. Signer for carefully reading
the manuscript and useful comments. This work is supported in part by
the Research Executive Agency (REA) of the European Union under the
Grant No. PITN-GA-2012-316704 (Higgstools).
References
[1]
G. Aad et al. [ATLAS Collaboration],
Phys. Lett. B716 (2012) 1;
S. Chatrchyan et al. [CMS Collaboration],
Phys. Lett. B716 (2012) 30.
[2]
G. Aad et al. [ATLAS and CMS Collaborations],
arXiv:1606.02266 [hep-ex].
[3]
P. W. Higgs,
Phys. Lett. 12 (1964) 132,
Phys. Rev. Lett. 13 (1964) 508
and
Phys. Rev. 145 (1966) 1156;
F. Englert and R. Brout,
Phys. Rev. Lett. 13 (1964) 321;
G. S. Guralnik, C. R. Hagen and T. W. Kibble,
Phys. Rev. Lett. 13 (1964) 585.
[4]
G. ’t Hooft,
Nucl. Phys. B 35 (1971) 167;
G. ’t Hooft and M. J. G. Veltman,
Nucl. Phys. B 44 (1972) 189.
[5]
H. M. Georgi, S. L. Glashow, M. E. Machacek and D. V. Nanopoulos,
Phys. Rev. Lett. 40 (1978) 692.
[6]
A. Djouadi, M. Spira and P. M. Zerwas,
Phys. Lett. B 264 (1991) 440;
S. Dawson,
Nucl. Phys. B 359 (1991) 283;
S. Dawson and R. Kauffman,
Phys. Rev. D 49 (1994) 2298.
[7]
R. V. Harlander and W. B. Kilgore,
Phys. Rev. D 64 (2001) 013015
and
Phys. Rev. Lett. 88 (2002) 201801;
C. Anastasiou and K. Melnikov,
Nucl. Phys. B 646 (2002) 220;
V. Ravindran, J. Smith and W. L. van Neerven,
Nucl. Phys. B 665 (2003) 325;
S. Marzani, R. D. Ball, V. Del Duca, S. Forte and A. Vicini,
Nucl. Phys. B 800 (2008) 127.
[8]
T. Gehrmann, M. Jaquier, E. W. N. Glover and A. Koukoutsakis,
JHEP 1202 (2012) 056;
C. Anastasiou, C. Duhr, F. Dulat and B. Mistlberger,
JHEP 1307 (2013) 003;
C. Anastasiou, C. Duhr, F. Dulat, F. Herzog and B. Mistlberger,
JHEP 1312 (2013) 088;
W. B. Kilgore,
Phys. Rev. D 89 (2014) 7, 073008;
Y. Li, A. von Manteuffel, R. M. Schabinger and H. X. Zhu,
Phys. Rev. D 91 (2015) 036008;
C. Anastasiou, C. Duhr, F. Dulat, E. Furlan, T. Gehrmann, F. Herzog
and B. Mistlberger,
JHEP 1503 (2015) 091;
C. Anastasiou, C. Duhr, F. Dulat, F. Herzog and B. Mistlberger,
Phys. Rev. Lett. 114 (2015) 21, 212001;
C. Anastasiou, C. Duhr, F. Dulat, E. Furlan, T. Gehrmann, F. Herzog,
A. Lazopoulos and B. Mistlberger,
JHEP 1605 (2016) 058
[9]
M. Spira, A. Djouadi, D. Graudenz and P. M. Zerwas,
Nucl. Phys. B 453 (1995) 17.
[10]
D. Graudenz, M. Spira and P. M. Zerwas,
Phys. Rev. Lett. 70 (1993) 1372;
R. Harlander and P. Kant,
JHEP 0512 (2005) 015;
C. Anastasiou, S. Bucherer and Z. Kunszt,
JHEP 0910 (2009) 068.
[11]
R. V. Harlander and K. J. Ozeren,
Phys. Lett. B 679 (2009) 467
and
JHEP 0911 (2009) 088;
A. Pak, M. Rogal and M. Steinhauser,
Phys. Lett. B 679 (2009) 473
and
JHEP 1002 (2010) 025.
[12]
R. Mueller and D. G. Öztürk,
arXiv:1512.08570 [hep-ph].
[13]
M. Krämer, E. Laenen and M. Spira,
Nucl. Phys. B 511 (1998) 523;
M. Spira,
Fortsch. Phys. 46 (1998) 203.
[14]
S. Catani, D. de Florian, M. Grazzini and P. Nason,
JHEP 0307 (2003) 028;
S. Moch and A. Vogt,
Phys. Lett. B 631 (2005) 48;
V. Ravindran,
Nucl. Phys. B 746 (2006) 58
and
Nucl. Phys. B 752 (2006) 173;
A. Idilbi, X. d. Ji, J. P. Ma and F. Yuan,
Phys. Rev. D 73 (2006) 077501;
V. Ahrens, T. Becher, M. Neubert and L. L. Yang,
Eur. Phys. J. C 62 (2009) 333;
D. de Florian and M. Grazzini,
Phys. Lett. B 674 (2009) 291;
D. de Florian, J. Mazzitelli, S. Moch and A. Vogt,
JHEP 1410 (2014) 176.;
S. Catani, L. Cieri, D. de Florian, G. Ferrera and M. Grazzini,
Nucl. Phys. B 888 (2014) 75;
M. Bonvini and L. Rottoli,
Phys. Rev. D 91 (2015) 5, 051301.
[15]
D. de Florian and M. Grazzini,
Phys. Lett. B 718 (2012) 117;
M. Bonvini and S. Marzani,
JHEP 1409 (2014) 007;
T. Schmidt and M. Spira,
Phys. Rev. D 93 (2016) no.1, 014022.
[16]
J. R. Ellis, M. K. Gaillard and D. V. Nanopoulos,
Nucl. Phys. B 106 (1976) 292;
M. A. Shifman, A. I. Vainshtein, M. B. Voloshin and V. I. Zakharov,
Sov. J. Nucl. Phys. 30 (1979) 711
[Yad. Fiz. 30 (1979) 1368];
B. A. Kniehl and M. Spira,
Z. Phys. C 69 (1995) 77.
[17]
S. Dawson, S. Dittmaier and M. Spira,
Phys. Rev. D 58 (1998) 115012.
[18]
D. de Florian and J. Mazzitelli,
Phys. Lett. B 724 (2013) 306
and
Phys. Rev. Lett. 111 (2013) 201801.
[19]
J. Grigo, K. Melnikov and M. Steinhauser,
Nucl. Phys. B 888 (2014) 17.
[20]
D. Y. Shao, C. S. Li, H. T. Li and J. Wang,
JHEP 1307 (2013) 169;
D. de Florian and J. Mazzitelli,
JHEP 1509 (2015) 053.
[21]
S. Borowka, N. Greiner, G. Heinrich, S. P. Jones, M. Kerner,
J. Schlenk, U. Schubert and T. Zirke,
arXiv:1604.06447 [hep-ph].
[22]
J. Grigo, J. Hoff, K. Melnikov and M. Steinhauser,
Nucl. Phys. B 875 (2013) 1;
R. Frederix, S. Frixione, V. Hirschi, F. Maltoni, O. Mattelaer,
P. Torrielli, E. Vryonidou and M. Zaro,
Phys. Lett. B 732 (2014) 142;
F. Maltoni, E. Vryonidou and M. Zaro,
JHEP 1411 (2014) 079.
[23]
W. Bernreuther and W. Wetzel,
Nucl. Phys. B 197 (1982) 228
Erratum: [Nucl. Phys. B 513 (1998) 758];
[24]
K. G. Chetyrkin, B. A. Kniehl and M. Steinhauser,
Nucl. Phys. B 510 (1998) 61.
[25]
Y. Schröder and M. Steinhauser,
JHEP 0601 (2006) 051;
K. G. Chetyrkin, J. H. Kühn and C. Sturm,
Nucl. Phys. B 744 (2006) 121.
[26]
P. A. Baikov, K. G. Chetyrkin and J. H. Kühn,
arXiv:1606.08659 [hep-ph].
[27]
T. Luthe, A. Maier, P. Marquard and Y. Schröder,
arXiv:1606.08662 [hep-ph].
[28]
N. Gray, D. J. Broadhurst, W. Grafe and K. Schilcher,
Z. Phys. C 48 (1990) 673;
K. G. Chetyrkin and M. Steinhauser,
Phys. Rev. Lett. 83 (1999) 4001;
K. G. Chetyrkin and M. Steinhauser,
Nucl. Phys. B 573 (2000) 617;
K. Melnikov and T. v. Ritbergen,
Phys. Lett. B 482 (2000) 99;
P. Marquard, A. V. Smirnov, V. A. Smirnov and M. Steinhauser,
Phys. Rev. Lett. 114 (2015) no.14, 142002.
[29]
T. Inami, T. Kubota and Y. Okada,
Z. Phys. C 18 (1983) 69;
K. G. Chetyrkin, B. A. Kniehl and M. Steinhauser,
Phys. Rev. Lett. 79 (1997) 353.
[30]
A. G. Grozin, M. Höschele, J. Hoff and M. Steinhauser,
JHEP 1109 (2011) 066.
[31]
C. Anastasiou, R. Boughezal and E. Furlan,
JHEP 1006 (2010) 101. |
Millimeter Wave Communications with Reconfigurable Antennas
Biao He
and Hamid Jafarkhani
This work was supported in part by the NSF Award ECCS-1642536.
Center for Pervasive Communications and Computing,
University of California, Irvine, CA, USA
Email: {biao.he, hamidj}@uci.edu
Abstract
The highly sparse nature of propagation channels and the restricted use of radio frequency (RF)
chains at transceivers limit the performance of millimeter wave (mmWave) multiple-input multiple-output (MIMO) systems.
Introducing reconfigurable antennas to mmWave can offer an additional degree of freedom on designing mmWave MIMO systems.
This paper provides a theoretical framework for studying the mmWave MIMO with reconfigurable antennas.
We present an architecture of reconfigurable mmWave MIMO with beamspace hybrid analog-digital beamformers and reconfigurable antennas at both the transmitter and the receiver.
We show that employing reconfigurable antennas can provide throughput gain for the mmWave MIMO. We derive the expression for the average throughput gain of using reconfigurable antennas, and further simplify the expression by considering the case of large number of reconfiguration states.
In addition, we propose a low-complexity algorithm for the reconfiguration state and beam selection, which achieves nearly the same throughput performance as the optimal selection of reconfiguration state and beams by exhaustive search.
I Introduction
The ubiquitous use of wireless devices in modern life is creating a capacity crisis in wireless communications.
Exploring the millimeter wave (mmWave) band for commercial wireless networks is regarded as a promising solution to the crisis, since the large available bandwidth may offer multiple-Gbps data rates [1].
A major difference between low-frequency communications and mmWave communications is the huge increase in carrier frequencies, which results in propagating challenges for mmWave communications, such as large pathloss and severe shadowing [2]. Meanwhile, the small wavelength enables a large number of antennas to be closely packed to form mmWave large multiple-input multiple-output (MIMO) systems, which can be utilized to overcome the propagation challenges and provide reasonable signal to noise ratios (SNRs) [3].
However, the performance of mmWave MIMO is still considerably limited due to the high sparsity of the channels and the stringent constraint of using radio frequency (RF) chains in mmWave transceivers.
The directional propagations and clustered scattering make the mmWave paths to be highly sparse [1]. More importantly, the high cost and power consumption of RF components and data converters preclude the adoption of fully digital processing for mmWave MIMO to achieve large beamforming gains [1, 4], and low-complexity transceivers relying heavily on analog or hybrid (analog-digital) processing are often adopted [5, 6, 7].
The limited beamforming capability and performance of mmWave MIMO motivate us to investigate the potential benefits of employing reconfigurable antennas for mmWave MIMO in this work.
Different from conventional antennas with fix radiation characteristics, reconfigurable antennas can dynamically change their radiation patterns [8, 9], and offer an additional degree of freedom on designing mmWave MIMO systems.
The radiation characteristics of an antenna is directly determined by the distribution of its current [10], and the mechanism of reconfigurable antennas is to control the current flow in the antenna by altering the antenna’s physical configuration, so that the radiation pattern, polarization, and/or frequency can be modified. The study of reconfigurable antennas for traditional low-frequency MIMO has received considerable attention, e.g., [11, 12, 13] from the perspective of practical antenna design and [9, 14] from the perspective of theoretical performance analysis.
More recently, reconfigurable antennas for communications at mmWave frequencies have been designed and realized, e.g., [15, 16, 17].
The design of space-time codes for a $2\times 2$ mmWave MIMO with reconfigurable transmit antennas was investigated in [18] and [19], and the diversity gain and coding gain were demonstrated.
Due to the simple structure of a $2\times 2$ MIMO, neither the important sparse nature of mmWave channels nor the transceivers with low-complexity beamforming were considered in [18] and [19].
In this work, we provide a theoretical framework for studying the reconfigurable antennas in mmWave MIMO systems.
We take the sparse nature of mmWave channels into account, and present a practical architecture of the mmWave MIMO with low-complexity beamformers and reconfigurable antennas.
We derive the expression for the average throughput gain, which involves an infinite integral of the error function. We further consider the case of large number of reconfiguration states, and derive the simplified expression for the average throughput gain. To the best of our knowledge, the throughput gain of employing reconfigurable antennas have never been derived in the literature, even in the case of low-frequency systems.
Moreover, we propose a fast algorithm for selecting the reconfiguration state of the antennas and the beams for the beamspace hybrid beamformers. Taking advantage of the sparse nature of mmWave channels,
the proposed algorithm significantly reduces the complexity of the reconfiguration state and beam selection, and achieves nearly the same throughput performance as the optimal selection of reconfiguration state and beams by exhaustive search.
Notations:
$\mathbf{X}^{T}$ and $\mathbf{X}^{H}$ denote the transpose and conjugate transpose of $\mathbf{X}$, respectively,
$\mathbf{X}\left(m,n\right)$ denotes the entry of $\mathbf{X}$ in the $m$-th row and $n$-th column,
$\mathrm{Tr}(\mathbf{X})$ denotes the trace of $\mathbf{X}$,
$\left|\mathbf{X}\right|$ denotes the determinant of $\mathbf{X}$,
$\left\|\mathbf{X}\right\|_{F}$ denotes the Frobenius norm of $\mathbf{X}$,
$\mathrm{Re}[x]$ and $\mathrm{Im}[x]$ denote the real and imaginary parts of $x$, respectively,
$\odot$ denotes the Hadamard (element-wise) product,
$\left|\mathcal{X}\right|$ denotes the cardinality of set $\mathcal{X}$,
$\mathrm{sgn}(\cdot)$ denotes the sign function,
$\mathrm{erf}(\cdot)$ denotes the error function, $\mathrm{erf}^{-1}(\cdot)$ denotes the inverse error function, $\mathbb{E}\{\cdot\}$ denotes the expectation operation,
$\mathbb{P}(\cdot)$ denotes the probability measure,
$\mathbf{I}_{n}$ denotes the identity matrix of size $n$,
$\mathcal{CN}(\mu,\sigma^{2})$ denotes the complex Gaussian distribution with mean $\mu$ and variance $\sigma^{2}$,
and $\mathcal{CN}(\mathbf{a},\mathbf{A})$ denotes the distribution of a
circularly symmetric complex Gaussian random vector with mean $\mathbf{a}$ and covariance matrix $\mathbf{A}$.
II System Model
We consider a mmWave system where a transmitter with $N_{t}$ antennas sends messages to a receiver with $N_{r}$ antennas.
We assume that both the transmitter and the receiver are reconfigurable simultaneously, and the total number of possible combinations in which the transmit and receive ports can be reconfigured is $\Psi$.
We refer to each one of these combinations as a reconfiguration state, and
refer to the $\psi$-th reconfiguration state as reconfiguration state $\psi$.
We consider the narrowband block-fading channels.
Denote the transmitted signal vector from the transmitter as $\mathbf{x}\in\mathbb{C}^{N_{t}\times 1}$ with a transmit power constraint $\mathrm{Tr}\left(\mathbb{E}\{\mathbf{x}\mathbf{x}^{H}\}\right)=P$. The received signal at the receiver with reconfiguration state $\psi$ is given by
$$\mathbf{y}=\mathbf{H}_{\psi}\mathbf{x}+\mathbf{n},$$
(1)
where $\mathbf{H}_{\psi}\in\mathbb{C}^{N_{r}\times N_{t}}$ denotes the channel matrix corresponding to the reconfiguration state $\psi$ and $\mathbf{n}\sim\mathcal{CN}(\mathbf{0};\sigma^{2}_{n}\mathbf{I}_{N_{r}})$ denotes the additive white Gaussian noise (AWGN) vector at the receive antennas.
Note that $\mathbf{H}_{\psi}(i,j)$
represents the channel coefficient that contains
the gain and phase information of the path between the $i$-th
transmit antenna and the $j$-th receive antenna in the reconfiguration state $\psi$.
We assume that the channel matrices for different reconfiguration states are independent [9, 14, 18, 19], and have the same average channel power such that $\mathbb{E}\{\left\|\mathbf{H}_{1}\right\|^{2}_{F}\}=\cdots=\mathbb{E}\{\left\|%
\mathbf{H}_{\Psi}\right\|^{2}_{F}\}=N_{r}N_{t}$. We further assume that the channel state information (CSI) of all reconfiguration states is perfectly known at the receiver [9, 14, 18, 19]. The full CSI is not necessarily known at the transmitter.
II-A Channel Model
In the following, we present the channel model of mmWave MIMO systems with reconfigurable antennas.
II-A1 Physical Channel Representation
The mmWave MIMO channel can be characterized by physical multipath models. In particular, the clustered channel representation is usually adopted as a practical model for mmWave channels. The channel matrix for reconfiguration state $\psi$ is contributed by $N_{\psi,\mathrm{cl}}$ scattering clusters, and each cluster contains $N_{\psi,\mathrm{ry}}$ propagation paths.
The 2D physical multipath model for the channel matrix $\mathbf{H}_{\psi}$ is given by
$$\mathbf{H_{\psi}}=\sum^{N_{\psi,\mathrm{cl}}}_{i=1}\sum^{N_{\psi,\mathrm{ry}}}%
_{l=1}\alpha_{\psi,i,l}\mathbf{a}_{R}\left(\theta^{r}_{\psi,i,l}\right)\mathbf%
{a}_{T}^{H}\left(\theta^{t}_{\psi,i,l}\right),$$
(2)
where
$\alpha_{\psi,i,l}$ denotes the path gain, $\theta^{r}_{\psi,i,l}$ and $\theta^{t}_{\psi,i,l}$ denote the angle of arrival (AOA) and the angle of departure (AOD), respectively,
$\mathbf{a}_{R}\left(\theta^{r}_{\psi,i,l}\right)$ and $\mathbf{a}_{T}^{H}\left(\theta^{t}_{\psi,i,l}\right)$ denote the steering vectors of the receive antenna array and the transmit antenna array, respectively.
In this work, we consider the 1D uniform linear array (ULA) at both the transmitter and the receiver. The steering vectors are given by
$\mathbf{a}_{R}\left(\theta^{r}_{\psi,i,l}\right)=\left[1,e^{-j2\pi\vartheta^{r%
}_{\psi,i,l}},\cdots,e^{-j2\pi\vartheta^{r}_{\psi,i,l}(N_{r}-1)}\right]^{T}$
and
$\mathbf{a}_{T}\left(\theta^{t}_{\psi,i,l}\right)=\left[1,e^{-j2\pi\vartheta^{t%
}_{\psi,i,l}},\cdots,e^{-j2\pi\vartheta^{t}_{\psi,i,l}(N_{t}-1)}\right]^{T},$
where $\vartheta$ denotes the normalized spatial angle. The normalized spatial angle is related to the physical AOA or AOD $\theta\in\left[-\pi/2,\pi/2\right]$ by
$\vartheta=d\sin(\theta)/\lambda,$
where $d$ denotes the antenna spacing and $\lambda$ denotes the wavelength.
We assume that $N_{1,\mathrm{cl}}=\cdots=N_{\Psi,\mathrm{cl}}$ and $N_{1,\mathrm{ry}}=\cdots=N_{\Psi,\mathrm{ry}}$, which implies that the sparsity of the mmWave MIMO channel remains the same for all reconfiguration states.
II-A2 Virtual Channel Representation
The virtual (beamspace) representation is a natural choice for modelling mmWave MIMO channels due to the highly directional nature of propagation [20].
The virtual model characterizes the physical channel by coupling between the
spatial beams in fixed virtual transmit and receive directions, and represents the channel in beamspace domain.
The virtual channel representation of $\mathbf{H_{\psi}}$ in (2) is given by [21, 22]
$$\mathbf{H_{\psi}}\!=\!\sum^{N_{r}}_{i=1}\sum^{N_{t}}_{j=1}\!H_{\psi,V}(i,j)%
\mathbf{a}_{R}\!\left(\!\ddot{\theta}_{R,i}\!\right)\mathbf{a}_{T}^{H}\!\left(%
\!\ddot{\theta}_{T,j}\!\right)\!=\!\mathbf{A}_{R}\mathbf{H}_{\psi,V}\mathbf{A}%
_{T}^{H},$$
(3)
where $\ddot{\theta}_{R,i}=\arcsin\left(\lambda\ddot{\vartheta}_{R,i}/d\right)$ and $\ddot{\theta}_{T,j}=\arcsin\left(\lambda\ddot{\vartheta}_{T,j}/d\right)$
are fixed virtual receive and transmit angles corresponding to uniformly spaced spatial angles111Without loss of generality, we here assume that $N_{r}$ and $N_{t}$ are odd.
$\ddot{\vartheta}_{R,i}=\frac{i-1-(N_{r}-1)/2}{N_{r}}$
and
$\ddot{\vartheta}_{T,j}=\frac{j-1-(N_{t}-1)/2}{N_{t}},$
respectively,
$\mathbf{A}_{R}=\frac{1}{\sqrt{N_{r}}}\left[\mathbf{a}_{R}\left(\ddot{\theta}_{%
R,1}\right),\cdots,\mathbf{a}_{R}\left(\ddot{\theta}_{R,N_{r}}\right)\right]^{T}$
and
$\mathbf{A}_{T}=\frac{1}{\sqrt{N_{t}}}\left[\mathbf{a}_{T}\left(\ddot{\theta}_{%
T,1}\right),\cdots,\mathbf{a}_{T}\left(\ddot{\theta}_{T,N_{t}}\right)\right]^{T}$
are unitary DFT matrices, and $\mathbf{H}_{\psi,V}\in\mathbb{C}^{N_{r}\times N_{t}}$ is the virtual channel matrix.
Since $\mathbf{A}_{R}\mathbf{A}_{R}^{H}=\mathbf{A}_{R}^{H}\mathbf{A}_{R}=\mathbf{I}_{%
N_{r}}$ and $\mathbf{A}_{T}\mathbf{A}_{T}^{H}=\mathbf{A}_{T}^{H}\mathbf{A}_{T}=\mathbf{I}_{%
N_{t}}$, the virtual channel matrix and the physical channel matrix are unitarily equivalent, such that
$\mathbf{H}_{\psi,V}=\mathbf{A}_{R}^{H}\mathbf{H}_{\psi}\mathbf{A}_{T}.$
II-A3 Low-Dimensional Virtual Channel Representation
The link capacity of a MIMO system is directly related to the rank of the channel matrix $\mathbf{H}_{\psi}$, which depends on the amount of scattering and reflection in the multipath environment.
An important property of the mmWave MIMO channel is its highly sparse structure, i.e., $\mathrm{rank}\left\{\mathbf{H}_{\psi}\right\}\ll\min\left\{N_{r},N_{t}\right\}$.
In the clustered scattering environment of mmWave MIMO, the dominant channel power is expected to be captured by a few rows and columns of the virtual channel matrix, i.e., a low-dimensional submatrix of $\mathbf{H}_{\psi,V}$.
The discussion above motivates the development of low-dimensional virtual representation of mmWave MIMO channels and the corresponding low-complexity beamforming designs for mmWave MIMO transceivers [23, 24, 25, 26].
Specifically, a low-dimensional virtual channel matrix, denoted by $\widetilde{\mathbf{H}}_{\psi,V}\in\mathbb{C}^{L_{r}\times L_{t}}$,
is obtained by beam selection from $\mathbf{H}_{\psi,V}$, such that $\widetilde{\mathbf{H}}_{\psi,V}$ captures $L_{t}$ dominant transmit beams and $L_{r}$ dominant receive beams of the full virtual channel matrix.
The low-dimensional virtual channel matrix is defined by
$$\widetilde{\mathbf{H}}_{\psi,V}=\left[\mathbf{H}_{\psi,V}(i,j)\right]_{i\in{%
\mathcal{M}_{\psi,r}},j\in\mathcal{M}_{\psi,t}},$$
(4)
where $\mathcal{M}_{\psi,r}=\left\{i:(i,j)\in\mathcal{M}_{\psi}\right\}$, $\mathcal{M}_{\psi,t}=\left\{j:(i,j)\in\mathcal{M}_{\psi}\right\}$, and $\mathcal{M}_{\psi}$ is the beam selection mask.
The beam selection mask $\mathcal{M}$ is related to the criterion of beam selection.
II-B Transceiver Architecture
We adopt the reconfigurable beamspace hybrid beamformer as the architecture of low-complexity transceivers for mmWave MIMO systems with reconfigurable antennas.
At the transmitter, the symbol vector $\mathbf{s}\in\mathbb{C}^{N_{s}\times 1}$ is first processed by a low-dimensional digital precoder $\mathbf{F}\in\mathbb{C}^{L_{t}\times N_{s}}$, where $L_{t}$ denotes the number of RF chains at the transmitter. The obtained $L_{t}\times 1$ signal vector is denoted by $\widetilde{\mathbf{x}}_{V}=\mathbf{F}\mathbf{s}$, which is then converted to analog signals by $L_{t}$ digital-to-analog converters (DACs). Next, the $L_{t}$ signals go through the beam selector to obtain the $N_{t}\times 1$ (virtual) signal vector $\mathbf{x}_{V}$.
For a given beam selection mask $\mathcal{M}$, $\mathbf{x}_{V}$ is constructed by $\left[\mathbf{x}_{V}(j)\right]_{j\in\mathcal{M}_{t}}=\widetilde{\mathbf{x}}_{V}$ and $\left[\mathbf{x}_{V}(j)\right]_{j\notin\mathcal{M}_{t}}=\mathbf{0}$, where $\mathcal{M}_{t}=\left\{j:(i,j)\in\mathcal{M}\right\}$.
The beam selector can be easily realized by switches in practice. $\mathbf{x}_{V}$ is further processed by the DFT analog precoder $\mathbf{A}_{T}\in\mathbb{C}^{N_{t}\times N_{t}}$, and the obtained signal vector is given by $\mathbf{x}=\mathbf{A}_{T}\mathbf{x}_{V}$.
Note that $\mathrm{Tr}\left(\mathbb{E}\{\widetilde{\mathbf{x}}_{V}\widetilde{\mathbf{x}}_%
{V}^{H}\}\right)=\mathrm{Tr}\left(\mathbb{E}\{\mathbf{x}_{V}\mathbf{x}_{V}^{H}%
\}\right)=\mathrm{Tr}\left(\mathbb{E}\{\mathbf{x}\mathbf{x}^{H}\}\right)=P$.
Finally, the transmitter sends $\mathbf{x}$ with the reconfigurable antennas.
The received signal vector at the receive antennas with a given reconfiguration state $\psi$ is given by
$$\mathbf{y}=\mathbf{H}_{\psi}\mathbf{x}+\mathbf{n}=\mathbf{A}_{R}\mathbf{H}_{%
\psi,V}\mathbf{A}_{T}^{H}\mathbf{x}+\mathbf{n}=\mathbf{A}_{R}\mathbf{H}_{\psi,%
V}\mathbf{x}_{V}+\mathbf{n}.$$
(5)
At the receiver side, $\mathbf{y}$ is first processed by the IDFT analog decoder $\mathbf{A}_{R}^{H}\in\mathbb{C}^{N_{r}\times N_{r}}$, and the obtained (virtual) signal vector is given by
$\mathbf{y}_{V}=\mathbf{A}_{R}^{H}\mathbf{y}=\mathbf{H}_{\psi,V}\mathbf{x}_{V}+%
\mathbf{n}_{V},$
where the distribution of $\mathbf{n}_{V}=\mathbf{A}_{R}^{H}\mathbf{n}$ is $\mathcal{CN}(\mathbf{0};\sigma^{2}_{n}\mathbf{I}_{N_{r}})$.
According to the given beam selection mask $\mathcal{M}$, the receiver then uses the beam selector to obtain the low-dimensional $L_{r}\times 1$ signal vector $\widetilde{\mathbf{y}}_{V}=\left[\mathbf{y}_{V}(i)\right]_{i\in\mathcal{M}_{r}}$, where $L_{r}$ denotes the number of the RF chains at the receiver and $\mathcal{M}_{r}=\left\{i:(i,j)\in\mathcal{M}\right\}$.
The low-dimensional virtual system representation for a given reconfiguration state $\psi$ is formulated as
$$\widetilde{\mathbf{y}}_{V}=\widetilde{\mathbf{H}}_{\psi,V}\widetilde{\mathbf{x%
}}_{V}+\widetilde{\mathbf{n}}_{V},$$
(6)
where $\widetilde{\mathbf{H}}_{\psi,V}=\left[\mathbf{H}_{\widehat{\psi},V}(i,j)\right%
]_{i\in{\mathcal{M}_{r}},j\in\mathcal{M}_{t}}$, $\widetilde{\mathbf{n}}_{V}=\left[\mathbf{n}_{V}(i)\right]_{i\in\mathcal{M}_{r}}$,
and $\widetilde{\mathbf{n}}_{V}\sim\mathcal{CN}(\mathbf{0};\sigma^{2}_{n}\mathbf{I}%
_{L_{r}})$.
The analog signals are finally converted to digital signals by $L_{r}$ analog-to-digital converters (ADCs) for the low-dimensional digital signal processing.
As mentioned earlier, we assume that the full CSI is perfectly known at the receiver, and a limited feedback is available from the receiver to the transmitter to enable the beam selection and the reconfiguration state selection. Per the number of all possible combinations of selected beams and reconfiguration states, the number of the feedback bits is equal to $\log_{2}\left(\Psi\right)+\log_{2}\left(\binom{N_{t}}{L_{t}}\binom{N_{r}}{L_{r%
}}\right)$.
We assume that $N_{s}=L_{t}\leq L_{r}$ to maximize the multiplexing gain of the system.
The digital precoder at the transmitter is then given by $\mathbf{F}=\mathbf{I}_{N_{s}}$ with equal power allocation between the $N_{s}$ data streams, since the transmitter does not have the full CSI.
At the receiver, the digital decoder is the joint ML decoder for maximizing the throughput.
With the aforementioned transceiver architecture and CSI assumptions, the system throughput with a selected $\widetilde{\mathbf{H}}_{\psi,V}$ is given by [22]
$$R_{\widetilde{\mathbf{H}}_{\psi,V}}=\log_{2}\left|\mathbf{I}_{L_{r}}+\frac{%
\rho}{L_{t}}\widetilde{\mathbf{H}}_{\psi,V}\widetilde{\mathbf{H}}_{\psi,V}^{H}%
\right|,$$
(7)
where $\rho=P/\sigma^{2}_{n}$ denotes the transmit power to noise ratio.
III Throughput Gain of Employing Reconfigurable Antennas
In this section, we analyze the performance gain of employing the reconfigurable antennas in terms of the average throughput.
With the optimal reconfiguration state selection, the instantaneous system throughput is given by
$$R_{\widehat{\psi}}=\max_{\psi\in\left\{1,\cdots,\Psi\right\}}R_{\psi},$$
(8)
where
$$\displaystyle R_{\psi}$$
$$\displaystyle=\log_{2}\left|\mathbf{I}_{L_{r}}+\frac{\rho}{L_{t}}\widehat{%
\widetilde{\mathbf{H}}}_{\psi,V}\widehat{\widetilde{\mathbf{H}}}_{\psi,V}^{H}\right|$$
$$\displaystyle=\max_{\widetilde{\mathbf{H}}_{\psi,V}\in\left\{\tilde{\mathcal{H%
}}_{\psi}\right\}}\log_{2}\left|\mathbf{I}_{L_{r}}+\frac{\rho}{L_{t}}%
\widetilde{\mathbf{H}}_{\psi,V}\widetilde{\mathbf{H}}_{\psi,V}^{H}\right|$$
(9)
represents the maximum achievable throughput under the reconfiguration state $\psi$,
$\widehat{\widetilde{\mathbf{H}}}_{\psi,V}$ denotes the optimal low-dimensional virtual channel of $\mathbf{H}_{\psi,V}$,
and $\tilde{\mathcal{H}}_{\psi}$ denotes the set of all possible $L_{r}\times L_{t}$ submatrices of $\mathbf{H}_{\psi,V}$.
The average throughput gain of employing the reconfigurable antennas is given by
$$G_{\bar{R}}=\frac{\bar{R}_{\widehat{\psi}}}{\bar{R}_{\psi}},$$
(10)
where $\bar{R}_{\widehat{\psi}}=\mathbb{E}\{R_{\widehat{\psi}}\}$, $\bar{R}_{\psi}=\mathbb{E}\{R_{\psi}\}$, and the expectation is over different channel realizations.
As mentioned before, we assume that the channel matrices for different reconfiguration states have the same average channel power, and hence, $\bar{R}_{1}=\cdots=\bar{R}_{\Psi}$.
Note that each entry of $\widetilde{\mathbf{H}}_{\psi,V}$ is associated with a set of physical paths [21], and it is approximated equal to the sum of the complex gains of the corresponding paths [25]. When the number of distinct paths associated with $\widetilde{\mathbf{H}}_{\psi,V}(i,j)$ is sufficiently large, we note from the central limit theorem that $\widetilde{\mathbf{H}}_{\psi,V}(i,j)$ tends toward a complex Gaussian random variable.
Different from the rich scattering environment, the associated groups of paths to different entries of $\widetilde{\mathbf{H}}_{\psi,V}$ may be correlated in the mmWave environment. As a result, the entries of $\widetilde{\mathbf{H}}_{\psi,V}$ can be correlated, and the entries of $\widetilde{\mathbf{H}}_{\psi,V}$ are then approximated by correlated zero-mean complex Gaussian variables.
In the literature, it has been shown that the instantaneous capacity of a MIMO system whose channel matrix has correlated zero-mean complex Gaussian entries can be approximated by a Gaussian variable. Based on the discussion above,
the distribution of $R_{\psi}$ is approximated by a Gaussian distribution,
and the approximated pdf of $R_{\psi}$ is given by
$$f_{R_{\psi}}(x)=\left\{\begin{array}[]{ll}\frac{1}{\sqrt{2\pi\sigma^{2}_{R_{%
\psi}}}}\exp\left(-\frac{x-\bar{R}_{\psi}}{2\sigma^{2}_{R_{\psi}}}\right)\;,&%
\mbox{if}~{}x\geq 0\\
0\;,&\mbox{otherwise},\end{array}\right.$$
(11)
where $\bar{R}_{\psi}$ and $\sigma^{2}_{R_{\psi}}$ denote the mean and the variance of $R_{\psi}$, respectively.
The average throughput gain of employing the reconfigurable antennas is then given in the following proposition.
Proposition 1
The average throughput gain of employing the reconfigurable antennas with $\Psi$ distinct reconfiguration states is approximated by
$$G_{\bar{R}}\approx\int_{0}^{\infty}\frac{1}{\bar{R}_{\psi}}-\frac{1}{2^{\Psi}%
\bar{R}_{\psi}}\left(1+\mathrm{erf}\left(\frac{x-\bar{R}_{\psi}}{\sqrt{2\sigma%
^{2}_{R_{\psi}}}}\right)\right)^{\Psi}\mathrm{d}x.$$
(12)
Proof:
With the Gaussian approximation of the distribution of $R_{\psi}$, we can obtain the approximated $R_{\widehat{\psi}}$ as the maximum of $\Psi$ i.i.d. Gaussian random variables, and
$\bar{R}_{\widehat{\psi}}\approx\int_{0}^{\infty}1-\left(F_{R_{\psi}}(x)\right)%
^{\Psi}\mathrm{d}x,$
where
$F_{R_{\psi}}(x)$
denotes the approximated cdf of $R_{\psi}$.
Substituting the approximated $R_{\widehat{\psi}}$ into (10) completes the proof.
∎
We now consider the case that the number of reconfiguration states is large, and derive the simplified expression for $G_{\bar{R}}$ in the following proposition.
Proposition 2
When $\Psi$ is large, the average throughput gain of employing the reconfigurable antennas is approximated by
$$G_{\bar{R}}\!\approx\!1+\frac{\sqrt{2\sigma^{2}_{R_{\psi}}}}{\bar{R}_{\psi}}\!%
\left(\!\left(1\!-\!\beta\right)\mathrm{erf}^{-1}\!\!\left(\!1\!-\!\frac{2}{%
\Psi}\!\right)\!+\!\beta\mathrm{erf}^{-1}\!\!\left(\!1\!-\!\frac{2}{e\Psi}\!%
\right)\!\right),$$
(13)
where $\beta$ denotes the Euler’s constant.
Proof:
Proposition 2 can be proved with the aid of Fisher-âTippett theorem to approximate the distribution of the maximum of $\Psi$ independent standard normal random variables as a Gumbel distribution [27, Chapter 10]. The detailed proof is omitted here.
∎
IV Fast Reconfiguration State and Beam Selection
The objective of selecting the optimal reconfiguration state and beams is to obtain the corresponding optimal $\widetilde{\mathbf{H}}_{\psi,V}$ that maximizes the system throughput given in (7).
The design problem of selecting $\widetilde{\mathbf{H}}_{\psi,V}$
is formulated as
$$\max_{\psi\in\left\{1,\cdots,\Psi\right\}}\max_{\widetilde{\mathbf{H}}_{\psi,V%
}\in\left\{\tilde{\mathcal{H}}_{\psi}\right\}}\left|\mathbf{I}_{L_{r}}+\frac{%
\rho}{L_{t}}\widetilde{\mathbf{H}}_{\psi,V}\widetilde{\mathbf{H}}_{\psi,V}^{H}%
\right|.$$
(14)
A straightforward method to obtain the optimal $\widetilde{\mathbf{H}}_{\psi,V}$ is the exhaustive search among all possible selections of $\widetilde{\mathbf{H}}_{\psi,V}$. That is, we first search for the optimal beam selection for each reconfiguration state to obtain $\widehat{\widetilde{\mathbf{H}}}_{\psi,V}$, i.e., the optimal low-dimensional virtual channel of $\mathbf{H}_{\psi,V}$.
Then, we compare the obtained $\widehat{\widetilde{\mathbf{H}}}_{\psi,V}$ among all reconfiguration states to complete the selection of optimal $\widetilde{\mathbf{H}}_{\psi,V}$, denoted by $\widehat{\widetilde{\mathbf{H}}}_{\widehat{\psi},V}$.
The total number of possible selections to search is given by
$N_{\mathrm{total}}=\Psi\binom{N_{t}}{L_{t}}\binom{N_{r}}{L_{r}}.$
When $N_{t}\gg L_{t}$, $N_{r}\gg L_{r}$, and $\Psi\gg 1$, the total number to search, $N_{\mathrm{total}}$, would be too large for practical applications due to the high complexity.
Thus, in what follows, we propose a low-complexity design to obtain $\widehat{\widetilde{\mathbf{H}}}_{\widehat{\psi},V}$ which achieves the near optimal throughput performance.
As discussed earlier, the mmWave MIMO channel has a sparse nature such that $\mathrm{rank}\left\{\mathbf{H}_{\psi}\right\}\ll\min\left\{N_{r},N_{t}\right\}$. Also, the rank of the channel matrix is directly related to the number of non-vanishing rows and columns of its virtual representation.
Now let us consider an extreme scenario such that all of the non-vanishing entries of $\mathbf{H}_{\psi,V}$ are contained in the low-dimensional submatrix, and $\mathbf{H}_{\psi,V}$ is approximated by
$$\mathbf{M}\odot\mathbf{H}_{\psi,V},$$
(15)
where
$\mathbf{M}(i,j)=1$ if $(i,j)\in\widehat{\mathcal{M}}_{\psi}$, $\mathbf{M}(i,j)=0$ if $(i,j)\notin\widehat{\mathcal{M}}_{\psi}$,
and $\widehat{\mathcal{M}}_{\psi}$ is the beam selection mask corresponding to $\widehat{\widetilde{\mathbf{H}}}_{\psi,V}$. With (15), we have
$\left|\mathbf{I}_{L_{r}}+\frac{\rho}{L_{t}}\widehat{\widetilde{\mathbf{H}}}_{%
\psi,V}\widehat{\widetilde{\mathbf{H}}}_{\psi,V}^{H}\right|\approx\left|%
\mathbf{I}_{N_{r}}+\frac{\rho}{L_{t}}\mathbf{H}_{\psi,V}\mathbf{H}_{\psi,V}^{H%
}\right|=\left|\mathbf{I}_{N_{r}}+\frac{\rho}{L_{t}}\mathbf{H}_{\psi}\mathbf{H%
}_{\psi}^{H}\right|.$
Thus, we find that a fast selection of reconfiguration state can be achieved by directly comparing their (full) physical channel matrices. Instead of finding the optimal beam selection of each reconfiguration state first, we can directly determine the optimal reconfiguration state by
$$\widehat{\psi}=\arg\max_{\psi\in\left\{1,\cdots,\Psi\right\}}\left|\mathbf{I}_%
{N_{r}}+\frac{\rho}{L_{t}}\mathbf{H}_{\psi}\mathbf{H}_{\psi}^{H}\right|.$$
(16)
To reduce the complexity of beam selection, we utilize some techniques for MIMO antenna selection. Again due to the high sparsity of mmWave channels, the dominant beams usually significantly outperform the other beams, and they can be easily selected by the fast beam selection scheme.
Note that the transmitter does not have the full CSI in the considered system, and hence, the existing beamspace selection schemes in, e.g., [24], with the requirement of full CSI on the beamspace channel at the transmitter are not applicable in our work.
Our fast beam selection method is explained next.
The beam selection problem in fact includes both transmit and receive beam selections. We adopt a separable transmit and receive beam selection technique [28] for first selecting the best $L_{r}$ receive beams and then selecting the best $L_{t}$ transmit beams. For both the receive and transmit beam selections, a technique based on the incremental successive selection algorithm (ISSA) [29] is utilized. We start from the empty set of selected beams and then add one beam at each step to this set. In each step, the objective is to select one of the unselected beams that leads to the highest increase of the throughput.
The proposed fast reconfiguration state and beam selection methods are summarized in
Algorithm 1. The outputs of the algorithm are the optimal reconfiguration state, the indices of the selected receive beams, the indices of the selected transmit beams, and the selected low-dimensional virtual channel, denoted by $\widehat{\psi}$, $\mathcal{M}_{r}$, $\mathcal{M}_{t}$, and $\widehat{\widetilde{\mathbf{H}}}_{\widehat{\psi},V}$, respectively.
V Numerical Results
For all simulation results in this work, we adopt the clustered multipath channel model in (2) to generate the channel matrix. We assume that $\alpha_{\psi,i,l}$ are i.i.d. $\mathcal{CN}\left(0,\sigma^{2}_{\alpha,\psi,i}\right)$, where $\sigma^{2}_{\alpha,\psi,i}$ denotes the average power of the $i$-th cluster,
and
$\sum_{i=1}^{N_{\psi,c}}\sigma^{2}_{\alpha,\psi,i}=\gamma_{\psi}$, where $\gamma_{\psi}$ is a normalization parameter to ensure that $\mathbb{E}\{\left\|\mathbf{H}_{\psi}\right\|^{2}_{F}\}=N_{r}N_{t}$.
We also assume that $\theta^{r}_{\psi,i,l}$ are uniformly distributed with
mean $\theta_{\psi,i}^{r}$ and a constant angular spread (standard deviation) $\sigma_{\theta^{r}}$. $\theta^{t}_{\psi,i,l}$ are uniformly distributed with
mean $\theta_{\psi,i}^{t}$ and a constant angular spread (standard deviation) $\sigma_{\theta^{t}}$. We further assume that $\theta_{\psi,i}^{r}$ and $\theta_{\psi,i}^{t}$ are uniformly distributed within the range of $[-\pi/2,\pi/2]$. The system parameters are $N_{r}=N_{t}=17,L_{r}=L_{t}=5,N_{\psi,\mathrm{cl}}=3,N_{\psi,\mathrm{ry}}=10,$ $\sigma_{\theta^{r}}=\sigma_{\theta^{t}}=4^{\circ}$, and $d/\lambda=1/2$.
All average results are over 5000 randomly generated channel realizations.
We first demonstrate the accuracy of the Gaussian approximated pdf of $R_{\psi}$.
Figure 1 plots the simulated pdf and the Gaussian approximated pdf of $R_{\psi}$.
As presented in the figure, the Gaussian approximations match precisely the simulated pdfs for both $\rho=0$ dB and $\rho=10$ dB.
These observations confirm that the distribution of $R_{\psi}$ can be well approximated by the Gaussian distribution.
We then show the average throughput gain of employing the reconfigurable antennas. Figure 2 plots the average throughput gain, $G_{\bar{R}}$, versus the number of reconfiguration states, $\Psi$. The illustrated results are for the actual gain by simulations,
the theoretical approximation in (12), and the simplified theoretical approximation for large $\Psi$ in (13).
As depicted in the figure, the derived theoretical approximations match precisely the simulated results.
From both the simulation and the approximations, we find that employing the reconfigurable antennas provides average throughput gains compared with the conventional system without the reconfigurable antennas.
In addition, we find that the growth of $G_{\bar{R}}$ with $\Psi$ is fast when $\Psi$ is small, while it becomes slow when $\Psi$ is relatively large. This finding indicates that the dominant average throughput gain of employing the reconfigurable antennas can be achieved by having a few number of reconfiguration states.
Finally, we examine the performance of the proposed algorithm for fast reconfiguration state and beam selection by evaluating the average throughput loss ratio, which is defined by
$\Delta_{R}=\left(\bar{R}_{\mathrm{opt}}-\bar{R}_{\mathrm{fast}}\right)/\bar{R}%
_{\mathrm{opt}},$
where $\bar{R}_{\mathrm{opt}}$ denotes the average throughput achieved by the exhaustive search and $\bar{R}_{\mathrm{fast}}$ denotes the average throughput achieved by the proposed fast selection algorithm.
Figure 3 plots the throughput loss ratio, $\Delta_{R}$, versus the transmit power to noise ratio, $\rho$. Systems with different numbers of reconfiguration states are considered, i.e., $\Psi=2$, $\Psi=4$, and $\Psi=8$. As shown in the figure, the proposed fast selection algorithm always achieves the near optimal throughput performance. Although $\Delta_{R}$ increases as $\Psi$ increases, the throughput loss ratio is less than $0.9\%$ even when $\Psi=8$.
VI Conclusions
In this paper, we have presented a framework for the theoretical study of the mmWave MIMO with reconfigurable antennas, where the low-complexity transceivers and the sparse channels are considered.
We have shown that employing reconfigurable antennas can provide the throughput gain for mmWave MIMO systems. The approximated expression for the average throughput gain have been derived. Based on the highly sparse nature of mmWave channels, we have further developed a fast algorithm for the reconfiguration state and beam selection. The accuracy of our derived approximations and the performance of the developed algorithm have been verified by simulation. We have found from the results that the dominant throughput gains by employing the reconfigurable antennas can be achieved by having a few number of reconfiguration states.
References
[1]
Z. Pi and F. Khan, “An introduction to millimeter-wave mobile broadband
systems,” IEEE Commun. Mag., vol. 49, no. 6, pp. 101–107, June 2011.
[2]
T. S. Rappaport, R. W. Heath, R. C. Daniels, and J. N. Murdock,
Millimeter Wave Wireless Communications. Prentice Hall, 2014.
[3]
T. S. Rappaport, S. Sun, R. Mayzus, H. Zhao, Y. Azar, K. Wang, G. N. Wong,
J. K. Schulz, M. Samimi, and F. Gutierrez, “Millimeter wave mobile
communications for 5G cellular: It will work!” IEEE Access, vol. 1,
pp. 335–349, May 2013.
[4]
C. H. Doan, S. Emami, D. A. Sobel, A. M. Niknejad, and R. W. Brodersen,
“Design considerations for 60 GHz CMOS radios,” IEEE Commun.
Mag., vol. 42, no. 12, pp. 132–140, Dec. 2004.
[5]
V. Venkateswaran and A. J. van der Veen, “Analog beamforming in MIMO
communications with phase shift networks and online channel estimation,”
IEEE Trans. Signal Process., vol. 58, no. 8, pp. 4131–4143, Aug.
2010.
[6]
O. E. Ayach, S. Rajagopal, S. Abu-Surra, Z. Pi, and R. W. Heath, “Spatially
sparse precoding in millimeter wave MIMO systems,” IEEE Transactions
on Wireless Communications, vol. 13, no. 3, pp. 1499–1513, Mar. 2014.
[7]
L. Liu and H. Jafarkhani, “Space-time trellis codes based on channel-phase
feedback,” IEEE Trans. Commun., vol. 54, no. 12, pp. 2186–2198, Dec.
2006.
[8]
B. A. Cetiner, H. Jafarkhani, J.-Y. Qian, H. J. Yoo, A. Grau, and F. D.
Flaviis, “Multifunctional reconfigurable MEMS integrated antennas for
adaptive MIMO systems,” IEEE Commun. Mag., vol. 42, no. 12, pp.
62–70, Dec. 2004.
[9]
A. Grau, H. Jafarkhani, and F. D. Flaviis, “A reconfigurable multiple-input
multiple-output communication system,” IEEE Trans. Wireless Commun.,
vol. 7, no. 5, pp. 1719–1733, May 2008.
[10]
C. A. Balanis, Antenna Theory: Analysis and Design, 3rd ed. Wiley, 2005.
[11]
C. G. Christodoulou, Y. Tawk, S. A. Lane, and S. R. Erwin, “Reconfigurable
antennas for wireless and space applications,” Proc. IEEE, vol. 100,
no. 7, pp. 2250–2261, July 2012.
[12]
R. L. Haupt and M. Lanagan, “Reconfigurable antennas,” IEEE Antennas
Propag. Mag., vol. 55, no. 1, pp. 49–61, Feb. 2013.
[13]
S. Pendharker, R. K. Shevgaonkar, and A. N. Chandorkar, “Optically controlled
frequency-reconfigurable microstrip antenna with low photoconductivity,”
IEEE Antennas Wireless Propag. Lett., vol. 13, pp. 99–102, Jan. 2014.
[14]
F. Fazel, A. Grau, H. Jafarkhani, and F. D. Flaviis, “Space-time-state block
coded MIMO communication systems using reconfigurable antennas,”
IEEE Trans. Wireless Commun., vol. 8, no. 12, pp. 6019–6029, Dec.
2009.
[15]
S. F. Jilani, B. Greinke, Y. Hao, and A. Alomainy, “Flexible millimetre-wave
frequency reconfigurable antenna for wearable applications in 5G
networks,” in Proc. URSI EMTS, Aug. 2016, pp. 846–848.
[16]
B. Ghassemiparvin and N. Ghalichechian, “Reconfigurable millimeter-wave
antennas using paraffin phase change materials,” in Proc. EuCAP, Apr.
2016, pp. 1–4.
[17]
I. F. da Costa, A. C. S., D. H. Spadoti, L. G. da Silva, J. A. J. Ribeiro, and
S. E. Barbin, “Optically controlled reconfigurable antenna array for mm-wave
applications,” IEEE Antennas Wireless Propag. Lett., vol. 16, pp.
2142–2145, May 2017.
[18]
V. Vakilian, H. Mehrpouyan, and Y. Hua, “High rate space-time codes for
millimeter-wave systems with reconfigurable antennas,” in Proc. IEEE
WCNC, Mar. 2015, pp. 591–596.
[19]
V. Vakilian, H. Mehrpouyan, Y. Hua, and H. Jafarkhani, “High-rate space coding
for reconfigurable 2x2 millimeter-wave MIMO systems,” 2015. [Online].
Available: https://arxiv.org/abs/1505.06466
[20]
R. W. Heath, N. González-Prelcic, S. Rangan, W. Roh, and A. M. Sayeed, “An
overview of signal processing techniques for millimeter wave MIMO
systems,” IEEE J. Sel. Signal Process., vol. 10, no. 3, pp. 436–453,
Apr. 2016.
[21]
A. M. Sayeed, “Deconstructing multiantenna fading channels,” IEEE
Trans. Signal Process., vol. 50, no. 10, pp. 2563–2579, Oct. 2002.
[22]
D. Tse and P. Viswanath, Fundamentals of Wireless Communication. Cambridge University Press, 2005.
[23]
J. Brady, N. Behdad, and A. M. Sayeed, “Beamspace MIMO for millimeter-wave
communications: System architecture, modeling, analysis, and measurements,”
IEEE Trans. Antennas Propag., vol. 61, no. 7, pp. 3814–3827, July
2013.
[24]
P. V. Amadori and C. Masouros, “Low RF-complexity millimeter-wave
beamspace-MIMO systems by beam selection,” IEEE Trans. Commun.,
vol. 63, no. 6, pp. 2212–2223, June 2015.
[25]
A. M. Sayeed and V. Raghavan, “Maximizing MIMO capacity in sparse multipath
with reconfigurable antenna arrays,” IEEE J. Sel. Signal Process.,
vol. 1, no. 1, pp. 156–166, June 2007.
[26]
V. Raghavan and A. M. Sayeed, “Sublinear capacity scaling laws for sparse
MIMO channels,” IEEE Trans. Inf. Theory, vol. 57, no. 1, pp.
345–364, Jan. 2011.
[27]
H. A. David and H. N. Nagaraja, Order Statistics, 3rd ed. Wiley, 2005.
[28]
S. Sanayei and A. Nosratinia, “Capacity maximizing algorithms for joint
transmit-receive antenna selection,” in Proc. ACSSC, vol. 2, Nov.
2004, pp. 1773–1776.
[29]
M. Gharavi-Alkhansari and A. B. Gershman, “Fast antenna subset selection in
MIMO systems,” IEEE Trans. Signal Process., vol. 52, no. 2, pp.
339–347, Feb. 2004. |
Consistency and convergence of simulation schemes in Information field dynamics
Martin Dupont
Max Planck Institute for Astrophysics, Karl-Schwarzschild-Str. 1, 85741 Garching, Germany
Torsten Enßlin
Max Planck Institute for Astrophysics, Karl-Schwarzschild-Str. 1, 85741 Garching, Germany
Abstract
We explore a new simulation scheme for partial differential equations (PDE’s) called Information Field Dynamics (IFD). Information field dynamics attempts to improve on existing simulation schemes by incorporating Bayesian field inference, which seeks to preserve the maximum amount of information about the field being simulated. The field inference is truly Bayesian and thus depends on a notion of prior belief. Here, we analytically prove that a restricted subset of simulation schemes in IFD are consistent, and thus deliver valid predictions in the limit of high resolutions. This has not previously been done for any IFD schemes. This restricted subset is roughly analogous to traditional fixed-grid numerical PDE solvers, given the additional restriction of translational symmetry. Furthermore, given an arbitrary IFD scheme modelling a PDE, it is a-priori not obvious to what order the scheme is accurate in space and time. For this subset of models, we also derive an easy rule-of-thumb for determining the order of accuracy of the simulation. As with all analytic consistency analysis, an analysis for nontrivial systems is intractable, thus these results are intended as a general indicator of the validity of the approach, and it is hoped that the results will generalize.
I Introduction
Information Field Dynamics (IFD) is a new framework for constructing numerical simulation schemes for partial differential equations (PDE’s), first proposed in IFD , and further developed in IFDMath and ReimarTowards . The basic premise of IFD is that the gridpoints of traditional simulation schemes have been replaced with a generalized notion of data about the field being simulated. In this framework, the data can represent any finite measurement of the field, and IFD proposes a scheme for updating this data in time such that it optimally represents the true time evolution of the field. This is achieved using the framework of Information Field Theory (IFT) IFT ; Lemm . While the mathematical formalism has already been laid out in IFD , what has not yet been done, is show that schemes constructed using IFD reflect the true behaviour of the field being simulated (that is to say, that the schemes are consistent. It has also not been shown that the numerical error in IFD codes is bounded and approaches zero in the limit of high resolutions. Resolving these two issues is the main focus of this paper.
The idea of supplementing deterministic simulations with probabilistic and statistical techniques has existed for a long time. Some of the earliest papers on this topic are Diaconis ; Skilling1 ; oHagan ; Kadane , which treat the problem of solving PDE’s as a Bayesian inference problem. This approach has also received attention in recent years, with Briol ; Schober ; Conrad using Bayesian uncertainty to find probabilistic estimates for numerical error. Bayesian methods have not just been used to supplement differential equation solvers; Ref.Hennig uses Gaussian posterior beliefs to obtain numerical error estimates for solutions to linear algebra problems, and Snoek uses Bayesian logic to optimize machine learning algorithms. In a similar vein, Raissi uses machine learning to directly infer solutions to linear differential equations based on data. One paper which proposes a scheme similar to IFD is Archambeau , which solves stochastic differential equations by incorporating posterior beliefs, and a notion of observations, as IFD does.
To give a brief overview of the IFD framework, we first restrict ourself to the “linear case”, for simplicity. The linear case makes a number of simplifying assumptions. We first assume the goal is to simulate a PDE of the form
$$\partial_{t}\phi(t,x)=L\phi(t,x)$$
(1)
For $L$ some linear differential operator. This equation has a formal solution given by $\phi(t)=U(t)(\phi_{0})$, for
$U(t)=\exp(tL)$.
The scheme is then as follows: It is assumed that there is some data $d_{i}$ taken at some point in time, $t_{i}$, which is interpreted as being some coarse-grained representation of the true field $\phi(t_{i},x)$, which is obtained by some measurement (real or fictitious). For the “linear case”, we restrict ourselves to linear measurements of the form:
$$d_{i}=R_{i}\phi+n$$
(2)
Where $R_{i}$ is a linear measurement from the space of fields to a finite-dimensional data space $\mathbb{R}^{m}$ performed at time $t_{i}$, and $n$ is some vector-valued additive Gaussian noise, with covariance matrix $N$. The space of all possible fields will be referred to interchangeably as field space or signal space throughout this paper.
To run the simulation, i.e. obtain the data at $t_{i+1}$ from that at $t_{i}$, IFT is used to reconstruct the posterior probability distribution of the field, $\mathcal{P}(\phi(t_{i},x)|d_{i})$, given the initial data, $d_{i}$. This probability distribution is then evolved from $t_{i}$ to $t_{i+1}$ using the analytic time evolution operator $U$. To obtain the data at the next timestep, a second measurement, $R_{i+1}$, is postulated, which is used to construct a second posterior distribution. The new data is then chosen as to minimize the information loss between the evolved and unevolved posterior distributions, using the Kullback-Leibler divergence (KL divergence).
IFD is a Bayesian theory, and as such the field reconstructions rely on having a prior probability distribution over the space of fields. One further requirement of the linear case in IFD is that this prior distribution must be Gaussian: $\mathcal{P}(\phi)\propto\exp(-\frac{1}{2}\bra{\phi}\Phi\ket{\phi})$, where $\Phi$ is the prior covariance matrix. This ensures that the simulation equations are linear whenever the underlying PDE is.
It was noted by Reimar ; ReimarTowards that in the original derivation of the IFD update equation, the KL divergence was taken in the wrong direction. The new data update equation (eqn 55 in IFD ), taking the data at time $t_{i}$ to $t_{i+1}$ must be modified to 111Note that we discarded the mean field term, and changed the usage of the $D$ operators with respect to IFD .:
$$\boxed{d_{i+1}=(R_{i+1}W_{i+1})^{-1}R_{i+1}UW_{i}d_{i}}$$
(3)
Where the notation is the same as in IFD . The subscripts denote time indices, as the response and prior covariance are allowed to vary between timesteps. $W$ is the Wiener filter and is given by $W=\Phi R^{\dagger}(R\Phi R^{\dagger}+N)^{-1}$. For simplicity, we will refer to the above operator as the “update operator” and denote it by $T_{i}=(R_{i+1}W_{i+1})^{-1}R_{i+1}UW_{i}$. To achieve a practical simulation scheme, the time evolution $U$ must be truncated to some finite order, which we denote by $\bar{U}=\sum_{k=0}^{\alpha}(\Delta tL)^{k}/k!$ for some order $\alpha$.
The goal of this paper is to show that in a restricted setting, the IFD equations are consistent. This is a valuable goal, for the Lax Equivalence theorem
Lax states, if a scheme is consistent, then it converges to the true solution if and only if it is stable. We state the (paraphrased) definition of consistency:
Definition I.1 (Consistency).
For an operator $T(\Delta t,\Delta x)$ which approximates $U(t)$, with $U(t)$ being the analytic time evolution operator corresponding to $L(t)$,
the approximation is said to be consistent, if for some set of solutions, $\Omega$, to the differential equation, then for any $\phi\in\Omega$,
$$\lim_{\Delta t,\Delta x\to 0}\bigg{\|}\big{(}T(\Delta t,\Delta x)-U(\Delta t)%
\big{)}\phi(t,x)\bigg{\|}=0$$
(4)
uniformly in $t$.
Note that the above definition involves comparing operators which are defined on different spaces: $T(\Delta t,\Delta x)$ acts on a discrete space, yet $U(\Delta t)$ acts on a continuous space. Lax assumes that there is some sufficient level of smoothness such that Taylor series expansions or smooth interpolation etc. may be used to bridge the gap between spaces. We discuss the nature of this bridging for IFD later.
The other goal of this paper is to analyse the numerical error of IFD schemes, and how such error scales as the spatial and temporal resolutions $\Delta x$ and $\Delta t$ become arbitrarily fine. Typically, the error is simply the distance in norm of the simulated field, and the actual field at the simulated gridpoint Numerics . For IFD, defining a notion of error becomes somewhat more tricky, as the data $d$ can be an arbitrary transformation of the field $\phi$ being simulated. This discussion must also be postponed until after the proof of consistency.
Before moving on to proving consistency, there is a useful simplification to the IFD update operator for the linear case (Gaussians and linear responses).
Namely, we can always assume that we are in the no-noise case.
Lemma I.2.
The equations of motion for linear IFD are independent of the noise up to a simple equivalence.
Proof.
For a simulation scheme with timesteps ${t_{i}}$ for $i\in\{1,...n\}$, responses $\{R_{i}\}$, priors $\{\Phi_{i}\}$, noises $\{N_{i}\}$, Wiener filters $\{W_{i}=\Phi_{i}R_{i}^{\dagger}(R_{i}\Phi_{i}R_{i}^{\dagger}+N_{i})^{-1}\}$, and linear time evolution operators $\bar{U}_{i}=\mathbbm{1}+\Delta tL_{i}+...$, the data update equations are given by:
$$\displaystyle d_{i+1}=(R_{i+1}W_{i+1})^{-1}R_{i+1}\bar{U}_{i}W_{i}d_{i}\\
\displaystyle=\big{[}R_{i+1}\Phi_{i+1}R_{i+1}^{\dagger}(R_{i+1}\Phi_{i+1}R_{i+%
1}^{\dagger}+N_{i+1})^{-1}\big{]}^{-1}\\
\displaystyle\cdot R_{i+1}\bar{U}_{i}\Phi_{i}R_{i}^{\dagger}(R_{i}\Phi_{i}R_{i%
}^{\dagger}+N_{i})^{-1}$$
(5)
The second line is obtained by inserting the definition of the Wiener filter. We rename the terms: $(R_{i}\Phi_{i}R_{i}^{\dagger}+N_{i})=C_{i}$ , $(R_{i}\Phi_{i}R_{i}^{\dagger})=B_{i}$ and $R_{i+1}\bar{U}_{i}\Phi_{i}R_{i}^{\dagger}=A_{i}$,
yielding:
$$d_{i+1}=(B_{i+1}C_{i+1}^{-1})^{-1}A_{i}C_{i}^{-1}d_{i}=C_{i+1}B_{i+1}^{-1}A_{i%
}C_{i}^{-1}d_{i}$$
(6)
The update equations are then iterated $n$ times, yielding:
$$d_{n}=C_{n}\big{(}\prod_{i=0}^{n}B_{i+1}^{-1}A_{i}\big{)}C_{0}^{-1}d_{0}$$
(7)
The only noise-dependent terms were the $C$ terms and therefore, up to a change of basis at the beginning and end of the simulation, the equations of motion are independent of the noise. In the infinite-noise limit, $C\to N$, and in the zero noise limit $C\to B$.
∎
Given the equivalence, from here on we will
always work in the no-noise limit. Under this assumption, the update operator becomes: $T_{i}=R_{i+1}\bar{U}_{i}\Phi_{i}R_{i}(R_{i}\Phi_{i}R_{i}^{\dagger})^{-1}$
II Formalism
We now begin the work of proving consistency by defining the type of models we will be working on. We restrict our focus to PDE’s for which $L$ is translation-invariant. This case can already be solved analytically by Fourier analysis. This practice is however entirely normal in numerical methods, as many advanced schemes are too complicated to permit an analytic analysis (Godunov, , ch. 7). This is the case in IFD; as the codes are typically nonlocal, meaning the algebraic equations tend to be dependent on the global geometry of the simulation domain. Thus, the best that we can do is prove convergence for the analytically solvable case, and then hope that these conclusions hold in the non-analytically solvable case. This should be kept in mind at all times. The results presented here are for the one-dimensional case, it is strongly suspected, but not proven, that they generalize easily.
We now select the signal and data spaces. The simulated space inside the computer must always be of finite extent. For this reason, we choose the field solution space to be $\mathcal{L}^{2}([0,l])$. We apply periodic boundary conditions to render the analytic equations tractable.
Now the prior must be selected. If the PDE under consideration is translation-invariant, then one should choose a prior belief which is also translation-invariant.
Thus the prior covariance will also have a diagonal representation in Fourier space. The positivity and self-adjointness conditions on the prior covariance ensure that the eigenvalues in momentum space will be everywhere positive and greater than zero, and symmetric about the origin. Priors of this form are generally referred to as smoothness priors. Using $k$ to denote momentum, a prior $\Phi(k)$ whose values fall to zero as $k\to\infty$ essentially states that rapid oscillations in the signal are deemed unlikely; the field is smooth.
Simple examples of a prior include power laws in momentum, i.e. $|k|^{-\beta}$ for some integer $\beta$, often supplemented by a regularizing mass term: $\Phi(k)=1/(|k|^{\beta}+m^{\beta})$.
We pick the responses by assuming that we have $N$ spatial points which will be labelled with the index $j$. The responses are chosen to be constant in time, and the subscripts $R_{j}$ now denote spatial indices. The most natural and naive response is to choose the index $j$ to label a regular grid of positions. We define $\Delta x=l/N$. We let the response be any response which measures the field by integrating over some function $B(x)$ on $\mathcal{L}^{2}([0,l])$ localized at the point $x_{j}$:
$$(R\phi)_{j}=\int_{0}^{l}dxB(x-x_{j})\phi(x)\equiv\int_{0}^{l}dxB_{j}(x)\phi(x)$$
(8)
where $x_{j}$ is the $x$-position of the $j$-th gridpoint, i.e. $x_{j}=\Delta x\cdot j$. A simple example could let $B(x)$ be a box, and thus the response is an average of the field around that point. If the $x_{j}$’s are evenly spaced, we refer to any response of this form as a translation-invariant response. The $B(x)$ functions will be referred to as the response bins or just bins.
We now begin to calculate the update operator, starting with the computation of $(R\Phi R^{\dagger})^{-1}$. Since both the responses and prior covariance are invariant under translations of multiples of $\Delta x$, we can make a very general statement:
Lemma II.1.
Given a signal space of the form $\mathcal{L}^{2}([0,l])$ with periodic boundary conditions, a translation invariant response $R_{j}$ whose bin function $B(x)$ has a Fourier series representation, as well as a prior covariance $\Phi$ which is diagonal in momentum space, $(R\Phi R^{\dagger})_{jl}$ will be of the form:
$$\sum_{k}\Phi(k)|\widehat{B}(k)|^{2}e^{ik(x_{j}-x_{l})},$$
(9)
where $\widehat{B}(k)$ is the Fourier coefficient of $B(x)$.
Proof.
By the shift property of the Fourier transform, $\widehat{R}_{j,k}=e^{-ikx_{j}}\widehat{R}_{0,k}=e^{-ikx_{j}}\widehat{B}(k)$. Therefore $(R\Phi R^{\dagger})_{jl}$ is
$$\displaystyle(R\Phi R^{\dagger})_{jl}=\sum_{k}\sum_{q}e^{ikx_{j}}\widehat{B}(k%
)\Phi(k)\delta_{kq}\widehat{B}^{*}(q)e^{-iqx_{l}}$$
$$\displaystyle=\sum_{k}\Phi(k)|\widehat{B}(k)|^{2}e^{ik(x_{j}-x_{l})}$$
(10)
as desired.
∎
The generalization to higher dimensions would take $x$ and $k$ to vectors. It must be stressed that we are not demanding that the simulation is carried out in Fourier space, it is rather that the operator will always have such a representation. From now on, any simulation scheme which satisfies the criteria of the previous lemma, and in addition has a translation-invariant time evolution operator $U$, will be referred to as a translation invariant scheme.
The $R\Phi R^{\dagger}$ matrix now needs to be inverted, however the inverse is not equal to the inverse of the Fourier coefficients.
Observe that the spatial gridpoints are both finite and discrete, which means
that terms such as $\sum_{j}e^{ix_{j}(k-q)}$ do not form Kroenecker deltas $\delta_{kq}$.
The sum equals $N$, not only when $k=q$ but also when $(n-m)/N$ is an integer, i.e. $n=m\mod(N)$.
The reason for this, is that data space is a discrete periodic interval, which has a discrete Fourier transform (DFT). For a DFT, the momentum values $k$ are the same as those for the continuous interval, albeit with a highest uniquely resolvable frequency known as the Nyquist frequency, which is equal to half of the sampling frequency. In this case, the Nyquist frequency is $\frac{\pi}{\Delta x}$ and is denoted by $f_{N}$. Given that the matrix is indeed translation-invariant in data space, it must have some diagonal representation in the discrete Fourier transform, i.e. some scalar function of $k$, for $k$ now less than $f_{N}$.
This representation can be found by resumming over multiples of the Nyquist frequency:
Lemma II.2.
Given a regular, discrete grid of points $\{x_{j}\}$ for $j\in\{1,\ldots,N\}$ on a periodic interval, and a matrix of the form:
$$A_{lj}=\sum^{\infty}_{k=-\infty}f(k)e^{ik(x_{l}-x_{j})}$$
(11)
for $f(k)$ some function of $k$, it has a diagonal representation in the DFT Fourier space, given by:
$$A_{lj}=\sum^{f_{N}}_{|k|}\bigg{(}\sum_{b\in 2f_{N}\mathbb{Z}}f(k+b)\bigg{)}e^{%
ik(x_{l}-x_{j})}=\sum^{f_{N}}_{|k|}g(k)e^{ik(x_{l}-x_{j})}$$
(12)
where the new diagonal function $g(k)$ denotes the sum $\sum_{b\in 2f_{N}\mathbb{Z}}f(k+b)$.
Proof.
We partition the infinite sum over $k$ in eqn. 11 into smaller sums shifted by multiples of the Nyquist frequency.222Note that depending on whether the number of data points is even or odd, the domain of $|k|<f_{N}$ changes. For odd $N$ we use the convention that $k\in[-(N-1)/2,(N-1)/2]$ and if it’s even we use $k\in[-N/2,N/2-1]$. For
any $x_{i}$ and $x_{j}$ separated by a multiple of $\Delta x$ and $b=2\pi n/\Delta x$, we have $(k+b)(x_{i}-x_{j})=k(x_{i}-x_{j})+2\pi n$. This factor of $2\pi$ then
disappears in the complex exponential, yielding the desired result.
This resummed function is a diagonal function of the DFT frequencies $k<f_{N}$, and so must be the desired operator.
∎
Due to the physical analogy with Brillouin zones, we refer to the procedure of summing over multiples of the Nyquist frequency as the sum over Brillouin zones.
Now that we have obtained a representation of the operator which is diagonal in the DFT space, inverting follows easily by taking the inverse of the DFT Fourier coefficients:
$$(R\Phi R^{\dagger})_{lj}^{-1}=\frac{1}{N}\sum^{f_{N}}_{|k|}\frac{e^{ik(x_{l}-x%
_{j})}}{\sum_{b\in 2f_{N}\mathbb{Z}}\Phi(k+b)|\widehat{B}(k+b)|^{2}}$$
(13)
The factor of $N$ comes from the different normalizations of the DFT and the regular Fourier transform. Fourier modes in the DFT are normalized as $\frac{1}{\sqrt{N}}e^{-ikx_{j}}$.
Now it is time to compute the second part of the update operator, $R\bar{U}\Phi R^{\dagger}$. Given that $\bar{U}$ is assumed to be diagonal in Fourier space, the previous lemma II.1 applies, and the operator will also be diagonal in the DFT space, with a sum over Brillouin zones. With this information, we may now write down the general form of the update operator $T=R\bar{U}\Phi R^{\dagger}(R\Phi R^{\dagger})^{-1}$ for translation-invariant systems:
$$T_{lj}=\sum^{f_{N}}_{|k|}\frac{\sum_{b\in 2f_{n}\mathbb{Z}}\bar{U}(k+b)\Phi(k+%
b)|\widehat{B}(k+b)|^{2}}{\sum_{\hat{b}\in 2f_{n}\mathbb{Z}}\Phi(k+\hat{b})|%
\widehat{B}(k+\hat{b})|^{2}}e^{ik(x_{l}-x_{j})}$$
(14)
The factor of $1/N$ is cancelled by a factor of $N$ coming from the sum over spatial indices.
II.1 Consistency
This can be shown in the translation-invariant case by adding some light restrictions: the response bins $B(x)$ are compactly supported with bounded Fourier transform, and $\widehat{B}(0)\neq 0$. We also require that $U(k)\Phi(k)\to 0$ as $k\to\pm\infty$.
The bounded Fourier transform requirement will almost always be true for any reasonable response. It holds for all smooth, compactly-supported functions, by the
Paley-Wiener theorem Reedsimon .
To prove consistency, we ask if $T(k)\to U(k)$ in the limit of high resolution. In the Fourier representation, comparing the action of $T$ and $U$ is simple, despite the fact that they technically act on different spaces. The definition (def. I.1) requires that the operators converge for any chosen function in signal space, but not that they converge at the same rate for all functions. Pick a basis of signal space consisting of Fourier modes, then pick out a single mode of frequency $k$. As the spatial resolution increases, eventually the Nyquist frequency ($f_{N}=\pi/\Delta x$) will be greater than $k$. Past this resolution, $T(k)$ and $U(k)$ can both be thought of as acting on the same space. $T(k)$ contains a time-order approximation $\bar{U}(k)$ to $U(k)$. If we can show that as $\Delta x\to 0$, $T(k)\to\bar{U}(k)$, then in the joint limit of time and space resolution going to infinity, then $T$ approaches $U$.
Hence we need that for each fixed $k$, $T(k)\to\bar{U}(k)$ in $\Delta x$, but the convergence does not need to be uniform in $k$.
For the translation-invariant responses, increasing spatial resolution requires increasing the number of bins while simultaneously decreasing their width.
Given some initial resolution $\Delta x_{0}$ for which all the bins fit evenly inside the interval, we pick an integer $\lambda$ ranging from 1 to infinity, then set $\Delta x=\Delta x_{0}/\lambda$. This guarantees that the new set of scaled bins $B(\lambda x)=B_{\lambda}(x)$ fits evenly inside the interval.
The compact support property of the bins allows us to exploit the
fact that up to a normalization constant, the coefficients $\widehat{B}(k)$ of the discrete values of $k$ in the Fourier series of the bins are the same as the values at $k$
in the continuous Fourier transform of $B(x)$. Because if a function is compact, the integrals over a sufficiently large finite interval, and an infinite interval are the same.
This in turn allows us to exploit the scaling property of the Fourier transform $\widehat{B_{\lambda}}(k)=\frac{1}{\lambda}\widehat{B}(k/\lambda)$. The normalization constant $\lambda$ and the constant from the differing normalizations of the Fourier transform cancel due to the division in eqn. 14.
Now observe the sum over the Brillouin zones. We sum over $b\in 2\mathbb{Z}f_{N}$ where $f_{N}=\pi/\Delta x$ and thus $f_{N}^{\lambda}=\pi\lambda/\Delta x_{0}$ and $b^{\lambda}=2\pi n\lambda/\Delta x_{0}$ for $n\in\mathbb{Z}$, and observe the upper term of eqn. 14 :333The prior is not scaled with $\lambda$ (one’s beliefs about the system should not change depending on the resolution of their equipment)
$$\sum_{n\in\mathbb{Z}}\bar{U}(k+\frac{2\pi n\lambda}{\Delta x_{0}})\Phi(k+\frac%
{2\pi n\lambda}{\Delta x_{0}})\big{|}\widehat{B}(\frac{1}{\lambda}(k+\frac{2%
\pi n\lambda}{\Delta x_{0}}))\big{|}^{2}$$
(15)
The $\lambda$ term inside $\widehat{B}$ can be absorbed to give:
$$|\widehat{B}(\frac{1}{\lambda}(k+\frac{2\pi n\lambda}{\Delta x_{0}}))|^{2}=|%
\widehat{B}(\frac{k}{\lambda}+\frac{2\pi n}{\Delta x_{0}}))|^{2}$$
(16)
We expect that in the limit of $\Delta x\to\infty$, the higher terms in the sum vanish, leaving only terms in the first Brillouin zone. That is to say, we can express the numerator as:
$$\displaystyle\bar{U}(k)\Phi(k)|\widehat{B}(\frac{k}{\lambda})|^{2}+$$
(17)
$$\displaystyle\underbrace{\sum_{n\neq 0}\bar{U}(k+\frac{2\pi n\lambda}{\Delta x%
_{0}})\Phi(k+\frac{2\pi n\lambda}{\Delta x_{0}})|\widehat{B}(\frac{k}{\lambda}%
+\frac{2\pi n}{\Delta x_{0}})|^{2}}_{\equiv\delta(k,\lambda)\to 0}$$
Where we rename the sum as $\delta(k,\lambda)$ to denote that the term (hopefully) vanishes as $\lambda\to\infty$. The denominator is expanded similarly.
In the limit, the $\delta(k,\lambda)$ terms in the numerator and denominator of eqn. 14 would then cancel, leaving just $\bar{U}$, i.e. we want
$$\displaystyle\frac{\lim_{\lambda\to\infty}\sum_{n\in\mathbb{Z}}\bar{U}(k+\frac%
{2\pi n\lambda}{\Delta x_{0}})\Phi(k+\frac{2\pi n\lambda}{\Delta x_{0}})|%
\widehat{B}(\frac{k}{\lambda}+\frac{2\pi n}{\Delta x_{0}})|^{2}}{\lim_{\lambda%
\to\infty}\sum_{m\in\mathbb{Z}}\Phi(k+\frac{2\pi m\lambda}{\Delta x_{0}})|%
\widehat{B}(\frac{k}{\lambda}+\frac{2\pi m}{\Delta x_{0}})|^{2}}$$
$$\displaystyle=\frac{\bar{U}(k)\Phi(k)|\widehat{B}(\frac{k}{\lambda})|^{2}+%
\delta(k,\lambda)}{\Phi(k)|\widehat{B}(\frac{k}{\lambda})|^{2}+\delta^{\prime}%
(k,\lambda)}$$
$$\displaystyle\approx\frac{\bar{U}(k)\Phi(k)|\widehat{B}(\frac{k}{\lambda})|^{2%
}}{\Phi(k)|\widehat{B}(\frac{k}{\lambda})|^{2}}=\bar{U}(k)$$
(18)
This works provided $\widehat{B}(0)\neq 0$, so that the above denominator remains nonzero, and the equation remains well-defined. We can expect that for all terms with $n\neq 0$, in the limit of $\lambda\to\infty$ each term goes to zero because $\bar{U}(k)\Phi(k)$ goes to zero at large $|k|$, by assumption. Therefore we want to swap the limit and the infinite sum. This is possible if and only if the sequence of above functions converges uniformly, which we now prove.
We discard the $n=0$ term, and consider the positive
$n$ half of the sum; the negative $n$ half of the sum follows trivially.
The goal is for the functions to converge to zero, so we state: a function converges uniformly to zero if for any positive $\epsilon$,
there is an $N$ such that $\forall n\geq N$, $|f_{n}(\lambda)|<\epsilon$ for all values of $\lambda$.
For our purposes, $f_{n}(\lambda)=\bar{U}(k+\frac{2\pi n\lambda}{\Delta x_{0}})\Phi(k+\frac{2\pi n%
\lambda}{\Delta x_{0}})|\widehat{B}(\frac{k}{\lambda}+\frac{2\pi n}{\Delta x_{%
0}})|^{2}$. We bound the function $|\widehat{B}(\frac{k}{\lambda}+\frac{2\pi n}{\Delta x_{0}})|^{2}<C$
for some constant $C$, which we may do by assumption. The bin terms do not vanish in the limit of large $\lambda$, because as the bins become narrower, their Fourier transforms widen out, at the exact same rate as the Nyquist frequency is increasing. Hence the condition $\Phi(k)\bar{U}(k)\to 0$ as $k\to\infty$ is required to guarantee convergence.
This condition means that for $|k|$ large enough $\bar{U}(k)\Phi(k)$ can be bounded by some monotonically decreasing function of $|k|$, call it $g(|k|)$. We start by finding a bound for
$\lambda=1$, and then show that this bound holds for all $\lambda$. For $\lambda=1$, and the desired $\epsilon$ bound, we can pick some $n$ large enough such that we are in this decreasing regime, hence $|\bar{U}(k+\frac{2\pi n\lambda}{\Delta x_{0}})\Phi(k+\frac{2\pi n\lambda}{%
\Delta x_{0}})|C<g(k+\frac{2\pi n\lambda}{\Delta x_{0}})<\epsilon$. For higher $\lambda$ and large $n$, $|k+\frac{2\pi n}{\Delta x_{0}}|<|k+\frac{2\pi n\lambda}{\Delta x_{0}}|$, and since we have taken $n$ to be large enough that we are in the decreasing regime, the $g(k)$ bound also holds. Thus the bound holds for all $\lambda$. The sequence of functions is therefore uniformly convergent, and eqn. II.1 holds. We can now state:
Theorem II.3.
For a 1-D translationally-invariant IFD scheme, whose response bins are compactly supported with bounded Fourier transform and $\widehat{B}(0)\neq 0$, and some time-order approximation $\bar{U}(k)$ to $U(k)$, where $k$ denotes momentum, then the scheme is consistent provided $\lim_{k\to\infty}\bar{U}(k)\Phi(k)=0$.
Important to note is that we only require $\bar{U}(k)\Phi(k)\to 0$, not $U(k)\Phi(k)$. For derivative operators s.t. $U=\exp(\Delta t\partial_{x})=\exp(i\Delta tk)$ or similar, this would require that the prior covariance, $\Phi(x)$, is infinitely differentiable, a.k.a smooth. Using the approximated time expansion, the prior covariance only needs to be as many-times differentiable as the order of the expansion dictates.
There is a subtlety in the above proof that was not mentioned, namely that the bridge between the simulation space and the field space was through the frequency component. However, given that the responses may have a very general form, we have not checked if plane waves with momentum $k$ in data and signal space actually represent the same field. Fortunately, they do. Pick a signal space plane wave $\phi(x)=e^{-ikx}$, and compute
$$R_{j}\phi=\int B(x-x_{j})e^{-ikx}dx=\widehat{B}(k)e^{-ikx_{j}}$$
(19)
Thus, regardless of the shape of the response bins, plane waves in signal space show up as plane waves in data space, as long as the grid is translation-invariant.
The plane wave will be scaled by a constant factor $\widehat{B}(-k)$ which depends on its momentum, but it is still a plane wave. The preceding proof shows that the $k$ dependence will cease to matter in the limit of infinite resolution. Note that if $k$ was above the Nyquist frequency, the $e^{ikx_{j}}$ term with the discrete $x_{j}$ coordinates implies that the plane wave appears in the data with a frequency below the Nyquist, as expected.
II.2 Error scaling
We seek an estimate of the one-step error (OSE), which is the error induced in a single timestep of the simulation, assuming zero initial error (Numerics, , p.593). One way of phrasing the OSE in the IFD framework, would be to take initial data $d_{i}$ that is a perfect measurement of the field $R_{i}\phi(t_{i})$ at time $t_{i}$ and then find $E=|d_{i+1}-R_{i+1}\phi(t_{i+1})|$ using the update operator etc. This would involve switching back and forth between different spaces. Fortunately, the work of the previous section allows us to avoid this by noticing that for every frequency, there is a certain resolution beyond which data space and signal space become essentially the same. Thus we can get a good estimate on the error by just calculating the difference in the operator norm:
$$E=|T(\Delta t,\Delta x)-U|$$
(20)
and analysing the rate of convergence in terms of $\mathcal{O}(\Delta x)$ and $\mathcal{O}(\Delta t)$. We reuse formula 14 and take the same limit as before, giving:
$$\displaystyle E(k)=$$
$$\displaystyle\bigg{|}\frac{\sum_{n\in\mathbb{Z}}\bar{U}(k+\frac{2\pi n\lambda}%
{\Delta x_{0}})\Phi(k+\frac{2\pi n\lambda}{\Delta x_{0}})|\widehat{B}(\frac{k}%
{\lambda}+\frac{2\pi n}{\Delta x_{0}})|^{2}}{\sum_{m\in\mathbb{Z}}\Phi(k+\frac%
{2\pi m\lambda}{\Delta x_{0}})|\widehat{B}(\frac{k}{\lambda}+\frac{2\pi m}{%
\Delta x_{0}})|^{2}}-U(k)\bigg{|}$$
(21)
We use the expansion $\bar{U}=\sum_{p=0}^{\alpha}(\Delta tL)^{p}/p!$ to find the error in terms of powers of $L$.
$$\displaystyle E(k)\leq\sum_{p=0}^{\alpha}\frac{\Delta t^{p}}{p!}\times$$
(22)
$$\displaystyle\bigg{|}\frac{\sum_{n\in\mathbb{Z}}L^{p}(k+\frac{2\pi n\lambda}{%
\Delta x_{0}})\Phi(k+\frac{2\pi n\lambda}{\Delta x_{0}})|\widehat{B}(\frac{k}{%
\lambda}+\frac{2\pi n}{\Delta x_{0}})|^{2}}{\sum_{m\in\mathbb{Z}}\Phi(k+\frac{%
2\pi m\lambda}{\Delta x_{0}})|\widehat{B}(\frac{k}{\lambda}+\frac{2\pi m}{%
\Delta x_{0}})|^{2}}-L^{p}(k)\bigg{|}$$
Note that each term inside the absolute value is not dependent on the time resolution. We analyse the scaling of each of these spatial terms individually, by starting with the first order $\Delta t$ term, then generalizing. In the limit of high resolutions, we can expand the numerator in the same way that we did in eqn. 17: $L(k)\Phi(k)|\widehat{B}(\frac{k}{\lambda})|^{2}+\epsilon(k,\lambda)$ for some function $\epsilon$, which goes to zero as $\lambda\to\infty$. We expand the denominator as $\Phi(k)|\widehat{B}(\frac{k}{\lambda})|^{2}+\delta(k,\lambda)$, with $\delta$ being some other small vanishing function.
The strategy is then to find an expression for the fraction in terms of $\epsilon$ and $\delta$, then bound each term and analyse how fast they approach zero.
We use the Taylor expansion for $1/(1-x)\approx 1+x+x^{2}+\cdots$ to expand the denominator in eqn. 22 into:
$$\displaystyle\frac{1}{\Phi(k)|\widehat{B}(\frac{k}{\lambda})|^{2}+\delta(k,%
\lambda)}=$$
(23)
$$\displaystyle\frac{1}{\Phi(k)|\widehat{B}(\frac{k}{\lambda})|^{2}}-\frac{%
\delta(k,\lambda)}{(\Phi(k)|\widehat{B}(\frac{k}{\lambda})|^{2})^{2}}+\cdots$$
We then multiply the numerator by the denominator:
$$\displaystyle\bigg{(}L(k)\Phi(k)|\widehat{B}(\frac{k}{\lambda})|^{2}+\epsilon(%
k,\lambda)\bigg{)}\times$$
(24)
$$\displaystyle\bigg{(}\frac{1}{\Phi(k)|\widehat{B}(\frac{k}{\lambda})|^{2}}-%
\frac{\delta(k,\lambda)}{(\Phi(k)|\widehat{B}(\frac{k}{\lambda})|^{2})^{2}}%
\bigg{)}$$
$$\displaystyle=L(k)+\frac{\epsilon(k,\lambda)}{\Phi(k)|\widehat{B}(\frac{k}{%
\lambda})|^{2}}-\frac{L(k)\delta(k,\lambda)}{\Phi(k)|\widehat{B}(\frac{k}{%
\lambda})|^{2}}+\cdots$$
Calculating the power-law scaling of the above terms is complicated by the fact that each has a $|\widehat{B}(\frac{k}{\lambda})|^{2}$ in the denominator, which has it’s own scaling with $\lambda$. Exploiting the fact that $\widehat{B}(0)\neq 0$, allows us to write out each of these as a Taylor series, and then reuse the $1/(1-x)$ expansion:
$$\displaystyle\frac{1}{\Phi(k)|\widehat{B}(\frac{k}{\lambda})|^{2}}=\frac{1}{%
\Phi(k)|\widehat{B}(0)|^{2}+\mathcal{O}(1/\lambda)+\cdots}$$
$$\displaystyle=\frac{1}{\Phi(k)|\widehat{B}(0)|^{2}}+\mathcal{O}(1/\lambda)+\cdots$$
(25)
We then see that however fast $\epsilon(k,\lambda)$ goes to zero, $\epsilon(k,\lambda)\mathcal{O}(1/\lambda)$ goes to zero faster. Since only the worst (slowest) converging terms are of interest, we can replace $\frac{1}{\Phi(k)|\widehat{B}(\frac{k}{\lambda})|^{2}}$ with $\frac{1}{\Phi(k)|\widehat{B}(0)|^{2}}$ without any adverse effects.
The scaling of the $\epsilon$ and $\delta$ terms can only be estimated if the scaling behaviour of the prior and $L(k)$ are known. To this end, suppose that as $|k|$ becomes large, $\Phi(k)$ can be bounded by some decreasing power law in $k$, $|k|^{-\beta}$ for $\beta$ positive. We also assume that $L(k)$ can be bounded by some $|k|^{\gamma}$ for $\gamma$ positive, as $L$ will typically be a derivative operator, with $\partial_{x}^{n}=(ik)^{n}$. There will be constants of proportionality, but they are irrelevant with respect to the scaling.
Using the uniform bound $C$ from before, we can bound the $\epsilon$ term by:
$$\displaystyle|\epsilon(k,\lambda)|=\bigg{|}\sum_{n\neq 0}L(k+\frac{2\pi n%
\lambda}{\Delta x_{0}})\Phi(k+\frac{2\pi n\lambda}{\Delta x_{0}})|\widehat{B}(%
\frac{k}{\lambda}+\frac{2\pi n}{\Delta x_{0}})|^{2}\bigg{|}$$
$$\displaystyle\leq C^{2}\sum_{n\neq 0}\bigg{|}L(k+\frac{2\pi n\lambda}{\Delta x%
_{0}})\Phi(k+\frac{2\pi n\lambda}{\Delta x_{0}})\bigg{|}$$
$$\displaystyle\leq C^{2}\sum_{n\neq 0}\bigg{|}\frac{2\pi n\lambda}{\Delta x_{0}%
}\bigg{|}^{\gamma-\beta}=\lambda^{\gamma-\beta}C^{2}\sum_{n\neq 0}\bigg{|}%
\frac{2\pi n}{\Delta x_{0}}\bigg{|}^{\gamma-\beta}$$
(26)
The term inside the sum is independent of the scaling. Therefore, this bound scales as $\mathcal{O}(\lambda^{\gamma-\beta})$, which we identify with $\mathcal{O}(\Delta x^{\beta-\gamma})$, since $\Delta x=\Delta x_{0}/\lambda$. We repeat the argument with the denominator, and obtain a term proportional to $\mathcal{O}(\Delta x^{\beta})$.
Thus eqn. 24 scales as:
$$\displaystyle=L(k)+\mathcal{O}(\Delta x^{\beta})+\mathcal{O}(\Delta x^{\beta-%
\gamma})$$
$$\displaystyle=L(k)+\boxed{\mathcal{O}(\Delta x^{\beta-\gamma})}$$
(27)
yielding a total scaling of $\mathcal{O}(\Delta t)\mathcal{O}(\Delta x^{\beta-\gamma})$. The other $\mathcal{O}$ term vanishes because only the term with the worst scaling (lowest power) contributes. For a term of order $\Delta t^{p}$ in eqn. 22, we repeat the argument and obtain $E\propto\mathcal{O}(\Delta t^{p}\Delta x^{\beta-p\gamma})$.
The total error scaling in eqn. 22 is determined by the sum of the individual $p$ terms:
$$\boxed{E\propto\sum_{p=0}^{N}\mathcal{O}(\Delta t^{p}\Delta x^{\beta-p\gamma})}.$$
(28)
Though the error will be bounded by the worst scaling of any of the individual terms. We see from this formula that going to higher orders in $\Delta t$ decreases the spatial order. This is fine for $L=\partial_{x}$, because the total order remains the same, but for higher derivatives, the spatial order decreases faster in $p$ than the time order increases. If $\Delta x$ and $\Delta t\to 0$ at a proportional rate, this will decrease the total order and making the overall scaling worse.
This can be thought of in the following way: if the prior covariance drops off as some power $\beta$, then it is only $\beta$ times differentiable, so it is not smooth. Going to higher orders in the expansion $\bar{U}=\sum_{p=0}^{\alpha}(\Delta tL)^{p}/p!$ involves taking derivatives of ever-higher order, and thus at some point the $L\Phi(x)$ term in the update operator can no longer be taken. The bin functions do not appear in the above expression, because in the limit of high resolutions, they tend to approximate delta functions, and their exact form becomes irrelevant.
The consequences of this formula deserve some thought, particularly the troubling implication that going to higher orders in time can in fact decrease the quality of the simulation. First, it should be noted that higher-order schemes are not necessarily better, depending on the task. For example, according to the Godunov Theorem(Godunov, , p. 280), higher order schemes have a tendency to develop spurious oscillations around shocks. It should also be noted that the above formula applies in the high resolution (and thus high-$k$) limit. One could conceivably introduce a prior covariance which has a cutoff at high $k$, or perhaps one whose value drops of exponentially with $k$. An exponentially-falling prior covariance would then raise the prospect of a scheme with intermediate error scaling. The implications of such a scheme are however not yet clear.
III Conclusions
We have now proved consistency, and found an estimate of the error scaling for IFD schemes, using a set of strong simplifying assumptions, which we grouped together under the name of a translation-invariant scheme. These assumptions were:
•
“Linear case” of IFD: linear differential equation, linear measurements with additive noise, and Gaussian prior distribution of the fields.
•
Translation invariance of all the above quantities.
•
One-dimensional simulation space with periodic boundary conditions.
•
Bin functions $B(x)$ which are compactly supported, have bounded Fourier transform, and whose Fourier transform $\widehat{B}(k)$ has $\widehat{B}(0)\neq 0$.
•
$\bar{U}(k)\Phi(k)\to 0$ as $k\to\infty$.
To obtain an estimate of the scaling, we needed to assume:
•
The operators $L(k)$ and $\Phi(k)$ may be bounded by power-laws $|k|^{\gamma}$ and $|k|^{-\beta}$ for $\beta,\gamma>0$ respectively, at large values of $k$.
These restrictions mean that the results in this paper are only directly applicable to a very small subset of the simulation schemes that may be constructed using IFD. Given the immense amount of freedom inherent in the IFD framework, it is doubtful that a general analytic proof of consistency will be achievable. This paper should be instead taken as a general indication that IFD is at least a sensible methodology.
That being said, it is expected that the above assumptions could be easily weakened in order to obtain a much stronger result. The most obvious step is to remove the restriction of a 1-D simulation space. Generalizing to higher dimensions would simply involve taking $x$ and $k$ to vectors, and negotiating the multiple infinite sums over spatial and momentum coordinates. Furthermore, the fact that the data update operators can be expressed using a sum over Brillouin zones immediately suggests that these results could be extended to a simulation over any periodic lattice of data points, not just rectangular domains.
The restrictions on the bin functions are relatively weak. Given that a simulation domain must always be of finite extent, the bin functions are forced to be compactly supported. The requirement that $\widehat{B}(0)\neq 0$ deserves some discussion however. This requirement, rather than being physically motivated, was inserted solely to avoid the occurrence of $0/0$ terms in the limit of high resolutions. It may however, reflect a physical requirement. Take, for example, a bin function $B(x)$ which is everywhere positive, and is symmetric and peaked about zero. It will satisfy $\widehat{B}(0)\neq 0$, and in the limit of high resolutions, this bin will approach a delta function, and will represent a sample of the field value at that point. In contrast, take $xB(x)$; this function is now odd, and in the limit of high resolutions, this will approach something that represents a point sample of the derivative of the field about that point. Attempting to apply IFD to reconstructions of the derivative of the field may give nonsensical results, which is what the $\widehat{B}(0)\neq 0$ requirement may be implying.
Removing the translation-invariance requirements would be extremely desirable, however would be much more difficult. The main reason the Fourier approach was necessary, was the inversion of the $(R\Phi R^{\dagger})_{ij}$ matrices. With a reasonable smoothness prior, these matrices tend to be relatively local in the spatial indices. However, inverting is a nonlocal problem, which makes the inverses of these matrices dependent on the global geometry of the simulation domain, and makes them very difficult to analyse analytically. The use of periodic boundary conditions allowed us to sidestep this consideration. Any proof seeking to show consistency and convergence in the non-translationally-invariant case would probably have to use a different approach to what we have done here.
Finally, an extension of these results outside of the linear regime is self-evidently desirable, but may present some significant challenges. In particular, IFT inference problems including nonlinear responses and non-Gaussian priors on the fields typically result in a need to calculate Feynmann diagrams. Integrating these into an analytic proof of consistency will be challenging to say the least.
References
[1]
Torsten A. Enßlin.
Information field dynamics for simulation scheme construction.
Phys. Rev. E, 87:013308, Jan 2013.
[2]
Christian Münch.
Mathematical foundation of Information Field dynamics.
Master’s thesis, Technische Universität München, Germany,
2014.
Retrieved from:
http://wwwmpa.mpa-garching.mpg.de/~ensslin/research/research_IFT.html#Publications.
[3]
R. H. Leike and T. A. Enßlin.
Towards information optimal simulation of partial differential
equations.
ArXiv e-prints, September 2017.
[4]
Torsten A. Enßlin, Mona Frommert, and Francisco S. Kitaura.
Information field theory for cosmological perturbation reconstruction
and nonlinear signal analysis.
Phys. Rev. D, 80:105005, Nov 2009.
[5]
J. C. Lemm.
Bayesian Field Theory: Nonparametric Approaches to Density
Estimation, Regression, Classification, and Inverse Quantum Problems.
ArXiv Physics e-prints, December 1999.
[6]
Persi Diaconis.
Bayesian numerical analysis.
Statistical decision theory and related topics IV, 1:163–175,
1988.
[7]
J. Skilling.
Bayesian solution of ordinary differential equations.
Maximum Entropy and Bayesian Methods, Seattle, 1991.
[8]
Anthony O’Hagan.
Some bayesian numerical analysis.
Bayesian Statistics, 4(345–363):4–2, 1992.
[9]
J. B. Kadane and G. W. Wasilkowski.
Average case epsilon-complexity in computer science: A Bayesian
view.
In Bayesian Statistics 2, Proceedings of the Second Valencia
International Meeting, number July, pages 361–374, 1985.
[10]
F.-X. Briol, C. J. Oates, M. Girolami, M. A. Osborne, and
D. Sejdinovic.
Probabilistic Integration: A Role in Statistical Computation?
ArXiv e-prints, December 2015.
[11]
Michael Schober, David K Duvenaud, and Philipp Hennig.
Probabilistic ode solvers with runge-kutta means.
In Z. Ghahramani, M. Welling, C. Cortes, N. D. Lawrence, and K. Q.
Weinberger, editors, Advances in Neural Information Processing Systems
27, pages 739–747. Curran Associates, Inc., 2014.
[12]
P. R. Conrad, M. Girolami, S. Särkkä, A. Stuart, and
K. Zygalakis.
Probability Measures for Numerical Solutions of Differential
Equations.
ArXiv e-prints, June 2015.
[13]
Philipp Hennig.
Probabilistic interpretation of linear solvers.
SIAM Journal on Optimization, 25(1):234–260, 2015.
[14]
Jasper Snoek, Hugo Larochelle, and Ryan P Adams.
Practical bayesian optimization of machine learning algorithms.
In F. Pereira, C. J. C. Burges, L. Bottou, and K. Q. Weinberger,
editors, Advances in Neural Information Processing Systems 25, pages
2951–2959. Curran Associates, Inc., 2012.
[15]
Maziar Raissi, Paris Perdikaris, and George Em Karniadakis.
Inferring solutions of differential equations using noisy
multi-fidelity data.
Journal of Computational Physics, 335:736 – 746, 2017.
[16]
Cedric Archambeau, Dan Cornford, Manfred Opper, and John Shawe-Taylor.
Gaussian process approximations of stochastic differential equations.
In Neil D. Lawrence, Anton Schwaighofer, and Joaquin Quiñonero
Candela, editors, Gaussian Processes in Practice, volume 1 of Proceedings of Machine Learning Research, pages 1–16, Bletchley Park, UK,
12–13 Jun 2007. PMLR.
[17]
Reimar H. Leike and Torsten A. Enßlin.
Optimal belief approximation.
Entropy, 19(8), 2017.
[18]
P. D. Lax and R. D. Richtmyer.
Survey of the stability of linear finite difference equations.
Communications on Pure and Applied Mathematics, 9(2):267–293,
1956.
[19]
R. Corless and N. Fillion.
A Graduate Introduction to Numerical Methods: From the Viewpoint
of Backward Error Analysis.
SpringerLink : Bücher. Springer New York, 2013.
[20]
C. Hirsch.
Numerical Computation of Internal and External Flows:
Fundamentals of Computational Fluid Dynamics.
Butterworth-Heinemann. Elsevier/Butterworth-Heinemann, 2007.
[21]
M. Reed and B. Simon.
Methods of Modern Mathematical Physics.
Number v. 2 in Methods of Modern Mathematical Physics. Academic
Press, 1975. |
Optimistic Optimization for Statistical Model Checking with Regret Bounds
Extended Abstract
Negin Musavi${}^{1}$, Dawei Sun${}^{1}$, Sayan Mitra${}^{1}$, Geir Dullerud${}^{1}$, and Sanjay Shakkottai${}^{2}$
{nmusavi2,daweis2,mitras,dullerud}@illinois.edu
${}^{1}$University of Illinois at Urbana Champaign
sanjay.shakkottai@utexas.edu
${}^{2}$University of Texas at Austin
Abstract
We explore application of multi-armed bandit algorithms to statistical model checking (SMC) of Markov chains initialized to a set of states.
We observe that model checking problems requiring maximization of probabilities of sets of execution over all choices of the initial states, can be formulated as a multi-armed bandit problem, for appropriate costs and rewards. Therefore, the problem can be solved using multi-fidelity hierarchical optimistic optimization (MFHOO).
Bandit algorithms, and MFHOO in particular, give (regret) bounds on the sample efficiency which rely on the smoothness and the near-optimality dimension of the objective function, and are a new addition to the existing types of bounds in the SMC literature.
We present a new SMC tool—HooVer—built on these principles and our experiments suggest that: Compared with exact probabilistic model checking tools like Storm, HooVer scales better; compared with the statistical model checking tool PlasmaLab, HooVer can require much less data to achieve comparable results.
1 Introduction
The multi-armed bandit problem is an idealized mathematical model for sequential decision making in unknown random environments and it has been used to study exploration-exploitation trade-offs.
In the problem setup, each arm $x\in\mathcal{X}$ of the bandit is associated with a cost $\lambda_{x}$ of playing and an unknown reward distribution $M_{x}$. In order to maximize the final reward with a given cost budget, the algorithm plays some arm, collects the stochastically generated reward, and decides on the next arm, until the cost budget is exhausted.
Starting from the motivation of designing clinical trials in the 1930s [1, 2, 3], there has been major developments in the Bandit theory over the last few decades (see, for example the books [4, 5, 6]).
Several different strategies have addressed this problem and
strong connections have been drawn with other fields such as online learning.
In this paper, we explore how Bandit algorithms can be used for model checking of stochastic systems.
A requirement $R$ for a stochastic system $\mathcal{M}$ usually checks whether the measure of executions of $\mathcal{M}$ satisfying certain temporal formulas cross certain thresholds [7, 8].
Model checking for such requirements can be solved by calculating the exact measure of the executions that satisfy the relevant subformulas of $R$ [9, 10, 11, 12].
In this paper, we focus on the alternative statistical model checking (SMC)
approach which samples some executions of $\mathcal{M}$ and uses hypothesis testing to infer whether the samples provide
statistical evidence for the satisfaction (or violation) of $R$ [7, 13, 14].
Execution data is a costly resource111Generating execution data involves running simulations or performing tests., therefore, a number of SMC approaches minimize the expected number of samples needed for verification, for example, using sequential probability ratio tests, Chernoff bound, and Student’s t-distribution.
Several SMC tools have been developed (for example, Ymer [15], VESTA [16], MultiVesta [17], PlasmaLab [18], MODES [19], UPPAAL [20], and MRMC [21]
),
and they have been used to verify many systems [22, 23, 24, 25, 26, 27, 28, 29].
Most SMC algorithms crucially rely on fully stochastic models that never make nondeterministic choices. Although recent progress has been made towards verifying Markov Decision Processes with restricted types of schedulers [30, 31, 32],
SMC for MDPs remain a challenge problem (see [33] and [34] for recent surveys).
We will focus on stochastic models that are essentially Discrete Time Markov Chains, except that they are initialized from a (possibly very large) set of states.
In other words, these are Markov Decision Processes (MDPs) where the adversary gets to initialize222
Finite number nondeterministic action choices can be encoded in the choice of the initial state..
Further, we restrict our attention to safety requirements333All the results in the paper generalize to bounded time properties of the form $P_{\geq\theta}(\psi)$ where $\theta$ is a threshold constant and $\psi$ is a path formula. Generalizing to nested probabilistic operators and unbounded time properties will require further research.. That is, we study problems that require maximizing (or minimizing) the probability of hitting certain unsafe states, starting from any initial state.
Further, this class of models and requirements capture many practical problems like online monitoring where the initialization has to consider worst case error bounds in state estimation, for example, from sensing and perception.
We observe that this optimization of a probability measure over a set of initial choices, can coincide with the multi-armed bandit problem for appropriately defined costs and rewards.
By building the connection with the Bandit literature, we not only gain algorithmic ideas, but also new types of theoretical (regret) bounds on the sample efficiency of the algorithms. These bounds rely on the smoothness and the near-optimality dimension of the objective function, and are fundamentally different from the existing performance bounds in the SMC literature.
Hierarchical optimistic optimization (HOO) [5] is a bandit algorithm that builds a tree on a search space $\mathcal{X}$ by using the so called principle of optimism in the face of uncertainty. It is a black-box optimization method that applies an upper confidence bound (UCB) on a tree search method for finding the optimal points over the uncertain domain. The UCB in the tree search approach takes care of the trade-off between exploiting the most promising parts of the domain and exploring the most uncertain parts of the domain. Multi-fidelity hierarchical optimistic optimization(MFHOO) [35] is a multi-fidelity HOO based method that allows noisy and biased observations from the uncertain domain. The performance of MFHOO is measured by how the regret—the gap between the actual maximum and the computed—scales with the number of samples. A key feature of these algorithms is that they can work with black-box functions and the regret guarantees only rely on certain smoothness parameters and the near-optimality dimension of the problem (see Definition 3.1).
The standard theoretical assumptions required by off-the-shelf bandit algorithms in order to get performance guarantees do not precisely fit our verification problem, and that in-depth analysis and modification is required to get these guarantees in our setting; In addition, to apply these algorithms several functions need to be judiciously determined, a priori, and are at the heart of how the algorithms will perform. These choices are non-trivial and multi-faceted, and we develop and provide such functions explicitly in the context of our SMC problem in order to demonstrate successful application.
The key contributions of the paper are as follows.
First, we show how the MFHOO algorithm, can be used for statistical model checking with provable regret bounds. In the process, we define an appropriate notions of fidelity, bias-functions, and also modify the existing near-optimality dimension required for regret bounds of MFHOO to accommodate the non-smoothness of the typical functions we have to optimize for SMC.
Second, we have built a new SMC tool called HooVer using MFHOO [35].
We have created a practically inspired [36, 37] suite of benchmark NiMC models that can be useful for safety analysis of driver assistance features in vehicles for standards such as ISO26262 [38].
Using the benchmarks we have carried out a detailed performance analysis of HooVer and our results suggest that the proposed approach can indeed make use of simulations more effectively than existing SMC approaches.
A fair comparison of HooVer with other discrete-state model checkers like Prism [39], Storm [40], and PlasmaLab [41] is complicated as it relies on a continuous state models. We created discretized models for comparison, and observed that: Compared with exact probabilistic model checking tools like Storm, HooVer is faster, more memory efficient and scales better, and thus it can be used to check models with very large initial state space; Compared with statistical model checking tools like PlasmaLab, HooVer requires much less data to achieve comparable results.
Finally, to our knowledge, this is the first work connecting statistical model checking with the Bandits theory; specifically, the hierarchical tree search using the principle of optimism in the face of uncertainty. Thus we believe that the exposition of these algorithms (Section 3) engender new applications in verification and synthesis algorithms.
2 Model and problem statement
Consider a Euclidean space $\mathcal{X}={\mathbb{R}}^{m}$ and let ${{\mathbb{R}}_{\geq 0}}$ denote the non-negative real numbers.
For any real-valued function $p$ of $\mathcal{X}$ its support is the set ${\rm{supp}\mathit{(p)}}\coloneqq\{x\in\mathcal{X}\ |\ p(x)\neq 0\}.$
A discrete probability distribution over $\mathcal{X}$ is a function $p:\mathcal{X}\rightarrow[0,1]$ such that ${\rm{supp}\mathit{(p)}}$ is countable, and $\sum_{x\in{\rm{supp}\mathit{(p)}}}p(x)=1.$
We use ${\mathbb{P}}(\mathcal{X})$ to denote the set of discrete probability distributions over $\mathcal{X}$.
For a finite set $\mathcal{S}$, $|\mathcal{S}|$ denotes the cardinality of $\mathcal{S}$.
2.1 Nondeterministically initialized Markov chains
Definition 2.1
A Nondeterministically initialized Markov chains (NiMC) $\mathcal{M}$ is defined by a triple $(\mathcal{X},\Theta,P)$, where:
(i) $\mathcal{X}={\mathbb{R}}^{m}$is the state space;
(ii) $\Theta\subseteq\mathcal{X}$is the set of possible initial states; and
(iii) $P:\mathcal{X}\rightarrow{\mathbb{P}}(\mathcal{X})$is the probability transition function.
That is, from state $x\in\mathcal{X}$, the next state is chosen according to the discrete distribution $P(x)$. The probability of transitioning from state $x$ to state $x^{\prime}\in\mathcal{X}$ is $P(x)(x^{\prime})$, which we write more compactly as ${P}_{{x,x^{\prime}}}$.
An execution $\alpha$ of length $k$ for the NiMC $\mathcal{M}$ is a sequence of states $\alpha=\{x_{0},x_{1},\ldots,x_{k}\}$, where $x_{0}\in\Theta$, and for each $i>0$, ${P}_{{x_{i-1},x_{i}}}>0.$
We denote the set of all length $k$ executions of $\mathcal{M}$ starting from $x_{0}$ as ${{\rm{Execs}}_{\mathit{x_{0}}}}(k)$.
The probability of an execution $\alpha$, given $x_{0}$, is
$\prod_{i=1}^{k}{P}_{{x_{i-1},x_{i}}}=:p(\alpha)$.
Given a set $\mathcal{U}\subseteq\mathcal{X}$, we say an execution $\alpha$ hits $\mathcal{U}$ if there exists $x\in\alpha$ such that $x\in\mathcal{U}$. We denote the subset of executions starting from $x_{0}$, of length $k$, that hit $\mathcal{U}$ by ${{\rm{Execs}}_{x_{0}}}(k,\mathcal{U})$.
From a given initial state $x_{0}\in\Theta$ the probability of $\mathcal{M}$ hitting an unsafe state within $k$ steps is given by:
$$\displaystyle{p_{k,\mathcal{U}}{(x_{0})}}=\sum_{\alpha\in{{\rm{Execs}}_{x_{0}}%
}(k,\mathcal{U})}p(\alpha).$$
(1)
Note that if $x_{0}\in\mathcal{U}$ then ${{\rm{Execs}}_{x_{0}}}(k,\mathcal{U})={{\rm{Execs}}_{\mathit{x_{0}}}}(k)$ and ${p_{k,\mathcal{U}}{(x_{0})}}=1$.
We are interested in finding the worst case probability of hitting unsafe states from any initial state of $\mathcal{M}$. This can be regarded as determining, for each $k$, the value
$$\displaystyle\max_{x_{0}\in\Theta}{p_{k,\mathcal{U}}{(x_{0})}}.$$
(2)
Importantly, we would like to solve this optimization problem without relying on detailed information about the probability transition function $P$. Further, our solution should not rely on precisely computing ${p_{k,\mathcal{U}}{(x_{0})}}$ for a given $x_{0}\in\Theta$, but instead only the use of noisy observations.
2.2 Example: Single-lane platoon with two speeds ($\mathsf{SLplatoon2}$)
We present an NiMC of a platoon of $m$ cars on a single lane ($\mathsf{SLplatoon2}$). Variations of this model are used in all our experiments later in Section 5. Each car probabilistically decides to “cruise” or “brake” based on its current gap with the predecessor.
These types of models are used for risk analysis of Automatic Emergency Braking (AEB) systems [36, 37].
The probabilistic parameters of the model are derived from data collected from overhead traffic enforcement cameras on roads. The uncertainty in the initial positions (and gaps) arise from perception inaccuracies, which are modeled as worst-case ranges.
Let $s_{i}$ be the position of $i$th car in the sequence. Initially, $s_{i}$ takes a value in an interval on the $x$-axis such that $s_{1}>s_{2}>\ldots>s_{m}$.
The pseudocode in Figure 1 specifies the probabilistic transition rule that updates the position of all the cars synchronously. Car $1$ always moves at a constant breaking speed of $\mathsf{vc}$.
The variable $\mathit{gap}_{i}$ is the distance of $i$ to the predecessor $i-1$, for each $i=2,...,m$.
If $\mathit{gap}_{i}$ is less than the constant threshold $\mathsf{near}$, then $i$ continues to cruise with probability $\mathsf{pnear}$ and it brakes with probability $1-\mathsf{pnear}.$
Similarly, if $\mathit{gap}_{i}$ greater than $\mathsf{near}$, then $i$ continues to cruise with probability $\mathsf{pfar}$ and brakes with probability $1-\mathsf{pfar}.$
It is straightforward to connect the above description to a formal definition of a NiMC. The state space $\mathcal{X}=\mathbb{R}^{m}$. The set of initial states $\Theta$ is a hyperrectangle in $\mathcal{X}$ (such that $s_{1}>s_{2}\ldots>s_{m}$). For any state, $x\in\mathcal{X}$ the probability transition function is given by the equations in lines 1, 1, 1, 1 and 1.
We define the set of unsafe states $\mathcal{U}=\{(s_{1},s_{2},...,s_{m})\in\mathcal{X}\ |\ \exists i\in\{2,...,m%
\},\ \mbox{ such that }\mathit{gap}_{i}\leq\delta\}\subseteq\mathcal{X}$ for some constant collision threshold $\delta$.
Given that cars start their motion at any initial state from $\Theta$, the goal is to find the maximum probability of hitting the unsafe set $\mathcal{U}$.
For $m=2$,
$\mathsf{vb}=1$,
$\mathsf{vc}=2$,
$\mathsf{pfar}=0.85$
$\mathsf{pnear}=0.15$,
$\mathsf{near}=3$,
$\delta=1$,
and initial $s_{1}\in(20,23),s_{2}\in(0,2)$,
Figure 1 shows estimates of probabilities of hitting the unsafe set from different initial separations between cars.
As our intuition suggests, for large enough time horizons the probability of hitting the unsafe set approaches $1$ from all initial states, but, for smaller time horizons the maximum probability of unsafety arises when the initial gap is smaller.
3 Background: Hierarchical Optimistic Optimization
Multi-Fidelity Hierarchical Optimistic Optimization (MFHOO) [35] is an black-box optimization algorithm from the multi-armed bandits literature [4, 5, 6]. The setup is the following: suppose we want to maximize the function $f:\mathcal{X}\rightarrow\mathbb{R}$, which is assumed to have a unique global maximum. Let $f^{*}=\underset{x\in\mathcal{X}}{\sup}$ $f(x)$. MFHOO allows the choice of evaluating $f$ at different fidelities with different costs.
This flexibility matters for SMC because it will be beneficial to evaluate the probability of unsafety ${p_{k,x_{0}}{(\mathcal{U})}}$ for certain initial states more precisely, for example, with longer number of simulations, while for other initial states a less precise evaluation may be adequate.
Thus, MFHOO has access to a biased function $f_{z}(x)$ that depends on fidelity parameter $z\in[0,1]$.
Setting $z=0$ gives the lowest fidelity (and lowest cost) and $z=1$ corresponds to full fidelity (and highest cost).
At full fidelity, $f_{1}(x)=f(x)$, and the evaluation is unbiased. More generally, $|f_{z}(x)-f(x)|\leq\zeta(z)$ and evaluating $f_{z}(x)$ costs $\lambda(z)$, where the functions $\zeta,\lambda:[0,1]\rightarrow\mathbb{R}_{>0}$ are respectively, non-increasing and non-decreasing, and called the bias and the cost functions [35].
A bandit algorithm chooses a sequence of sample points (arms) $x_{1},x_{2},\ldots\in\mathcal{X}$, evaluates them at fidelities $z_{1},z_{2},\ldots$, and receives the corresponding sequence of noisy observations (rewards) $y_{1},y_{2},\ldots$.
We assume that each $y_{j}$ is drawn from a unknown distribution $M_{z_{j},x_{j}}$ satisfying $f_{z_{j}}(x_{j})=\int udM_{z_{j},x_{j}}(u)$. Further the distribution has a sub-Gaussian component, with variance $\sigma^{2}$, which captures uncertainty in the observations.
The algorithm actively chooses $x_{j+1}$ based on past choices $x_{1},\ldots,x_{j}$ and observations $y_{1},\ldots,y_{j}$. When the budget $\Lambda$ is exhausted, the algorithm decides the optimal point $\bar{x}_{n(\Lambda)}\in\mathcal{X}$ and the optimal value $f(\bar{x}_{n(\Lambda)})$ with the aim of minimizing regret, which is defined as
$$R(\Lambda)=f^{*}-f(\bar{x}_{n(\Lambda)}).$$
The MFHOO algorithm (Algorithm 1) for selecting $x_{j+1}$ estimates $f^{*}$ by building a binary tree in which each height in the tree represents a partition of the state space $\mathcal{X}$. The algorithm maintains estimates of an upper-bound on $f$ for each partition subset, and uses the principle of optimism for choosing the next sample $x_{j+1}$. That is, it chooses the samples in those partitions where the estimated upper-bounds are the highest.
Each node in the constructed tree is labeled by a pair of integers $(h,i)$, where $h$ is the height of the node in the tree, and $i$ satisfying $0\leq i\leq 2^{h}$ is its position within height level $h$. The root is labeled $(0,1)$, and each
node $(h,i)$ can have two children $(h+1,2i-1)$ and $(h+1,2i)$.
Node $(h,i)$ is associated with subset a of $\mathcal{X}$, denoted by ${\mathcal{P}_{h,i}}$, where ${\mathcal{P}_{h,i}}={\mathcal{P}_{h+1,2i-1}}\cup{\mathcal{P}_{h+1,2i}}$, and for each $h$ these disjoint subsets satisfy $\cup_{i=1}^{2^{h}}{\mathcal{P}_{h,i}}=\mathcal{X}$. Therefore, larger values of $h$ represent finer partitions of $\mathcal{X}$.
For each node $(h,i)$ in the tree, the algorithm maintains the following information:
(i) ${\mathit{count}_{h,i}}$: the number of times the node is visited;
(ii) $\hat{f}_{h,i}$: the empirical mean of observations over points visited in ${\mathcal{P}_{h,i}}$;
(iii) $U_{h,i}$: an initial estimate of the maximum of $f$ over ${\mathcal{P}_{h,i}}$; and
(iv) $B_{h,i}$: a tighter and optimistic upper bound on the maximum of $f$ over ${\mathcal{P}_{h,i}}.$
The algorithm proceeds as follows. The $\mathit{tree}$ starts out with a single node, the root $(0,1)$, initializes the $B$-values of its two children $B_{1,1}$ and $B_{1,2}$ to $+\infty$, and initializes the cost $C$ to $0$. At a high-level, in each iteration of the while loop (line 3), the algorithm adds a new node $(\mathit{hnew},\mathit{inew})$ in the $\mathit{tree}$ and updates all of the above quantities for several nodes in $\mathit{tree}$. In more detail, first a $\mathit{path}$ from the root to a leaf is found by traversing the child with the higher $B$-value (with ties broken arbitrarily). Let the child with the higher $B$-value of the traversed leaf be $(\mathit{hnew},\mathit{inew})$ (line 4). An arbitrary point $x$ in the partition of this node ${\mathcal{P}_{\mathit{hnew},\mathit{inew}}}$ is chosen (line 5). Then, this point is evaluated at fidelity $z_{hnew}=\zeta^{-1}(\nu\rho^{hnew})$ and a reward $y$ is received (line 6). Next, $\mathit{tree}$ is extended by inserting $(\mathit{hnew},\mathit{inew})$ in the $\mathit{tree}$ (line 8) and for all the nodes $(h,i)$ in $\mathit{path}$ including $(\mathit{hnew},\mathit{inew})$, the ${\mathit{count}_{h,i}}$ and the empirical mean $\hat{f}_{h,i}$ are updated (line 9). Finally, in line 13, for all nodes $(h,i)$ in $\mathit{tree}$, $U_{h,i}$ and $B_{h,i}$ are updated using the smoothness parameters $\nu_{1}>0$ and $\rho\in(0,1)$ which will discussed later in Section 3.1 and the parameter $\sigma$. Once the sampling budget $\Lambda$ is exhausted, a leaf with maximum $B$-value is returned by the Algorithm 1 [35].
3.1 Regret bounds for MFHOO
In this section, we summarize the assumptions and results from [35]. In order to analyze the regret bounds for the MFHOO algorithm, the following assumption on the smoothness of $f(x)$ [35] is made.
Assumption 1
There exist $\nu>0$ and $\rho\in(0,1)$ such that for all $\mathcal{P}_{h,i}$ satisfying $f^{*}-\underset{x\in\mathcal{P}_{h,i}}{\sup}\ f(x)\leq c\nu\rho^{h}$ (for a constant $c\geq 0$), we have that $f(x)\geq f^{*}-\max\{2c,c+1\}\nu\rho^{h}$ , for all $x\in\mathcal{P}_{h,i}$.
This assumption connects the function $f$ to the partitioning rule of the binary tree.
We now define the concept of near-optimality dimension which plays an important role in the analysis of black-box optimization algorithms [5, 35, 42, 43]. It measures the dimension of sets that are close to optimal.
The regret bound for MFHOO uses this notion.
First, given a partitioning scheme ${\mathcal{P}_{h,i}}$ over $\mathcal{X}$, we define $\mathcal{N}_{h}(\epsilon)$ as the number of $\epsilon$-near optimal partitions, that is, the number of partitions $\mathcal{P}_{h,i}$ such that satisfies $\sup_{{x\in\mathcal{P}_{h,i}}}f(x)\geq f(x^{*})-\epsilon$.
Definition 3.1
The near-optimality dimension of $f$ with respect to parameters $(\nu,\rho)$ is:
$d(\nu,\rho)=\inf\{d^{\prime}\in\mathbb{R}_{>0}:\exists B>0,\ s.t.\ \forall h%
\geq 0,\ \mathcal{N}_{h}(2\nu\rho^{h})\leq B\rho^{-d^{\prime}h}\}.$
With Assumption 1, the regret bound for MFHOO is proved in [35].
Theorem 1
If Algorithm 1 runs with parameters $\nu$ and $\rho$ that satisfy Assumption 1 and a cost budget of $\Lambda$, then the simple regret is bounded
$$\displaystyle R(\Lambda)=O\bigg{(}(\frac{B\log n(\Lambda)}{n(\Lambda)})^{\frac%
{1}{d(\nu,\rho)+2}}\bigg{)},$$
where $n(\Lambda)=\max\{n:\sum_{h=1}^{n}\lambda(z_{h})\leq\Lambda\}$.
Here, $z_{h}=\zeta^{-1}(\nu\rho^{h})$.
According to the Theorem 1, regret is minimized if the near-optimality dimension $d(\nu,\rho)$ is minimized. If the smoothness parameters that minimize the near-optimlaity dimension $d(\cdot,\cdot)$ are known, then MFHOO achieves the minimum regret of Theorem 1.
4 Statistical Model Checking with Optimistic Optimization
We aim to solve the statistical model checking problem of maximizing ${p_{k,\mathcal{U}}{(x)}}$ of Equation (2) for a given NiMC $\mathcal{M}$ and a time horizon $k$, using MFHOO.
In order to apply the MFHOO algorithm, one has to make several critical choices regarding the objective function, the budget, the cost, the parameters for fidelity and smoothness, and the multi-fidelity bias function. In this section we discuss the rationale behind our choices.
4.1 Objective function, budget, cost, and fidelity
Fidelity parameter $z$.
Consider a NiMC $\mathcal{M}=(\mathcal{X},\Theta,P)$ with the unsafe set $\mathcal{U}\subseteq\mathcal{X}$. We have to maximize ${p_{k_{\mathit{max}},\mathcal{U}}{(x)}}$ over all initial states $x\in\Theta$, and for a long time horizon $k_{\mathit{max}}$.
Given $x\in\Theta$, the fidelity of evaluating ${p_{k_{\mathit{max}},\mathcal{U}}{(x)}}$ will depend on the actual length of the simulations drawn for creating the observation $y$ for the state $x$.
Suppose we fix $k_{\mathit{min}}$ as the shortest simulation to be used. We define the fidelity of an observation (or evaluation) with simulations of length $k\in[k_{\mathit{min}},k_{\mathit{max}}]$ as $z=(k-k_{min})/(k_{min}-k_{max})$.
Objective function $f$ and observations.
A natural choice for the objective function would be to define $f_{z=1}(x):={p_{k_{max},\mathcal{U}}{(x)}}$, for any initial state $x\in\Theta$.
Computing this probability, however, is infeasible when the probability transition function $P_{\mathcal{M}}$ is unknown. Even if $P_{\mathcal{M}}$ is known, calculating ${p_{k,\mathcal{U}}{(x)}}$ involves summing over many executions (as in (1)).
Instead, we take advantage of the fact that MFHOO can work with noisy observations. For any initial $x\in\Theta$, and execution $\alpha\in{{\rm{Execs}}_{x}}(k)$ we define the observation:
$$\displaystyle Y=1\ \mbox{if}\ \alpha\in{{\rm{Execs}}_{x}}(k,\mathcal{U}),\ %
\mathit{and}\ =0\ \mathit{otherwise}.$$
(3)
Recall that for a fixed initial states $x$, $\mathcal{M}$ is a Markov chain and defines a probability distribution over the set of executions ${{\rm{Execs}}_{x}}(k)$ as given by Equation (1).
Thus, given an initial state $x$,
$Y=1$ with probability ${p_{k,\mathcal{U}}{(x)}}$, and $Y=0$ with probability $1-{p_{k,\mathcal{U}}{(x)}}$.
That is, $Y$ is a Bernoulli random variable with mean ${p_{k,\mathcal{U}}{(x)}}$ at fidelity $z$.
In MFHOO, once an initial state $x\in{\mathcal{P}_{\mathit{hnew},\mathit{inew}}}$ is chosen (line 5), we simulate $\mathcal{M}$ upto $k$ steps several times starting from $x$ and calculate the empirical mean of $Y$, which serves as the noisy observation $y$ at fidelity $z$.
Cost $\lambda(z)$ and budget $\Lambda$.
In our setup the cost function $\lambda(z)$ for any $z$ is the computational time required to simulate an execution $\alpha\in{{\rm{Execs}}_{x}}(k)$ for any $x\in\Theta$, where $k$ is the execution length corresponding to the fidelity $z$. The budget $\Lambda$ is a computational cost budget.
The next proposition states that with these choices, our goal to maximize ${p_{k_{\mathit{max}},\mathcal{U}}{(x)}}$ over $\Theta$ can be achieved using MFHOO.
Proposition 1
Given smoothness parameters $\rho$ and $\nu$ satisfying Assumption 1 and budget $\Lambda$,
suppose Algorithm 1 returns $\bar{x}_{\Lambda}\in\Theta$. Then,
$R(\Lambda)={p_{k_{\mathit{max}},\mathcal{U}}{(x^{*})}}-{p_{k_{\mathit{max}},%
\mathcal{U}}{(\bar{x}_{\Lambda})}}$, where $R(\Lambda)$ bound is given by Theorem 1.
Thus, the point $\bar{x}_{\Lambda}\in\Theta$ returned by Algorithm 1 is such that the gap between its probability ${p_{k,\mathcal{U}}{(\bar{x})}}$ of hitting $\mathcal{U}$ and the true maximum probability ${p_{k,\mathcal{U}}{(x^{*})}}$ is bounded by Theorem 1 in terms of the available budget $\Lambda$.
4.2 Multi-fidelity bias function
Recall that the bias function $\zeta(z)$ gives an upper bound $|f(x)-f_{z}(x)|\leq\zeta(z)$, over all $x\in\Theta$ and for any fidelity $z\in[0,1]$.
We will derive a bias function satisfying $|{p_{k_{\mathit{max}},\mathcal{U}}{(x_{0})}}-{p_{k,\mathcal{U}}{(x_{0})}}|\leq%
\zeta(z)$. Of course, a guarantee like this is only possible for known models. Therefore, for this section we will assume that the NiMC $\mathcal{M}$ is known. We also assume $\mathcal{X}$ is finite and $\mathcal{U}$ is the absorbing set for $\mathcal{M}$ (i.e., all other states are transient; $x\in\mathcal{U}$ if and only if $P_{xx}=1$).
Fixing an initial state $x$, $\mathcal{M}$ is a reducible Markov chain.
Let $q$ be the number of transient states and $u=|\mathcal{U}|$.
Then, the probability transition function can be represented by the $(q+u)\times(q+u)$ matrix $P$:
$$\displaystyle P=\left(\begin{array}[]{c|c}Q&R\\
\hline{\bf 0}&I\end{array}\right)\ \mathit{and}\ P^{t}=\left(\begin{array}[]{c%
|c}Q^{t}&\sum_{k=0}^{t-1}Q^{k}R\\
\hline{\bf 0}&I\end{array}\right).$$
(4)
Here $Q$ is an $q\times q$ matrix, $I$ is an $u\times u$ identity matrix, ${\bf 0\/}$ is an $u\times q$ zero matrix, and $R$ is a nonzero $q\times u$ matrix.
The following standard result gives the absorption probabilities in terms of $Q$.
Proposition 2 ( [44])
Suppose $B=NR$ is a $q\times u$ matrix, where $N=\sum_{j=0}^{\infty}Q^{j}=(I-Q)^{-1}$ and $R$ is as defined in (4).
Then $B_{ij}$ is the probability, that starting from state $s_{i}$, the chain $\mathcal{M}$ is absorbed in $s_{j}\in\mathcal{U}$.
We now state and prove the theorem that defines a multi-fidelity bias function.
Theorem 2
For any $x\in\Theta$ and any time horizon $k<k_{max}$,
$$|{p_{k_{\mathit{max}},\mathcal{U}}{(x)}}-{p_{k,\mathcal{U}}{(x)}}|\leq\frac{%
\kappa_{\infty}(E)(k_{\mathit{max}}-k)}{(k_{\mathit{max}}-(k_{\mathit{max}}-1)%
\lambda_{\mathit{max}})(k-(k-1)\lambda_{\mathit{max}})},$$
where $\lambda_{max}$ is the maximum eigenvalue,
$E$ is the matrix of eigenvectors of the matrix $Q$, and $\kappa_{\infty}(E)$ is the ($\infty$-norm) condition number of $E$.
This theorem can be proved using the spectral decomposition of matrix $Q$ and $\infty$-norm bounds (for detailed analysis please refer to Appendix B).
Remark 1
Using the fidelity parameter $z=(k_{max}-k)/(k_{max}-k_{min})$ in the upper bound given by Theorem 2, the bias function can be rewritten as:
$$\displaystyle\zeta(z)$$
$$\displaystyle=\frac{g_{1}(1-z)}{g_{2}z+g_{3}},\mathit{where}\ g_{1}=\kappa_{%
\infty}(E)(k_{max}-k_{min})$$
(5)
$$\displaystyle g_{2}$$
$$\displaystyle=((1-\lambda_{max})k_{max}-\lambda_{max})(k_{max}-k_{min})(1-%
\lambda_{max})$$
$$\displaystyle g_{3}$$
$$\displaystyle=((1-\lambda_{max})k_{max}-\lambda_{max})((1-\lambda_{max})k_{min%
}-\lambda_{max}).$$
This bias function upper bounds the gap between the probability of hitting the unsafe set for a point in
the initial set for different time horizons. Thus, it can be used in the analysis of the
theoretical regret bound of the Algorithm 1 given in
Theorem 1. If the transition matrix $P$ is known, then the bias function can be used in updating the $U$-values of nodes in the Algorithm 1. Note, this bias function depends on parameters $\kappa_{\infty}(E)$ and $\lambda_{max}$. In the case where the full transition model of the NiMC is unknown,
but there is access to $\kappa_{\infty}(E)$ and $\lambda_{max}$, this function can be utilized in the algorithm. More generally, in problems with unknown $P$ and no access to the parameters
$\kappa_{\infty}(E)$ and $\lambda_{max}$, we consider a linear parameterized bias function $\zeta(z)=b(1-z)$ with an unknown parameter $b$, and adaptively estimate the parameter $b$ [35].
4.3 Non-smoothness and non-uniqueness of hitting probabilities
In this section, we start with the observation that in general the function ${p_{k,\mathcal{U}}{(x)}}$ over $x\in\Theta$ is not a continuous function and has infinitely many maxima. Thus this function does not satisfy the Assumption 1 and the assumption of finite number of maxima, that are required for the regret bounds of Theorem 1. However, a modified definition of near-optimality dimension we show that the bounds given by Theorem 1 will hold. Finally, we will compare the theoretical regret with the actual regret achieved by MFHOO.
Consider a discrete time linear system with state space $\mathcal{X}=\mathbb{R}^{m}$, set of initial states $\Theta\subseteq{\mathcal{X}}$ and set of unsafe states given by $\mathcal{U}:=\{x\in\mathcal{X}\ |\ c^{T}x\leq a\}$ for some vector $c\in{\mathbb{R}}^{m}$ and $a\in{\mathbb{R}}$. The system evolves according to:
$$x_{t+1}=Ax_{t}+BW_{t},$$
(6)
where $x_{t}\in\mathcal{X}$ is the state vector at time $t$. $W_{t}\in{\mathbb{R}}^{r}$ is a discrete random vector with probability mass distribution $p_{W}$ at sequence $t$, and
$A$ and $B$ are $m\times m$ and $m\times r$ matrices, respectively.
Consider two initial states $x_{0},x_{0}^{\prime}\in\Theta,$ such that $x_{0}^{\prime}=x_{0}+\epsilon$, for some small $\epsilon$. Writing out the probabilities of hitting unsafe states of this system explicitly for the two initial states and finding the gap between these probabilities we can observe that, for small enough $\epsilon$, the gap between the probability is zero444
The full analysis is given in Appendix D.. By increasing the $\epsilon$, the gap can take a nonzero value. This means that the probability of hitting the unsafe set of the system (13) is not a continuous function in terms of the initial states and has infinitely many maxima. This is because random variable $W_{t}$ is a discrete-type random variable. $\mathsf{SLplatoon2}$ is a concrete example of this kind.
Assumption 1 asserts that there exist smoothness parameters such that for all $h\geq 0$ in the tree, and all the partitions that are near optimal, the gap between $f^{*}$ and the value of the function over those partitions should be bounded. From the above discussion we see that this may not hold for all depths $h\geq 0$. In addition, the regret bounds in Theorem 1 are for function with finite number of maxima. If there are infinitely many maxima, then for given smoothness parameters the number of $2\nu\rho^{h}$-near-optimal partitions $\mathcal{N}_{h}(2\nu\rho^{h})$ would not be bounded for all $h\geq 0$. However, we observe that any instance of MFHOO runs on a finite budget and the final constructed tree has a maximum height $h_{max}$. For this $h_{\mathit{max}}$, there exist smoothness parameters $\nu$ and $\rho$ that satisfy the Assumption 1, and we can modify the near-optimality dimension in the Definition 3.1 as:
Definition 4.1
$h_{\mathit{max}}$-bounded near-optimality dimension of $f$ with respect to $(\nu,\rho)$ is:
$d_{m}(\nu,\rho)=\inf\{d^{\prime}\in\mathbb{R}_{>0}:\exists B>0,\ \forall h\in[%
0,h_{max}],\mathcal{N}_{h}(2\nu\rho^{h})\leq B\rho^{-d^{\prime}h}\}$.
With this modified definition, there exists a $B$ satisfying the above condition and the corresponding $d_{m}$ can be used to recover the regret bound of the Theorem 1.
Regret in theory and in practice.
Given a budget $\Lambda$, let $\bar{x}_{\Lambda}\in\Theta$ be the point that the Algorithm 1 returns after the budget is exhausted. Then the regret $R(\Lambda)$ would be ${p_{k_{max},\mathcal{U}}{(x^{*})}}-{p_{k_{max},\mathcal{U}}{(\bar{x}_{\Lambda}%
)}}$. We compare the theoretical regret bound and the actual regret for the $\mathsf{SLplatoon2}$ model. Consider an instance of $\mathsf{SLplatoon2}$ with $m=2$, initial states $s_{1}\in(50,70)$ and $s_{2}\in(0,20)$. Intuitively, the partition corresponding to $s_{1}\in(50,51)$ and $s_{2}\in(19,20)$ would maximize the probability of hitting $\mathcal{U}$. In Figure 2 (top-left), the mean of actual regret obtained by running MFHOO for various smoothness parameters is presented.
As the budget increases, as expected, the actual regret decreases monotonically and approximately approaches to $0$.
For budget $\Lambda=30$, the partitioning stops at $h_{max}=15$. We can derive the number of $2\nu\rho^{h}$-near-optimal partitions for any $\nu$ and $\rho$. Setting $\nu=0.08$ $\rho=0.77$ (actual values used to generate regret results), we see that for $h\in[9,15]$, the maximum number of $2\nu\rho^{h}$-near-optimal partitions $\mathcal{N}_{h}(2\nu\rho^{h})=\lceil 2^{h}/400\rceil$.
This is because the total number of partitions at depth $h$ is equal to $2^{h}$, and for $h\in[9,15]$, the $2\nu\rho^{h}$-near-optimal partitions belong to the area corresponding to $s_{1}\in(50,51)$ and $s_{2}\in(19,20)$, whose area is $1/400$. According to Definition 4.1, for different values of $B$, different $h_{\mathit{max}}$-bounded near-optimality dimensions $d_{m}$ can be obtained. We are looking for the pair of $(d_{m},B)$ values that minimize the theoretical regret bound of Theorem 1. Figure 2 (top-right) represents the number of $2\nu\rho^{h}$-near-optimal partitions for $h\in[9,15]$ that are upper bounded by $B\rho^{-d_{m}h}$ for different values of $d_{m}$ and their corresponding $B$. Figure 2 (bottom) also, represent the theoretical upper bound vs. $d_{m}$ and their corresponding $B$ for different values of smoothness parameters used to generate the actual regret. As it is seen, the best theoretical regret bounds that can be achieved is approximately $0.08$.
5 HooVer tool and experimental evaluation
We have implemented a prototype tool called
HooVer which uses MFHOO for solving the SMC problem of (2). We compare the performance of HooVer with that of Prism [39], Storm [40], and PlasmaLab [41] on several benchmarks we have created.
5.1 Benchmark models
We have created several NiMC models for evaluation of probabilistic and statistical model checking tools. The benchmarks are variants of $\mathsf{SLplatoon2}$ (Section 2.2). The complete models are described in Appendix C. The executable models are available from [45].
$\mathsf{SLplatoon3}$ models a sequence of $m$ cars on a single lane. At each step, each car can choose to move with one of three speeds: $\mathsf{vbrake}$, $\mathsf{vcruise}$, and $\mathsf{vspeedup}$.
The first vehicle moves at a constant speed. For all the others, the speed is chosen probabilistically, according to distributions that depend on their distance to the preceding vehicle. For example, if the longitudinal distance $s_{i-1}-s_{i}$ is less than a threshold $\mathsf{th\_close}$ then the speed for vehicle $i$ is chosen according to a probability distribution $\mathsf{pclose}$, and so on.
The state variables, the initial states, and the unsafe set are defined in the same way as in $\mathsf{SLplatoon2}$.
$\mathsf{MLplatoon}$ models $m$ cars on $\ell$ lanes. At every step, each car probabilistically chooses to either move forward like a vehicle in $\mathsf{SLplatoon3}$, or it changes to its left or right lane. These actions are chosen probabilistically, according to probability distributions that depend on their distances to the vehicles on its current lane, as well as left and right lanes.
In addition to the longitudinal position $s_{i}$, each vehicle $i$ has a second state variable $y_{i}\in\{1,\ldots,\ell\}$ which keeps track of its current lane.
The initial value of each $s_{i}$ is chosen as in the case of
$\mathsf{SLplatoon2}$, and $y_{i}$ is set to 1.
The unsafe set is defined based on the distance to the preceding car on the same lane.
5.2 HooVer implementation and metrics
Our implementation of the HooVer tool uses the MFHOO implementation presented in [35] to solve the model checking problem of Equation (2). It works in two stages: First, it uses MFHOO to find the best partition ${\mathcal{P}_{h,i}}$ and a putative “best” (most unsafe) initial point $x\in{\mathcal{P}_{h,i}}$ with the maximum estimate for the probability of hitting the unsafe set $\mathcal{U}$. In the second stage, HooVer uses additional simulations to do a Monte Carlo estimation of the probability ${p_{k,\mathcal{U}}{(x)}}$. In the experiments reported below, a constant number of $26$K simulations are used in all experiments in the second stage.
To achieve the theoretically optimal performance, MFHOO requires the smoothness parameters $\rho$ and $\nu$ which are unknown for our benchmarks. To circumvent this HooVer chooses several parameter configurations ($3$ sets in our experiments), runs an instance of MFHOO for each, and returns the result with the highest hitting probability. For each instance of MFHOO, we set a time budget which is the maximum time allowed to be consumed by the simulator.
Metrics.
We report the regret of HooVer. In order to calculate the regret, first we have to calculate the actual maximum hitting probability for each benchmark. This is computed using Prism [39] which uses numerical and symbolic analysis. The regret is the difference between the exact probability and the estimated probability. Then, we report the running time and memory usage. The memory usage is measured by calculating the total size of the Python object which contains the constructed tree and all other data of MFHOO. We also report the number of queries for each method, which is total number of simulations used in both stage 1 and 2.
All the experiments are conducted on a Linux workstation with two Xeon Silver 4110 CPUs and 32 GB RAM.
Table 1 shows the running time, the memory usage, the number of queries, and the resulting actual regret for HooVer using MFHOO as well as HooVer(1) using HOO.
On every benchmark, HooVer gives low regrets.
With the same simulation budget, HooVer devotes longer simulations in the interesting parts $\Theta$, as a consequence, it is usually faster than HooVer(1) as shown in Figure 3.
5.3 Comparison with other model checkers
We compare the performance of HooVer with other model checkers. Prism [39] and Storm [40] are leading probabilistic model checkers for Markov chains and MDPs and compute exact probability of reaching the unsafe states. As Storm has the same functionality as Prism and we found it to be much more efficient than Prism in all our experiments, we only compare HooVer with Storm.
PlasmaLab [41] is one of the few statistical model checkers that can handle MDPs. For probability estimation problems, PlasmaLab uses smart sampling algorithm [46] to efficiently assign the simulation budgets to each scheduler and then estimates the probability for the putative “best” scheduler.
Discretizing and scaling the benchmarks.
The theory for MFHOO is based on a continuous state spaces, however, most model checking tools, including the ones mentioned above, are designed for discrete state space models and they do not support floats as state variables. Therefore, a direct comparison of the approaches on the same model is infeasible, and we created equivalent, discretized (quantized) versions of the benchmarks.
In HooVer, the algorithm keeps partitioning $\Theta$ hierarchically and stops at a depth $h$, which can be considered as searching over all the $2^{h}$ partitions at depth $h$. In order to make a fair comparison, we make sure that the discrete version of the benchmark has $2^{h}$ initial states, i.e. $|\overline{\Theta}|=2^{h}$ where $\overline{\Theta}$ is the discretized initial state space.
Before stating the discretizing process, we give two key observations of the benchmarks. First, considering the nature of our benchmarks, it is obvious that if we scale the velocities, distance thresholds and initial distances by a constant factor, the probability of hitting unsafe set doesn’t change. Second, taking $\mathsf{SLplatoon2}$ as an example, we set all the constants (velocities and distance thresholds) in the model as integers, which leads to the function ${p_{k_{\mathit{max}},\mathcal{U}}{(x)}}$ shown in Figure 1. It’s clear that for any state $x\in\mathcal{X}$, there exists an integer state $\overline{x}$ such that ${p_{k_{\mathit{max}},\mathcal{U}}{(x)}}={p_{k_{\mathit{max}},\mathcal{U}}{(%
\overline{x})}}$. If we make sure that for each integer interval in the original continuous space, the discretized space has an value in that interval, then the maximum probability doesn’t change. With these observations, we discretize the benchmark as follows. First, we sample $2^{h}$ points uniformly in the continuous initial state space. Due to the second observation mentioned above, if $2^{h}$ is larger than the number of integer points in the original space, the maximum probability doesn’t change. Then, we find an integer scaling factor $c$ such that by multiplying $c$ all the $2^{h}$ points become integer points. Other constants in the model are also multiplied by $c$. According to the first observation, scaling with a constant doesn’t change the probability. Thus, we now have a model where all state variables are integers and it has the same maximum probability as the original model. Then we evaluate Storm and PlasmaLab on this new model.
All of the model checkers mentioned above support the Prism [39] language. Thus, we implement each benchmark in Prism language, and then we check the equivalence between the Prism implementation and the Python implementation by calculating and comparing the probability ${p_{k_{\mathit{max}},\mathcal{U}}{(x)}}$ at all integer points $x$.
For Storm, we report the running time and the memory usage. These metrics are directly measured by the software itself. For PlasmaLab, we report the running time. We do not report the memory usage of PlasmaLab because it doesn’t provide an interface for that and it is hard to track the actual memory used by the algorithm inside the JAVA virtual machine. We also report the regret for PlasmaLab. We use the term “regret” here just for simplicity, which also refers to the difference between the estimated probability and the exact probability.
The results of HooVer, PlasmaLab and Storm are summarized in Table 1. We show in Figure 3 how the performance of different methods changes as the $|\overline{\Uptheta}|$ increases. As the size of the initial state space increases, the memory usage of Storm grows quickly, which limits its application on large models. In contrast, HooVer and PlasmaLab scale well. The running time of PlasmaLab is determined by the parameters of the smart sampling [46] algorithm. We use the same parameters regardless of $|\overline{\Uptheta}|$, and thus the running time of PlasmaLab is almost constant.
As shown in Table 1, Storm fails to check the $\mathsf{MLplatoon}$ model due to the memory limitation. Compared with PlasmaLab, HooVer requires more running time.
However, we note that PlasmaLab is a considerably more mature tool than HooVer. As shown in Table 1, compared to PlasmaLab, HooVer requires much fewer queries to reach comparable regrets, which attests to the sample efficiency of our proposed method.
6 Conclusions
In this paper, we formulated the statistical model checking problem for a special type of MDP models as a multi-armed bandit problem and showed how a hierarchical optimistic optimization algorithm from [35] can be used to solve it. In the process, we modified the existing definition of near-optimality dimension to accommodate the non-smoothness of the typical functions we have to optimize and recovered the regret bounds of the algorithm. We created several benchmarks, developed a SMC tool HooVer, and experimentally established the sample efficiency and scalability of the method.
In order to enlarge the area of application of this method, it is necessary to study more general temporal properties for NiMC beyond bounded time safety. In this paper, we focused on a very special type of MDP models with nondeterminism only in the initial set. The results suggest that it will be interesting to study black-box optimization algorithms for model checking more general MDP models.
References
[1]
William R Thompson.
On the theory of apportionment.
American Journal of Mathematics, 57(2):450–456, 1935.
[2]
William R Thompson.
On the likelihood that one unknown probability exceeds another in
view of the evidence of two samples.
Biometrika, 25(3/4):285–294, 1933.
[3]
Herbert Robbins.
Some aspects of the sequential design of experiments.
Bulletin of the American Mathematical Society, 58(5):527–535,
1952.
[4]
Rémi Munos.
From Bandits to Monte-Carlo Tree Search: The Optimistic Principle
Applied to Optimization and Planning.
Technical report, 2014.
130 pages.
[5]
Sébastien Bubeck, Rémi Munos, Gilles Stoltz, and Csaba Szepesvári.
X-armed bandits.
J. Mach. Learn. Res., 12:1655–1695, July 2011.
[6]
Sébastien Bubeck and Nicolò Cesa-Bianchi.
Regret analysis of stochastic and nonstochastic multi-armed bandit
problems.
Foundations and Trends in Machine Learning, 5(1):1–122, 2012.
[7]
Håkan L. S. Younes and Reid G. Simmons.
Probabilistic verification of discrete event systems using acceptance
sampling.
In International Conference on Computer Aided Verification
(CAV2002), pages 223–235. Springer, 2002.
[8]
Daniel Gburek and Christel Baier.
Bisimulations, logics, and trace distributions for stochastic systems
with rewards.
In Proceedings of the 21st International Conference on Hybrid
Systems: Computation and Control (part of CPS Week), HSCC 2018, Porto,
Portugal, April 11-13, 2018, pages 31–40, 2018.
[9]
Doron Bustan, Sasha Rubin, and Moshe Y. Vardi.
Verifying omega-regular properties of markov chains.
In Computer Aided Verification, 16th International Conference,
CAV 2004, Boston, MA, USA, July 13-17, 2004, Proceedings, pages 189–201,
2004.
[10]
Holger Hermanns, Björn Wachter, and Lijun Zhang.
Probabilistic CEGAR.
In Computer Aided Verification, 20th International Conference,
CAV 2008, Princeton, NJ, USA, July 7-14, 2008, Proceedings, pages
162–175, 2008.
[11]
David N. Jansen, Joost-Pieter Katoen, Marcel Oldenkamp, Mariëlle
Stoelinga, and Ivan S. Zapreev.
How fast and fat is your probabilistic model checker? an experimental
performance comparison.
In Hardware and Software: Verification and Testing, Third
International Haifa Verification Conference, HVC 2007, Haifa, Israel,
October 23-25, 2007, Proceedings, pages 69–85, 2007.
[12]
Jan J. M. M. Rutten, Marta Z. Kwiatkowska, Gethin Norman, David Parker, and
Prakash Panangaden.
Mathematical techniques for analyzing concurrent and
probabilistic systems, volume 23 of CRM monograph series.
American Mathematical Society, 2004.
[13]
Koushik Sen, Mahesh Viswanathan, and Gul Agha.
On statistical model checking of stochastic systems.
In Proceedings of the 17th International Conference on Computer
Aided Verification, CAV’05, pages 266–280, Berlin, Heidelberg, 2005.
Springer-Verlag.
[14]
Håkan L. S. Younes.
Probabilistic verification for ”black-box” systems.
In Computer Aided Verification, 17th International Conference,
CAV 2005, Edinburgh, Scotland, UK, July 6-10, 2005, Proceedings, pages
253–265, 2005.
[15]
Ymer. 2015. ymer tool.
Retrieved from http://www.tempastic.org/ymer/.
[16]
Koushik Sen, Mahesh Viswanathan, and Gul A. Agha.
VESTA: A statistical model-checker and analyzer for probabilistic
systems.
In QEST, pages 251–252, 2005.
[17]
Multivesta. 2015. multivesta tool.
Retrieved from http://sysma.imtlucca.it/tools/multivesta/.
[18]
Benoît Boyer, Kevin Corre, Axel Legay, and Sean Sedwards.
PLASMA-lab: A flexible, distributable statistical model checking
library.
In Proceedings of the 10th International Conference on
Quantitative Evaluation of Systems, QEST’13, pages 160–164, Berlin,
Heidelberg, 2013. Springer-Verlag.
[19]
Modes. 2006. modes tool.
Retrieved from http://www.modestchecker.net.
[20]
Uppaal 2015. uppaal.
Retrieved from http://www.uppaal.org.
[21]
Mrmc. 2011. mrmc tool.
Retrieved from http://www.mrmc-tool.org.
[22]
Alexandre David, Kim G. Larsen, Axel Legay, Marius Mikucionis, Danny Bøgsted
Poulsen, Jonas van Vliet, and Zheng Wang.
Statistical model checking for networks of priced timed automata.
In Formal Modeling and Analysis of Timed Systems - 9th
International Conference, FORMATS 2011, Aalborg, Denmark, September 21-23,
2011. Proceedings, pages 80–96, 2011.
[23]
Christel Baier, Boudewijn R. Haverkort, Holger Hermanns, and Joost-Pieter
Katoen.
Automated performance and dependability evaluation using model
checking.
In Performance Evaluation of Complex Systems: Techniques and
Tools, Performance 2002, Tutorial Lectures, pages 261–289, 2002.
[24]
Sumit Kumar Jha, Edmund M. Clarke, Christopher James Langmead, Axel Legay,
André Platzer, and Paolo Zuliani.
A bayesian approach to model checking biological systems.
In Computational Methods in Systems Biology, 7th International
Conference, CMSB 2009, Bologna, Italy, August 31-September 1, 2009.
Proceedings, pages 218–234, 2009.
[25]
David Kyle, Jeffery P. Hansen, and Sagar Chaki.
Statistical model checking of distributed adaptive real-time
software.
In Runtime Verification - 6th International Conference, RV
2015 Vienna, Austria, September 22-25, 2015. Proceedings, pages 269–274,
2015.
[26]
Sucheendra K. Palaniappan, Benjamin M. Gyori, Bing Liu, David Hsu, and P. S.
Thiagarajan.
Statistical model checking based calibration and analysis of
bio-pathway models.
In Computational Methods in Systems Biology - 11th International
Conference, CMSB 2013, Klosterneuburg, Austria, September 22-24, 2013.
Proceedings, pages 120–134, 2013.
[27]
Paolo Zuliani, André Platzer, and Edmund M. Clarke.
Bayesian statistical model checking with application to
stateflow/simulink verification.
Formal Methods in System Design, 43(2):338–367, 2013.
[28]
José Meseguer and Raman Sharykin.
Specification and analysis of distributed object-based stochastic
hybrid systems.
In Proceedings of the 9th International Conference on Hybrid
Systems: Computation and Control, HSCC’06, pages 460–475, Berlin,
Heidelberg, 2006. Springer-Verlag.
[29]
João Martins, André Platzer, and João Leite.
Statistical model checking for distributed probabilistic-control
hybrid automata with smart grid applications.
In Proceedings of the 13th International Conference on Formal
Methods and Software Engineering, ICFEM’11, pages 131–146, Berlin,
Heidelberg, 2011. Springer-Verlag.
[30]
Richard Lassaigne and Sylvain Peyronnet.
Approximate planning and verification for large markov decision
processes.
International Journal on Software Tools for Technology
Transfer, 17(4):457–467, 2015.
[31]
Arnd Hartmanns and Holger Hermanns.
The modest toolset: An integrated environment for quantitative
modelling and verification.
In Tools and Algorithms for the Construction and Analysis of
Systems - 20th International Conference, TACAS 2014, Held as Part of the
European Joint Conferences on Theory and Practice of Software, ETAPS 2014,
Grenoble, France, April 5-13, 2014. Proceedings, pages 593–598, 2014.
[32]
David Henriques, Joao G Martins, Paolo Zuliani, André Platzer, and Edmund M
Clarke.
Statistical model checking for markov decision processes.
In 2012 Ninth International Conference on Quantitative
Evaluation of Systems, pages 84–93. IEEE, 2012.
[33]
Gul Agha and Karl Palmskog.
A survey of statistical model checking.
ACM Trans. Model. Comput. Simul., 28(1):6.1–6.39, January
2018.
[34]
Axel Legay and Mahesh Viswanathan.
Statistical model checking: Challenges and perspectives.
Int. J. Softw. Tools Technol. Transf., 17(4):369–376, August
2015.
[35]
Rajat Sen, Kirthevasan Kandasamy, and Sanjay Shakkottai.
Noisy blackbox optimization using multi-fidelity queries: A tree
search approach.
In The 22nd International Conference on Artificial Intelligence
and Statistics, pages 2096–2105, 2019.
[36]
Simone Fabris.
Method for hazard severity assessment for the case of undemanded
deceleration.
TRW Automotive, Berlin, 2012.
[37]
Chuchu Fan, Bolun Qi, and Sayan Mitra.
Data-driven formal reasoning and their applications in safety
analysis of vehicle autonomy features.
IEEE Design & Test, 35(3):31–38, 2018.
[38]
ISO.
ISO 26262:road vehicles – functional safety.
Norm ISO 26262, 2011.
[39]
A. Hinton, M. Kwiatkowska, G. Norman, and D. Parker.
PRISM: A tool for automatic verification of probabilistic systems.
In H. Hermanns and J. Palsberg, editors, Proc. 12th
International Conference on Tools and Algorithms for the Construction and
Analysis of Systems (TACAS’06), volume 3920 of LNCS, pages 441–444.
Springer, 2006.
[40]
Christian Dehnert, Sebastian Junges, Joost-Pieter Katoen, and Matthias Volk.
A storm is coming: A modern probabilistic model checker.
In International Conference on Computer Aided Verification,
pages 592–600. Springer, 2017.
[41]
Axel Legay, Sean Sedwards, and Louis-Marie Traonouez.
Plasma lab: a modular statistical model checking platform.
In International Symposium on Leveraging Applications of Formal
Methods, pages 77–93. Springer, 2016.
[42]
Jean-Bastien Grill, Michal Valko, and Rémi Munos.
Black-box optimization of noisy functions with unknown smoothness.
In Advances in Neural Information Processing Systems, pages
667–675, 2015.
[43]
Michal Valko, Alexandra Carpentier, and Rémi Munos.
Stochastic simultaneous optimistic optimization.
In International Conference on Machine Learning, pages 19–27,
2013.
[44]
Charles M Grinstead and James Laurie Snell.
Introduction to probability.
American Mathematical Soc., 2012.
[45]
Negin Musavi, Dawei Sun, Sayan Mitra, Geir Dullerud, and Sanjay Shakkottai.
Hoover: a statistical model checking tool with optimistic
optimization, October 2019.
Available from https://github.com/sundw2014/HooVer.
[46]
Pedro D’Argenio, Axel Legay, Sean Sedwards, and Louis-Marie Traonouez.
Smart sampling for lightweight verification of markov decision
processes.
International Journal on Software Tools for Technology
Transfer, 17(4):469–484, 2015.
Appendix A Multi-Fidelity Parallel Optimistic Optimization (MFPOO)
The optimal smoothness parameters are generally not known. Algorithm 2 [35] adaptively searches for the optimal smoothness parameters by releasing several MFHOO instances with various $\nu$ and $\rho$ values.
The following theorem gives the upper bound for the algorithm 2.
Theorem 3
If Algorithm 2 is run with parameters $(\nu_{max}\geq\nu^{*})$, $(\rho_{max}\geq\rho^{*})$ and given a total cost budget $\Lambda$, then the simple regret of at least one of the MFHOO instances spawned by Algorithm 2 is bounded a follows
$$\displaystyle R(\Lambda)=O\left((\dfrac{\nu_{max}}{\nu^{*}})^{D_{max}}(\dfrac{%
B\log n(\Lambda/\log{\Lambda})}{n(\Lambda/\log{\Lambda})})^{\frac{1}{d(\nu^{*}%
,\rho^{*})+2}}\right).$$
It is noted that if the Algorithm 1 is run at full fidelity case $z=1$, then it will be equivalent to a budgeted HOO algorithm. At this case the simple regret bound will be:
$$R(\Lambda)=O\bigg{(}(\dfrac{\log{(\Lambda/\lambda(1))}}{{\Lambda/\lambda(1)}})%
^{\frac{1}{d(\nu,\rho)+2}}\bigg{)}.$$
(7)
Appendix B Proofs
Proof 1
(Theorem 2)
We define a new Markov chain $\mathcal{M}(k)$ that starts from $x$ (like $\mathcal{M}$) and stops by going entering a new absorbing state called Stop, in $k$ steps, in expectation.
That is, from any transient state $s_{i}$, the one step transition probability of hitting $\mathit{Stop}$ is $\frac{1}{k}$. For any other transient state $s_{j}$, the one step probability of transitioning from $s_{i}$ to $s_{j}$ is $(1-\frac{1}{k})P_{ij}$. Therefore, the probability transition matrix $P_{k}$ for the new Markov chain $\mathcal{M}(k)$ is as follows.
$$\displaystyle P_{k}=\left(\begin{array}[]{c|cc}Q_{k}&R_{k}&\frac{1}{k}\textbf{%
1}\\
\hline{\bf 0}&I&{\bf 0}\\
{\bf 0}&{\bf 0}&1\\
\end{array}\right)=\left(\begin{array}[]{c|cc}(1-\frac{1}{k})Q&(1-\frac{1}{k})%
R&\frac{1}{k}\textbf{1}\\
\hline{\bf 0}&I&{\bf 0}\\
{\bf 0}&{\bf 0}&1\\
\end{array}\right),$$
where 1 is a $q\times 1$ vector whose elements are $1$, and the $\mathit{Stop}$ state is the last state.
$P_{k}$ is a $(q+u+1)\times(q+u+1)$ row stochastic matrix. Then after $t$ steps, $P_{k}^{t}$ would be
$$\displaystyle P_{k}^{t}=\left(\begin{array}[]{c|cc}Q_{k}^{t}&\sum_{j=0}^{t-1}Q%
_{k}^{j}R_{k}&\frac{1}{k}\sum_{j=0}^{t-1}Q_{k}^{j}\textbf{1}\\
\hline{\bf 0}&I&{\bf 0}\\
{\bf 0}&{\bf 0}&1\\
\end{array}\right).$$
For the Markov chain $\mathcal{M}(k)$, ${p_{k,\mathcal{U}}{(s_{i})}}$ is the probability that the chain is absorbed to any of the states in the set $\mathcal{U}$, given that chain starts at a transient state $s_{i}$. Using Theorem 2, ${p_{k,\mathcal{U}}{(s_{i})}}$ and ${p_{k_{\mathit{max}},\mathcal{U}}{(s_{i})}}$
can be written for $\mathcal{M}$ and $\mathcal{M}(k_{\mathit{max}})$, respectively:
$$\displaystyle{p_{k,\mathcal{U}}{(s_{i})}}=c^{T}N_{k}R_{k}\textbf{1},\ \ {p_{k_%
{\mathit{max}},\mathcal{U}}{(s_{i})}}=c^{T}N_{k_{\mathit{max}}}R_{k_{\mathit{%
max}}}\textbf{1},$$
where $c$ is a selection $m\times 1$ vector whose $i$th entry is $1$ and all other entries are $0$.
$N_{k}=(I-Q_{k})^{-1}$ and 1 is a $u\times 1$ vector of $1$’s.
Subtracting, we have
$$\displaystyle|{p_{k_{max},\mathcal{U}}{(s_{i})}}-{p_{k,\mathcal{U}}{(s_{i})}}|%
=|c^{T}(N_{k_{max}}R_{k_{max}}-N_{k}R_{k})\textbf{1}|.$$
(8)
Without loss of generality we can assume that $Q$ is diagonalizable555Even if $Q$ is not diagonalizable, we can use Jordan decomposition and that will not affect the analysis beyond this point.. Let $D$ the diagonal matrix of eigenvalues of $Q$, i.e. $D=\mbox{diag}(\lambda_{i})$, where $\lambda_{i}$ are the eigenvalues of $Q$ and let $E$ be the matrix such that $Q=EDE^{-1}$. Using this and definition of $Q_{k}$, we can rewrite $N_{k}=(I-Q_{k})^{-1}$ as
$$\displaystyle N_{k}=E(I-(1-\frac{1}{k})D)^{-1}E^{-1}=E\ \mbox{diag}\bigg{(}%
\frac{k}{k-(k-1)\lambda_{i}}\bigg{)}\ E^{-1}.$$
(9)
Rewriting $N_{k_{\mathit{max}}}$ in the same way the right hand side of (8) can be expressed as
$$\displaystyle=|\ c^{T}\ E\ \mbox{diag}\bigg{(}\frac{k_{max}-k}{(k_{max}-(k_{%
max}-1)\lambda_{i})(k-(k-1)\lambda_{i})}\bigg{)}\ E^{-1}\ R\ \textbf{1}\ |$$
(10)
$$\displaystyle\leq\frac{\|c\|_{\infty}\|E\|_{\infty}\|E^{-1}\|_{\infty}\|R\|_{%
\infty}\|\textbf{1}\|_{\infty}(k_{max}-k)}{(k_{max}-(k_{max}-1)\lambda_{max})(%
k-(k-1)\lambda_{max})}$$
(11)
$$\displaystyle\leq\frac{\kappa_{\infty}(E)(k_{max}-k)}{(k_{max}-(k_{max}-1)%
\lambda_{max})(k-(k-1)\lambda_{max})}.$$
(12)
In deriving the last two inequalities, we used $\|c\|_{\infty}=1$, $\kappa_{\infty}(E)=\|E\|_{\infty}\|E^{-1}\|_{\infty}$ and $\|\textbf{1}\|_{\infty}=1$. In addition, we have used the fact that $R$ is a row-stochastic matrix, that is, its rows sum up to less than unity, and thus, $\|R\|_{\infty}\leq 1$.
Appendix C Benchmarks
All our benchmarks are variants of $\mathsf{SLplatoon2}$ discussed in Section 2.2. The executable models are available from [45].
$\mathsf{SLplatoon3}$ models a platoon of $m$ cars where in every time step, each car can choose to move with one of three speeds: $\mathsf{vbrake}$, $\mathsf{vcruise}$, and $\mathsf{vspeedup}$.
The first vehicle always moves at some constant speed. For all the others, these three actions are chosen probabilistically, according to probability distributions that depend on their distance to the preceding vehicle.
For example, if the distance $s_{i-1}-s_{i}$ is less than a threshold constant $\mathsf{th\_close}$ then the speed is chosen according to a probability distribution $\mathsf{pclose}$.
The state variables of the model are defined as follows: for the $i^{th}$ car, we denote $s_{i}\in\mathbb{R}$ as the position along the lane. With out loss of generality, we assume $s_{1}>s_{2}>\ldots>s_{m}$. We also define an auxiliary variable $\mathit{gap}_{i}$ for all $i>1$ as the distance to the preceding car, i.e. $\mathit{gap}_{i}=s_{i-1}-s_{i}$. The set of initial states and the unsafe set have the same definition as in $\mathsf{SLplatoon2}$.
The constants in the model are defined as follows: $\mathsf{th\_far}$ and $\mathsf{th\_close}$ are some distance thresholds; $\mathsf{vbrake}$, $\mathsf{vcruise}$ and $\mathsf{vspeedup}$ are some velocities; $\mathsf{pclose}$, $\mathsf{pfine}$ and $\mathsf{pfar}$ are probability distributions for different modes. For example, $\mathsf{pclose}$ is the probability distribution over three actions, “brake”, “cruise” and “speed up”, and we denote $\Pr(v=\mathsf{vbrake})=\mathsf{pclose}[brake]$ as the probability of choosing action “brake” when this probability distribution is used.
With these variables, the behavior of each car at every time step is described in Fig. 4.
$\mathsf{MLplatoon}$ models a platoon of $m$ cars on $\ell$ lanes where in every time step, each car can choose to move with one of five actions: moving forward with speed $\mathsf{vbrake}$, $\mathsf{vcruise}$ or $\mathsf{vspeedup}$, or chaging to the left or right lane. These actions are chosen probabilistically, according to probability distributions that depend on their distances to the vehicles on its current lane, left lane or the right lane.
The state variables of the model are defined as follows: for the $i^{th}$ car, we denote $s_{i}\in\mathbb{R}$ as the position along the lane and $y_{i}\in\{1,\ldots,\ell\}$ as the ID of the current lane. Then, we define some auxiliary variables that can be derived from the state variables. We denote $\mathit{d\_ahead}[i]$ as the distance to the preceding car on the same lane. If the $i^{th}$ car is the leading car on its current lane, then $\mathit{d\_ahead}[i]=\infty$. Then, we denote $\mathit{d\_left}[i]$ as the minimal $s$-distance (i.e. only considering the difference of the $s$ variables) to the cars on the left lane. If there is no car on the left lane, then $\mathit{d\_left}[i]=\infty$. If the $i^{th}$ car is on the left-most lane, i.e. $y_{i}=1$, then $\mathit{d\_left}[i]=-\infty$. Similarly, we define $\mathit{d\_right}[i]$.
The definition of the initial state space of this model is a little bit different due to the $y$ variables. When choosing the initial state, all the $y$ variables are set to 1, i.e. all cars start from the left-most lane. Then, $(s_{1},\ldots,s_{m})$ is picked from a rectangle in $\mathbb{R}^{n}$ just as what we have done in $\mathsf{SLplatoon2}$ and $\mathsf{SLplatoon3}$. Finally, we define the unsafe set $\mathcal{U}=\{(s_{1},\ldots,s_{m})\ |\ \exists i\in\{1,\ldots,m\},\mathit{d\_%
ahead}[i]<\mathsf{unsafe\_rule}\}$.
The constants in the model are defined as follows: $\mathsf{th\_far}$, $\mathsf{th\_close}$ and $\mathsf{th\_clear}$ are some thresholds; $\mathsf{pturn}$ is the probability of chaning to the target lane if allowed to do that; $\mathsf{vbrake}$, $\mathsf{vcruise}$, $\mathsf{vspeedup}$, $\mathsf{pclose}$, $\mathsf{pfine}$ and $\mathsf{pfar}$ are defined with the same meaning as in $\mathsf{SLplatoon3}$.
With these variables, the behavior of each car at every time step is described in Fig. 5.
Appendix D Example illustrating non-smoothness of hitting probabilities
In this section, we present an illustrative example to show that in general the function ${p_{k,\mathcal{U}}{(x)}}$ over $x\in\Theta$ is not a continuous function and has infinitely many maxima.
Consider a discrete time linear system with state space $\mathcal{X}=\mathbb{R}^{m}$, set of initial states $\Theta\subseteq{\mathcal{X}}$ and set of unsafe states given by $\mathcal{U}:=\{x\in\mathcal{X}\ |\ c^{T}x\leq a\}$ for some vector $c\in{\mathbb{R}}^{m}$ and $a\in{\mathbb{R}}$. The system evolves according to:
$$x_{t+1}=Ax_{t}+BW_{t},$$
(13)
where $x_{t}\in\mathcal{X}$ is the state vector at time $t$. $W_{t}\in{\mathbb{R}}^{r}$ is a discrete random vector with probability mass distribution $p_{W}$ at sequence $t$, and
$A$ and $B$ are $m\times m$ and $m\times r$ matrices, respectively. Starting from a initial state $x_{0}\in\Theta$, the state of the system at any time $t$ can be written as
$$x_{t}=A^{t}x_{0}+\sum_{i=0}^{t-1}A^{i}BW_{i}.$$
(14)
Suppose the system in (13) hits the unsafe set first at sequence $t_{h}$, i.e. $t_{h}=min\{t\geq 0:x_{t}\in\mathcal{U}\}$.
Other trajectories may reach the unsafe set at a time after $t_{h}$ or may not reach the unsafe set at all. Assume that the trajectories reaching the unsafe set will stay there after. Starting from $x_{0}$, suppose we are interested in the probability of hitting the unsafe set for the system in (13) at sequence $n$, where $t_{h}<n$. Then the probability of reaching the unsafe set at sequence $n$ starting from $x_{0}$ can be expressed as
$$\displaystyle\mbox{Pr}\{x_{n}\in\mathcal{U}\}$$
$$\displaystyle=\mbox{Pr}\{c^{T}x_{n}\leq a\}=\mbox{Pr}\{c^{T}A^{n}x_{0}+\sum_{i%
=0}^{n-1}c^{T}A^{i}BW_{i}\leq a\}$$
(15)
$$\displaystyle=\mbox{Pr}\{\sum_{i=0}^{n-1}c^{T}A^{i}BW_{i}\leq a-c^{T}A^{n}x_{0%
}\}.$$
Since $W_{i}$s are discrete-type random variable, $\sum_{i=0}^{n-1}c^{T}A^{i}BW_{i}$ is so, i.e. there is a finite set of values $\{v_{i}:i\in J\subset{\mathbb{N}}\}$ such that $\mbox{Pr}\{\sum_{i=0}^{n-1}c^{T}A^{i}BW_{i}\in\{v_{i}:i\in J\}\}=1$. Therefore,
$$\displaystyle\mbox{Pr}\{x_{n}\in\mathcal{U}\}$$
$$\displaystyle=\sum_{k:v_{k}\leq a-c^{T}A^{n}x_{0}}\mbox{Pr}\{\sum_{i=0}^{n-1}c%
^{T}A^{i}BW_{i}=v_{k}\}.$$
(16)
Similarly the probability of reaching the unsafe set at sequence $n$ starting from initial state $x_{0}^{{}^{\prime}}=x_{0}+\epsilon\in\Theta$ for some $\epsilon\in\mathbb{R}^{m}$ can be expressed as
$$\displaystyle\mbox{Pr}\{x^{\prime}_{n}\in\mathcal{U}\}$$
$$\displaystyle=\sum_{k:v_{k}\leq a-c^{T}A^{n}x_{0}-c^{T}A^{n}\epsilon}\mbox{Pr}%
\{\sum_{i=0}^{n-1}c^{T}A^{i}BW_{i}=v_{k}\}.$$
(17)
Let $v_{min}=min(a-c^{T}A^{n}x_{0}-c^{T}A^{n}\epsilon,\ a-c^{T}A^{n}x_{0})$ and $v_{max}=max(a-c^{T}A^{n}x_{0}-c^{T}A^{n}\epsilon,\ a-c^{T}A^{n}x_{0})$, then we can write
$$\displaystyle|\mbox{Pr}\{x_{n}\in\mathcal{U}\}-\mbox{Pr}\{x^{\prime}_{n}\in%
\mathcal{U}\}|$$
$$\displaystyle=\sum_{k:v_{\mathit{min}}<v_{k}\leq v_{\mathit{max}}}\mbox{Pr}\{%
\sum_{i=0}^{n-1}c^{T}A^{i}BW_{i}=v_{k}\}.$$
(18)
Based on (18), for small enough $\epsilon$, in a norm sense, $v_{max}-v_{min}$ could be small such that, there exists no $v_{k}$ in the interval $(v_{min},\ v_{max})$ to make $|\mbox{Pr}(x_{n}\in\mathcal{U})-\mbox{Pr}(x^{\prime}_{n}\in\mathcal{U})|\neq 0$. In other words, for small enough $\epsilon$, $|\mbox{Pr}(x_{n}\in\mathcal{U})-\mbox{Pr}(x^{\prime}_{n}\in\mathcal{U})|=0$. By increasing the $\epsilon$, once the gap $v_{max}-v_{min}$ is large enough, there exist a $v_{k}$ in the interval $(v_{min},\ v_{max})$ with nonzero $\mbox{Pr}(\sum_{i=0}^{n-1}c^{T}A^{i}BW_{i}=v_{k})$. This means that there will be discontinuities in the function of probability of hitting the unsafe set of the system (13) in terms of the initial states. Here, discontinuities origin in the type of the random variable $W_{t}$, which is a discrete-type random variable. In addition, the hitting probability function would have infinitely many maxima. An example of this kind of function can be seen in Figure 1. For the systems of this kind, neither the local smoothness Assumption 1, nor the assumption of finite number of maxima (or unique maximum) does not hold.
Appendix E Regret: Theory and practice for single car example ( $\mathsf{Singlecar}$)
We define the NiMC of a single car. Consider a single car that moves in $\mathbb{R}^{2}$. Let $(s_{1},s_{2})$ be the position vector of the car in $\mathbb{R}^{2}$. The transition model of the car is as:
$$(s_{1},s_{2})\leftarrow\left\{\begin{array}[]{ll}(s_{1}+1,s_{2}+1)&\ \ \mbox{w%
.p.}\ \ p\\
(s_{1}+1,s_{2}-1)&\ \ \mbox{w.p.}\ \ 1-p,\\
\end{array}\right.$$
(19)
where $p\in[0,1]$ and is unknown. While the state space $\mathcal{X}=\mathbb{R}^{2}$, the set of initial states $\Theta=\{(s_{1},s_{2})\in\mathcal{X}\ |\ s_{1}=0\ and\ s_{2}\in(-b,b)\}$, where $b\in\mathbb{R}_{>0}$. Let set of unsafe states be $\mathcal{U}=\{(s_{1},s_{2})\in\mathcal{X}\ |\ |s_{2}|\geq b\}$. Given that car starts its motion at a initial state in $\Theta$, the goal is to find the maximum probability of hitting the unsafe set $\mathcal{U}$.
Figure 6 represents the probability of hitting the unsafe set for states in the initial set through different time horizons for $b=10$ and $p=0.5$. For large enough time horizons the probability of hitting the unsafe set for the initial states of set $\Theta$ approaches $1$, but, for smaller time horizons the maximum probability of hitting the unsafe set arises for initial states that are close to the unsafe set, meaning all initial states such that $s_{1}=0,\ s_{2}\in(-10,-9]$ or $s_{1}=0,\ s_{2}\in[9,10)$. The probability of hitting the unsafe set, as it is seen in the Figure 6, is not continuous and it has infinitely many maxima.
Now, Consider $\mathsf{Singlecar}$ model with parameters $k=100$, $b=10$ and $p=0.5$. Clearly any $s2\in[9,10)\cup(-10,-9]$ maximize the probability of hitting the unsafe set. In Figure 2 (top-left) the mean of actual regret obtained by running the Algorithm 1 with various smoothness parameters is presented. As budget increases the actual regret obtained by the algorithm approaches to $0$. For $\Lambda=37.5$, the partitioning stops at $h_{max}=13$. We can derive the number of $2\nu\rho^{h}$-near-optimal partitions for any $\nu$ and $\rho$. Let’s continue the analysis by setting $\nu=0.02$ $\rho=0.94$ which are used in generating the results of actual regret. Then for $h\in[6,13]$, the number of $2\nu\rho^{h}$-near-optimal partitions $\mathcal{N}_{h}(2\nu\rho^{h})=\lceil 2^{h}/10\rceil$. This is because the total number of partitions at depth $h$ is equal to $2^{h}$ and for $h\in[6,13]$, the $2\nu\rho^{h}$-near-optimal partitions belong to the interval $[9,10)\cup(-10,9]$ whose length is $1/10$. According to the Definition 4.1, for different values of $B$, different near-optimality dimensions can be obtained. We are looking for the pair of values $d_{m}$ and $B$ that minimizes the theoretical regret bound in Theorem 1. In Figure 2 (top-right) the number of $2\nu\rho^{h}$-near-optimal partitions are presented for $h\in[6,13]$ that are upper bounded by $B\rho^{-d_{m}h}$ for different values of $d_{m}$ and their corresponding $B$. In Figure 2 (bottom) the theoretical upper bound vs. $d_{m}$ and their corresponding $B$ for different values of smoothness parameters used in generating the actual regret are represented. As it is seen, the Best theoretical regret bounds that can be achieved is approximately $0.12$. |
Entangled microwave photons from quantum dots
C. Emary, B. Trauzettel, and C. W. J. Beenakker
Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506,
2300 RA Leiden, The Netherlands
(May 26th, 2005)
Abstract
We describe a mechanism for the production of polarisation-entangled
microwaves using intra-band transitions in a pair of quantum dots.
This proposal relies neither on spin-orbit coupling nor on control over
electron-electron interactions. The quantum correlation of microwave
polarisations is obtained from orbital degrees of freedom in an
external magnetic field.
We calculate
the concurrence of emitted microwave photon pairs, and show that
a maximally entangled Bell pair is obtained
in the limit of weak inter-dot coupling.
pacs: 42.65.Lm, 78.67.Hc, 78.70.Gq
Entangled photons at optical frequencies are routinely produced by
non-linear optical effects in macroscopic crystals man95 .
The use of semiconductor nanostructures to produce these states
promises both a greater frequency range and a closer integration with
quantum electronics.
One way to produce entangled photons is to start with entangled
electrons and then transfer this entanglement cer04 . However,
one can also start with non-entangled particles, and most work in this area
has focused on the decay of bi-excitonic states in quantum dots
ben00 ; sta03 ; gyw02 ; per04 .
In
the original proposal ben00 , a bi-exciton is formed in a
single dot by electrical pumping,
and then subsequently decays by emitting a
pair of photons
via one of two cascades. The polarisations of these two photons are
linked to the cascade by which the bi-exciton decays, and thus, if
these cascades proceed coherently, one produces
polarisation-entangled photons.
All these proposals cer04 ; ben00 ; sta03 ; gyw02 ; per04
involve inter-band transitions between valence and conduction bands
of a quantum dot, so that the output photons have frequencies in the
visible range.
In this paper, we propose the use of intra-band
transitions of conduction band electrons to generate
entangled microwave photons. Our proposal, illustrated in
Fig. 1, can be seen as the real-space analogue of the
bi-exciton decay cascade in energy of Ref. ben00 .
The microwaves originate from spontaneous
downward transitions between single-particle levels in a quantum dot.
That these transitions couple to microwaves has been demonstrated in
photon-assisted tunnelling experiments wie02 .
Our entanglement scheme requires four quantum dots
as shown in Fig. 1: two dots ($L$,$R$) to provide unique initial
and final states for the electron, and two more dots ($A$,$B$) to provide the
two decay paths.
For real symmetric tunnel couplings, an electron tunnels through the single level in dot $L$ into
an equal superposition
$2^{-1/2}(|A^{*}\rangle+|B^{*}\rangle)$ of upper levels in dots $A$ and $B$.
It decays to the ground state with the emission of two photons. The dots
are configured such that the same process gives rise to two left-circularly-polarized (CP+) photons in dot
$A$ and two right-circularly-polarized (CP$-$) photons in dot $B$. (We will describe later how this can be done.)
The resulting state
$|\Psi\rangle=2^{-1/2}(|A_{\mathrm{G}}\rangle|++\rangle+|B_{\mathrm{G}}\rangle|%
--\rangle)$
encodes the state of the quantum dot onto pairs of photons with
left or right circular polarisation.
To disentangle the photons from the electrons we couple dots $A$ and
$B$ symmetrically to a fourth dot $R$, drained by an electron
reservoir. This construction effectively projects the electronic
state in dots $A$ and $B$ onto the even combination
$|\Phi_{+}\rangle=2^{-1/2}(|A_{G}\rangle+|B_{G}\rangle)$, since
destructive interference prevents the odd combination
$|\Phi_{-}\rangle=2^{-1/2}(|A_{G}\rangle-|B_{G}\rangle)$ from tunneling
coherently into dot $R$. Defining also the even and odd combinations
$|\Psi_{\pm}\rangle=2^{-1/2}(|++\rangle\pm|--\rangle)$ of the photon
states, we may write
$|\Psi\rangle=2^{-1/2}(|\Phi_{+}\rangle|\Psi_{+}\rangle+|\Phi_{-}\rangle|\Psi_{%
-}\rangle)$.
The projection of $|\Psi\rangle$ onto $|\Phi_{+}\rangle$ thus produces
the required entangled photon pair $|\Psi_{+}\rangle$.
Whilst not strictly necessary for the entangling mechanism, it is
helpful to assume that the quantum dots are inserted into a cylindrical
microwave resonator able to support both polarisations at the resonant
frequency $\Omega$.
Recent results for double quantum dots and superconducting transmission line
resonators have shown the possibility of extremely large
dot-microwave couplings bla04 ; chi04 .
The cavity ensures that microwave, and not phonon, transitions dominate,
and also serves to counteract any slight non-idealities
in the quantum dot emission frequencies that might otherwise render the two decay paths
distinguishable.
But most importantly, the cavity allows operation of the device without postselection.
In the absence of the cavity, it is necessary to detect whether the
electron escapes into the right reservoir after having produced a
photon pair, in order to effectuate the projection onto
$|\Phi_{+}\rangle$. If the electron remains trapped in the state
$|\Phi_{-}\rangle$, the photon pair should be discarded. This
postselection discards one out of two attempts — even under ideal
conditions. As we will show in what follows, it is possible to
entirely avoid the postselection by using a microwave resonator to
evolve $|\Phi_{-}\rangle|\Psi_{-}\rangle$ into
$|\Phi_{+}\rangle|\Psi_{+}\rangle$. The electron will then always escape
into the reservoir and the required state $|\Psi_{+}\rangle$ is
produced at each and every attempt. The resultant entanglement could
be detected via the violation of a Bell inequality. This is a
routine experiment for visible light; the analogous experiment at
microwave frequencies is an experimental challenge, with some recent
progress gab04 .
After these qualitative considerations, we now turn to a
quantitative description.
The two emitting quantum dots,
$A$ and $B$, are assumed to have cylindrically symmetric,
parabolic confining potentials.
A perpendicular field $B_{0}$ is applied to control
the microwave polarisations.
The dots are set in a small area such that the charging
energy is sufficient to prevent
more than one electron being in the four-dot system at any time.
Dots $A$ and $B$ are in tunnelling contact with
the single levels in dots $L$ and $R$.
The state $E_{L}$ of dot $L$ is aligned with the levels $E_{1}$ in
dots $A$ and $B$,
and the single level $E_{R}$ in dot $R$ is aligned with
the ground states at $E_{3}$ (cf. Fig. 2). The
chemical potentials of the electron reservoirs
are adjusted such that on the left
$\mu_{L}\gg E_{L}$ and on the right $\mu_{R}\ll E_{R}$.
Since we never occupy the dots
with more than one electron at a time, a spinless single-particle picture
is appropriate. The Hamiltonian of quantum dot $Y=A,B$ is
$$\displaystyle H_{Y}=\omega^{Y}_{+}{a^{Y}_{+}}^{\dagger}a^{Y}_{+}+\omega^{Y}_{-%
}{a^{Y}_{-}}^{\dagger}a^{Y}_{-}=\sum_{i}\varepsilon^{Y}_{i}|Y_{i}\rangle%
\langle Y_{i}|.$$
(1)
The excitation energies are $\omega^{Y}_{\pm}=\sqrt{{\omega^{Y}_{0}}^{2}+\omega_{c}^{2}/4}\pm\omega_{c}/2$, with $\omega_{c}=eB_{0}/m$ the
cyclotron frequency and $\omega^{Y}_{0}$ the confinement energy of dot
$Y$ (we set $\hbar=1$). This so-called Fock-Darwin spectrum
kow01 is thus determined by the two quantum numbers $n^{Y}_{\pm}$
FDnote . Electric dipole transitions between dot levels with
photon emission along the symmetry axis satisfy the selection rule
$|\Delta n_{\pm}|=1$. Transitions in which $n_{-}$ decreases emit CP$+$
photons, and transitions in which $n_{+}$ decreases emit CP$-$
photons.
Figure 2 shows the six lowest levels $|Y_{i}\rangle$ in dots $Y=A,B$,
labelled $i=1$ to 6 from the bottom up. We take $B_{0}$
small enough that there are no
level crossings in the spectrum.
Since $\omega_{\pm}$ depends on the confinement energy $\omega_{0}$,
the spectra of
the dots can be tuned by electrostatically changing their sizes.
We set $\omega_{-}$ of dot $A$ equal to $\omega_{+}$ of dot $B$ by choosing
the ratio of the linear sizes to be
$l_{A}/l_{B}\approx 1-l_{B}^{2}/2l_{c}^{2}$ for $l_{c}\gg l_{B}$,
with $l_{c}=\sqrt{\hbar/eB_{0}}$ the magnetic length.
Fixing $\omega_{-}^{A}=\omega_{+}^{B}=\Omega$, only one free parameter
$\omega_{c}\ll\Omega$ remains to specify the two spectra.
We calculate the coupled dynamics of electron and photons
from the master equation sto96 ; gur98 ; bra04 ,
$$\displaystyle\frac{d}{dt}\rho=-i\left[H,\rho\right]+{\cal L}\left[\rho\right].$$
(2)
The commutator describes the coherent evolution of the density
matrix $\rho$ under the action of the Hamiltonian $H$ of the
dot-cavity system. The operator ${\cal L}\left[\rho\right]$
describes the coupling of dot $R$ to the electron reservoir on the
right. As we are interested in the entanglement produced by the
passage of a single electron through the device, we initialise the
system with an electron in dot $L$. The left reservoir serves only
to populate this level initially and is then decoupled, while the
coupling to the right reservoir is permanent and acts as a sink for
the electron.
The Hamiltonian is
$H=\sum_{Y}H_{Y}+H_{T}+H_{+}+H_{-}+H_{\mu}$, with the sum $Y$ taken
over all four dots. The Hamiltonians of the dots $A$ and $B$
contain six levels each, according to Eq. (1).
For dots $L$ and $R$ we have $H_{L}=E_{L}|L\rangle\langle L|$ and $H_{R}=E_{R}|R\rangle\langle R|$.
We set $E_{L}=E_{1}$ and $E_{R}=E_{3}$.
The dots are connected via the tunnelling Hamiltonian
$$\displaystyle H_{T}$$
$$\displaystyle=$$
$$\displaystyle\sum_{i=1}^{6}\bigl{\{}T_{LA_{i}}|L\rangle\langle A_{i}|+T_{RA_{i%
}}|R\rangle\langle A_{i}|$$
(3)
$$\displaystyle~{}~{}~{}~{}+T_{LB_{i}}|L\rangle\langle B_{i}|+T_{RB_{i}}|R%
\rangle\langle B_{i}|\bigr{\}}+\mathrm{H.c.},$$
where $T_{XY_{i}}$ are tunnel amplitudes.
The microwave photons have the Hamiltonians
$H_{+}=\omega^{A}_{-}b_{+}^{\dagger}b_{+}$ and $H_{-}=\omega^{B}_{+}b_{-}^{\dagger}b_{-}$,
with $b_{\pm}$ the field operators of CP$\pm$ microwaves.
Since we are on resonance, $\omega_{-}^{A}=\omega_{+}^{B}=\Omega$.
In the rotating wave approximation, the emission of microwaves is
governed by the Hamiltonian
$$\displaystyle H_{\mu}$$
$$\displaystyle=$$
$$\displaystyle g^{A}\bigl{\{}|A_{3}\rangle\langle A_{5}|+|A_{2}\rangle\langle A%
_{4}|+|A_{1}\rangle\langle A_{2}|\bigr{\}}b^{\dagger}_{+}$$
(4)
$$\displaystyle\!\!\!\!\!\!\!\!\!\!\mbox{}+g^{B}\bigl{\{}|B_{3}\rangle\langle B_%
{6}|+|B_{2}\rangle\langle B_{5}|+|B_{1}\rangle\langle B_{3}|\bigr{\}}b^{%
\dagger}_{-}+\mathrm{H.c.},$$
neglecting off-resonant transitions.
The coupling of the right electron reservoir
to dot $R$ is incorporated into the
master equation through the Lindblad operator
${\cal L}\left[\rho\right]=D\rho D^{\dagger}-\frac{1}{2}D^{\dagger}D\rho-\frac{%
1}{2}\rho D^{\dagger}D$.
The jump operator is $D=\sqrt{\Gamma_{R}}|0_{\mathrm{dot}}\rangle\langle R|$,
with $\Gamma_{R}$ the tunnelling rate and
$|0_{\mathrm{dot}}\rangle$ denoting all four dots empty.
In the following, we will for simplicity set $g^{A}=g^{B}=g$ and
$T_{XA_{i}}=T_{A}$, $T_{XB_{i}}=T_{B}$ for $X=L,R$ and $i=1,\cdots,6$.
Results of a numerical integration of the master equation (2)
are plotted in Figs. 3 and 4.
The density matrix of the field is obtained by tracing
out the dot degrees of freedom from the density matrix $\rho(t)$.
We denote its (unnormalised) two-photon projection as $\chi(t)$. It has
the form
$$\displaystyle\chi$$
$$\displaystyle=$$
$$\displaystyle r_{+}|++\rangle\langle++|+r_{-}|--\rangle\langle--|$$
(5)
$$\displaystyle\mbox{}+r_{c}|++\rangle\langle--|+r_{c}^{*}|--\rangle\langle++|.$$
The mean number of photon pairs in the cavity is given by
$N_{2}=\mathrm{Tr}~{}\chi=r_{+}+r_{-}$. This is plotted as a function of time
in Fig. 3.
The time scale on which
the electron is transmitted sequentially
through the elements of the system is
$\tau=\Gamma_{R}^{-1}+2g^{-1}+T_{A}^{-1}+T_{B}^{-1}$.
We see that for times $t\gg\tau\approx 70\,\Omega^{-1}$,
the number of photon pairs $N_{2}$
approaches a stationary value, $N_{2}^{\infty}$.
The degree of entanglement (concurrence) ${\cal C}$ of the photon pair
can be calculated using Wootter’s formula woo98 for the
concurrence of the density matrix $\chi(t)$, which in general
describes a mixed state. We find
$$\displaystyle{\cal C}=\frac{2|r_{c}|}{r_{+}+r_{-}}.$$
(6)
The time-dependent concurrence ${\cal C}(t)$ is shown in
Fig. 3, and this is seen to saturate for $t\gg\tau$. In
Fig. 4 (upper panel) we plot this long-time limit ${\cal C}^{\infty}$ as a function of the coupling asymmetry $T_{A}/T_{B}$ for
several values of $T_{A}$.
An analytical solution is possible for $T_{A},T_{B}\ll\omega_{c}$.
Since the top three levels in dots $A$ and $B$
are each separated by $\omega_{c}$, when $T_{A},T_{B}\ll\omega_{c}$
the electron tunnels from dot $L$ only into the resonant levels at $E_{1}$,
producing a photon-pair before leaving the dots.
In this limit, the concurrence ${\cal C}^{\infty}$ of the final state
is easily calculated, as the relative amplitudes for the generation of each of
the two photon pairs are proportional to the product of the individual
coupling amplitudes along each of the two paths. We find
$$\displaystyle{\cal C}=2\frac{|T_{A}T_{B}g_{A}g_{B}|^{2}}{|T_{A}g_{A}|^{4}+|T_{%
B}g_{B}|^{4}}.$$
(7)
The numerical weak coupling results in Fig. 4 (crosses),
are very close to this analytic result (solid curve). For weak,
symmetric inter-dot couplings, the concurrence approaches unity,
corresponding to the production of the Bell state $2^{-1/2}(|++\rangle+|--\rangle)$. The negative effects of level broadening
induced by the inter-dot couplings are slight provided $T_{A},T_{B}\lesssim\omega_{c}$.
Successful
operation of the device requires that, after the passage of the electron,
the cavity is left with a pair of microwave photons. The
probability of successful operation is thus given by the asymptotic
mean number of photon pairs $N_{2}^{\infty}$. This probability is plotted
in Fig. 4 (lower panel). The success probability tends to unity in the weak-coupling limit.
In the final part of this paper we consider two different mechanisms
by which the efficiency of the device is reduced. The first
mechanism is direct inelastic transitions from upper levels in dots
$A$ and $B$ into dot $R$. This reduces the number of photon pairs
produced, but provided that the rate of these inelastic processes is
smaller than the dot-cavity coupling, photon-pair production will
still dominate. Moreover, provided that these inelastic processes
give no which-way information on the path of the electron, affecting
the two dots roughly equally, they have little effect on the degree
of entanglement of the photon pairs that are emitted.
The second mechanism is decoherence of the two spatially separated
decay paths. The decoherence typically results when the charge on
one of the dots couples to other charges in the environment, thereby
providing which-way information. We model this charge noise by
adding to the master equation (2) jump operators
$$D_{Y}=\sqrt{\Gamma_{\phi}}\left(\sum_{i}|Y_{i}\rangle\langle Y_{i}|\right)%
\otimes\openone_{\rm photon}$$
(8)
which measure the charge on each of the four dots $Y=A,B,L,R$. Here
the sum over $i$ ranges over all states $|Y_{i}\rangle$ in quantum
dot $Y$, the symbol $\openone_{\rm photon}$ denotes the identity for
the photon degrees of freedom, and $\Gamma_{\phi}$ is the
decoherence rate. The resulting degradation of the concurrence is
plotted in Fig. 5, for two values of $\Gamma_{\phi}$. More
extensive data indicates that the degradation is algebraic, ${\cal C}\propto(\tau^{\ast}\Gamma_{\phi})^{-1}$, with $\tau^{\ast}$ a
characteristic time scale of the device that takes on it smallest
value when all transition rates $g,T_{A},T_{B},\Gamma_{R}$ are close
to each other. In Fig. 5 we vary the electron-photon
coupling constant $g$, and indeed observe that the concurrence is
maximized when $g$ is neither much smaller nor much larger than the
other rates. Because the degradation of the concurrence is only
algebraic, rather than exponential, a reasonable amount of charge
noise can be tolerated.
In summary, we have described a mechanism for the production of
entangled microwave photon pairs using intra-band transitions in
quantum dots. Our calculations indicate that a four-dot device is
capable of the output of highly entangled pairs with useful success
probability. The entangler may be thought of as an electron
interferometer in which each of the two paths is coupled to a
different photon-pair producing process. Apart from the different
frequency range, it differs from the celebrated bi-exciton entangler
ben00 in that the interfering paths are in real space, rather
than in energy space, and also in that the correlated polarisations
result from orbital selection rules — rather than from spin-orbit
coupling.
This work was supported by the Dutch Science Foundation NWO/FOM.
References
(1)
L. Mandel and E. Wolf,
Optical Coherence and Quantum Optics (Cambridge, 1995).
(2)
V. Cerletti, O. Gywat, and D. Loss, cond-mat/0411235;
M. Titov, B. Trauzettel, B. Michaelis, and C.W.J. Beenakker,
cond-mat/0503676.
(3)
O. Benson, C. Santori, M. Pelton, and Y. Yamamoto,
Phys. Rev. Lett. 84, 2513 (2000).
(4)
O. Gywat, G. Burkard, and D. Loss,
Phys. Rev. B 65, 205329 (2002).
(5)
T. M. Stace, G. J. Milburn, and C. H. W. Barnes,
Phys. Rev. B 67, 085317 (2003).
(6)
J. I. Perea and C. Tejedor, cond-mat/0409745.
(7)
W.G. van der Wiel, S. de Franceschi, J. M. Elzerman,
T. Fujisawa, S. Tarucha, and L. P. Kouwenhoven,
Rev. Mod. Phys. 75, 1 (2003).
(8)
A. Blais, R.-S. Huang, A. Wallraff,
S. M. Girvin, and R. J. Schoelkopf, Phys. Rev. A 69, 062320 (2004).
(9)
L. Childress, A. S. Sørensen, and M. D. Lukin,
Phys. Rev. A 69, 042302 (2004).
(10)
J. Gabelli, L.-H. Reydellet, G. Fève,
J.-M. Berroir, B. Plaçais, P. Roche, and D. C. Glattli,
Phys. Rev. Lett. 93, 056801 (2004).
(11)
L. P. Kouwenhoven, D. G. Austing, and S. Tarucha,
Rep. Prog. Phys. 64, 701 (2001).
(12)
The quantum numbers $n^{Y}_{\pm}$ are related to the set
$\left\{n,l\right\}$ of Ref. kow01 via $n=\mathrm{Min}(n_{+},n_{-})$
and $l=n_{-}-n_{+}$.
(13)
T. H. Stoof and Yu. V. Nazarov, Phys. Rev. B 53,
1050 (1996).
(14)
S. A. Gurvitz, Phys. Rev. B 57, 6602 (1998).
(15)
T. Brandes, Phys. Rep. 408, 315 (2005).
(16)
W. K. Wootters, Phys. Rev. Lett. 80, 2245 (1998). |
Singularity-free dark energy star
Farook Rahaman
farook_rahaman@yahoo.com
Department of
Mathematics, Jadavpur University, Kolkata 700 032, West Bengal,
India
Anil Kumar Yadav
abanilyadav@yahoo.co.in
Department of
Physics, Anand Engineering College, Keetham, Agra -282 007, India
Saibal Ray
saibal@iucaa.ernet.in
Department of Physics,
Government College of Engineering & Ceramic Thechnology, Kolkata
700 010, West Bengal, India
Raju Maulick
rajuspinor@gmail.com
Department of
Mathematics, Jadavpur University, Kolkata 700 032, West Bengal,
India
Ranjan Sharma
rsharma@iucaa.ernet.in
Department of Physics,
P. D. Women’s College, Jalpaiguri 735101, India.
(November 20, 2020)
Abstract
We propose a model for an anisotropic dark energy star where we
assume that the radial pressure exerted on the system due to the
presence of dark energy is proportional to the isotropic perfect
fluid matter density. We discuss various physical features of our
model and show
that the model satisfies all the regularity conditions and is stable as well as singularity-free.
pacs: 04.40.Nr, 04.20.Jb, 04.20.Dw
I Introduction
Current cosmological observations of the accelerated expansion of
the universe strongly suggest that about 96% of the total energy
content of the universe is exotic in nature out of which $73\%$ is
believed to be gravitationally repulsive in nature popularly
called dark energy and the remaining $23\%$ is attractive in
nature and exists in the form of dark matter
Perlmutter (1998); Riess (2004). Consequently, cosmological models
based on dark energy either in the form of a cosmological constant
or in some other exotic forms of matter have got tremendous
attention in the recent past. From the astrophysical perspective,
if it is fundamentally impossible to get any observational
evidence for the existence of an event horizon in our universe
Abramowicz (2002) (though our current understanding of the
general theory of relativity strongly favour the existence of
strong gravitational regions induced by compact objects covered
under the event horizon), one is tempted to look for alternative
models which may serve as alternatives to black holes. A dark
energy star is, in particular, interesting in this scenario
Lobo (2006).
Any interior solution to the vacuum Schwarzschild exterior
comprising a fluid distribution governed by an equation of
state(EOS) of the form $p=-\frac{1}{3}\rho$, may be considered
as a dark energy star Chapline (2005). In the past, various
model specific dark energy stars have been proposed (see for
example, Ref.
Chapline (2005); Mazur (2002); Lobo (2006); Chan (2008); Ghezzi (2005) and
Ref. Padmanabhan (2008) for a recent review). In the present work,
we propose a model for an anisotropic dark energy star where we
assume that the radial pressure exerted on the system due to the
presence of dark energy is proportional to the isotropic perfect
fluid matter density. The stellar configuration comprises two
fluids - an ordinary baryonic perfect fluid together with an yet
unknown form of matter (dark energy) which is repulsive in nature.
We also assume that the two fluids are non interacting amongst
each other.
To describe the energy-momentum tensor for such a hybrid model, we
assume that our resulting composition is anisotropic in nature,
i.e. $p_{r}\neq p_{t}$, where $p_{r}$ and $p_{t}$ correspond to radial
and tangential pressure, respectively. Ever since the pioneering
works of Bowers and Liang Bowers (1974), anisotropic
relativistic stellar models have played an inportant role in the
description of compact stellar objects (see Herrera (1995) for
a recent review). At the microscopic level, a variety of reasons
such as the existence of type $3A$ superfluid, phase transition,
pion condensation, rotation, magnetic field, mixture of two
fluids, bosonic composition etc., may give rise to anisotropic
pressures inside a stellar object. Recent observations on highly
compact astrophysical objects like X ray pulsar Her X-1, X ray
buster 4U 1820-30, millisecond pulsar SAX J 1808.4 - 3658, X ray
sources 4U 1728 - 34, etc., also strongly favour an anisotropic
matter distribution since the density inside such an ultra-compact
object is expected to be beyond nuclear matter density. In our
model, we assume that the ansisotropy is generated due to two
kinds of fluid distributions. In a repent paper
Yazadjiev (2011), an exterior solution corresponding to a two
fluid stellar model composed of non-interacting phantom scalar
field describing the dark energy and ordinary matter has been
reported which reduces to Schwarzschild solution in the absence of
the dark energy. In our paper, we match the interior solution to
the Schwarzschild exterior solution at the boundary.
Our paper is organized as follows: In Section II we have provided
the basic equations in connection to the proposed model for dark
energy star. Sections. III - VIII are dealt, respectively, with
the boundary conditions, TOV equation, energy conditions,
stability, mass-radius relation and junction conditions for the
solutions under consideration. Some concluding remarks are made in
the Section IX.
II Basic Equations and Their Solutions
To describe the space-time of the dark energy stellar
configuration, we take the Krori and Barua Krori (1975) metric
(henceforth KB) given by
$$ds^{2}=-e^{\nu(r)}dt^{2}+e^{\lambda(r)}dr^{2}+r^{2}(d\theta^{2}+sin^{2}\theta d%
\phi^{2}),$$
(1)
with $\lambda(r)=Ar^{2}$ and $\nu(r)=Br^{2}+C$ where $A$, $B$ and
$C$ are arbitrary constants to be determined on physical grounds.
The energy-momentum of the two fluids are such that
$$\displaystyle T^{0}_{0}\equiv(\rho)_{eff}$$
$$\displaystyle=$$
$$\displaystyle\rho+\rho_{de},$$
(2)
$$\displaystyle T^{1}_{1}\equiv-(p_{r})_{eff}$$
$$\displaystyle=$$
$$\displaystyle-(p+p_{der}),$$
(3)
$$\displaystyle T^{2}_{2}\equiv T^{3}_{3}\equiv-(p_{t})_{eff}$$
$$\displaystyle=$$
$$\displaystyle-(p+p_{det}),$$
(4)
where $\rho$ and $p$ correspond to the energy density and pressure
of the baryonic matter, respectively, and $\rho_{de}$, $p_{der}$
and $p_{det}$ are the ‘dark’ energy density, radial pressure and
tangential pressure, respectively. The left hand sides of
equations (2)-(4) are the effective energy-density
and two pressures, respectively, of the composition.
The Einstein’s field equations for the metric (1) are then
obtained as (we assume $G=c=1$ under geometrized relativistic
units)
$$\displaystyle 8\pi\left(\rho+\rho_{de}\right)$$
$$\displaystyle=$$
$$\displaystyle e^{-\lambda}\left(\frac{\lambda^{\prime}}{r}-\frac{1}{r^{2}}%
\right)+\frac{1}{r^{2}},$$
(5)
$$\displaystyle 8\pi\left(p+p_{der}\right)$$
$$\displaystyle=$$
$$\displaystyle e^{-\lambda}\left(\frac{\nu^{\prime}}{r}+\frac{1}{r^{2}}\right)-%
\frac{1}{r^{2}},$$
(6)
$$\displaystyle 8\pi\left(p+p_{det}\right)$$
$$\displaystyle=$$
$$\displaystyle\frac{e^{-\lambda}}{2}\left[\frac{{\nu^{\prime}}^{2}-\lambda^{%
\prime}\nu^{\prime}}{2}+\frac{\nu^{\prime}-\lambda^{\prime}}{r}+\nu^{\prime%
\prime}\right].$$
(7)
To solve the above set of equations, we assume that the dark
energy radial pressure is proportional to the dark energy density,
i.e.,
$$p_{der}=-\rho_{de},$$
(8)
and the dark energy density is proportional to the matter density, i.e.,
$$\rho_{de}=\alpha\rho,$$
(9)
where $\alpha>0$ is a proportionality constant. In connection to
the ansatz (8) it is worthwhile to mention that the
equation of state of this type which implies that the matter
distribution under consideration is in tension is available in
literature and hence the matter is known as a ‘false vacuum’ or
‘degenerate vacuum’ or
‘$\rho$-vacuum’ Davies (1984); Blome (1984); Hogan (1984); Kaiser (1984).
Now, from the metric (1) we get $\lambda^{\prime}=2Ar$,
$\nu^{\prime}=2Br$ and $e^{-\lambda}=e^{-Ar^{2}}$ and substituting
these values in equations (5) - (7), together with
our assumptions as given in equations (8) and (9),
we get
$$\displaystyle 8\pi\rho$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{(1+\alpha)}\left[e^{-Ar^{2}}\left(2A-\frac{1}{r^{2}}%
\right)+\frac{1}{r^{2}}\right],$$
(10)
$$\displaystyle 8\pi(\rho+p)$$
$$\displaystyle=$$
$$\displaystyle 2e^{-Ar^{2}}\left(A+B\right).$$
(11)
Subtracting equation (10) from (11), we get
$$\displaystyle 8\pi p=e^{-Ar^{2}}\left(2A+2B\right)-\\
\displaystyle\frac{1}{(1+\alpha)}\left[e^{-Ar^{2}}\left(2A-\frac{1}{r^{2}}%
\right)+\frac{1}{r^{2}}\right].$$
(12)
The equation (12), alongwith (7), then provides
$$\displaystyle 8\pi p_{det}=e^{-Ar^{2}}\left[B^{2}r^{2}-ABr^{2}-3A\right]\\
\displaystyle+\frac{1}{(1+\alpha)}\left[e^{-Ar^{2}}\left(2A-\frac{1}{r^{2}}%
\right)+\frac{1}{r^{2}}\right].$$
(13)
Thus the effective energy density $(\rho)_{eff}$, effective radial
pressure $(p_{r})_{eff}$ and the effective tangential pressure
$(p_{t})_{eff}$ are obtained as
$$\displaystyle(\rho)_{eff}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{8\pi}\left[e^{-Ar^{2}}\left(2A-\frac{1}{r^{2}}\right)+%
\frac{1}{r^{2}}\right],$$
(14)
$$\displaystyle(p_{r})_{eff}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{8\pi}\left[e^{-Ar^{2}}\left(2B+\frac{1}{r^{2}}\right)-%
\frac{1}{r^{2}}\right],$$
(15)
$$\displaystyle(p_{t})_{eff}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{8\pi}\left[e^{-Ar^{2}}\left(B^{2}r^{2}+2B-ABr^{2}-A%
\right)\right].$$
(16)
Using equations (16) - (18) the
equation of state (EOS) corresponding to radial and
transverse directions may be written as
$$\omega_{r}(r)=\frac{\left[e^{-Ar^{2}}\left(2B+\frac{1}{r^{2}}\right)-\frac{1}{%
r^{2}}\right]}{\left[e^{-Ar^{2}}\left(2A-\frac{1}{r^{2}}\right)+\frac{1}{r^{2}%
}\right]}$$
(17)
$$\omega_{t}(r)=\frac{\left[e^{-Ar^{2}}\left(B^{2}r^{2}+2B-ABr^{2}-A\right)%
\right]}{\left[e^{-Ar^{2}}\left(2A-\frac{1}{r^{2}}\right)+\frac{1}{r^{2}}%
\right]}$$
(18)
It is interesting to note that the effective energy density
$(\rho_{eff})$, effective radial pressure $(p_{r~{}eff})$ and
effective tangential pressure $(p_{t~{}eff})$ are independent of
$\alpha$.
We also note that
$$\frac{d\rho_{eff}}{dr}=-\frac{1}{8\pi}\left[\left(4A^{2}r-\frac{2A}{r}-\frac{2%
}{r^{3}}\right)e^{-Ar^{2}}+\frac{2}{r^{3}}\right]<0,$$
and
$$\frac{dp_{r~{}eff}}{dr}<0.$$
We impose the following conditions for our anisotropic fluid
configuration to be physically acceptable:
•
The density is positive definite and its gradient is negative everywhere within the fluid distribution.
•
The radial and tangential pressures are positive definite and the radial pressure gradient is negative definite.
The above results and Figs. 1-2 are in agreement with these
conditions.
Note that, at $r=0$, our model provides
$$\frac{d\rho_{eff}}{dr}=0,~{}~{}\frac{dp_{r~{}eff}}{dr}=0,$$
$$\frac{d^{2}\rho_{eff}}{dr^{2}}=-\frac{A^{2}}{\pi}<0,$$
and
$$\frac{d^{2}p_{r~{}eff}}{dr^{2}}<0,$$
which indicate maximality of central density and central pressure.
Interestingly, similar to an ordinary matter distribution, the
bound on the effective EOS in this construction is given by $0<\omega_{i}(r)<1$, (see Fig. 3) despite the fact that star
is constituted by the combination of ordinary matter and dark
energy.
The parameter $\Delta=\frac{2}{r}\left(p_{t~{}eff}-p_{r~{}eff}\right)$ representing the
‘force’ due to the local anisotropy is obtained as
$$\displaystyle\Delta=\frac{1}{4\pi r}\left[e^{-Ar^{2}}\left(B^{2}r^{2}-ABr^{2}-%
3A\right)\right]\\
\displaystyle+\frac{1}{8\pi}\left[e^{-Ar^{2}}\left(2A-\frac{1}{r^{2}}\right)+%
\frac{1}{r^{2}}\right].$$
(19)
This ‘force’ will be directed outward when $P_{t}>P_{r}$ i.e.
$\Delta>0$, and inward if $P_{t}<P_{r}$ i.e. $\Delta<0$. As it
is apparent from the Fig. 4 of our model with a repulsive
‘anisotropic’ force ($\Delta>0$) allows the construction of more
massive distributions.
III Boundary Conditions
We match the interior metric to the Schwarzschild exterior
$$\displaystyle ds^{2}=-\left(1-\frac{2M}{r}\right)dt^{2}+\left(1-\frac{2M}{r}%
\right)^{-1}dr^{2}\\
\displaystyle+r^{2}(d\theta^{2}+\sin^{2}\theta d\phi^{2}),$$
(20)
at the boundary $r=R$. Assuming continuity of the metric functions $g_{tt}$, $g_{rr}$ and $\frac{\partial g_{tt}}{\partial r}$ at the boundary surface $S$, we get
$$\displaystyle 1-\frac{2M}{R}$$
$$\displaystyle=$$
$$\displaystyle e^{BR^{2}+C},$$
(21)
$$\displaystyle\left(1-\frac{2M}{R}\right)^{-1}$$
$$\displaystyle=$$
$$\displaystyle e^{AR^{2}},$$
(22)
$$\displaystyle\frac{M}{R^{2}}$$
$$\displaystyle=$$
$$\displaystyle BRe^{BR^{2}+C}.$$
(23)
By solving Eqs. (21)-(23), we have
$$\displaystyle A$$
$$\displaystyle=$$
$$\displaystyle-\frac{1}{R^{2}}\ln\left[1-\frac{2M}{R}\right],$$
(24)
$$\displaystyle B$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{R^{2}}\left[\frac{M}{R}\right]\left[1-\frac{2M}{R}\right%
]^{-1},$$
(25)
$$\displaystyle C$$
$$\displaystyle=$$
$$\displaystyle\ln\left[1-\frac{2M}{R}\right]-\frac{\frac{M}{R}}{\left[1-\frac{2%
M}{R}\right]}.$$
(26)
We also impose the boundary conditions that at the boundary
$(p_{r})_{eff}(r=R)=0$ and $\rho_{~{}eff}(r=0)=b$ ($=a$
constant), where $b$ is the central density. Thus,
$$\displaystyle A$$
$$\displaystyle=$$
$$\displaystyle\frac{8\pi b}{3},$$
(27)
$$\displaystyle B$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{2R^{2}}\left[e^{\frac{8\pi b}{3}R^{2}}-1\right].$$
(28)
Combining, equations (24) and (27), we get
$$A=\frac{8\pi b}{3}=-\frac{1}{R^{2}}\ln\left[1-\frac{2M}{R}\right].$$
(29)
Note that the values of $B$ obtained from equations
(25) and (28) are identical.
At this juncture, to get an insight of our model, let us first
evaluate some reasonable set for values of $A$, $B$, and $C$.
According to Buchdahl Buchdahl (1959), the maximum allowable
compactness (mass-radius ratio) for a fluid sphere is given by
$\frac{2M}{R}<\frac{8}{9}$. Accordingly, let us assume that we
have a dark energy star whose mass and radius are such that
$\frac{M}{R}=0.3999052$. Due to highly compact nature of the
star, we set the radius of the star at $R=8~{}$km. With these
specifications, we obtain the values of the constants $A$, $B$ and
$b$ as $A=.025$, $B=.030883$, $b=.002984$. Later, we have
shown that these values of $A$ and $B$ are justified since the
energy conditions imply $2A\geq B\geq 0$ (see Sec. V).
IV TOV equation
For an anisotropic fluid distribution, the generalized TOV equation is given by
$$\frac{d(p_{r~{}eff})}{dr}+\nu^{\prime}\left(\rho_{eff}+p_{r~{}eff}\right)+%
\frac{2}{r}\left(p_{r~{}eff}-p_{t~{}eff}\right)=0.$$
(30)
Following Ponce de León leon (1993), we write
the above TOV equation as
$$\displaystyle-\frac{M_{G}\left(\rho_{eff}+p_{r~{}eff}\right)}{r^{2}}e^{\frac{%
\lambda-\nu}{2}}-\frac{dp_{r~{}eff}}{dr}\\
\displaystyle+\frac{2}{r}\left(p_{t~{}eff}-p_{r~{}eff}\right)=0,$$
(31)
where $M_{G}=M_{G}(r)$ is the effective gravitational mass inside a
sphere of radius $r$ and is given by
$$M_{G}(r)=\frac{1}{2}r^{2}e^{\frac{\nu-\lambda}{2}}\nu^{\prime},$$
(32)
which can easily be derived from the Tolman-Whittaker formula and
the Einstein’s field equations. Obviously, the modified TOV
equation describes the equilibrium condition for the dark star
subject to gravitational and hydrostatic plus another force due to
the anisotropic nature of the stellar object. Using equations
(14) - (16), the above equation can be written as
$$F_{g}+F_{h}+F_{a}=0,$$
(33)
where,
$$\displaystyle F_{g}$$
$$\displaystyle=$$
$$\displaystyle-Br\left(\rho_{eff}+p_{r~{}eff}\right),$$
(34)
$$\displaystyle F_{h}$$
$$\displaystyle=$$
$$\displaystyle-\frac{dp_{r~{}eff}}{dr},$$
(35)
$$\displaystyle F_{a}$$
$$\displaystyle=$$
$$\displaystyle\frac{2}{r}\left(p_{t~{}eff}-p_{r~{}eff}\right).$$
(36)
The profiles of $F_{g}$, $F_{h}$ and $F_{a}$ for our chosen source are
shown in Fig. 5. The figure indicates that the static
equilibrium is attainable due to pressure anisotropy,
gravitational and hydrostatic forces.
V Energy Conditions
In this section, we verify whether our particular choices of the
values of mass and radius leading to solutions for the unknownn
parameters, satisfy the following conditions through out the
configuration:
$$\rho_{eff}\geq 0,$$
$$\rho_{eff}+p_{r~{}eff}\geq 0,$$
$$\rho_{eff}+p_{t~{}eff}\geq 0,$$
$$\rho_{eff}+p_{r~{}eff}+2p_{t~{}eff}\geq 0,$$
$$\rho_{eff}>|p_{r~{}eff}|,$$
$$\rho_{eff}>|p_{t~{}eff}|.$$
Note that all the energy conditions namely, the null energy
condition (NEC), weak energy condition (WEC), strong energy
condition (SEC) and dominant energy condition (DEC), for our
particular choices of the values of mass and radius, are satisfied
as shown in Fig. 6. It is interesting to note here that
the model satisfies the strong energy condition, which implies
that the space-time does contain a black hole region.
The anisotropy, as expected, vanishes at the centre i.e.,
$p_{t~{}eff}=p_{r~{}eff}=p_{0~{}eff}=\frac{2B-A}{8\pi}$ at r=0. The
effective energy density and the two pressures are also well
behaved in the interior of the stellar configuration.
Employing the energy conditions at the centre ($r=0$), we may get
a bound on the constants $A$ and $B$ as follows:
(i) NEC: $p_{0~{}eff}+\rho_{0~{}eff}\geq 0$ $\Rightarrow$ $A+B\geq 0$,
(ii) WEC: $p_{0~{}eff}+\rho_{0~{}eff}\geq 0$ $\Rightarrow$ $A+B\geq 0$,
$\rho_{0~{}eff}\geq 0$ $\Rightarrow$ $A\geq 0$,
(iii) SEC: $p_{0~{}eff}+\rho_{0~{}eff}\geq 0$ $\Rightarrow$ $A+B\geq 0$,
$3p_{0~{}eff}+\rho_{0~{}eff}\geq 0$ $\Rightarrow$ $B\geq 0$,
(iv) DEC: $\rho_{0~{}eff}>|p_{0~{}eff}|$ $\Rightarrow$ $2A\geq B$.
VI Stability
For a physically acceptable model, one expects that the velocity of sound
should be within the range $0\leq v_{s}=(\frac{dp}{d\rho})\leq 1$ Herrera (1992); Abreu (2007).
In our anisotropic model, we define sound speeds as
$$\displaystyle v_{sr}^{2}=\frac{dp_{r~{}eff}}{d\rho_{eff}}\\
\displaystyle=-1+\frac{4Are^{-Ar^{2}}(A+B)}{\left(2A-\frac{1}{r^{2}}\right)2%
Are^{-Ar^{2}}+\frac{2}{r^{3}}\left(1-e^{-Ar^{2}}\right)},$$
(37)
$$\displaystyle v^{2}_{st}=\frac{dp_{t~{}eff}}{d\rho_{eff}}\\
\displaystyle=\frac{e^{-Ar^{2}}\left[2Ar\left(B^{2}r^{2}+2B-ABr^{2}-A\right)+2%
Br(A-B)\right]}{\left(2A-\frac{1}{r^{2}}\right)2Are^{-Ar^{2}}+\frac{2}{r^{3}}%
\left(1-e^{-Ar^{2}}\right)}.$$
(38)
We plot the radial and transverse sound speeds in Fig. 7
and observe that these parameters satisfy the inequalities $0\leq v_{sr}^{2}\leq 1$ and $0\leq v_{st}^{2}\leq 1$ everywhere within the
stellar object.
Equations (37) and (38) lead to
$$\displaystyle v^{2}_{st}-v^{2}_{sr}\\
\displaystyle=1-\frac{e^{-Ar^{2}}\left[2A^{2}Br^{3}+6A^{2}r+2Br^{2}-2ABr-2AB^{%
2}r^{3}\right]}{\left(2A-\frac{1}{r^{2}}\right)2Are^{-Ar^{2}}+\frac{2}{r^{3}}%
\left(1-e^{-Ar^{2}}\right)}.$$
(39)
From equation (39), we note that $v^{2}_{st}-v^{2}_{sr}\leq 1$
. Since, $0\leq v_{sr}^{2}\leq 1$ and $0\leq v_{st}^{2}\leq 1$,
therefore, $\mid v_{st}^{2}-v_{sr}^{2}\mid\leq 1$. In
Fig. 8, we have plotted $\mid v_{st}^{2}-v_{sr}^{2}\mid$.
Now, to examine the stability of local anisotropic matter
distribution, we use Herrera’s Herrera (1992) cracking (or
overturning) concept which states that the region for which radial
speed of sound is greater than the transverse speed of sound is a
potentially stable region. Thus, if the difference of the two
sound speeds $v_{st}^{2}-v_{sr}^{2}$ retains the same sign
everywhere within a matter distribution, no cracking will occur.
In our case, Fig. 9 indicates that there is no change of
sign for the term $v_{st}^{2}-v_{sr}^{2}$ within the specific
configuration since the difference is negative throughout the
distribution. Therefore, we conclude that our dark energy star
model is stable.
VII Mass-Radius relation
In this section, we study the maximum allowable mass-radius ratio
in our model. For a static spherically symmetric perfect fluid
star, Buchdahl Buchdahl (1959) showed that the maximally
allowable mass-radius ratio is given by $\frac{2M}{R}<\frac{8}{9}$ (for a more generalized expression for the same see
Ref. Mak (2001)). In our model, the effective gravitational
mass in terms of the effective energy density $\rho_{eff}$ can be
expressed as
$$\displaystyle M_{eff}=4\pi\int^{R}_{0}\left(\rho+\rho_{de}\right)r^{2}dr=\frac%
{1}{2}R\left(1-e^{-AR^{2}}\right).$$
(40)
In Fig. 10, we plot this mass-radius relation. We have
also plotted $\frac{M_{eff}}{R}$ against $R$ (see
Fig. 11) which shows that the ratio $\frac{M_{eff}}{R}$
is an increasing function of the radial parameter. We note that a
constraint on the maximum allowed mass-radius ratio in our case is
similar to the isotropic fluid sphere, i.e., $\frac{M}{R}<\frac{4}{9}$ as obtained earlier. The compactness of the star is
given by
$$u=\frac{M_{eff}(R)}{R}=\frac{1}{2}\left(1-e^{-AR^{2}}\right).$$
(41)
The surface redshift ($Z_{s}$) corresponding to
the above compactness ($u$) is obtained as
$$Z_{s}=(1-2u)^{-\frac{1}{2}}-1,$$
(42)
where
$$Z_{s}=e^{\frac{A}{2}R^{2}}.$$
(43)
Thus, the maximum surface redshift for an anisotropic star of
radius $8~{}$km turns out to be $Z_{s}=2.225541$.
VIII Junction Condition
One of the issues in connection with a static anisotropic matter
distribution is that, though the radial pressure at the boundary
of the star must vanish, the tangential pressure is not
necessarily zero at the boundary. This forces us to examine the
junction conditions of a static anisotropic star in closer
details. We propose here a shell type envelope at the boundary
surface so as to address this issue. Note that the fundamental
junction condition for a static star is that there has to be a
smooth matching between the interior solution and Schwarzschild
exterior at the boundary. Now, though the metric coefficients must
be continuous at the junction surface $S$ where $r=R$, their
derivatives may not be continuous at the junction. In other words,
the affine connections may be discontinuous at the boundary
surface. This can be taken care of if we consider the second
fundamental forms of the boundary shell. The second fundamental
forms associated with the two sides of the shell
Israel (1966); Usmani (2010); Rahaman (2010, 2011) are given by
$$K_{ij}^{\pm}=-n_{\nu}^{\pm}\ \left[\frac{\partial^{2}X_{\nu}}{\partial\xi^{i}%
\partial\xi^{j}}+\Gamma_{\alpha\beta}^{\nu}\frac{\partial X^{\alpha}}{\partial%
\xi^{i}}\frac{\partial X^{\beta}}{\partial\xi^{j}}\right]_{|_{S}},$$
(44)
where $n_{\nu}^{\pm}\ $ are the unit normals to $S$ and can be
writtten as
$$n_{\nu}^{\pm}=\pm\left|g^{\alpha\beta}\frac{\partial f}{\partial X^{\alpha}}%
\frac{\partial f}{\partial X^{\beta}}\right|^{-\frac{1}{2}}\frac{\partial f}{%
\partial X^{\nu}},$$
(45)
with $n^{\mu}n_{\mu}=1$. In Eq. (45), $\xi^{i}$ are the
intrinsic coordinates on the shell with $f=0$ being the
parametric equation of the shell $S$ and $-$ and $+$ correspond to
interior and exterior (Schwarzschild) metrices. Note that radial
pressure on the shell is zero. By using Lanczos equations
Israel (1966); Usmani (2010); Rahaman (2010, 2011), the surface
energy term $\Sigma$ and surface tangential pressures $p_{\theta}=p_{\phi}\equiv p_{t}$ may be obtained as
$$\displaystyle\Sigma$$
$$\displaystyle=$$
$$\displaystyle-\frac{1}{4\pi R}\left[\sqrt{e^{-\lambda}}\right]_{-}^{+},$$
(46)
$$\displaystyle p_{t}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{8\pi R}\left[\left(1+\frac{R\nu^{\prime}}{2}\right)\sqrt%
{e^{-\lambda}}\right]_{-}^{+}.$$
(47)
Since the metric functions are continuous on $S$, we have
$$\Sigma=0,$$
(48)
and
$$\displaystyle p_{t}=\frac{1}{8\pi R}\left[\frac{\left(\frac{M}{R}\right)}{{%
\sqrt{1-\frac{2M}{R}}}}-BR^{2}\sqrt{e^{-AR^{2}}}\right].$$
(49)
Therefore, we are now in a position to match our interior solution
to the Schwarzschild exterior in the presence of a thin shell.
From Fig. 13, we note that the effective transverse
pressure at the boundary ($R=8~{}$km) is positive though the
transverse pressure in the shell is negative. This clearly
indicates that the static equilibrium may be attained due to
positive $p_{t~{}eff}$ and negative $p_{t~{}shell}$. In this figure,
we have plotted $p_{t~{}eff}$ and $p_{t~{}shell}$ together by
assuming that the width of the shell is in between $8-15~{}$km. We
observe that at $r=15~{}$km, both $p_{t~{}eff}$ and $p_{t~{}shell}$
vanish simultaneously. Thus the thickness of the shell in this
case is $7~{}$km. Physically, this suggests that anisotropic matter
is confined within $8~{}$km from the centre of the star and the
outer region contains a thick shell extending up to $7~{}$km. The
thick shell is characterized by zero energy density and non zero
transverse pressure though the shell does not exert any radial
pressure.
IX Conclusion
Dark energy stellar models have found astrophysical relevance for
various reasons, one particular reason being it’s importance as an
alternative candidate to a black hole. The model developed here
satisfies all the physical requirements and is horizon free and,
therefore, can potentially describe a compact object which is
neither a neutron stars nor a quark star.
Similar to many model proposed earlier, our model also requires an
envelope just outside the baryonic matter for smooth matching with
the Schwarzschild exterior space-time. For matching we have firstly, assumed continuity of the metric functions $g_{tt}$,
$g_{rr}$ and $\frac{\partial g_{tt}}{\partial r}$ at the boundary
surface $S$, and secondly imposed the boundary conditions
that at the boundary $(p_{r})_{eff}(r=R)=0$ and $\rho_{~{}eff}(r=0)=b$ ($=a$ constant), where $b$ is the central density. Thus
we get a set of expressions for $A$, $B$ and $C$ which, due to the
maximum allowable compactness for a fluid sphere
Buchdahl (1959), eventually can be worked out as $A=.025$,
$B=.030883$, $b=.002984$ for the assumed mass-radius ratio as
$\frac{M}{R}=0.3999052$. Later on, in Sec. V, it has been shown
that these values of $A$ and $B$ are justified since the energy
conditions imply $2A\geq B\geq 0$.
Regarding stability of local anisotropic matter distribution, we
use cracking concept of Herrera Herrera (1992) which states
that the region for which radial speed of sound is greater than
the transverse speed of sound is a potentially stable region. It
is observed from Fig. 9 that there is no change of sign
for the term $v_{st}^{2}-v_{sr}^{2}$ within the specific
configuration and hence advocating in favour of stability of our
dark energy star model.
Let us now concentrate on some of the other works on KB analysis,
especially the works by Varela et al. Varela (2010) and
Farook et al. Farook (2010). In both the works static,
spherically symmetric, Einstein-Maxwell spacetime have been
considered with a fluid source of anisotropic stresses whereas the
present investigation is neutral one with anisotropic fluid
source. However, a common feature of all these KB-models is
singularity-free, stable configurations. Valera et
al.Varela (2010) in their work have found out a link of their
construction with a charged strange quark star as well as models
of dark matter including massive charged particles. Farook et
al.Farook (2010), by using a Chaplygin-type EOS, predicted
the possible existence of a Chaplygin charged dark energy star or
a strange quark star of radius about 8 km.
Therefore, it is interesting to note that present work deals with
a singularity-free spherically symmetric body of radius $r=15$ km
such that both $p_{t~{}eff}$ and $p_{t~{}shell}$ vanish
simultaneously. The model physically contains anisotropic matter
which is confined within $8~{}$km from the centre of the star and
the outer region contains a thick shell extending up to $7~{}$km.
Here the thick shell is characterized by zero energy density and
non-zero transverse pressure though the shell does not exert any
radial pressure. Note that in a recently proposed toy model
Dzhunushaliev (2011), a relativistic stellar configuration has
been developed where the core of the star is characterized by a
wormhole like solution for some kind of exotic matter violating
the weak/null energy condition and is surrounded by some ordinary
matter satisfying a polytropic EOS. This kind of theoretical
modelling would get observational support in the future.
We hope our model inspires observational workers to search this
type of stars. That is, we mean, the stars containing anisotropic
matter which is confined within certain radius from the centre of
the star and the outer region contains a thick shell extending up
to several kilometer.
Acknowledgments
FR, SR and RS gratefully acknowledge support from IUCAA, Pune,
India under Visiting Associateship under which a part of this work
was carried out. FR is also thankful to PURSE for providing
financial support. We thank the anonymous referee for drawing our
attention to a couple of references relevant to our studies.
References
Perlmutter (1998)
S. Perlmutter et al, Nature 391, 51 (1998).
Riess (2004)
A. G. Riess et al, Astrophys. J. 607, 665 (2004).
Abramowicz (2002)
M. A. Abramowicz, W. Kluzniak and J. P. Lasota, Astron. Astrophys. 396, L31 (2002).
Lobo (2006)
F. S. N. Lobo, Class. Quantum Grav. 23, 1525 (2006), 24, 1069 (2007).
Chapline (2005)
G. Chapline, Proceedings of the Texas Conference on Relativistic Astrophysics, Stanford, CA, December, (2004).
Mazur (2002)
P. O. Mazur and E. Mottola,
arXiv:gr-qc/0109035.
Chan (2008)
R. Chan, M. F. A. da Silva and J. F. V. da Rocha,
Gen. Relativ. Gravit. 41, 1835 (2009).
Ghezzi (2005)
C. R. Ghezzi, Phys. Rev. D 72 104017 (2005).
Padmanabhan (2008)
T. Padmanabhan, Gen. Relativ. Gravit. 40, 529 (2008).
Bowers (1974)
R. L. Bowers and E. P. T. Liang, Astrophys. J., 188, 657 (1974).
Herrera (1995)
L. Herrera and N. O. Santos, Astrophys. J., 438, 308 (1995).
Yazadjiev (2011)
Stoytcho S. Yazadjie, arXiv:1104.1865v2 [gr-qc].
Krori (1975)
K. D. Krori and J. Barua, J. Phys. A.: Math. Gen. 8,
508 (1975).
Davies (1984)
C. W. Davies, Phys. Rev. D30, 737 (1984).
Blome (1984)
J. J. Blome and W. Priester, Naturwissenshaften 71, 528 (1984).
Hogan (1984)
C. Hogan, Nature 310, 365 (1984).
Kaiser (1984)
N. Kaiser and A. Stebbins, Nature 310, 391 (1984).
Buchdahl (1959)
H. A. Buchdahl, Phys. Rev. 116, 1027 (1959).
leon (1993)
J. Ponce de León, Gen. Relativ. Gravit.
25, 1123 (1993).
Herrera (1992)
L. Herrera, Phys. Lett. A, 165 206, (1992).
Abreu (2007)
H. Abreu, H. Hernandez and L. A. Nunez, Class. Quantum Gravit. 24, 4631 (2007).
Mak (2001)
M. K. Mak, P. N. Dobson and T. Harko, Europhys. Lett. 55, 310 (2001).
Israel (1966)
W. Israel, Nuo. Cim. B 44, 1 (1966);
erratum - ibid. 48B, 463 (1967).
Usmani (2010)
A. A. Usmani, Z. Hasan, F. Rahaman, Sk. A.
Rakib, S. Ray, P. K. F. Kuhfittig, Gen. Relativ. Gravit. 42, 2901 (2010).
Rahaman (2010)
F. Rahaman, K. A. Rahman, Sk. A Rakib, P. K. F.
Kuhfittig, Int. J. Theor. Phys. 49, 2364 (2010).
Rahaman (2011)
F. Rahaman, P. K. F. Kuhfittig, M. Kalam, A. A. Usmani and S. Ray, arXiv:gr-qc/1011.3600.
Varela (2010)
V. Varela, F. Rahaman, S. Ray, K. Chakraborty and M. Kalam, Phys. Rev. D 82, 044052 (2010).
Farook (2010)
F. Rahaman, S. Ray, A. K. Jafry and K. Chakraborty, Phys. Rev. D 82, 104055 (2010).
Dzhunushaliev (2011)
Vladimir Dzhunushaliev et al, arXiv:1102.4454v3 [astro-ph.GA] |
The evolution of the mass–size relation for
early type galaxies from $z\sim 1$ to the present: dependence on environment, mass–range and detailed morphology
M. Huertas-Company${}^{1,2}$, S. Mei${}^{1,2}$, F. Shankar${}^{1}$, L. Delaye${}^{1,9}$, A. Raichoor${}^{3}$,
G. Covone${}^{4,5}$, A. Finoguenov${}^{6}$, J.P. Kneib${}^{7}$ , O. Le Fèvre${}^{7}$, M. Povic̀${}^{8}$
${}^{1}$GEPI, Paris Observatory, 77 av. Denfert Rochereau, 75014 Paris, France
${}^{2}$University Denis Diderot, 4 Rue Thomas Mann, 75205 Paris, France
${}^{3}$ INAF – Osservatorio Astronomico di Brera, via Brera 28, 20121 Milan, Italy
${}^{4}$Dipartimento di Scienze Fisiche, Università di Napoli ”Federico II”, Via Cinthia, I-80126 Napoli, Italy
${}^{5}$INFN Sez. di Napoli, Compl. Univ. Monte S. Angelo, Via Cinthia, I-80126 Napoli, Italy
${}^{6}$Department of Physics, University of Helsinki, Gustaf Hällströmin katu2a, FI-00014 Helsinki, Finland
${}^{7}$ Laboratoire d’Astrophysique de Marseille, CNRS-Université
${}^{8}$ Instituto de astrofísica de Andalucía(IAA-CSIC), C/ Glorieta de la Astronomía s/n, 18008 Granada, Spain
${}^{9}$ CEA, IRFU, SAp, F-91191 Gif-sur-Yvette, France
E-mail:
marc.huertas@obspm.fr
(Accepted 1988 December 15. Received 1988 December 14; in original form 1988 October 11)
Abstract
We study the dependence of the galaxy size evolution on morphology, stellar mass and large scale environment for a sample of 298 group and 384 field quiescent early-type galaxies from the COSMOS survey, selected from $z\sim 1$ to the present, and with masses $log(M/M_{\odot})>10.5$.
From a detailed morphological analysis we infer that $\sim 80\%$ of passive galaxies with mass $log(M/M_{\odot})>10.5$ have an early-type morphology and that this fraction does not evolve over the last 6 Gyr. However the relative abundance of lenticular and elliptical galaxies depends on stellar mass. Elliptical galaxies dominate only at the very high mass end – $log(M/M_{\odot})>11$ – while S0 galaxies dominate at lower stellar masses – $10.5<log(M/M_{\odot})<11$.
The galaxy size growth depends on galaxy mass range and early–type galaxy morphology, e.g., elliptical galaxies evolve differently than lenticular galaxies. At the low mass end – $10.5<Log(M/M_{\odot})<11$, ellipticals do not show strong size growth from $z\sim 1$ to the present (10% to 30% depending on the morphological classification). On the other end, massive ellipticals – $log(M/M_{\odot})>11.2$ – approximately doubled their size.
Interestingly, lenticular galaxies display different behavior: they appear more compact on average and they do show a size growth of $\sim 60\%$ since $z=1$ independent of stellar mass range.
We compare our results with state-of-the art semi-analytic models. While major and minor mergers can account for most of the galaxy size growth, we find that with present data and the theoretical uncertainties in the modeling we cannot state clear evidence favoring either merger or mass loss via quasar and/or stellar winds as the primary mechanism driving the evolution.
The galaxy mass-size relation and size growth do not depend on environment
in the halo mass range explored in this work (field to group mass $log(M_{h}/M_{\odot})<14)$,
i.e., group and field galaxies follow the same trends. At low redshift, where we examine both SDSS and COSMOS groups, this result is at variance with
predictions from some current hierarchical models that show a clear dependence of size
growth on halo mass for massive ellipticals ( $log(M_{*}/M_{\odot})>11.2$). In future work we will analyze in detail if this result is specific of the observations and model used in this work.
BCG and satellite galaxies lie on the same mass-size relation, at variance with
predictions from hierarchical models, which predict that BCGs should have larger
sizes than satellites because they experience more mergers in groups over the halo
mass range probed.
keywords:
galaxies: evolution, galaxies: elliptical and lenticular, cD, galaxies: groups
††pagerange: The evolution of the mass–size relation for
early type galaxies from $z\sim 1$ to the present: dependence on environment, mass–range and detailed morphology–B††pubyear: 2002
1 Introduction
The fact that massive quiescent galaxies experienced a strong size evolution in the last 10 Gyrs is now a commonly accepted picture since first works on this topic were published (Daddi et al. 2005; Trujillo et al. 2006).
Many independent groups using different datasets and selections have come up with similar conclusions, i.e. massive galaxies roughly doubled their size from $z\sim 1$ and probably increased it by $3\sim 5$ from $z\sim 2$ (e.g. van der Wel et al. 2008; van Dokkum et al. 2008; Buitrago et al. 2008; Damjanov et al. 2011; Cimatti et al. 2012) even though there might be a population with larger sizes already in place at high redshift (e.g Mancini et al. 2010; Saracco et al. 2011)
As several works pointed out since the first publications came out, the result could be biased by a wrong estimate of stellar masses (which is usually done through SED fitting, e.g. Raichoor et al. 2011) and/or an under-estimate of galaxy sizes at very high redshifts since surface brightness dimming could cause the loss of the very outer parts of galaxies (e.g. Bezanson et al. 2009; Hopkins et al. 2009). Independent measurements based on dynamical masses have however confirmed the compactness of several objects (e.g. Martinez-Manso et al. 2011; van de Sande et al. 2011; Newman et al. 2012) and it now seems clear that there is indeed a population of compact objects at high redshift.
Some works also pointed out that we cannot exclude that the current galaxy selections might be biased and we are missing those compact objects in the nearby universe, or selecting the most compact objects at high redshift (Valentinuzzi et al. 2010). This would be at variance with the work of Trujillo et al. (2009) that showed that there is not a strong evidence for a significant fraction of compact objects in the local Universe.
From the theoretical point of view, two main mechanisms have been proposed to increase galaxy size. Fan et al. (2008) proposed AGN and supernovae feedback as the main responsible of galaxy expansion while Hopkins et al. (2006) and Naab et al. (2009) argued that minor dry mergers are the most efficient way to grow sizes since they affect the outer parts of the galaxy without significantly modifying the stellar mass nor the star formation (see also Shankar et al. 2011).
Several recent observational works have tried to disentangle the two scenarios. As a result, minor dry merging seems to emerge as the most plausible explanation for the size evolution (e.g. Trujillo et al. 2011) at least from $z\sim 1$, leading to an inside-out growth of galaxies (e.g. Tiret et al. 2011; Patel et al. 2012) even though there is still some debate. Newman et al. (2012) showed by carefully counting small companions in very deep images from the CANDELS survey (Grogin et al. 2011; Koekemoer et al. 2011) that minor mergers are roughly enough to account for the size evolution from $z\sim 1$ (provided that a short merger timescale is assumed) and a combination of minor merging and star formation quenching can explain the galaxy growth from $z\sim 2$. Using morphological merging indicators, Bluck et al. (2011) reached a similar conclusion and even argue that the problem is close to be solved (see also McLure et al. (2012) for similar considerations). On the other hand, López-Sanjuan et al. (2011, 2012) point out that for galaxies with masses $log(M/M_{\odot})>11$ minor mergers are not the only process responsible for size growth since $z\sim 1$. These authors propose a scenario for which minor and major mergers contribute to $\sim 55\%$, while the remaining $\sim 45\%$ to $\sim 25\%$ is due to other processes and specially to younger galaxies (hence larger) arriving at later epochs (progenitor bias).
When measuring the galaxy mass–size relation and mass growth, different works use different sample selections though and this might lead to different conclusions. It is very rare in fact to find two works dealing with size evolution which apply the same criteria to select their galaxy sample and the same methodology to analyze the data. Some are based on star-formation only (Papovich et al. 2012) others on morphology (Raichoor et al. 2012; Cooper et al. 2011b) or on a combination of both (e.g. van der Wel et al. 2008; Damjanov et al. 2011). Finally, many works combine a stellar mass selection with a quiescence criterion (e.g. van Dokkum et al. 2008; Newman et al. 2012) and generally the mass cuts are not always the same. See Damjanov et al. (2011) and Cimatti et al. (2012) for two different compilations of recent results. These different selections are based on the general idea that almost all massive galaxies are passive and have an early-type morphology, which is not always true as shown in recent works (e.g. van Dokkum et al. 2011; Trujillo et al. 2012)
Another recent point of discussion in the literature is the role of environment. Recent observational studies show controversial results.
Raichoor et al. (2012) studied the mass size relation for morphologically selected early-type galaxies at $z\sim 1.2$ in three different environments (field, cluster, groups) and find that, on average, for masses $10<log(M/M_{\odot})<11.5$ cluster galaxies appear to be smaller at fixed stellar mass than field galaxies. Interestingly, in the same stellar mass range but at lower redshift, Cooper et al. (2011b) find exactly the opposite trend using DEEP3 (Cooper et al. 2011a). Larger sizes in the cluster environment are also observed at $z=1.62$ by Papovich et al. (2012) with CANDELS data for passive galaxies with stellar masses larger than $log(M/M_{\odot})\sim 10.5$. However, other two works (Maltby et al. 2010; Rettura et al. 2010) do not find any trend with environment at $z<0.4$ and $z\sim 1.2$ respectively. The differences between these works are somehow puzzling but might come from the different sample selections (e.g.: based on color or Sérsic index vs visual morphology classification) and/or the way environment is measured (local vs. global) and/or low statistics (high redshift results indeed rely on some tens of galaxies).
We will address here these questions by carefully selecting a statistically significant sample of galaxies from the recently published COSMOS X-ray detected group galaxy sample (George et al. 2011; Finoguenov et al. 2007) as compared to the field. Within this sample we will dissect the properties of passive galaxies and look at the effects on the size growth of the galaxies with different morphology in different environments.
The paper is organized as follows: in section 2 we describe the datasets used in this work and the derived quantities (morphologies, sizes and stellar masses) and in sections 3 and 4 we present and discuss our main results.
Throughout the paper we consider a standard $\Lambda$CDM cosmology ($\Omega_{M}=0.3$, $\Omega_{\Lambda}=0.7$).
2 Data and analysis
2.1 Datasets
We use two samples of galaxies from the COSMOS survey belonging to two different environments: groups and field.
The group sample is composed of groups in the COSMOS field that have been detected as extended X-ray emitters (Finoguenov et al. 2007), which is a strong signature of virialized structures, and have several spectroscopic confirmed members. The full sample contains groups with halo masses from $M_{200C}/M_{\odot}\sim 10^{13}$ to $\sim 10^{14}$ as measured by weak-lensing (Leauthaud et al. 2010) and spans the redshift range $0.2<z<1.0$. Group members have been selected based on photometric redshifts (and spectroscopic redshifts when available) derived from the extensive COSMOS multi-wavelength imaging (Ilbert et al. 2009). For this work, we use two group samples, for which details on the galaxy selection can be found in George et al. (2011) who carried out a careful analysis of potential biases and contaminations.
1.
We call the first sample central which includes only galaxies within $0.5R_{200C}$ and a probability of being a group member greater than 0.5. We expect then a contamination of $\sim 15\%$ , and a completeness of $\sim 90\%$ (George et al. 2011).
2.
The second larger sample includes galaxies within within R200C and is mainly used to increase statistics, using always the central sample to control contamination effects if environment is discussed. For the larger selection the contamination is $\sim 30\%$ and the completeness $\sim 90\%$ (George et al. 2011).
To both samples, we apply a magnitude cut $I814(AB)<24$ mag required for size estimates and morphology classification as explained in sections 2.2 and 2.3.
Field galaxies are selected in the COSMOS field in the same redshift range as group members and with the same magnitude cut but making sure that they do not belong to any detected group (e.g. with $GROUP_{ID}=-1$ in the George et al. (2011) group catalog). We assume that these galaxies lie in a dark matter halo of mass $M_{200C}/M_{\odot}<10^{13.2}$, otherwise they should have been detected as groups members – see Fig 1 from George et al. (2011). The field sample contains 3760 galaxies.
2.2 Sizes and masses
Sizes of all galaxies have been estimated on the COSMOS HST/ACS F814W images (Mosaic v2.0, Koekemoer et al. 2007) using galapagos (Barden et al. 2012) which is an IDL based pipeline to run Sextractor (Bertin & Arnouts 1996) and Galfit (Peng et al. 2002) together. We basically fit every galaxy in the field and in the group sample with a 2D Sérsic profile (Sersic 1968) using the default galapagos parameters as described and tested in Häussler et al. (2007). Our number of failures (i.e. fits that do not converge) is less than $5\%$. The point spread function (PSF) used for the fitting is taken from Rhodes et al. (2007) who computed spatially-varying model PSFs for the COSMOS survey taking into account variations in the effective HST focus positions. For this work we used a single PSF (estimated at the average focus position for the COSMOS survey ($\Delta=-2\mu$m)) for all the galaxies after checking that it does not introduce significant biases (see below).
The reliability of our size estimates is estimated by placing mock galaxies ($1<n<8$, $0.1^{"}<r_{e}<1.5^{"}$, $17<I<24$ mag) in a real background. In our simulations we also explore the possible biases due to local over densities by dropping the mock galaxies in the same positions as in real high redshift clusters. We also explore in the simulations the possible effects of a variable PSF by using a different PSF for simulating and for fitting. Both PSFs are taken from the Rhodes et al. (2007) PSFs models using the tabulation of the positional dependence of the PSF.We find that our size measurements are unbiased ($|<(r_{e,out}-r_{e,in})/r_{e,in}>|<0.1$) with a reasonable scatter ($\sqrt{Var[(r_{e,out}-r_{e,in})/r_{e,in}]}<0.2$) up to $I_{814}(AB)<24$ mag, $r_{e,in}<1.0$ arcsec and $\mu_{814W}<24$ $mag\times arcsec^{-2}$(Fig. 1 and Delaye et al., in prep). All galaxies in our sample have surface brightness brighter than $\mu_{814W}=24$ $mag\times arcsec^{-2}$ so our size estimates are reliable according to the simulations.
Since we are using the same wavelength to estimate sizes up to $z\sim 1$, we estimate sizes in different rest-frames at different redshifts (e.g. from the r-band rest-frame at $z\sim 0.2$ to the B-band rest-frame at $z\sim 1$).
Recent work by Damjanov et al. (2011) and Cassata et al. (2011) shows that sizes in the ultraviolet and optical rest–frame strongly correlate, with sizes in the UV rest-frame being only $\sim 10\%$ smaller than those in the optical (see also Cimatti et al. 2012). Therefore, an additional (small) artificial size difference could be created when estimating the evolution between low and high redshift data. However, as shown in section 3 and in figure 9, the size evolution we measure in our sample for early-type galaxies is fully consistent with previous published results, specially with the ones of Newman et al. (2012) who computed sizes in a unique rest-frame optical filter. Therefore we do not expect a significant bias due to this issue.
Throughout the paper we will use circularized effective radii as primary size estimator, i.e. $r_{e}^{circ}=r_{e}^{fit}\times\sqrt{b/a}$.
Stellar masses are computed through spectral energy distribution (SED) fitting using the Bayesian code described in Bundy et al. (2006) (KB06) which employs Bruzual & Charlot (2003) (BC03) models with a Chabrier IMF, and published in George et al. (2011) catalogs. More details can be found in section 2.3 of Leauthaud et al. (2010).
2.3 Morphology
Galaxy morphologies in the group and field samples are also computed on the HST/ACS F814W images with galSVM, a code for automated morphological classification based on support vector machines specially designed for high redshift data (Huertas-Company et al. 2008, 2009). The code requires a visually classified training set which is used to simulate galaxies at higher redshift. The training set used in this work is a combination of the Nair & Abraham (2010) sample of $\sim 14000$ galaxies and the EFIGI project (Baillard et al. 2011) both from the Sloan Digital Sky Survey DR4. Our final training set therefore contains $\sim 20.000$ galaxies.111An updated and stable version of the code as well as the training set used for this work are available at http://gepicom04.obspm.fr/galSVM/Home.html
We follow the bayesian approach presented in Huertas-Company et al. (2011) to associate to every galaxy 4 probabilities of being in 4 morphological classes as defined in the local universe by Nair & Abraham (2010), i.e. ellipticals ($E0,E+$), lenticulars ($S0-,S0,S0+,S0/a$), early spirals ($Sa,Sab,Sb,Sbc$) or late spirals ($Sc,Sd,Sdm,Sm,Im$): $P(E),P(S0),P(Sab),P(Scd)$. Errors in probabilities are computed by bootstrapping, i.e. we repeat the classification 10 times with randomly selected training sets from the main sample and keep the average probability for each galaxy as the final classification (see Huertas-Company et al. 2011 for more details).
We then select as elliptical and lenticular galaxies the galaxies for which $max(P(E),P(S0),P(Sab),P(Scd))=P(E)$ and $max(P(E),P(S0),P(Sab),P(Scd))=P(S0)$ respectively. Early-type galaxies are then defined as the combination of both populations. With this selection, the completeness for early-type galaxies is $\sim 95\%$ and the contamination rate is expected to be less than 7% as stated by detailed comparisons with visual morphological classifications at $z\sim 1.3$ from (Mei et al. 2009, 2012) in the Lynx super cluster at $z=1.26$. Our classification is therefore very close to a visual classification. This point is also confirmed by the fact that the axis ratio distribution of our early-type sample is very close to the one reported by Buitrago et al. (2011) from a visually classified sample (see Fig. 2 and their figure 2).
To compare our morphological classification with those often used in the literature, we estimated the contamination that would suffer a selection of early-type galaxies, based on a simple $n>2.5$ cut. We find that such a sample would be contaminated by approximately $50\%$ of early spirals (see also Mei et al. 2012). We refer to section 3 for a detailed analysis of the implications of such a selection.
Separating ellipticals from lenticulars is extremely challenging even by eye, so our classification is necessarily more contaminated. Simulations show that we are close to $30\%$ uncertainties, similar to those observed in other works (e.g. by visual morphological classifications by Postman et al. 2005). Our visually trained probabilistic approach for galaxy classification allows us to distinguish galaxies based on a quantitative separation in the parameter space that corresponds to that usually used to separate S0s and ellipticals in the local Universe (e.g. by visual classification). Indeed, the stellar mass and axis ratios distributions of the two populations shown in Fig. 2 present different behaviors, as stated by Kuiper tests ($P<0.6$), confirming that we are seeing separate populations and that the classification is not random. Ellipticals appear rounder, more massive on average, with slightly higher Sérsic indices and larger sizes. In order to double check our separation between lenticulars and ellipticals we performed two additional tests.
First, we took the visually classified sample of Nair & Abraham (2010) in the SDSS, cross–correlated it with the Sérsic decompositions recently published by Simard et al. (2011) and looked at the same properties we studied for our high redshift sample. Results are shown in Fig. 3. We do clearly observe the same trends, i.e. lenticulars are also more elongated and slightly more compact than ellipticals, even though it is less pronounced than in the high redshift sample. This test confirms that our E/S0 classification is consistent with the local visual classification, as expected.
As a second step, two of us (MHC and SM) made visual classifications of our two automatically defined classes. We find that the two classification of E/S0s agree at $\sim 80\%$ at $z<0.5$ and $\sim 70\%$ at $z>0.5$ which is fully consistent with the results from simulations and also with the differences found between independent human classifiers (e.g. Postman et al. 2005). However, in order to make sure that our results are not biased because of the automated method used, we double check our main results using visual classifications (see appendix A) and comment in our analysis in subsequent sections whenever there is a significant difference.
We deduce that the population of elliptical galaxies selected through our method is dominated by pure bulge systems whereas an important fraction of what we call lenticulars have a disk component (but not observed arms) even if they are still bulge dominated. We also notice that a simple Sérsic based selection of any kind would not allow the separation between these two populations.
2.4 Completeness
Completeness of our sample is mainly driven by the apparent magnitude cut ($I_{814}(AB)<24$ mag) required to properly estimate morphologies and sizes. Our main results in the following are shown as a function of stellar mass. Therefore it is important to understand how this magnitude selection is translated in terms of stellar mass completeness to estimate how much the results might be affected by selection effects. Since in this work we focus on passive ETGs (see section 3.1), all the completeness values are given for that population.
We used an approach similar to Pozzetti et al. 2010 and Giodini et al. 2012. For each passive galaxy we compute its limiting stellar mass ($M_{*}^{lim}$) which is the stellar mass it would have if its apparent magnitude was equal to the limiting magnitude of the survey ($I_{814}(AB)=24$ mag in our case). This value is given by the relation $Log(M_{*}^{lim})=Log(M_{*})-0.4(I-I_{lim})$ following Pozzetti et al. (2010). We compute this limiting mass for the $20\%$ faintest galaxies in each redshift bin and estimate the 95% completeness as the 95th percentile of the resulting distribution. Following this approach we obtain a 95% completeness for galaxies with stellar masses greater than $log(M_{*}/M_{\odot})\sim 9.56$ at $z\sim 0.2$ and $log(M_{*}/M_{\odot})\sim 10.57$ at $z\sim 1$ (fig. 4).
We notice that the mass completeness for the COSMOS sample has been largely discussed in the literature (e.g. Tasca et al. 2009, Giodini et al. 2012, Pozzetti et al. 2010, Meneux et al. 2009) finding similar values.
In order to further check that low surface brightness objects are not lost, we carried a new set of simulations close to the ones performed to calibrate the size recovery. We modeled 2000 ETGs ($B/T>0.6$) with effective radii varying from $0.1$ to $1.5$ arcsec and magnitudes at the faint end of the survey detection limit ($23<I_{814}<26$) , dropped them in real background and computed the detection rate as a function of magnitude and size (fig 5). Galaxies with $I_{814}<24$ of all sizes (and hence all surface brightness) are all detected thus confirming that our analysis sample is complete .
3 Results
3.1 Sample selection
We investigate in this section how the sample selection affects the observed mass-size relation and size evolution of selected galaxies. We compare first three selections usually found in other works:
1.
a passive selection. This selection includes all quiescent galaxies with $log(M/M_{\odot})>10.5$. The mass cut ensures that we are selecting a volume limited sample without being affected by incompleteness (see section 2.4). Throughout this work, we will select quiescent or passive galaxies based on the M(NUV)-M(R) dust corrected rest–frame color as computed by Ilbert et al. (2009) by means of SED fitting. Ilbert et al. (2009) have shown that a $M(NUV)-M(R)>3.5$ selection results in a good separation of passive and star-forming galaxies. See also Giodini et al. (2012) for a discussion on this selection on the same sample as the one used for this work.
2.
a Sérsic based selection, i.e. we apply a simple $n>2.5$ cut as usually done as well as the same stellar mass selection ($log(M/M_{\odot})>10.5$).
3.
an ETG selection in which we select all early type galaxies (ellipticals and lenticulars) with $log(M/M_{\odot})>10.5$ from our morphological classification detailed in section 2.3 independently of the star formation activity.
We show in Figs. 6 to 8 the mass size relations for samples (i), (ii) and (iii) respectively in different environments and redshifts and we summarize the best fit parameters computed through a standard chi square minimization in table 1. Also are shown in the figures, with different symbols the morphologies of the objects belonging to the given selection as well as the star formation activity (quiescent or active) color coded for selection (ii) and (iii). We also show the position of the brightest group galaxies (BGGs), defined as the most massive galaxies within an NFW scale radius of the X-ray position, from the George et al. (2011) catalog. Notice that some of them are morphologically classified as early-spirals. This is probably because of the extended halo usually found in these objects which could be interpreted as a disk component by the automated algorithms.
Despite the fact that we see some differences between the different selections, the fit parameters are roughly consistent within $1\sigma$. Since a Sérsic index based selection contains an important fraction of star-forming galaxies ($\sim 50\%$) (a population which presents, on average, higher sizes than ETGs; e.g. Mei et al. 2012), this population tends to increase the scatter of the relation.
A similar behavior is observed if we select galaxies just based on the morphology (ETGs, fig. 8). The scatter of the relation might be increased by the presence of a star-forming population of galaxies with early-type morphology. Even if the number densities of these objects are small in the local universe (e.g Kannappan et al. 2009) they tend to increase at high z (e.g Huertas-Company et al. 2010) and might have a consequence on the size measurements.
We quantify the possible effects of galaxy selection in Fig. 9 where we plot the mass-normalized radius ($R_{e}\times(10^{11}M_{\odot}/M_{*})$) evolution for different galaxy selections (ETG, passive ETGs, $n>2.5$, passive galaxies) with $log(M/M_{\odot})>10.7$ (the mass selection for this figure is chosen to match the one from Newman et al. 2012). We observe that surprisingly, the differences between the different selection in terms of size evolution are negligible and also consistent with recent results (i.e. Cimatti et al. 2012; Damjanov et al. 2011; Newman et al. 2012). We can conclude that these different selections usually found in the literature lead to similar results.
In this work we have an additional ingredient though, usually lacking in previous published results which is the morphological dissection of passive galaxies as explained in section 2.3. As a matter of fact, the population of passive galaxies is not a homogeneous population of objects. We show indeed some example stamps of massive ($log(M/M_{\odot})>10.5$) passive galaxies with different morphologies in Fig. 10. It is easy to notice by simple visual inspection that not all passive galaxies are bulge dominated. The relative abundance of each morphological type as a function of stellar mass is quantified in Fig. 11, for all our passive galaxy sample.
As expected, early-type morphologies dominate the population of passive galaxies up to $z\sim 1$ at all stellar masses, being about 80% of the total population at all redshifts. The remaining 20% is populated by early-type spirals while late-type spiral fractions are negligible. At all redshifts up to $z\sim 1$, elliptical galaxies dominate the ETG population at masses $log(M/M_{\odot})>11-11.2$. At $z<0.5$, galaxies with masses $log(M/M_{\odot})<11$ are around half ellipticals and half lenticular, while lenticulars dominate the low–mass fractions at $z>0.5$. In other words, an important fraction of passive galaxies from $z\sim 1$ (if not the majority) could have a disk component (see also Bundy et al. 2010, Mei et al. 2012, van der Wel et al. 2011). We also show in appendix, a version of figure 11 obtained with the two independent visual classifications performed in this work. One of the classifiers (SM) finds visually more Sas than S0s with respect to the automated classification, specially at high redshift. However the separation between galaxies with disk (Sas and S0s) and without disk (ellipticals) is very similar to what is obtained with the automated classification, except that elliptical galaxies are not dominant at the high mass end in the highest redshift bin, which is also found by the second classifier (MHC).
Studying the size evolution of passive galaxies all together (as often done in several works), mixes not only ETGs with early-spirals (Sa-b), but also ETG different morphological populations (e.g. ellipticals and lenticulars) for which the evolution is not necessarily of the same nature.
Differences in the mass-size relation of ellipticals and lenticulars are visible in Fig. 6 to 8 and Table 1. S0 galaxies tend to be not only less massive but also more compact with respect to ellipticals.
In the next sections, we will focus on the sample of passive galaxies (sample (i)) and will quantify these differences by studying the dependence of the size evolution of passive galaxies on different morphological types (namely ellipticals and lenticulars). The final sample contains 404 group (232 within $0.5\times R_{200C}$ and with $P_{MEM}>0.5$) and 459 field galaxies.
3.2 Size growth of passive galaxies with different morphologies from $z\sim 1$
Throughout this section we normalize sizes using the local reference derived by Bernardi et al. (2010) on the SDSS using a clean sample of elliptical galaxies (excluding S0s). We payed special attention in properly calibrating the local reference to be as consistent as possible with our high redshift measurements. We checked for that purpose that our low redshift ($z<0.3$) data are consistent at $1\sigma$ with their local mass-size relation which confirms that both relations are well calibrated. Sizes in each redshift bin are computed by fitting a 1-D gaussian function to the size ratio distributions and keeping the position of the peak as the adopted value. Results remain however unchanged if we use instead a classical median or a three sigma-clipped average. Uncertainties are computed through bootstrapping, i.e. we repeat the computation of each value 1000 times removing one element each time and compute the error as the $1-\sigma$ error of all the measurements.
In this section, since we are mainly interested in the effects of different morphologies, we mix group (larger selection) and field galaxies in a single population to improve statistics. Both populations will be considered separately in section 3.3 in which we study the effects of environment in the mass–size relation.
3.2.1 Ellipticals
In fig. 12, we show the size evolution of ellipticals as compared to lenticulars in two mass bins ($10.5<Log(M/M_{\odot})<11$ and $11.0<Log(M/M_{\odot})<11.5$). The upper limit is chosen because there are almost no S0s with masses greater then $10^{11.5}$ solar masses (see fig. 2) and we want to compare the evolution of the two morphological types in the same mass ranges.
Ellipticals in the mass range $11.0<Log(M/M_{\odot})<11.5$ experienced a $\sim 40\%\pm 10\%$ size growth from $z\sim 1$ to present. We find an $\alpha$ value of $\alpha=0.8\pm 0.28$ when fitting the evolution with a power-law ($r_{e}\propto(1+z)^{-\alpha}$). However, the most relevant feature is that, as shown in the left panel of fig. 12, below $\sim 10^{11}$ solar masses, elliptical galaxies do not experience a significant size growth, e.g their size does not evolve in a significant way with respect to local ellipticals from the SDSS ($r_{e}\propto(1+z)^{-0.34\pm 0.17}$) (see also Raichoor et al. 2012 for similar results at higher redshift). The mass dependence is even more clear in figure 13 in which we show in the left column, the observed size evolution for ellipticals in three bins of increasing stellar mass ($10.5<log(M/M_{\odot})<11$, $11<log(M/M_{\odot})<11.2$ and $11.2<log(M/M_{\odot})<12$). The derived $\alpha$ value parameterizing the evolution is $\alpha=0.34\pm 0.17$, $\alpha=0.63\pm 0.18$ and $\alpha=0.98\pm 0.18$ from low to high mass, showing a clear increasing trend with stellar mass. In appendix A, we show that when using visual morphological classifications the evolution in the low mass bin is stronger ($\sim 30\%$) making the difference between the two first mass bins less pronounced. The trend is still preserved though (see table 3), specially between the two last bins. One possible source of error though is that the stellar mass bins used are comparable to the expected error of the stellar mass (i.e. 0.2 dex), so the different behaviors found can be affected by contaminations of objects with lower/higher stellar masses. We have run Monte Carlo simulations to check if the trends found are preserved with typical errors expected on the stellar mass. For that purpose, we added to every stellar mass a random shift within $3\sigma$ of the expected error in stellar mass and recomputed the median sizes 1000 times. The values found are consistent at $1\sigma$ level so we do not expect a significant contribution of this effect in our measurements.
Interestingly, this mass dependence is less pronounced when studying all ETGs or passive galaxies as a whole (right column of fig. 13). The best fit relations for this whole sample are $r_{e}\propto(1+z)^{-1.01\pm 0.23}$ for ETGs with stellar masses $10.5<Log(M/M_{\odot})<11$ and $r_{e}\propto(1+z)^{-1.21\pm 0.22}$ and $r_{e}\propto(1+z)^{-1.19\pm 0.18}$ for $11<Log(M/M_{\odot})<11.2$ and $11.2<Log(M/M_{\odot})<12$ respectively which are fully consistent within $1\sigma$.
The mass dependence has been discussed in the recent literature providing different results. Williams et al. (2010) and Ryan et al. (2012) measured a mass dependence similar to the one reported here for ellipticals, i.e. with the evolution being stronger at higher stellar masses, while other works like Damjanov et al. (2011) suggest the slope of the mass-size relation is mass independent. We show here that the results might depend on how the selection is performed. When selecting pure passive bulges a mass dependent evolution seems to emerge. As a matter of fact, Williams et al. (2010) select passive galaxies based on the specific star formation rate (instead of the red sequence for other works). They might be removing from their sample more disky galaxies (also removed from our elliptical sample and not from an ETG sample) which could explain that we find similar results. Notice also that for galaxies with $log(M_{*}/M_{\odot})>11.2$, the contribution from disky galaxies is almost zero and therefore selecting ETGs or ellipticals has almost no effect in the size evolution. The strong size evolution found for these galaxies is therefore robust to galaxy selections and will be discussed in detail in section 4.
3.2.2 Lenticulars and early spirals
The bottom panels of Fig. 12 show the size evolution of lenticular galaxies in two mass bins normalized to the same local relation. They appear on average more compact than ellipticals, in a similar way as in the local Universe, as shown in section 2.3 and confirming the fact already pointed out by van der Wel et al. (2011) that the most compact galaxies at high redshift have a disk component (this effect is preserved even when not circularizing the radii). Since we normalize with the same local relation as for ellipticals, we also plot, for reference, the ratio of lenticular sizes over those of the ellipticals in the SDSS (notice that all fractions are normalized to the local SDSS elliptical sizes).
What is interesting is that the size growth of lenticulars does not seem to depend significantly on stellar mass given the large uncertainties, contrary to the behavior shown for the elliptical population: $r_{e}\propto(1+z)^{-0.67\pm 0.30}$ and $r_{e}\propto(1+z)^{-1.02\pm 0.25}$ for low mass and massive lenticulars, respectively.
In order to further investigate what is driving the size increase in these galaxies, we made a simple exercise which is to compare the observed evolution to the one expected in a star-forming disk dominated population. We selected for that purpose, galaxies in our sample with $M(NUV)-M(R)<3.5$ and a spiral-like morphology ($max(P(E),P(S0),P(Sab),P(Scd))=P(Sab)$ or $max(P(E),P(S0),P(Sab),P(Scd))=P(Scd)$), computed the size growth as a function of redshift as we did for the S0s. Since the overall sizes of late–type galaxies are larger than ETGs (see also Mei et al. 2012), we divide the overall relation by a factor of $1.3$ so that the values at $z=0$ agree (see fig. 12). We just plot the relation for the low mass end because there are not enough high mass late-type to derive a reliable relation at high masses.
Even though the normalization for disks is still slightly higher, the trends of the size growth for the two populations are very similar ($r_{e}\propto(1+z)^{-0.67\pm 0.30}$ for lenticulars and $r_{e}\propto(1+z)^{-0.54\pm 0.44}$ for star forming disks) which suggests that the size growth of a large part of the early–type galaxies and that of the late–type galaxies evolve at similar rates. If we add to the passive S0 population, also passive early-spirals ($max(P(E),P(S0),P(Sab),P(Scd))=P(Sab)$ and $M(NUV)-M(R)>3.5$) the trend is preserved.
3.3 Environment
We now study the effect of environment on the mass–size relation of galaxies paying special attention to the uncertainties due to the galaxy sample selection.
Figure 14 shows the evolution of mass-normalized radii $\gamma$ for $log(M_{*}/M_{\odot}>10.7)$ passive ETGs in groups and in the field. In the right panel we show only group members with a probability greater than 0.5 and within $0.5\times R_{200C}$ from the group center (central selection) to make sure that the signal is not washed out by interlopers. No significant differences are observed between the central and the larger sample. Our main result is therefore that the mass–size relation does not depend on environment for field and groups with halo masses $M_{200C}/M_{\odot}<10^{14.22}$. Fig. 15 also shows the evolution of the size of field and group passive galaxies divided into the same morphological and stellar mass bins than in the previous section. Relations are more scattered because of the Poisson noise (especially for massive lenticulars) but we do recognize the same trends as for the mixed population.
A point to take into account when normalizing with the local relation is the fact that groups and field galaxies in the local universe could follow different mass-size relations. We have checked this point by comparing the median mass-size relation of early-type galaxies in groups from the Yang et al. (2007) catalog to the full DR7 sample. Yang et al. (2007) put together a catalog of $\sim 300000$ groups detected in the SDSS DR4 using an automated halo-based group finder. For this work, we restricted to groups with more than 2 members and removed those objects affected by edge effects ($f_{edge}<0.6$). According to their figure 2, $\sim 80\%$ of the groups have $\sim 20\%$ contamination which is comparable to the expected contamination in our sample. We use as halo mass estimate, HM1, which is based on the characteristic luminosity of the group but results remain unchanged when using an halo mass estimate based on the characteristic stellar mass. The group catalog has been correlated with the catalog of 2D sersic decompositions by Simard et al. (2011) to get a size estimate for all our galaxies (circularized effective radius of the best fit) as well as with the morphological catalog by Huertas-Company et al. (2011) to select ETGs ($P_{early}>0.8$). The mass-size relations that we obtained for the field and group galaxies are fully consistent within $1\sigma$, we therefore use the same local relation to normalize group and field sizes.
In Raichoor et al. (2012), in a sample dominated by galaxies with $10<log(M/M_{\odot})<11$, we have noticed that while differences in median/average sizes cannot distinguish early–type populations in different environments, a test on both the mean and scatter of their distribution can point out environmental differences.
We extend Kuiper and Kolmogorov Smirnoff tests analysis to our group and field passive early-type galaxy size distribution in different mass bins to test environmental dependences on their distribution scatter that might distinguish size evolution in the groups and the field. We divide our sample in 3 redshift bins ($0.2<z<0.5$, $0.5<z<0.8$ and $z>0.8$) and 2 mass bins ($10.5<log(M/M_{\odot})<11$ and $log(M/M_{\odot})>11$) and performed the tests for the two group selections.
Results of the analysis are summarized in table 2 and the different size distributions for several sample selections are shown in Fig. 16. Same results for the central selection are shown in the appendix. For all redshift and mass bins, both the KS and Kuiper test results almost always give a probability $>80\%$ that field and groups size ratios are drawn from the same distribution even when considering only group members with $P_{MEM}>0.5$ and close to the group center.
Recent works at similar redshifts have found that galaxy size evolution depends on environment. Cooper et al. (2011a) find indeed that denser environments at $z<1$ tend to be populated by larger galaxies. Differences with this work can come from the way environment is measured and/or how galaxy selection is performed. Cooper and collaborators selection is indeed based exclusively on the Sérsic index, which as shown in Fig. 7 leads to the inclusion of some star-forming galaxies which tend to increase the scatter of the field population. As a matter of fact, we find that KS and Kuiper tests applied to the size distributions of group and field galaxies selected on the basis of the Sérsic index do show that the distributions have probabilities $<50\%$ to be drawn from the same distributions. However, median sizes do not change significantly so this fact cannot fully explain the difference between the two works. Concerning the environment measurement, we use here the DM halo mass as primary environment estimator whereas Cooper and collaborators use the local density based on neighbors.
3.3.1 BGGs
Brightest group galaxies (BGGs) deserve a particular mention since they have been more studied because of their very particular position at the center of massive structures and their high surface brightness which make them easy to detect and analyze. Current works, though, do not agree about the BCG size evolution and how BCG sizes compare to those of field and satellite galaxies.
Based on SDSS data, Bernardi (2009) found that BCGs are larger than field and satellite galaxies at fixed stellar mass and that there is a steep evolution of their size from $z\sim 0.3$ to present. The author also argues that minor dry mergers are the most probable mechanism to explain the build-up of these objects. Also in the SDSS, Weinmann et al. (2009) did not find a significant difference between the sizes of centrals and satellites in groups.
At higher z, Ascaso et al. (2011) find a significant size evolution from $z\sim 0.6$ to present, but do not detect any evolution in their profile. They interpret this fact as a signature of feedback instead of merging. (Stott et al. 2011) studied a sample of high redshift BCGs and found a very mild evolution of the BCG size from $z\sim 1$.
From the modeling point of view, Shankar et al. (2011) showed that BCGs should evolve much faster than satellite galaxies.
We use the BGGs definition as the most massive galaxies within an NFW scale radius of the X-ray position, from the George et al. (2011) catalog. Fig. 17 shows the size evolution for passive BGGs and satellite group members with similar stellar mass ($log(M/M_{\odot}>11$). We find that the two population evolve in a similar way within the error bars, e.g. we do not observe significant differences in the size evolution of BCGs as compared to satellite group members with similar stellar mass.
4 Discussion
To understand why not all populations evolve in the same way, we consider different scenarios of galaxy growth. There are two main, well distinct, physical mechanisms proposed so far in the literature
to puff up massive bulges from high redshifts
to the local Universe: galaxy mergers and mass loss
via quasar and/or stellar wind.
4.1 Mergers
Hierarchical models of structure formation
envisage the growth in size of massive galaxies
via a sequence of major and minor
mergers. While (mainly gas-rich) major mergers are believed to happen at high redshifts
and may be responsible for forming the galaxy, minor dry merger happen
on cosmological timescales and tend to impact mainly
the external regions of the galaxy, thus increasing its size,
but leaving its inner regions mostly unaltered.
In this work we consider the predictions of several, representative,
hierarchical galaxy evolution models that
differ in terms of underlying techniques and physical assumptions.
We adopt, more precisely, Bower et al. (2006), Hopkins et al. (2009), Guo et al. (2010) and Shankar et al. (2011). All model predictions have been computed for the range of stellar masses
222All predictions have been corrected to a common Chabrier IMF.
We however expect differences in the IMF
to have a minor impact on the rate of size evolution of massive spheroids.
and galaxy type of interest to this paper. We mainly considered galaxies with $B/T>0.5$
when comparing to early-type galaxies (though we checked that our
results are practically unchanged when restricting to $B/T>0.7$), and galaxies with $0.3<B/T<0.7$
when comparing to S0s.
Both the Bower et al. (2006) and Guo et al. (2010) models follow the hierarchical
growth of galaxies along the merger trees of the Millennium simulation (Springel et al. 2005).
Galaxy progenitors are initially disk-like and after a major merger the
remnant is considered to be an elliptical (though disk regrowth can happen).
Minor mergers instead tend to preserve the initial morphology of the most massive
progenitor but tend to increase the mass of the bulge and disk components
via the aggregation of old stars and newly formed ones during the merger.
Half-mass sizes are then updated at each merger event assuming energy conservation.
Despite being built on the same dark matter simulation,
the subhalo/galaxy merger rates of these two models differ due to the different corrections in dynamical
friction timescales. In both models, bulges can also grow via disk instabilities.
However, the implementation of the latter physical ingredient substantially differs in the two
models, with Bower et al. (2006) assuming a much stronger bulge growth via disk instabilities
with respect to the Guo et al. (2010) model (see discussion in Shankar et al. 2012).
Most importantly for size evolution, both models do not consider gas dissipation during gas-rich major mergers,
a feature that has instead been included by Shankar et al. (2011) by properly adapting the
Guo et al. (2010) model.
The Hopkins et al. (2009) model follows the analytic mass accretion histories of haloes
and at each time step initializes central galaxies and infalling satellites
according to empirical correlations inspired by halo occupation techniques and
high-redshift data. Equivalently to the models discussed above,
at each merger the half-mass radius is updated following energy conservation
arguments, with also gas dissipation (see also
Nipoti et al. 2012 for a more recent work adopting similar techniques).
Hopkins et al. (2009) have mainly focused on early-type galaxies (not lenticulars)
with stellar masses above $M_{*}>10^{10}{\,\rm M_{\odot}}$.
4.1.1 The Size Growth of Ellipticals compared to model predictions
In Figs. 18, we compare our results with predictions from Shankar et al. (2011), Bower et al. (2006) and Guo et al. (2010). For ellipticals, we focus here on the lowest and highest stellar mass bins of fig 13, where the differences are stronger (i.e. $10.5<log(M/M_{\odot})<11$ and $log(M/M_{\odot})>11.2$).
The first relevant feature arising
from the comparison is that most of the theoretical predictions are
in agreement with the (mild) size evolution of ETG galaxies with
stellar masses $M_{*}<10^{11}$ (left panel of Fig. 18) (specially taking into account the uncertainties due to morphological classifications). Merger models are therefore successful to predict the size evolution of these lower mass ellipticals.
More interestingly, despite the significant variance in the input parameters and/or physics most merger models seem to be unable to reproduce completely the
fast drop in sizes for quiescent ellipticals with stellar masses $Log(M_{*}/M_{\odot})>11.2$, especially at $z>0.5$ (right panel of Fig. 18).
This is in line, and further complements, the recent claims of a possible
inefficiency of the puffing-up via mergers pointed out by Shankar et al. (2011), Cimatti et al. (2012), and Nipoti et al. (2012). Notice that in this stellar mass bin, the contribution of S0 galaxies is almost negligible (see fig. 13), so a similar behavior is found when selecting all ETGs with $log(M_{*}/M_{\odot})>11.2$ and should be independent of the morphological selection.
Nipoti et al. (2012) have recently shown the results of a merger model initialized
via halo occupation techniques that,
by neglecting dissipative processes and assuming only mergers
with spheroids, maximizes the evolution in surface density.
They conclude that minor and major mergers
may not be sufficient to explain the observed size growth of early type galaxies (see also López-Sanjuan et al. 2012). An other work where the size growth since $z\sim 1$ is explained almost completely by mergers (Newman et al. 2012) has to assume very short merger time scales ($1$ Gyr) and steep growth efficiency, optimistic with respect to observations (notice in addition that the size evolution they measure is less steep than the one reported here though, because of the selection used, as shown in sections 3.1 and 3.2). Oser et al. (2012) have recently analyzed 40 cosmological re-simulations of individual massive
galaxies with final $M_{*}>6.3\times 10^{10}{M_{\odot}}$, out of which 25 appear quiescent early-type galaxies.
While they claim a strong size evolution in the cumulative distribution
of galaxies with present stellar mass $M_{*}>6.3\times 10^{10}{M_{\odot}}$,
a close inspection of their Figure 1 (left panel) reveals that galaxies above
$M_{*}>2\times 10^{11}{M_{\odot}}$ have indeed had a rather mild
size evolution at fixed stellar mass of about $\sim 30-40\%$ at $z<1$. Nevertheless, this may not necessarily reflect a failure of hierarchical models, as part of
the discrepancy could simply arise from the specific underlying assumptions made. For example, the early semi-analytic model by Khochfar & Silk (2006)
predicts a stronger size evolution for the very
massive galaxies with $M_{*}>5\times 10^{11}{\times M_{\odot}}$,
in better agreement with observations.
Overall, most merger models have some difficulties in fully reproducing the size evolution of the most massive early-type galaxies.
Part of the apparent evolution in sizes may be driven
by relatively younger larger galaxies formed at later epochs (progenitor bias) even if the relevance of this effect is still unclear. A recent empirical model by López-Sanjuan et al. (2012), based on merger observations in the COSMOS field, shows in fact that taking into consideration major and minor mergers observed in COSMOS can explain $\sim 55\%$ of the size evolution of massive ($>10^{11}{M_{\odot}}$) ETGs. If the progenitor bias of massive ETGs accounts for a factor 1.25, this work can explain $\sim 75\%$ of the size evolution. We also show, for completeness, in figure 18 the expected evolution taking into account this effect from that work (notice however that the selection in terms of morphology and star-formation is not exactly the same than the one used in this work). However, Whitaker et al. (2012) showed that the recently quenched galaxies at $1<z<2$ are not significantly larger than their older counterparts, suggesting that the effect of the so called progenitor bias is limited. Moreover, fig. 11 also shows that the number densities (fractions) of massive ellipticals do not evolve significantly from $z\sim 1$, pointing again towards a reduced effect of newly formed galaxies.
There are also other possible tensions
between merger models and current observations. A puffing-up of the external regions
with no inner density variation as suggested by mergers (e.g., Naab et al. 2009),
naturally produces an increase in the Sérsic index n of an ideally light profile fitted to
the projected density profile. In Fig. 19, we plot the median Sérsic index for massive elliptical galaxies.
The median Sérsic index of the overall population does not evolve since $z\sim 1$. These results
are in possible tension with this prediction
(similar conclusions were found by Stott et al. (2011) for instance).
Another prediction of merger models is that galaxies residing at the centre
of more massive haloes should, at fixed stellar mass, experience more mergers
and thus be larger than their counterparts in less massive haloes
, at least above $M_{h}>5\times 10^{12}M_{\odot}$, according
to the analysis of Shankar et al. (2011). The latter, in fact, showed that BCGs should evolve
much faster
at $z<2$, and also end up being larger
than other galaxies of similar mass. We do not observe this trend.
4.1.2 Environmental dependencies compared to model predictions
Concerning the dependence of galaxy sizes and their evolution on environment,
we compare our observations to predictions from Shankar et al. (2012) in figure 20. We plot the mass–normalized radius as a function of halo mass for two redshift bins from COSMOS ($0.5<z<0.8$ and $0.8<z<1.0$), as well as for the local Universe from SDSS, for two mass ranges. All radii are shown in units of the mass–normalized radius measured in the field.
Uncertainties in the observations have been calculated by bootstrapping 1000 times and recomputing the median of the distribution each time, as in all previously shown plots.
As discussed in Section 3 and summarized in Figs. 15, 14 and 20, our COSMOS analysis does not show any significant dependence of median galaxy size on large-scale environment, e.g., galaxy sizes have similar medians in the field and in the groups. We observe the same in the SDSS sample, comparing measurements in galaxy groups from Yang et al. (2007) with the field.
The top panels of figure 20 compares observations to model predictions with no accounting for our sample size and observational uncertainties. In this case, the Shankar et al. (2012) model predicts that our most massive halos should have mass-normalized radii of close to twice that of the field.
The middle and lower panels demonstrate that this strong difference is diluted when taking into account uncertainties intrinsic to our sample: the number of galaxies in each bin, the uncertainty due to photometric redshift estimation, and that on the estimate of the halo mass. In the middle panel, to properly compare observations with model predictions at each redshift and halo mass interval of interest, we perform
1000 Monte Carlo realizations in which we draw subsamples of galaxies from the Shankar et al. (2011) catalog with numbers equal to those in the SDSS and COSMOS samples.
Galaxies are selected to have $B/T>0.5$ and to share the
same stellar and halo mass intervals as in the observations, and sizes have been normalized to the local mass-size relation Shankar et al. (2011).
To each mock subsample we substitute 30% (when comparing to COSMOS, George et al. 2011) and
20% (when comparing to SDSS, Yang et al. 2007) of members with galaxies of the same stellar mass residing
in the halo bin with the lowest mass in order to mimic contamination from the field (e.g. because of photometric redshift uncertainties). As expected, this tends to reduce the increase of mean size with halo mass by up to 20%. For each Monte Carlo realization, we compute the mean and then extract, from the full distribution of means, the final mean and its 1-sigma error. In the lower panel, we add the further uncertainty due to halo mass estimation. We included in the simulations a Gaussian scatter of $0.3$ dex width to reproduce the average uncertainties in the halo mass (Yang et al. 2007; Leauthaud et al. 2010).
When taking into account all these sources of uncertainties (lower panels of figure 20), the model predicts a much smaller difference in size between the most massive halos and the field. Even if the trend observed in the upper panels still holds, when comparing our results obtained with COSMOS to the model, they are consistent at $1\sigma$, i.e., the mass-normalized radius does not depend on environment.
However, the larger SDSS sample shows that the model predicts sizes in groups to be about 1.5 times larger than those in the
field, at variance with the observations at more that 3 $\sigma$. In future work, we will investigate if this result is specific to the particular model we consider (Shankar et al., in preparation) and/or to the observational data we have used (Huertas–Company et al., in preparation).
4.1.3 The Size Growth of Lenticulars compared to model predictions
For what concerns lenticular galaxies (Fig. 21), model galaxies have been selected to have $0.4<B/T<0.7$.
For this range of $B/T$, disk instabilities also play
a non-negligible role in building bulges (see, e.g., Shankar et al. 2011, and references therein).
Despite the significantly different procedures
to grow bulges in the models, from mild instabilities (Guo et al. 2010) to violent ones (Bower et al. 2006), the result in the median size evolution is similar to the elliptical results.
It is interesting to note that, analogously to what inferred with respect to
more bulge dominated galaxies, hierarchical models are able to reproduce the trend
in size evolution for lenticulars below $M_{*}<10^{11}{\,\rm M_{\odot}}$, but they tend
to predict a shallower evolution above this mass.
4.2 Expansion
An alternative model
for efficiently puffing up sizes of massive, early-type
galaxies, considers the galaxy expansion consequent to significant mass loss
via quasar and/or stellar winds (Fan et al. 2008, 2010; Damjanov et al. 2009).
This model envisages that all massive ($M_{*}>3\times 10^{10}{\,\rm M_{\odot}}$) early-type galaxies forming at $z>1$
went through a rapid expulsion of large amounts of mass,
possibly caused by a powerful quasar wind, that caused the galaxy to expand.
Numerical experiments in favor of this physical scenario have
been performed by Ragone-Figueroa & Granato (2011), who showed that even in the presence
of large amounts of dark matter, massive galaxies can significantly
expand their sizes by a factor approximately proportional
to the fraction of mass loss.
Fan et al. (2008) predict that most of the size evolution for the massive
spheroidal galaxies should be delayed with respect to the peak of quasar activity by about 0.5-1 Gyr.
The model also predicts a milder and possibly longer size evolution for early-type galaxies with stellar
mass below $M_{*}<2\times 10^{10}{\,\rm M_{\odot}}$, for which
the dominant energy input in the interstellar medium comes from
supernovae explosions.
Clearly, the typical evolutionary timescales to fully evolve
a galaxy onto the local size-mass relation in the Fan et al. (2010)
model are in general quite shorter than the cosmological ones
required by a merger scenario. While a fast evolution
in the size growth of massive ellipticals is supported by our data,
their model predicts strong size evolution
mainly at $z>1$, while our data show strong size
evolution also at $z<1$, at least for massive ellipticals.
Fan et al. (2010) also point out that a fast size evolution necessarily should lead to
the co-existence of large and compact quiescent galaxies at
any redshift $z>1$, and thus a large dispersion up to a factor
of $<5-6$ in the size distribution at fixed stellar mass. The
dispersion should then significantly reduce below $z<1$, as
most of the galaxies should be already evolved and their formation rate, which parallels the one of quasars peaking at $z=2$ (Lapi et al. 2006), should progressively drop at late times (see their Fig. 1).
The dispersion in sizes for quiescent, massive early-type galaxies that we measure from COSMOS seems instead to be rather contained, within a factor of two at fixed redshift and stellar mass. We also note, however, that our lower mass,
quiescent ETG that have larger sizes might be, at least in part, be composed by
the population of high–redshift, already evolved early-type
galaxies predicted by Lapi et al. (2006).
The expansion model predicts that compact
galaxies should be relatively young at the time of observation, being close to their formation epoch because the size evolution occurs only $\sim 20$ Myr after the expulsion via quasar feedback (Ragone-Figueroa & Granato 2011). This
may be possibly at variance with the rather old ages that usually characterize
massive ellipticals (see also Trujillo et al. 2011 for similar considerations based on observations).
On the other hand, Ragone-Figueroa & Granato (2011) also found in their numerical tests that the galaxy mass profiles
should not change after the blow-out, if the mass loss is contained to a factor of two or so.
This might explain the nearly constancy with redshift of the Sérsic index in our sample.
Also, being galaxy expansion an in-situ physical mechanism, it should be largely independent
of environment, as suggested by our data (Figs. 15 and 20).
In summary, while merger models can explain most of the size growth size $z\sim 1$,
we find evidence that both proposed scenarios (mergers and mass loss
via quasar and/or stellar winds) suffer from some shortcomings
with respect to our results and more advances in modeling are clearly needed to deeply understand how the two different scenario shape the galaxy size evolution.
5 Summary and conclusions
We have studied a sample of 3146 group galaxies and 3760 field galaxies from the COSMOS survey, and selected 298 group and 384 field galaxies as passive galaxies (based on their M(NUV)-M(R) dust corrected rest-frame color) with $log(M/M_{\odot})>10.5$. We show, for the first time, how the mass-size relation and size growth depend on the detailed morphology of the quiescent population (mainly ellipticals and lenticulars) as well as on large-scale environment defined by the dark matter halo mass.
Our main results are:
1.
A detailed morphological dissection of the passive population up to $z\sim 1$ reveals that:
•
About $80\%$ of all passive galaxies have an early-type morphology at all stellar masses and at all redshifts from $z\sim 1$. The remaining $20\%$ are essentially early-type spirals.
•
Early–type galaxies are both ellipticals and S0s. At all redshifts up to $z\sim 1$, elliptical galaxies dominate the ETG population at masses $log(M/M_{\odot})>11-11.2$. At $z<0.5$, galaxies with masses $log(M/M_{\odot})<11$ are around half ellipticals and half lenticular, while lenticulars dominate the low–mass fractions at $z>0.5$. An important fraction of group and field low-mass passive galaxies in the redshift range $0.5<z<1$ are lenticular galaxies, e.g. have a disk component (see also Bundy et al. 2010, Mei et al. 2012)
Therefore, studying the population of passive galaxies as a whole mixes different morphological populations which do not necessarily share the same evolutions.
2.
When separating the ellipticals from the lenticulars, we show that galaxy size evolution strongly depends on mass range and ETG morphology:
•
Massive ellipticals ($log(M_{*}/M_{\odot})>11.2$) do experience a very strong size evolution from $z\sim 1$ to present ($r_{e}\propto(1+z)^{-0.98\pm 0.18}$). Even though the trend is somehow steeper than the one predicted by most of published semi-analytical models, minor dry mergers remain the most plausible explanation to the expansion. However, the Sérsic index of this population is not significantly evolving with time as expected from hierarchical models. We cannot exclude from our data that expansion via feedback might also play a role on the size evolution of this population.
•
Less massive ellipticals ($10.5<log(M_{*}/M_{\odot})<11$) do not evolve much from $z\sim 1$ (from $\sim 10\%$ to $\sim 30\%$ depending on which morphological classification we use). This behavior is well reproduced by most of current semi-analytic models from which the size growth is mainly due to major and minor mergers.
•
The evolution observed in the lenticular population does not change significantly with stellar mass and they do show a size growth of $55\%\pm 10\%$ since $z=1$.
3.
Finally, we studied environmental effects on the size evolution by dividing our sample into field and group galaxies:
•
We do not detect any significant evidence that the evolution of field and group galaxies ($13<log(M_{h}/M_{\odot})<14.2$) with stellar masses $log(M_{*}/M_{\odot})>11.2$ are different. This is observed both in the local Universe, when comparing SDSS groups from Yang et al. (2007) with field galaxies, and in our COSMOS group and field samples. This is also true for massive central ellipticals, where models clearly predict a major impact from mergers Shankar et al. (2011) .
This result is at variance with predictions from the standard hierarchical model (e.g., from Shankar et al. (2011) ), which predicts instead that the mass-normalized radius in our group mass range should be $\approx$ 2 times larger than in the field. When taking into account uncertainties due our sample size, photometric redshift estimates and halo masses, this prediction remains, although the effect is reduced where our galaxy sample size is limited. Model predictions are consistent with our COSMOS results at 1 $\sigma$, mainly because of the sample size. However, for our SDSS sample, the difference between the observed and predicted mass-normalized radii is at variance at $>3\sigma$. While the observations do not show any dependence of the mass-normalized radius with environment, the Shankar et al. (2012) predicts that galaxy mass-normalized radii over our group mass range should be $\sim 1.5$ larger than that in the field.
In future work, we will investigate the dependence of this result on the specific model (Shankar et al., in preparation) and on the properties of SDSS sample (Huertas–Comany et al., in preparation) used here.
•
BGGs and satellite galaxies of similar stellar mass ($log(M_{*}/M_{\odot})>11$) evolve in a similar way. This is in disagreement with hierarchical models, that also predict a difference between centrals and satellites living in halos with similar masses than our groups ($log(M_{*}/M_{\odot})\sim 13.4-13.6$). In fact, Shankar et al. (2011) has shown that galaxies residing at the centre of more massive haloes should, at fixed stellar mass, experience more mergers and thus be larger than their counterparts in less massive haloes.
Future work will involve a detailed detailed discussion of environmental effects on models (Shankar et al. 2012, in prep) as well as an extension to higher redshifts by studying the sizes of ETGs in distant clusters of galaxies (Delaye et al. 2012, in prep).
The authors thank Boris Haussler for his great help with galapagos and M.R. George for putting together the main catalog used in this paper. We are also thankful to P. Capak, O. Ilbert as well as to K. Bundy and A. Leauthaud for interesting discussions. FS acknowledges support from a Marie Curie grant.
References
Ascaso et al. (2011)
Ascaso, B., Aguerri, J. A. L., Varela, J., et al. 2011, ApJ, 726, 69
Baillard et al. (2011)
Baillard, A., Bertin, E., de Lapparent, V., et al. 2011, A&A, 532, A74
Barden et al. (2012)
Barden, M., Häußler, B., Peng, C. Y., McIntosh, D. H., &
Guo, Y. 2012, MNRAS, 422, 449
Bernardi (2009)
Bernardi, M. 2009, MNRAS, 395, 1491
Bernardi et al. (2010)
Bernardi, M., Shankar, F., Hyde, J. B., et al. 2010, MNRAS, 404, 2087
Bertin & Arnouts (1996)
Bertin, E. & Arnouts, S. 1996, A&As, 117, 393
Bezanson et al. (2009)
Bezanson, R., van Dokkum, P. G., Tal, T., et al. 2009, ApJ, 697, 1290
Bluck et al. (2011)
Bluck, A. F. L., Conselice, C. J., Buitrago, F., et al. 2011, ArXiv
e-prints
Bower et al. (2006)
Bower, R. G., Benson, A. J., Malbon, R., et al. 2006, MNRAS, 370, 645
Bruzual & Charlot (2003)
Bruzual, G. & Charlot, S. 2003, MNRAS, 344, 1000
Buitrago et al. (2008)
Buitrago, F., Trujillo, I., Conselice, C. J., et al. 2008, ApJl, 687,
L61
Buitrago et al. (2011)
Buitrago, F., Trujillo, I., Conselice, C. J., & Haeussler, B. 2011,
ArXiv e-prints
Bundy et al. (2006)
Bundy, K., Ellis, R. S., Conselice, C. J., et al. 2006, ApJ, 651, 120
Bundy et al. (2010)
Bundy, K., Scarlata, C., Carollo, C. M., et al. 2010, ApJ, 719, 1969
Cassata et al. (2011)
Cassata, P., Giavalisco, M., Guo, Y., et al. 2011, ApJ, 743, 96
Cimatti et al. (2012)
Cimatti, A., Nipoti, C., & Cassata, P. 2012, MNRAS, L424
Cooper et al. (2011a)
Cooper, M. C., Aird, J. A., Coil, A. L., et al. 2011a,
ApJs, 193, 14
Cooper et al. (2011b)
Cooper, M. C., Griffith, R. L., Newman, J. A., et al.
2011b, ArXiv e-prints
Daddi et al. (2005)
Daddi, E., Renzini, A., Pirzkal, N., et al. 2005, ApJ, 626, 680
Damjanov et al. (2011)
Damjanov, I., Abraham, R. G., Glazebrook, K., et al. 2011, ApJl, 739,
L44
Damjanov et al. (2009)
Damjanov, I., McCarthy, P. J., Abraham, R. G., et al. 2009, ApJ, 695,
101
Fan et al. (2010)
Fan, L., Lapi, A., Bressan, A., et al. 2010, ApJ, 718, 1460
Fan et al. (2008)
Fan, L., Lapi, A., De Zotti, G., & Danese, L. 2008, ApJl, 689, L101
Finoguenov et al. (2007)
Finoguenov, A., Guzzo, L., Hasinger, G., et al. 2007, ApJs, 172, 182
George et al. (2011)
George, M. R., Leauthaud, A., Bundy, K., et al. 2011, ArXiv e-prints
Giodini et al. (2012)
Giodini, S., Finoguenov, A., Pierini, D., et al. 2012, A&A, 538, A104
Grogin et al. (2011)
Grogin, N. A., Kocevski, D. D., Faber, S. M., et al. 2011, ArXiv
e-prints
Guo et al. (2010)
Guo, Q., White, S., Li, C., & Boylan-Kolchin, M. 2010, MNRAS, 404,
1111
Häussler et al. (2007)
Häussler, B., McIntosh, D. H., Barden, M., et al. 2007, ApJs, 172,
615
Hopkins et al. (2009)
Hopkins, P. F., Bundy, K., Murray, N., et al. 2009, MNRAS, 398, 898
Hopkins et al. (2006)
Hopkins, P. F., Hernquist, L., Cox, T. J., et al. 2006, ApJs, 163, 1
Huertas-Company et al. (2011)
Huertas-Company, M., Aguerri, J. A. L., Bernardi, M., Mei, S., &
Sánchez Almeida, J. 2011, A&A, 525, A157
Huertas-Company et al. (2010)
Huertas-Company, M., Aguerri, J. A. L., Tresse, L., et al. 2010, A&A,
515, A3
Huertas-Company et al. (2008)
Huertas-Company, M., Rouan, D., Tasca, L., Soucail, G., & Le
Fèvre, O. 2008, A&A, 478, 971
Huertas-Company et al. (2009)
Huertas-Company, M., Tasca, L., Rouan, D., et al. 2009, A&A, 497, 743
Ilbert et al. (2009)
Ilbert, O., Capak, P., Salvato, M., et al. 2009, ApJ, 690, 1236
Kannappan et al. (2009)
Kannappan, S. J., Guie, J. M., & Baker, A. J. 2009, AJ, 138, 579
Khochfar & Silk (2006)
Khochfar, S. & Silk, J. 2006, ApJL, 648, L21
Koekemoer et al. (2007)
Koekemoer, A. M., Aussel, H., Calzetti, D., et al. 2007, ApJs, 172, 196
Koekemoer et al. (2011)
Koekemoer, A. M., Faber, S. M., Ferguson, H. C., et al. 2011, ApjS,
197, 36
Lapi et al. (2006)
Lapi, A., Shankar, F., Mao, J., et al. 2006, ApJ, 650, 42
Leauthaud et al. (2010)
Leauthaud, A., Finoguenov, A., Kneib, J.-P., et al. 2010, ApJ, 709, 97
López-Sanjuan et al. (2011)
López-Sanjuan, C., Le Fèvre, O., de Ravel, L., et al. 2011,
A&A, 530, A20
López-Sanjuan et al. (2012)
López-Sanjuan, C., Le Fèvre, O., Ilbert, O., et al. 2012, ArXiv
e-prints
Maltby et al. (2010)
Maltby, D. T., Aragón-Salamanca, A., Gray, M. E., et al. 2010,
MNRAS, 402, 282
Mancini et al. (2010)
Mancini, C., Daddi, E., Renzini, A., et al. 2010, MNRAS, 401, 933
Martinez-Manso et al. (2011)
Martinez-Manso, J., Guzman, R., Barro, G., et al. 2011, ApJl, 738, L22
McLure et al. (2012)
McLure, R. J., Pearce, H. J., Dunlop, J. S., et al. 2012, ArXiv
e-prints
Mei et al. (2009)
Mei, S., Holden, B. P., Blakeslee, J. P., et al. 2009, ApJ, 690, 42
Mei et al. (2012)
Mei, S., Stanford, S. A., Holden, B. P., et al. 2012, ApJ, 754, 141
Meneux et al. (2009)
Meneux, B., Guzzo, L., de la Torre, S., et al. 2009, A&A, 505, 463
Naab et al. (2009)
Naab, T., Johansson, P. H., & Ostriker, J. P. 2009, ApJl, 699, L178
Nair & Abraham (2010)
Nair, P. B. & Abraham, R. G. 2010, ApJs, 186, 427
Newman et al. (2012)
Newman, A. B., Ellis, R. S., Bundy, K., & Treu, T. 2012, ApJ, 746, 162
Nipoti et al. (2012)
Nipoti, C., Treu, T., Leauthaud, A., et al. 2012, MNRAS, 2680
Oser et al. (2012)
Oser, L., Naab, T., Ostriker, J. P., & Johansson, P. H. 2012, ApJ,
744, 63
Papovich et al. (2012)
Papovich, C., Bassett, R., Lotz, J. M., et al. 2012, ApJ, 750, 93
Patel et al. (2012)
Patel, S. G., van Dokkum, P. G., Franx, M., et al. 2012, ArXiv e-prints
Peng et al. (2002)
Peng, C. Y., Ho, L. C., Impey, C. D., & Rix, H.-W. 2002, AJ, 124, 266
Postman et al. (2005)
Postman, M., Franx, M., Cross, N. J. G., et al. 2005, ApJ, 623, 721
Pozzetti et al. (2010)
Pozzetti, L., Bolzonella, M., Zucca, E., et al. 2010, A&A, 523, A13
Ragone-Figueroa & Granato (2011)
Ragone-Figueroa, C. & Granato, G. L. 2011, MNRAS, 414, 3690
Raichoor et al. (2011)
Raichoor, A., Mei, S., Nakata, F., et al. 2011, ApJ, 732, 12
Raichoor et al. (2012)
Raichoor, A., Mei, S., Stanford, S. A., et al. 2012, ApJ, 745, 130
Rettura et al. (2010)
Rettura, A., Rosati, P., Nonino, M., et al. 2010, ApJ, 709, 512
Rhodes et al. (2007)
Rhodes, J. D., Massey, R. J., Albert, J., et al. 2007, ApJs, 172, 203
Ryan et al. (2012)
Ryan, Jr., R. E., McCarthy, P. J., Cohen, S. H., et al. 2012, ApJ, 749,
53
Saracco et al. (2011)
Saracco, P., Longhetti, M., & Gargiulo, A. 2011, MNRAS, 412, 2707
Sersic (1968)
Sersic, J. L. 1968, Atlas de galaxias australes
Shankar et al. (2011)
Shankar, F., Marulli, F., Bernardi, M., et al. 2011, ArXiv e-prints
Shankar et al. (2012)
Shankar, F., Marulli, F., Mathur, S., Bernardi, M., & Bournaud, F.
2012, A&A, 540, A23
Simard et al. (2011)
Simard, L., Mendel, J. T., Patton, D. R., Ellison, S. L., &
McConnachie, A. W. 2011, ApJs, 196, 11
Springel et al. (2005)
Springel, V., White, S. D. M., Jenkins, A., et al. 2005, Nature, 435,
629
Stott et al. (2011)
Stott, J. P., Collins, C. A., Burke, C., Hamilton-Morris, V., &
Smith, G. P. 2011, MNRAS, 414, 445
Tasca et al. (2009)
Tasca, L. A. M., Kneib, J.-P., Iovino, A., et al. 2009, A&A, 503, 379
Tiret et al. (2011)
Tiret, O., Salucci, P., Bernardi, M., Maraston, C., & Pforr, J.
2011, MNRAS, 411, 1435
Trujillo et al. (2012)
Trujillo, I., Carrasco, E. R., & Ferré-Mateu, A. 2012, ApJ, 751, 45
Trujillo et al. (2009)
Trujillo, I., Cenarro, A. J., de Lorenzo-Cáceres, A., et al. 2009,
ApJL, 692, L118
Trujillo et al. (2011)
Trujillo, I., Ferreras, I., & de La Rosa, I. G. 2011, MNRAS, 415, 3903
Trujillo et al. (2006)
Trujillo, I., Förster Schreiber, N. M., Rudnick, G., et al. 2006,
ApJ, 650, 18
Valentinuzzi et al. (2010)
Valentinuzzi, T., Fritz, J., Poggianti, B. M., et al. 2010, ApJ, 712,
226
van de Sande et al. (2011)
van de Sande, J., Kriek, M., Franx, M., et al. 2011, ApJl, 736, L9
van der Wel et al. (2008)
van der Wel, A., Holden, B. P., Zirm, A. W., et al. 2008, ApJ, 688, 48
van der Wel et al. (2011)
van der Wel, A., Rix, H.-W., Wuyts, S., et al. 2011, ApJ, 730, 38
van Dokkum et al. (2011)
van Dokkum, P. G., Brammer, G., Fumagalli, M., et al. 2011, ApJl, 743,
L15
van Dokkum et al. (2008)
van Dokkum, P. G., Franx, M., Kriek, M., et al. 2008, ApJl, 677, L5
Weinmann et al. (2009)
Weinmann, S. M., Kauffmann, G., van den Bosch, F. C., et al. 2009,
MNRAS, 394, 1213
Whitaker et al. (2012)
Whitaker, K. E., Kriek, M., van Dokkum, P. G., et al. 2012, ApJ, 745,
179
Williams et al. (2010)
Williams, R. J., Quadri, R. F., Franx, M., et al. 2010, ApJ, 713, 738
Yang et al. (2007)
Yang, X., Mo, H. J., van den Bosch, F. C., et al. 2007, ApJ, 671, 153
Appendix A Visual morphological classifications
Some results presented in this paper are based on an automated morphological classification of galaxies at high redshift. Despite of the extensive tests performed to properly assess the accuracy of our classification, the separation of ellipticals from lenticulars might still remain a challenge.
In this appendix, we want to make sure that our results do not change while using an alternative classical visual classification. Two of us (SM and MHC) performed independent detailed visual classifications of all passive galaxies previously selected as early-type by galSVM (we assume that the separation between early and late is good enough). SM classified galaxies into three morphological classes (Ellipticals, Lenticulars and early-spirals) while MHC followed a slightly different criterion and separated galaxies into two classes (disk, no disk).
In Figs. 23 and 24 we reproduce figures 12 and 13 of the main text using the 3 morphological classifications (2 visual and 1 automated). While there are some differences between the three estimates, the main trends discussed in the text do hold and the results are consistent within the error bars. This confirms that our morphological classification is robust and that the major results presented in the paper are not biased by the automated classification. Most of the differences between visual and automated classifications are seen in the low mass elliptical population for which visual classifications show a steeper evolution than that estimated while using an automated classification. When the evolution of low mass ellipticals is compared to their massive counterpart, it is still less steep as discussed in section 3.2.1 and shown in table 3.
Appendix B Environment and galaxy selection
We include here the equivalent of Fig. 16 but only with galaxies having a probability $P_{MEM}>0.8$ to be a group member and being at a distance $d<0.5\times R_{200}$ of the cluster center. Our results do not change and remain basically the same as the ones shown in table 2. The size distribution in mass and redshift bins does not depend on environment. |
[
Julia Westermayr
[
Philipp Marquetand
[
philipp.marquetand@univie.ac.at
Abstract
Electronically excited states of molecules are at the heart of photochemistry, photophysics, as well as photobiology and also play a role in material science. Their theoretical description requires highly accurate quantum chemical calculations, which are computationally expensive. In this review, we focus on how machine learning is employed not only to speed up such excited-state simulations but also how this branch of artificial intelligence can be used to advance this exciting research field in all its aspects. Discussed applications of machine learning for excited states include excited-state dynamics simulations, static calculations of absorption spectra, as well as many others. In order to put these studies into context, we discuss the promises and pitfalls of the involved machine learning techniques. Since the latter are mostly based on quantum chemistry calculations, we also provide a short introduction into excited-state electronic structure methods, approaches for nonadiabatic dynamics simulations and describe tricks and problems when using them in machine learning for excited states of molecules.
keywords: machine learning, deep learning, kernel ridge regression, excited states, nonadiabatic dynamics, nonadiabatic couplings, spin-orbit couplings, intersystem crossing, internal conversion, excited-state dynamics, surface hopping, photochemistry, spectra
Vienna]Institute of Theoretical Chemistry, Faculty of Chemistry, University of Vienna, Währinger Str. 17, 1090 Vienna
Vienna]Institute of Theoretical Chemistry, Faculty of Chemistry, University of Vienna, Währinger Str. 17, 1090 Vienna
\alsoaffiliation[ViRAPID]Vienna Research Platform on Accelerating Photoreaction Discovery, University of Vienna, Währinger Str. 17, 1090 Vienna, Austria.
\alsoaffiliation[DS]Data Science @ Uni Vienna, University of Vienna, Währinger Str. 29, 1090 Vienna, Austria.
Title]
Machine learning for electronically excited states of molecules
\abbreviationsML,QC,SchNarc
Contents
1 Introduction
1.1 From Foundations to Applications
1.2 Scope and Philosophy of this Review
2 General Background: From the Ground State to the Excited States
3 Quantum Chemical Theory and Methods
3.1 Electronic Structure Theory for Excited States
3.1.1 Wave Function Theory (WFT)
3.1.2 Density Functional Theory
3.2 Bases
3.2.1 Adiabatic (Spin-Diabatic) Basis
3.2.2 Diabatic Basis
3.2.3 Diagonal Basis
3.3 Excited-State Dynamics Simulations
3.3.1 Quantum Nuclear Dynamics
3.3.2 Mixed Quantum-Classical Molecular Dynamics
3.4 Dipole Moments and Spectra
4 Data Sets for Excited States
4.1 Choosing the Right Reference Method for Excited-State Data
4.2 Phase of the Wave Function
4.2.1 Phase Correction of Adiabatic Data
4.2.2 ML-Based Internal Phase Correction
4.3 Training Set Generation
4.3.1 Basic Sampling Techniques and Existing Databases
4.3.2 Active Learning
5 ML Models
5.1 ML Models: Type of Regressor
5.2 Descriptors and Features
6 Application of ML for Excited States
6.1 Parameters for Quantum Chemistry
6.2 ML of Primary Outputs
6.3 ML of Secondary Outputs
6.3.1 ML in the Diabatic Basis
6.3.2 ML in the Adiabatic Basis
6.4 ML of Tertiary Outputs
6.5 ML-Assisted Analysis
7 Conclusion and Future Perspectives
1 Introduction
1.1 From Foundations to Applications
In recent years, machine learning (ML) has become a pioneering field of research and has an increasing influence on our daily lives. Today it is a component of almost all applications we use. For example, when we talk to Siri or Alexa, we interact with a voice assistant and make use of natural language processing 1, 2. ML is applied for refugee integration 3, for playing board games 4, in medicine 5, for example, for image recognition 6 or for autonomous driving 7. A short historical overview over general ML is provided in ref 8.
Recently, ML has also gained increasing interest in the field of quantum chemistry 9, 10.
The power of (big) data-driven science is even seen as the "fourth paradigm of science" 11, which has the potential to accelerate and enable quantum chemical simulations that were considered unfeasible just a few years ago. In general, the field of ML in quantum chemistry is progressing faster and faster. In this review, we focus on an emerging part of this field, namely ML for electronically excited states. In doing so, we concentrate on singlet and triplet states of molecular systems, since almost all existing approaches of ML for the excited states focus on singlet states and only a few studies consider triplet states. 12, 13, 14, 15
We note that electron detachment or uptake further leads to doublet and quartet states, and even higher spin multiplicities, such as quintets, sextets, etc. are common in transition metal complexes, where an important task is to identify which multiplicity yields the lowest energy and is thus the ground state 15. refs 16, 17, 18, 19
give a good overview of such processes.
The theoretical study of the excited states of molecules is crucial to complement experiments and to shed light on many fundamental processes of life and nature 20. For example, photosynthesis, human vision, photovoltaics or photodamage of biologically relevant molecules are a results of light-induced reactions 21, 22, 23, 24, 25, 26, 27, 28, 20, 29, 30, 31, 32, 33, 34, 35, 36, 37. Experimental techniques like UV/visible spectroscopy or photoionization spectroscopy 38, 39, 40, 41, 42, 43, 44, 45 lack the ability to directly describe the exact electronic mechanisms of photo-induced reactions. The theoretical simulation of the corresponding experiments can go hand-in-hand with experimental results and can provide the missing details of photodamage and -stability of molecules 20, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 27, 35, 44, 60, 61, 62, 63, 64, 65, 66, 67.
However, the computation of the excited states is highly complex, costly, and often necessitates expert knowledge 68.
As ML models have only recently been applied in the field of photochemistry, keeping track of the approaches is still possible and this field is still in its initial stage.
Due to the multi-faceted photochemistry of molecular systems, ML models can target this research field in many different ways, which are summarized in Figure 1. For example, the choice of relevant molecular orbitals for active space selections can be assisted with ML. 70 The fundamentals of quantum chemistry, e.g., to obtain an optimal solution to the Schrödinger equation or Density Functional Theory, can be central ML applications. For the ground state, ML approximations to the molecular wave function 71, 72, 73, 74, 75, 76, 77, 78, 79 or the density (functional) of a system exist. 80, 81, 82, 83, 69, 84, 85, 86, 87, 79, 88 Obtaining a molecular wave function from ML can be seen as the most powerful approach in many perspectives, as any property we wish to know could be derived from it. Unfortunately, such models for the excited states are lacking and have yet only been investigated for a one-dimensional system 89, leaving much room for improvement.
Most ML studies instead focus on predicting the output of a quantum chemical calculation, the so-called "secondary-output" 69. Hence they fit a manifold of energetic states of different spin multiplicities, their derivatives and properties thereof. With respect to different spin states of molecular systems only a few studies exist, which predict spins of transition metal complexes 15 or singlet and triplet energies of carbenes 12 of different composition or focus on the conformational changes within one molecular system 13, 90, 91 for the sake of improving molecular dynamics (MD) simulations. The energies of a system in combination with its properties, i.e., the derivatives, the coupling values between them, and the permanent and transition dipole moments 92, 13, 93, 14, 90, 94, 95, 96, 91, 97, can be used for MD simulations to study the temporal evolution of a system in the ground-state 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135 and in the excited states. 136, 91, 137, 138, 139, 140, 141, 142, 92, 14, 130, 143, 144, 145, 146, 90, 143, 147, 147, 148, 149, 13
With energies and different properties, tertiary outputs can be computed, such as absorption, ionization or X-ray spectra, 150, 151, 152, 153 gaps between HOMO (highest occupied molecular orbital) and LUMO (lowest occupied MO) or vertical excitation energies. 154, 155, 156, 157
In addition, quantum chemical outputs can also be analyzed or fitted in a direct way, e.g., reaction kinetics as results of dynamics simulations can be mapped to a set of molecular geometries and can be predicted with ML models. 158 Excitation energy transfer properties can be learned, 159, 160, and structure-property correlations can be explored to design materials with specific properties. 76, 161, 162, 163, 164, 165, 131, 166, 167, 168, 169, 170, 16, 152
1.2 Scope and Philosophy of this Review
ML for the excited states is developing at a slower pace than the exploding field of ML for the electronic ground state 171, 172, 173, 168. The reason is in our opinion mainly a result of the complexity and high expenses of the underlying reference calculations and the associated complexity of the corresponding ML models. Simulation techniques to understand the excited-state processes are not yet viable for many applications at an acceptable cost and accuracy. Therefore, within this review we also want to highlight the existing problems of quantum chemical approaches that might be solvable with ML and put emphasis on identifying challenges and limitations that hamper the application of ML for the excited states.
The young age of this research field leaves much room for improvement and new methods.
This review is structured as follows:
(1) Throughout this review, we will start (non-exhaustively) discussing ground state processes, since they are inherently linked to the excited state processes and should also be considered here.
We will therefore start by discussing the differences between the ground-state potential energy hypersurfaces (PESs) and the excited-state PESs and will also emphasize the difference in their properties in section 2.
(2) Section 3.1 gives an overview of the theoretical methods that can be used to describe the excited states of molecules. In the forthcoming discussion, we will describe different reference methods with a view to their application in time-dependent simulations, namely MD simulations 172, 27. It is worth mentioning, that unlike for the ground state, where a lot of different methods can provide reliable reference computations for training, choosing a proper quantum chemistry method for the treatment of excited states is a challenge on its own. Many methods require expert knowledge, prohibiting their use further 37, 174. In addition, not any method can provide the necessary properties for any type of application.
Subsequently, we aim to review the different flavours of excited-state MD simulations with focus on those methods that have been enhanced with ML models lately.
(3) After having provided the basic theoretical background, we will discuss how to generate a comprehensive, yet full-fledged training set for the excited states from the quantum chemistry data. We will summarize the existing approaches that are applied to create a comprehensive training set and put emphasis on the bottlenecks of existing methods that can limit also the application of ML. This will provide the reader with the knowledge about starting points for future research questions and clarify where method development is needed.
It further provides the basis for the discussion of ML models for the excited states of molecular systems.
(4) A summary of state-of-the-art ML methods for photochemistry follows. We will differentiate between single-state and multi-state ML models and single-property and multi-property ML models 93. As mentioned before, ML models can tackle a quantum chemical calculation in many different ways, see Figure 1.
The different ML models will be classified in the ways they enhance quantum chemical simulations. Most approaches aim at providing an ML-based force field for the excited states, so most focus will be put on this topic.
At last, the prospects of ML models to revolutionize this field of research and future avenues for ML will be highlighted.
Noteworthy, we focus on the excited states of molecules, as the excited electronic states in the condensed phase are challenging to fit and are thus often not explicitly considered in conventional approaches 175, 176, 177, 178, 179, 180.
In solid state physics for example, the electronic states are usually treated as continua. The density of states at the Fermi level, 181 band gaps, 182, 183, 184 and electronic friction tensors 123, 185, 186 have been described with ML models up to date and especially the electronic friction tensor is useful to study the indirect effects of electronic excitations in materials. 187, 188, 189, 190, 191, 192 Electron transfer processes as a result of electron-hole-pair excitations can be further investigated along with multi-quantum vibrational transitions by discretizing the continuum of electronic states and fitting them (often manually) to reproduce experimental or quantum chemical data in a model Hamiltonian. 193, 194, 195, 178, 196, 197, 198 Yet, to the best of our knowledge, the excited electronic states in the condensed phase have not been fitted with ML. A recent review on reactive and inelastic scattering processes and the use of ML for quantum dynamics reactions in the gas phase and at a gas-phase interface can be found in ref 199.
Besides the electronic excitations that take place in molecules after light excitation, ML models have successfully entered research fields, which focus on other types of excitations as well. Those are for example vibrational or rotational excitations giving rise to Raman spectra or Infrared spectra, 43, 200, 201, 202, 109, 203, 204 nuclear magnetic resonance, 205 or magnetism 206, 207, which we will also not consider in this review.
2 General Background: From the Ground State to the Excited States
Figure 2 gives an overview of the excited state processes that will be discussed within this review.
It shows a schematic one-dimensional representation of the potential energy curves for the ground and excited states as a function of molecular coordinates. Figure 2 illustrates that the ground state potential energy curve, given by a dark-blue solid line, is mostly a smooth function of the reaction coordinate and gives information about several local minima.
In the ground state, many methods exist to describe the physico-chemical properties of molecules and materials reasonably well, ranging from small systems up to proteins, DNA or nanoparticles. For small system sizes, highly accurate ab-initio methods can be applied, while more crude approximations have to be used for larger systems. The unfavorable scaling of many quantum chemical methods with the size of system under investigation requires this compromise between accuracy and system size. Crude approximations for systems that are larger than several 100s of atoms become inevitable 37, 24, 20.
The chemistry we are interested in, however, is not static, but rather depends to a large extent on the changes that matter undergoes. In this regard, it is more intuitive to study the temporal evolution of a system. Much effort has been devoted to develop methods to study the temporal evolution of matter in the ground state potential. As an example, physical functions can be obtained with conventional force fields, such as AMBER 208, CHARMM 209 or GROMOS 210, 211.
The first ones already date back to the 1940s-1950s. Such force fields enable the study of large and complex systems, protein dynamics or binding-free energies on time scales up to a couple of nanoseconds 212, 213, 214, 175, 215, 216, 217, 218, 219, 220.
However, their applicability is restricted by the limited accuracy and inability to describe bond formation and breaking. Novel approaches, such as reactive force fields exist, but have not yet entered the mainstream and still face the problem of generally low accuracy. 221
The accuracy of ab-initio methods can be combined with the efficiency of conventional force fields with ML models. The latter have shown to advance simulations in the ground state considerably and allow for the fitting of almost any input-output relation. 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 222, 172, 135 Accurate and reactive PESs of molecules in the ground state can be obtained with a comprehensive reference data set, which contains the energies, forces and ground-state properties of a system under investigation. Proper training of an ML model then guarantees that the accuracy of the reference method is retained, while inferences can be made much faster. In this way, they allow for a description of reactions and can overcome the limitations of existing force fields. 133, 223, 224, 225, 226, 227, 171
Regarding the excited states, processes become much more complex and the computation of excited state PESs is far more difficult than the computation of the ground state PESs. As can be seen in Figure 2, a lot of different classes of excited states, e.g. singlet states as shown by continuous blueish lines or triplet states as shown by dashed reddish lines, have to be accounted for, which are characterized by several transition states, local minima, and crossing points. This complexity makes a separate treatment of each electronically excited state inaccurate and leads to further challenges that prohibit the straight-forward and large-scale use of many existing quantum chemical methods and consequently also existing ML models for the ground state.
Additionally, computations of the excited states suffer from being generally less efficient. To name only one central problem: The larger the system becomes, the closer the electronic states lie in energy, and the more excited-state processes can usually take place. The necessary consideration of an increasing number of excited states increases the already substantial computational expenses even more and restricts the use of accurate methods to systems containing only a few dozens of atoms in a reasonable amount of time with current computers. This increasing complexity makes not only the reference computations, but also the application of ML models for the excited states more complicated than for the ground state. At the same time, the application of ML models for the excited states might also be more promising, because higher speed-ups can be achieved.
For the excited states, methods similar to force fields, like the linear vibronic coupling (LVC) approach 228, 229, are usually limited to small regions of conformational space and restricted to a single molecule. General force fields that are valid for different molecules in the excited states do not exist. Also the ML analogue, so-called transferable ML models, to fit the excited state PESs of molecules throughout chemical compound space are unavailable up to date. Nevertheless, it is out of question that an ML model, which is capable of describing the photochemistry of several different molecular systems, e.g., different amino acids or DNA bases of different sizes, is highly desirable. A lot remains to be done in order to achieve this goal and yet, to the best of our knowledge, no more than a maximum of about 20 atoms and 3 electronic states with a distinct multiplicity have been fitted accurately with ML models 94, 136, 91, 137, 138, 139, 140, 141, 142, 92, 14, 130, 143, 144, 144, 145, 146, 90, 143, 147, 147, 148, 149, 13.
Whether or not the excited states of a molecular system become populated depends on the ability of a molecule to absorb energy in the form of light, or more generally, electromagnetic radiation of a given wavelength.
Usually, the so-called resonance condition has to be fulfilled, i.e., the energy gap between two electronic states has to be equivalent to the photon energy of the incident light. Note however that also multi-photon processes can occur, where several photons have to be absorbed at once to bridge the energy difference between two electronic states 230, 231, 232
Further, the absorption of light does not only provide access to one, but most often to a manifold of energetically close-lying states. The number of states that can be excited is related to the range of photon energies that is contained in the electromagnetic radiation. This energy range is inversely proportional to the duration of the electric field, e.g., of a laser pulse, due to the Fourier relation of energy and time 233.
However, the energy range of the photons and the energy difference between the electronic states are not the only factors influencing the absorption of light, which gives rise to questions like:
Is the molecule able to absorb light of a considered wavelength? Which of the excited states is populated with the highest probability?
An answer to these questions can be obtained from an analysis of the oscillator strength. In order to make an electronic transition possible, an oscillating dipole must be induced as a result of the interaction of the molecule with light. The oscillator strength, $f^{osc}_{ij}$, between two electronic states, i and j, is proportional in atomic units (a.u.) to the respective transition dipole moment, $\mu{ij}$, and the respective energy difference, $\Delta E_{ij}$: 234
$$f^{osc}_{ij}=\frac{2}{3}\Delta E_{ij}\mid\mu_{ij}\mid^{2}.$$
(1)
If the transition dipole moment between two states is zero, no transition is allowed. The reasons can be that a change of the electronic spin would be required, and the transition is thus spin forbidden. Another reason can be the molecular symmetry, leading to symmetry forbidden transitions. The latter are common in molecules that carry an inversion centre and transitions that conserve parity are forbidden 235.
An energetic state is called dark, if the transition dipole moment is very small or zero. In contrast, a state is called bright, if the transition dipole moment is large. Most often, studies that target the photochemistry of molecules focus on excitation to the lowest brightest singlet state, i.e., the state that absorbs most of the incident energy.
After an excitation process, the molecule is considered to move on the excited-state PESs and is expected to undergo further conversions. The excess of energy a molecule carries – as a result of the initial absorption of energy – is most often converted into heat, light, such as fluorescence or phosphorescence, or into chemical energy. If the molecule returns to its original state, then the molecule is photostable. Otherwise, either photodamage, such as decomposition, or useful photochemical reactions including bond breaking/formation occur. In all cases, heat or light can be emitted, which can also be harnessed in light-emission applications 27, 236, 237, 238.
With respect to photo-stability, ultrafast transitions, in the range of femto- to picoseconds (10${}^{-15}$–10${}^{-12}$ seconds) take place and lead the molecule back to the ground state. This means, the electronic energy is converted into vibrations of the molecule and the molecule is termed hot. This heat is usually dissipated into the environment, a procedure that is often neglected in excited-state simulations due to the cost of describing surrounding molecules.
Radiationless transitions from one electronic state to another take place in so-called critical regions of the PESs. As the name already suggests, critical regions are crucial for the dynamics of a molecule, but are also challenging to model accurately. A transition from one state to another that conserves the spin-multiplicity is called internal conversion. Furthermore, states of different spin-multiplicities may be accessible via intersystem crossing.
The critical points, where transitions are most likely to occur, are called conical intersections and are illustrated in Figure 2.
At these crossing points, PESs computed with quantum chemistry can show discontinuities. These discontinuities can occur also in other excited-state properties and pose an additional challenge for an ML model when fitting excited-state quantities.
In addition to the aforementioned complications of treating a manifold of excited states, also the probability of a radiationless transition between them has to be computed somehow. This probability is usually determined by couplings between two approaching PESs. Between states of the same spin multiplicity, nonadiabatic couplings (NACs) arise, and spin-orbit couplings (SOCs) give rise to the transition probability between states of different spin multiplicities. These couplings are intimately linked to the excited-state PESs and therefore should also be considered with ML. However, only a handful of publications describe couplings with ML, 138, 94, 92, 13, 93, 90, 147, 143, 144, 239 which highlights the difficulty of providing the necessary reference data as well as the challenges of accurately fitting them. New methods are constantly needed to further enhance this exciting research field.
3 Quantum Chemical Theory and Methods
In this section, we present some key aspects of quantum theory for excited states because (i) the outcome of the corresponding calculations serve as training data for ML and (ii) to clarify the employed nomenclature. We put special emphasis on describing the differences of excited-state computations to computations in the ground state and the challenges that arise due to the treatment of a manifold of excited states. These challenges also point at issues that are problematic for ML. These explanations will provide the groundwork to evaluate different quantum chemical methods for their use to generate a training set for ML and to use it for different types of applications, such as excited-state MD simulations. Naturally, we can only provide a general idea of this field and refer the interested reader to pertinent textbooks and reviews, such as Refs. 37, 29, 240, 241, 242, 36, 26, 243, 244, 245, 246, 247, 248.
In order to follow a consistent notation within this review, we try to explain all basic concepts with notations that are frequently used in literature. Currently, a zoo of different notations for the same property can be found. For example, the NACs, or derivative couplings, are sometimes referred to as so-called interstate couplings, i.e. couplings between two states multiplied with the corresponding energy gap between those two states 142, while in other works interstate couplings refer to off-diagonal elements of the Hamiltonian in another basis, where the potential energies are no eigenvalues of the electronic Schrödinger equation. We want to avoid a confusion of the different notations and thus provide a consistent definition below. For the excited states, a number of different electronic states is required. Throughout this review, we adopt the following labelling convention for different electronic states: The lower case Latin letters, $i$, $j$, etc. will be used to denote different electronic states. The abbreviations $N_{S}$, $N_{M}$, and $N_{A}$ will indicate the number of states, molecules and atoms, respectively.
The foundation for the following sections is a separation of electronic and nuclear degrees of freedom, which is based on the work of Born and Oppenheimer 249. However, the famous Born-Oppenheimer approximation is later on (partly) lifted and the coupling between electrons and nuclei is taken into account in nonadiabatic dynamics simulations.
3.1 Electronic Structure Theory for Excited States
The main goal when carrying out an electronic structure calculation is usually to compute the potential energy and other physico-chemical properties of a compound.
We distinguish between two overarching theories to achieve this goal: Wave Function Theory (WFT) and Density Functional Theory (DFT) – as outlined, e.g., by Kohn in his Nobel lecture 250.
The basis of WFT, as for any electronic structure calculation, is the electronic Schrödinger equation 251, 252 with the electronic Hamilton operator, $\hat{H}_{el}$, and the N-electron wave function $\Psi_{i}(\mathbf{R},\mathbf{r})$ of electronic state i, which is dependent on the electronic coordinates $\mathbf{r}$ and parametrically dependent on the nuclear coordinates, $\mathbf{R}$:
$$\hat{H_{el}}(\mathbf{R},\mathbf{r})\mid\Psi_{i}(\mathbf{R},\mathbf{r})\rangle=%
E_{i}\mid\Psi_{i}(\mathbf{R},\mathbf{r})\rangle.$$
(2)
From the wave function, the eigenvector of this eigenvalue equation, any property of the system under investigation can be derived.
How to solve the electronic Schrödinger equation exactly to obtain the potential energy of an electronic state i, $E_{i}$, is known in theory. However, from a practical point of view, the computation is infeasible for molecules that are more complex than for example H${}_{2}$, He${}_{2}^{+}$, and similar systems. 253 In order to make the computation of larger and more complex systems viable, approximated wave functions are introduced.
In contrast to WFT, DFT reformulates the energy of a system in terms of the ground state electron density rather than the N-electron wave function and the energy is expressed as a functional thereof. The advantage of DFT over WFT is a rather high accuracy for a rather low computational cost. If DFT is applied properly, it is considered as one of the most efficient ways to obtain reliable and reasonably accurate results of molecules up to 100s of atoms. In solid state physics, DFT is even the workhorse of most studies aiming to describe ground state properties 254. However, the problem is that the equations to be solved are unknown. The missing piece is the exact exchange-correlation functional of a system. Up to date, researchers have come up with many different approximations to this functional that can be used to treat specific problems, but a universal functional capable of describing different problems equally accurately has not yet been found. Moreover, there is no systematic way to improve a density functional. The results obtained with DFT therefore critically depend on the choice of the functional. 255, 253
In the following sections, we will describe both theories in the light of excited states of molecules. We will start to cover ab-initio methods, which means that they are derived from first principles without parametrization.
3.1.1 Wave Function Theory (WFT)
The basis of all discussed ab-initio methods is the Hartree-Fock method. The N-electron wave function is represented by a single Slater Determinant, $\phi_{0}$,
which makes N coupled one-electron problems out of the N-body problem. This Slater determinant is the anti-symmetric product of one-electron wave functions, the spin orbitals, which can be atomic, molecular or crystal orbitals, depending on the system. In the case of molecular (or also crystal) orbitals, they are usually expanded as a linear combination of atomic orbitals, where the expansion coefficients are optimized during the calculation. In order to do so efficiently, the atomic orbitals are themselves expanded with the help of a basis set. The N-electron wave function is therefore obtained as a double expansion. Two approximations are applied, which is the use of a finite basis set to represent the atomic orbitals and in turn also the molecular orbitals on the one hand and the use of a single Slater Determinant on the other hand. This usually gives a poor description of a system under investigation, due to a lack of electronic correlation.
Electronic correlation describes how much the motion of an electron is influenced by all other electrons. Since the Hartree-Fock method can be seen as a mean-field theory, where an electron ”feels” only the average of the other electrons, correlation is quantified by the correlation energy, which is the difference between the Hartree-Fock energy and the exact energy of a system.
Unsurprisingly, all further discussed quantum chemical methods aim at improving the Hartree-Fock method. They can be seen as different flavors of the same solution to the problem: They all include more determinants in one way or another. Accordingly, the wave function is expanded as a linear combination of determinants, where a determinant consists of molecular orbitals, which are expanded in atomic orbitals. This ansatz contains two types of coefficients that can be optimized, the ones for the determinants and the ones yielding the molecular orbitals. If the latter are kept the same for different determinants, we speak of a single-reference wave function. If both types of coefficients are adapted, we speak of a multi-reference wave function. Similarly, the electron correlation is also divided into two parts, termed dynamic correlation and static correlation. Single-reference methods improve on the dynamic correlation, while a multi-reference wave function allows for static correlation. However, the separation is not so strict, as can be seen by the following fact: Both the aforementioned single-reference variant and the multi-reference variant become equivalent when including an infinite number of terms and deliver the exact solution to the Schrödinger equation if also an infinite basis set is used.
Configuration Interaction
In the case of single-reference methods, the orbitals obtained from the reference calculation (usually Hartree-Fock) are kept fixed. Since usually more orbitals than the number of electrons in the system are calculated, the possibility of constructing different Slater determinants from these orbitals exist, which can be used for expanding the actual wave function: 256, 257
$$\mid\Psi_{i}\rangle=\sum_{I}c_{I}\mid\phi_{I}\rangle$$
(3)
Each Slater Determinant is weighted by a coefficient, $c_{I}$. These coefficients can be obtained variationally by minimizing the total energy under the constraint of fixed orbitals, ending up in the Configuration Interaction (CI) methods. $\Psi_{0}$ is the reference, Hartree-Fock, wave function. In principle, the exact solution can be obtained by considering all possible Slater Determinants in combination with a complete basis set. The use of all possible configurations is called Full-CI and represents the case, when all electrons are arranged in all possible ways. This approach is infeasible for almost all molecular systems, more complex than e.g. He, and truncated methods are needed. Those are for example, CIS (CI Singles) or CISD (CIS and Doubles), where only single excitations or additionally double excitations are accounted for, respectively. Figure 3 gives a schematic overview of the improvements of CI that one can apply. A huge advantage of these methods is, that how to obtain the exact solution is known, and that they are systematically improvable. However, truncated CI does not scale correctly with the system size and is therefore not size-extensive and also not size-consistent (i.e., the energy of two fragments A and B at large distance computed together, $E(A+B)$, is not equal to the sum of the energies of the fragments from separate calculations, $\neq E(A)+E(B)$). 258
The CI scheme can be employed to improve the ground-state wave function by mixing the Hartree-Fock determinant and determinants of different electron configurations. In the same way, also wave functions of excited states can be computed. Then, the coefficients $C_{I}$ are optimized for higher eigenvalues of the electronic Hamiltonian instead of the first one. Beginners in the field then often get confused by terms like single excitation in comparison to first excited state. A single excitation determinant (see Fig. 3) can be part of the wave function for the first excited state but can also be a part of the ground-state wave function.
Electron Propagator Methods
Another class of methods that we shortly want to mention here are electron propagator methods, that are based on one electron Green’s function and are another variant of perturbation theory schemes.
One popular method that is based on Green’s function one electron propagator approach is the algebraic diagrammatic construction scheme to second order perturbation theory (ADC(2)). 259
ADC(2) is a single-reference method and can be used to efficiently compute excited states of molecules. It offers a good compromise between computational efficiency and accuracy, while being systematically improvable (higher order variants like ADC(2)-x or ADC(3) exist). The time evolution of a systems polarizability is obtained by applying the polarization propagation, which contains information on a system’s excited states. 256, 260, 261, 262, 263
The ground-state energy of ADC(2) is based on Møller-Plesset perturbation theory of second order, 264, 265 MP2, where the latter can formally be shown to include double excitations for the improvement of Hartree-Fock, see Ref. 256. The dependence of ADC(2) on MP2 gives rise to instabilities in regions, where excited states come close to the ground state or homolytic dissociation takes place.
The excited states of bound molecules are described with reasonable accuracy. Compared to multi-reference CI methods (see below), the black box behaviour of ADC(2) is a clear advantage. 259
Coupled Cluster
The gold standard of ab-initio methods for the ground state is the family of Coupled Cluster (CC) methods. CC is often referred to as the size-extensive and size-consistent version of CI. The different electronic configurations accounting for single or double excitations (such as in CIS and CISD for example) are obtained by applying an excitation operator, $\hat{T}$: 266
$$\begin{split}\displaystyle\mid\phi_{CC}\rangle=e^{\hat{T}}\mid\Psi_{0}\rangle=%
&\\
\displaystyle(\hat{T}=1+\hat{T}+\frac{1}{2!}\hat{T}^{2}+\frac{1}{3!}\hat{T}^{3%
}+...)\mid\Psi_{0}\rangle.&\end{split}$$
(4)
Similarly to CI, this operator can be truncated. If $\hat{T}=\hat{T}_{1}+\hat{T}_{2}$, single and double excitations are accounted for.
Excited states can be computed in a single-reference approach by equation-of-motion-CC (EOM-CC), where the excited-state wave function is written as an excitation operator times the ground-state wave function. For further details, see, e.g., the reviews 267, 268.
CASSCF
The problem of missing static correlation in the Hartree-Fock approach is tackled by a multi-reference ansatz for the wave function. 255
This treatment is important for many excited-state problems, but also some transition metal complexes in their ground state, transition states or homolytic bond-breaking with the dissociation of the N${}_{2}$ molecule being a notoriously difficult example. 269, 270
The multi-configurational self-consistent field (MCSCF) method can be seen as the multi-reference counterpart to the Hartree-Fock method. 271
One of the most popular variants of MCSCF methods is the Complete Active Space SCF (CASSCF), 272, 273 where important atomic orbitals and electrons are selected giving rise to an active space. An example is shown in Figure 4.
According to this scheme, the orbitals are split into an inactive, doubly occupied part, an active part and an inactive, empty part. Within the active space a FCI computation is carried out. The active space has to be chosen manually by selecting a number of active electrons and active orbitals. CASSCF is no black box method and a meaningful active space selection is the full responsibility of the user. As an advantage, CASSCF can describe static correlation well, which is necessary in systems with nearly degenerated configurations with respect to the reference Slater determinant. For completeness, state-averaging (i.e. SA-CASSCF) is most often applied, where states belonging to the same symmetry are averaged. Another variant of MCSCF methods is restricted active space SCF (RASSCF), which is very similar to CASSCF, but within RASSCF the active space is restricted and no FCI computation is carried out. 256
MR-CI
Even higher accuracy can be obtained with multi-reference CI methods 274, 29, 275, such as MR-CISD, that additionally add single and double excitations out of the active space and are therefore based on CASSCF wave functions. With this approach electronic correlation, i.e. static and dynamic correlation, can be treated.
CASPT2
Alternatively, complete-active-space perturbation theory of second order, CASPT2, 276, 277, 278 can correct electronic correlation effects via treating multi-reference problems with perturbation theory. This variant of multi-reference perturbation theory methods uses the CASSCF wave function as the zeroth order wave function. CASPT2 can be applied to each state separately (single-state (SS)-CASPT2) or correlated states can be mixed at second order resulting in a multi-state perturbation treatment (MS-CASPT2). 276, 277, 278 Other perturbation approaches for multi-reference problems exist, like the n-electron valence state perturbation theory (NEVPT2).279, 280, 281
MRCC
In addition to multi-reference methods based on CI, multi-reference variants of CC approaches exist. A relatively efficient implementation is for example the Mk-MRCC approach of Mukherjee and co-workers 282 or the Brillouin-Wigner approach 283, which is however not size extensive. Noticeably, the development of multi-reference CC approaches is a rather young research field compared to other excited-state methods and the computation of properties and forces is not well explored. Many studies therefore focus on the simulation of energies of low-lying states with MRCC methods. Additionally, such methods suffer from algebraic complexity and numerical instabilities. Interested readers that seek for a more extensive summary of existing MRCC methods are referred to Refs. 284, 285, 29.
Challenges
The probably biggest drawback of the aforementioned multi-reference methods is that their protocols are very demanding. Finding a proper active space is a tedious task that often requires expert knowledge. Too small active spaces can lead to inaccurate energies and problems with so-called intruder states are common. Those are electronic states, that are high in energy at a reference molecular geometry, but become very low in energy at another molecular geometry, that is visited along a reaction coordinate. The active space then changes along this path. This behavior can result in inconsistent potential energies. In case of CASPT2, the configurations of intruder states can lead to large contributions in the second-order energy, making the assumption of small perturbations invalid. Especially for describing molecular systems with many energetically close-lying states and for the generation of a training set for ML, such inconsistencies are problematic. Figure 5 shows an example of potential energy curves of 3 singlet states and 4 triplet states of tyrosine computed with (a) CASSCF(12,11) and (b) CASPT2(12,11),
where 12 refers to the number of active electrons and 11 to the number of active orbitals. We used OpenMolcas286 to compute an unrelaxed scan along the reaction coordinate, which is a stretching of the O-H bond located at the phenyl-ring of tyrosine.
Intruder states are no exception. Actually, they are quite common in small to medium sized organic molecules. A large enough reference space can mitigate this problem, but makes computations almost infeasible.
The computational costs increase exponentially with the number of active orbitals. In many cases, the improved accuracy due to a larger active space cannot justify the considerably higher expenses. At its best and with massively parallel simulations, an active space of about 20 electrons in 20 orbitals can be treated, 288 which is impracticable for many applications, such as dynamics simulations. For medium-sized molecules, the active space that would be required for a given simulation might even be way to large to be feasible for calculations in a static picture.
Worth mentioning at this point are also Rydberg states, that often need to be considered in small to medium sized molecules. Rydberg states can be strongly interlaced with valence excited states. In such cases, the active space needs to be large enough to treat both, the valence and Rydberg molecular orbitals. Additionally, the one electron basis set should be flexible enough to describe both types of orbitals. This increases the computational costs additionally. More details on the inclusion of Rydberg states in simulations can be found in refs 289, 290, 291, 292.
A promising tool to eliminate the complex choice of active orbitals is autoCAS 293, 294, 295. It provides a measure of the entanglement of molecular orbitals that is based on the density matrix renormalization group (DMRG). A DMRG-SCF calculation is similar to a CASSCF calculation, but instead of a FCI solution of the active space, an approximated solution with DMRG is obtained to avoid the exponential scaling of the computational costs with the number of active orbitals 296, 297, 298, 299, 300, 301. As an alternative, ML can be used to determine an active space.70
3.1.2 Density Functional Theory
A complementary view on how to obtain the energy of a system is provided by DFT. DFT dates back to 1964, when it was formulated by Hohenberg and Kohn 302 entirely in terms of the electron density, $\eta(\vec{r})$. A one-to-one correspondence between this density and an external potential, $v(\vec{r})$, exists and the potential acts on the electron density. The energy can be formulated in terms of a universal functional, $F[\eta(\vec{r})]$, of the electron density, which is independent of the external potential. In this way, the energy of a system’s ground state can be computed with the following equation:
$$E[\eta(\vec{r})]=\int v(\vec{r})\eta(\vec{r})d\vec{r}+F[\eta(\vec{r})]$$
(5)
The most widely used implementations of DFT rely on the Kohn-Sham approach. 303 In fact, Kohn-Sham DFT is so successful that it is often simply referred to as DFT. In this approach, an auxiliary wave function in the form of a Slater determinant is employed. Since a single Slater determinant is the exact solution for a system of noninteracting electrons, this DFT approach can be seen as describing a system of noninteracting electrons that are forced to behave as if they were interacting. The latter effect can be achieved only by an unknown modification of the Hamiltonian or rather of the aforementioned functional. In other words, a Slater determinant as wave function ansatz is exact but the Hamiltonian can only be approximated, in contrast to Hartree-Fock, where the true electronic Hamiltonian is used but the Slater determinant is only an approximate wave function.
The functional $F[\eta(\vec{r})]$ can be separated into Coulombic interactions and a non-Coulombic part. The latter can further be divided into two terms: the kinetic energy of the noninteracting electrons and the exchange-correlation part, which describes the interaction of electrons and thus also corrects the kinetic energy by the difference of the real kinetic energy and the kinetic energy of the fictitious system of noninteracting electrons. The exchange-correlation functional is the part of DFT that is unknown and finding the exchange-correlation functional remains the holy grail of DFT.
In principle, if the exact functional was known, the exact ground-state energy of a system could be computed. Unfortunately, it is not known and the success of a DFT calculation critically depends on the approximation that is used to the unknown exchange-correlation functional.
For completeness, KS-DFT is often used for closed-shell systems. In case of open-shell systems, two spin densities are distinguished, resulting in spin-polarized KS theory. 304
As explained above, the electron density is computed from a single reference Kohn-Sham wave function, i.e., the one of noninteracting electrons with the density of the real system. This single-reference wave function makes DFT a single-reference method. In fact, most failures of DFT are a consequence of an improper description of static correlation. 255
In order to describe excited states, the time-dependent (TD) version of DFT, namely TDDFT, can be used. The foundation of this theory was laid in the 1980s with the Runge-Gross theorems 305, which can be regarded as analogies to the Hohenberg-Kohn theorems. They are based on the assumption that a one-to-one correspondence exists also between a time-dependent potential and a time-dependent electron density in this potential. A system can therefore be completely described by its time-dependent density. Also in the time-dependent case, the variational principle for the density is proposed.
The most widely used approach of TDDFT is linear response TDDFT (LR-TDDFT). Again, often TDDFT is used synonymously with LR-TDDFT due to its extensive use. Within this theory and the KS approximation, no time dependent density is necessary to compute excitation energies and excited state properties. Linear response theory can be directly applied to the ground state density 306, 307. Casida’s formulation of this theory is the most popular one and gives rise to random-phase approximation pseudo-eigenvalue equations, which are also known as the Casida equations. Within the adiabatic approximation, they are implemented efficiently in many existing electronic structure programs. The Tamm-Dancoff approximation 308, 309 further simplifies the equations to an eigenvalue problem, resulting in the counterpart to CIS 310. Especially in cases, when the time evolution of a system is studied, the Tamm-Dancoff approximation is beneficial, since it leads to more stable computations close to critical regions of the PESs 253, 304.
The advantage of LR-TDDFT is its computational efficiency. The reasonable accuracy if a proper functional is chosen makes this approach often the method of choice to study the photochemistry of medium-sized to large and complex systems, which are not feasible to treat with costly multi-reference WFT based methods. 311, 29, 312
Shortcomings of LR-TDDFT are the incorrect dimensionality of conical intersections, which are, however, one of the most important regions during nonadiabatic MD simulations 313, 314, 315.
The incorrect dimensionality of conical intersections with standard TDDFT implementations leads to a qualitatively incorrect description of such critical regions. The missing couplings can be corrected for example with the CI-corrected Tamm-Dancoff approximation 316 or the hole-hole Tamm-Dancoff approximation,317
which can recover the missing couplings and provide correct dimensionality at conical intersections.
In addition, one should be aware that by definition, double excitations cannot be accounted for with LR-TDDFT. The computation of double excitations can be achieved by using a frequency dependent exchange kernel, which is known as dressed TDDFT 318, 319. Alternatively, spin-flip TDDFT 320, 321 can be used, where a triplet state is taken as a reference state and single excitations are treated with a flip in the electron’s spin.
However, spin-contamination is quite common within these methods. In general, the description of double excitations from a multi-reference state would be more favorable, although spin-flip TDDFT is often considered to be a multi-reference method. In order to compute specific orbital occupations and consequently excitations and charge-transfer states, an alternative approximation exists, which is known as the $\Delta$-SCF approach. In this theory, the electrons are forced into specific KS orbitals. The SCF is applied to converge the energy with respect to this configuration 322, 323, 324.
Other multi-reference variants of TDDFT exist too. However, their description is beyond the scope of this review and we refer the reader to a review covering this topic in much more detail. 29
Last but not least, we shortly want to discuss the most critical part of a DFT calculation, which is the proper choice of the exchange-correlation functional. In case of excited states, the treatment of valence excitations, Rydberg states and long-range charge transfer excitations on the same footing is highly problematic. While hybrid (meta-) generalized gradient approximation (GGA) or range-separated hybrid functionals 325 are for example well suited for vertical excitations and the latter also for Rydberg states, global hybrid meta GGA or range-separated hybrid GGA functionals are better to describe charge transfer 326, 253. Most often, functionals are accurate for one specific problem, but they fail to describe others. Although much effort has been devoted to develop functionals, finding a universal functional for DFT is still far from being achieved 304, 253, 175, 29.
In summary, it should be stressed that, in general, there is not only one single solution to a particular problem, but that many possible ways can be considered which lead to an equivalent description of a particular problem. Considering the excited states of molecules, it should be mentioned that it is of utmost importance to think carefully about the photochemical processes that may occur in order to find the most appropriate method for most of the assumed reactions. It often happens that within the same molecular system, one method can describe a certain photochemical reaction quite well, while another reaction can be described better with another method. However, the mixing of methods is not practicable for standard applications. Recently, studies on ML models have emerged that combine the different strengths of several methods, e.g. $\Delta$-learning techniques 327, 328 or transfer learning 329.
These methods could be well-suited solutions for many future applications to overcome the current limitations of existing quantum mechanical methods for the excited states. Even more than for ground state properties, the quality of the excited states depends critically on the ability of a method to describe the different possible reactions - as a consequence of the larger accessible configuration space of a molecular system. Even for medium-sized systems it should be clear that a suitable method may already be computationally impracticable and a balance between accuracy and computational effort has to be found.
3.2 Bases
The potentials computed with the aforementioned methods for different nuclear geometries can be represented in different bases, which are connected by unitary transformations. An example of five states in different bases are given in Figure 6. Note that often a system in a certain basis is also referred to as being in a certain picture or representation; here we will not use the term representation in order to not confuse the reader with molecular representations used in ML. As it is visible in the figure, we focus on three types of bases: (a) the diabatic basis, (b) the adiabatic (spin-diabatic) basis, i.e., the direct output of standard electronic structure programs, (c) the diagonalized version of (a) and (b), i.e., the spin-adiabatic basis. Throughout literature, different names are given to these bases, which are summarized in Table 1.
They stem from a partition of the total wave function into a sum of electronic and nuclear contributions, which can be written for all bases as:
$$\Psi(r,\mathbf{R},t)=\sum_{i}\psi_{i}^{\text{basis}}(r,\mathbf{R})\chi_{i}^{%
\text{basis}}(\mathbf{R},t).$$
(6)
In a similar way as the number 20 can be factored into $4\cdot 5$ or $4.5\cdot 4.\bar{4}$, the total wave function can be expanded in the different bases. Here, $\psi_{i}^{\text{basis}}(r,\mathbf{R})$ corresponds to the eigenfunctions of the electronic Hamiltonian only for one of the bases (namely the one from column B of table 1). Associated with these functions are the corresponding potentials, depicted for a model system in Fig. 6.
Note that a different approach is taken in the exact factorization method,330
where the total wave function is expanded only in a single product, i.e., without the sum in eq. 6, giving rise to only one (time-dependent) potential.
3.2.1 Adiabatic (Spin-Diabatic) Basis
The direct output of an electronic structure calculation usually provides the eigenenergies and eigenfunctions of the electronic Hamiltonian. In many cases, only one spin multiplicity is calculated. If this procedure is repeated along a nuclear coordinate, potential curves result that are termed adiabatic. Adiabatic means ”not going through” (from greek a=not, dia=through, batos=passable) and, indeed, the potentials never cross when considering one multiplicity. This situation is schematically illustrated in Figure 6(b) for singlet S${}_{i}$ and singlet S${}_{j}$.
Within one multiplicity, 3N${}_{A}$-dimensional adiabatic PESs are obtained that are strictly ordered by energy. Hence, the states are usually denominated with the first letter of the multiplicity and a number as subscript, e.g., S${}_{0}$, S${}_{1}$, etc. For states of the same multiplicity, critical points and seams exist. These regions of the PESs are referred to as conical intersection (seams), in which the corresponding states become degenerate.
Such features make adiabatic PESs non-smooth functions of the atomic coordinates, which make them difficult to predict with the intrinsically smooth regressors of ML. At a conical intersection, the approaching potential energy curves form a cone and the NACs, denoted as $C_{\text{NAC}_{ij}}$, between them show singularities as a result of the inverse proportionality to the vanishing energy gap 335, 274:
$$\begin{array}[]{lr}C_{\text{NAC}_{ij}}\approx\langle\Psi_{i}\mid\frac{\partial%
}{\partial\mathbf{R}}\Psi_{j}\rangle=\\
\frac{1}{E_{i}-E_{j}}{\langle\Psi_{i}\mid\frac{\partial H_{el}}{\partial%
\mathbf{R}}\mid\Psi_{j}\rangle}~{}~{}~{}\text{for}~{}i\neq j,\end{array}{}$$
(7)
Second order derivatives are neglected here, as is done in many quantum chemistry programs that compute NAC vectors. The blue dashed curve in panel (b) of Fig. 6 illustrates the norm of the NAC vector, C${}_{ij}^{NAC}$, that couples the states S${}_{i}$ and S${}_{j}$. At the avoided crossing points of the states, the NAC norm shows a sharp spike, but is almost vanishing elsewhere.
If more than one multiplicity is considered, the term adiabatic is not adequate anymore, because potentials of different multiplicity might cross through each other. This situation is then called diabatic with respect to the spin multiplicities, or spin-diabatic in short. For example, singlets are adiabatic among each other, triplets are adiabatic among each other but singlets are diabatic with respect to triplets. However, also the diabatic basis (see Fig. 6(a) and also below) qualifies as spin-diabatic. Due to this nomenclature issue, which even gets experts confused sometimes, we refer to this basis as MCH (Molecular Coulomb Hamiltonian) because it is obtained from the eigenfunctions and eigenvalues of the non-relativistic electronic Hamiltonian, where only Coulomb interactions are considered.
As an example, a crossing of a singlet state and a triplet state is shown in Fig. 6(b). As it is visible, the triplet components, which are defined by different magnetic quantum numbers, are degenerate. The states are coupled by SOCs (denoted as $C_{\text{SOC}_{ij}}$), which are usually obtained as smooth potential couplings
with standard quantum chemistry programs 242, 334, 28:
$$C_{\text{SOC}_{jk}}=\langle\Psi_{j}\mid\hat{H}^{SO}\mid\Psi_{k}\rangle.$$
(8)
These couplings are single real-valued or complex-valued properties 34, 336. Whether they are complex or not depends on the electronic structure program employed, but they can be converted into each other. 337, 242, 36, 34
$\hat{H}^{SO}$ in eq. 8 is the spin-orbit Hamilton operator, which describes the relativistic effect due to interactions of the electron-spin with the orbital angular momentum, allowing states of different spin-multiplicities to couple. 338, 339, 337
Note that also SOCs between different states of the same multiplicity exist except for singlets.
No exact expression on how to include relativistic effects into the many-body equations has been found, yet. Among the most popular approximations used is the Breit equation 340, applying an adapted Hamiltonian instead of the electronic Hamiltonian, which comprises, among other terms, a relativistic part. This additional part of the Hamiltonian accounts for spin-orbit effects and is proportional to the atomic charge, 340, 339, 337, 36, 34 leading to the belief that SOCs would only be relevant in systems with heavy atoms. 341, 342 Today it is known, that spin-orbit effects also play a crucial role in many other molecular systems and are important for intersystem crossing between states of different spin multiplicities. 343, 344, 345, 37
The states in the MCH basis can also be coupled via external electric-magnetic fields, e.g., by sunlight or a laser. The corresponding couplings stem from the transition dipole moments multiplied with the electric field. Since the effect of the field is not included in the potentials but as off-diagonal potential couplings, the MCH basis is also called field-free. 331, 332, 333, 346 However, also the diabatic basis qualifies as field-free.
3.2.2 Diabatic Basis
In the diabatic basis, the electronic wave function is not parametrically dependent on the nuclear coordinates. Note that such a strictly diabatic basis for polyatomic systems does not exist in practice and only approximated, so called, quasi-diabatic, PESs can be fit. In literature, quasi-diabatic PESs are most often referred to as diabatic ones, so we will also use this notation here. Further, diabatic potentials usually need to be determined from adiabatic potentials and are not unique, i.e., they rely on the method and the reference point, which is chosen in the adiabatic basis to fit diabatic potentials. 228, 242
An example of a system in the diabatic basis as given in panel (a) of Figure 6 and commonly used notations can be found in Table 1 in the first column.
In regions, where an avoided crossing is present in the adiabatic basis, the coupled diabatic potential energy curves cross. Since the electronic wave function of a state is ideally independent of the nuclear coordinates, its character is conserved. Consequently the states are labeled according to their character and multiplicity, e.g., as ${}^{1}\pi\pi\ast$ or according to symmetry labels. Similar to the character, also spectroscopically important quantities like the dipole moment are mostly conserved or vary smoothly along the nuclear coordinates. Therefore, spectroscopic experiments can easily be interpreted when using the diabatic basis, which is thus sometimes also called spectroscopic basis. Note that sometimes labels like S${}_{1}$, etc. are used also when referring to the diabatic basis, especially in experimental papers when an identification of the wave function’s character has not been carried out and only one geometry is considered. However, at a different geometry, the energetic order of the states might have changed such that a state previously labeled as S${}_{2}$ might now be lower in energy than a state previously labeled as S${}_{1}$. Furthermore, this labeling scheme in the diabatic basis can lead to confusion with the labels from the MCH basis, and we suggest to reserve it only for the MCH basis.
Due to the mostly conserved characters and the crossing of states, diabatic potentials are smooth functions of the nuclear coordinates, in contrast to adiabatic potentials. A diabatic PES is thus highly favorable for several numerical applications including ML.
The MCH and diabatic bases can be interconverted by a unitary transformation
$$\Psi^{MCH}(\mathbf{r},\mathbf{R})=\mathbf{U}(\mathbf{R})\Psi^{diab}(\mathbf{r}%
,\mathbf{R})$$
(9)
with a unitary matrix, $\mathbf{U}$, that is determined up to an arbitrary sign (as a result of the arbitrary sign of the wave function, which will be discussed in detail in section 4.2).
In the case of two states, $\mathbf{U}$, is a rotation matrix:
$$\mathbf{U}=\left({\begin{array}[]{cc}cos\theta(\mathbf{R})&-sin\theta(\mathbf{%
R})\\
sin\theta(\mathbf{R})&cos\theta(\mathbf{R})\\
\end{array}}\right)$$
(10)
and is dependent on the rotation angle, $\theta$.
Accordingly, the peaky NACs, which are obtained as derivative couplings (also called kinetic couplings) in the MCH basis, are converted to smooth potential couplings in the diabatic basis. The smooth SOCs from the MCH basis become even smoother (ideally constant) in the diabatic basis.
While one can straightforwardly apply diagonalization to convert diabatic PESs to adiabatic PESs (and similarly adiabatic PESs to diagonal PESs), a dilemma arises when one wants to take the inverse way to obtain diabatic PESs from adiabatic ones (and similarly adiabatic PESs from diagonal ones). In fact, finding diabatic PESs is highly complex and most often requires expert knowledge. Up to date, only small molecules could be represented with accurate diabatic potentials and developing a method to automatically generate diabatic PESs remains an active field of research. Existing methods to obtain diabatic potentials require human input and are mostly applicable to small systems and certain reaction coordinates. Early pioneering works can be found in refs 347, 228. Today, a lot more variants exist. Examples are the propagation diabatization procedure 348, diabatization by localization 349, Procrustes diabatization 239 or diabatization by ansatz 350, 140. Further, methods can be based on couplings or other properties 351, 352, 353, 354, configuration uniformity 355, block-diagonalization 356, 357, CI vectors 358 or (partly) on ML 359, 360, 361, 362, 350, 140, 141.
3.2.3 Diagonal Basis
As the name indicates, the diagonal basis can be obtained by a diagonalization from the MCH or diabatic bases. In this case, a strictly adiabatic picture is obtained, where states never cross. 242 Accordingly, the concept of multiplicity for a single state is lost because the state might be of singlet character in one region and of triplet character in another region. Therefore, the basis is also called spin-mixed or spin-adiabatic. 336, 36, 363 The states are strictly ordered by energy and can be labeled simply with numbers (see Fig. 6(c)). The resulting wave functions are eigenfunctions of the relativistic electronic Hamiltonian.242, 344, 36 These eigenfunctions as well as the eigenenergies can be also obtained directly with e.g. relativistic two-component or four-component calculations,364 instead of via diagonalization.
In this basis, the effect of the SOCs are incorporated into the PESs to a large extent. What remains are localized kinetic couplings, which are similar in nature to the NACs in the MCH basis.
An example is given in Fig. 6(c). The parts of the potentials that correspond to the different triplet components in the MCH basis are split energetically in the diagonal basis. In the case of small SOCs, the diagonal potentials look similar to the MCH potentials. However, if the SOCs are strong, potentials that are degenerate in the MCH basis can be easily shifted apart by 1 eV in the diagonal basis. Such splittings are then also experimentally observable, and the diagonal basis yields a more intuitive interpretation of these experiments. 365, 45, 366
As mentioned above, the states in the MCH basis can also be coupled via electromagnetic fields. A diagonalization of the potential matrix then yields so-called field-dressed states or light-induced potentials, which can also be termed field-adiabatic.367, 331, 368, 369, 346 Since the fields are usually time-dependent, the most important axis along which the potentials in this field-dressed basis need to be plotted is time. 346
In principle, all these bases are equivalent but only if an infinite number of terms is considered in eq. (6). In practice, potentials represented in different bases have different advantages for dynamics simulations, especially in combination with different approximations made in the different dynamics methods as outlined below.
3.3 Excited-State Dynamics Simulations
In order to investigate the temporal evolution of an isolated molecular system in the excited states, the time-dependent Schrödinger equation has to be solved:240
$$i\hbar\frac{\partial\Psi(\mathbf{r},\mathbf{R},t)}{\partial t}=\hat{H}_{el}(%
\mathbf{r},\mathbf{R})\Psi(\mathbf{r},\mathbf{R},t).$$
(11)
From a technical point of view, a sequence of time steps is computed, where in every step the electronic problem is solved to yield potentials, which determine the forces acting on the nuclei such that the nuclear equations of motion can be solved for the current time step.
Ideally, the nuclei are treated quantum mechanically. In this case, the PESs are usually computed in advance and either interpolated or stored on a grid for later use. The hope is that ML can improve the interpolation of potentials drastically. Such global PESs are needed because a wave function is employed for the nuclei, which extends over a range of nuclear coordinates at the same time (see Fig. 7(a)). An overview over corresponding dynamics methods is given in section 3.3.1.
The nuclear dynamics can also be approximated classically while quantum potentials are used, i.e., mixed quantum classical dynamics (MQCD) simulations are carried out. Such methods is discussed in section 3.3.2. Since the classical nuclear trajectories are defined only at one nuclear geometry at a time (see Fig. 7(b)), on-the-fly calculations of the potential energies are possible. An on-the-fly scheme is computationally advantageous, if the number of visited geometries during the dynamics is smaller than the number of points needed to represent the conformational space on a grid or via interpolation. 314, 370, 313, 344, 242, 26, 28
No fitting of PESs is necessary in an on-the-fly approach but fitted PESs can still be used as an alternative. Since ML approaches provide such interpolated potentials, the amount of training points generated with quantum chemistry must be less than the number of points needed in an on-the-fly approach in order to be advantageous. This demand is satisfied, e.g., for long time scales or if many trajectories are necessary.
In the following, we will shortly discuss the different types of nuclear motion and the opportunities of ML models to enhance the respective dynamics simulations.
3.3.1 Quantum Nuclear Dynamics
The computational cost of an exact nuclear dynamics simulation scales exponentially with the nuclear degrees of freedom. Hence simulations are limited to small systems, typically containing less than 5 atoms 34, 27, 371.
Still, the calculation of the PESs of the molecule can be a rather expensive part of the whole scheme and the use of ML algorithms is advisable even for such small systems.
To treat larger systems, approximations have to be invoked. A prominent approach that can be converged to the exact solution is the
multi-configurational time-dependent Hartree (MCTDH) approach 372, 49, 373, 374.
Its high efficiency stems from the use of time-dependent basis functions to represent the nuclear wave functions. Nonetheless, the computations are computationally costly and the nuclear degrees of freedom are often reduced to only a few important key coordinates,228, 375
where classical simulations can help identifying the latter.376
Whether quantum dynamics of such reduced-dimensionality models are better than using classical dynamics of a full-dimensional system is still under debate and probably depends on the system. The potentials need to be presented to the algorithm in the diabatic basis, mostly due to numerical stability (e.g., smooth couplings are easier to integrate than singular ones). Since more than 20 years, (modified) Shepard interpolation is used to fit diabatic potentials 377, 149, 378, 379, 380. Notably, the grow algorithm 149 can be used to efficiently generate the database of points upon which the interpolation is based. However, it is clearly desirable to treat larger systems, and ML models like neural networks (NNs) promise higher performance or more flexibility in such cases. 359, 360, 361, 348, 362, 147, 141, 145, 144
More recently, on-the-fly methods addressing quantum dynamics have been developed. 381, 382, 383, 143 They mostly rely on a combination of Gaussians to represent the nuclear wave function.26 For example, the variational multi-configuration Gaussian method (dd-vMCG) 384 offers a variational and thus accurate solution for the equations of motion. Also full multiple spawning46, 371, 385 can be regarded as fully quantum mechanically by describing the wave function with a number of time-dependent Gaussian functions, that follow classical trajectories with quantum mechanically determined time-dependent coefficients. In its more affordable ab-initio multiple spawning variant, more approximations are introduced such that the results sometimes draw near the classical solutions.386, 387
Further related methods exist, like the ab-initio multiple cloning method,388 or the thawed Gaussian approximation.389
Another class of dynamics methods are semi-classical approaches, which allow the inclusion of quantum effects in the classical dynamics of nuclei, such as quantum mechanical tunnelling or coherence. 390 Note that these methods, where the nuclear dynamics is treated semi-classically, should not be confused with the MQCD approaches (see below) that are also often termed semi-classical (because the nuclei are treated classically and the electrons quantum-mechanically). The semi-classical dynamics methods range from the initial value representation, 391, 392
adapted with the Zhu-Nakamura approach leading to the Zhu-Nakamura-Herman-Kluk initial value representation, 393 to path integral approaches.394
The path integral formalism is especially interesting when the quantum and classical degrees of freedom should be coupled in a dynamically consistent manner. By using so-called ring-polymers, i.e., replica of the original classical system, a deviation of the nuclear dynamics from the classical path can be obtained and the time evolution of a system including nuclear quantum effects can be investigated. However, ring-polymer dynamics suffer from high computational efforts as a consequence of the large number of replica required. Accelerated formalism exist, which are for example implemented in the Python wrapper i-PI, 395, 396
which allows to interface path-integral methods with programs that provide PESs, but are mostly dedicated to the electronic ground state. Up to date, only a few implementations of semi-classical methods in atomistic simulation software are available. Compared to classical mechanics, the computational costs increase by a factor of about 10 to 100. 390, 397, 398
3.3.2 Mixed Quantum-Classical Molecular Dynamics
While semi-classical methods are promising to simulate the dynamics of molecular systems containing up to tens of atoms highly accurately, the study of larger systems is still dominated by computationally cheaper MQCD methods, where the nuclear motion is treated fully classically. 27, 399, 397, 398 In contrast to quantum dynamics, the motion of the nuclei can be computed very fast using classical mechanics, and the computation of the PESs, on which the nuclei are assumed to move, remains the time limiting step. In this sense, ML models have a huge potential to enhance MQCD simulations by providing the electronic PESs and enabling the investigation of reactions that are not feasible with conventional approaches. 400, 13, 401, 402
In fact, most studies that describe photochemistry with ML up to date aim to replace the quantum chemical calculation of the PESs in MQCD approaches.
The most popular MQCD method is trajectory surface hopping, 403, 404, 405 schematically represented in Figure 7(b).
A manifold of independent trajectories is required to obtain statistically relevant results and to mimic the extended nuclear wave functions. For a single trajectory, the nuclei move classically on one of the quantum potentials, hence only one state is considered to be active, but transitions between different states are allowed.406
Different approaches exist to determine the probability of such a transition, also called hop or jump in surface hopping methods. To this aim, different quantities are needed that are commonly provided in the MCH basis, as it is the direct outcome of a quantum chemical simulation. One of the first implementations to compute the hopping probability is based on the Landau-Zener formalism. 407, 408 Based on the Landau-Zener formula, the potential energy differences are used to determine the hopping probability. No information about couplings is required, which implies that the approach must fail for states that do not couple but lie close in energy. Very similar to this approach is the Zhu-Nakamura theory 409, 410, 411, 412. Also here, the computation of couplings is omitted and only information about PESs is used. Among the mostly used hopping algorithm is Tully’s fewest switches algorithm 403, which is valid for many cases and based on the NACs between different PESs. An extension to other couplings is provided e.g. in the SHARC (surface hopping including arbitrary couplings) method.344 When couplings are considered, an internal transformation from the MCH basis to the diagonal basis is most advantageous because the localized couplings of the diagonal picture precisely indicate, where the few switches of the fewest switches approach should take place. In cases, where the PESs are fit in advance, either with ML models or other types of analytical functions, the use of a diabatic basis is favorable (because of the Berry phase, see below) but should be transformed to the diagonal picture for the calculation of hopping probabilities. Other flavors to account for transitions exist. However, they have not been applied in simulations with ML algorithms yet. Interested readers are therefore referred to refs 403, 413, 410, 314, 414, 415, 344, 416, 417, 48, 242, 36 for further information.
The bottleneck of approaches that require NACs is that the computation of the couplings remains one of the most expensive part of a quantum chemical calculation. The computational effort to compute a NAC vector is comparable to that of a force calculation. However, more NACs are present than there are forces, i.e. $N_{S}\times(N_{S}-1)/2$ NACs need to be computed, whereas $N_{S}$ forces are needed (respectively with entries for the Cartesian coordinates of each nucleus). Note that in case of fitted PESs with ML, all of these vectors have to be computed for each data point. Conventional approaches with an ab-initio on-the-fly evaluation of the PESs can make use of the fact, that only one active state needs to be considered at a certain time step. Many MD programs therefore only require a computation of the forces of the active state and the respective couplings arising from this state.
Note that despite the benefits of MQCD simulations, they obey micro-reversibility only approximately418
and effects due to coherences or tunneling necessitate additional considerations as a consequence of the classical treatment of nuclear motion.419
A more approximate approach is the Ehrenfest dynamics method, also referred to as mean-field trajectory method. It is often used for large systems and also frequently in material science. 178, 189 The Ehrenfest method is based on the approximation that nuclei move classically on an average potential, rather than switching from one specific state to another.420, 314, 421 Due to the treatment of each electronic state separately, surface hopping methods allow the accurate bifurcation into different reaction channels, while such effects are neglected in a mean-field treatment of PESs.
The main limitation of MQCD approaches are the expensive evaluation of ab-initio potentials, which allows dynamics simulations only for up to a couple of picoseconds. In addition, rare reaction channels are hardly explored as a result of usually bad statistics 422, 423, 36. In this sense, MQCD simulations offer a perfect place for ML to enter this field of research and advance it significantly. The fast evaluation of the ML PESs can help to explore different reaction channels and to obtain accurate reaction kinetics. Observables and macroscopic properties can be computed directly or with post-processing as well as analysis runs, and offer another fulcrum for ML. The computed observables should then be directly compared to experiments.
3.4 Dipole Moments and Spectra
An important property for comparing experiment and theory is the dipole moment. The permanent dipole moment of the ground state is a frequent target of studies with ML 424, 425, 426, 109, 427, 428, 429, 430, 431, 432, 433, 434, 435. The permanent dipole moment, $\mu_{i}$ (or $\mu_{ii}$) , of a state $i$ can be obtained via the dipole moment operator (see eq. (12) below) or as the sum of partial charges, $q_{a,i}$ of atom $a$ in state $i$, and the vector that describes the distance of the position of atom $a$ to the center of mass of the molecule, $r_{\alpha}$: $\mu_{i}=\sum_{a}^{N_{A}}=q_{a,i}r_{\alpha}$.
It can be used for the computation of infrared spectra with MD simulations. The spectrum is then obtained as the Fourier transform of the time auto-correlation function of the time derivative of the dipole moment 436.
In contrast to the ground state, excited-state simulations often make use of the transition dipole moments, which are computed from the dipole moment operator within many quantum chemistry programs:
$$\mu_{ij}=\langle\Psi_{i}\mid\hat{\mu}\mid\Psi_{j}\rangle.$$
(12)
The ground state dipole moment can differ strongly from those in the excited states, due to a frequency shift and altered electron distribution upon light-excitation 437.
Transition and permanent dipole moments can be fit with the charge model of ref. 109, where point charges are never learned directly, but instead are inferred as latent variables by an NN dipole model making use of r${}_{\alpha}$.435
Noticeably, the computation of absolute values of permanent and transition dipole moments is very challenging even when highly accurate quantum chemistry methods are employed and experimental values are hardly reproduced 438, 93. However, also experimental studies provide absolute values only in few cases. Most computational studies therefore do not aim to reproduce the absolute values of transition dipole moments but rather use relative values to obtain reasonably accurate absorption spectra, which can be compared to experiments. 240, 439, 56, 440, 242, 441 Since many molecules absorb in the UV, the terms UV spectra and absorption spectra are often used interchangeably. However, absorption can take place in many regions of the electromagnetic spectrum, including, e.g., X-rays, where rather core electrons than valence electrons are excited.31
As already mentioned shortly, absorption spectra can be obtained from a calculation of excited-state energies and oscillator strengths, which are proportional to the squared transition dipole moments. Noticeably, the transition dipole moment is only defined up to an arbitrary sign as a result of the arbitrary phase of the wave function (see section 4.2). To circumvent this ill-definition, oscillator strengths or the lengths of dipole vectors can be fitted with ML. However, this workaround can be problematic if explicit field-dipole interactions should be considered with ML models.
4 Data Sets for Excited States
The basis of any successful ML model is a comprehensive and accurate training set that can describe the required conformational space of a molecule comprehensively and accurately with as little noise as possible 442.
While electronic structure theory for ground state problems is almost free of noise, the same cannot be said so easily for problems in the excited states. "Bad points with abrupt changes" 14 within ab-initio calculations for the excited states are frequently observed, which can occur even far away from any critical point of the PESs and are difficult to detect 92, 13, 14.
The amount of noise in the reference data does not only depend on the chosen method (and in case of multi-reference methods on the selected active space), but also on the number of electronic states considered and the photochemistry of the molecule under investigation.
4.1 Choosing the Right Reference Method for Excited-State Data
Many existing training sets for ML in quantum chemistry are based on DFT 103, 443, 444, 445, 101, 446, 110.
The ease of use and low computational costs of DFT-based methods make them suitable to treat large systems with acceptable accuracy. In fact, DFT is the workhorse of many studies solving ground-state problems. In contrast, TDDFT has not yet managed to equal DFT for the treatment of excited-state problems. Consequently, training sets for the excited states are less frequently computed with TDDFT 96, 91, 327, 160, 95, 447 and rely most often on multi-reference methods. Examples of applied methods are CASSCF 147, 137, 139, 138, 143, 144, 158, 13 or MR-CI schemes 448, 449, 450, 140, 451, 452, 453, 142, 92, 93, 12, 90, 14, 13, 94, where the latter method is more expensive than the former and therefore limited to describe small systems.
In general, the computation of excited-state PESs is much more expensive than the computation of the ground state potential of the same molecule. Not only highly accurate ab-initio methods have to be applied for many systems, but also forces and couplings are required for the considered states. A high density of electronic states present in a molecular system can thus increase the costs of a calculation considerably. In this regard, an active, efficient and meaningful training set generation is indispensable, especially when photodynamics simulations are the target of a study.
Keeping in mind, that the quality of the reference data confines the quality of an ML model, several key questions can be identified when designing a study based on ML potentials.
We believe the following questions to be important for the selection of a suitable reference method:
1) What is the goal of an ML model and what properties must it predict in order to benefit from the advantages that ML can offer? Are only energy gaps of different electronic states to the electronic ground state necessary or are gaps between other states and couplings between them also relevant? Especially the description of couplings requires further consideration, as they cannot be calculated with all quantum chemistry methods and additionally face the problem of random sign jumps along different reaction coordinates 454, 92, 90.
2) How many excited states are relevant and which method is computationally affordable to treat the amount of states required?
A comparison with experiment and the computation of vertical excitation spectra with reference methods can help to obtain an answer to this question.
3) How large is the system under investigation and how complex are the excited state processes that are considered to be important?
This question is important in order to identify if single reference methods like LR-TDDFT or ADC(2) make sense for certain reactions that might occur. While large and flexible molecules with a lot of energetically close-lying states can give rise to a multifaceted photochemistry including dissociation, homolytic bond-breaking, and bond-formation, the dynamics of rigid molecules might only be dominated by one main reaction channel and lose the additional energy in form of molecular vibrations. The complexity of the excited-state processes can help to estimate the number of necessary data points to describe the relevant configurational space of the molecule.
In case multi-reference methods are necessary to describe many different excited-state processes of a molecule, the training set generation can become infeasible. For example, 356 data points were computed for the 15-atom cyclopentoxy molecule with MR-CISD(5,3)/cc-pVD(T)Z. 94 Respective calculations comprised 19,302,445 configuration state functions and one reaction coordinate could be fitted in the diabatic basis. We also ran into a similar problem when fitting the excited states of the amino acid tyrosine containing 24 atoms, which also requires a multi-reference treatment. The size of the active space and the number of states needed for an accurate description made multi-reference methods like CASSCF or CASPT2 computationally too expensive, see Fig. 5.
In these cases, the computation of an ample training set is far too expensive with multi-reference methods and the quantum chemistry calculations remain the bottleneck even when using ML.
In addition to the aforementioned intricacies to build up a meaningful, yet accurate training set for the excited-states, the process is further complicated by the arbitrary phase of the wave function. As a consequence, excited-state properties resulting from two different electronic states, such as transition dipole moments or couplings between different electronic states 454, 92, 93, 13, 90, 14, are not uniquely defined and cannot simply be fitted with conventional ML models. Either an additional data preprocessing or an adaption of the learning algorithm has to be incorporated to render data learnable with ML models.
4.2 Phase of the Wave Function
In contrast to ground state properties, excited-state properties such as transition dipole moments, NACs or SOCs arise from two different electronic states. As a consequence of the arbitrary phase of the wave function of each electronic state, properties resulting from two different states carry an arbitrary sign, which makes them generally double-valued.
In case of vectorial properties, such as dipole moments or coupling vectors, the whole vector can be multiplied by +1 or -1 and is still a valid solution. Similarly, single valued properties, such as SOCs obtained from electronic structure programs, can be multiplied by +1 or -1 and are equally correct. This additional complexity prohibits that conventional ML algorithms learn such raw data of quantum chemistry and hampers the training process to find a proper relation between a molecular geometry and the excited-state property. 454, 92
A one-dimensional example of this problem is illustrated for the NAC (exemplified using one single value along the reaction coordinate) that couples an excited singlet state, S${}_{i}$, and a second excited singlet state, S${}_{j}$, in Figure 8. A positively signed function of atomic coordinates is shown by dashed blue lines with a cusp at the point at which the two singlet states are degenerate. Such a smooth function (besides the sharp spike at the conical intersection) is highly desirable when fitting with ML models is aimed for. It is worth mentioning that a consistent negative sign (light-blue dashed line) along this reaction coordinate is equally correct and that it is desirable to seek for one global sign. However, the direct output of a quantum chemistry program along this reaction coordinate looks more similar to the dashed magenta line in-between the blue curves. As one can imagine, no proper training can be guaranteed with these inconsistent data. Note that existing MD programs for the excited states usually track such phase jumps within electronic wave functions in order to account for nonadiabatic transitions correctly.242
The idea of phase tracking can also be applied in ML in order to thwart the problems due to the arbitrariness within coupling or dipole elements. Some algorithms have been developed to remove the arbitrary sign jumps and provide smooth functions of atomic coordinates 92, 13, 14, 455. Noticeably, the properties obtained after a transformation to the diabatic basis are already smoothly varying functions of atomic coordinates 336. However, the challenges arising due to the arbitrary phase of the wave function still persist, because the inconsistencies within adiabatic properties have to be removed in order to make the diabatization process feasible. 14, 90
It is worth mentioning at this point that also another kind of phase exists that cannot be eliminated in the aforementioned way. It is called Berry phase or geometric phase. After performing a loop in space around a conical intersection and returning to the original point, a change in the phase of the wave function of $\pi$ can be observed, i.e., the same point is only reached after two loops around the conical intersection. Neglecting this effect can lead to false transition probabilities, depending on the dynamics method and the system. While in most cases in MQCD the Berry phase can be safely neglected, this is not possible in quantum dynamics simulations. A diabatic basis is advantageous in this case, because the Berry phase is absent in this picture. However, the Berry phase has to be kept in mind, when fitting diabatic potentials 456, 457, 458, 459, 460.
4.2.1 Phase Correction of Adiabatic Data
First ML studies on dynamics in the adiabatic basis omitted a preprocessing and were unable to reproduce reference results based on ML alone,138 or avoided the phase problem by using the Zhu-Nakamura method.137, 139
Evidently, potentials and forces can be learned with conventional ML approaches but adaptations or a preprocessing of data is necessary to learn coupling elements or transition dipole moments.
Independent of the purpose – the fitting of adiabatic quantities 454, 92 or the diabatization of adiabatic data with property-dependent diabatization schemes 14 – the adiabatic data has to be corrected to remove the arbitrary sign jumps that are due to the arbitrary phase of the wave function. Several ways for these corrections exist, which have been shown to work well for different excited-state problems.
One possibility is to preprocess data according to the wave function overlap – betweem the wave functions from a geometry of interest and a reference geometry – for each electronic state. This process is termed phase correction 454, 242 and has been applied by us in order to generate a training set for three singlet states of CH${}_{2}$NH${}_{2}^{+}$ 92 and 2 singlet and 2 triplet states of CSH${}_{2}$. SOCs 13, NACs 92, 93, 13, and transition dipole moments 92, 93 could be fitted in the adiabatic basis with deep NNs and kernel ridge regression (KRR) 92, 13, 93. Very recently, Zhang et al. 95 applied this procedure to describe transition dipole moments of N-methylacetamide.
The wave function overlap matrix, $\mathbf{S}$, with size N${}_{S}\times$N${}_{S}$, is computed between two molecular geometries $\alpha$ and $\beta$ 461:
$$\boldsymbol{S}=\langle\Psi_{\alpha}\mid\Psi_{\beta}\rangle.$$
(13)
In many cases along a given reaction path, the off-diagonal elements of the overlap matrix are very close to zero and the diagonal elements are very close to +1 or -1, indicating whether the phase of a state has changed along this path or not. Whenever a new state enters along the reaction path or adiabatic states switch their character, which is common after passing through a conical intersection for example, the off-diagonal elements provide the relevant phase information instead of the diagonal elements. Taking all these effects into account, a phase vector, $\mathbf{p}$, can be derived for each given molecular geometry. A property resulting from electronic state i and j has to be multiplied by the corresponding phase factors of these states 92.
An advantage of this algorithm is that it does not require any manual fitting of data. However, this procedure has to be carried out for every data point included in the training set with respect to one pre-defined reference wave function. This reference wave function can be for example the wave function of the ground-state equilibrium structure of the molecule and needs to be identified to guarantee an almost globally consistent sign of elements.
During a photo-initiated simulation, it is common that geometries quickly start to differ from the reference geometry. The wave function overlap then tends to zero and cannot provide information about the correct sign of a certain electronic state. In this case, the phase must be propagated from the reference geometry on with $n$ interpolation steps. The phase vector applicable for the correction of the data point to be included in the training set is then obtained by multiplication with all previously obtained phase vectors, $p_{0}$ to $p_{n-1}$:
$$\boldsymbol{p}=\prod_{\alpha=0}^{n-1}p_{\alpha}.$$
(14)
Intruder states prohibit a proper tracking because their wave function is absent at the earlier geometries. Hence, a phase correction may be rendered infeasible for systems with a high density of states.
In order to obtain the correct phase, more states can be included in the simulations, which however increases the computational cost. A solution is to take many electronic states into account only close to the reference geometry. The amount of states can then be reduced along a given reaction coordinate and relevant states can be disentangled from irrelevant ones. Further, it makes sense to save the already phase-corrected wave functions of several geometries in addition to the reference geometry. Whenever a new data point should be included into the training set, the distance to each saved data point can be computed in order to find the closest available structure and reduce the amount of interpolation steps 92, 400.
This problem has also been recognized by Robertson et al. 358 for a diabatization process, where a sufficiently large vector space of the CAS wave function is required for proper diabatization. The overlaps of electronic states can be maximized by rotation of CI vectors of CAS wave function states. A similar version to use the information of CI vectors for diabatization was applied by Williams et al. 140, who used NNs to assist the diabatization process of adiabatic NO${}_{3}$ potentials.
Another way to correct the sign of data points was carried out by Guan et al. 14, who fitted diabatic 1,2${}^{1}$A PESs and dipole moment surfaces of NH${}_{3}$ from MR-CISD/aug-cc-pVTZ data with NNs.
The diabatic PESs were taken from a previous study and obtained with the Zhu-Yarkony diabatization procedure 462, 463, 464. By diagonalization, the rotation matrix defined in eq 10 could be obtained, which connects the diabatic and the adiabatic basis (see eq. (9)). The adiabatic dipole moments, $\mu^{MCH}$, could then be transformed into the diabatic basis using the unitary matrix, $\mathbf{U}$:
$$\mu^{diab}=\mathbf{U}\mu^{MCH}\mathbf{U}^{\dagger}.$$
(15)
As the unitary matrix $\mathbf{U}$ is only defined up to an arbitrary sign, the signs of the diabatic dipole moments have to be corrected in order to provide a consistent diabatic dipole moment surface. This correction has been done with a so-called cluster growing algorithm 455.
The cluster growing algorithm requires an initial set of phase corrected data points. In this work, 347 data points were adjusted manually for this purpose. Subsequently, a Gaussian process regression (GPR) model 465 was fitted to these data points. The signs of the rest of the data points to be corrected were then adjusted with the GPR model. Several iterations were carried out, where each iteration aims for the inclusion of close-lying points to the cluster, leading to the name "cluster growing" algorithm 146.
The singularities in regions close to conical intersections can make this algorithm fail. Therefore, data points in such regions have been removed by setting a threshold. Data points with energy gaps lower than this threshold were excluded from the cluster. The regions around conical intersections could not be fitted as comprehensively as other regions of the PESs.
As another drawback, the authors note that the initial manual fitting of the signs is a tedious task, especially when larger systems and more dimensions are described.
Two of the authors also fitted diabatic PESs of two singlet states and one triplet state as well as the SOCs between singlets and triplets of formaldehyde, CH${}_{2}$O, with NNs. 90 The electronic structure reference method was MR-CISD/cc-pVTZ. The diabatic potentials were obtained using an adapted version of the Boys localization 351. The energy differences between two states are incorporated in the equations in order to remove earlier identified diabolic singularities 146. The range of $\pi$, which the rotation angle for the diabatization covers, guarantees a proper treatment of the Berry phase. The diabatization procedure further requires consistent transition dipole moments, which were adjusted manually for this purpose. The diabatic SOCs were then obtained as a linear combination of the adiabatic SOCs by applying the same rotation matrix as for the energies. One separate NN function was used to fit each coupling value and electronic state separately.
It becomes clear that only a small number of works on this topic exist. At the moment, many problems remain unsolved for generating a training set that properly accounts for both types of phases, the arbitrary phase and the Berry phase, and is applicable for large systems with many states. An automatic phase correction procedure without the need of manual input would be very advantageous, especially when larger and more flexible systems are treated. Further developments are needed.
4.2.2 ML-Based Internal Phase Correction
One step towards a routine application of ML for photochemical studies and an easier training set generation with quantum chemistry is an ML-based internal phase correction, which has been implemented by us into the SchNarc approach for photodynamics simulations 13. In contrast to the phase correction algorithm to correct the training data, this procedure renders the learning of inconsistent quantum chemical data possible. A modification of the training process, termed phase-free training, is required for this purpose. 13
We implemented this training algorithm in a combination of the deep continuous-filter convolutional-layer NN SchNet 428, 432, adapted for excited states, and the MD program SHARC 344, 242, 466
Similar to standard training algorithms, parameters of an ML model are optimized in order to minimize a cost function. Most frequently, the L${}_{1}$ or L${}_{2}$ loss functions are applied, which take the mean absolute error or mean squared error between predicted and reference data into account. The phase-free training algorithm uses a phase-less loss function, which includes all trained properties at once and additionally removes the influence of the random phase switches. In this way, the computational costs for the training set generation can be reduced.
Compared to the previously reported ML models for photochemistry, where each state was fitted independently 139, 90, 14, SchNarc is capable of describing all PESs at once, including the elements resulting from different pairs of states. This results in an overall loss function with several terms, where each term is weighted with a different trade-off value, $t$, that can be defined manually:
$$\begin{array}[]{ll}L_{ph}=\\
t_{E}\mid\mid E^{QC}-E^{ML}\mid\mid^{2}\\
+t_{F}\mid\mid F^{QC}-F^{ML}\mid\mid^{2}\\
+t_{SOC}\cdot L_{SOC}\\
+t_{NAC}\cdot L_{NAC}\end{array}$$
(16)
If only energies (E) and forces (F) are fitted, then the loss function is equal to a linear combination of L${}_{2}$ loss functions for energies and forces 109, 13. The parts of the SOCs and NACs are
$$\begin{array}[]{ll}L_{SOC}=min(\mid\varepsilon_{SOC}^{\kappa}\mid)\\
\text{with}~{}0\leq\kappa\leq 2^{N_{S}-1}\end{array}$$
(17)
and
$$\begin{array}[]{ll}L_{NAC}=min(\mid\varepsilon_{NAC}^{\kappa}\mid)\\
\text{with}~{}0\leq\kappa\leq 2^{N_{S}-1},\end{array}$$
(18)
respectively. The error for SOCs and NACs that enters the loss function is the minimum error that can be achieved when trying out all possible combinations of phases for each pair of states, i.e., 2${}^{N_{S}-1}$ possible solutions. The algorithm takes into account that the signs of SOCs and NACs coupling different pairs of states depend on each other.
The error function containing all possible solutions for SOCs, $\varepsilon_{SOC}^{phase}$, and NACs, $\varepsilon_{NAC}^{phase}$, can be obtained as follows:
$$\begin{array}[]{ll}\varepsilon_{SOC}^{\kappa}=\\
\frac{1}{N_{S}^{2}}\sum_{i=1}^{N_{S}}\sum_{i\neq j}^{N_{S}}\mid\mid C^{QC}_{%
SOC_{ij}}-C^{ML}_{SOC_{ij}}\cdot p_{i}^{\kappa}\cdot p_{j}^{\kappa}\mid\mid^{2%
}\\
\text{with}~{}0\leq\kappa\leq 2^{N_{S}-1}\end{array}$$
(19)
$$\begin{array}[]{ll}\varepsilon_{NAC}^{\kappa}=\\
\frac{1}{N_{S}^{2}}\sum_{i=1}^{N_{S}}\sum_{i\neq j}^{N_{S}}\frac{1}{N_{A}}\sum%
_{a=1}^{N_{A}}\\
\mid\mid C^{QC}_{NAC_{ij,a}}-C^{ML}_{NAC_{ij,a}}\cdot p_{i}^{\kappa}\cdot p_{j%
}^{\kappa}\mid\mid^{2}\\
\text{with}~{}0\leq\kappa\leq 2^{N_{S}-1}\end{array}$$
(20)
This phase-less loss procedure does not require any preprocessing of training data. Quantum chemistry calculations can be directly fitted with this adaption of the loss function. The power of this approach is that, once a given phase vector for a data point has been found, it can be directly applied to correct the arbitrary signs of other properties, such as transition dipole moments. If other properties are targeted, the loss function applied for NACs can be similarly used for other vectorial properties, and the loss function applied for SOCs can be used for any other single- or complex-valued element of arbitrary sign 13.
However, as a consequence of the higher complexity of the loss function, the training process is generally more expensive. The computational effort required for training can be reduced if only one type of coupling is treated within MD simulations. In these cases, a simpler adaption of the phase-free loss is also applicable. 13
4.3 Training Set Generation
The requirements and desirable specifications for a training set can vary strongly, dependent on the type of application: When the focus of a study is the investigation of the huge chemical space and the search for certain patterns thereof or the design of new molecules with targeted properties, usually the training set should be as large as possible to cover as many molecules as possible. In the best case, the data points are computed with high accuracy and this reference method is accurate for the excited states of many different types of systems. In terms of accuracy and general applicability, ab-initio methods are more suitable, as they do not require the selection of a density functional, which might be accurate for some cases, but fail for others. However, the costs and complexity of highly accurate multi-reference ab-initio methods limit their applicability, so that TDDFT remains the method of choice when making predictions throughout chemical compound space 327, 150, 467. The most widely applied approach to generate a training set for this purpose is to start from an existing (ground-state) data base that already covers a large chemical space of certain types of molecules. In this way, not much effort has to be devoted into the exploration of chemical space and structure optimizations to get the most stable conformations of different molecules.
For the purpose of ML-based excited-state dynamics simulations, things look quite different. Note that for photodynamics simulations, only molecule-specific ML model exist until now, which can potentially develop into a universal excited-state force field, but much remains to be done to achieve this goal. Indeed, the generalization of the excited state PESs and corresponding couplings is expected to be a highly complex task, especially due to the problematic generalization of excited states 92. A comparison of the isoelectronic molecules CH${}_{2}$NH${}_{2}^{+}$ and C${}_{2}$H${}_{4}$ can serve as an example. Their conical intersection between the first excited singlet state and the ground state is accompanied by a rotation along the dihedral angle, which could lead to very similar photo-initiated processes. However, higher-lying excited states are ordered completely different in both molecules and excitation leads to completely different photodynamics. 468, 469, 470, 471, 472, 473, 474, 475, 476, 477, 478, 92
As it stands, existing ML models for photodynamics simulations are developed to investigate the photo-initiated processes of one specific molecule.
Overall, we arrive at the following wish list for the training set, which has been identified also for MD in the ground state 479, 101, 480, 115: 1) The training set should be as small as possible to keep the number of reference calculations at a minimum. 2) At the same time, the relevant conformational space of the molecule that is required for the reaction under investigation should be sampled comprehensively 480, 115, 92, 442, 328.
Keeping this in mind, an efficient procedure to obtain relevant molecular structures has to be applied. A large number of schemes to achieve this goal have been proposed, which are mainly based on two different strategies: One approach is to simulate MD in the ground and excited states with the reference method and putting much effort into covering critical regions of the PESs comprehensively 137, 138, 139. Structure-based sampling or subsequent clustering is beneficial in this case 138, 137, 481, 482, 483. The other strategy is to use an active learning approach, which decreases the number of necessary reference calculations considerably, but is usually more time-consuming 480. Noticeable, within ML for quantum chemistry, active learning often refers to an approach, where an initial training set is used to fit an ML model and this previously learned information is applied to expand the training set 402.
The latter approach is often carried out with the help of MD simulations, but has also recently been adapted in a trajectory-free way 484, 402.
4.3.1 Basic Sampling Techniques and Existing Databases
To find patterns within certain groups of molecules, to explore chemical space and to develop new methods that can fit for example different properties of molecules, such as the valence density used in DFT 81, or large molecules from small building blocks 107, a good starting point is often considered to be an already existing data base. Prominent examples are the QM data bases, namely QM7, QM7b, QM8, and QM9 424, which have been used in a large number of publications up to date and provide a benchmark for many ML studies 485, 433, 486, 446, 428, 487, 488, 489, 12, 150, 434.
Especially the QM9 424 data set containing more than 133k small organic molecular structures and corresponding DFT energies, enthalpies, harmonic frequencies, and dipole moments (to name only a few properties) is very popular among the scientific community and has also been used in challenges on kaggle, where researcher and layperson all over the world can compete against each other to find the most suitable solution to a given task. Prices up to several thousand dollar are quite common 490.
In a similar spirit, the QM9 IPAM ML 2016 challenge requires to predict the energies of QM9 from only 100 training points within chemical accuracy (error of $\approx$0.05 eV).491
All aforementioned data bases originate from GDB data bases 492, 493, 494, and are often a subset thereof. The chemical universe GDB data bases have been designed using molecular graphs to sample a comprehensive space of molecular structures for the search of new lead compounds in drug design 494.
One of the first data bases available for the scientific community to treat the excited states of molecules is most probably the QM7b 495 data set, that contains the excitation energies computed with TDDFT for a total amount of $>$14k molecules with atoms C, N, O, H, S, and Cl. This data set is based on the molecular geometries of the QM7 494, 100 data set plus an additional amount of 7211 molecules containing a chlorine atom. The excitation energies of the first singlet state and other properties were recomputed for each optimized molecular geometry. Very similar, the QM8 327 data base was developed, based on the GDB-17 data base 496. This data set can be used for the computation of vertical excitation spectra. It hence includes not only the vertical excitation energies of the first excited singlet state, but also the corresponding oscillator strengths. Oscillator strengths are also reported in an auto-generated data set for optoelectronic materials with DFT. 467 Note that the oscillator strength is computed from the squared transition dipole moment, hence an arbitrary phase factor cancels out and the data does not have to be preprocessed. In addition to the TDDFT energies, CCSD energies are reported, having enabled the development of the so-called $\Delta$-learning approach - a powerful way to obtain the accuracy of highly accurate ab-initio methods with only a small amount of respective reference calculations. Two ML models are trained in this approach, one on a less accurate method and another one on the difference between the less accurate and higher sophisticated method 497. This scheme can also be applied multiple times to achieve increasing accuracy with little additional computational effort 328 and has been adapted for spectroscopy in the condensed phase as well 151.
The QM9 data set has further been the basis of a very recently constructed data set for singlet and triplet states of $>$13k carbene structures, termed QMspin 12. 4,000 geometries from the QM9 data set were randomly selected, hydrogen atoms were subtracted and singlet and triplet states were optimized using CASSCF(2,2)/cc-pVDZ-F12 and open-shell restricted KS-DFT with the B3LYP 498, 499 functional, respectively. The MR-CI method was subsequently used to compute the electronic energies of singlet and triplet states. This data set has been used to investigate structural and electronic relationships in carbenes, which are important intermediates in many organic reaction networks 12.
The OE62 500 data base, a benchmark data set applicable for spectroscopy, is another descent of several existing data sets, such as the QM8 and QM9 data sets. It consists of $>$61k organic molecules able to form crystals including up to 174 non-hydrogen atoms. Reported are the orbital energies of molecules computed with DFT/PBE 501.
Another database, which also contains excited state data, is the PubChemQC data base. 502 It contains over three million molecules, whose structures are reported along with the energies at DFT/B3LYP/6-31G* level of theory. In addition, the excitation energies of at least three million structures are reported for the 10 energetically lowest-lying singlet states at TDDFT/B3LYP/6-31G* level of theory.
A simple strategy was carried out by Kolb et al., 503 who used an existing analytical PES to create an ML potential: They randomly sampled data points, trained an ML model and added more points in regions with deviations from the original PES. Other strategies have been carried out mainly for the fitting of ground state potentials and for materials, which are however also relevant to consider for the excited states. One novel, suitable strategy is for example "de novo exploration" of PESs using a similarity measure provided by ML models. 504 At least for material discovery, this method can be used to omit any additional active learning procedure to converge PESs. Another way to build a training set is to employ molecule-generating ML models,164, 505, 506
such as the recently developed Gschnet. 507 Alternatively, MD simulations with the reference method can provide a good starting point for training. 486, 508, 120
Ye et al. 509 sampled 70k conformations for N-methylacetamide via MD simulations with the OPLS force field510
within GROMACS 511 for subsequent UV spectra calculations.
We have applied a similar scheme to generate a training set of SO${}_{2}$ based on an LVC model 229. Surface hopping MD simulations with the SHARC method 344, 242, 466 were carried out with the reference method LVC(MR-CISD) ending up in $>$200k data points of different conformations of SO${}_{2}$ 13. Due to the crude sampling and low cost of the reference method, no emphasis was put on clustering the training set into a smaller, still comprehensive set.
90k data points were required in an ML-based surface hopping study of CH${}_{2}$NH with the Zhu-Nakamura method. Reference data for the ground and first excited singlet state, S${}_{0}$ and S${}_{1}$, were generated with CASSCF(2,2)/6-31G via ground-state and surface hopping MD simulations. The latter method was applied to sample the regions around conical intersections between the S${}_{0}$ and S${}_{1}$ state. 139
Similarly, Hu et al. 137 sampled 200k data points of 6-aminopyrimidine using ground-state and surface hopping MD with CASSCF(10,8)/6-31G*. State-averaging over three singlet states was applied. In addition, structures that led to hops between different states were used as starting points to find minimum energy conical intersections and clustering was carried out to reduce the amount of data for training.
One way to select data points more efficiently is a structure-based sampling scheme, as proposed for instance by Ceriotti et al. with sketch map, 512, 513, 481 an algorithm for dimensionality reduction of atomistic MD simulations or enhanced sampling simulations.
Likewise, Dral et al. 138 applied a grid-based sampling method to construct PESs of a model spin-boson Hamiltonian to execute surface hopping MD with KRR. The energetically low-lying regions of the PESs were first sampled via an inexpensive method and subsequently the distances between the molecular structures were computed. In this way, 10,000 data points were obtained 482, 138. ML models trained on only 1,000 data points were accurate enough to reproduce reference dynamics. This approach was compared with random sampling for the methyl chloride molecule and was shown to reduce the amount of training data needed up to 90% for static calculations 482, 483.
4.3.2 Active Learning
As shown in the previous section, training sets with the respective equilibrium structure of a large number of molecules are very powerful for investigating the huge chemical space or for the design of new molecules. However, the usefulness of such training sets for photodynamics is rather questionable. The reason for this deficiency is that, especially in MD simulations in the excited states, the excess of energy carried by a molecule very quickly leads to conformations that are far beyond the equilibrium structure and most likely far away from originally sampled structures. The formation and breaking of bonds is quite common in photodynamics simulations and is usually only accessible from an excited, dissociative state. The use of photodynamics simulations with the reference method could solve this problem, but are not feasible if specific reactions occur on a rather slow time scale or if many different processes take place. 28, 27, 36, 37, 171, 400 As previous studies have shown, inefficient sampling techniques lead to a huge amount of data, which still does not guarantee that the training set is comprehensive enough for excited-state MLMD simulations. In fact, ML models fail dramatically in under-sampled and extrapolative regions of the PESs.
A smarter sampling technique is advantageous in these cases in order to efficiently identify such under-sampled regions and build trustworthy ML models.
Active learning, where ML ”asks” for its training data, is one solution to create a data set more efficiently. An example from chemistry is the adaption of an initially generated training set due to an uncertainty measure for ML models trained on this initial training set. This concept has already been introduced in 1992 as query by committee 514 and has been adapted for quantum chemistry quite fast due to the required fitting and interpolation of PESs for grid-based quantum dynamics simulations.
Pioneering works by Collins and co-workers 377, 149, 148, 515 applied modified Shepard interpolation to fit PESs and iteratively adapt them in out-of-confidence regions using the GROW algorithm 515, 516. Since then, several sampling techniques have been developed that are based on MD and an extension of data bases using interpolation moving least squares 517, 518, permutation invariant polynomial fitting 519, 520, and different ML models for the ground state 521, 522, 523, 99, 524, 525, 526, 479, 101, 480, 527, 528, 109, 529, 530, 531, 532, 533 and also excited states 138, 92, 13.
As active learning starts from already trained ML models, an initial training set has to be provided. Some strategies to provide this initial reference data set will be discussed, following strategies applied to adapt this initial training set. Note that all previously discussed methods can be similarly applied to generate an initial training set.
Initial training set
In general, an initial training set can be obtained in many different ways. As photo-initiated MD simulations usually start from vertical excitation of the ground state equilibrium geometry, this structure is commonly used as the starting point and reference geometry for the training set generation.
In principle, any technique can be applied to then add conformations to obtain a preliminary training set.
A good starting guess is to use normal modes of a molecule, as they are generally important for dynamics. In two recent works, we carried out scans along different normal modes and combinations thereof to sample conformations of small molecules. 92, 13 Normal modes are also sampled for generating ANI-1 NN PESs. 113 For the excited states, it is favorable to include critical regions of the molecule in the initial training set by carrying out optimization of these geometries and including the calculations into the training set 92, 137.
When small molecules are targeted, this initial training set can already be comprehensive to start the training of ML models and adapt the training set based on an uncertainty measure provided by the ML models 92.
In case more flexible and larger molecules are studied that give rise to a complex photochemistry and a high density of states including different spin multiplicities, a small initial training set might not be sufficient and a larger conformational space of the molecule needs to be sampled. This can be done for example via Wigner sampling 534 and also with MD simulations in the ground state 535, 536. Suitable methods are for example umbrella sampling 537, trajectory-guided sampling 538, enhanced sampling 539 or metadynamics 540 in combination with a cheap electronic structure method like the semi-empirical tight-binding based quantum chemistry method GFN2-xTB 541 or existing ground-state force fields. A large amount of different geometries can be created very fast and inexpensively, which then can be clustered to exclude similar conformations of the molecule to keep the number of reference simulations at a minimum. The selected data points for the training set can then be computed with the chosen reference method, whose accuracy is targeted with ML.
Additionally, if certain reaction coordinates have been shown to be important in experiments or previous studies, then it is favorable to include data from scans along these reaction coordinates. 400, 94
As soon as meaningful ML models can be obtained from the initial training set, active learning techniques can be applied to enlarge the set. What number of data points turns out to be sufficient for the initial training set is dependent on a lot of different factors, such as the size and flexibility of the molecule under investigation, the number of excited electronic states described, and the ML model and descriptor applied 92, 93. In order to give a ballpark figure, we note that we used approximately 1000 data points as initial training set for small molecules in recent studies using deep multi-layer feed-forward NNs. 92, 13
Strategies for actively expanding the training set
The next step in active learning is to expand the initial training set by adding points from out-of-confidence regions.
The detection of these undersampled regions can be done in many different ways, whereby most approaches rely on MD simulations.
Among the most popular strategy is the iterative sampling scheme of Behler 480, originally developed for fitting ground-state PESs. Today, it is widely used, see for example refs 479, 101, 542, and has been modified as a so-called adaptive sampling approach.109 The latter has been adapted by us for the generation of a training set for the excited state PESs of molecules including couplings 92.
The basis of almost any iterative or adaptive sampling scheme is a similarity measure to judge whether a molecular geometry can be predicted reliably with ML models or not. While kernel methods intrinsically provide a measure of similarity for each molecular geometry, NNs do not. Therefore, adaptive sampling with NNs requires at least two ML models. In case of KRR or GPR, two ML models can be used as well, but are not necessarily needed. Indeed, the statistical uncertainty estimate of the predictions remains a huge advantage of GPR models. 543, 442
The adaptive sampling scheme for the excited states is illustrated in Figure 9 and exemplified with two ML models. The whole process starts with an initial training set, which is used to train the two (or more) preliminary ML models. These models differ in their initial weights or model parameters. The resulting dissimilar ML architectures guarantee that the ML models do not predict the exact same number for a given molecular input. The hypothesis underlying this scheme is that inferences of different ML models trained on the same training set will be similar to each other as long as an interpolative regime is given. The inferences of the ML models are inaccurate and should differ from each other to a much larger extent if a molecular input lies in an unknown or under-sampled region of the PESs.
In order to find such regions, sampling steps are carried out, e.g., by running (excited-state) MD simulations based on
the mean of the inferences made by the different ML models for energies, $\overline{E}^{ML}$, forces, $\overline{F}^{ML}$, and if required also couplings, $\overline{C}^{ML}$. In each sampling step, the variances for each predicted property are computed. In the present example, energies and forces are treated together as $\sigma_{E+F}^{ML}$ (but can also be used separately), separately from variance of the couplings $\sigma_{C}^{ML}$. If a variance exceeds a pre-defined threshold, the ML models diverge and the predictions are deemed untrustworthy. $N_{ML}$ refers to the number of different ML models, $\zeta$, used for adaptive sampling:
$$\begin{array}[]{ll}\sigma_{E+F}^{ML}=\\
\frac{1}{N_{S}}\sum_{i}^{N_{S}}\left(\sqrt{\frac{1}{N_{ML}-1}\sum_{\zeta=1}^{N%
_{ML}}\left(E^{ML}_{\zeta}-\overline{E}^{ML}\right)^{2}}\color[rgb]{1,1,1}%
\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}%
\pgfsys@color@gray@fill{1}\right)+\\
\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}%
\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}\left(\color[rgb]{0,0,0}%
\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}%
\pgfsys@color@gray@fill{0}\sqrt{\frac{1}{N_{ML}-1}\sum_{\zeta=1}^{N_{ML}}\left%
(\frac{1}{3N_{A}}\sum_{a}^{3N_{A}}\left(F^{ML}_{\zeta,a}-\overline{F}^{ML}_{a}%
\right)^{2}\right)}\right)\end{array}$$
(21)
$$\begin{array}[]{lr}\sigma_{C}^{ML}=\\
\frac{1}{2N_{S}^{2}}\sum_{i}^{N_{S}}\sum_{j}^{N_{S}}\sqrt{\frac{1}{N_{ML}-1}%
\sum_{\zeta=1}^{N_{ML}}\left(C^{ML}_{\zeta}-\overline{C}^{ML}\right)^{2}}\end{array}$$
(22)
Note that the variance is averaged over all states for energies and forces and over all pairs of states for couplings, that are described with the ML models. As a variant, each state could also be treated separately. However, as the different electronic states are not independent of each other, a mean-treatment is assumed to be advantageous 93.
Each data point that is predicted with a variance larger than the pre-defined threshold for a given property, is recomputed with the reference quantum chemistry method and added to the training set. In this way, undersampled or generally unknown regions of the PESs are identified. Whenever the variance of each property is within the range that is thought to be reliable, the mean of the inferences is forwarded to the MD program to propagate the nuclei and continue MLMD simulations.
The name adaptive sampling is based on the recommendation to choose a rather large threshold in the beginning of the adaptive sampling procedure and to adapt this threshold to smaller values as the ML models become more accurate and robust 109. A first estimate for the initial value of a threshold can be obtained from the MAE of the corresponding ML model on the initial training set.
In principle, adaptive sampling can be carried out for every property, that should be represented with ML potentials, and is not restricted to energies, forces, and couplings. Similarly, it does not need to be executed with excited-state dynamics, but could also be done with ground-state MD or any sampling method that is considered to be suitable.
As a negative side effect, this procedure is generally more time-consuming than many other sampling techniques, because ML models have to be trained each time a new data point is added to the training set. To apply adaptive sampling in a more efficient way, it is advantageous to execute not only one ML trajectory, but many hundred trajectories in parallel, as it is usually done in MD simulations. The ML models should then only be retrained, when all ML-based trajectories have reached an undersampled conformational region 480, 109, 92.
Despite the higher complexity of adaptive sampling compared to random sampling, it can reduce the number of required data points for MLMD simulations substantially. In this regard, also the computational costs for the training set generation can be kept at a minimum.
Adaptive sampling was carried out successfully to generate a training set of 4,000 data points of CH${}_{2}$NH${}_{2}^{+}$ containing three singlet states and couplings. ML-based surface hopping MD simulation could be carried out on long time scales using the average of two deep NNs. The concept of iterative sampling also proved beneficial for the long MD simulation to guarantee accurate ML potentials throughout the production run. Here, the threshold was not adapted anymore and the MD was continued from the current geometry after a training cycle was completed 92. In addition, the average of more NNs turned out to be more accurate than the prediction of only one NN, which was also shown in Ref. 109.
Another quality control besides the property-based one proposed by Behler can be obtained by comparing the molecular structures at each time step as done by Dral et al. 138, 482 and Ceriotti et al. 481.
A combination of a structure-based and property-based detection of sparsely sampled regions of the PESs has been done by Zhang et al. and Guo et al. 524, 362, 544, 545, 546 Very recently, an alternative approach has been applied with NNs by Lin et al. 402 that does not require MD simulations. It is based on the finding that the negative of the squared difference surface obtained from NNs approaches zero in regions, where no data points are available. 518
Therefore, new points can be computed at the minima of the negative squared difference surfaces of at least two NNs (or, equivalently, at local maxima of the squared difference surface). This method is supposed to be very efficient in cases, where different conformations are separated by large energy barriers or strongly stabilized local minima are common. MD simulations would take a long time to overcome the potential barriers and reach the region of unknown molecular structures. 402
The idea behind this technique is similar to previous works with GPR. A measure of confidence can be provided with GPR models that enables the search of regions with large variance in ML predictions. In these regions, data points can be added to build up a training set .547, 548, 484, 549 Similarly, Bayesian Optimisation Structure Search (BOSS) has been proposed for constructing energy landscapes of organic and inorganic interfaces. 550
A combination of different approaches has also been applied by Häse et al., 160 who fitted TDDFT excited-state energies of a light-harvesting system. Given a large enough, error-free, and comprehensive data set, ML has the potential to determine known and unknown (un)physical laws within the data. 551
5 ML Models
Besides the training set, which defines the highest possible accuracy an ML model can attain, the type of regressor and the descriptor to represent a molecule to the ML model play also important roles.552 Improper choices of regressors and descriptors can result in inaccurate ML models.
5.1 ML Models: Type of Regressor
Given the vast number of ML algorithms applied in the field of computational chemistry, one might ask which one to use or adapt for photochemistry. As recent studies applying ML for quantum chemistry have shown, many possible choices of ML approaches exist and there is no single solution. Nevertheless, a trend can be observed: Many studies that use ML in the research field of quantum chemistry employ labelled data sets, i.e., supervised learning techniques. Within supervised learning, one can distinguish between regression and classification. Classification aims at finding patterns and at grouping data into certain clusters. 553 Those types of ML models are often used e.g. in spam filters, in medicine to diagnose diseases 554, 555, or in food research, e.g. to guarantee a certain wine quality or origin. 556 Examples of applied classification models in the field of computational chemistry are for example support vector machines, random forests or decision trees used, e.g., to classify enzymes 557 or for the selection of an active space 70, 558.
More often than classification models, regression models are applied to assist the search for a solution of a quantum chemical problem. Regression is used to fit functions that can relate a molecular input, $X$, to a quantum chemical output, $Y$.
The simplest relation that can be assumed is linear. Although many quantum chemical problems cannot be accurately described with a linear function as given in eq. 23, it can serve as a baseline model to evaluate the minimum accuracy one can obtain. 553, 171, 559, 560, 92
$$Y=b+w\cdot X$$
(23)
The regression coefficients, also known as weights, $w$, and biases, $b$, are tailored for a given problem under investigation. In case of linear regression, ordinary least squares regression can be applied to find these coefficients. The process of finding the optimal relation between $X$ and $Y$ is termed training. The coefficients are optimized by minimizing a so-called loss function, $L$, which monitors the error between the original property, $Y^{QC}$, and the predicted property by the ML model, $Y^{ML}$, with respect to the training instances. Most often, the L${}_{1}$ loss or the L${}_{2}$ loss is used as an indicator for the training convergence. The L${}_{1}$ monitors the mean average error (MAE) and the L${}_{2}$ loss the mean squared error (MSE) of predictions:
$$L_{2}=\frac{1}{N_{M}}\sum_{\beta}^{N_{M}}\left(Y_{\beta}^{ML}-Y_{\beta}^{QC}%
\right).$$
(24)
The Greek letter $\beta$ runs over all molecules, N${}_{M}$, inside the training set. In principle, any error estimate can be used to train an ML model and find suitable regression coefficients.
An example specifically developed for excited-state problems is the aforementioned phase-less loss (see section 4.2.2).13
Such adapted loss functions and also conventional ones are employed in different types of ML models. In the following, we focus on the two most widely used models for the description of the excited states: Kernel methods and NNs.
Kernel methods
Kernel methods 561 are based on a similarity measure between data points. Examples are KRR or GPR, which go beyond linear regression by applying the kernel trick and ridge regression. Ridge regression is used to find the weights, which differs from linear regression by a regularization term, $\lambda$:
$$w=(K+\lambda\mathbf{1})^{-1}Y^{QC}$$
(25)
Y${}^{QC}$ refers to the training data and $K$ to the kernel matrix.
The kernel trick makes it possible to apply ridge regression to non-linearly separable data by mapping them into a higher-dimensional feature space, in which the data points are linearly separable.
Therefore, a kernel function, $k$, e.g. a Gaussian or Laplacian, is placed on each compound to measure the distance to all of the other compounds in the training set. The kernel function defines the non-linearity of the model. A property of a query compound, $\alpha$, can be obtained as the weighted sum of regression coefficients and kernel instances:
$$Y^{ML}(X_{\alpha})=\sum_{\beta}^{N_{M}}w_{\beta}K(X_{\alpha},X_{\beta}).$$
(26)
The size of the kernel matrix is dependent on the number of training points and hence the depth of the model is inherently linked to the size of the training set, which is why they are called ”non-parametric”.553, 562
An advantage of kernel methods is that they mainly contain two hyperparameters, i.e., internal model parameters, which need to be optimized for proper training.
Most important are the width of the non-linear kernel function, $\sigma$, and the regularization. The latter is used to prevent the model from overfitting – the case when the model fits training data including noise almost exactly and fails to accurately predict data points not included in the training set but stemming from an interpolative regime. As quantum chemical data is most often noise-free, the regularization term is usually small.
As the optimization of hyperparameters is often a tedious task, kernel methods with their few hyperparameters are easier to use than, e.g., NNs with many hyperparameters. Nonetheless, kernel methods can provide almost exact solutions of problems under investigation. 125 A drawback is, however, that the inversion of the kernel matrix can become expensive and even be rendered infeasible on current computers due to increasing memory requirements with increasing training set size. 93
Further, kernel methods are usually defined to only map an input to a single output. Therefore, they can treat only one electronic state at a time in standard implementations and, thus, can be referred to as single-state models. A single-state treatment requires a separate ML model for each electronic state or for each property resulting of a pair of states, whereas a multi-state ML model describes all electronic states and properties resulting from different pairs of states at once. 400, 93 Hence in their standard implementation, the treatment of several excited states necessitates the use of several kernel models, which is commonly done in the research field of quantum chemistry. 563, 564, 147, 138, 137 The description of forces is possible for the ground state or a single excited state and is implemented, e.g., in the QML toolkit using KRR and the Faber-Christensen-Huang-Lilienfeld (FCHL) representation, 433 in the symmetric gradient domain ML (sGDML) 120, 120 method or with smooth overlaps of atomic positions (SOAP) 565 for GPR. 119
Neural Networks
Another prominent approach in ML is the use of NNs as highly flexible parametric functions, which can fit huge amounts of data and can map a molecular input to many quantum chemical outputs. 93
The simplest form of NNs are multi-layer feed-forward NNs, which are schematically represented in Fig. 10.
As it is visible in Fig. 10, the width of the model is dependent on the number of nodes, $n_{r}^{t}$, which are connected to each other using weights, $w_{rs}^{tu}$. The indices refer to a connection between node r and node s from layer t and layer u, respectively. The number of nodes and hidden layers can be chosen independently of the training set size.
Due to the highly flexible functional form of NNs, highly complex relationships can be fit, but an analytical solution to find the weights is not available (in contrast to KRR). A numerical solution can be obtained with stochastic gradient algorithms, which are frequently applied to obtain a step-wise update of the weights:
$$w_{k+1}=w_{k}-l_{r}\nabla L_{2}(w).$$
(27)
The gradient of the loss function as given in eq. (24) with respect to the weights is multiplied with a so-called learning rate, $l_{r}$. This hyperparameter is deemed one of the most important hyperparameters used for training. 9, 566 In order to obtain an optimal solution, the learning rate needs to be chosen properly.
Algorithms such as AdaGrad 567 or Adam 568 can automatically adapt the learning rate during training. Further, the second-order derivatives can be included into algorithms, which is for instance done in the global extended Kalman filter 569, in its parallel variant,570
or the element-decoupled variant 103. The loss function can be adapted so that more than only one property can be trained at once. This is often done to include the forces in the training process.
In general, NNs possess various hyperparameters like the learning rate, regularizers, number of nodes, etc. As a consequence, an extensive hyperparameter search complicates the use of NNs and makes them more complex to apply than kernel methods.
Besides simple multi-layer feed-forward NNs, high-dimensional variants exist. These networks comprise several atomic NNs, which represent atoms in their chemical and structural environment and are thus also called atomistic NNs.
Each local atomic contribution, $E_{a}$, can be summed up to provide the energy of the whole system, $E$, which is well known to work for the ground state PESs:
$$E=\sum_{a=1}^{N_{A}}E_{a},$$
(28)
and was originally implemented by Behler to construct high-dimensional NN potentials. 571
Embedded-atom NNs 533 are similar to high-dimensional NNs in their way of constructing the energy of a system. They differ in the underlying descriptors to the ones of Behler. Atomic contributions to the energy are dependent on the embedded density of atoms and are summed up according to eq 28. These embedded density-like descriptors are approximated from atomic orbitals.
Independent of a simple or an atomistic architecture, the model can be used to fit a single output or a vector of many outputs at the same time. For ground state problems, a single-state model is usually used, which maps an input to a single output, e.g. the PES of the ground state. Oftentimes, this single-state fashion is adapted to fit different excited states with different NN models. 14, 199, 139, 509 However, it has been shown that including more excited-states in one model can be advantageous 93, as the excited-states are inherently linked to each other and so are the excited-state properties. 37 Treating many excited states can be referred to as multi-state model and the inclusion of more properties can result in a multi-property model. 93, 400, 76, 95, 186 The different properties can be weighted with respect to their magnitudes or importance for a given chemical problem under investigation, such that the best possible accuracy can be obtained. 13
Another type of networks are convolutional NNs, which are most often applied in image or speech recognition 572, 573, 574, but can also be adapted to process a molecular input and identify an optimal molecular descriptor. This type of network can be combined in an end-to-end fashion with an architecture, which fits this generated molecular representation to a query output. 575, 508, 576, 428, 432
An important ingredient of all these ML models is the descriptor, which is mapped to the output. In most studies, the descriptor is one of many different possibilities to represent a molecule, which will be discussed in the next section.
5.2 Descriptors and Features
Electronic structure methods can process and uniquely identify molecules using e.g. Cartesian coordinates. In contrast, such types of inputs are not optimal for ML models as the same molecular geometry, but translated or rotated, could only be mapped to the same output with great effort and unnecessary computational cost. Hence, a molecular descriptor should fulfill the following requirements: It should be translationally, rotationally, and permutationally invariant as well as differentiable. 102 It should also be unique with respect to the relative spatial arrangement of atoms, universally applicable for any kind of system, and computationally efficient.552
However, a descriptor can be more than that; it can already include a part of the mapping, e.g., from a molecular structure to an energy. It can thus ease the task of the regressor and help to attain the best possible accuracy for a given training set.
The ways to represent a molecule to an ML model can be classified roughly into two categories: molecule-wise descriptors, which represent the molecule as a whole to the ML model, and atom-wise descriptors, which represent atoms in their chemical and structural environment and build up a property using local contributions. 480, 102 Both ways in describing a molecular system have their merits and pitfalls and will be discussed along with their applications in recent studies for the excited states in the following.
Molecule-wise descriptors
The distance matrix is one of the simplest descriptors that preserves rotational and translational invariance. Most often it is used in its inverse form with distances between atoms a and b,
$$D_{ab}=\frac{1}{\mid\mid r_{a}-r_{b}\mid\mid},$$
(29)
giving rise to the symmetric inverse distance matrix, $\mathbf{D}$. Due to the ill-definition of diagonal elements, which are not differentiable, the diagonal elements are excluded and only the upper or lower triangular matrix is used to represent a molecule to an ML model. 564 Since the Hamiltonian contains distances rather in the denominator, it makes sense to also use the matrix of inverse distances.92 The matrix of inverse distances is very similar to the Coulomb Matrix, $\mathbf{C}$: 100
$$C_{ab}=\begin{cases}0.5Z_{a}^{2.4}~{}~{}\text{if}~{}~{}a=b\\
\frac{Z_{a}Z_{b}}{\mid\mid r_{a}-r_{b}\mid\mid}\end{cases}$$
(30)
but the Coulomb matrix additionally considers the atomic charges, Z. These types of descriptors are frequently used in ML studies for the excited states. For example, MLMD simulations in the excited states could be advanced using these simple descriptors 138, 137, 92, 93 and were also accurate enough to fit NNs and KRR models for excited-state properties 509, 93, 92, 327, 160, 563. Distance based descriptors are further implemented in several program packages that have been used for photodynamics simulations with KRR. For example, MLAtom 577 contains the Coulomb Matrix and a representation that includes all nuclear pairs in form of normalized inverted internuclear distances 482. The QML toolkit 578 includes the Coulomb matrix in addition to other representations, such as bag of bonds 579. Another variant are polynomials formed from inverse distances 92.
These molecule-wise descriptors have the advantage of being easy to use and implement. Especially for small molecular systems and with regard to the training of an ML model, they are cheap. However, they might miss some important information based on angular distributions. Currently, it is also investigated, whether representations based on two-body or three-body terms are accurate enough to uniquely identify a molecule. 580
A problematic issue of the aforementioned types of distance-based molecular descriptors is that they are not permutationally invariant. 480, 102, 576, 400 This problem can be mitigated by data augmentation, i.e., randomly permutation of atoms by mixing of matrix rows, which results in more data points for the same molecular input. The additional amount of data increases rapidly with the system size and could lead to long training times. 480, 576 Alternatively, another metric than the commonly used L${}_{1}$ or L${}_{2}$ norms can be employed, the so-called Wasserstein metric, which was tested with the Coulomb matrix. 581
Permutation invariant polynomials (PIPs), introduced by Bowman and co-workers, 519, 582, 520 are frequently applied in a PIP-NN approach by Guo and coworkers to investigate photochemical problems 359, 360, 361, 362, 141, 145, 142. The advantage of these polynomials is that they are invariant to permutation of atoms and inversion. 145 They comprise single-valued functions, $p_{ab}$, such as logarithmic or Morse like functions, which incorporate internuclear distances, $r_{ab}$. The PIP vector, $\mathbf{G}$ is obtained applying a symmetrization operator, $\hat{S}$, accounting for possible permutation operations:
$$\mathbf{G}=\hat{S}\prod_{a<b}^{N_{A}}p_{ab}$$
(31)
with an example of p${}_{ab}$:
$$p_{ab}=e^{-cr_{ab}}.$$
(32)
Evidently, additional hyperparameters such as $c$ have to be optimized and the choice of PIPs is generally not unique. 360, 543
Another negative aspect of molecule-wise descriptors is that they can only treat one molecular system, because the input size is fixed. The input dimension could, in principle, be defined according to the largest system included in the training set, but this would lead to unnecessarily large input vectors for smaller systems, which would then contain many zero values. 576, 571 The training of more ML models, each for one specific system size, is one possible solution 160, but obviously necessitates the training and evaluation of more than one ML model.
Atom-wise descriptors
In contrast, atom-wise representations allow for a fitting of molecules of arbitrary size and composition. Such descriptors are state-of-the-art for ground-state problems with commonly used examples being the SOAP 565, atom-centered symmetry functions (ACSF) 571, weighted ACSFs 446, 583 or the FCHL representation. 125, 487 These representations describe atoms in their chemical and structural local environment and usually rely on a cut-off function. This cut-off function defines the sphere around an atom, which is deemed to be important and is therefore considered when modelling the atomic local environment. Radial distribution functions, so-called second-order terms, account for inter-atomic distances and are often used together with angular distribution functions, i.e., third-order terms. It is further beneficial to include first-order terms, i.e., the stoichiometry of atoms. 125, 428, 446, 583 Most often, higher order terms than third-order terms are not included due to increasing costs and little improvements in accuracy. 576
The description of PESs from atomic contributions is beneficial in order to treat systems of arbitrary sizes and to use systematic molecular fragmentation methods. 107
Admittedly, the validity of this approach is not so clear for the excited-states and consequently, such representations are less frequently used in ML studies targeting the excited states. Up to day, only small molecules have been fitted with atom-wise representations, which are too small to prove the validity of excited-state PESs, which are constructed from local atomic contributions. To the best of our knowledge, the largest molecule fitted with atom-wise descriptors contained 12 atoms and was N-methylacetamide. 95
Other molecules were CH${}_{2}$NH${}_{2}^{+}$ 93, 13, CH${}_{2}$NH 139, SO${}_{2}$ 13 or CSH${}_{2}$ 13. Further studies are needed to demonstrate whether an atom-wise construction of excited-state properties and PESs is possible or not. Nevertheless, this approach is most powerful for studies that aim to describe large and complex systems, which could potentially be described from smaller building blocks. For instance, the construction of a DNA double strand or a peptide could be, at least in principle, constructed from ML models that are trained on their smaller subsystems, i.e., DNA bases and amino acids, respectively. Unfortunately, we are far away from having achieved a description of large molecular systems for the excited states, let alone the construction of accurate PESs of medium-sized molecular systems, such as DNA bases or amino acids.
Other types of descriptors
Besides the benefits high-dimensional ML models offer for the fitting of PESs of molecules, descriptors are not restricted to the aforementioned examples. In general, any type of descriptor might be suitable for a given problem. Applied descriptors range from topological and binary features generated from SMILES strings 584 to normal modes, which are often used as a coordinate system and descriptors to fit diabatic PESs. 143, 134, 144, 97, 585, 143, 147, 359, 360, 361, 362, 141, 145, 14 Other types of molecular features besides structure-based ones, e.g. electronegativity, bond-order, oxidation states, …, 70, 15 are also used.
Automatically generated descriptors
The selection of an optimal descriptor and the optimization of the related parameters for this descriptor is no trivial task and requires expert knowledge in many cases. 576
A way to circumvent an extensive parameter search is offered by the aforementioned message passing NNs 575, which include the descriptor parameters in the network architecture. In this way, they automatically fit the optimal parameters of a descriptor for a given problem, i.e., training set under investigation. Such tailored descriptors can guarantee highly accurate solutions if the NN model is trained properly. PhysNet, 586 HIP-NN 587 or Deep Tensor NN (DTNN) 508, which forms the basis of the deep learning model SchNet, 428, 432, which in turn is used within the SchNarc approach for excited states,13 are examples of such NNs.
6 Application of ML for Excited States
In this chapter, we review ML studies of excited states and their properties. We aim to show how they have been employed to improve static and dynamics calculations and focus on the used type of regressor, descriptor, training set, and property. We will classify the approaches according to Figure 1.
6.1 Parameters for Quantum Chemistry
At the current state of research, the user must decide whether a multi-reference method is necessary or a single reference method is sufficient to describe a chemical problem. It would be helpful if ML models could suggest a suitable reference method, e.g. based on a literature search. Unfortunately, such a tool is not yet available, but ML can help to select an active space for multi-reference methods. Jeong et. al 70 developed an ML protocol for classification based on XGBoost 558 to allow for a ”black box” use of many multi-reference methods by automatically selecting the relevant active space for molecular systems. The tedious selection of active orbitals and active electrons can thus be avoided. The accuracy of this approach was demonstrated for diatomic molecules in the dissociation limit and the molecules were represented via the molecular orbital bond order and the average electronegativity of the system.
6.2 ML of Primary Outputs
To the best of our knowledge, no ML models for providing primary outputs of quantum chemistry exist for excited states (see Figure 1). Targeting the primary output of a quantum chemistry simulation, i.e., the N-electron wave function, or providing ML density (functionals) is far from trivial even for ground-state problems. 71, 72, 73, 74, 75, 76, 77, 78, 79, 89, 588, 81, 589, 590, 591 However, such an approach for excited states could solve many problems and allow for wave function analysis, providing additional insights like the excited state characters.592
Therefore, we expect such models to appear in the near future.
6.3 ML of Secondary Outputs
In the following, we summarize the contributions of ML models that fit the secondary output of quantum chemical calculations, i.e., PESs, SOCs, NACs, and transition as well as permanent dipole moments in the adiabatic and diabatic basis (Figure 1).
The prediction of the manifold quantities (see Fig. 2) can be done in two ways, i.e., in a single-state fashion and in a multi-state fashion. 93
The applicability of such ML models to the simulation of photodynamics will be discussed.
6.3.1 ML in the Diabatic Basis
Diabatic PESs are fitted with ML and related methods since more than 25 years. 377, 149 An advantage of diabatic PESs is their smoothness, which is perfectly matched by ML models built upon smooth functions. However, the tedious procedure to generate diabatic PESs remains. Some effort is therefore devoted to develop ML-assisted diabatization procedures and eliminate this limiting step.
Diabatization
Williams et. al 140 incorporated NNs into diabatization by ansatz and fit diabatic NO${}_{3}$ PESs.
Recently, Shen and Yarkony 94 fit two diabatic potentials of the cyclopentoxy radical, C${}_{5}$H${}_{9}$O, and one state of cyclopentoxide, C${}_{5}$H${}_{9}$O${}^{-}$, with 356 data points sampled from scans along different reaction coordinates. The diabatization was assisted with NNs. Due to the high dimensionality of the system,
the authors resort to application of regularization in the fitting algorithm and an adapted loss function to obtain an accurate representation of two-state diabatic PESs with NNs. This novel strategy is envisioned for the computation of the photoelectron spectrum of cyclopentoxide 94. Fitting 39 degrees of freedom in the diabatic basis is a huge improvement in this research field.
The authors further note that a comprehensive sampling of the full relevant PESs in such high dimensional space is problematic.
Due to the aforementioned problems, a description of medium-sized to large molecules with diabatic potentials is often done with more crude approximations. 140, 376 An example is the LVC model 228, with its one-shot variant 229, or the exciton model. 593, 177 For more details on this topic, the reader is referred to refs 594, 228, 595, 313, 63, 596.
The Frenkel exciton Hamiltonian can be used to describe light-harvesting systems or charge-transfer. 177, 593 Such a Hamiltonian was constructed for the investigation of the excited state energies of bacteriochlorophylls of the Fenna-Matthews-Olson complex. Multi-layer feed-forward NNs with the Coulomb matrix as a molecular descriptor could accelerate the construction of such Hamiltonians for the prediction of excited-state energies. 160 The effective Hamiltonian of the whole complex was subsequently used to predict excitation energy transfer times and efficiencies. Therefore, Häse et al. used exciton Hamiltonians as an input. 159
Fitting diabatic potentials and properties
Given diabatic PESs, ML models can be used to fit them. KRR models are often employed for this task, due to their ease of use and ability to provide accurate predictions, as mentioned above.
Recent studies by Habershon and co-workers focus on interpolation of diabatic PESs and their use for grid-based quantum dynamics methods, i.e., variational Gaussian wavepackets and MCTDH. The butatriene cation has been investigated in two-dimensions comprising two electronic states. 147 The description of this molecule has been recently advanced with a new diabatization scheme, namely Procrustes diabatization. The method was evaluated with two-state direct-dynamics MCTDH (DD-MCTDH) simulations of LiF and applied to four electronic states of butatriene. 239
Some of the authors also carried out DD-MCTDH 4-mode/2-state 143 and subsequently 12-mode/2-state dynamics of pyrazine 144. The investigation of the higher-dimensional space of pyrazine could be achieved by systematic tensor decomposition of KRR and advances conventional MCTDH simulations considerably with respect to accuracy and computational efficiency. Further, the method was applied to investigate the ultrafast photodynamics of mycosporine-like amino acids, which are suitable as ingredients in sunscreens due to their photochemical properties and photostability. 597 However, the reduced 6-dimensional and 14-dimensional DD-MCTDH simulations with KRR interpolated PESs were unable to reproduce the expected ultrafast photodynamics, which had been observed in previously performed surface hopping calculations and is typical for sunscreen ingredients.
The authors note that the inclusion of more adiabatic states for the diabatization procedure and the consideration of additional relevant modes can lead to more accurate results. All of the reference simulations were carried out at the CASSCF level of theory with KRR fitted diabatic PESs.
In addition to KRR models, NNs were also used to describe diabatic PESs. Seminal works include PIP-based NNs by Guo, Yarkony and co-workers. Absorption spectra and the dynamics of excited states of NH${}_{3}$ and H${}_{2}$O could be studied by fitting potential energy matrix elements. 359, 360, 361, 362, 141, 145, 543 Subsequently, some of the authors fit the dipole moments corresponding to the diabatic 1,2${}^{1}$A surface of NH${}_{3}$. 14
SOCs of formaldehyde were learned with NNs in the diabatic picture. 90 341 data points were used for training of SOCs. A singlet and a triplet state in the adiabatic basis were transformed to diabatic states using Boys localization 351. Since this diabatization is based on transition dipole moments, the respective properties of the excited states had to be phase corrected. The authors proved the accuracy of their fitted PESs and emphasized the usability of the ML models to describe full-dimensional quantum dynamics. 14, 90, 543
Very recently, they investigated the $OH+H_{2}$ reaction, i.e., the nonadiabatic quenching of the hydroxyl radical colliding with molecular hydrogen. Four diabatic potentials including forces and couplings were fitted using a least squares fitting procedure. 1345 data points of 1,2,3 ${}^{2}A$ adiabatic PESs were computed with MR-CISD. 543
The aforementioned ML models are single-state models. Each energetic state and each coupling or dipole moment value resulting from different pairs of states is fitted with a separate ML model. While this yields justifiable accuracy for energies and diabatic coupling values 93, dipole moments are vectorial properties and need to preserve rotational covariance. 95
As the aforementioned studies show, ML models are generally powerful to advance quantum dynamics simulations for the excited states and can also assist the construction of effective Hamiltonians.
However currently, diabatic PESs cannot simply be fit for systems with arbitrary size and arbitrary complexity. The diabatization remains a methodological bottleneck, where additional developments are needed.
The investigation of medium-sized to larger molecular systems, especially the investigation of their temporal evolution, is more often carried out in the adiabatic basis using on-the-fly simulations. An increasing number of recent studies focus on fitting such adiabatic PESs. The inconsistencies in adiabatic properties make such quantities generally more challenging to fit, which is why this field of research gained a lot of attention relatively late, i.e., only in the last 3 years.
6.3.2 ML in the Adiabatic Basis
Surface hopping MD
Probably, the first ML models for MQCD calculations date back to the year 2008. 96 Nonadiabatic MD simulations were carried out with NN-interpolated PESs to investigate O${}_{2}$ scattered from Al(III). Symmetry functions were used as descriptors. 598 A spin-unpolarized singlet and a spin-polarized triplet state at DFT level of theory were fitted with 3768 data points. 599, 598 This two-state spin-diabatic problem allowed for evaluation of coupling values and singlet-triplet transitions with the fewest switches surface hopping approach. 403, 404 In a later study, another adiabatic spin-polarized PES was included and coupling values were computed between singlets and triplets 600 and evaluated from constructed Hamiltonian matrices. 91 MD simulations were executed using a manifold of ML-fitted PESs according to different spin-configurations. The studies showed that singlet-triplet transitions are highly probable during the scattering event of O${}_{2}$ on Au(III).96, 91
After these two seminal studies, the interest in advancing MQC photodynamics simulations in the adiabatic basis increased mainly in the last three years. One of the first works during this time was conducted by Hu et. al 137, who investigated the nonadiabatic dynamics of 6-aminopyrimidine with KRR and the Coulomb matrix.
Due to the many degrees of freedom of the molecule and including three singlet states, a large amount of training data was required ($>$ 65k data points). Coupling values were not fitted but, instead, the Zhu-Nakamura approach was used to compute hopping probabilities.
Later, Dral et al. 138 applied KRR models to accurately fit a two-state spin-Boson Hamiltonian and reproduce reference dynamics using 1,000 and 10,000 data points. NAC vectors were fit in a single-state fashion.
During dynamics simulations, conformations close to critical regions were computed with the reference method instead of the ML model in order to allow for accurate transitions.
In another study, Chen et al.139 used two separate deep NNs to fit the energies and forces of two adiabatic singlet states of CH${}_{2}$NH. About 90k data points were used to generate these single-state models. Using the Zhu-Nakamura approach to account for hopping probabilities, the reference dynamics could be reproduced and quantum chemical calculations were replaced completely during the dynamics.
Cui and coworkers 601 further developed a multi-layer energy-based fragmentation method to study the excited-state dynamics and photochemistry of larger systems. This scheme composes a molecular system into a photochemically active (inner) region and a photochemically inert (outer) region. In the original scheme, the active region and the interactions with the outer region are described with the multi-reference method CASSCF, whereas the outer region is treated with DFT. This decomposition of the total energy of a system allows to treat larger systems, which cannot be described fully with CASSCF. The approach is similar to QM/MM (quantum mechanics/molecular mechanics) schemes in the mechanical embedding framework. The authors simulated two-state photodynamics of CH${}_{3}$N=NCH${}_{3}$ (inner region) including five water molecules (outer region) without the use of ML. The Zhu-Nakamura approximation to model hopping probabilities in nonadiabatic MD simulations was applied. 601 In order to make the simulations more efficient, the authors replaced the DFT calculations with deep multi-layer feed-forward NNs using a distance-based descriptor 123, hence they describe the ground state energies and forces of the photochemically inert region with ML and describe the S${}_{1}$ and S${}_{0}$ state of the inner region with CASSCF. The hybrid ML multi-layer energy-based fragmentation method can reproduce the photodynamics of the system. 443 Subsequently, the deep NNs were replaced with embedded-atom NNs 533 and accurate second derivatives could be computed efficiently. 444
Recently, we sought to fit NACs and transition and permanent dipole moments in addition to energies and forces of three singlet states of the methylenimmonium cation, CH${}_{2}$NH${}_{2}^{+}$, using deep NNs and the matrix of inverse distances as a molecular descriptor. 92 We were able to perform ML-enhanced excited-state MD simulations with hopping probabilities based on ML-fitted NACs. NNs could replace the reference method MR-CISD completely during the dynamics. Long time scale photodynamics simulations for 1 ns were achieved using the mean of 2 NN models in approximately two months, whereas the reference method would have taken an estimated 19 years to compute the dynamics for 1 ns on the same computer. This study demonstrated the possibility of MLMD simulations to go beyond time scales of conventional methods. As another benefit of the ML models, it was shown that a large ensemble of trajectories could be calculated, still at lower cost than a few trajectories with the reference method. 92
With the same training set, we further assessed the performance of KRR together with von Lilienfeld and co-workers. 93 The operator formalism 602 and the FCHL representation 125, 487 were used to fit the three singlet states of CH${}_{2}$NH${}_{2}^{+}$. A single-state treatment and a multi-state treatment for predicting energies were compared. To this aim, a multi-state KRR approach as developed with an additional kernel that encodes the quantum energy levels. The accuracy of KRR models could be improved using this extended approach. 93
The KRR models were further compared to deep NN models regarding their ability to predict dipole moments and NACs.
While NNs yielded slightly higher accuracy at the largest available training set size, KRR models exhibited a steeper learning curve, hence more efficient learning.
The different performance of NNs and KRR models was proposed to be a result of the parametric dependence of the depth of NNs and the non-parametric dependence of the depth of KRR models.
Results further suggested that small differences between the reference method and ML models, especially in critical regions of the PESs, can lead to completely wrong photodynamics simulations. 93
Nevertheless, multi-reference quantum chemical potential energy curves could be faithfully reproduced with KRR models and NN models for the three singlet energies of CH${}_{2}$NH${}_{2}^{+}$.
In order to omit the extensive hyperparameter search of the descriptor and regressor, we further developed the SchNarc approach for photodynamics 13, which is based on SchNet. 428, 432 SchNarc allows for (1) a description of SOCs, (2) an NAC approximation based on ML-fitted PESs, their first and second derivatives with respect to Cartesian coordinates, and (3) a phase-free training algorithm to enable a training of raw quantum chemical data. The SchNarc approach is based on the message passing NN SchNet 428, 432, which was adapted by us for the treatment of a manifold of excited electronic states. Additionally, this model can describe dipole moments using the charge model of ref 109, also adapted for excited-states.
All excited-state properties can be described in one ML model in a multi-state fashion. The performance of SchNarc was evaluated with surface hopping dynamics: Three singlet and three triplet states of SO${}_{2}$ were computed with ML models for 700 fs and the underlying PESs were based on an "one-shot" LVC(MR-CISD) model 229. CSH${}_{2}$ was investigated using 2 singlets and 2 triplet states for 3 ps at CASSCF level of theory representing slow population transfer, and the performance of SchNarc to reproduce ultrafast transitions during dynamics was assessed using CH${}_{2}$NH${}_{2}^{+}$ with the aforementioned training set. The hopping probabilities were computed according to ML-fitted SOCs and NACs – the latter being fitted in a rotationally covariant way as derivatives of virtual ML properties and approximated from ML PESs.
In all cases, excellent agreement with the reference method could be achieved.
Noticeably, all the aforementioned photodynamics studies with ML models 137, 138, 139, 13, 93, 92 make use of Tully’s fewest switches surface hopping approach with hopping probabilities based on coupling values or approximated schemes. 403, 404
Exemplary timings for MLMD, LVC dynamics, and MQCD
The speed-up of simulations is one of the main arguments employed for promoting ML in quantum chemistry. In order to get an idea about the computational time used in different calculations, we provide an example here. The timings of surface hopping MD with analytical PESs (from LVC), quantum chemical PESs, and ML-fitted PESs based on fitted and approximated NACs from Hessians can be found for three exemplary molecules in Table 2.
Obviously, crude excited-state force fields like the LVC model are faster than ML models, e.g., for SO${}_{2}$. We note that even such force field implementations can probably still be streamlined for speed but will always be more expensive than ground-state MD simulations, where it would take approximately 0.005 seconds to simulate 100 fs for the gas-phase methylenimmonium cation, CH${}_{2}$NH${}_{2}^{+}$, using a state-of-the-art program like Amber.208
However, dynamics based on highly accurate quantum chemical calculations can be accelerated significantly with ML-fitted PESs, e.g., SchNarc models for CH${}_{2}$NH${}_{2}^{+}$ based on MR-CISD/aug-cc-pVDZ.13 The speedup is higher if NACs are learned directly (MLMD1) compared to when they are approximated from Hessians (MLMD2). A lot of Hessian evaluations are required in this example because ultrafast transitions occur in CH${}_{2}$NH${}_{2}^{+}$. The second-order derivatives reduce the efficiency by a factor of about ten. Nevertheless, Hessian calculations of ML-PESs can be accelerated by a factor of about 5-10 using a GPU (dependent on the molecule and GPU used).
Table 2 further shows that a cheaper underlying reference method, such as CASSCF(6,5)/def2-SVP used for CSH${}_{2}$, does not allow for such a significant speed-up. In this example however, the difference between simulations with learned NACs and approximated NACs is small because the dynamics of CSH${}_{2}$ is characterized by slow population transfer. Hence, less Hessian evaluations are required to estimate the hopping probabilities.
The time required to train a SchNarc model on a GeForce GTX 1080 Ti GPU is approximately 11 hours for energies and forces of 3 singlet states with 3,000 data points of CH${}_{2}$NH${}_{2}^{+}$, about 13 hours for energies, forces, and SOCs of 2 singlet and 2 triplet states using 4,000 data points of CSH${}_{2}$ and about 4 hours for energies and forces of 3 singlet states of SO${}_{2}$ using 5,000 data points.
Dipole Moments
In addition to the investigation of the temporal evolution of some systems in the excited states, permanent and transition dipole moments have been computed with ML models. As mentioned before, in our earlier approaches, we fitted permanent and transition dipole moments as single values with NNs and KRR – strictly speaking we were neglecting the rotational covariance of the vectors (since rotations were negligible in these simulations). 92, 93 The SchNarc model improved on this description by treating dipole moments as vectorial properties.
The NN and KRR models for dipole moments have been evaluated and compared to quantum chemical reference dipole moments using learning curves and MAEs. Their potential to compute UV spectra was emphasized.
The use of dipole moments to actually simulate UV spectra was demonstrated by Jiang, Mukamel, and co-workers using
N-methylacetamide, a model system to investigate peptide bonds. 509, 95 They evaluated the ability of ML to describe transition dipole moments at TDDFT level of theory. In a first attempt,509 the authors predicted dipole vectors as independent values. 14 internal coordinates in combination with multi-layer feed-forward NNs were used to predict transition energies of N-methylacetamide. Xyz representations served as an input for fitting ground state dipole moments. The Coulomb matrix was employed to fit transition dipole moments for the n$\pi^{\ast}$ and $\pi\pi^{\ast}$ transitions, but did not lead to sufficiently accurate results. Higher accuracy was obtained by replacing the atomic charges in the Coulomb matrix (eq 30) with charges from natural population analysis. The choice of descriptors was justified by screening different types of descriptors for prediction of different properties.
In a later work, some of the authors used embedded-atom NNs to predict transition dipole moments from atomic contributions in a rotationally covariant way. The dipole moment vector between two states i and j was obtained as a linear combination of three contributions:
$$\mu_{ij}=\mu_{T}^{i}+\mu_{T}^{j}+\mu_{T}^{3}$$
(33)
$\mu_{T}^{i}$ and $\mu_{T}^{j}$ were modeled using the charge model of ref 109. A third contribution, $\mu_{T}^{3}$, was obtained as the cross product of $\mu_{T}^{i}$ and $\mu_{T}^{j}$:
$$\mu_{T}^{3}=\sum_{a}^{N_{A}}q_{a}^{3}(\mu_{T}^{i}\times\mu_{T}^{j})$$
(34)
$\mu_{T}^{i}$, $\mu_{T}^{j}$ and $q_{a}^{3}$ were outputs of the same embedded-atom NN.
6.4 ML of Tertiary Outputs
The secondary outputs, such as dipole moments or excited state energies can be used to calculate oscillator strengths (eq 1) and energy gaps (Fig. 1(d)). These properties can serve for the modelling of UV absorption spectra. UV spectra were computed in the previously described studies of N-methylacetamid with the ML fitted transition dipole moments. Jiang, Mukamel and co-workers 509 applied the transition dipole moment and additionally fitted n$\pi^{\ast}$ and $\pi\pi^{\ast}$ excitation energies to compute UV spectra this molecule with NNs. Subsequently, some of the authors 95 used these excitation energies and the transition dipole moments to model a Frenkel exciton Hamiltonian for proteins using amino acid residues and peptide bonds. This effective Hamiltonian could subsequently be used to approximate UV spectra of proteins. The interaction between amino acid residues and peptides was neglected so only the isolated peptide excitation energies, i.e., those of N-methylacetamid, and the respective transition dipole moments were needed to construct the Hamiltonian. The authors made use of the dipole-dipole approximation 603 and applied embedded-atom NNs.
Ramakrishnan et. al 327 predicted excitation energies of the lowest-lying two excited singlet states, S${}_{1}$ and S${}_{2}$, as well as corresponding oscillator strengths obtained from TDDFT calculations with KRR. The QM8 496 data base was used consisting of 20k organic molecules. With the $\Delta$-learning approach, CC2 accuracy could be obtained.
Very recently, Xue et al. 563 assessed the performance of KRR models with the normalized inverse distances as a molecular descriptor to predict absorption spectra of benzene and a derivative of acridine containing 38 atoms. Therefore, the authors learned the excited-state energy gaps of several states and the corresponding oscillator strengths in a single-state fashion. Applying Gaussian broadening, the absorption cross sections could be computed at TDDFT accuracy.
Pronobis et al. 156 compared 2-body, 3-body and automatically designed descriptors to learn TDDFT HOMO-LUMO gaps as well as first and second vertical excitation energies.
More than 20k molecules of the QM9 data base 496, 424 were selected for this purpose and learning curves were used to evaluate the learning behaviour of different ML models.
While atom-wise descriptors worked well for HOMO-LUMO gaps, the authors concluded that the accuracy of predicted transition energies is not sufficiently accurate and suggested that advanced non-local descriptors might be necessary to achieve higher accuracy. They further proposed the idea of encoding information about the electronic state in the ML model. 156 Indeed, our recent study, in which we compared the performance of KRR and NN models with atom-wise and molecule-wise descriptors demonstrated that encoding of the energy level is advantageous. 93
Recently, Kang et. al 584 used 500,000 molecules of the PubChemQC 502 data base to train a random forest model on the excitation energy and the oscillator strength corresponding to the electronic state with the highest oscillator strength. 10 singlet states, as available in the PubChemQC data base, were evaluated for that purpose. The authors used SMILES (simplified molecular-input line-entry system) strings and converted them into descriptors. The descriptors comprised several topological 604 and binary 605 fingerprints, which were calculated with the help of the RDkit library 606.
The authors compared the prediction accuracy to the aforementioned models and stated that their model outperformed previous ML models in the task of predicting accurate oscillator strengths and excitation energies for the most probable transition in organic molecules. Analysis of important features led the authors identify that nitrogen-containing heterocycles are important for high oscillator strengths in molecules. The authors concluded that their study could serve the design of new fluorophores with high oscillator strengths. 584
Ghosh et. al 150 used multi-layer feed-forward NNs, convolutional NNs and DTNNs to fit 16 highest occupied orbital energies from DFT, i.e., the respective eigenvalues, for the computation of molecular spectra with a full width at half maximum of 0.5 eV for Gaussian broadening.
Geometries from the QM7b 494, 495 and QM9 496, 424 data base were used for training and molecular spectra were tested using 10k additional diastereomers, which were also used by Ramakrishnan et. al 327 to evaluate the $\Delta$-learning approach. The convolutional NNs with the Coulomb matrix and DTNNs with an automatically generated representation outperformed the simpler NNs. Overall, good agreement to reference DFT spectra could be achieved. 150
Markland and co-workers 447 trained NNs with atom-centered Chebyshev polynomial descriptors 108 on the TDDFT/CAM-B3LYP/6-31+G* S${}_{0}$-S${}_{1}$ energy gap of the deprotonated trans-thiophenyl-p-coumarate (chromophore of yellow protein) in water and Nile red chromophore in water and benzene. Farthest point sampling 121 was used to select about 2,000 data points from a larger set of 36,000 data points and was compared to random sampling.
The authors assessed the performance of three different ML approaches to compute absorption spectra, spectral densities and 2-dimensional electronic spectra. One model (hidden solvation) completely ignored any environmental effects and only described the chromophore, another model (indirect solvation) incorporated environmental effects within a 5Å cutoff of the atomistic descriptor for the chromophore and a third model (direct solvation) treated the whole system, i.e., the chromophore and the atoms of the solvent, explicitly. As expected, the hidden solvation model turned out to be insufficiently accurate for systems with strong solvent-chromophore interactions, but was comparable to the hidden solvation model when describing Nile red chromophore in benzene. The indirect solvation and direct solvation models were comparable to each other, but with respect to the computational efficiency, the indirect solvation model was beneficial. This model could reproduce reference linear absorption spectra, spectral densities, and could capture spectral diffusion of 2-dimensional electronic spectra of all treated chromophores. 447
Penfold and co-workers 153 applied deep multi-layer feed-forward NNs to proof the ability of ML to predict X-ray absorption spectra (XAS), which provide a wealth of information on the geometry and electronic structure of chemical systems, especially in the near-edge structure region. Note that X-Ray free-electron laser spectroscopy can further be used to generate ultrashort X-ray pulses to investigate photodynamics simulations in real-time. The training set for the prediction of Fe K-edge X-ray near-edge structure spectra contained 9040 data points. The inputs for NNs were generated using local radial distributions around the Fe absorption site of arbitrary systems taken from the Materials Project Database. 607 Qualitatively accurate peak positions and intensities could be obtained computationally efficient and the structural refinement of nitrosylmyoglobin and [Fe(bpy)${}_{3}$]${}^{2+}$ was assessed with NNs.
The authors noted that future development is needed to accurately capture structures far from equilibrium as well as irregularities in the bulk.
Another study was executed by Aarva et al., 608 who focused on XAS and X-ray photoelectron spectra of functionalized amorphous carbonaceous materials. By clustering of DFT data with unsupervised ML techniques average fingerprint spectra of distinct functionalized surfaces could be obtained. The authors use GPR. Similarly to the aforementioned state encoding, 93 the authors encoded the electronic structure, i.e., the $\Delta$-Kohn Sham values (core-electron binding energies), in a Gaussian kernel. This kernel was then linearly combined with a structure-based kernel based on the SOAP 609 descriptor. The spectra computed from the different clusters were used to fit experimental spectra allowing for an approximation to the composition of experimental samples on a semi-quantitative level. The so-called fingerprint spectra, which enabled the differentiation of the spectral signatures, were assessed in a previous study using different models for amorphous carbon 610, among them an ML fitted PES using GPR 110, 611.
Kulik and co-workers 15 used deep NNs to predict the spin-state ordering in transition metal complexes to determine the spin of the lowest lying energetic state in open-shell systems. The determination of spin states is important to evaluate catalytic and material properties of metal complexes. Descriptors based on a selection of empirical features were used to capture the bonding in inorganic molecular systems. The performance of descriptors including different features was assessed for a set of octahedral complexes with first-row transition metals. The most important features were identified to be the atom, which connects the ligand to the metal, its environment and its electronegativity, the metal identity and its oxidation state, as well as the formal charge and denticity of the ligand. 612 The ML models were tested on spin-crossover complexes and could assign the correct spin in most cases. Additionally, ML models were applied for the discovery of inorganic complexes 613, 614, 615, 616
The inverse design of molecules with specific properties was further targeted by Schütt et. al, 76 who developed SchNOrb, a deep NN model based on SchNet. The automatically generated descriptor was extended with a description of atom pairs in their chemical and structural environment. An analytic representation of the electronic structure of a molecular system was obtained in a local atomic orbital representation. The analytic derivatives of the electronic structure allowed for optimization of electronic properties. This was demonstrated by minimizing and maximizing the HOMO-LUMO gap of malonaldehyde 486. Besides, the ML method was used to predict the lowest 20 molecular orbitals of ethanol at DFT level of theory, to investigate proton transfer in malonaldehyde using ground-state dynamics and to analyze bond order and partial charges of uracil.
Bayesian NN models were applied by Häse et. al 158 to relate molecular geometries to the outcome of nonadiabatic MD simulations obtained with CASSCF. Normal modes with and without velocities of initial conditions served as an input for NN models.
Velocities in addition to normal modes as descriptors improved the accuracy of ML models slightly, pointing out that normal modes contain already enough information for the sake of their study. The dissociation times of 1,2-dioxetane obtained from nonadiabatic MD simulations was the targeted output. The NNs could faithfully reproduce dissociation times and further provided a measure of uncertainty. The authors noted that their method could be particularly interesting for analysis of MLMD simulations.
6.5 ML-Assisted Analysis
The aforementioned studies have shown that ML enables the simulation of MD simulations and spectra predictions at low computational costs. The computational efficiency allows for enhanced statistics, i.e., in case of MD simulations a huge number of trajectories and the simulations on long time scales. 92, 13 Therefore, subsequent analyses of production runs can become a time limiting step of studies. This problem was identified in the aforementioned study on the dissociation times of 1,2-dioxetane by Häse et. al. 158 Therefore, the authors further used their method to interpret the outcomes of nonadiabatic MD simulations.
1,2-dioxetane is the target of their study as it is the smallest molecule known to show chemilumiescence after nonadiabatic transitions from an excited state to the ground state. The chemiluminescent properties of this compound were related to its decomposition rate into two formaldehyde molecules. By analysis of the ML models that fit the dissociation times, correlations could be observed between the normal modes and the dissociation times. For example, the modes corresponding to C-C bond stretching and C-O bond stretching were relevant for the accurate prediction of dissociation times.
It was further emphasized by the authors that although the findings of NNs were expected and obey physical laws, ML models were helpful to extract relevant information of large amount of data and could potentially serve as an inspiration to humans.
Time-resolved experimental photoluminescence spectra could be analyzed with the LumiML software developed by Ðorđević et. al, 617 who applied linear regression models to learn from computer-generated photoluminescence data. The software was employed to predict decay rate distributions 618 of perovskite nanocrystals from data generated with femtosecond broadband fluorescence upconversion spectroscopy. 619 The authors highlighted the applicability of their method to enhance studies on the optimization and design of optical devices and further noted that their approach can also be used to analyze transient absorption spectra.
Aspuru-Guzik and co-workers 152 applied Bayesian NNs to find correlations of nanoaggregates with electronic coupling in semiconducting materials using absorption spectra. In general, the analysis of experimental spectra and the inverse design of compounds is most frequently applied in the research field of material science. Their description goes beyond the scope of this review and the reader is referred to Refs 169, 163, 164, 165, 166, 167.
7 Conclusion and Future Perspectives
In the last few years, machine learning (ML) has started to slowly enter the research field of photochemistry, especially the photochemistry of molecular systems. Although this field of research is rather young compared to ML for the electronic ground-state, some groundbreaking works have already shown the potential of ML models to significantly accelerate and improve existing simulation techniques. So far, most studies provide a proof of concept using small molecular systems or model systems. Different applications are targeted and will also be aimed at in the future, ranging from dynamics with excited-state ML potentials via absorption spectra to the interpretation of data, see Fig. 1.
Analysing the different studies reviewed here, some trends in the choice of reference methods, ML models, and descriptors can be observed. These trends are illustrated in Figure 11.
The pie chart in panel 11(a) shows the used reference methods for the computation of a training set to describe the excited states or excited-state properties of molecules. As can be seen, about half of the training sets are computed with multi-reference methods. 12, 140, 158, 70, 14, 90, 94, 92, 13, 93, 137, 139, 239, 144, 143, 147, 362, 141, 359, 142, 360, 145, 597 The employed single-reference approaches are exclusively based on DFT. 150, 95, 509, 76, 156, 15, 467, 327, 96, 91, 584, 447, 610
Analytical methods or experimental data are also applied. 617, 138, 152, 160, 159
When restricting the analysis to studies targeting dynamics, the fraction that employs multi-reference methods even increases. About 70% of all dynamics studies use multi-reference methods to compute the training data for ML models. 15% of the studies use single-reference methods and an equally large portion apply model Hamiltonians or analytical potentials. This shows that most chemical problems for the investigation of the excited states of molecules require multi-reference accuracy.
Recent studies of ML-based photodynamics simulations have shown that many thousands of data points are necessary to describe a few excited-state potentials of small molecular systems. To the best of our knowledge, the dynamics in the excited states with ML for molecules with more than 12 atoms in full dimensions has not yet been investigated. 137, 144, 143 Especially the huge number of data points is concerning in this case, as larger molecules with more energetic states and a complex photochemistry could require many more data points. A meaningful training set generation, which can be achieved with active learning, adaptive sampling and structure-based sampling techniques, is thus essential for dynamics simulations. 480, 109, 92, 479 Clustering of molecular geometries obtained from dynamics simulations with a cheap method further is beneficial for selecting important reference geometries. 138, 137, 481, 482, 483
Still, the high costs and the complexity of multi-reference methods to compute an ample training set for ML also hampers the application of ML models to fit the excited states of larger polyatomic systems, whose accurate photochemical description is often additionally complicated by a high density of electronic states.
Single reference methods, such as time-dependent DFT, are advantageous with respect to the computational costs of the training set, but suffer from qualitatively incorrect PESs in some conformational regions of molecules, such as dissociative regions. In principle, these conformational regions could be excluded from the training set and the remaining conformational space could be interpolated using ML, but the training set would then remain incomplete and so would the dynamics. Schemes like the $\Delta-learning$ approach 327 or transfer learning 329 could be helpful in this regard. These approaches might be useful to let ML models learn from single-reference data and adjust their accuracy according to multi-reference methods. The direct use of approximated methods, such as time-dependent DFT-based tight binding, is most likely not suitable for photodynamics on long time scales, because such approaches might easily be quantitatively incorrect. Of particular concern is then the accumulation of quantitatively tiny errors in the underlying potentials toward wrong dynamics trends. At the current stage of research, it is not clear whether such approximate potentials can provide qualitatively correct trends for reaction dynamics. 400
In addition to the aforementioned problems, the training set generation is complicated by the arbitrariness of the signs of coupling values and properties resulting from two different electronic states. 14, 90, 92, 93, 13, 95 This arbitrariness has to be removed in order to make data learnable with conventional methods. Such a correction scheme is termed phase correction and has been applied to correct coupling values and dipole moments. 92, 454, 14, 90, 95 An alternative phase correction training algorithm has been shown to be beneficial with respect to the costs of the training set generation and has enabled the learning of raw quantum chemical data. 13
Figure 11(b) shows which ML models are applied in the discussed studies. About two thirds rely on NNs, whereby simple multi-layer feed-forward NNs are most often employed. Several research fields were advanced with NN-fitted functions: photodynamics simulations 92, 13, 93, 138, 139, 362, 141, 359, 142, 360, 145, 96, 91, spectra predictions and analysis, 563, 153, 150, 447, 509, 95
excited-state properties, 14, 90, 93, 13, 15, 95, 509 diabatization procedures, 140, 94 interpretation of reaction outcomes, 158, 617 and the prediction of HOMO-LUMO gaps or gaps between energetic states. 156, 150, 76 KRR methods were mainly applied to interpolate diabatic potentials 239, 144, 143, 147, 597 and in studies focusing on more than one molecular systems. 327 In general, only a few studies focused on extrapolation throughout chemical compound space in the excited states. Yet only the energies, HOMO-LUMO gaps or spectra based on fitted oscillator strengths could be predicted using a single ML model for different molecules. 15, 153, 327, 156
Decision trees were used to select an active space for diatomic molecules. 70
One drawback of recently developed ML models is that they are molecule-specific and thus not universal. In part, this issue is related to the used molecular descriptors. As can be seen in panel (c) in Figure 11, most studies apply descriptors that capture molecules as a whole. The few studies, which describe PESs and properties of molecular systems from atomic contributions, either treat small molecular systems 13, 93, 95 or predict properties related to the ground-state equilibrium structure of a molecular system or to electronic ground state calculations, e.g. the HOMO-LUMO gaps. 156, 150, 76 Due to the limited transferability of existing ML models to predict the excited state PESs and properties of different molecular systems, an extrapolation throughout chemical compound space is hindered in many cases.
In order to fully exploit the advantages that ML models offer and to achieve the aforementioned goal of a transferable ML model for the excited states, a highly versatile descriptor is required, which can describe atoms in their chemical and structural environment and enables an ML model to treat molecules of arbitrary size and composition. It would be highly desirable, if an ML model could then describe the photochemistry of large systems, which are too expensive to compute with precise multi-reference methods, using only small building blocks, i.e., small enough ones to describe their electronic structure accurately. For example, the excited states of proteins or DNA strands could potentially be predicted from contributions of amino acids or DNA bases, respectively, which is most often done using effective model Hamiltonians up to date.56 A local description of the excited-state PESs and their properties derived from the ML-fitted PESs, could further provide a way toward excited-state ML/MM simulations alike QM/MM (quantum mechanics/molecular mechanics) techniques. 400, 601, 443 Unfortunately, it is not yet known whether the excited-state PESs and properties can be constructed from atomic contributions or not. 400
In studies comparing different ML models, it was even suggested that non-local descriptors might be needed or that the electronic state has to be encoded explicitly in the molecular representation to enable a transferable description of the excited states with ML 156, 93.
To conclude, the reviewed studies focus on almost all aspects of excited-state quantum chemistry and improve them successfully: ML models can help to choose a proper active space for multi-reference methods, they predict secondary and tertiary outputs of quantum chemical calculations and help in the interpretation of theoretical studies. ML models push the boundaries of computed time scales 92 and are used to investigate and analyze the huge amount of data we produce every day in experiments or with high-performance computers. 617, 158
It should be emphasized once more that the recent studies show that the goal of ML is not to replace existing methods completely, but to provide a way to improve them. In fact, ML models for the excited states at their current stage are far from replacing existing quantum chemical methods, and they are also far from being routine. Without human intervention, ML cannot solve existing problems and much remains to be done to describe systems beyond single, isolated molecules.
To the best of our knowledge, what is still missing is the proof that ML can provide an approximation to the multi-reference wave function of a molecular system. Such an achievement would be a great advancement in the research field of photochemistry, as any property we wish to know could possibly be derived from the ML wave function. An ML representation of the electronic structure would further be beneficial to allow for an inverse design of molecules with specific properties, which has been shown to be feasible for the ground state of a molecular system 76. The optimization of photochemical properties with respect to molecular geometries would be useful for many exciting research fields, e.g. photocatalysis 165, photosensitive drug design 620 or photovoltaics 621, 609.
The multi-faceted photochemistry offers a perfect playground for ML models.
It may be important to highlight that, despite the negative image ML has suffered in some research communities, it cannot be denied that it opens up many new ways and possibilities to improve simulations and make studies feasible that were considered unattainable only a few years, if not only months ago. 442 The computational efficiency and high flexibility of deep learning models can lead this research field toward simulations of long time and large length scales. The possibilities ML models offer are far from being being exhausted.
Considering the enormous chemical space, estimated to consist of more than $10^{60}$ molecules 622, and the desire to develop methods, which could develop into a universal approximator, make ML models perfectly suited to advance this research field.
The possibility of deep ML models to process a huge amount of data can even assist the interpretation and analysis 158, 617 of many photochemical studies and can help to explore unknown physical relations and be a source of potential human inspiration.
{acknowledgement}
This work was financially supported by the Austrian Science Fund, W 1232 (MolTag) and the uni:docs program of the University of Vienna (J.W.). P. M. thanks the University of Vienna for continuous support, also in the frame of the research platform ViRAPID. We thank P. A. Sánchez-Murcia for help in setting up the quick Amber simulation for MD timings.
References
Këpuska and Bohouta 2018
Këpuska, V.; Bohouta, G. Next-Generation of Virtual Personal Assistants
(Microsoft Cortana, Apple Siri, Amazon Alexa and Google Home). 2018 IEEE
8th Annual Computing and Communication Workshop and Conference (CCWC). 2018;
pp 99–103.
Hoy 2018
Hoy, M. B. Alexa, Siri, Cortana, and More: An Introduction to Voice
Assistants. Med. Ref. Serv. Q. 2018, 37, 81–88.
Silver et al. 2016
Silver, D. et al. Mastering the Game of Go with Deep Neural Networks
and Tree Search. Nature 2016, 529, 484–489.
Bansak et al. 2018
Bansak, K.; Ferwerda, J.; Hainmueller, J.; Dillon, A.; Hangartner, D.;
Lawrence, D.; Weinstein, J. Improving Refugee Integration through
Data-Driven Algorithmic Assignment. Science 2018,
359, 325–329.
Leung et al. 2016
Leung, M. K. K.; Delong, A.; Alipanahi, B.; Frey, B. J. Machine
Learning in Genomic Medicine: A Review of Computational Problems and Data
Sets. Proc. IEEE 2016, 104, 176–197.
Shen et al. 2017
Shen, D.; Wu, G.; Suk, H.-I. Deep Learning in Medical Image Analysis.
Annu. Rev. Biomed. Eng. 2017, 19, 221–248.
Chen et al. 2015
Chen, C.; Seff, A.; Kornhauser, A.; Xiao, J. DeepDriving: Learning Affordance
for Direct Perception in Autonomous Driving. The IEEE International
Conference on Computer Vision (ICCV). 2015.
Yang et al. 2019
Yang, X.; Wang, Y.; Byrne, R.; Schneider, G.; Yang, S. Concepts of Artificial
Intelligence for Computer-Assisted Drug Discovery. Chem. Rev.
2019, 119, 10520–10594.
Goodfellow et al. 2016
Goodfellow, I.; Bengio, Y.; Courville, A. Deep Learning; MIT Press,
2016.
Gómez-Bombarelli and Aspuru-Guzik 2018
Gómez-Bombarelli, R.; Aspuru-Guzik, A. In Handbook of Materials
Modeling : Methods: Theory and Modeling; Andreoni, W., Yip, S., Eds.;
Springer International Publishing: Cham, 2018; pp 1–24.
Agrawal and Choudhary 2016
Agrawal, A.; Choudhary, A. Perspective: Materials Informatics and Big Data:
Realization of the âFourth Paradigmâ of Science in Materials Science.
APL Mat. 2016, 4, 053208.
Schwilk et al. 2020
Schwilk, M.; Tahchieva, D. N.; von Lilienfeld, O. A. Large yet Bounded: Spin
Gap Ranges in Carbenes. arXiv 2020, 2004.10600.
Westermayr et al. 2020
Westermayr, J.; Gastegger, M.; Marquetand, P. Combining SchNet and SHARC: The
SchNarc Machine Learning Approach for Excited-State Dynamics. J. Phys.
Chem. Lett. 2020, 11, 3828–3834.
Guan et al. 2020
Guan, Y.; Guo, H.; Yarkony, D. R. Extending the Representation of Multistate
Coupled Potential Energy Surfaces to Include Properties Operators using
Neural Networks: Application to the 1,2${}^{1}$A States of Ammonia. J.
Chem. Theory Comput. 2020, 16, 302–313.
Taylor et al. 2020
Taylor, M. G.; Yang, T.; Lin, S.; Nandy, A.; Janet, J. P.; Duan, C.;
Kulik, H. J. Seeing Is Believing: Experimental Spin States from Machine
Learning Model Structure Predictions. J. Phys. Chem. A 2020,
124, 3286–3299.
Kulik 2020
Kulik, H. J. Making Machine Learning a Useful Tool in the Accelerated Discovery
of Transition Metal Complexes. WIREs Comput. Mol. Sci. 2020,
10, e1439.
Power 2012
Power, P. P. Stable Two-Coordinate, Open-Shell (d1âd9) Transition Metal
Complexes. Chem. Rev. 2012, 112, 3482–3507.
Bousseksou et al. 2004
Bousseksou, A.; Molnár, G.; Matouzenko, G. Switching of Molecular Spin States
in Inorganic Complexes by Temperature, Pressure, Magnetic Field and Light:
Towards Molecular Devices. Eur. J. Inorg. chem. 2004,
2004, 4353–4369.
Li et al. 2014
Li, H.; Feng, H.; Sun, W.; Fan, Q.; King, R. B.; Schaefer, H. F. First-Row
Transition Metals in Binuclear Cyclopentadienylmetal Derivatives of
Tetramethyleneethane: $\eta^{3},\eta^{3}$ versus $\eta^{4},\eta^{4}$ LigandâMetal
Bonding Related to Spin State and MetalâMetal Bonds.
Organometallics 2014, 33, 3489–3499.
Matsika and Krylov 2018
Matsika, S.; Krylov, A. I. Introduction: Theoretical Modeling of Excited State
Processes. Chem. Rev. 2018, 118, 6925–6926.
Cohen et al. 2004
Cohen, B.; Crespo-Hernández, C. E.; Hare, P. M.; Kohler, B. Ultrafast
Excited-State Dynamics in DNA and RNA Polymers; Elsevier: Amsterdam, 2004;
Chapter Ultrafast Excited-State Dynamics in DNA and RNA Polymers, pp
463–470.
Levine and Martínez 2007
Levine, B. G.; Martínez, T. J. Isomerization Through Conical
Intersections. Annu. Rev. Phys. Chem. 2007, 58,
613–634.
Turro et al. 2009
Turro, N. J.; Ramamurthy, V.; Scaiano, J. C. Principles of Molecular
Photochemistry: An Introduction. 2009.
Yarkony 2012
Yarkony, D. R. Nonadiabatic Quantum Chemistry - Past, Present, and Future.
Chem. Rev. 2012, 112, 481–498.
Barbatti et al. 2014
Barbatti, M.; Borin, A. C.; Ullrich, S. Photoinduced Phenomena in Nucleic
Acids I; Topics in Current Chemistry; Springer Berlin Heidelberg, 2014; Vol.
355; pp 1–32.
Ibele et al. 2020
Ibele, L. M.; Nicolson, A.; Curchod, B. F. E. Excited-State Dynamics of
molecules with classically driven trajectories and Gaussians. Mol.
Phys. 2020, 118, e1665199.
Nelson et al. 2020
Nelson, T. R.; White, A. J.; Bjorgaard, J. A.; Sifain, A. E.; Zhang, Y.;
Nebgen, B.; Fernandez-Alberti, S.; Mozyrsky, D.; Roitberg, A. E.; Tretiak, S.
Non-adiabatic Excited-State Molecular Dynamics: Theory and Applications for
Modeling Photophysics in Extended Molecular Materials. Chem. Rev.
2020, 120, 2215–2287.
Mai and González 2020
Mai, S.; González, L. Molecular Photochemistry: Recent Developments in Theory.
Angew. Chem. Int. Ed. 2020, n/a.
Lischka et al. 2018
Lischka, H.; Nachtigallová, D.; Aquino, A. J. A.; Szalay, P. G.;
Plasser, F.; Machado, F. B. C.; Barbatti, M. Multireference Approaches for
Excited States of Molecules. Chem. Rev. 2018, 118,
7293–7361.
Ghosh et al. 2018
Ghosh, S.; Verma, P.; Cramer, C. J.; Gagliardi, L.; Truhlar, D. G. Combining
Wave Function Methods with Density Functional Theory for Excited States.
Chem. Rev. 2018, 118, 7249–7292.
Norman and Dreuw 2018
Norman, P.; Dreuw, A. Simulating X-Ray Spectroscopies and Calculating
Core-Excited States of Molecules. Chem. Rev. 2018,
118, 7208–7248.
Casanova 2018
Casanova, D. Theoretical Modeling of Singlet Fission. Chem. Rev.
2018, 118, 7164–7207.
Hestand and Spano 2018
Hestand, N. J.; Spano, F. C. Expanded Theory of H- and J-Molecular Aggregates:
The Effects of Vibronic coupling and Intermolecular Charge Transfer.
Chem. Rev. 2018, 118, 7069–7163.
Penfold et al. 2018
Penfold, T. J.; Gindensperger, E.; Daniel, C.; Marian, C. M. Spin-Vibronic
Mechanism for Intersystem Crossing. Chem. Rev. 2018,
118, 6975–7025.
Vacher et al. 2018
Vacher, M.; Fdez. Galván, I.; Ding, B.-W.; Schramm, S.;
Berraud-Pache, R.; Naumov, P.; Ferré, N.; Liu, Y.-J.; Navizet, I.;
Roca-Sanjuán, D.; Baader, W. J.; Lindh, R. Chemi- and Bioluminescence of
Cyclic Peroxides. Chem. Rev. 2018, 118,
6927–6974.
Crespo-Otero and Barbatti 2018
Crespo-Otero, R.; Barbatti, M. Recent Advances and Perspectives on Nonadiabatic
Mixed QuantumâClassical Dynamics. Chem. Rev. 2018,
118, 7026–7068.
González and Lindh 2020
González, L.; Lindh, R. Quantum Chemistry and Dynamics of Excited
States : Methods and Applications; John Wiley and Sons Ltd, 2020.
Harris and Bertolucci 1989
Harris, D. C.; Bertolucci, M. D. Symmetry and Spectroscopy: an
Introduction to Vibrational and Electronic Spectroscopy; New York: Dover
Publications, 1989.
Ng 1991
Ng, C.-Y. Vacuum Ultraviolet Photoionization and Photodissociation of
Molecules and Clusters; World Scientific, 1991.
Zewail 1994
Zewail, A. H. Femtochemistry: Ultrafast Dynamics of the Chemical Bond;
World Scientific, 1994; pp 3–22.
Brixner et al. 2005
Brixner, T.; Pfeifer, T.; Gerber, G.; Wollenhaupt, M.; Baumert, T. In
Femtosecond Laser Spectroscopy; Hannaford, P., Ed.; Springer-Verlag:
New York, 2005; pp 225–266.
Iqbal and Stavros 2010
Iqbal, A.; Stavros, V. G. Active Participation of $1\pi\sigma$* States in the
Photodissociation of Tyrosine and its Subunits. J. Phys. Chem. Lett.
2010, 1, 2274–2278.
Kowalewski et al. 2017
Kowalewski, M.; Fingerhut, B. P.; Dorfman, K. E.; Bennett, K.; Mukamel, S.
Simulating Coherent Multidimensional Spectroscopy of Nonadiabatic Molecular
Processes: From the Infrared to the X-Ray Regime. Chem. Rev.
2017, 117, 12165–12226.
Soorkia et al. 2020
Soorkia, S.; Jouvet, C.; Grégoire, G. UV Photoinduced Dynamics of
Conformer-Resolved Aromatic Peptides. Chem. Rev. 2020,
120, 3296â3327.
Yusong Liu, et al. 2020
Yusong Liu, et al., Spectroscopic and Structural Probing of Excited-State
Molecular Dynamics with Time-Resolved Photoelectron Spectroscopy and
Ultrafast Electron Diffraction. Phys. Rev. X 2020, 10,
021016.
Martínez 2006
Martínez, T. J. Insights for Light-Driven Molecular Devices from Ab
Initio Multiple Spawning Excited-State Dynamics of Organic and Biological
Chromophores. Acc. Chem. Res. 2006, 39, 119–126.
Barbatti et al. 2008
Barbatti, M.; Sellner, B.; Aquino, A. J. A.; Lischka, H. In Radiation
Induced Molecular Phenomena in Nucleic Acids; Shukla, M., Leszczynski, J.,
Eds.; Challenges and Advances in Computational Chemistry and Physics;
Springer Netherlands, 2008; Vol. 5; pp 209–235.
Subotnik et al. 2016
Subotnik, J. E.; Jain, A.; Landry, B.; Petit, A.; Ouyang, W.; Bellonzi, N.
Understanding the Surface Hopping View of Electronic Transitions and
Decoherence. Annu. Rev. Phys. Chem. 2016, 67,
387–417.
Curchod and Martínez 2018
Curchod, B. F. E.; Martínez, T. J. Ab Initio Nonadiabatic Quantum
Molecular Dynamics. Chem. Rev. 2018, 118,
3305–3336.
Ashfold et al. 2017
Ashfold, M. N. R.; Bain, M.; Hansen, C. S.; Ingle, R. A.; Karsili, T. N. V.;
Marchetti, B.; Murdock, D. Exploring the Dynamics of the Photoinduced
Ring-Opening of Heterocyclic Molecules. J. Phys. Chem. Lett.
2017, 8, 3440–3451.
Tajti et al. 2009
Tajti, A.; Fogarasi, G.; Szalay, P. G. Reinterpretation of the UV Spectrum of
Cytosine: Only Two Electronic Transitions? ChemPhysChem 2009,
10, 1603–1606.
Barbatti et al. 2011
Barbatti, M.; Szymczak, J. J.; Aquino, A. J. A.; Nachtigallová, D.;
Lischka, H. The Decay Mechanism of Photoexcited Guanine – A Nonadiabatic
Dynamics Study. J. Chem. Phys. 2011, 134, 014304.
Lu et al. 2014
Lu, Y.; Lan, Z.; Thiel, W. Photoinduced Phenomena in Nucleic Acids II;
Topics in Current Chemistry; Springer Berlin Heidelberg, 2014; Vol. 356; pp
89–122.
Ruckenbauer et al. 2016
Ruckenbauer, M.; Mai, S.; Marquetand, P.; González, L. Photoelectron
Spectra of 2-Thiouracil, 4-Thiouracil, and 2,4-Dithiouracil. J. Chem.
Phys. 2016, 144, 074303.
Manathunga et al. 2016
Manathunga, M.; Yang, X.; Luk, H. L.; Gozem, S.; Frutos, L. M.; Valentini, A.;
Ferrè, N.; Olivucci, M. Probing the Photodynamics of Rhodopsins with
Reduced Retinal Chromophores. J. Chem. Theory Comput. 2016,
12, 839–850.
Nogueira et al. 2017
Nogueira, J. J.; Plasser, F.; González, L. Electronic Delocalization,
Charge Transfer and Hypochromism in the UV Absorption Spectrum of Polyadenine
Unravelled by Multiscale Computations and Quantitative Wavefunction
Analysis. Chem. Sci. 2017, 8, 5682–5691.
Mai et al. 2018
Mai, S.; Mohamadzade, A.; Marquetand, P.; González, L.; Ullrich, S.
Simulated and Experimental Time-Resolved Photoelectron Spectra of the
Intersystem Crossing Dynamics in 2-Thiouracil. Molecules
2018, 23, 2836.
Rauer et al. 2018
Rauer, C.; Nogueira, J. J.; Marquetand, P.; González, L. Stepwise
photosensitized thymine dimerization mediated by an exciton intermediate.
Monatsh. Chem. 2018, 149, 1â9.
Zobel et al. 2018
Zobel, J. P.; Heindl, M.; Nogueira, J. J.; González, L. Vibrational
Sampling and Solvent Effects on the Electronic Structure of the Absorption
Spectrum of 2-Nitronaphthalene. J. Chem. Theory Comput. 2018,
14, 3205–3217.
Pathak et al. 2019
Pathak, S. et al. Tracking the Ultraviolet Photochemistry of
Thiophenone During and Beyond the Initial Ultrafast Ring Opening. 2019.
Maria Teresa Neves-Petersen and Gajula 2012
Maria Teresa Neves-Petersen, S. P.; Gajula, G. P. UV Light Effects on
Proteins: From Photochemistry to Nanomedicine, Molecular Photochemistry -
Various Aspects; IntechOpen, 2012; Chapter 7.
Cadet et al. 2014
Cadet, J.; Grand, A.; Douki, T. Photoinduced Phenomena in Nucleic Acids
II; Topics in Current Chemistry; Springer Berlin Heidelberg, 2014; Vol. 356;
pp 249–275.
Segatta et al. 2019
Segatta, F.; Cupellini, L.; Garavelli, M.; Mennucci, B. Quantum Chemical
Modeling of the Photoinduced Activity of Multichromophoric Biosystems.
Chem. Rev. 2019, 119, 9361–9380.
Landry and Subotnik 2014
Landry, B. R.; Subotnik, J. E. Quantifying the Lifetime of Triplet Energy
Transfer Processes in Organic Chromophores: A Case Study of
4-(2-Naphthylmethyl)benzaldehyde. J. Chem. Theory Comput.
2014, 10, 4253–4263.
Schultz et al. 2003
Schultz, T.; Quenneville, J.; Levine, B.; Toniolo, A.; Martínez, T. J.;
Lochbrunner, S.; Schmitt, M.; Shaffer, J. P.; Zgierski, M. Z.; Stolow, A.
Mechanism and Dynamics of Azobenzene Photoisomerization. J. Am. Chem.
Soc. 2003, 125, 8098–8099.
Toniolo et al. 2004
Toniolo, A.; Olsen, S.; Manohar, L.; Martínez, T. J. Conical Intersection
Dynamics in Solution: The Chromophore of Green Fluorescent Protein.
Faraday Discuss. 2004, 127, 149–163.
Domcke et al. 2011
Domcke, W.; Yarkony, D.; Köppel, H. Conical Intersections: Theory,
Computation and Experiment; Advanced Series in Physical Chemistry; World
Scientific Publishing Company, 2011.
Serrano-Andrés and Merchán 2005
Serrano-Andrés, L.; Merchán, M. Quantum chemistry of the excited state:
2005 overview. J. Mol. Struc.-THEOCHEM 2005, 729, 99
– 108.
Chandrasekaran et al. 2019
Chandrasekaran, A.; Kamal, D.; Batra, R.; Kim, C.; Chen, L.; Ramprasad, R.
Solving the Electronic Structure Problem with Machine Learning. npj
Comput. Mater. 2019, 5, 22.
Jeong et al. 2020
Jeong, W.; Stoneburner, S. J.; King, D.; Li, R.; Walker, A.; Lindh, R.;
Gagliardi, L. Automation of Active Space Selection for Multireference Methods
via Machine Learning on Chemical Bond Dissociation. J. Chem. Theory
Comput. 2020, 16, 2389–2399.
Carleo and Troyer 2017
Carleo, G.; Troyer, M. Solving the Quantum Many-Body Problem with Artificial
Neural Networks. Science 2017, 355, 602–606.
Saito 2017
Saito, H. Solving the BoseâHubbard Model with Machine Learning. J.
Phys. Soc. Jpn. 2017, 86, 093001.
Nomura et al. 2017
Nomura, Y.; Darmawan, A. S.; Yamaji, Y.; Imada, M. Restricted Boltzmann Machine
Learning for solving strongly correlated quantum systems. Phys. Rev. B
2017, 96, 205152.
Han et al. 2019
Han, J.; Zhang, L.; E, W. Solving Many-Electron Schrödinger Equation using
Deep Neural Networks. J. Comput. Phys. 2019, 399,
108929.
Townsend and Vogiatzis 2019
Townsend, J.; Vogiatzis, K. D. Data-Driven Acceleration of the coupled-Cluster
Singles and Doubles Iterative Solver. J. Phys. Chem. Lett.
2019, 10, 4129–4135.
Schütt et al. 2019
Schütt, K. T.; Gastegger, M.; Tkatchenko, A.; Müller, K.-R.;
Maurer, R. J. Unifying Machine Learning and quantum chemistry with a deep
neural network for molecular wavefunctions. Nat. Commun.
2019, 10, 5024.
Pfau et al. 2019
Pfau, D.; Spencer, J. S.; de G. Matthews, A. G.; Foulkes, W. M. C. Ab-Initio
Solution of the Many-Electron Schrödinger Equation with Deep Neural
Networks. 2019.
Hermann et al. 2019
Hermann, J.; Schätzle, Z.; Noé, F. Deep Neural Network Solution of the
Electronic Schrödinger Equation. 2019.
Gastegger et al. 2020
Gastegger, M.; McSloy, A.; Luya, M.; Schütt, K. T.; Maurer, R. J. A Deep
Neural Network for Molecular Wave Functions in Quasi-Atomic Minimal Basis
Representation. https://arxiv.org/abs/2005.06979 2020,
Hegde and Bowen 2017
Hegde, G.; Bowen, R. C. Machine-Learned Approximations to Density Functional
Theory Hamiltonians. Sci. Rep. 2017, 7, 42669.
Brockherde et al. 2017
Brockherde, F.; Vogt, L.; Li, L.; Tuckerman, M. E.; Burke, K.;
Müller, K.-R. Bypassing the Kohn-Sham Equations with Machine Learning.
Nat. Commun. 2017, 8, 872.
Gastegger et al. 2019
Gastegger, M.; González, L.; Marquetand, P. Exploring Density Functional
Subspaces with Genetic Algorithms. Monatsh. Chem. 2019,
150, 173–182.
Nelson et al. 2019
Nelson, J.; Tiwari, R.; Sanvito, S. Machine Learning Density Functional Theory
for the Hubbard Model. Phys. Rev. B 2019, 99,
075132.
Cheng et al. 2019
Cheng, L.; Welborn, M.; Christensen, A. S.; Miller, T. F. A Universal Density
Matrix Functional from Molecular Orbital-Based Machine Learning:
Transferability Across Organic Molecules. J. Chem. Phys.
2019, 150, 131103.
Lei and Medford 2019
Lei, X.; Medford, A. J. Design and Analysis of Machine Learning
Exchange-Correlation Functionals via Rotationally Invariant Convolutional
Descriptors. Phys. Rev. Materials 2019, 3,
063801.
Zhou et al. 2019
Zhou, Y.; Wu, J.; Chen, S.; Chen, G. Toward the Exact ExchangeâCorrelation
Potential: A Three-Dimensional Convolutional Neural Network Construct.
J. Phys. Chem. Lett. 2019, 10, 7264–7269.
Kolb et al. 2017
Kolb, B.; Lentz, L. C.; Kolpak, A. M. Discovering Charge Density Functionals
and Structure-Property Relationships with PROPhet: A General Framework for
Coupling Machine Learning and First-Principles Methods. Sci. Rep.
2017, 7.
Willatt et al. 2019
Willatt, M. J.; Musil, F.; Ceriotti, M. Atom-Density Representations for
Machine Learning. J. Chem. Phys. 2019, 150,
154110.
Choo et al. 2018
Choo, K.; Carleo, G.; Regnault, N.; Neupert, T. Symmetries and Many-Body
Excitations with Neural-Network Quantum States. Phys. Rev. Lett.
2018, 121, 167204.
Guan and Yarkony 2020
Guan, Y.; Yarkony, D. R. Accurate Neural Network Representation of the Ab
Initio Determined SpinâOrbit Interaction in the Diabatic Representation
Including the Effects of Conical Intersections. J. Phys. Chem. Lett.
2020, 11, 1848–1858.
Carbogno et al. 2010
Carbogno, C.; Behler, J.; Reuter, K.; Groß, A. Signatures of Nonadiabatic
O${}_{2}$ Dissociation at Al(111): First-Principles Fewest-Switches Study.
Phys. Rev. B 2010, 81, 035410.
Westermayr et al. 2019
Westermayr, J.; Gastegger, M.; Menger, M. F. S. J.; Mai, S.; González, L.;
Marquetand, P. Machine Learning Enables Long Time Scale Molecular
Photodynamics Simulations. Chem. Sci. 2019, 10,
8100–8107.
Westermayr et al. 2020
Westermayr, J.; Faber, F. A.; Christensen, A. S.; von Lilienfeld, O. A.;
Marquetand, P. Neural Networks and Kernel Ridge Regression for Excited States
Dynamics of CH${}_{2}$NH${}_{2}^{+}$: From Single-State to Multi-State
Representations and Multi-Property Machine Learning Models. Mach.
Learn.: Sci. Technol. 2020, 1, 025009.
Shen and Yarkony 2020
Shen, Y.; Yarkony, D. R. Construction of Quasi-diabatic Hamiltonians That
Accurately Represent Ab Initio Determined Adiabatic Electronic States
Coupled by Conical Intersections for Systems on the Order of 15 Atoms.
Application to Cyclopentoxide Photoelectron Detachment in the Full 39 Degrees
of Freedom. J. Phys. Chem. A 2020, 124,
4539–4548.
Zhang et al. 2020
Zhang, Y.; Ye, S.; Zhang, J.; Jiang, J.; Jiang, B. Efficient and Accurate
Spectroscopic Simulations with Symmetry-Preserving Neural Network Models for
Tensorial Properties. arXiv 2020, 2004.13605.
Carbogno et al. 2008
Carbogno, C.; Behler, J.; Groß, A.; Reuter, K. Fingerprints for
Spin-Selection Rules in the Interaction Dynamics of O${}_{2}$ at
Al(111). Phys. Rev. Lett. 2008, 101, 096104.
Polyak et al. 2019
Polyak, I.; Richings, G. W.; Habershon, S.; Knowles, P. J. Direct Quantum
Dynamics using Variational Gaussian Wavepackets and Gaussian Process
Regression. J. Chem. Phys. 2019, 150, 041101.
Hobday et al. 1999
Hobday, S.; Smith, R.; Belbruno, J. Applications of Neural Networks to Fitting
Interatomic Potential Functions. Modell. Simul. Mater. Sci. Eng.
1999, 7, 397.
Bartók et al. 2010
Bartók, A. P.; Payne, M. C.; Kondor, R.; Csányi, G. Gaussian
Approximation Potentials: The Accuracy of Quantum Mechanics, without the
Electrons. Phys. Rev. Lett. 2010, 104, 136403.
Rupp et al. 2012
Rupp, M.; Tkatchenko, A.; Müller, K.-R.; von Lilienfeld, O. A. Fast and
Accurate Modeling of Molecular Atomization Energies with Machine
Learning. Phys. Rev. Lett. 2012, 108, 058301.
Li et al. 2015
Li, Z.; Kermode, J. R.; De Vita, A. Molecular Dynamics with On-the-Fly Machine
Learning of Quantum-Mechanical Forces. Phys. Rev. Lett. 2015,
114, 096405.
von Lilienfeld et al. 2015
von Lilienfeld, O. A.; Ramakrishnan, R.; Rupp, M.; Knoll, A. Fourier Series
of Atomic Radial Distribution Functions: A Molecular Fingerprint for Machine
Learning Models of Quantum Chemical Properties. Int. J. Quantum Chem.
2015, 115, 1084–1093.
Gastegger and Marquetand 2015
Gastegger, M.; Marquetand, P. High-Dimensional Neural Network Potentials for
Organic Reactions and an Improved Training Algorithm. J. Chem. Theory
Comput. 2015, 11, 2187–2198.
Rupp et al. 2015
Rupp, M.; Ramakrishnan, R.; von Lilienfeld, O. A. Machine Learning for Quantum
Mechanical Properties of Atoms in Molecules. J. Phys. Chem. Lett.
2015, 6, 3309–3313.
Behler 2016
Behler, J. Perspective: Machine Learning Potentials for Atomistic Simulations.
J. Chem. Phys. 2016, 145, 170901.
Artrith and Urban 2016
Artrith, N.; Urban, A. An implementation of artificial neural-network
potentials for atomistic materials simulations: Performance for TiO${}_{2}$.
computational Materials Science 2016, 114, 135 –
150.
Gastegger et al. 2016
Gastegger, M.; Kauffmann, C.; Behler, J.; Marquetand, P. Comparing the Accuracy
of High-Dimensional Neural Network Potentials and the Systematic Molecular
Fragmentation Method: A Benchmark Study for All-Trans Alkanes. J. Chem.
Phys. 2016, 144.
Artrith et al. 2017
Artrith, N.; Urban, A.; Ceder, G. Efficient and Accurate Machine-Learning
Interpolation of Atomic Energies in Compositions with Many Species.
Phys. Rev. B 2017, 96, 014112.
Gastegger et al. 2017
Gastegger, M.; Behler, J.; Marquetand, P. Machine Learning Molecular Dynamics
for the Simulation of Infrared Spectra. Chem. Sci. 2017,
8, 6924–6935.
Deringer and Csányi 2017
Deringer, V. L.; Csányi, G. Machine Learning Based Interatomic Potential
for Amorphous Carbon. Phys. Rev. B: Condens. Matter Mater. Phys.
2017, 95, 094203.
Botu et al. 2017
Botu, V.; Batra, R.; Chapman, J.; Ramprasad, R. Machine Learning Force Fields:
Construction, Validation, and Outlook. J. Phys. Chem. C 2017,
121, 511–522.
Glielmo et al. 2017
Glielmo, A.; Sollich, P.; De Vita, A. Accurate Interatomic Force Fields via
Machine Learning with Covariant Kernels. Phys. Rev. B 2017,
95, 214302.
Smith et al. 2017
Smith, J. S.; Isayev, O.; Roitberg, A. E. ANI-1: An Extensible Neural Network
Potential with DFT Accuracy at Force Field Computational Cost. Chem.
Sci. 2017, 8, 3192–3203.
Fujikake et al. 2018
Fujikake, S.; Deringer, V. L.; Lee, T. H.; Krynski, M.; Elliott, S. R.;
Csányi, G. Gaussian Approximation Potential Modeling of Lithium
Intercalation in Carbon Nanostructures. J. Chem. Phys. 2018,
148, 241714.
Behler 2017
Behler, J. First Principles Neural Network Potentials for Reactive Simulations
of Large Molecular and Condensed Systems. Angew. Chem. Int. Edit.
2017, 56, 12828–12840.
Zong et al. 2018
Zong, H.; Pilania, G.; Ding, X.; Ackland, G. J.; Lookman, T. Developing an
Interatomic Potential for Martensitic Phase Transformations in Zirconium by
Machine Learning. npj comput Mater 2018, 4.
Wood and Thompson 2018
Wood, M. A.; Thompson, A. P. Extending the Accuracy of the SNAP Interatomic
Potential Form. J. Chem. Phys. 2018, 148, 241721.
Chen et al. 2018
Chen, X.; Jørgensen, M. S.; Li, J.; Hammer, B. Atomic Energies from a
Convolutional Neural Network. J. Chem. Theory Comput. 2018,
14, 3933–3942.
Bartók et al. 2018
Bartók, A. P.; Kermode, J.; Bernstein, N.; Csányi, G. Machine
Learning a General-Purpose Interatomic Potential for Silicon. Phys.
Rev. X 2018, 8, 041048.
Chmiela et al. 2018
Chmiela, S.; Sauceda, H. E.; Müller, K.-R.; Tkatchenko, A. Towards Exact
Molecular Dynamics Simulations with Machine-Learned Force Fields. Nat.
Commun. 2018, 9, 3887.
Imbalzano et al. 2018
Imbalzano, G.; Anelli, A.; Giofré, D.; Klees, S.; Behler, J.; Ceriotti, M.
Automatic Selection of Atomic Fingerprints and Reference Configurations for
Machine-Learning Potentials. J. Chem. Phys. 2018, 148,
241730.
Zhang et al. 2018
Zhang, L.; Han, J.; Wang, H.; Saidi, W. A.; Car, R.; Weinan, E. End-to-end
Symmetry Preserving Inter-atomic Potential Energy Model for Finite and
Extended Systems. Proceedings of the 32Nd International conference on Neural
Information Processing Systems. USA, 2018; pp 4441–4451.
Zhang et al. 2018
Zhang, L.; Han, J.; Wang, H.; Car, R.; E, W. Deep Potential Molecular Dynamics:
A Scalable Model with the Accuracy of Quantum Mechanics. Phys. Rev.
Lett. 2018, 120, 143001.
Chan et al. 2019
Chan, H.; Narayanan, B.; Cherukara, M. J.; Sen, F. G.; Sasikumar, K.;
Gray, S. K.; Chan, M. K. Y.; Sankaranarayanan, S. K. R. S. Machine Learning
Classical Interatomic Potentials for Molecular Dynamics from First-Principles
Training Data. J. Phys. Chem. C 2019, 123,
6941–6957.
Faber et al. 2018
Faber, F. A.; Christensen, A. S.; Huang, B.; von Lilienfeld, O. A. Alchemical
and Structural Distribution Based Representation for Universal Quantum
Machine Learning. J. Chem. Phys. 2018, 148,
241717.
Wang and Yang 2019
Wang, H.; Yang, W. Toward Building Protein Force Fields by Residue-Based
Systematic Molecular Fragmentation and Neural Network. J. Chem. Theory
Comput. 2019, 15, 1409–1417.
Gerrits et al. 2019
Gerrits, N.; Shakouri, K.; Behler, J.; Kroes, G.-J. Accurate Probabilities for
Highly Activated Reaction of Polyatomic Molecules on Surfaces Using a
High-Dimensional Neural Network Potential: CHD${}_{3}$ + Cu(111). J. Phys.
Chem. Lett. 2019, 10, 1763–1768.
Chmiela et al. 2019
Chmiela, S.; Sauceda, H. E.; Poltavsky, I.; Müller, K.-R.; Tkatchenko, A.
sGDML: Constructing Accurate and Data Efficient Molecular Force Fields using
Machine Learning. Comput. Phys. Commun. 2019, 240, 38
– 45.
Carleo et al. 2019
Carleo, G.; Cirac, I.; Cranmer, K.; Daudet, L.; Schuld, M.; Tishby, N.;
Vogt-Maranto, L.; Zdeborová, L. Machine Learning and the Physical
Sciences. Rev. Mod. Phys. 2019, 91, 045002.
Krems 2019
Krems, R. V. Bayesian Machine Learning for Quantum Molecular Dynamics.
Phys. Chem. Chem. Phys. 2019, 21, 13392–13410.
Deringer et al. 2019
Deringer, V. L.; Caro, M. A.; Csányi, G. Machine Learning Interatomic
Potentials as Emerging Tools for Materials Science. Adv. Mat.
2019, 31, 1902765.
Ward et al. 2019
Ward, L.; Blaiszik, B.; Foster, I.; Assary, R. S.; Narayanan, B.; Curtiss, L.
Machine Learning Prediction of Accurate Atomization Energies of Organic
Molecules from Low-Fidelity Quantum Chemical Calculations. MRS Commun.
2019, 9, 891â899.
Noé et al. 2020
Noé, F.; Tkatchenko, A.; Müller, K.-R.; Clementi, C. Machine Learning
for Molecular Simulation. Annu. Rev. Phys. Chem. 2020,
71, 361–390.
Alborzpour et al. 2016
Alborzpour, J. P.; Tew, D. P.; Habershon, S. Efficient and Accurate Evaluation
of Potential Energy Matrix Elements for Quantum Dynamics using Gaussian
Process Regression. J. Chem. Phys. 2016, 145,
174112.
Cheng et al. 2020
Cheng, Z.; Zhao, D.; Ma, J.; Li, W.; Li, S. An On-the-Fly Approach to Construct
Generalized Energy-Based Fragmentation Machine Learning Force Fields of
Complex Systems. J. Phys. Chem. A 2020, 124,
5007–5014.
Behler et al. 2008
Behler, J.; Reuter, K.; Scheffler, M. Nonadiabatic Effects in the Dissociation
of Oxygen Molecules at the Al(111) Surface. Phys. Rev. B
2008, 77, 115421.
Hu et al. 2018
Hu, D.; Xie, Y.; Li, X.; Li, L.; Lan, Z. Inclusion of Machine Learning Kernel
Ridge Regression Potential Energy Surfaces in On-the-Fly Nonadiabatic
Molecular Dynamics Simulation. J. Phys. Chem. Lett. 2018,
9, 2725–2732.
Dral et al. 2018
Dral, P. O.; Barbatti, M.; Thiel, W. Nonadiabatic Excited-State Dynamics with
Machine Learning. J. Phys. Chem. Lett. 2018, 9,
5660–5663.
Chen et al. 2018
Chen, W.-K.; Liu, X.-Y.; Fang, W.-H.; Dral, P. O.; Cui, G. Deep Learning for
Nonadiabatic Excited-State Dynamics. J. Phys. Chem. Lett.
2018, 9, 6702–6708.
Williams and Eisfeld 2018
Williams, D. M. G.; Eisfeld, W. Neural Network Diabatization: A New Ansatz for
Accurate High-Dimensional Coupled Potential Energy Surfaces. J. Chem.
Phys. 2018, 149, 204106.
Xie et al. 2018
Xie, C.; Zhu, X.; Yarkony, D. R.; Guo, H. Permutation Invariant Polynomial
Neural Network Approach to Fitting Potential Energy Surfaces. IV. Coupled
Diabatic Potential Energy Matrices. J. Chem. Phys. 2018,
149, 144107.
Guan et al. 2019
Guan, Y.; Zhang, D. H.; Guo, H.; Yarkony, D. R. Representation of Coupled
Adiabatic Potential Energy Surfaces using Neural Network Based Quasi-Diabatic
Hamiltonians: 1,2 ${}^{2}$A’ States of LiFH. Phys. Chem. Chem. Phys.
2019, 10.1039/C8CP06598E.
Richings and Habershon 2018
Richings, G. W.; Habershon, S. MCTDH on-the-Fly: Efficient Grid-Based Quantum
Dynamics without Pre-Computed Potential Energy Surfaces. J. Chem.
Phys. 2018, 148, 134116.
Richings et al. 2019
Richings, G. W.; Robertson, C.; Habershon, S. Improved on-the-Fly MCTDH
Simulations with Many-Body-Potential Tensor Decomposition and Projection
Diabatization. J. Chem. Theory Comput. 2019, 15,
857–870.
Guan et al. 2019
Guan, Y.; Guo, H.; Yarkony, D. R. Neural Network Based Quasi-Diabatic
Hamiltonians with Symmetry Adaptation and a Correct Description of Conical
Intersections. J. Chem. Phys. 2019, 150, 214101.
Wang et al. 2019
Wang, Y.; Xie, C.; Guo, H.; Yarkony, D. R. A Quasi-Diabatic Representation of
the 1,2${}^{1}$A States of Methylamine. J. Phys. Chem. A 2019,
123, 5231–5241.
Richings and Habershon 2017
Richings, G. W.; Habershon, S. Direct Grid-Based Quantum Dynamics on Propagated
Diabatic Potential Energy Surfaces. Chem. Phys. Lett. 2017,
683, 228 – 233.
Netzloff et al. 2006
Netzloff, H. M.; collins, M. A.; Gordon, M. S. Growing Multiconfigurational
Potential Energy Surfaces with Applications to X+H${}_{2}$ (X=C,N,O) Reactions.
J. Chem. Phys. 2006, 124, 154104.
Bettens and Collins 1999
Bettens, R. P. A.; Collins, M. A. Learning to Interpolate Molecular Potential
Energy Surfaces with Confidence: A Bayesian Approach. J. Chem. Phys.
1999, 111, 816–826.
Ghosh et al. 2019
Ghosh, K.; Stuke, A.; Todorović, M.; Jørgensen, P. B.; Schmidt, M. N.;
Vehtari, A.; Rinke, P. Deep Learning Spectroscopy: Neural Networks for
Molecular Excitation Spectra. Adv. Sci. 2019, 6,
1801367.
Kananenka et al. 2019
Kananenka, A. A.; Yao, K.; Corcelli, S. A.; Skinner, J. L. Machine Learning for
Vibrational Spectroscopic Maps. J. Chem. Theory Comput. 2019,
15, 6850–6858.
Roch et al. 2020
Roch, L. M.; Saikin, S. K.; Häse, F.; Friederich, P.; Goldsmith, R. H.;
León, S.; Aspuru-Guzik, A. From Absorption Spectra to Charge Transfer in
Nanoaggregates of Oligomers with Machine Learning. ACS Nano
2020, in press, doi:10.1021/acsnano.0c00384.
Rankine et al. 2020
Rankine, C. D.; Madkhali, M. M. M.; Penfold, T. J. A Deep Neural Network for
the Rapid Prediction of X-Ray Absorption Spectra. J. Phys. Chem. A
2020, 124, 4263–4270.
Pereira et al. 2017
Pereira, F.; Xiao, K.; Latino, D. A. R. S.; Wu, C.; Zhang, Q.; Aires-de
Sousa, J. Machine Learning Methods to Predict Density Functional Theory B3LYP
Energies of HOMO and LUMO Orbitals. J. Chem. Inf. Model.
2017, 57, 11–21.
Isayev et al. 2017
Isayev, O.; Oses, c.; Toher, c.; Gossett, E.; Curtarolo, S.; Tropsha, A.
Universal Fragment Descriptors for Predicting Properties of Inorganic
Crystals. Nat. Commun. 2017, 8, 15679.
Pronobis et al. 2018
Pronobis, W.; Schütt, K. R.; Tkatchenko, A.; Müller, K.-R. Capturing
Intensive and Extensive DFT/TDDFT Molecular Properties with Machine Learning.
Eur. Phys. J. B 2018, 91, 178.
Stuke et al. 2019
Stuke, A.; Todorović, M.; Rupp, M.; Kunkel, C.; Ghosh, K.; Himanen, L.;
Rinke, P. Chemical Diversity in Molecular Orbital Energy Predictions with
Kernel Ridge Regression. J. Chem. Phys. 2019, 150,
204121.
Häse et al. 2019
Häse, F.; Fdez. Galván, I.; Aspuru-Guzik, A.; Lindh, R.; Vacher, M. How
Machine Learning can Assist the Interpretation of Ab Initio Molecular
Dynamics Simulations and Conceptual Understanding of Chemistry. Chem.
Sci. 2019, 10, 2298–2307.
Häse, Florian and Kreisbeck, Christoph and Aspuru-Guzik,
Alán 2017
Häse, Florian and Kreisbeck, Christoph and Aspuru-Guzik, Alán,
Machine Learning for Quantum Dynamics: Deep Learning of Excitation Energy
Transfer Properties. Chem. Sci. 2017, 8,
8419–8426.
Häse et al. 2016
Häse, F.; Valleau, S.; Pyzer-Knapp, E.; Aspuru-Guzik, A. Machine Learning
Exciton Dynamics. Chem. Sci. 2016, 7, 5139–5147.
OâBoyle et al. 2011
OâBoyle, N. M.; Campbell, C. M.; Hutchison, G. R. Computational Design and
Selection of Optimal Organic Photovoltaic Materials. J. Phys. Chem. C
2011, 115, 16200–16210.
Teunissen et al. 2017
Teunissen, J. L.; De Proft, F.; De Vleeschouwer, F. Tuning the HOMOâLUMO
Energy Gap of Small Diamondoids Using Inverse Molecular Design. J.
Chem. Theory Comput. 2017, 13, 1351–1365.
Liu et al. 2018
Liu, D.; Tan, Y.; Khoram, E.; Yu, Z. Training Deep Neural Networks for the
Inverse Design of Nanophotonic Structures. ACS Photonics
2018, 5, 1365–1369.
Elton et al. 2019
Elton, D. C.; Boukouvalas, Z.; Fuge, M. D.; Chung, P. W. Deep Learning for
Molecular Design – A Review of the State of the Art. Mol. Syst. Des.
Eng. 2019, 4, 828–849.
Sanchez-Lengeling and Aspuru-Guzik 2018
Sanchez-Lengeling, B.; Aspuru-Guzik, A. Inverse Molecular Design using Machine
Learning: Generative Models for Matter Engineering. Science
2018, 361, 360–365.
Goldsmith et al. 2018
Goldsmith, B. R.; Esterhuizen, J.; Liu, J.-X.; Bartel, C. J.; Sutton, C.
Machine Learning for Heterogeneous Catalyst Design and Discovery. AIChE
J. 2018, 64, 2311–2323.
Davies et al. 2018
Davies, D. W.; Butler, K. T.; Isayev, O.; Walsh, A. Materials Discovery by
Chemical Analogy: Role of Oxidation States in Structure Prediction.
Faraday Discuss. 2018, 211, 553–568.
Anatole von Lilienfeld et al. 2020
Anatole von Lilienfeld, O.; Müller, K.-R.; Tkatchenko, A. Exploring
Chemical compound Space with Quantum-Based Machine Learning. Nat. Rev.
Chem. 2020,
Freeze et al. 2019
Freeze, J. G.; Kelly, H. R.; Batista, V. S. Search for Catalysts by Inverse
Design: Artificial Intelligence, Mountain Climbers, and Alchemists.
Chemical Reviews 2019, 119, 6595–6612.
Lee 2020
Lee, M.-H. Robust Random Forest Based Non-Fullerene Organic Solar Cells
Efficiency Prediction. Org. Electron. 2020, 76,
105465.
Cartwright 2020
Cartwright, H. M., Ed. Machine Learning in Chemistry; Theoretical and
Computational Chemistry Series; The Royal Society of Chemistry, 2020.
Gastegger and Marquetand 2020
Gastegger, M.; Marquetand, P. In Machine Learning Meets Quantum Physics;
Schütt, K. T., Chmiela, S., von Lilienfeld, O. A., Tkatchenko, A.,
Tsuda, K., Müller, K.-R., Eds.; Springer International Publishing: Cham,
2020; pp 233–252.
Schütt et al. 2020
Schütt, K. T., Chmiela, S., von Lilienfeld, O. A., Tkatchenko, A.,
Tsuda, K., Müller, K.-R., Eds. Machine Learning Meets Quantum
Physics; Springer International Publishing, 2020.
Park et al. 2020
Park, J. W.; Al-Saadon, R.; MacLeod, M. K.; Shiozaki, T.; Vlaisavljevich, B.
Multireference Electron Correlation Methods: Journeys along Potential Energy
Surfaces. Chem. Rev. 2020, in press, null.
Akimov and Prezhdo 2015
Akimov, A. V.; Prezhdo, O. V. Large-Scale Computations in Chemistry: A Bird’s
Eye View of a Vibrant Field. Chem. Rev. 2015, 115,
5797–5890.
Frutos et al. 2007
Frutos, L. M.; Andruniów, T.; Santoro, F.; Ferré, N.; Olivucci, M.
Tracking the Excited-State Time Evolution of the Visual Pigment with
Multiconfigurational Quantum Chemistry. Proceedings of the National
Academy of Sciences 2007, 104, 7764–7769.
Menger et al. 2018
Menger, M. F. S. J.; Plasser, F.; Mennucci, B.; González, L. Surface
Hopping within an Exciton Picture. An Electrostatic Embedding Scheme.
J. Chem. Theory Comput. 2018, 14, 6139–6148.
Dou and Subotnik 2020
Dou, W.; Subotnik, J. E. Nonadiabatic Molecular Dynamics at Metal Surfaces.
J. Phys. Chem. A 2020, 124, 757–771.
Dou et al. 2015
Dou, W.; Nitzan, A.; Subotnik, J. E. Frictional Effects Near a Metal Surface.
J. Chem. Phys. 2015, 143, 054103.
Tavernelli 2006
Tavernelli, I. Electronic Density Response of Liquid Water using Time-Dependent
Density Functional Theory. Phys. Rev. B 2006, 73,
094204.
Schütt et al. 2014
Schütt, K. T.; Glawe, H.; Brockherde, F.; Sanna, A.; Müller, K. R.;
Gross, E. K. U. How to Represent Crystal Structures for Machine Learning:
Towards Fast Prediction of Electronic Properties. Phys. Rev. B
2014, 89, 205118.
Lee et al. 2016
Lee, J.; Seko, A.; Shitara, K.; Nakayama, K.; Tanaka, I. Prediction Model of
Band Gap for Inorganic Compounds by Combination of Density Functional Theory
Calculations and Machine Learning Techniques. Phys. Rev. B
2016, 93, 115104.
Zhuo et al. 2018
Zhuo, Y.; Mansouri Tehrani, A.; Brgoch, J. Predicting the Band Gaps of
Inorganic Solids by Machine Learning. J. Phys. Chem. Lett.
2018, 9, 1668–1673.
Pilania et al. 2017
Pilania, G.; Gubernatis, J.; Lookman, T. Multi-Fidelity Machine Learning Models
for Accurate Bandgap Predictions of Solids. Comput. Mat. Sci.
2017, 129, 156 – 163.
Spiering et al. 2019
Spiering, P.; Shakouri, K.; Behler, J.; Kroes, G.-J.; Meyer, J.
Orbital-Dependent Electronic Friction Significantly Affects the Description
of Reactive Scattering of N${}_{2}$ from Ru(0001). J. Phys. Chem.
Lett. 2019, 10, 2957–2962.
Zhang et al. 2020
Zhang, Y.; Maurer, R. J.; Jiang, B. Symmetry-Adapted High Dimensional Neural
Network Representation of Electronic Friction Tensor of Adsorbates on Metals.
J. Phys. Chem. C 2020, 124, 186–195.
Zhang et al. 2019
Zhang, Y.; Maurer, R. J.; Guo, H.; Jiang, B. Hot-Electron Effects during
Reactive Scattering of H${}_{2}$ from Ag(111): The Interplay between
Mode-Specific Electronic Friction and the Potential Energy Landscape.
Chem. Sci. 2019, 10, 1089–1097.
Head-Gordon and Tully 1995
Head-Gordon, M.; Tully, J. C. Molecular Dynamics with Electronic Frictions.
J. Chem. Phys. 1995, 103, 10137–10145.
Douglas-Gallardo et al. 2019
Douglas-Gallardo, O. A.; Berdakin, M.; Frauenheim, T.; Sánchez, C. G.
Plasmon-Induced Hot-Carrier Generation Differences in Gold and Silver
Nanoclusters. Nanoscale 2019, 11, 8604–8615.
Yin et al. 2019
Yin, R.; Zhang, Y.; Jiang, B. Strong Vibrational Relaxation of NO Scattered
from Au(111): Importance of the Adiabatic Potential Energy Surface. J.
Phys. Chem. Lett. 2019, 10, 5969–5974.
Rittmeyer et al. 2018
Rittmeyer, S. P.; Bukas, V. J.; Reuter, K. Energy dissipation at metal
surfaces. Adv. Phys-X 2018, 3, 1381574.
Therrien et al. 2019
Therrien, A. J.; Kale, M. J.; Yuan, L.; Zhang, C.; Halas, N. J.;
Christopher, P. Impact of Chemical Interface Damping on Surface Plasmon
Dephasing. Faraday Discuss. 2019, 214, 59–72.
Wodtke et al. 2004
Wodtke, A. M.; Tully, J. C.; Auerbach, D. J. Electronically Non-Adiabatic
Interactions of Molecules at Metal Surfaces: Can we Trust the
BornâOppenheimer Approximation for Surface Chemistry? Int. Rev. Phys.
Chem. 2004, 23, 513–539.
Park et al. 2019
Park, G. B.; Krüger, B. C.; Borodin, D.; Kitsopoulos, T. N.; Wodtke, A. M.
Fundamental Mechanisms for Molecular Energy Conversion and Chemical Reactions
at Surfaces. Rep. Prog. Phys. 2019, 82, 096401.
Jiang and Guo 2019
Jiang, B.; Guo, H. Dynamics in Reactions on Metal Surfaces: A Theoretical
Perspective. J. Chem. Phys. 2019, 150, 180901.
Shenvi et al. 2009
Shenvi, N.; Roy, S.; Tully, J. C. Nonadiabatic Dynamics at Metal Surfaces:
Independent-Electron Surface Hopping. J. Chem. Phys. 2009,
130, 174107.
Shenvi et al. 2009
Shenvi, N.; Roy, S.; Tully, J. C. Dynamical Steering and Electronic Excitation
in NO Scattering from a Gold Surface. Science 2009,
326, 829–832.
Dou et al. 2018
Dou, W.; Schinabeck, C.; Thoss, M.; Subotnik, J. E. A broadened classical
master equation approach for treating electron-nuclear coupling in
non-equilibrium transport. J. Chem. Phys. 2018, 148,
102317.
Jiang et al. 2020
Jiang, B.; Li, J.; Guo, H. High-Fidelity Potential Energy Surfaces for Gas
Phase and Gas-Surface Scattering Processes from Machine Learning. J.
Phys. Chem. Lett. 2020, 11, 5120â5131.
Buhrke and Hildebrandt 2020
Buhrke, D.; Hildebrandt, P. Probing Structure and Reaction Dynamics of Proteins
Using Time-Resolved Resonance Raman Spectroscopy. Chem. Rev.
2020, 120, 3577–3630.
Raimbault et al. 2019
Raimbault, N.; Grisafi, A.; Ceriotti, M.; Rossi, M. Using Gaussian Process
Regression to Simulate the Vibrational Raman Spectra of Molecular Crystals.
New J. Phys. 2019, 21, 105001.
Hu et al. 2019
Hu, W.; Ye, S.; Zhang, Y.; Li, T.; Zhang, G.; Luo, Y.; Mukamel, S.; Jiang, J.
Machine Learning Protocol for Surface-Enhanced Raman Spectroscopy. J.
Phys. Chem. Lett. 2019, 10, 6026–6031.
Lussier et al. 2020
Lussier, F.; Thibault, V.; Charron, B.; Wallace, G. Q.; Masson, J.-F. Deep
Learning and Artificial Intelligence Methods for Raman and Surface-Enhanced
Raman Scattering. TrAC, Trends Anal. Chem. 2020, 124,
115796.
Fu and Hopkins 2018
Fu, W.; Hopkins, W. S. Applying Machine Learning to Vibrational Spectroscopy.
J. Phys. Chem. A 2018, 122, 167–171.
Aires-de Sousa et al. 2002
Aires-de Sousa, J.; Hemmer, M. C.; Gasteiger, J. Prediction of ${}^{1}$H NMR
Chemical Shifts Using Neural Networks. Anal. Chem. 2002,
74, 80–90.
Taguchi et al. 2019
Taguchi, A. T.; Evans, E. D.; Dikanov, S. A.; Griffin, R. G. Convolutional
Neural Network Analysis of Two-Dimensional Hyperfine Sublevel Correlation
Electron Paramagnetic Resonance Spectra. J. Phys. Chem. Lett.
2019, 10, 1115–1119.
Cobas 2020
Cobas, C. NMR Signal Processing, Prediction, and Structure Verification with
Machine Learning Techniques. Magn. Reson. Chem. 2020,
58, 512–519.
Salomon-Ferrer et al. 2013
Salomon-Ferrer, R.; Case, D. A.; Walker, R. C. An overview of the Amber
biomolecular simulation package. WIREs Computational Molecular Science
2013, 3, 198–210.
B. r. Brooks, et al. 2009
B. r. Brooks, et al., CHARMM: The Biomolecular Simulation Program. J.
Comput. Chem. 2009, 30, 1545–1614.
Eichenberger et al. 2011
Eichenberger, A. P.; Allison, J. R.; Dolenc, J.; Geerke, D. P.; Horta, B.
A. C.; Meier, K.; Oostenbrink, C.; Schmid, N.; Steiner, D.; Wang, D.; van
Gunsteren, W. F. GROMOS++ Software for the Analysis of Biomolecular
Simulation Trajectories. J. Chem. Theory Comput. 2011,
7, 3379–3390.
Reif et al. 2012
Reif, M. M.; Hünenberger, P. H.; Oostenbrink, C. New Interaction Parameters
for Charged Amino Acid Side Chains in the GROMOS Force Field. J. Chem.
Theory Comput. 2012, 8, 3705–3723.
Perthold et al. 2020
Perthold, J. W.; Petrov, D.; Oostenbrink, C. Towards Automated Free Energy
Calculation with Accelerated Enveloping Distribution Sampling (A-EDS).
J. Chem. Inf. Model. 2020, in press,
doi:10.1021/acs.jcim.0c00456.
Öhlknecht et al. 2020
Öhlknecht, C.; Lier, B.; Petrov, D.; Fuchs, J.; Oostenbrink, C. Correcting
Electrostatic Artifacts due to Net-Charge Changes in the Calculation of
Ligand Binding Free Energies. J. Comput. Chem. 2020,
41, 986–999.
Michlits et al. 2020
Michlits, H.; Lier, B.; Pfanzagl, V.; Djinović-Carugo, K.;
Furtmüller, P. G.; Oostenbrink, C.; Obinger, C.; Hofbauer, S.
Actinobacterial Coproheme Decarboxylases Use Histidine as a Distal Base to
Promote Compound I Formation. ACS Catal. 2020, 10,
5405–5418.
Brunk and Rothlisberger 2015
Brunk, E.; Rothlisberger, U. Mixed Quantum Mechanical/Molecular Mechanical
Molecular Dynamics Simulations of Biological Systems in Ground and
Electronically Excited States. Chem. Rev. 2015, 115,
6217–6263.
Bedrov et al. 2019
Bedrov, D.; Piquemal, J.-P.; Borodin, O.; MacKerell, A. D.; Roux, B.;
Schröder, C. Molecular Dynamics Simulations of Ionic Liquids and
Electrolytes Using Polarizable Force Fields. Chem. Rev. 2019,
119, 7940–7995.
Sosso et al. 2016
Sosso, G. C.; Chen, J.; Cox, S. J.; Fitzner, M.; Pedevilla, P.; Zen, A.;
Michaelides, A. Crystal Nucleation in Liquids: Open Questions and Future
Challenges in Molecular Dynamics Simulations. Chem. Rev.
2016, 116, 7078–7116.
Venable et al. 2019
Venable, R. M.; Krämer, A.; Pastor, R. W. Molecular Dynamics Simulations of
Membrane Permeability. Chem. Rev. 2019, 119,
5954–5997.
Marrink et al. 2019
Marrink, S. J.; Corradi, V.; Souza, P. C.; Ingólfsson, H. I.; Tieleman, D. P.;
Sansom, M. S. Computational Modeling of Realistic Cell Membranes.
Chemical Reviews 2019, 119, 6184–6226.
G. 2013
G., G. In Biomolecular Simulations. Methods in Molecular Biology (Methods
and Protocols); Monticelli, L., Salonen, E., Eds.; Humana Press, Totowa, NJ,
2013; Vol. 924.
Thomas P. Senftle, et al. 2016
Thomas P. Senftle, et al., The ReaxFF Reactive Force-Field: Development,
Applications and Future Directions. npj Comput. Mater. 2016,
2.
Sauceda et al. 2020
Sauceda, H. E.; Chmiela, S.; Poltavsky, I.; Müller, K.-R.; Tkatchenko, A.
In Machine Learning Meets Quantum Physics; Schütt, K. T.,
Chmiela, S., von Lilienfeld, O. A., Tkatchenko, A., Tsuda, K.,
Müller, K.-R., Eds.; Springer International Publishing: Cham, 2020; pp
277–307.
Noé 2020
Noé, F. In Machine Learning Meets Quantum Physics;
Schütt, K. T., Chmiela, S., von Lilienfeld, O. A., Tkatchenko, A.,
Tsuda, K., Müller, K.-R., Eds.; Springer International Publishing: Cham,
2020; pp 331–372.
Glielmo et al. 2020
Glielmo, A.; Zeni, C.; Fekete, Á.; De Vita, A. In Machine Learning
Meets Quantum Physics; Schütt, K. T., Chmiela, S., von
Lilienfeld, O. A., Tkatchenko, A., Tsuda, K., Müller, K.-R., Eds.;
Springer International Publishing: Cham, 2020; pp 67–98.
Abbott et al. 2019
Abbott, A. S.; Turney, J. M.; Zhang, B.; Smith, D. G. A.; Altarawy, D.;
Schaefer, H. F. PES-Learn: An Open-Source Software Package for the Automated
Generation of Machine Learning Models of Molecular Potential Energy Surfaces.
J. Chem. Theory Comput. 2019, 15, 4386–4398.
Hellström and Behler 2020
Hellström, M.; Behler, J. In Machine Learning Meets Quantum Physics;
Schütt, K. T., Chmiela, S., von Lilienfeld, O. A., Tkatchenko, A.,
Tsuda, K., Müller, K.-R., Eds.; Springer International Publishing: Cham,
2020; pp 253–275.
Vargas-Hernández and Krems 2020
Vargas-Hernández, R. A.; Krems, R. V. In Machine Learning Meets
Quantum Physics; Schütt, K. T., Chmiela, S., von Lilienfeld, O. A.,
Tkatchenko, A., Tsuda, K., Müller, K.-R., Eds.; Springer International
Publishing: Cham, 2020; pp 171–194.
Köppel et al. 2004
Köppel, H.; Domcke, W.; Cederbaum, L. S. in: Conical Intersections
(W. Domcke, D. R. Yarkony, H. Köppel, Eds.); World Scientific: New York,
2004.
Plasser et al. 2019
Plasser, F.; Gómez, S.; Menger, M. F. S. J.; Mai, S.; González, L. Highly
Efficient Surface Hopping Dynamics using a Linear Vibronic Coupling Model.
Phys. Chem. Chem. Phys. 2019, 21, 57–69.
He et al. 2008
He, G. S.; Tan, L.-S.; Zheng, Q.; Prasad, P. N. Multiphoton Absorbing
Materials: Molecular Designs, Characterizations, and Applications.
Chem. Rev. 2008, 108, 1245–1330.
Marquetand et al. 2014
Marquetand, P.; Weinacht, T.; Rozgonyi, T.; González-Vazquez, J.;
GeiÃler, D.; González, L. In Advances in Multiphoton Processes and
Spectroscopy; Fujimura, Y., Ed.; World Scientific, Singapore, 2014; Vol. 21;
pp 1–54.
Tagliamonti et al. 2016
Tagliamonti, V.; Sándor, P.; Zhao, A.; Rozgonyi, T.; Marquetand, P.;
Weinacht, T. Nonadiabatic Dynamics and Multiphoton Resonances in Strong-Field
Molecular Ionization with Few-Cycle Laser Pulses. Phys. Rev. A
2016, 93, 051401.
M. Wollenhaupt and Baumert 2007
M. Wollenhaupt, A. A.; Baumert, T. In Springer Handbook of Lasers and
Optics; Träger, F., Ed.; Springer Science and Business Media, LLC New
York, 2007; Chapter 12, pp 937–983.
Hilborn 1982
Hilborn, R. C. Einstein Coefficients, Cross Sections, f Values, Dipole Moments,
and All That. Am. J. Phys. 1982, 50, 982–986.
Andrews 2014
Andrews, D. L. Molecular Photophysics and Spectroscopy; 2053-2571;
Morgan & Claypool Publishers, 2014; pp 9–1 to 9–4.
Silva et al. 2007
Silva, G. L.; Ediz, V.; Yaron, D.; Armitage, B. A. Experimental and
Computational Investigation of Unsymmetrical Cyanine Dyes: Understanding
Torsionally Fluorogenic Dyes. J. Am. Chem. Soc. 2007,
129, 5710–5718.
Hartschuh et al. 2003
Hartschuh, A.; Pedrosa, H. N.; Novotny, L.; Krauss, T. D. Simultaneous
Fluorescence and Raman Scattering from Single Carbon Nanotubes.
Science 2003, 301, 1354–1356.
Terenziani et al. 2008
Terenziani, F.; Katan, C.; Badaeva, E.; Tretiak, S.; Blanchard-Desce, M.
Enhanced Two-Photon Absorption of Organic Chromophores: Theoretical and
Experimental Assessments. Adv. Mater. 2008, 20,
4641–4678.
Richings and Habershon 2020
Richings, G. W.; Habershon, S. A New Diabatization Scheme for Direct Quantum
Dynamics: Procrustes Diabatization. J. Chem. Phys. 2020,
152, 154108.
Tannor 2006
Tannor, D. Introduction to Quantum Mechanics: A Time-Dependent
Perspective; University Science Books: Sausalito, 2006.
Weinacht and Pearson 2019
Weinacht, T.; Pearson, B. Time-Resolved Spectroscopy: An Experimental
Perspective; CRC Press: New York, 2019.
Mai et al. 2018
Mai, S.; Marquetand, P.; González, L. Nonadiabatic Dynamics: The SHARC
Approach. WIREs Comput. Mol. Sci. 2018, 8,
e1370.
Yonehara et al. 2012
Yonehara, T.; Hanasaki, K.; Takatsuka, K. Fundamental Approaches to
Nonadiabaticity: Toward a Chemical Theory beyond the Born–Oppenheimer
Paradigm. Chem. Rev. 2012, 112, 499–542.
Casida and Huix-Rotllant 2012
Casida, M.; Huix-Rotllant, M. Progress in Time-Dependent Density-Functional
Theory. Annu. Rev. Phys. Chem. 2012, 63,
287–323.
Maitra 2016
Maitra, N. T. Perspective: Fundamental Aspects of Time-Dependent Density
Functional Theory. J. Chem. Phys. 2016, 144,
220901.
Szalay et al. 2012
Szalay, P. G.; Müller, T.; Gidofalvi, G.; Lischka, H.; Shepard, R.
Multiconfiguration Self-Consistent Field and Multireference Configuration
Interaction Methods and Applications. Chem. Rev. 2012,
112, 108–181.
Helgaker et al. 2014
Helgaker, T.; Jørgensen, P.; Olsen, J. Molecular
ElectronicâStructure Theory; John Wiley & Sons, Ltd, 2014.
Roos et al. 2016
Roos, B. O.; Lindh, R.; Malmqvist, P. Å.; Veryazov, V.; Widmark, P.
Multiconfigurational Quantum Chemistry; John Wiley & Sons, Ltd,
2016.
Born and Oppenheimer 1927
Born, M.; Oppenheimer, R. Zur Quantentheorie der Molekeln. Ann. Phys.
1927, 389, 457–484.
Kohn 1999
Kohn, W. Nobel Lecture: Electronic Structure of Matter – Wave Functions and
Density Functionals. Rev. Mod. Phys. 1999, 71,
1253–1266.
Schrödinger 1926
Schrödinger, E. An Undulatory Theory of the Mechanics of Atoms and
Molecules. Phys. Rev. 1926, 28, 1049–1070.
252
Erwin-Schrödinger – Nobel Lecture.
https://www.nobelprize.org/prizes/phy-sics/1933/schrodinger/lecture/.
Yu et al. 2016
Yu, H. S.; Li, S. L.; Truhlar, D. G. Perspective: Kohn-Sham Density
Functional Theory Descending a Staircase. J. Chem. Phys.
2016, 145, 130901.
Maurer et al. 2019
Maurer, R. J.; Freysoldt, C.; Reilly, A. M.; Brandenburg, J. G.;
Hofmann, O. T.; Björkman, T.; Lebègue, S.; Tkatchenko, A. Advances in
Density-Functional Calculations for Materials Modeling. Annual Review
of Materials Research 2019, 49, 1–30.
Benavides-Riveros et al. 2017
Benavides-Riveros, C. L.; Lathiotakis, N. N.; Marques, M. A. L. Towards a
Formal Definition of Static and Dynamic Electronic Correlations. Phys.
Chem. Chem. Phys. 2017, 19, 12655–12664.
Szabo and Ostlund 2012
Szabo, A.; Ostlund, N. Modern Quantum Chemistry: Introduction to
Advanced Electronic Structure Theory; Dover Books on Chemistry; Dover
Publications, 2012.
Helgaker et al. 2014
Helgaker, T.; Jørgensen, P.; Olsen, J. Molecular
ElectronicâStructure Theory; John Wiley & Sons, Ltd, 2014; Chapter 10, pp
433–522.
Helgaker et al. 2014
Helgaker, T.; Jørgensen, P.; Olsen, J. Molecular
ElectronicâStructure Theory; John Wiley & Sons, Ltd, 2014; Chapter 11, pp
523–597.
Dreuw and Wormit 2015
Dreuw, A.; Wormit, M. The algebraic diagrammatic construction scheme for the
polarization propagator for the calculation of excited states. WIREs
Comput. Mol. Sci. 2015, 5, 82–95.
von Niessen et al. 1984
von Niessen, W.; Schirmer, J.; Cederbaum, L. Computational Methods for the
One-Particle Green’s Function. Comp. Phys. Rep. 1984,
1, 57 – 125.
Linderberg and Öhrn 2005
Linderberg, J.; Öhrn, Y. Propagators in Quantum Chemistry; John
Wiley & Sons, Ltd, 2005; Chapter 2, pp 3–6.
Melin et al. 2005
Melin, J.; Ayers, P.; Ortiz, J. The Electron-Propagator Approach to Conceptual
Density-Functional Theory. J. Chem. Sci. 2005, 117,
387–400.
Corzo and Ortiz 2017
Corzo, H. H.; Ortiz, J. V. In Löwdin Volume; Sabin, J. R.,
Brändas, E. J., Eds.; Advances in Quantum Chemistry; Academic Press,
2017; Vol. 74; pp 267 – 298.
Möller and Plesset 1934
Möller, C.; Plesset, M. S. Note on an Approximation Treatment for
Many-Electron Systems. Phys. Rev. 1934, 46,
618–622.
Bartlett 1981
Bartlett, R. J. Many-Body Perturbation Theory and Coupled Cluster Theory for
Electron Correlation in Molecules. Annu. Rev. Phys. Chem.
1981, 32, 359–401.
Helgaker et al. 2014
Helgaker, T.; Jørgensen, P.; Olsen, J. Molecular
ElectronicâStructure Theory; John Wiley & Sons, Ltd, 2014; Chapter 13, pp
648–723.
Izsák 2020
Izsák, R. Single-Reference Coupled Cluster Methods for Computing Excitation
Energies in Large Molecules: The Efficiency and Accuracy of Approximations.
WIREs Comput. Mol. Sci. 2020, 10, e1445.
Krylov 2008
Krylov, A. I. Equation-of-Motion Coupled-Cluster Methods for Open-Shell and
Electronically Excited Species: The Hitchhiker’s Guide to Fock Space.
Annu. Rev. Phys. Chem. 2008, 59, 433–462.
Parrill and Lipkowitz 2018
Parrill, A.; Lipkowitz, K. Reviews in Computational Chemistry, Volume
31; Reviews in Computational Chemistry; Wiley, 2018.
Pacifici L. 2013
Pacifici L., L. A., Verdicchio M. In Computational Science and Its
Applications â ICCSA 2013; B. Murgante, et. al,, Ed.; Springer,
Berlin, Heidelberg, 2013; Vol. 7971.
Helgaker et al. 2014
Helgaker, T.; Jørgensen, P.; Olsen, J. Molecular
ElectronicâStructure Theory; John Wiley & Sons, Ltd, 2014; Chapter 12, pp
598–647.
Roos et al. 1980
Roos, B. O.; Taylor, P. R.; Siegbahn, P. E. A Complete Active Space SCF Method
(CASSCF) using a Density Matrix Formulated Super-CI Approach. Chem.
Phys. 1980, 48, 157–173.
Roos and Siegbahn 1980
Roos, B. O.; Siegbahn, P. E. M. A Direct CI Method with a Multiconfigurational
Reference State. Int. J. Quantum Chem. 1980, 17,
485–500.
Lischka et al. 2004
Lischka, H.; Dallos, M.; Szalay, P. G.; Yarkony, D. R.; Shepard, R. Analytic
Evaluation of Nonadiabatic coupling Terms at the MR-CI Level. I.
Formalism. J. Chem. Phys. 2004, 120,
7322–7329.
Lischka et al. 2020
Lischka, H. et al. The Generality of the GUGA MRCI Approach in
COLUMBUS for Treating Complex Quantum Chemistry. J. Chem. Phys.
2020, 152, 134110.
Andersson et al. 1990
Andersson, K.; Malmqvist, P. A.; Roos, B. O.; Sadlej, A. J.; Wolinski, K.
Second-Order Perturbation Theory with a CASSCF Reference Function. J.
Phys. Chem. 1990, 94, 5483–5488.
Andersson et al. 1992
Andersson, K.; Malmqvist, P.; Roos, B. O. SecondâOrder Perturbation Theory
with a Complete Active Space SelfâConsistent Field Reference Function.
J. Phys. Chem. 1992, 96, 1218–1226.
Finley et al. 1998
Finley, J.; Malmqvist, P.-A.; Roos, B. O.; Serrano-Andrés, L. The
Multi-State {CASPT2} Method. Chem. Phys. Lett. 1998,
288, 299 – 306.
Angeli et al. 2001
Angeli, C.; Cimiraglia, R.; Evangelisti, S.; Leininger, T.; Malrieu, J.-P.
Introduction of N-Electron Valence States for Multireference Perturbation
Theory. J. Chem. Phys. 2001, 114, 10252–10264.
Roemelt et al. 2016
Roemelt, M.; Guo, S.; Chan, G. K.-L. A Projected Approximation to Strongly
Contracted N-Electron Valence Perturbation Theory for DMRG Wavefunctions.
J. Chem. Phys. 2016, 144, 204113.
Guo et al. 2017
Guo, Y.; Sivalingam, K.; Valeev, E. F.; Neese, F. Explicitly Correlated
N-Electron Valence State Perturbation Theory (NEVPT2-F12). J. Chem.
Phys. 2017, 147, 064110.
Maitra et al. 2012
Maitra, R.; Sinha, D.; Mukherjee, D. Unitary Group Adapted State-Specific
Multi-Reference Coupled Cluster Theory: Formulation and Pilot Numerical
Applications. J. Chem. Phys. 2012, 137, 024105.
Máşik and Hubaç 1998
Máşik, J.; Hubaç, I. In Multireference Brillouin-Wigner
Coupled-Cluster Theory. Single-Root Approach.; Sabin, J. R., Zerner, M. C.,
Brändas, E., Wilson, S., Maruani, J., Smeyers, Y., Grout, P.,
McWeeny, R., Eds.; Advances in Quantum Chemistry; Academic Press, 1998;
Vol. 31; pp 75 – 104.
Musiał et al. 2011
Musiał, M.; Perera, A.; Bartlett, R. J. Multireference Coupled-Cluster
Theory: The Easy Way. J. Chem. Phys. 2011, 134,
114108.
Evangelista 2018
Evangelista, F. A. Perspective: Multireference Coupled Cluster Theories of
Dynamical Electron Correlation. J. Chem. Phys. 2018,
149, 030901.
Fdez. Galván et al. 2019
Fdez. Galván, I. et al. OpenMolcas: From Source Code to Insight.
J. Chem. Theory Comput. 2019, 15, 5925–5964.
Roos et al. 2004
Roos, B.; Lindh, R.; Malmqvist, P.-Å.; Veryazov, V.; Widmark, P.-O. Main
Group Atoms and Dimers Studied with a new Relativistic ANO Basis Set.
J. Phys. Chem. A 2004, 108, 2851–2858.
Vogiatzis et al. 2017
Vogiatzis, K. D.; Ma, D.; Olsen, J.; Gagliardi, L.; de Jong, W. A. Pushing
Configuration-Interaction to the limit: Towards Massively Parallel MCSCF
Calculations. J. Chem. Phys. 2017, 147, 184111.
Kato and Baba 1995
Kato, H.; Baba, M. Dynamics of Excited Molecules: Predissociation. Chem.
Rev. 1995, 95, 2311–2349.
Merer and Mulliken 1969
Merer, A. J.; Mulliken, R. S. Ultraviolet Spectra and Excited States of
Ethylene and its Alkyl Derivatives. Chem. Rev. 1969,
69, 639–656.
Ashfold and Langford 1999
Ashfold, M. N. R.; Langford, S. R. In The Role of Rydberg States in
Spectroscopy and Photochemistry: Low and High Rydberg States;
Sándorfy, C., Ed.; Springer Netherlands: Dordrecht, 1999; pp 23–56.
Merkt 1997
Merkt, F. Molecules in High Rydberg States. Annu. Rev. Phys. Chem.
1997, 48, 675–709.
Stein and Reiher 2016
Stein, C. J.; Reiher, M. Automated Selection of Active Orbital Spaces. J.
Chem. Theory Comput. 2016, 12, 1760–1771.
Stein and Reiher 2017
Stein, C. J.; Reiher, M. Measuring Multi-Configurational Character by Orbital
Entanglement. Mol. Phys. 2017, 115, 2110–2119.
Stein and Reiher 2017
Stein, C. J.; Reiher, M. Automated Identification of Relevant Frontier
Orbitals for Chemical Compounds and Processes. CHIMIA 2017,
71, 170–176.
Chan and Van Voorhis 2005
Chan, G. K.-L.; Van Voorhis, T. Density-Matrix Renormalization-Group Algorithms
with Nonorthogonal Orbitals and Non-Hermitian Operators, and Applications to
Polyenes. J. Chem. Phys. 2005, 122, 204101.
Zgid and Nooijen 2008
Zgid, D.; Nooijen, M. The Density Matrix Renormalization Group Self-Consistent
Field Method: Orbital Optimization with the Density Matrix Renormalization
Group Method in the Active Space. J. Chem. Phys. 2008,
128, 144116.
Keller et al. 2015
Keller, S.; Dolfi, M.; Troyer, M.; Reiher, M. An Efficient Matrix Product
Operator Representation of the Quantum Chemical Hamiltonian. J. Chem.
Phys. 2015, 143, 244118.
Knecht et al. 2016
Knecht, S.; Keller, S.; Autschbach, J.; Reiher, M. A Nonorthogonal
State-Interaction Approach for Matrix Product State Wave Functions. J.
Chem. Theory Comput. 2016, 12, 5881–5894.
Freitag et al. 2017
Freitag, L.; Knecht, S.; Angeli, C.; Reiher, M. Multireference Perturbation
Theory with Cholesky Decomposition for the Density Matrix Renormalization
Group. J. Chem. Theory Comput. 2017, 13,
451–459.
Freitag et al. 2019
Freitag, L.; Ma, Y.; Baiardi, A.; Knecht, S.; Reiher, M. Approximate
Analytical Gradients and Nonadiabatic Couplings for the State-Average Density
Matrix Renormalization Group Self-Consistent-Field Method. J. Chem.
Theory Comput. 2019, 15, 6724–6737.
Hohenberg and Kohn 1964
Hohenberg, P.; Kohn, W. Inhomogeneous Electron Gas. Phys. Rev.
1964, 136, B864–B871.
Kohn and Sham 1965
Kohn, W.; Sham, L. J. Self-Consistent Equations Including Exchange and
Correlation Effects. Phys. Rev. 1965, 140,
A1133–A1138.
Casida 2009
Casida, M. E. Time-Dependent Density-Functional Theory for Molecules and
Molecular Solids. J. Mol. Struc.-Theochem 2009, 914, 3
– 18.
Runge and Gross 1984
Runge, E.; Gross, E. K. U. Density-Functional Theory for Time-Dependent
Systems. Phys. Rev. Lett. 1984, 52, 997–1000.
Zangwill and Soven 1980
Zangwill, A.; Soven, P. Density-Functional Approach to Local-Field Effects in
Finite Systems: Photoabsorption in the Rare Gases. Phys. Rev. A
1980, 21, 1561–1572.
Chong 1995
Chong, D. P. Recent Advances in Density Functional Methods; World
Scientific, 1995.
Tamm 1945
Tamm, I. Relativistic Interaction of Elementary Particles. J. Phys.
(Moscow) 1945, 9, 449.
Dancoff 1950
Dancoff, S. M. Non-Adiabatic Meson Theory of Nuclear Forces. Phys. Rev.
1950, 78, 382–385.
Hirata and Head-Gordon 1999
Hirata, S.; Head-Gordon, M. Time-Dependent Density Functional Theory within the
TammâDancoff Approximation. Chem. Phys. Lett. 1999,
314, 291 – 299.
Cordova et al. 2007
Cordova, F.; Doriol, L. J.; Ipatov, A.; Casida, M. E.; Filippi, C.; Vela, A.
Troubleshooting Time-Dependent Density-Functional Theory for Photochemical
Applications: Oxirane. J. Chem. Phys. 2007, 127,
164111.
Goerigk and Casanova-Paéz 2020
Goerigk, L.; Casanova-Paéz, M. The Trip to the Density Functional Theory Zoo
Continues: Making a Case for Time-Dependent Double Hybrids for Excited-State
Problems. Aust. J. Chem. 2020, in press,
DOI:10.1071/CH20093.
Worth and Cederbaum 2004
Worth, G. A.; Cederbaum, L. S. Beyond Born-Oppenheimer: Molecular Dynamics
Through a conical Intersection. Annu. Rev. Phys. Chem. 2004,
55, 127–158.
Doltsinis 2006
Doltsinis, N. L. Molecular Dynamics Beyond the Born-Oppenheimer
Approximation: Mixed Quantum-Classical Approaches; NIC Series; John von
Neuman Institut for computing, 2006; Vol. 31; pp 389–409.
Jacquemin and Adamo 2016
Jacquemin, D.; Adamo, C. In Density-Functional Methods for Excited
States; Ferré, N., Filatov, M., Huix-Rotllant, M., Eds.; Springer
International Publishing: Cham, 2016; pp 347–375.
Li et al. 2014
Li, S. L.; Marenich, A. V.; Xu, X.; Truhlar, D. G. Configuration
Interaction-Corrected TammâDancoff Approximation: A Time-Dependent Density
Functional Method with the Correct Dimensionality of Conical Intersections.
J. Phys. Chem. Lett. 2014, 5, 322–328.
Bannwarth et al. 2020
Bannwarth, C.; Yu, J. K.; Hohenstein, E. G.; Martínez, T. J. Hole-Hole
Tamm-Dancoff-Approximated Density Functional Theory: A Highly Efficient
Electronic Structure Method Incorporating Dynamic and Static Correlation.
ChemRxiv 2020,
Maitra et al. 2004
Maitra, N. T.; Zhang, F.; Cave, R. J.; Burke, K. Double Excitations within
Time-Dependent Density Functional Theory Linear Response. J. Chem.
Phys. 2004, 120, 5932–5937.
Elliott et al. 2011
Elliott, P.; Goldson, S.; Canahui, C.; Maitra, N. T. Perspectives on
Double-Excitations in TDDFT. Chem. Phys. 2011, 391,
110 – 119.
Katriel et al. 2001
Katriel, J.; Zahariev, F.; Burke, K. Symmetry and Degeneracy in Density
Functional Theory. Int. J. Quantum Chem. 2001, 85,
432–435.
Shao et al. 2003
Shao, Y.; Head-Gordon, M.; Krylov, A. I. The SpinâFlip Approach within
Time-Dependent Density Functional Theory: Theory and Applications to
Diradicals. J. Chem. Phys. 2003, 118, 4807–4818.
Gavnholt et al. 2008
Gavnholt, J.; Olsen, T.; Engelund, M.; Schiøtz, J. $\Delta$
Self-Consistent Field Method to obtain Potential Energy Surfaces of Excited
Molecules on Surfaces. Phys. Rev. B 2008, 78,
075441.
Maurer and Reuter 2011
Maurer, R. J.; Reuter, K. Assessing Computationally Efficient Isomerization
Dynamics: $\Delta$-SCF Density-Functional Theory Study of Azobenzene
Molecular Switching. J. Chem. Phys. 2011, 135,
224303.
Maurer and Reuter 2013
Maurer, R. J.; Reuter, K. Excited-State Potential-Energy Surfaces of
Metal-Adsorbed Organic Molecules from Linear Expansion
$\Delta$-Self-Consistent Field Density-Functional Theory
($\Delta$SCF-DFT). J. Chem. Phys. 2013, 139,
014708.
Chai and Head-Gordon 2008
Chai, J.-D.; Head-Gordon, M. Systematic Optimization of Long-Range Corrected
Hybrid Density Functionals. J. Chem. Phys. 2008, 128,
084106.
Tozer and Handy 2000
Tozer, D. J.; Handy, N. C. On the Determination of Excitation Energies using
Density Functional Theory. Phys. Chem. Chem. Phys. 2000,
2, 2117–2121.
Ramakrishnan et al. 2015
Ramakrishnan, R.; Hartmann, M.; Tapavicza, E.; von Lilienfeld, O. A. Electronic
Spectra from TDDFT and Machine Learning in Chemical Space. J. Chem.
Phys. 2015, 143, 084111.
Dral et al. 2020
Dral, P. O.; Owens, A.; Dral, A.; Csányi, G. Hierarchical Machine Learning
of Potential Energy Surfaces. J. Chem. Phys. 2020,
152, 204110.
Smith et al. 2019
Smith, J. S.; Nebgen, B. T.; Zubatyuk, R.; Lubbers, N.; Devereuz, C.;
Barros, K.; Tretiak, S.; Isayev, O.; Roitberg, A. E. Approaching Coupled
Cluster Accuracy with a General-Purpose Neural Network Potential Through
Transfer Learning. Nat. Commun. 2019, 10.
Abedi et al. 2010
Abedi, A.; Maitra, N. T.; Gross, E. K. U. Exact Factorization of the
Time-Dependent Electron-Nuclear Wave Function. Phys. Rev. Lett.
2010, 105, 123002.
Thachuk et al. 1996
Thachuk, M.; Ivanov, M. Y.; Wardlaw, D. M. A Semiclassical Approach to
IntenseâField AboveâThreshold Dissociation in the Long Wavelength Limit.
J. Chem. Phys. 1996, 105, 4094–4104.
Mitrić
et al. 2009
Mitrić, R.; Petersen, J.; Bonači ć Koutecký, V.
Laser-Field-Induced Surface-Hopping Method for the Simulation and Control of
Ultrafast Photodynamics. Phys. Rev. A 2009, 79,
053416.
Mitrić et al. 2011
Mitrić, R.; Petersen, J.; Wohlgemuth, M.; Werner, U.;
Bonaçić-Koutecký, V. Field-Induced Surface Hopping Method for
Probing Transition State Nonadiabatic Dynamics of Ag${}_{3}$. Phys. Chem.
Chem. Phys. 2011, 13, 8690–8696.
Granucci et al. 2012
Granucci, G.; Persico, M.; Spighi, G. Surface Hopping Trajectory Simulations
with Spin-Orbit and Dynamical Couplings. J. Chem. Phys. 2012,
137, 22A501.
Baer 2002
Baer, M. Introduction to the Theory of Electronic Non-Adiabatic Coupling Terms
in Molecular Systems. Phys. Rep. 2002, 358,
75–142.
Kryachko and Yarkony 2000
Kryachko, E. S.; Yarkony, D. R. Diabatic Bases and Molecular Properties.
Int. J. Quantum Chem. 2000, 76, 235–243.
Marian 2012
Marian, C. M. SpinâOrbit Coupling and Intersystem Crossing in Molecules.
WIREs Comput. Mol. Sci. 2012, 2, 187–203.
K. G. Dyall 2007
K. G. Dyall, K. F. Introduction to Relativistic Quantum Chemistry;
Oxford University Press, 2007.
M. Reiher 2009
M. Reiher, A. W. Relativistic Quantum Chemistry; Wiley VCH Verlag
Weinheim, 2009.
H. A. Bethe 1957
H. A. Bethe, E. E. S. Quantum Mechanics of One- and Two-Electron
Atoms; Springer, Berlin, 1957.
Pyykko 1988
Pyykko, P. Relativistic Effects in Structural Chemistry. Chem. Rev.
1988, 88, 563–594.
Neese et al. 2007
Neese, F.; Petrenko, T.; Ganyushin, D.; Olbrich, G. Advanced Aspects of
Ab Initio Theoretical Optical Spectroscopy of Transition Metal
complexes: Multiplets, spin-orbit coupling and resonance Raman intensities.
Coord. Chem. Rev. 2007, 251, 288 – 327.
Neese 2005
Neese, F. Efficient and Accurate Approximations to the Molecular Spin-Orbit
Coupling Operator and their Use in Molecular G-Tensor Calculations. J.
Chem. Phys. 2005, 122, 034107.
Richter et al. 2011
Richter, M.; Marquetand, P.; González-Vázquez, J.; Sola, I.;
González, L. SHARC: Ab Initio Molecular Dynamics with Surface Hopping
in the Adiabatic Representation Including Arbitrary couplings. J. Chem.
Theory Comput. 2011, 7, 1253–1258.
Mai et al. 2015
Mai, S.; Marquetand, P.; González, L. A General Method to Describe
Intersystem Crossing Dynamics in Trajectory Surface Hopping. Int. J.
Quantum Chem. 2015, 115, 1215–1231.
Mai et al. 2018
Mai, S.; Plasser, F.; Marquetand, P.; González, L. Attosecond Molecular
Dynamics; The Royal Society of Chemistry, 2018; pp 348–385.
Köppel et al. 2001
Köppel, H.; Gronki, J.; Mahapatra, S. Construction Scheme for Regularized
Diabatic States. J. Chem. Phys. 2001, 115,
2377–2388.
Richings and Worth 2015
Richings, G. W.; Worth, G. A. A Practical Diabatisation Scheme for Use with the
Direct-Dynamics Variational Multi-configuration Gaussian Method. J.
Phys. Chem. A 2015, 119, 12457–12470.
Accomasso et al. 2019
Accomasso, D.; Persico, M.; Granucci, G. Diabatization by Localization in the
Framework of Configuration Interaction Based on Floating Occupation Molecular
Orbitals (FOMO-CI). ChemPhotoChem 2019, 3,
933–944.
Lenzen and Manthe 2017
Lenzen, T.; Manthe, U. Neural Network Based Coupled Diabatic Potential Energy
Surfaces for Reactive Scattering. J. Chem. Phys. 2017,
147, 084105.
Subotnik et al. 2008
Subotnik, J. E.; Yeganeh, S.; Cave, R. J.; Ratner, M. A. Constructing Diabatic
States from Adiabatic States: Extending Generalized MullikenâHush to
Multiple Charge Centers with Boys Localization. J. Chem. Phys.
2008, 129, 244101.
Hoyer et al. 2016
Hoyer, C. E.; Parker, K.; Gagliardi, L.; Truhlar, D. G. The DQ and DQØ
Electronic Structure Diabatization Methods: Validation for General
Applications. J. Chem. Phys. 2016, 144, 194101.
Wittenbrink et al. 2013
Wittenbrink, N.; Ndome, H.; Eisfeld, W. Toward SpinâOrbit coupled Diabatic
Potential Energy Surfaces for Methyl Iodide Using Effective Relativistic
Coupling by Asymptotic Representation. J. Phys. Chem. A 2013,
117, 7408–7420.
Varga et al. 2018
Varga, Z.; Parker, K. A.; Truhlar, D. G. Direct Diabatization Based on
Nonadiabatic Couplings: The N/D Method. Phys. Chem. Chem. Phys.
2018, 20, 26643–26659.
Nakamura and Truhlar 2002
Nakamura, H.; Truhlar, D. G. Direct Diabatization of Electronic States by the
Fourfold Way. II. Dynamical Correlation and Rearrangement Processes. J.
Chem. Phys. 2002, 117, 5576–5593.
Cave and Stanton 2014
Cave, R. J.; Stanton, J. F. Block Diagonalization of the Equation-of-Motion
Coupled Cluster Effective Hamiltonian: Treatment of Diabatic Potential
Constants and Triple Excitations. J. Chem. Phys. 2014,
140, 214112.
Venghaus and Eisfeld 2016
Venghaus, F.; Eisfeld, W. Block-Diagonalization as a Tool for the Robust
Diabatization of High-Dimensional Potential Energy Surfaces. J. Chem.
Phys. 2016, 144, 114110.
Robertson et al. 2019
Robertson, C.; González-Vázquez, J.; corral, I.; Díaz-Tendero, S.;
Díaz, C. Nonadiabatic Scattering of NO off Au3 Clusters: A Simple and
Robust Diabatic State Manifold Generation Method for Multiconfigurational
Wavefunctions. J. Comput. Chem. 2019, 40,
794–810.
Li et al. 2013
Li, J.; Jiang, B.; Guo, H. Permutation Invariant Polynomial Neural Network
Approach to Fitting Potential Energy Surfaces. II. Four-Atom Systems.
J. Chem. Phys. 2013, 139, 204103.
Jiang and Guo 2013
Jiang, B.; Guo, H. Permutation Invariant Polynomial Neural Network Approach to
Fitting Potential Energy Surfaces. J. Chem. Phys. 2013,
139, 054112.
Jiang and Guo 2014
Jiang, B.; Guo, H. Permutation Invariant Polynomial Neural Network Approach to
Fitting Potential Energy Surfaces. III. Molecule-Surface Interactions.
J. Chem. Phys. 2014, 141, 034109.
Jiang et al. 2016
Jiang, B.; Li, J.; Guo, H. Potential Energy Surfaces from High Fidelity Fitting
of Ab Initio Points: The Permutation Invariant Polynomial - Neural
Network Approach. Int. Rev. Phys. Chem. 2016, 35,
479–506.
Mai et al. 2014
Mai, S.; M. Richter, M.; Marquetand, P.; González, L. Excitation of
Nucleobases from a Computational Perspective II: Dynamics. 2014,
Liu 2020
Liu, W. Essentials of Relativistic Quantum Chemistry. J. Chem. Phys.
2020, 152, 180901.
Horton et al. 2019
Horton, S. L.; Liu, Y.; Forbes, R.; Makhija, V.; Lausten, R.; Stolow, A.;
Hockett, P.; Marquetand, P.; Rozgonyi, T.; Weinacht, T. Excited state
dynamics of CH${}_{2}$I${}_{2}$ and CH${}_{2}$BrI studied with UV pump VUV probe
photoelectron spectroscopy. J. Chem. Phys. 2019, 150,
174201.
Horton et al. 2018
Horton, S. L.; Liu, Y.; Chakraborty, P.; Marquetand, P.; Rozgonyi, T.;
Matsika, S.; Weinacht, T. Strong-Field- Versus Weak-Field-Ionization
Pump-Probe Spectroscopy. Phys. Rev. A 2018, 98,
053416.
Sussman et al. 2006
Sussman, B. J.; Townsend, D.; Ivanov, M. Y.; Stolow, A. Dynamic Stark Control
of Photochemical Processes. Science 2006, 314,
278–281.
Marquetand et al. 2011
Marquetand, P.; Richter, M.; González-Vázquez, J.; Sola, I.;
González, L. Nonadiabatic Ab Initio Molecular Dynamics Including Spin-Orbit
Coupling and Laser Fields. Faraday Discuss. 2011, 153,
261–273.
Bajo et al. 2012
Bajo, J. J.; González-Vázquez, J.; Sola, I.; Santamaria, J.; Richter, M.;
Marquetand, P.; González, L. Mixed Quantum-Classical Dynamics in the
Adiabatic Representation to Simulate Molecules Driven by Strong Laser Pulses.
J. Phys. Chem. A 2012, 116, 2800–2807.
Köppel et al. 1984
Köppel, H.; Domcke, W.; Cederbaum, L. S. Multimode Molecular Dynamics
Beyond the Born-Oppenheimer Approximation. Adv. Chem. Phys.
1984, 57, 59–246.
Ben-Nun and Martínez 2002
Ben-Nun, M.; Martínez, T. J. Advances in Chemical Physics; John
Wiley & Sons, Ltd, 2002; pp 439–512.
Beck et al. 2000
Beck, M.; Jäckle, A.; Worth, G.; Meyer, H.-D. The Multiconfiguration
Time-Dependent Hartree (MCTDH) Method: A Highly Efficient Algorithm for
Propagating Wavepackets. Phys. Rep. 2000, 324,
1–105.
Yeager and Jørgensen 1979
Yeager, D. L.; Jørgensen, P. A Multiconfigurational Time-Dependent
Hartree-Fock Approach. Chem. Phys. Lett. 1979, 65,
77–80.
Manthe 2017
Manthe, U. Wavepacket Dynamics and the Multi-Configurational Time-Dependent
Hartree Approach. J. Phys.: Condens. Matter 2017, 29,
253001.
Eng et al. 2015
Eng, J.; Gourlaouen, C.; Gindensperger, E.; Daniel, C. Spin-Vibronic Quantum
Dynamics for Ultrafast Excited-State Processes. Acc. Chem. Res.
2015, 48, 809–817.
Gómez et al. 2019
Gómez, S.; Heindl, M.; Szabadi, A.; González, L. From Surface Hopping
to Quantum Dynamics and Back. Finding Essential Electronic and Nuclear
Degrees of Freedom and Optimal Surface Hopping Parameters. J. Phys.
Chem. A 2019, 123, 8321–8332.
Ischtwan and Collins 1994
Ischtwan, J.; Collins, M. A. Molecular Potential Energy Surfaces by
Interpolation. J. Chem. Phys. 1994, 100,
8080–8088.
Evenhuis and collins 2004
Evenhuis, C. R.; collins, M. A. Interpolation of Diabatic Potential Energy
Surfaces. J. Chem. Phys. 2004, 121, 2515–2527.
Evenhuis and Martínez 2011
Evenhuis, C.; Martínez, T. J. A Scheme to Interpolate Potential Energy
Surfaces and Derivative Coupling Vectors without Performing a Global
Diabatization. J. Chem. Phys. 2011, 135, 224110.
Mukherjee et al. 2013
Mukherjee, S.; Bandyopadhyay, S.; Paul, A. K.; Adhikari, S. Construction of
Diabatic Hamiltonian Matrix from Ab Initio Calculated Molecular
Symmetry Adapted Nonadiabatic Coupling Terms and Nuclear Dynamics for the
Excited States of Na${}_{3}$ Cluster. J. Phys. Chem. A 2013,
117, 3475–3495.
Worth et al. 2008
Worth, G.; Robb, M.; Lasorne, B. Solving the Time-Dependent Schrödinger
Equation for Nuclear Motion in One Step: Direct Dynamics of Non-Adiabatic
Systems. Mol. Phys. 2008, 106, 2077–2091.
Persico and Granucci 2014
Persico, M.; Granucci, G. An Overview of Nonadiabatic Dynamics Simulations
Methods, with Focus on the Direct Approach Versus the Fitting of Potential
Energy Surfaces. Theor. Chem. Acc. 2014, 133,
1526.
Komarova et al. 2018
Komarova, K. G.; Remacle, F.; Levine, R. On the Fly Quantum Dynamics of
Electronic and Nuclear Wave Packets. Chem. Phys. Lett. 2018,
699, 155 – 161.
Lasorne et al. 2007
Lasorne, B.; Robb, M. A.; Worth, G. A. Direct Quantum Dynamics using
Variational Multi-Configuration Gaussian Wavepackets. Implementation Details
and Test Case. Phys. Chem. Chem. Phys. 2007, 9,
3210–3227.
Ben-Nun and Martínez 2000
Ben-Nun, M.; Martínez, T. J. Photodynamics of Ethylene: Ab Initio
Studies of Conical Intersections. Chem. Phys. 2000,
259, 237 – 248.
Curchod et al. 2016
Curchod, B. F. E.; Rauer, C.; Marquetand, P.; González, L.;
Martínez, T. J. Communication: GAIMSâGeneralized Ab Initio
Multiple Spawning for Both Internal Conversion and Intersystem Crossing
Processes. J. Chem. Phys. 2016, 144, 101102.
Mignolet and Curchod 2018
Mignolet, B.; Curchod, B. F. E. A Walk Through the Approximations of Ab
Initio Multiple Spawning. J. Chem. Phys. 2018, 148,
134110.
Freixas et al. 2018
Freixas, V. M.; Fernandez-Alberti, S.; Makhov, D. V.; Tretiak, S.;
Shalashilin, D. An Ab Initio Multiple Cloning Approach for the
Simulation of Photoinduced Dynamics in Conjugated Molecules. Phys.
Chem. Chem. Phys. 2018, 20, 17762–17772.
Tomislav Beguşić and Aurélien Patoz and Miroslav Şulc and
Jiŗí Vaníçek 2018
Tomislav Beguşić and Aurélien Patoz and Miroslav Şulc and
Jiŗí Vaníçek, On-the-Fly Ab Initio Three Thawed
Gaussians Approximation: A Semiclassical Approach to Herzberg-Teller Spectra.
Chem. Phys. 2018, 515, 152 – 163.
Markland and Ceriotti 2018
Markland, T.; Ceriotti, M. Nuclear Quantum Effects Enter the Mainstream.
Nat. Rev. Chem. 2018, 2.
Miller 1970
Miller, W. H. Classical S Matrix: Numerical Application to Inelastic
Collisions. J. Chem. Phys. 1970, 53, 3578–3587.
Ceotto et al. 2009
Ceotto, M.; Atahan, S.; Shim, S.; Tantardini, G. F.; Aspuru-Guzik, A.
First-Principles Semiclassical Initial Value Representation Molecular
Dynamics. Phys. Chem. Chem. Phys. 2009, 11,
3861–3867.
Nakamura et al. 2016
Nakamura, H.; Nanbu, S.; Teranishi, Y.; Ohta, A. Development of Semiclassical
Molecular Dynamics Simulation Method. Phys. Chem. Chem. Phys.
2016, 18, 11972–11985.
Gao et al. 2020
Gao, X.; Saller, M. A. C.; Liu, Y.; Kelly, A.; Richardson, J. O.; Geva, E.
Benchmarking Quasiclassical Mapping Hamiltonian Methods for Simulating
Electronically Nonadiabatic Molecular Dynamics. J. Chem. Theory
Comput. 2020, 16, 2883–2895.
Ceriotti et al. 2014
Ceriotti, M.; More, J.; Manolopoulos, D. E. i-PI: A Python Interface for
Ab Initio Path Integral Molecular Dynamics Simulations.
Comput. Phys. Commun. 2014, 185, 1019 – 1026.
Kapil et al. 2019
Kapil, V. et al. i-PI 2.0: A Universal Force Engine for Advanced
Molecular Simulations. Comput. Phys. Commun. 2019,
236, 214–223.
Thoss et al. 2000
Thoss, M.; Miller, W. H.; Stock, G. Semiclassical Description of Nonadiabatic
Quantum Dynamics: Application to the S${}_{1}$âS${}_{2}$ Conical Intersection in
Pyrazine. J. Chem. Phys. 2000, 112, 10282–10292.
Lee et al. 2016
Lee, M. K.; Huo, P.; Coker, D. F. Semiclassical Path Integral Dynamics:
Photosynthetic Energy Transfer with Realistic Environment Interactions.
Annu. Rev. Phys. Chem. 2016, 67, 639–668.
Stock and Thoss 1997
Stock, G.; Thoss, M. Semiclassical Description of Nonadiabatic Quantum
Dynamics. Phys. Rev. Lett. 1997, 78, 578–581.
Westermayr and Marquetand 2020
Westermayr, J.; Marquetand, P. Machine Learning and Excited-State Molecular
Dynamics. Mach. Learn.: Sci. Technol. 2020, in press,
doi:10.1088/2632–2153/ab9c3e.
Weinreich et al. 2020
Weinreich, J.; Römer, A.; Paleico, M. L.; Behler, J. Properties of
$\alpha$-Brass Nanoparticles. 1. Neural Network Potential Energy Surface.
J. Phys. Chem. C 2020, in press.
Lin et al. 2020
Lin, Q.; Zhang, Y.; Zhao, B.; Jiang, B. Automatically Growing Global Reactive
Neural Network Potential Energy Surfaces: A Trajectory-Free Active Learning
Strategy. J. Chem. Phys. 2020, 152, 154104.
Tully 1990
Tully, J. C. Molecular Dynamics with Electronic Transitions. J. Chem.
Phys. 1990, 93, 1061–1071.
Tully 1991
Tully, J. C. Nonadiabatic Molecular Dynamics. Int. J. Quantum Chem.
1991, 40, 299–309.
C. Tully 1998
C. Tully, J. Mixed QuantumâClassical Dynamics. Faraday Discuss.
1998, 110, 407–419.
S. Mai 2020
S. Mai, L. G., P. Marquetand In Quantum Chemistry and Dynamics of Excited
States: Methods and Applications; González, L., Lindh, R., Eds.; Wiley,
2020; in press.
Zener 1932
Zener, C. Non-Adiabatic Crossing of Energy Levels. Proc. Roy. Soc. Lond.
A 1932, 137, 696–701.
Wittig 2005
Wittig, C. The Landau-Zener Formula. J. Phys. Chem. B 2005,
109, 8428–8430.
Zhu et al. 2001
Zhu, C.; Kamisaka, H.; Nakamura, H. Significant Improvement of the Trajectory
Surface Hopping Method by the ZhuâNakamura Theory. J. Chem. Phys
2001, 115, 11036–11039.
Zhu et al. 2002
Zhu, C.; Kamisaka, H.; Nakamura, H. New Implementation of the Trajectory
Surface Hopping Method with Use of the ZhuâNakamura Theory. II. Application
to the Charge Transfer Processes in the 3D DH${}_{2}$+ System. J. Chem.
Phys. 2002, 116, 3234–3247.
Oloyede et al. 2006
Oloyede, P.; Milânikov, G.; Nakamura, H. Generalized Trajectory Surface
Hopping Method Based on the Zhu-Nakamura Theory. J. Chem. Phys.
2006, 124, 144110.
Ishida et al. 2017
Ishida, T.; Nanbu, S.; Nakamura, H. Clarification of Nonadiabatic Chemical
Dynamics by the Zhu-Nakamura Theory of Nonadiabatic Transition: From
Tri-Atomic Systems to Reactions in Solutions. Int. Rev. Phys. Chem.
2017, 36, 229–286.
Zhu et al. 1995
Zhu, L.; Kleiman, V.; Li, X.; Lu, S. P.; Trentelman, K.; Gordon, R. J.
Ultrafast coherent control and Destruction of Excitons in Quantum Wells.
Phys. Rev. Lett. 1995, 75, 2598–2601.
Granucci and Persico 2007
Granucci, G.; Persico, M. Critical Appraisal of the Fewest Switching Algorithm
for Surface Hopping. J. Chem. Phys. 2007, 126,
134114.
Fabiano et al. 2008
Fabiano, E.; Keal, T.; Thiel, W. Implementation of Surface Hopping Molecular
Dynamics using Semiempirical Methods. Chem. Phys. 2008,
349, 334 – 347.
Malhado et al. 2014
Malhado, J. P.; Bearpark, M. J.; Hynes, J. T. Non-Adiabatic Dynamics Close to
Conical Intersections and the Surface Hopping Perspective. Front.
Chem. 2014, 2, 97.
Wang et al. 2016
Wang, L.; Akimov, A.; Prezhdo, O. V. Recent Progress in Surface Hopping:
2011-2015. J. Phys. Chem. Lett. 2016, 7,
2100–2112.
Subotnik and Rhee 2015
Subotnik, J. E.; Rhee, Y. M. On Surface Hopping and Time-Reversal. J.
Phys. Chem. A 2015, 119, 990–995.
HammesâSchiffer and Tully 1994
HammesâSchiffer, S.; Tully, J. C. Proton Transfer in Solution: Molecular
Dynamics with Quantum Transitions. J. Chem. Phys. 1994,
101, 4657–4667.
Sawada et al. 1985
Sawada, S.-I.; Nitzan, A.; Metiu, H. Mean-Trajectory Approximation for Charge-
and Energy-Transfer Processes at Surfaces. Phys. Rev. B 1985,
32, 851–867.
Li et al. 2005
Li, X.; Tully, J. C.; Schlegel, H. B.; Frisch, M. J. Ab Initio
Ehrenfest Dynamics. J. Chem. Phys. 2005, 123,
084106.
Mai et al. 2016
Mai, S.; Marquetand, P.; González, L. Intersystem Crossing Pathways in the
Noncanonical Nucleobase 2-Thiouracil: A Time-Dependent Picture. J.
Phys. Chem. Lett. 2016, 7, 1978–1983.
Mai et al. 2017
Mai, S.; Richter, M.; Marquetand, P.; González, L. The DNA Nucleobase
Thymine in Motion – Intersystem Crossing Simulated with Surface Hopping.
Chem. Phys. 2017, 482, 9 – 15.
Raghunathan Ramakrishnan and von
Lilienfeld 2014
Raghunathan Ramakrishnan, M. R., Pavlo O. Dral; von Lilienfeld, O. A. Quantum
Chemistry Structures and Properties of 134 Kilo Molecules. Sci. Data
2014, 1.
Artrith et al. 2011
Artrith, N.; Morawietz, T.; Behler, J. High-Dimensional Neural-Network
Potentials for Multicomponent Systems: Applications to Zinc Oxide.
Phys. Rev. B 2011, 83, 153101.
Huang and von Lilienfeld 2016
Huang, B.; von Lilienfeld, O. A. Communication: Understanding Molecular
Representations in Machine Learning: The Role of Uniqueness and Target
Similarity. J. Chem. Phys. 2016, 145, 161102.
Yao et al. 2018
Yao, K.; Herr, J. E.; Toth, D. W.; Mckintyre, R.; Parkhill, J. The
TensorMol-0.1 Model Chemistry: A Neural Network Augmented with Long-Range
Physics. Chem. Sci. 2018, 9, 2261–2269.
Schütt et al. 2018
Schütt, K. T.; Sauceda, H. E.; Kindermans, P.-J.; Tkatchenko, A.;
Müller, K.-R. SchNet – A Deep Learning Architecture for Molecules and
Materials. J. Chem. Phys. 2018, 148, 241722.
Nebgen et al. 2018
Nebgen, B.; Lubbers, N.; Smith, J. S.; Sifain, A. E.; Lokhov, A.; Isayev, O.;
Roitberg, A. E.; Barros, K.; Tretiak, S. Transferable Dynamic Molecular
Charge Assignment Using Deep Neural Networks. J. Chem. Theory Comput.
2018, 14, 4687–4698.
Sifain et al. 2018
Sifain, A. E.; Lubbers, N.; Nebgen, B. T.; Smith, J. S.; Lokhov, A. Y.;
Isayev, O.; Roitberg, A. E.; Barros, K.; Tretiak, S. Discovering a
Transferable Charge Assignment Model Using Machine Learning. J. Phys.
Chem. Lett. 2018, 9, 4495–4501.
Schütt et al. 2019
Schütt, K. T.; Gastegger, M.; Tkatchenko, A.; Müller, K.-R.
Explainable AI: Interpreting, Explaining and Visualizing Deep
Learning; Springer International Publishing, 2019; pp 311–330.
Schütt et al. 2019
Schütt, K. T.; Kessel, P.; Gastegger, M.; Nicoli, K. A.; Tkatchenko, A.;
Müller, K.-R. SchNetPack: A Deep Learning Toolbox For Atomistic Systems.
J. Chem. Theory Comput. 2019, 15, 448–455.
Christensen et al. 2019
Christensen, A. S.; Faber, F. A.; von Lilienfeld, O. A. Operators in Quantum
Machine Learning: Response Properties in Chemical Space. J. Chem.
Phys. 2019, 150, 064105.
Veit et al. 2020
Veit, M.; Wilkins, D. M.; Yang, Y.; Jr., R. A. D.; Ceriotti, M. Predicting
Molecular Dipole Moments by Combining Atomic Partial Charges and Atomic
Dipoles. arXiv 2020, 2003.12437.
Gastegger and Marquetand 2018
Gastegger, M.; Marquetand, P. Molecular dynamics with neural-network
potentials. arXiv:1812.07676 [physics.chem-ph] 2018,
Thomas et al. 2013
Thomas, M.; Brehm, M.; Fligg, R.; Vöhringer, P.; Kirchner, B. Computing
Vibrational Spectra from Ab Initio Molecular Dynamics. Phys.
Chem. Chem. Phys. 2013, 15, 6608–6622.
Wilke et al. 2016
Wilke, J.; Wilke, M.; Meerts, W. L.; Schmitt, M. Determination of Ground and
Excited State Dipole Moments via Electronic Stark Spectroscopy:
5-Methoxyindole. J. Chem. Phys. 2016, 144,
044201.
Tennyson 2016
Tennyson, J. Perspective: Accurate Ro-Vibrational Calculations on Small
Molecules. J. Chem. Phys. 2016, 145, 120901.
Marquetand et al. 2016
Marquetand, P.; Nogueira, J.; Mai, S.; Plasser, F.; González, L. Challenges
in Simulating Light-Induced Processes in DNA. Molecules 2016,
22, 49.
Nogueira and González 2018
Nogueira, J. J.; González, L. Computational Photophysics in the Presence
of an Environment. Annu. Rev. Phys. Chem. 2018, 69,
473–497.
Barbatti et al. 2007
Barbatti, M.; Granucci, G.; Persico, M.; Ruckenbauer, M.; Vazdar, M.;
Eckert-Maksić, M.; Lischka, H. The on-the-Fly Surface-Hopping Program
System Newton-X: Application to Ab Initio Simulation of the
Nonadiabatic Photodynamics of Benchmark Systems. J. Photochem.
Photobiol. A 2007, 190, 228–240.
Dral 2020
Dral, P. O. Quantum Chemistry in the Age of Machine Learning. J. Phys.
Chem. Lett. 2020, 11, 2336–2347.
Chen et al. 2019
Chen, W.-K.; Fang, W.-H.; Cui, G. Integrating Machine Learning with the
Multilayer Energy-Based Fragment Method for Excited States of Large Systems.
J. Phys. Chem. Lett. 2019, 10, 7836–7841.
Chen et al. 2020
Chen, W.-K.; Zhang, Y.; Jiang, B.; Fang, W.-H.; Cui, G. Efficient Construction
of Excited-State Hessian Matrices with Machine Learning Accelerated
Multilayer Energy-Based Fragment Method. The Journal of Physical
Chemistry A 2020, in press,
DOI:10.1021/acs.jpca.0c04117.
Behler et al. 2008
Behler, J.; Martoák, R.; Donadio, D.; Parrinello, M. Metadynamics
Simulations of the High-Pressure Phases of Silicon Employing a
High-Dimensional Neural Network Potential. Phys. Rev. Lett.
2008, 100, 185501.
Gastegger et al. 2018
Gastegger, M.; Schwiedrzik, L.; Bittermann, M.; Berzsenyi, F.; Marquetand, P.
wACSF – Weighted Atom-Centered Symmetry Functions as Descriptors in Machine
Learning Potentials. J. Chem. Phys. 2018, 148,
241709.
Chen et al. 2020
Chen, M. S.; Zuehlsdorff, T. J.; Morawietz, T.; Isborn, C. M.; Markland, T. E.
Exploiting Machine Learning to Efficiently Predict Multidimensional Optical
Spectra in Complex Environments. arXiv 2020,
2005.09776.
Koch and Zhang 2014
Koch, W.; Zhang, D. H. Communication: Separable Potential Energy Surfaces from
Multiplicative Artificial Neural Networks. J. Chem. Phys.
2014, 141, 021101.
He et al. 2016
He, D.; Yuan, J.; Li, H.; Chen, M. Global Diabatic Potential Energy Surfaces
and Quantum Dynamical Studies for the Li(2p) + H${}_{2}(X^{1}\Sigma^{+}_{g})\rightarrow$ LiH($X^{1}\Sigma^{+})$ + H Reaction. Sci. Rep.
2016, 6.
Guan et al. 2017
Guan, Y.; Fu, B.; Zhang, D. H. Construction of Diabatic Energy Surfaces for
LiFH with Artificial Neural Networks. J. Chem. Phys. 2017,
147, 224307.
Wang et al. 2018
Wang, S.; Yang, Z.; Yuan, J.; Chen, M. New Diabatic Potential Energy Surfaces
of the NaH${}_{2}$ System and Dynamics Studies for the Na(3p) + H${}_{2}$
$\rightarrow$ NaH + H Reaction. Sci. Rep. 2018,
8.
Yuan et al. 2018
Yuan, J.; He, D.; Wang, S.; Chen, M.; Han, K. Diabatic Potential Energy
Surfaces of MgH${}_{2}^{+}$ and Dynamic Studies for the Mg${}^{+}$(3p) +H${}_{2}$
$\rightarrow$ MgH${}^{+}$ + H Reaction. Phys. Chem. Chem. Phys.
2018, 20, 6638–6647.
Yin et al. 2019
Yin, Z.; Guan, Y.; Fu, B.; Zhang, D. H. Two-State Diabatic Potential Energy
Surfaces of ClH${}_{2}$ Based on Nonadiabatic Couplings with Neural Networks.
Phys. Chem. Chem. Phys. 2019, 21, 20372–20383.
Akimov 2018
Akimov, A. V. A Simple Phase correction Makes a Big Difference in Nonadiabatic
Molecular Dynamics. J. Phys. Chem. Lett. 2018, 9,
6096–6102.
Shu et al. 2019
Shu, Y.; Kryven, J.; Sampaio de Oliveira-Filho, A. G.; Zhang, L.; Song, G.-L.;
Li, S. L.; Meana-Pa$\tilde{n}$eda, R.; Fu, B.; Bowman, J. M.; Truhlar, D. G.
Direct Diabatization and Analytic Representation of Coupled Potential Energy
Surfaces and Couplings for the Reactive Quenching of the Excited ${}^{2}\Sigma^{+}$
State of OH by Molecular Hydrogen. J. Chem. Phys. 2019,
151, 104311.
Yarkony 1996
Yarkony, D. R. On the Consequences of Nonremovable Derivative Couplings. I.
The Geometric Phase and Quasidiabatic States: A Numerical Study. J.
Chem. Phys. 1996, 105, 10456–10461.
Yarkony 1996
Yarkony, D. R. On the Role of Conical Intersections in Photodissociation. V.
Conical Intersections and the Geometric Phase in the Photodissociation of
Methyl Mercaptan. J. Chem. Phys. 1996, 104,
7866–7881.
Ryabinkin et al. 2014
Ryabinkin, I. G.; Joubert-Doriol, L.; Izmaylov, A. F. When Do We Need to
Account for the Geometric Phase in Excited State Dynamics? J. Chem.
Phys. 2014, 140, 214116.
Gherib et al. 2015
Gherib, R.; Ryabinkin, I. G.; Izmaylov, A. F. Why Do Mixed Quantum-Classical
Methods Describe Short-Time Dynamics through Conical Intersections So Well?
Analysis of Geometric Phase Effects. J. Chem. Theory Comput.
2015, 11, 1375–1382.
Ryabinkin et al. 2017
Ryabinkin, I. G.; Joubert-Doriol, L.; Izmaylov, A. F. Geometric Phase Effects
in Nonadiabatic Dynamics Near Conical Intersections. Acc. Chem. Res.
2017, 50, 1785–1793.
Plasser et al. 2016
Plasser, F.; Ruckenbauer, M.; Mai, S.; Oppel, M.; Marquetand, P.;
González, L. Efficient and Flexible Computation of Many-Electron Wave
Function Overlaps. J. Chem. Theory Comput. 2016, 12,
1207.
Zhu and Yarkony 2010
Zhu, X.; Yarkony, D. R. Toward Eliminating the Electronic Structure Bottleneck
in Nonadiabatic Dynamics on the Fly: An Algorithm to Fit Nonlocal,
Quasidiabatic, Coupled Electronic State Hamiltonians Based on Ab
Initio Electronic Structure Data. J. Chem. Phys. 2010,
132, 104101.
Zhu and Yarkony 2012
Zhu, X.; Yarkony, D. R. Quasi-Diabatic Representations of Adiabatic Potential
Energy Surfaces Coupled by Conical Intersections including Bond Breaking: A
More General Construction Procedure and an Analysis of the Diabatic
Representation. J. Chem. Phys. 2012, 137, 22A511.
Zhu and Yarkony 2012
Zhu, X.; Yarkony, D. R. On the Representation of Coupled Adiabatic Potential
Energy Surfaces using Quasi-Diabatic Hamiltonians: A Distributed Origins
Expansion Approach. J. Chem. Phys. 2012, 136,
174110.
Rasmussen 2004
Rasmussen, C. E. In Advanced Lectures on Machine Learning: ML Summer
Schools 2003, Canberra, Australia, February 2 - 14, 2003, Tübingen,
Germany, August 4 - 16, 2003, Revised Lectures; Bousquet, O., von
Luxburg, U., Rätsch, G., Eds.; Springer Berlin Heidelberg: Berlin,
Heidelberg, 2004; pp 63–71.
Mai et al. 2018
Mai, S.; Richter, M.; Ruckenbauer, M.; Oppel, M.; Marquetand, P.;
González, L. SHARC2.0: Surface Hopping Including ARbitrary Couplings –
Program Package for Non-Adiabatic Dynamics. sharc-md.org, 2018.
Beard et al. 2019
Beard, E. J.; Sivaraman, G.; Vázquez-Mayagoitia, A.; Vishwanath, V.;
Cole, J. M. Comparative Dataset of Experimental and Computational Attributes
of UV/Vis Absorption Spectra. Sci. Data 2019, 6.
Barbatti et al. 2005
Barbatti, M.; Ruckenbauer, M.; Lischka, H. The Photodynamics of Ethylene: A
Surface-Hopping Study on Structural Aspects. J. Chem. Phys.
2005, 122, 174307.
Tapavicza et al. 2007
Tapavicza, E.; Tavernelli, I.; Rothlisberger, U. Trajectory Surface Hopping
within Linear Response Time-Dependent Density-Functional Theory. Phys.
Rev. Lett. 2007, 98, 023001.
Tavernelli et al. 2009
Tavernelli, I.; Tapavicza, E.; Rothlisberger, U. Nonadiabatic Coupling Vectors
within Linear Response Time-Dependent Density Functional Theory. J.
Chem. Phys. 2009, 130, 124107.
Tavernelli et al. 2009
Tavernelli, I.; Tapavicza, E.; Rothlisberger, U. Non-Adiabatic Dynamics using
Time-Dependent Density Functional Theory: Assessing the Coupling Strengths.
J. Mol. Struct.: THEOCHEM 2009, 914, 22 – 29.
Barbatti et al. 2006
Barbatti, M.; Aquino, A. J. A.; Lischka, H. Ultrafast Two-Step Process in the
Non-Adiabatic Relaxation of the CH${}_{2}$NH${}_{2}$ Molecule. Mol.
Phys. 2006, 104, 1053–1060.
Tao et al. 2011
Tao, H.; Allison, T. K.; Wright, T. W.; Stooke, A. M.; Khurmi, C.; van
Tilborg, J.; Liu, Y.; Falcone, R. W.; Belkacem, A.; Martinez, T. J. Ultrafast
internal conversion in ethylene. I. The excited state lifetime. J.
Chem. Phys. 2011, 134, 244306.
Allison et al. 2012
Allison, T. K.; Tao, H.; Glover, W. J.; Wright, T. W.; Stooke, A. M.;
Khurmi, C.; van Tilborg, J.; Liu, Y.; Falcone, R. W.; Martínez, T. J.;
Belkacem, A. Ultrafast internal conversion in ethylene. II. Mechanisms and
pathways for quenching and hydrogen elimination. J. Chem. Phys.
2012, 136, 124317.
Mori et al. 2012
Mori, T.; Glover, W. J.; Schuurman, M. S.; Martinez, T. J. Role of Rydberg
States in the Photochemical Dynamics of Ethylene. J. Phys. Chem. A
2012, 116, 2808–2818.
Sellner et al. 2013
Sellner, B.; Barbatti, M.; Müller, T.; Domcke, W.; Lischka, H. Ultrafast
Non-Adiabatic Dynamics of Ethylene including Rydberg States. Mol.
Phys. 2013, 111, 2439–2450.
Barbatti et al. 2012
Barbatti, M.; Lan, Z.; Crespo-Otero, R.; Szymczak, J. J.; Lischka, H.;
Thiel, W. Critical Appraisal of Excited State Nonadiabatic Dynamics
Simulations of 9H-Adenine. J. Chem. Phys. 2012, 117,
22A503.
Hollas et al. 2018
Hollas, D.; Šištík, L.; Hohenstein, E. G.; Martínez, T. J.;
Slavíček, P. Nonadiabatic Ab Initio Molecular Dynamics with
the Floating Occupation Molecular Orbital-Complete Active Space Configuration
Interaction Method. J. Chem. Theory Comput. 2018, 14,
339–350.
Botu and Ramprasad 2015
Botu, V.; Ramprasad, R. Adaptive Machine Learning Framework to Accelerate
Ab Initio Molecular Dynamics. Int. J. Quant. Chem.
2015, 115, 1074–1083.
Behler 2015
Behler, J. Constructing High-Dimensional Neural Network Potentials: A Tutorial
Review. Int. J. Quantum Chem. 2015, 115,
1032–1050.
Ceriotti et al. 2013
Ceriotti, M.; Tribello, G. A.; Parrinello, M. Demonstrating the Transferability
and the Descriptive Power of Sketch-Map. J. Chem. Theory Comput.
2013, 9, 1521–1532.
Dral et al. 2017
Dral, P. O.; Owens, A.; Yurchenko, S. N.; Thiel, W. Structure-Based Sampling
and Self-Correcting Machine Learning for Accurate Calculations of Potential
Energy Surfaces and Vibrational Levels. J. Chem. Phys. 2017,
146, 244108.
Sobol’ et al. 2011
Sobol’, I. M.; Asotsky, D.; Kreinin, A.; Kucherenko, S. Construction and
Comparison of High-Dimensional Sobol’ Generators. Wilmott
2011, 2011, 64–79.
Uteva et al. 2018
Uteva, E.; Graham, R. S.; Wilkinson, R. D.; Wheatley, R. J. Active Learning in
Gaussian Process Interpolation of Potential Energy Surfaces. J. Chem.
Phys. 2018, 149, 174114.
Hansen et al. 2013
Hansen, K.; Montavon, G.; Biegler, F.; Fazli, S.; Rupp, M.; Scheffler, M.; von
Lilienfeld, O. A.; Tkatchenko, A.; Müller, K.-R. Assessment and
Validation of Machine Learning Methods for Predicting Molecular Atomization
Energies. J. Chem. Theory Comput. 2013, 9,
3404–3419.
Chmiela et al. 2017
Chmiela, S.; Tkatchenko, A.; Sauceda, H. E.; Poltavsky, I.; Schütt, K. T.;
Müller, K.-R. Machine Learning of Accurate Energy-Conserving Molecular
Force Fields. Sci. Adv. 2017, 3.
Christensen et al. 2020
Christensen, A. S.; Bratholm, L. A.; Faber, F. A.; Anatole von Lilienfeld, O.
FCHL Revisited: Faster and More Accurate Quantum Machine Learning. J.
Chem. Phys. 2020, 152, 044107.
Kim et al. 2019
Kim, H.; Park, J.; Choi, S. Energy Refinement and Analysis of Structures in the
QM9 Database via a Highly Accurate Quantum Chemical Method. Sci. Data
2019, 6.
Glavatskikh et al. 2019
Glavatskikh, M.; Leguy, J.; Hunault, G.; Cauchi, T.; Da Mota, B. Dataset’s
Chemical Diversity Limits the Generalizability of Machine Learning
Predictions. J. Cheminform. 2019, 11.
490
https://www.kaggle.com/c/champs-scalar-coupling/, 2020-05-01.
von Lilienfeld 2018
von Lilienfeld, O. A. The QM9 challenge.
https://twitter.com/ProfvLilienfeld/status/1073179005854121984, 2018.
Fink et al. 2005
Fink, T.; Bruggesser, H.; Reymond, J.-L. Virtual Exploration of the
Small-Molecule Chemical Universe Below 160 Daltons. Angew. Chem., Int.
Ed. 2005, 44, 1504–1508.
Fink and Reymond 2007
Fink, T.; Reymond, J.-L. Virtual Exploration of the Chemical Universe up to 11
Atoms of C, N, O, F: Assembly of 26.4 Million Structures (110.9 Million
Stereoisomers) and Analysis for New Ring Systems, Stereochemistry,
Physicochemical Properties, Compound Classes, and Drug Discovery. J.
Chem. Inf. Model. 2007, 47, 342–353.
Blum and Reymond 2009
Blum, L. C.; Reymond, J.-L. 970 Million Druglike Small Molecules for Virtual
Screening in the Chemical Universe Database GDB-13. J. Am. Chem.
Soc. 2009, 131, 8732.
Montavon et al. 2013
Montavon, G.; Rupp, M.; Gobre, V.; Vazquez-Mayagoitia, A.; Hansen, K.;
Tkatchenko, A.; Müller, K.-R.; von Lilienfeld, O. A. Machine Learning of
Molecular Electronic Properties in Chemical Compound Space. New J.
Phys. 2013, 15, 095003.
Ruddigkeit et al. 2012
Ruddigkeit, L.; van Deursen, R.; Blum, L. C.; Reymond, J.-L. Enumeration of 166
Billion Organic Small Molecules in the Chemical Universe Database GDB-17.
J. Chem. Inf. Model. 2012, 52, 2864–2875.
Ramakrishnan et al. 2015
Ramakrishnan, R.; Dral, P. O.; Rupp, M.; von Lilienfeld, O. A. Big Data Meets
Quantum Chemistry Approximations: The $\Delta$-Machine Learning Approach.
J. Chem. Theory Comput. 2015, 11, 2087–2096.
Lee et al. 1988
Lee, C.; Yang, W.; Parr, R. G. Development of the Colle-Salvetti
Correlation-Energy Formula into a Functional of the Electron Density.
Phys. Rev. B 1988, 37, 785–789.
Becke 1988
Becke, A. D. Density-Functional Exchange-Energy Approximation with Correct
Asymptotic Behavior. Phys. Rev. A 1988, 38,
3098–3100.
Stuke et al. 2020
Stuke, A.; Kunkel, C.; Golze, D.; Todorović, M.; Margraf, J. T.;
Reuter, K.; Rinke, P.; Oberhofer, H. Atomic Structures and Orbital Energies
of 61,489 Crystal-Forming Organic Molecules. Sci. Data 2020,
7.
Perdew et al. 1996
Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized Gradient Approximation Made
Simple. Phys. Rev. Lett. 1996, 77, 3865–3868.
Nakata and Shimazaki 2017
Nakata, M.; Shimazaki, T. PubChemQC Project: A Large-Scale First-Principles
Electronic Structure Database for Data-Driven Chemistry. J. Chem. Inf.
Model. 2017, 57, 1300–1308.
Kolb et al. 2016
Kolb, B.; Zhao, B.; Li, J.; Jiang, B.; Guo, H. Permutation Invariant Potential
Energy Surfaces for Polyatomic Reactions using Atomistic Neural Networks.
J. Chem. Phys. 2016, 144, 224103.
Bernstein et al. 2019
Bernstein, N.; Csányi, G.; Deringer, V. L. De Novo Exploration and
Self-Guided Learning of Potential-Energy Surfaces. npj Comput. Mater.
2019, 5.
Yao et al. 2020
Yao, Z.; Sanchez-Lengeling, B.; Bobbitt, N. S.; Bucior, B. J.; Kumar, S. G. H.;
Collins, S. P.; Burns, T.; Woo, T. K.; Farha, O.; Snurr, R. Q.;
Aspuru-Guzik, A. Inverse Design of Nanoporous Crystalline Reticular
Materials with Deep Generative Models. ChemRxiv 2020,
DOI:10.26434/chemrxiv.12186681.v1.
Krenn et al. 2019
Krenn, M.; Häse, F.; Nigam, A.; Friederich, P.; Aspuru-Guzik, A.
SELFIES: A Robust Representation of Semantically Constrained Graphs with an
Example Application in Chemistry. arXiv 2019,
abs/1905.13741.
Gebauer et al. 2019
Gebauer, N. W. A.; Gastegger, M.; Schütt, K. T. Symmetry-Adapted Generation
of 3D Point Sets for the Targeted Discovery of Molecules. arXiv
2019, 1906.00957.
Schütt et al. 2017
Schütt, K. T.; Arbabzadah, F.; Chmiela, S.; Müller, K. R.;
Tkatchenko, A. Quantum-Chemical Insights from Deep Tensor Neural Networks.
Nat. Commun. 2017, 8, 13890 EP –.
Ye et al. 2019
Ye, S.; Hu, W.; Li, X.; Zhang, J.; Zhong, K.; Zhang, G.; Luo, Y.; Mukamel, S.;
Jiang, J. A Neural Network Protocol for Electronic Excitations of
N-Methylacetamide. Proc. Natl. Acad. Sci. 2019, 116,
11612–11617.
Jorgensen et al. 1996
Jorgensen, W. L.; Maxwell, D. S.; Tirado-Rives, J. Development and Testing of
the OPLS All-Atom Force Field on Conformational Energetics and Properties of
Organic Liquids. J. Am. Chem. Soc. 1996, 118,
11225–11236.
Van Der Spoel et al. 2005
Van Der Spoel, D.; Lindahl, E.; Hess, B.; Groenhof, G.; Mark, A. E.;
Berendsen, H. J. C. GROMACS: Fast, Flexible, and Free. J. Comp. Chem.
2005, 26, 1701–1718.
Ceriotti et al. 2011
Ceriotti, M.; Tribello, G. A.; Parrinello, M. Simplifying the Representation of
Complex Free-Energy Landscapes using Sketch-Map. Proc. Natl. Acad. Sci
2011, 108, 13023–13028.
Tribello et al. 2012
Tribello, G. A.; Ceriotti, M.; Parrinello, M. Using Sketch-Map Coordinates to
Analyze and Bias Molecular Dynamics Simulations. Proc. Natl. Acad. Sci
2012, 109, 5196–5201.
Seung et al. 1992
Seung, H. S.; Opper, M.; Sompolinsky, H. Query by Committee. Proceedings of the
Fifth Annual Workshop on Computational Learning Theory. New York, NY, USA,
1992; p 287â294.
Collins 2002
Collins, M. Molecular Potential-Energy Surfaces for Chemical Reaction Dynamics.
Theor. Chem. Acc. 2002, 108, 313â324.
Godsi et al. 2010
Godsi, O.; Collins, M. A.; Peskin, U. Quantum GrowâA Quantum Dynamics
Sampling Approach for Growing Potential Energy Surfaces and Nonadiabatic
Couplings. J. Chem. Phys. 2010, 132, 124106.
Dawes et al. 2007
Dawes, R.; Thompson, D. L.; Guo, Y.; Wagner, A. F.; Minkoff, M. Interpolating
Moving Least-Squares Methods for Fitting Potential Energy Surfaces: Computing
High-Density Potential Energy Surface Data from Low-Density Ab Initio
Data Points. J. Chem. Phys. 2007, 126, 184108.
Dawes et al. 2008
Dawes, R.; Thompson, D. L.; Wagner, A. F.; Minkoff, M. Interpolating Moving
Least-Squares Methods for Fitting Potential Energy Surfaces: A Strategy for
Efficient Automatic Data Point Placement in High Dimensions. J. Chem.
Phys. 2008, 128, 084107.
Braams and Bowman 2009
Braams, B. J.; Bowman, J. M. Permutationally Invariant Potential Energy
Surfaces in High Dimensionality. Int. Rev. Phys. Chem. 2009,
28, 577–606.
Qu et al. 2018
Qu, C.; Yu, Q.; Bowman, J. M. Permutationally Invariant Potential Energy
Surfaces. Annu. Rev. Phys. Chem. 2018, 69,
151–175.
Lorenz et al. 2004
Lorenz, S.; GroÃ, A.; Scheffler, M. Representing High-Dimensional
Potential-Energy Surfaces for Reactions at Surfaces by Neural Networks.
Chem. Phys. Lett. 2004, 395, 210 – 215.
Raff et al. 2005
Raff, L. M.; Malshe, M.; Hagan, M.; Doughan, D. I.; Rockley, M. G.;
Komanduri, R. Ab Initio Potential-Energy Surfaces for Complex,
Multichannel Systems using Modified Novelty Sampling and Feedforward Neural
Networks. J. Chem. Phys. 2005, 122, 084104.
Behler and Parrinello 2007
Behler, J.; Parrinello, M. Generalized Neural-Network Representation of
High-Dimensional Potential-Energy Surfaces. Phys. Rev. Lett.
2007, 98, 146401.
Chen et al. 2013
Chen, J.; Xu, X.; Xu, X.; Zhang, D. H. A Global Potential Energy Surface for
the H${}_{2}$ + OH $\leftrightarrow$ H${}_{2}$O + H Reaction using Neural
Networks. J. Chem. Phys. 2013, 138, 154301.
Jiang and Guo 2015
Jiang, B.; Guo, H. Dynamics of Water Dissociative Chemisorption on Ni(111):
Effects of Impact Sites and Incident Angles. Phys. Rev. Lett.
2015, 114, 166101.
Shen et al. 2015
Shen, X.; Chen, J.; Zhang, Z.; Shao, K.; Zhang, D. H. Methane Dissociation on
Ni(111): A Fifteen-Dimensional Potential Energy Surface using Neural Network
Method. J. Chem. Phys. 2015, 143, 144701.
Shao et al. 2016
Shao, K.; Chen, J.; Zhao, Z.; Zhang, D. H. Communication: Fitting Potential
Energy Surfaces with Fundamental Invariant Neural Network. J. Chem.
Phys. 2016, 145, 071101.
Cui and Krems 2016
Cui, J.; Krems, R. V. Efficient Non-Parametric Fitting of Potential Energy
Surfaces for Polyatomic Molecules with Gaussian Processes. J. Phys. B:
At., Mol. Opt. Phys. 2016, 49, 224001.
Kolb et al. 2017
Kolb, B.; Marshall, P.; Zhao, B.; Jiang, B.; Guo, H. Representing Global
Reactive Potential Energy Surfaces Using Gaussian Processes. J. Phys.
Chem. A 2017, 121, 2552–2557.
Kolb et al. 2017
Kolb, B.; Luo, X.; Zhou, X.; Jiang, B.; Guo, H. High-Dimensional Atomistic
Neural Network Potentials for MoleculeâSurface Interactions: HCl Scattering
from Au(111). J. Phys. Chem. Lett. 2017, 8,
666–672.
Huang et al. 2017
Huang, S.-D.; Shang, C.; Zhang, X.-J.; Liu, Z.-P. Material Discovery by
Combining Stochastic Surface Walking Global Optimization with a Neural
Network. Chem. Sci. 2017, 8, 6327–6337.
Zhou et al. 2017
Zhou, X.; Nattino, F.; Zhang, Y.; Chen, J.; Kroes, G.-J.; Guo, H.; Jiang, B.
Dissociative Chemisorption of Methane on Ni(111) using a Chemically Accurate
Fifteen Dimensional Potential Energy Surface. Phys. Chem. Chem. Phys.
2017, 19, 30540–30550.
Zhang et al. 2019
Zhang, Y.; Hu, C.; Jiang, B. Embedded Atom Neural Network Potentials: Efficient
and Accurate Machine Learning with a Physically Inspired Representation.
J. Phys. Chem. Lett. 2019, 10, 4962–4967.
Wigner 1932
Wigner, E. On The Quantum Correction for Thermodynamic Equilibrium. Phys.
Rev. 1932, 40, 749–750.
Bruccoleri and Karplus 1990
Bruccoleri, R. E.; Karplus, M. Conformational Sampling using High-Temperature
Molecular Dynamics. Biopolymers 1990, 29,
1847–1862.
Maximova et al. 2016
Maximova, T.; Moffatt, R.; Ma, B.; Nussinov, R.; Shehu, A. Principles and
Overview of Sampling Methods for Modeling Macromolecular Structure and
Dynamics. PLOS computational Biology 2016, 12,
1–70.
Kästner 2011
Kästner, J. Umbrella Sampling. Wiley Interdiscip. Rev. Comput. Mol.
Sci. 2011, 1, 932–942.
Tao 2019
Tao, G. Trajectory-Guided Sampling for Molecular Dynamics Simulation.
Theor. Chem. Acc. 2019, 138, 34.
Yang et al. 2019
Yang, Y. I.; Shao, Q.; Zhang, J.; Yang, L.; Gao, Y. Q. Enhanced Sampling in
Molecular Dynamics. J. Chem. Phys. 2019, 151,
070902.
Herr et al. 2018
Herr, J. E.; Yao, K.; McIntyre, R.; Toth, D. W.; Parkhill, J. Metadynamics for
Training Neural Network Model Chemistries: A Competitive Assessment. J.
Chem. Phys. 2018, 148, 241710.
Grimme 2019
Grimme, S. Exploration of Chemical compound, conformer, and Reaction Space with
Meta-Dynamics Simulations Based on Tight-Binding Quantum Chemical
Calculations. J. Chem. Theory Comput. 2019, 15,
2847–2862.
Smith et al. 2018
Smith, J. S.; Nebgen, B.; Lubbers, N.; Isayev, O.; Roitberg, A. E. Less is
More: Sampling Chemical Space with Active Learning. J. Chem. Phys.
2018, 148, 241733.
Malbon et al. 2020
Malbon, C. L.; Zhao, B.; Guo, H.; Yarkony, D. R. On the Nonadiabatic
Collisional Quenching of OH(A) by H${}_{2}$: A Four Coupled Quasi-Diabatic State
Description. Phys. Chem. Chem. Phys. 2020, –.
Xu et al. 2014
Xu, X.; Chen, J.; Zhang, D. H. Global Potential Energy Surface for the H+CH${}_{4}\leftrightarrow$ H${}_{2}$+CH${}_{3}$ Reaction using Neural Networks. Chin. J.
Chem. Phys. 2014, 27, 373–379.
Li and Guo 2015
Li, J.; Guo, H. Communication: An Accurate Full 15 Dimensional Permutationally
Invariant Potential Energy Surface for the OH + CH${}_{4}$ $\rightarrow$ H${}_{2}$O
+ CH${}_{3}$ Reaction. J. Chem. Phys. 2015, 143,
221103.
Jiang and Guo 2014
Jiang, B.; Guo, H. Six-Dimensional Quantum Dynamics for Dissociative
Chemisorption of H${}_{2}$ and D${}_{2}$ on Ag(111) on a Permutation Invariant
Potential Energy Surface. Phys. Chem. Chem. Phys. 2014,
16, 24704–24715.
Toyoura et al. 2016
Toyoura, K.; Hirano, D.; Seko, A.; Shiga, M.; Kuwabara, A.; Karasuyama, M.;
Shitara, K.; Takeuchi, I. Machine-Learning-Based Selective Sampling Procedure
for Identifying the Low-Energy Region in a Potential Energy Surface: A Case
Study on Proton Conduction in Oxides. Phys. Rev. B 2016,
93, 054112.
Guan et al. 2018
Guan, Y.; Yang, S.; Zhang, D. H. Construction of Reactive Potential Energy
Surfaces with Gaussian Process Regression: Active Data Selection. Mol.
Phys. 2018, 116, 823–834.
Vargas-Hernández et al. 2019
Vargas-Hernández, R. A.; Guan, Y.; Zhang, D. H.; Krems, R. V. Bayesian
Optimization for the Inverse Scattering Problem in Quantum Reaction Dynamics.
New J. Phys. 2019, 21, 022001.
Todorović et al. 2019
Todorović, M.; Gutmann, M. U.; Corander, J.; Rinke, P. Bayesian Inference
of Atomistic Structure in Functional Materials. npj Comput. Mater.
2019, 5.
Butler et al. 2018
Butler, K. T.; Davies, D. W.; Cartwright, H.; Isayev, O.; Walsh, A. Machine
Learning for Molecular and Materials Science. Nature 2018,
559, 547–555.
Haghighatlari et al. 2020
Haghighatlari, M.; Li, J.; Heidar-Zadeh, F.; Liu, Y.; Guan, X.;
Head-Gordon, T. Learning to Make Chemical Predictions: The Interplay of
Feature Representation, Data, and Machine Learning Methods. Chem
2020,
Bishop 2006
Bishop, C. M. Pattern Recognition and Machine Learning, 1st
ed.; Springer: New York, 2006.
Halama 2019
Halama, N. Machine Learning for Tissue Diagnostics in Oncology: Brave New
World. Br. J. Cancer 2019, 121, 431â433.
Bychkov et al. 2018
Bychkov, D.; Linder, N.; Turkki, R.; Nordling, S.; Kovanen, P. E.; Verrill, C.;
Walliander, M.; Lundin, M.; Haglund, C.; Lundin, J. Deep Learning Based
Tissue Analysis Predicts Outcome in Colorectal Cancer. Sci. Rep.
2018, 8, 3395.
Gómez-Meire et al. 2014
Gómez-Meire, S.; Campos, C.; Falqué, E.; Díaz, F.;
Fdez-Riverola, F. Assuring the Authenticity of Northwest Spain White Wine
Varieties using Machine Learning Techniques. Food Res. Int.
2014, 60, 230 – 240.
Watanabe et al. 2020
Watanabe, N.; Murata, M.; Ogawa, T.; Vavricka, C. J.; Kondo, A.; Ogino, C.;
Araki, M. Exploration and Evaluation of Machine Learning-Based Models for
Predicting Enzymatic Reactions. J. Chem. Inf. Model. 2020,
60, 1833–1843.
Chen and Guestrin 2016
Chen, T.; Guestrin, C. XGBoost: A Scalable Tree Boosting System. Proceedings of
the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data
Mining. New York, NY, USA, 2016; p 785â794.
Ahneman et al. 2018
Ahneman, D. T.; Estrada, J. G.; Lin, S.; Dreher, S. D.; Doyle, A. G. Predicting
Reaction Performance in C–N Cross-Coupling using Machine
Learning. Science 2018, 360, 186–190.
Atahan-Evrenk and Atalay 2019
Atahan-Evrenk, S.; Atalay, F. B. Prediction of Intramolecular Reorganization
Energy Using Machine Learning. J. Phys. Chem. A 2019,
123, 7855–7863.
Hofmann et al. 2008
Hofmann, T.; Schölkopf, B.; Smola, A. J. Kernel Methods in Machine Learning.
Ann. Statist. 2008, 36, 1171–1220.
Raschka and Mirjalili 2019
Raschka, S.; Mirjalili, V. Python Machine Learning, 3rd ed.; Packt
Publishing, 2019.
Xue et al. 2020
Xue, B.-X.; Barbatti, M.; Dral, P. O. Machine Learning for Absorption Cross
Sections. ChemRxiv 2020,
DOI:10.26434/chemrxiv.12594191.v1.
Ramakrishnan and von Lilienfeld 2017
Ramakrishnan, R.; von Lilienfeld, O. A. Reviews in Computational
Chemistry; John Wiley & Sons, Ltd, 2017; Chapter 5, pp 225–256.
Bartók et al. 2013
Bartók, A. P.; Kondor, R.; Csányi, G. On Representing Chemical
Environments. Phys. Rev. B 2013, 87, 184115.
Glorot and Bengio 2010
Glorot, X.; Bengio, Y. Understanding the Difficulty of Training Deep
Feedforward Neural Networks. Proceedings of the Thirteenth International
conference on Artificial Intelligence and Statistics. Chia Laguna Resort,
Sardinia, Italy, 2010; pp 249–256.
Duchi et al. 2011
Duchi, J.; Hazan, E.; Singer, Y. Adaptive Subgradient Methods for Online
Learning and Stochastic Optimization. J. Mach. Learn. Res.
2011, 12, 2121–2159.
Kingma and Ba 2014
Kingma, D. P.; Ba, J. Adam: A Method for Stochastic Optimization.
arXiv 2014, abs/1412.6980, 1412.6980.
Puskorius and Feldkamp 1991
Puskorius, G. V.; Feldkamp, L. A. Decoupled extended Kalman filter
training of feedforward layered networks. IJCNN-91-Seattle
International Joint Conference on Neural Networks. 1991; pp
771–777.
Singraber et al. 2019
Singraber, A.; Morawietz, T.; Behler, J.; Dellago, C. Parallel Multistream
Training of High-Dimensional Neural Network Potentials. J. Chem. Theory
Comput. 2019, 15, 3075–3092.
Behler 2011
Behler, J. Atom-Centered Symmetry Functions for Constructing High-Dimensional
Neural Network Potentials. J. Chem. Phys. 2011, 134,
074106.
LeCun and Bengio 1995
LeCun, Y.; Bengio, Y. The Handbook of Brain Theory and Neural Networks;
The MIT Press, Cambridge, MA, USA, 1995; pp 255–257.
Krizhevsky et al. 2012
Krizhevsky, A.; Sutskever, I.; Hinton, G. E. ImageNet Classification with Deep
Convolutional Neural Networks. 2012, 1097–1105.
Sainath et al. 2015
Sainath, T. N.; Kingsbury, B.; Saon, G.; Soltau, H.; rahman Mohamed, A.;
Dahl, G.; Ramabhadran, B. Deep Convolutional Neural Networks for Large-scale
Speech Tasks. Neural Networks 2015, 64, 39 –
48.
Gilmer et al. 2017
Gilmer, J.; Schoenholz, S. S.; Riley, P. F.; Vinyals, O.; Dahl, G. E. Neural
Message Passing for Quantum Chemistry. Proceedings of the 34th International
Conference on Machine Learning - Volume 70. 2017; p 1263â1272.
Schütt 2018
Schütt, K. Learning Representations of Atomistic Systems with Deep Neural
Networks. Doctoral Thesis, Technische Universität Berlin, Berlin,
2018.
Dral 2019
Dral, P. O. MLatom: A Program Package for Quantum Chemical Research Assisted by
Machine Learning. J. Comput. Chem. 2019, 40,
2339–2347.
Christensen et al. 2017
Christensen, A.; Faber, F.; Huang, B.; Bratholm, L.; Tkatchenko, A.;
Müller, K.; Lilienfeld, O. QML: A Python Toolkit for Quantum Machine
Learning. https://github.com/qmlcode/qml, 2017.
Hansen et al. 2015
Hansen, K.; Biegler, F.; Ramakrishnan, R.; Pronobis, W.; von Lilienfeld, O. A.;
Müller, K.-R.; Tkatchenko, A. Machine Learning Predictions of Molecular
Properties: Accurate Many-Body Potentials and Nonlocality in Chemical Space.
J. Phys. Chem. Lett. 2015, 6, 2326–2331.
Pozdnyakov et al. 2020
Pozdnyakov, S. N.; Willatt, M. J.; Bartók, A. P.; Ortner, C.;
Csányi, G.; Ceriotti, M. On the Completeness of Atomic Structure
Representations. arXiv 2020, 2001.11696.
Çaylak et al. 2020
Çaylak, O.; von Lilienfeld, O. A.; Baumeier, B. Wasserstein Metric for
Improved QML with Adjacency Matrix Representations. arXiv
2020, 2001.11005.
Bowman et al. 2011
Bowman, J. M.; Czakó, G.; Fu, B. High-Dimensional Ab Initio
Potential Energy Surfaces for Reaction Dynamics Calculations. Phys.
Chem. Chem. Phys. 2011, 13, 8094–8111.
Herr et al. 2019
Herr, J. E.; Koh, K.; Yao, K.; Parkhill, J. Compressing Physics with an
Autoencoder: Creating an Atomic Species Representation to Improve Machine
Learning Models in the Chemical Sciences. J. Chem. Phys.
2019, 151, 084103.
Kang et al. 2020
Kang, B.; Seok, C.; Lee, J. Prediction of Molecular Electronic Transitions
using Random Forests. ChemRxiv 2020,
DOI:10.26434/chemrxiv.12482840.v1.
Richings and Habershon 2017
Richings, G. W.; Habershon, S. Direct Quantum Dynamics Using Grid-Based Wave
Function Propagation and Machine-Learned Potential Energy Surfaces. J.
Chem. Theory Comput. 2017, 13, 4012–4024.
Unke and Meuwly 2019
Unke, O. T.; Meuwly, M. PhysNet: A Neural Network for Predicting Energies,
Forces, Dipole Moments, and Partial Charges. J. Chem. Theory Comput.
2019, 15, 3678–3693.
Lubbers et al. 2018
Lubbers, N.; Smith, J. S.; Barros, K. Hierarchical Modeling of Molecular
Energies using a Deep Neural Network. J. Chem. Phys. 2018,
148, 241715.
Zheng et al. 2019
Zheng, F.; Gao, X.; Eisfeld, A. Excitonic Wave Function Reconstruction from
Near-Field Spectra Using Machine Learning Techniques. Phys. Rev. Lett.
2019, 123, 163202.
Fabrizio et al. 2019
Fabrizio, A.; Grisafi, A.; Meyer, B.; Ceriotti, M.; Corminboeuf, C. Electron
Density Learning of Non-Covalent Systems. Chem. Sci. 2019,
10, 9424–9432.
Grisafi et al. 2019
Grisafi, A.; Fabrizio, A.; Meyer, B.; Wilkins, D. M.; Corminboeuf, C.;
Ceriotti, M. Transferable Machine-Learning Model of the Electron Density.
ACS Cent. Sci. 2019, 5, 57–64.
Fabrizio et al. 2020
Fabrizio, A.; Briling, K.; Grisafi, A.; Corminboeuf, C. Learning (from) the
Electron Density: Transferability, Conformational and Chemical Diversity.
CHIMIA Int. J. Chem. 2020, 74, 232–236.
Mai et al. 2018
Mai, S.; Plasser, F.; Dorn, J.; Fumanal, M.; Daniel, C.; González, L.
Quantitative Wave Function Analysis for Excited States of Transition Metal
Complexes. Coord. Chem. Rev. 2018, 361, 74 – 97.
Mennucci et al. 2009
Mennucci, B.; Cappelli, C.; Guido, C. A.; Cammi, R.; Tomasi, J. Structures and
Properties of Electronically Excited Chromophores in Solution from the
Polarizable Continuum Model Coupled to the Time-Dependent Density
Functional Theory. J. Phys. Chem. A 2009, 113,
3009–3020.
Jasper et al. 2004
Jasper, A. W.; Kendrick, B. K.; Mead, C. A.; Truhlar, D. G. Modern Trends
in Chemical Reaction Dynamics; World Scientific, 2004; pp 329–391.
Yarkony 2004
Yarkony, D. R. In conical Intersections; Domcke, W., Yarkony, D. R.,
Köppel, H., Eds.; Advanced Series in Physical Chemistry; World Scientific,
2004; Vol. 15.
Cupellini et al. 2020
Cupellini, L.; Bondanza, M.; Nottoli, M.; Mennucci, B. Successes & Challenges
in the Atomistic Modeling of Light-Harvesting and its Photoregulation.
Biochim. Biophys. Acta, Bioenerg. 2020, 1861,
148049.
Richings et al. 2019
Richings, G. W.; Robertson, C.; Habershon, S. Can We Use on-the-Fly Quantum
Simulations to Connect Molecular Structure and Sunscreen Action?
Faraday Discuss. 2019, 216, 476–493.
Behler et al. 2007
Behler, J.; Lorenz, S.; Reuter, K. Representing Molecule-Surface Interactions
with Symmetry-Adapted Neural Networks. J. Chem. Phys. 2007,
127, 014705.
Behler 2004
Behler, J. Dissociation of Oxygen Molecules on the Al(111) Surface. Ph.D. thesis, Technical University Berlin, 2004.
la Cour Jansen et al. 1999
la Cour Jansen, T.; Rettrup, S.; Sarma, C.; Snijders, J.; Palmieri, P. On the
Evaluation of Spin-Orbit Coupling Matrix Elements in a Spin-Adapted Basis.
Int. J. Quantum Chem. 1999, 23–27.
Chen et al. 2019
Chen, W.-K.; Fang, W.-H.; Cui, G. A Multi-Layer Energy-Based Fragment Method
for Excited States and Nonadiabatic Dynamics. Phys. Chem. Chem. Phys.
2019, 21, 22695–22699.
Christensen and von Lilienfeld 2019
Christensen, A. S.; von Lilienfeld, O. A. Operator Quantum Machine Learning:
Navigating the Chemical Space of Response Properties. CHIMIA
2019, 73, 1028–1031.
Kasha et al. 1965
Kasha, M.; Rawls, H. R.; El-Bayoumi, M. A. The Exciton Model in Molecular
Spectroscopy. Pure and Applied Chemistry 1965, 11, 371
– 392.
Rogers and Hahn 2010
Rogers, D.; Hahn, M. Extended-Connectivity Fingerprints. J. Chem. Inf.
Model. 2010, 50, 742–754.
Durant et al. 2002
Durant, J. L.; Leland, B. A.; Henry, D. R.; Nourse, J. G. Reoptimization of
MDL Keys for Use in Drug Discovery. J. Chem. Inf. Comput. Sci.
2002, 42, 1273–1280.
Landrum 2016
Landrum, G. RDKit: Open-Source Cheminformatics Software. 2016,
Jain et al. 2013
Jain, A.; Ong, S. P.; Hautier, G.; Chen, W.; Richards, W. D.; Dacek, S.;
Cholia, S.; Gunter, D.; Skinner, D.; Ceder, G.; Persson, K. A. Commentary:
The Materials Project: A Materials Genome Approach to Accelerating Materials
Innovation. APL Mater. 2013, 1, 011002.
Aarva et al. 2019
Aarva, A.; Deringer, V. L.; Sainio, S.; Laurila, T.; Caro, M. A. Understanding
X-Ray Spectroscopy of Carbonaceous Materials by Combining Experiments,
Density Functional Theory, and Machine Learning. Part II: Quantitative
Fitting of Spectra. Chem. Mat. 2019, 31,
9256–9267.
Bartók et al. 2017
Bartók, A. P.; De, S.; Poelking, C.; Bernstein, N.; Kermode, J. R.;
Csányi, G.; Ceriotti, M. Machine Learning Unifies the Modeling of
Materials and Molecules. Sci. Adv. 2017, 3.
Aarva et al. 2019
Aarva, A.; Deringer, V. L.; Sainio, S.; Laurila, T.; Caro, M. A. Understanding
X-ray Spectroscopy of Carbonaceous Materials by Combining Experiments,
Density Functional Theory, and Machine Learning. Part I: Fingerprint
Spectra. Chem. Mat. 2019, 31, 9243–9255.
Deringer et al. 2018
Deringer, V. L.; Caro, M. A.; Jana, R.; Aarva, A.; Elliott, S. R.; Laurila, T.;
Csányi, G.; Pastewka, L. Computational Surface Chemistry of Tetrahedral
Amorphous Carbon by Combining Machine Learning and Density Functional Theory.
Chem. Mat. 2018, 30, 7438–7445.
Janet and Kulik 2017
Janet, J. P.; Kulik, H. J. Predicting Electronic Structure Properties of
Transition Metal Complexes with Neural Networks. Chem. Sci.
2017, 8, 5137–5152.
Janet et al. 2019
Janet, J. P.; Duan, C.; Yang, T.; Nandy, A.; Kulik, H. J. A Quantitative
Uncertainty Metric Controls Error in Neural Network-Driven Chemical
Discovery. Chem. Sci. 2019, 10, 7913–7922.
Janet et al. 2017
Janet, J. P.; Gani, T. Z. H.; Steeves, A. H.; Ioannidis, E. I.; Kulik, H. J.
Leveraging Cheminformatics Strategies for Inorganic Discovery: Application to
Redox Potential Design. Ind. Eng. Chem. Res. 2017, 56,
4898–4910.
Janet et al. 2018
Janet, J. P.; Chan, L.; Kulik, H. J. Accelerating Chemical Discovery with
Machine Learning: Simulated Evolution of Spin Crossover Complexes with an
Artificial Neural Network. J. Phys. Chem. Lett. 2018,
9, 1064–1071.
Janet et al. 2020
Janet, J. P.; Ramesh, S.; Duan, C.; Kulik, H. J. Accurate Multiobjective
Design in a Space of Millions of Transition Metal Complexes with
Neural-Network-Driven Efficient Global Optimization. ACS Cent. Sci.
2020, 6, 513–524.
Ðorđević et al. 2018
Ðorđević, N.; Beckwith, J. S.; Yarema, M.; Yarema, O.;
Rosspeintner, A.; Yazdani, N.; Leuthold, J.; Vauthey, E.; Wood, V. Machine
Learning for Analysis of Time-Resolved Luminescence Data. ACS
Photonics 2018, 5, 4888–4895.
Abramavicius et al. 2010
Abramavicius, D.; Jiang, J.; Bulheller, B. M.; Hirst, J. D.; Mukamel, S.
Simulation Study of Chiral Two-Dimensional Ultraviolet Spectroscopy of the
Protein Backbone. Journal of the American Chemical Society
2010, 132, 7769–7775.
Zhang et al. 2011
Zhang, X.-X.; Würth, C.; Zhao, L.; Resch-Genger, U.; Ernsting, N. P.;
Sajadi, M. Femtosecond Broadband Fluorescence Upconversion Spectroscopy:
Improved Setup and Photometric Correction. Rev. Sci. Instrum.
2011, 82, 063108.
Ahmad et al. 2016
Ahmad, I.; Ahmed, S.; Anwar, Z.; Sheraz, M. A.; Sikorski, M. Photostability and
Photostabilization of Drugs and Drug Products. Int. J. Photoenergy
2016, 2016, 1–19.
Mathew et al. 2014
Mathew, S.; Yella, A.; Gao, P.; Humphry-Baker, R.; Curchod, B. F. E.;
Ashari-Astani, N.; Tavernelli, I.; Rothlisberger, U.; Nazeeruddin, M. K.;
Grätzel, M. Dye-Sensitized Solar Cells with 13% Efficiency Achieved
Through the Molecular Engineering of Porphyrin Sensitizers. Nat. Chem.
2014, 6, 242–247.
Dobson 2004
Dobson, C. M. Chemical Space and Biology. Nature 2004,
432, 824–828. |
Asymptotic normality of total least squares estimator in a multivariate errors-in-variables model $AX=B$
A.\fnmAlexanderKukush
alexander_
kukush@univ.kiev.ua
Ya.\fnmYaroslavTsaregorodtsev
777Tsar777@mail.ru
Taras Shevchenko National University of Kyiv, Kyiv, Ukraine
(2016; 11 February 2016; 7 March 2016; 11 March 2016)
Abstract
We consider a multivariate functional measurement error model
$AX\approx B$. The errors in $[A,B]$ are uncorrelated, row-wise
independent, and have equal (unknown) variances. We study the total
least squares estimator of $X$, which, in the case of normal errors,
coincides with the maximum likelihood one. We give conditions for
asymptotic normality of the estimator when the number of rows in $A$ is
increasing. Under mild assumptions, the covariance structure of the
limit Gaussian random matrix is nonsingular. For normal errors, the
results can be used to construct an asymptotic confidence interval for
a linear functional of $X$.
doi: 10.15559/16-VMSTA50
keywords:
Asymptotic normality\sepmultivariate errors-in-variables model\septotal least squares
\MSC[2010]
15A52\sep65F20\sep62E20\sep62S05\sep62F12\sep62H12
††volume: 3††issue: 1\DeclareMathOperator\M
E
\DeclareMathOperator\covcov
\DeclareMathOperator\trT
\DeclareMathOperator\II
\startlocaldefs
\urlstylerm
\endlocaldefs
\cortext
[cor1]Corresponding author.
\publishedonline
29 March 2016
1 Introduction
We deal with overdetermined system of linear equations $AX\approx B$,
which is common in linear parameter estimation problem [9]. If
the data matrix $A$ and observation matrix $B$ are contaminated with
errors, and all the errors are uncorrelated and have equal variances,
then the total least squares (TLS) technique is appropriate for solving
this system [9]. Kukush and Van Huffel [5] showed the
statistical consistency of the TLS estimator $\hat{X}_{\mathit{tls}}$ as the
number $m$ of rows in $A$ grows, provided that the errors in $[A,B]$
are row-wise i.i.d. with zero mean and covariance matrix proportional
to a unit matrix; the covariance matrix was assumed to be known up to a
factor of proportionality; the true input matrix $A_{0}$ was supposed to
be nonrandom. In fact, in [5] a more general, element-wise
weighted TLS estimator was studied, where the errors in $[A,B]$ were
row-wise independent, but within each row, the entries could be
observed without errors, and, additionally, the error covariance matrix
could differ from row to row. In [6], an iterative numerical
procedure was developed to compute the elementwise-weighted TLS
estimator, and the rate of convergence of the procedure was established.
In a univariate case where $B$ and $X$ are column vectors, the
asymptotic normality of $\hat{X}_{\mathit{tls}}$ was shown by Gallo [4]
as $m$ grows. In [7], that result was extended to mixing error
sequences. Both [4] and [7] utilized an explicit form of
the TLS solution.
In the present paper, we extend the Gallo’s asymptotic normality result
to a multivariate case, where $A$, $X$, and $B$ are matrices.
Now a closed-form solution is unavailable, and we work instead with the
cost function. More precisely, we deal with the estimating function,
which is a matrix derivative of the cost function. In fact, we show
that under mild conditions, the normalized estimator converges in
distribution to a Gaussian random matrix with nonsingular covariance
structure. For normal errors, the latter structure can be estimated
consistently based on the observed matrix $[A,B]$. The results can be
used to construct the asymptotic confidence ellipsoid for a vector
$Xu$, where $u$ is a column vector of the corresponding dimension.
The paper is organized as follows. In Section 2, we describe
the model, refer to the consistency result for the estimator, and
present the objective function and corresponding matrix estimating
function. In Section 3, we state the asymptotic normality of
$\hat{X}_{\mathit{tls}}$ and provide a nonsingular covariance structure for a
limit random matrix. The latter structure depends continuously on some
nuisance parameters of the model, and we derive consistent estimators
for those parameters. Section 4 concludes. The proofs are given
in Appendix. There we work with the estimating function and derive an
expansion for the normalized estimator using Taylor’s formula. The
expansion holds with probability tending to $1$.
Throughout the paper, all vectors are column ones, $\M$ stands for the
expectation and acts as an operator on the total product, $\cov(x)$
denotes the covariance matrix of a random vector $x$, and for a
sequence of random matrices $\{X_{m},m\geq 1\}$ of the same size, the
notation $X_{m}=O_{p}(1)$ means that the sequence $\{\|X_{m}\|\}$ is
stochastically bounded, and $X_{m}=o_{p}(1)$ means that $\|X_{m}\|\stackrel{{\scriptstyle\text{\rm P}}}{{\longrightarrow}}0$.
By $\I_{p}$ we denote the unit matrix of size $p$.
2 Model, objective, and estimating
2.1 The TLS problem
Consider the model $AX\approx B$. Here $A\in\mathbb{R}^{m\times n}$ and
$B\in\mathbb{R}^{m\times d}$ are observations, and $X\in\mathbb{R}^{n\times d}$ is a parameter of interest. Assume that
$$A=A_{0}+\tilde{A},\qquad B=B_{0}+\tilde{B},$$
(2.1)
and that there exists $X_{0}\in\mathbb{R}^{n\times d}$ such that
$$A_{0}X_{0}=B_{0}.$$
(2.2)
Here $A_{0}$ is the nonrandom true input matrix, $B_{0}$ is the true output
matrix, and $\tilde{A}$, $\tilde{B}$ are error matrices. The matrix
$X_{0}$ is the true value of the parameter.
We can rewrite the model (2.1)–(2.2) as a classical
functional errors-in-variables (EIV) model with vector regressor and
vector response [3]. Denote by $a_{i}^{\tr}$, $a_{0i}^{\tr}$,
$\tilde{a}_{i}^{\tr}$, $b_{i}^{\tr}$, $b_{0i}^{\tr}$, and $\tilde{b}_{i}^{\tr}$ the rows of $A$, $A_{0}$, $\tilde{A}$, $B$, $B_{0}$, and $\tilde{B}$,
respectively, $i=1,\dots,m$. Then the model considered is equivalent to
the following EIV model:
$$a_{i}=a_{0i}+\tilde{a}_{i},\qquad b_{i}=b_{0i}+\tilde{b}_{i},\qquad b_{oi}=X_{0}^{\tr}a_{0i},\quad i=1,\dots,m.$$
Based on observations $a_{i}$, $b_{i}$, $i=1,\dots,m$, we have to estimate
$X_{0}$. The vectors $a_{0i}$ are nonrandom and unknown, and the vectors
$\tilde{a}_{i}$, $\tilde{b}_{i}$ are random errors.
We state a global assumption of the paper.
(i)
The vectors $\tilde{z}_{i}$ with $\tilde{z}_{i}^{\tr}=[\tilde{a}_{i}^{\tr},\tilde{b}_{i}^{\tr}]$, $i=1,2,\dots$, are i.i.d., with zero mean and
variance–covariance matrix
$$S_{\tilde{z}}:=\cov(\tilde{z}_{1})=\sigma^{2}\I_{n+d},$$
(2.3)
where the factor of proportionality $\sigma^{2}$ is positive and unknown.
The TLS problem consists in finding the values of disturbances $\Delta\hat{A}$ and $\Delta\hat{B}$ minimizing the sum of squared corrections
$$\min_{(X\in\mathbb{R}^{n\times d},\>\Delta A,\>\Delta B)}\bigl{(}\|\Delta A\|_{F}^{2}+\|\Delta B\|_{F}^{2}\bigr{)}$$
(2.4)
subject to the constraints
$$(A-\Delta A)X=B-\Delta B.$$
(2.5)
Here in (2.4), for a matrix $C=(c_{ij})$, $\|C\|_{F}$ denotes the
Frobenius norm, $\|C\|_{F}^{2}=\sum_{i,j}c_{ij}^{2}$. Later on,
we will also use the operator norm $\|C\|=\sup_{x\neq 0}\tfrac{\|Cx\|}{\|x\|}$.
2.2 TLS estimator and its consistency
It may happen that, for some random realization, problem
(2.4)–(2.5) has no solution. In such a case, put $\hat{X}_{\mathit{tls}}=\infty$. Now, we give a formal definition of the TLS estimator.
Definition 1.
The TLS estimator $\hat{X}_{\mathit{tls}}$ of $X_{0}$ in the model (2.1)–(2.2) is a measurable mapping of the underlying
probability space into $\mathbb{R}^{n\times d}\cup\{\infty\}$, which
solves problem (2.4)–(2.5) if there exists a solution, and
$\hat{X}_{\mathit{tls}}=\infty$ otherwise.
We need the following conditions for the consistency of $\hat{X}_{\mathit{tls}}$.
(ii)
$\M\|\tilde{z}_{1}\|^{4}<\infty$, where $\tilde{z}_{1}$ satisfies condition
(i).
(iii)
$\tfrac{1}{m}A_{0}^{\tr}A_{0}\to V_{A}$ as $m\to\infty$, where $V_{A}$ is a
nonsingular matrix.
The next consistency result is contained in Theorem 4(a) of [5].
Theorem 2
Assume condition (i) to (iii). Then $\hat{X}_{\mathit{tls}}$ is
finite with probability tending to one, and $\hat{X}_{\mathit{tls}}$ tends to
$X_{0}$ in probability as $m\to\infty$.
2.3 The objective and estimating functions
Denote
$$\displaystyle q(a,b;X)=\bigl{(}a^{\tr}X-b^{\tr}\bigr{)}\bigl{(}\I_{d}+X^{\tr}X\bigr{)}^{-1}\bigl{(}X^{\tr}a-b\bigr{)},$$
(2.6)
$$\displaystyle Q(X)=\sum_{i=1}^{m}q(a_{i},b_{i};X),\quad X\in\mathbb{R}^{n\times d}.$$
(2.7)
The TLS estimator is known to minimize the objective function (2.7); see [8] or formula (24) in [5].
Lemma 3
The TLS estimator $\hat{X}_{\mathit{tls}}$ is finite iff there exists an
unconstrained minimum of the function (2.7), and then $\hat{X}_{\mathit{tls}}$ is a minimum point of that function.
Introduce an estimating function related to the loss function (2.6):
$$s(a,b;X):=a\bigl{(}a^{\tr}X-b^{\tr}\bigr{)}-X\bigl{(}\I_{d}+X^{\tr}X\bigr{)}^{-1}\bigl{(}X^{\tr}a-b\bigr{)}\bigl{(}a^{\tr}X-b^{\tr}\bigr{)}.$$
(2.8)
Corollary 4
(a)
Under conditions (i) to (iii), with probability tending to
one $\hat{X}_{\mathit{tls}}$ is a solution to the equation
$$\sum_{i=1}^{m}s(a_{i},b_{i};X)=0,\quad X\in\mathbb{R}^{n\times d}.\vspace{-3pt}$$
(b)
Under assumption (i), the function $s(a,b;X)$ is unbiased
estimating function, that is, for each $i\geq 1$, $\M_{X_{0}}s(a_{i},b_{i};X_{0})=0$.
Expression (2.8) as a function of $X$ is a mapping in $\mathbb{R}^{n\times d}$. Its derivative $s_{X}^{\prime}$ is a linear operator in
this space.
Lemma 5
Under condition (i), for each $H\in\mathbb{R}^{n\times d}$ and
$i\geq 1$, we have
$$\M_{X_{0}}\bigl{[}s_{X}^{\prime}(a_{i},b_{i};X_{0})\cdot H\bigr{]}=a_{0i}a_{0i}^{\tr}H.$$
(2.9)
Therefore, we can identify $\M_{X_{0}}s_{X}^{\prime}(a_{i},b_{i};X_{0})$ with the
matrix $a_{0i}a_{0i}^{\tr}$.
3 Main results
Introduce further assumptions to state the asymptotic normality of $\hat{X}_{\mathit{tls}}$. We need a bit higher moments compared with conditions (ii) and (iii) in order to use the Lyapunov CLT. Recall that
$\tilde{z}_{i}$ satisfies condition (i).
(iv)
For some $\delta>0$, $\M\|\tilde{z}_{1}\|^{4+2\delta}<\infty$.
(v)
For $\delta$ from condition (iv),
$$\frac{1}{m^{1+\delta/2}}\sum_{i=1}^{m}\|a_{0i}\|^{2+\delta}\to 0\quad\text{ as }m\to\infty.\vspace{-6pt}$$
(vi)
$\frac{1}{m}\sum_{i=1}^{m}a_{0i}\to\mu_{a}$ as $m\to\infty$,
where $\mu_{a}\in\mathbb{R}^{n\times 1}$.
(vii)
The distribution of $\tilde{z}_{1}$ is symmetric around the origin.
Introduce a random element in the space of systems consisting of five matrices:
$$W_{i}=\bigl{(}a_{0i}\tilde{a}_{i}^{\tr},a_{0i}\tilde{b}_{i}^{\tr},\tilde{a}_{i}\tilde{a}_{i}^{\tr}-\sigma^{2}\I_{n},\tilde{a}_{i}\tilde{b}_{i}^{\tr},\tilde{b}_{i}\tilde{b}_{i}^{\tr}-\sigma^{2}\I_{d}\bigr{)}.$$
(3.1)
Hereafter $\stackrel{{\scriptstyle\text{\rm d}}}{{\longrightarrow}}$ stands for the
convergence in distribution.
Lemma 6
Assume conditions (i) and (iii)–(vi). Then
$$\frac{1}{\sqrt{m}}\sum_{i=1}^{m}W_{i}\stackrel{{\scriptstyle\text{\rm d}}}{{\longrightarrow}}\varGamma=(\varGamma_{1},\dots,\varGamma_{5})\quad\text{ as }m\to\infty,$$
(3.2)
where $\varGamma$ is a Gaussian centered random element with matrix components.
Lemma 7
In assumptions of Lemma 6, replace condition (vi) with
condition (vii). Then the convergence (3.2) still holds
with independent components $\varGamma_{1},\dots,\varGamma_{5}$.
Now, we state the asymptotic normality of $\hat{X}_{\mathit{tls}}$.
Theorem 8
(a)
Assume conditions (i) and (iii)–(vi). Then
$$\sqrt{m}(\hat{X}_{\mathit{tls}}-X_{0})\stackrel{{\scriptstyle\text{\rm d}}}{{\longrightarrow}}V_{A}^{-1}\varGamma(X_{0})\quad\text{as }m\to\infty,$$
(3.3)
$$\varGamma(X):=\varGamma_{1}X-\varGamma_{2}+\varGamma_{3}X-\varGamma_{4}-X\bigl{(}\I_{d}+X^{\tr}X\bigr{)}^{-1}\bigl{(}X^{\tr}\varGamma_{3}X-X^{\tr}\varGamma_{4}-\varGamma_{4}^{\tr}X+\varGamma_{5}\bigr{)},$$
(3.4)
where $V_{A}$ satisfies condition (iii), and $\varGamma_{i}$ satisfy
relation (3.2).
(b)
In the assumption of part (a), replace condition (vi)
with condition (vii). Then the convergence (3.3) still
holds, and, moreover, the limit random matrix $X_{\infty}:=V_{A}^{-1}\varGamma(X_{0})$ has a nonsingular covariance structure, that
is, for each nonzero vector $u\in\mathbb{R}^{d\times 1}$, $\cov(X_{\infty}u)$ is a nonsingular matrix.
Remark 9.
Conditions of Theorem 8(a) are similar to Gallo’s
conditions [4] for the asymptotic normality in the univariate
case; see also, [9], pp. 240–243. Compared with Theorems 2.3
and 2.4 of [7], stated for univariate case with mixing errors,
we need not the requirement for entries of the true input $A_{0}$ to be
totally bounded.
In [7], Section 2, we can find a discussion of importance of the
asymptotic normality result for $\hat{X}_{\mathit{tls}}$. It is claimed there
that the formula for the asymptotic covariance structure of $\hat{X}_{\mathit{tls}}$ is computationally useless, but in case where the limit
distribution is nonsingular, we can use the block-bootstrap techniques
when constructing confidence intervals and testing hypotheses.
However, in the case of normal errors $\tilde{z}_{i}$, we can apply
Theorem 8(b) to construct the asymptotic confidence
ellipsoid, say, for $X_{0}u$, $u\in\mathbb{R}^{d\times 1}$, $u\neq 0$.
Indeed, relations 3.1, 3.2, 3.3 and 3.4 show that the nonsingular matrix
$$S_{u}:=\cov\bigl{(}\mathbf{V}_{A}^{-1}\varGamma(X_{0})u\bigr{)}$$
is a continuous function $S_{u}=S_{u}(X_{0},\mathbf{V}_{A},\sigma^{2})$ of
unknown parameters $X_{0}$, $\mathbf{V}_{A}$, and $\sigma^{2}$. (It is
important here that now the components $\varGamma_{j}$ of $\varGamma$ are
independent, and the covariance structure of each $\varGamma_{j}$ depends on
$\sigma^{2}$ and $\mathbf{V}_{A}$, not on some other limit characteristics
of $A_{0}$; see Lemma 6.) Once we possess consistent estimators $\hat{\mathbf{V}}_{A}$ and $\hat{\sigma}^{2}$ of $\mathbf{V}_{A}$ and $\sigma^{2}$,
the matrix
$\hat{S}_{u}:=S_{u}(\hat{X}_{\mathit{tls}},\hat{\mathbf{V}}_{A},\hat{\sigma}^{2})$ is
a consistent estimator for the covariance matrix $S_{u}$.
Hereafter, a bar means averaging for rows $i=1,\dots,m$, for example,
$\overline{ab^{\tr}}=m\frac{1}{m}\sum_{i=1}^{m}a_{i}b_{i}^{\tr}$.
Lemma 10
Assume the conditions of Theorem 2. Define
$$\hat{\sigma}^{2}=\frac{1}{d}\mathrm{tr}\bigl{[}\bigl{(}\overline{bb^{\tr}}-2\hat{X}_{\mathit{tls}}^{\tr}\overline{ab^{\tr}}+\hat{X}_{\mathit{tls}}^{\tr}\overline{aa^{\tr}}\hat{X}_{\mathit{tls}}\bigr{)}\bigl{(}\I_{d}+\hat{X}_{\mathit{tls}}^{\tr}\hat{X}_{\mathit{tls}}\bigr{)}^{-1}\bigr{]},$$
(3.5)
$$\hat{V}_{A}=\overline{aa^{\tr}}-\hat{\sigma}^{2}\I_{n}.$$
Then
$$\hat{\sigma}^{2}\stackrel{{\scriptstyle\text{\rm P}}}{{\longrightarrow}}\sigma^{2},\qquad\hat{V}_{A}\stackrel{{\scriptstyle\text{\rm P}}}{{\longrightarrow}}V_{A}.$$
(3.6)
Remark 11.
Estimator (3.5) is a multivariate analogue of the maximum
likelihood estimator (1.53) in [2] in the functional scalar EIV model.
Finally, for the case $\tilde{z}_{1}\sim N(0,\sigma^{2}\I_{n+d})$, based on
Lemma 10 and the relations
$$\sqrt{m}(\hat{X}_{\mathit{tls}}-X_{0})\stackrel{{\scriptstyle\text{\rm d}}}{{\longrightarrow}}N(0,S_{u}),\quad S_{u}>0,\ \hat{S}_{u}\stackrel{{\scriptstyle\text{\rm P}}}{{\longrightarrow}}S_{u},$$
we can construct the asymptotic confidence ellipsoid for the vector
$X_{0}u$ in a standard way.
Remark 12.
In a similar way, a confidence ellipsoid can be constructed for any
finite set of linear combinations of $X_{0}$ entries with fixed known
coefficients.
4 Conclusion
We extended the result of Gallo [4] and proved the asymptotic
normality of the TLS estimator in a multivariate model $AX\approx B$.
The normalized estimator converges in distribution to a random matrix
with quite complicated covariance structure. If the error distribution
is symmetric around the origin, then the latter covariance structure is
nonsingular. For the case of normal errors, this makes it possible to
construct the asymptotic confidence region for a vector $X_{0}u$, $u\in\mathbb{R}^{d\times 1}$, where $X_{0}$ is the true value of $X$.
In future papers, we will extend the result for the elementwise
weighted TLS estimator [5] in the model $AX\approx B$, where
some columns of the matrix $[A,B]$ may be observed without errors, and,
in addition, the error covariance matrix may differ from row to row.
Appendix
Proof of Corollary 4
(a) For any $n$ and $d$, the space $\mathbb{R}^{n\times d}$ is endowed
with natural inner product $\langle A,B\rangle=\mathrm{tr}(AB^{\tr})$ and
the Frobenius norm. The matrix derivative $q_{X}^{\prime}$ of the
functional (2.6) is a linear functional on $\mathbb{R}^{n\times d}$, which can be identified with certain matrix from $\mathbb{R}^{n\times d}$ based on the inner product.
Using the rules of matrix calculus [1], we have for $H\in\mathbb{R}^{n\times d}$:
$$\displaystyle\big{\langle}q_{X}^{\prime},H\big{\rangle}$$
$$\displaystyle=a^{\tr}H\bigl{(}\I_{d}+X^{\tr}X\bigr{)}^{-1}\bigl{(}X^{\tr}a-b\bigr{)}$$
$$\displaystyle\quad-\bigl{(}a^{\tr}X-b^{\tr}\bigr{)}\bigl{(}\I_{d}+X^{\tr}X\bigr{)}^{-1}\bigl{(}H^{\tr}X+X^{\tr}H\bigr{)}\bigl{(}\I_{d}+X^{\tr}X\bigr{)}^{-1}\bigl{(}X^{\tr}a-b\bigr{)}$$
$$\displaystyle\quad+\bigl{(}a^{\tr}X-b^{\tr}\bigr{)}\bigl{(}\I_{d}+X^{\tr}X\bigr{)}^{-1}H^{\tr}a.$$
Collecting similar terms, we obtain:
$$\displaystyle\frac{1}{2}\big{\langle}q_{X}^{\prime},H\big{\rangle}$$
$$\displaystyle=\bigl{(}a^{\tr}X-b^{\tr}\bigr{)}\bigl{(}\I_{d}+X^{\tr}X\bigr{)}^{-1}H^{\tr}a$$
$$\displaystyle\quad-\bigl{(}a^{\tr}X-b^{\tr}\bigr{)}\bigl{(}\I_{d}+X^{\tr}X\bigr{)}^{-1}H^{\tr}X\bigl{(}\I_{d}+X^{\tr}X\bigr{)}^{-1}\bigl{(}X^{\tr}a-b\bigr{)},$$
and
$$\displaystyle\dfrac{1}{2}\big{\langle}q_{X}^{\prime},H\big{\rangle}$$
$$\displaystyle=\mathrm{tr}\bigl{[}a\bigl{(}a^{\tr}X-b^{\tr}\bigr{)}\bigl{(}\I_{d}+X^{\tr}X\bigr{)}^{-1}H^{\tr}\bigr{]}$$
$$\displaystyle\quad-\mathrm{tr}\bigl{[}X\bigl{(}\I_{d}+X^{\tr}X\bigr{)}^{-1}\bigl{(}X^{\tr}a-b\bigr{)}\bigl{(}a^{\tr}X-b^{\tr}\bigr{)}\bigl{(}\I_{d}+X^{\tr}X\bigr{)}^{-1}H^{\tr}\bigr{]}.$$
Using the inner product in $\mathbb{R}^{n\times d}$, we get $\tfrac{1}{2}q_{X}^{\prime}=s(x)(\I_{d}+X^{\tr}X)^{-1}$, where $s(x)$ is the
left-hand side of (2.8). In view of Theorem 2 and Lemma
3, this implies the statement of Corollary 4(a).
(b) Now, we set
$$a=a_{0}+\tilde{a},\qquad b=b_{0}+\tilde{b},\qquad b_{0}=X^{\tr}a_{0},$$
(4.1)
where $a_{0}$ is a nonrandom vector, and, like in (2.3),
$$\cov\left(\left[\begin{array}[]{l}\tilde{a}\\
\tilde{b}\end{array}\right]\right)=\sigma^{2}\I_{n+d},\qquad\M\left[\begin{array}[]{l}\tilde{a}\\
\tilde{b}\end{array}\right]=0.$$
(4.2)
Then
$$\M_{X}a\bigl{(}a^{\tr}X-b^{\tr}\bigr{)}=\M a\bigl{(}\tilde{a}^{\tr}X-\tilde{b}^{\tr}\bigr{)}=\sigma^{2}X,$$
(4.3)
$$\M_{X}\bigl{(}X^{\tr}a-b\bigr{)}\bigl{(}a^{\tr}X-b^{\tr}\bigr{)}=\M\bigl{(}X^{\tr}\tilde{a}-\tilde{b}\bigr{)}\bigl{(}\tilde{a}^{\tr}X-\tilde{b}^{\tr}\bigr{)}=\sigma^{2}\bigl{(}\I_{d}+X^{\tr}X\bigr{)}.$$
(4.4)
Therefore (see (2.8)),
$$\M_{X}s(a,b;X)=\sigma^{2}X-\sigma^{2}X\bigl{(}\I_{d}+X^{\tr}X\bigr{)}^{-1}\bigl{(}\I_{d}+X^{\tr}X\bigr{)}=0.$$
This implies the statement of Corollary 4(b).
Proof of Lemma 5
The derivative $s_{X}^{\prime}$ of the function (2.8) is a linear
operator in $\mathbb{R}^{n\times d}$. For $H\in\mathbb{R}^{n\times d}$, we have:
$$\displaystyle s_{X}^{\prime}H$$
$$\displaystyle=aa^{\tr}H-H\bigl{(}\I_{d}+X^{\tr}X\bigr{)}^{-1}\bigl{(}X^{\tr}a-b\bigr{)}\bigl{(}a^{\tr}X-b^{\tr}\bigr{)}$$
(4.5)
$$\displaystyle\quad+X\bigl{(}\I_{d}+X^{\tr}X\bigr{)}^{-1}\bigl{(}H^{\tr}X+X^{\tr}H\bigr{)}\bigl{(}\I_{d}+X^{\tr}X\bigr{)}^{-1}\bigl{(}X^{\tr}a-b\bigr{)}$$
$$\displaystyle\quad\times\bigl{(}a^{\tr}X-b^{\tr}\bigr{)}-X\bigl{(}\I_{d}+X^{\tr}X\bigr{)}^{-1}\bigl{(}H^{\tr}a\bigl{(}a^{\tr}X-b^{\tr}\bigr{)}+\bigl{(}X^{\tr}a-b\bigr{)}a^{\tr}H\bigr{)}.$$
As before, we set (4.1), (4.2) and use relations (4.3), (4.4), and the relation$\M aa^{\tr}=a_{0}a_{0}^{\tr}+\sigma^{2}\I_{n}$. We obtain:
$$\displaystyle\M_{X}s_{X}^{\prime}H$$
$$\displaystyle=\bigl{(}a_{0}a_{0}^{\tr}+\sigma^{2}\I_{n}\bigr{)}H-\sigma^{2}H+\sigma^{2}X\bigl{(}\I_{d}+X^{\tr}X\bigr{)}^{-1}\bigl{(}H^{\tr}X+X^{\tr}H\bigr{)}$$
$$\displaystyle\quad-\sigma^{2}X\bigl{(}\I_{d}+X^{\tr}X\bigr{)}^{-1}\bigl{(}H^{\tr}H+X^{\tr}H\bigr{)}=a_{0}a_{0}^{\tr}H.$$
This implies (2.9).
Proof of Lemma 6
The random elements $W_{i}$, $i\geq 1$, in (3.1) are independent
and centered. We want to apply the Lyapunov CLT for the left-hand side
of (3.2).
(a) All the second moments of $m^{-\frac{1}{2}}\sum_{i=1}^{m}W_{i}$ converge to finite limits. For example, for the first
component, we have
$$\frac{1}{m}\sum_{i=1}^{m}\M\bigl{(}\big{\langle}a_{0i}\tilde{a}_{i}^{\tr},H_{1}\big{\rangle}\bigr{)}^{2}=\frac{1}{m}\sum_{i=1}^{m}\M\bigl{(}\text{tr }a_{0i}\tilde{a}_{1}^{\tr}H_{1}^{\tr}\bigr{)}^{2},$$
and this has a finite limit due to assumption (iii). Here $H_{1}\in\mathbb{R}^{n\times n}$, and we use the inner product introduced in the
proof of Corollary 4.
For the fifth component,
$$\frac{1}{m}\sum_{i=1}^{m}\M\bigl{(}\big{\langle}\tilde{b}_{i}\tilde{b}_{i}^{\tr}-\sigma^{2}\I_{d},H_{2}\big{\rangle}\bigr{)}^{2}=\M\bigl{[}\mathrm{tr}\bigl{(}\bigl{(}\tilde{b}_{1}\tilde{b}_{1}^{\tr}-\sigma^{2}\I_{d}\bigr{)}H_{2}\bigr{)}\bigr{]}^{2}<\infty,$$
because the fourth moments of $\tilde{b}_{i}$ are finite. Here $H_{2}\in\mathbb{R}^{d\times d}$.
For mixed moments of the first and fifth components, we have
$$\displaystyle\frac{1}{m}\sum_{i=1}^{m}\M\big{\langle}a_{0i}\tilde{a}_{i}^{\tr},H_{1}\big{\rangle}\cdot\big{\langle}\tilde{b}_{i}\tilde{b}_{i}^{\tr}-\sigma^{2}\I_{d},H_{2}\big{\rangle}$$
$$\displaystyle\quad=\M\Bigg{\langle}\Biggl{(}\frac{1}{m}\sum_{i=1}^{m}a_{0i}\Biggr{)}\tilde{a}_{1}^{\tr},H_{1}\Bigg{\rangle}\cdot\big{\langle}\tilde{b}_{1}\tilde{b}_{1}^{\tr}-\sigma^{2}\I_{d},H_{2}\big{\rangle},$$
(4.6)
and this, due to condition (vi), converges toward
$$\M\big{\langle}\mu_{a}\tilde{a}_{1}^{\tr},H_{1}\big{\rangle}\cdot\big{\langle}\tilde{b}_{1}\tilde{b}_{1}^{\tr}-\sigma^{2}\I_{d},H_{2}\big{\rangle}.$$
Other second moments can be considered in a similar way.
(b) The Lyapunov condition holds for each component of (3.1). Let
$\delta$ be the quantity from assumptions (iv), (v). Then
$$\frac{1}{m^{1+\delta/2}}\sum_{i=1}^{m}\M\big{\|}a_{0i}\tilde{a}_{i}^{\tr}\big{\|}^{2+\delta}\leq\frac{\M\|\tilde{a}_{1}\|^{2+\delta}}{m^{1+\delta/2}}\sum_{i=1}^{m}\|a_{0i}\|^{2+\delta}\to 0$$
as $m\to\infty$ by condition (v). For the fifth component,
$$\displaystyle\frac{1}{m^{1+\delta/2}}\sum_{i=1}^{m}\M\big{\|}\tilde{b}_{i}\tilde{b}_{i}^{\tr}-\sigma^{2}\I_{d}\big{\|}^{2+\delta}$$
$$\displaystyle=\frac{1}{m^{\delta/2}}\M\big{\|}\tilde{b}_{1}\tilde{b}_{1}^{\tr}-\sigma^{2}\I_{d}\big{\|}^{2+\delta}$$
$$\displaystyle\leq\frac{\mathrm{const}}{m^{\delta/2}}\M\|\tilde{b}_{1}\|^{4+2\delta}\to 0\quad\text{ as }m\to\infty.$$
The latter expectation is finite by condition (iv).
The Lyapunov condition for other components is considered similarly.
(c) Parts (a) and (b) of the present proof imply (3.2) by the
Lyapunov CLT.
Proof of Lemma 7
Under conditions (vii) and (i), all the five components of
$W_{i}$, which is given in (3.1), are uncorrelated (e.g., the
cross-correlation like (4.6) equals zero, and condition (vi) is not needed). As in proof of Lemma 6, the convergence
(3.2) still holds. The components $\varGamma_{1},\dots,\varGamma_{5}$ of
$\varGamma$ are independent because the components of $W_{i}$ are uncorrelated.
Proof of Theorem 8(a)
Our reasoning is typical for theory of generalized estimating
equations, with specific feature that a matrix parameter rather than
vector one is estimated.
By Corollary 4(a), with probability tending to $1$ we have
$$\sum_{i=1}^{m}s(a_{i},b_{i};\hat{X}_{\mathit{tls}})=0.$$
(4.7)
Now, we use Taylor’s formula around $X_{0}$ with the remainder in the
Lagrange form; see [1], Theorem 5.6.2. Denote
$$\hat{\Delta}=\sqrt{m}(\hat{X}_{\mathit{tls}}-X_{0}),\qquad y_{m}=\sum_{i=1}^{m}s(a_{i},b_{i};X_{0}),\qquad U_{m}=\sum_{i=1}^{m}s_{X}^{\prime}(a_{i},b_{i};X_{0}).$$
Then (4.7) implies the relation
$$\displaystyle\biggl{(}\frac{1}{m}U_{m}\biggr{)}\hat{\Delta}=-\frac{1}{\sqrt{m}}y_{m}+\mathit{rest}_{1},$$
(4.8)
$$\displaystyle\|\mathit{rest}_{1}\|\leq\|\hat{\Delta}\|\cdot\|\hat{X}_{\mathit{tls}}-X_{0}\|\cdot O_{p}(1).$$
Here $O_{p}(1)$ is a factor of the form
$$\frac{1}{m}\sum_{i=1}^{m}\sup_{(\|X\|\leq\|X_{0}\|+1)}\big{\|}s_{x}^{\prime\prime}(a_{i},b_{i};X)\big{\|}.$$
(4.9)
Relation (4.8) holds with probability tending to $1$ because, due
to Theorem 2, $\hat{X}_{\mathit{tls}}\stackrel{{\scriptstyle\text{\rm P}}}{{\longrightarrow}}X_{0}$; expression (4.9) is indeed $O_{p}(1)$ because the derivative $s_{x}^{\prime\prime}$
is quadratic in $a_{i}$, $b_{i}$ (cf. (4.5)), and the averaged second
moments of $[a_{i}^{\tr},b_{i}^{\tr}]$ are assumed to be bounded.
Now, $\|\mathit{rest}_{1}\|\leq\|\hat{\Delta}\|\cdot o_{p}(1)$. Next, by Lemma 5 and condition (iii),
$$\frac{1}{m}U_{m}=\frac{1}{m}\M U_{m}+o_{p}(1)=V_{A}+o_{p}(1).$$
Therefore, (4.8) implies that
$$\displaystyle V_{A}\hat{\Delta}=-\frac{1}{\sqrt{m}}y_{m}+\mathit{rest}_{2},$$
(4.10)
$$\displaystyle\|\mathit{rest}_{2}\|\leq\|\hat{\Delta}\|\cdot o_{p}(1).$$
(4.11)
Now, we find the limit in distribution of $y_{m}/\sqrt{m}$. The summands
in $y_{m}$ have zero expectation due to Corollary 4(b).
Moreover (see (2.8)),
$$s(a_{i},b_{i};X_{0})=(a_{0i}+\tilde{a}_{i})\bigl{(}\tilde{a}_{i}^{\tr}X_{0}-\tilde{b}_{i}^{\tr}\bigr{)}-X_{0}\bigl{(}\I_{d}+X_{0}^{\tr}X_{0}\bigr{)}^{-1}\bigl{(}X_{0}^{\tr}\tilde{a}_{i}-\tilde{b}_{i}\bigr{)}\bigl{(}\tilde{a}_{i}^{\tr}X_{0}-\tilde{b}_{i}^{\tr}\bigr{)},\vspace{-3pt}$$
$$\displaystyle s(a_{i},b_{i};X_{0})$$
$$\displaystyle=W_{i1}X_{0}-W_{i2}+W_{i3}X_{0}-W_{i4}-X_{0}\bigl{(}\I_{d}+X_{0}^{\tr}X_{0}\bigr{)}^{-1}$$
$$\displaystyle\quad\times\bigl{(}X_{0}^{\tr}W_{i3}X_{0}-X_{0}^{\tr}W_{i4}-W_{i4}^{\tr}X_{0}+W_{i5}\bigr{)}.$$
Here $W_{ij}$ are the components of (3.1). By Lemma 6 we
have (see (3.4))
$$\frac{1}{\sqrt{m}}y_{m}\stackrel{{\scriptstyle\text{\rm d}}}{{\longrightarrow}}\varGamma(X_{0})\quad\text{as }m\to\infty.$$
(4.12)
Finally, relations (4.10), (4.11), (4.12) and the
nonsingularity of $V_{A}$ imply that $\hat{\Delta}=O_{p}(1)$, and by
Slutsky’s lemma we get
$$V_{A}\hat{\Delta}\stackrel{{\scriptstyle\text{\rm d}}}{{\longrightarrow}}\varGamma(X_{0})\quad\text{as }m\to\infty.$$
(4.13)
By condition (iii) the matrix $V_{A}$ is nonsingular. Thus, the
desired relation (3.3) follows from (4.13).
Proof of Theorem 8(b)
The convergence (3.3) is justified as before, but using Lemma 7 instead of Lemma 6. It suffices to show that $\cov(\varGamma(X_{0})u)$ is nonsingular for $u\in\mathbb{R}^{d\times 1}$, $u\neq 0$.
Now, the components $\varGamma_{1},\dots,\varGamma_{5}$ are independent. Then
(see (3.4))
$$\displaystyle\cov\bigl{(}\varGamma(X_{0})u\bigr{)}$$
$$\displaystyle\geq\cov(\varGamma_{2}u)=\lim_{m\to\infty}\frac{1}{m}\sum_{i=1}^{m}\M\bigl{(}u^{\tr}\tilde{b}_{i}a_{0i}^{\tr}a_{0i}\tilde{b}_{i}^{\tr}u\bigr{)}$$
$$\displaystyle=\mathrm{tr}V_{A}\cdot\M\big{\|}\tilde{b}_{1}^{\tr}u\big{\|}^{2}=\sigma^{2}\rm{tr}V_{A}\cdot\|u\|^{2}>0.$$
Proof of Lemma 10
By condition (i) we have
$$\displaystyle\M a_{i}a_{i}^{\tr}$$
$$\displaystyle=a_{0i}a_{0i}^{\tr}+\sigma^{2}\I_{n},\qquad\M a_{i}b_{i}^{\tr}=a_{i0}a_{i0}^{\tr}X_{0},$$
$$\displaystyle\M b_{i}b_{i}^{\tr}$$
$$\displaystyle=X_{0}^{\tr}a_{0i}a_{0i}^{\tr}X_{0}+\sigma^{2}\I_{d},$$
$$\M b_{i}b_{i}^{\tr}-2X_{0}^{\tr}\M a_{i}b_{i}^{\tr}+X_{0}^{\tr}\bigl{(}\M a_{i}a_{i}^{\tr}\bigr{)}X_{0}=\sigma^{2}\bigl{(}\I_{d}+X_{0}^{\tr}X_{0}\bigr{)}.$$
(4.14)
Equality (4.14) implies the first relation in (3.6) because
$\hat{X}_{\mathit{tls}}\stackrel{{\scriptstyle\text{\rm P}}}{{\longrightarrow}}X_{0}$ and $\overline{aa^{\tr}}-\M\overline{aa^{\tr}}\stackrel{{\scriptstyle\text{\rm P}}}{{\longrightarrow}}0$, $\overline{ab^{\tr}}-\M\overline{ab^{\tr}}\stackrel{{\scriptstyle\text{\rm P}}}{{\longrightarrow}}0$,
$\overline{bb^{\tr}}-\M\overline{bb^{\tr}}\stackrel{{\scriptstyle\text{\rm P}}}{{\longrightarrow}}0$,
Finally,
$$\displaystyle\hat{V}_{A}=\M\overline{aa^{\tr}}+o_{p}(1)-\hat{\sigma}^{2}\I_{n}=\overline{a_{0}a_{0}^{\tr}}+(\sigma^{2}-\hat{\sigma}^{2})\I_{n}+o_{p}(1),$$
$$\displaystyle\quad\hat{V}_{A}\stackrel{{\scriptstyle\text{\rm P}}}{{\longrightarrow}}\displaystyle\lim_{m\to\infty}\overline{a_{0}a_{0}^{\tr}}=V_{A}.$$
References
[1]
{bbook}
\bauthor\bsnmCartan, \binitsH.:
\bbtitleDifferential Calculus.
\bpublisherHermann/Houghton Mifflin Co.,
\blocationParis/Boston, MA
(\byear1971).
\bcommentTranslated from French.
\bidmr=0344032
\OrigBibText{bbook}
\bauthor\bsnmCartan, \binitsH.:
\bbtitleDifferential Calculus.
\bpublisherHermann, Paris; Houghton Mifflin Co., Boston, Mass.
(\byear1971).
\bcommentTranslated from French.
\MR0344032
\endOrigBibText\bptokstructpyb\endbibitem
[2]
{bbook}
\bauthor\bsnmCheng, \binitsC.-L.,
\bauthor\bsnmVan Ness, \binitsJ.W.:
\bbtitleStatistical Regression with Measurement Error.
\bsertitleKendall’s Library of Statistics,
vol. \bseriesno6.
\bpublisherArnold,
\blocationLondon
(\byear1999).
\bcommentCo-published by Oxford University Press, New York.
\bidmr=1719513
\OrigBibText{bbook}
\bauthor\bsnmCheng, \binitsC.-L.,
\bauthor\bsnmVan Ness, \binitsJ.W.:
\bbtitleStatistical Regression with Measurement Error.
\bsertitleKendall’s Library of Statistics,
vol. \bseriesno6.
\bpublisherArnold, London; co-published by Oxford University Press,
New York
(\byear1999).
\MR1719513
\endOrigBibText\bptokstructpyb\endbibitem
[3]
{bbook}
\bauthor\bsnmFuller, \binitsW.A.:
\bbtitleMeasurement Error Models.
\bpublisherJohn Wiley & Sons, Inc.,
\blocationNew York
(\byear1987).
\biddoi=10.1002/9780470316665, mr=0898653
\OrigBibText{bbook}
\bauthor\bsnmFuller, \binitsW.A.:
\bbtitleMeasurement Error Models.
\bpublisherJohn Wiley & Sons, Inc., New York
(\byear1987).
doi:\doiurl10.1002/9780470316665.
\MR898653
\endOrigBibText\bptokstructpyb\endbibitem
[4]
{botherref}
\oauthor\bsnmGallo, \binitsP.P.:
Properties of estimators errors-in-variables models.
PhD thesis,
The University of North Carolina at Chapel Hill, NC
(1982).
\MR2632121
\OrigBibText{botherref}
\oauthor\bsnmGallo, \binitsP.P.:
Properties of estimators errors-in-variables models.
PhD thesis,
The University of North Carolina at Chapel Hill, NC
(1982).
\MR2632121
\endOrigBibText\bptokstructpyb\endbibitem
[5]
{barticle}
\bauthor\bsnmKukush, \binitsA.,
\bauthor\bsnmVan Huffel, \binitsS.:
\batitleConsistency of elementwise-weighted total least squares
estimator in a
multivariate errors-in-variables model $AX=B$.
\bjtitleMetrika
\bvolume59(\bissue1),
\bfpage75–\blpage97
(\byear2004).
\biddoi=10.1007/s001840300272, mr=2043433
\OrigBibText{barticle}
\bauthor\bsnmKukush, \binitsA.,
\bauthor\bsnmVan Huffel, \binitsS.:
\batitleConsistency of elementwise-weighted total least squares
estimator in a
multivariate errors-in-variables model $AX=B$.
\bjtitleMetrika
\bvolume59(\bissue1),
\bfpage75–\blpage97
(\byear2004).
doi:\doiurl10.1007/s001840300272.
\MR2043433
\endOrigBibText\bptokstructpyb\endbibitem
[6]
{barticle}
\bauthor\bsnmMarkovsky, \binitsI.,
\bauthor\bsnmRastello, \binitsM.L.,
\bauthor\bsnmPremoli, \binitsA.,
\bauthor\bsnmKukush, \binitsA.,
\bauthor\bsnmVan Huffel, \binitsS.:
\batitleThe element-wise weighted total least-squares problem.
\bjtitleComput. Stat. Data Anal.
\bvolume50(\bissue1),
\bfpage181–\blpage209
(\byear2006).
\biddoi=10.1016/j.csda.2004.07.014, mr=2196229
\OrigBibText{barticle}
\bauthor\bsnmMarkovsky, \binitsI.,
\bauthor\bsnmRastello, \binitsM.L.,
\bauthor\bsnmPremoli, \binitsA.,
\bauthor\bsnmKukush, \binitsA.,
\bauthor\bsnmVan Huffel, \binitsS.:
\batitleThe element-wise weighted total least-squares problem.
\bjtitleComput. Statist. Data Anal.
\bvolume50(\bissue1),
\bfpage181–\blpage209
(\byear2006).
doi:\doiurl10.1016/j.csda.2004.07.014.
\MR2196229
\endOrigBibText\bptokstructpyb\endbibitem
[7]
{barticle}
\bauthor\bsnmPešta, \binitsM.:
\batitleAsymptotics for weakly dependent errors-in-variables.
\bjtitleKybernetika
\bvolume49(\bissue5),
\bfpage692–\blpage704
(\byear2013).
\bidmr=3182634
\OrigBibText{barticle}
\bauthor\bsnmPešta, \binitsM.:
\batitleAsymptotics for weakly dependent errors-in-variables.
\bjtitleKybernetika (Prague)
\bvolume49(\bissue5),
\bfpage692–\blpage704
(\byear2013).
\MR3182634
\endOrigBibText\bptokstructpyb\endbibitem
[8]
{barticle}
\bauthor\bsnmSprent, \binitsP.:
\batitleA generalized least-squares approach to linear functional
relationships.
\bjtitleJ. R. Stat. Soc. B
\bvolume28,
\bfpage278–\blpage297
(\byear1966).
\bidmr=0230432
\OrigBibText{barticle}
\bauthor\bsnmSprent, \binitsP.:
\batitleA generalized least-squares approach to linear functional
relationships.
\bjtitleJ. Roy. Statist. Soc. Ser. B
\bvolume28,
\bfpage278–\blpage297
(\byear1966).
\MR0230432
\endOrigBibText\bptokstructpyb\endbibitem
[9]
{bbook}
\bauthor\bsnmVan Huffel, \binitsS.,
\bauthor\bsnmVandewalle, \binitsJ.:
\bbtitleThe Total Least Squares Problem.
\bsertitleFrontiers in Applied Mathematics,
vol. \bseriesno9.
\bpublisherSociety for Industrial and Applied Mathematics (SIAM),
\blocationPhiladelphia, PA
(\byear1991).
\biddoi=10.1137/1.9781611971002, mr=1118607
\OrigBibText{bbook}
\bauthor\bsnmVan Huffel, \binitsS.,
\bauthor\bsnmVandewalle, \binitsJ.:
\bbtitleThe Total Least Squares Problem.
\bsertitleFrontiers in Applied Mathematics,
vol. \bseriesno9.
\bpublisherSociety for Industrial and Applied Mathematics (SIAM),
Philadelphia, PA
(\byear1991).
doi:\doiurl10.1137/1.9781611971002.
\MR1118607
\endOrigBibText\bptokstructpyb\endbibitem |
Inflation in the Mixed Higgs-$R^{2}$ Model
[
[
[
Abstract
We analyze a two-field inflationary model consisting of the Ricci scalar squared ($R^{2}$) term and the standard Higgs field non-minimally coupled to gravity in addition to the Einstein $R$ term. Detailed analysis of the power spectrum of this model with mass hierarchy is presented, and we find that one can describe this model as an effective single-field model in the slow-roll regime with a modified sound speed. The scalar spectral index predicted by this model coincides with those given by the $R^{2}$ inflation and the Higgs inflation implying that there is a close relation between this model and the $R^{2}$ inflation already in the original (Jordan) frame. For a typical value of the self-coupling of the standard Higgs field at the high energy scale of inflation, the role of the Higgs field in parameter space involved is to modify the scalaron mass, so that the original mass parameter in the $R^{2}$ inflation can deviate from its standard value when non-minimal coupling between the Ricci scalar and the Higgs field is large enough.
1,2]Minxi He,
2,3]Alexei A. Starobinsky,
1,2,4]Jun’ichi Yokoyama
Inflation in the Mixed Higgs-$R^{2}$ Model
${}^{1}$
Department of Physics, Graduate School of Science,
The University of Tokyo, Tokyo 113-0033, Japan
${}^{2}$
Research Center for the Early Universe (RESCEU), Graduate School
of Science,
The University of Tokyo, Tokyo 113-0033, Japan
${}^{3}$
L. D. Landau Institute for Theoretical Physics, Moscow 119334, Russia
${}^{4}$
Kavli Institute for the Physics and Mathematics
of the Universe (Kavli IPMU), WPI, UTIAS,
The University of Tokyo, Kashiwa, Chiba 277-8568, Japan
E-mail: hemxzero@resceu.s.u-tokyo.ac.jp, alstar@landau.ac.ru, yokoyama@resceu.s.u-tokyo.ac.jp
Contents
1 Introduction
2 Lagrangian and Equations of Motion
3 Slow-Roll Inflation and Curvature Perturbations
3.1 Features of the Potential
3.2 Slow-Roll Inflation
3.3 Predictions for observations
4 To the Effective $R^{2}$ Inflation
4.1 Relation between $\xi$ and $M$
4.2 Explanation
5 Conclusion and Outlook
1 Introduction
A number of single-field models have been proposed [1, 2, 3, 4, 5, 6] since 1980s, some of them are in good agreement with the observation of cosmic microwave background (CMB) [7], such as the $R+R^{2}$ inflationary model (the $R^{2}$ one for brevity) [1] which is often called the Starobinsky model, and the original Higgs inflationary model[8, 9, 10]
in which the scalar field is strongly non-minimally coupled to the Ricci scalar 111See
[11] for a summary of all variants of the Higgs inflationary model. The $R^{2}$ added to the Einstein-Hilbert action yields an effective dynamical scalar field, scalaron realizing a quasi-de Sitter stage in the early universe while the Higgs boson in the standard model, with the help of non-minimal coupling to gravity, $\xi\chi^{2}R$, plays an essential role as an inflaton to drive inflation in the Higgs inflationary model. Both models produce the same spectral spectral index of primordial scalar (adiabatic density) perturbations which is supported by recent CMB observations. Meanwhile, the tensor-to-scalar ratio given by these two models has an amplitude though small, but still hopefully detectable in the future.
Due to the excellent performance of the $R^{2}$ inflation and the Higgs inflationary model, it is natural and more realistic to consider the extension of such single-field models to multi-field inflation by the combination of them which we consider in this paper. Multi-field inflation is a class of cosmological inflationary models with a de Sitter stage produced by more than one effective scalar fields among which two-field models constitute a special case. In multi-field inflationary models, only one linear combination of the scalar fields is responsible for the inflationary stage and consequently quantum fluctuations produced in this direction serve as adiabatic perturbations which finally grow to become the seeds of inhomogeneities seen in CMB temperature anisotropy and polarization and producing the large scale structure and compact objects in the universe. The other independent combinations are, on the other hand, responsible for production of isocurvature perturbations [12] and some other possible features [13]. Isocurvature modes represent the unique feature of multi-field models distinguishing them from single-field ones. They can survive to the present only under special conditions [14]. Also, in the presence of non-minimal coupling, recent research [15] points out that the preheating process after inflation becomes much more violent than the case without it.
In this paper, we investigate Higgs-$R^{2}$ inflation, namely the combination of the Higgs inflation and the $R^{2}$ inflation, in a certain part of the parameter space. For realistic values of the Higgs self coupling we find the presence of mass hierarchy and the appearance of effectively single-field slow-roll inflation in the original Jordan frame. We write down the effective single-field action to quadratic level for this model and use it to calculate the power spectrum of curvature perturbations. We find that this two-field model can be treated as an effective $R^{2}$ inflation with a modified scalaron mass. In Sec 2, we introduce the basic details of the model. We calculate the power spectrum in Sec 3 and discuss the effective $R^{2}$ inflation in Sec 4. Our conclusions and outlook are presented in Sec 5.
2 Lagrangian and Equations of Motion
The action considered here is given in the original Jordan frame, where the space-time metric is denoted as $\hat{g}_{\mu\nu}$, by
$$\displaystyle S_{\text{J}}$$
$$\displaystyle=\int d^{4}x\sqrt{-\hat{g}}\left[\frac{M_{p}^{2}}{2}\hat{R}+\frac%
{1}{2}\xi\chi^{2}\hat{R}+\frac{M_{p}^{2}}{12M^{2}}\hat{R}^{2}-\frac{1}{2}\hat{%
g}^{\mu\nu}\hat{\nabla}_{\mu}\chi\hat{\nabla}_{\nu}\chi-\frac{\lambda}{4}\chi^%
{4}\right]$$
(2.1)
$$\displaystyle=\int d^{4}x\sqrt{-\hat{g}}\left[F(\chi,\hat{R})-\frac{1}{2}\hat{%
g}^{\mu\nu}\hat{\nabla}_{\mu}\chi\hat{\nabla}_{\nu}\chi\right]$$
(2.2)
where $M_{p}\equiv(8\pi G)^{-1/2}$ and $\chi$ is a singlet scalar field, a simplified model of the Standard Model Higgs boson. We neglect its interaction to gauge fields. Here $F(\chi,\hat{R})$ is defined by
$$\displaystyle F(\chi,\hat{R})\equiv\frac{M_{p}^{2}}{2}\hat{R}+\frac{1}{2}\xi%
\chi^{2}\hat{R}+\frac{M_{p}^{2}}{12M^{2}}\hat{R}^{2}-\frac{\lambda}{4}\chi^{4}%
~{}.$$
(2.3)
This action was recently considered in [16, 17]. $\chi$ has a non-minimal coupling term with the Ricci scalar. We take the sign of the non-minimal coupling constant $\xi$ such that the conformal coupling corresponds to $\xi=-1/6$.
Defining the scalaron field as [18, 19]
$$\displaystyle\sqrt{\frac{2}{3}}\frac{\psi}{M_{p}}\equiv\ln(\frac{2}{M^{2}_{p}}%
\absolutevalue{\frac{\partial F}{\partial\hat{R}}})~{},$$
(2.4)
and performing a conformal transformation
$$\displaystyle g_{\mu\nu}(x)=e^{\sqrt{\frac{2}{3}}\frac{\psi(x)}{M_{p}}}\hat{g}%
_{\mu\nu}(x)~{},$$
(2.5)
we can transform the original action (2.1) into the one in the
Einstein frame and express the new action in terms of the new scalar fields as
$$\displaystyle S_{\text{E}}=\int d^{4}x\sqrt{-g}\left[\frac{M_{p}^{2}}{2}R-%
\frac{1}{2}g^{\mu\nu}\nabla_{\mu}\psi\nabla_{\nu}\psi-\frac{1}{2}e^{-\sqrt{%
\frac{2}{3}}\frac{\psi}{M_{p}}}g^{\mu\nu}\nabla_{\mu}\chi\nabla_{\nu}\chi-U(%
\psi,\chi)\right]~{},$$
(2.6)
where the potential is expressed as
$$\displaystyle U(\psi,\chi)\equiv\frac{\lambda}{4}\chi^{4}e^{-2\sqrt{\frac{2}{3%
}}\frac{\psi}{M_{p}}}+\frac{3}{4}M_{p}^{2}M^{2}e^{-2\sqrt{\frac{2}{3}}\frac{%
\psi}{M_{p}}}\left(e^{\sqrt{\frac{2}{3}}\frac{\psi}{M_{p}}}-1-\frac{1}{M_{p}^{%
2}}\xi\chi^{2}\right)^{2}~{}.$$
(2.7)
In addition to the metric and Higgs field $\chi$, $\psi$ is the third dynamical field which originates from the $R^{2}$ term. Note that $\psi$ shows up only in the exponent with an $\mathcal{O}(1)$ numerical factor, so that large values of $\psi$ will significantly suppress the terms with higher orders in $\exp(-\sqrt{2/3}\psi/M_{p})$.
The kinetic terms of the two scalar fields in the Einstein frame are coupled. This means that the field space spanned by these two fields is not flat. Following [20, 21], we introduce an induced metric of the field space and rewrite this system in a more compact way as
$$\displaystyle S_{\text{E}}=\int d^{4}x\sqrt{-g}\left[\frac{M_{p}^{2}}{2}R-%
\frac{1}{2}h_{ab}g^{\mu\nu}\nabla_{\mu}\phi^{a}\nabla_{\nu}\phi^{b}-U(\phi)%
\right],$$
(2.8)
where
$$\displaystyle\phi^{1}=\psi~{},\quad\phi^{2}=\chi~{},\quad h_{ab}=h_{ab}(\psi)=%
\begin{pmatrix}1&0\\
0&e^{-\sqrt{\frac{2}{3}}\frac{\psi}{M_{p}}}.\\
\end{pmatrix}$$
(2.9)
Here the Latin indices $a,b=1,2$ represent components in field space and the Greek indices $\mu,\nu=0,1,2,3$ denote space-time components.
We take the spatially flat Robertson-Walker metric $ds^{2}=-dt^{2}+a^{2}(t)\delta_{ij}dx^{i}dx^{j}$
as the background $i,j=1,2,3$) and split all fields into homogeneous background parts and small space-time-dependent perturbations,
$$\displaystyle\phi^{a}(\mathbf{x},t)=\phi^{a}_{0}(t)+\delta\phi^{a}(\mathbf{x},%
t)~{},$$
(2.10)
incorporating scalar metric perturbations in the spatially flat gauge.
Then equations of motion for both background and perturbations are given as follows.
$$\displaystyle H^{2}=\frac{1}{3M^{2}_{p}}\left[\frac{1}{2}h_{ab}\dot{\phi}^{a}%
\dot{\phi}^{b}+U(\phi)\right]~{},$$
(2.11)
$$\displaystyle\frac{D\dot{\phi}^{a}_{0}}{dt}+3H\dot{\phi}^{a}_{0}+h^{ab}U_{,b}=%
0~{},$$
(2.12)
$$\displaystyle\frac{D^{2}\delta\phi^{a}_{\mathbf{k}}}{dt^{2}}+3H\frac{D\delta%
\phi^{a}_{\mathbf{k}}}{dt}-R^{a}_{\;bcd}\dot{\phi}^{b}_{0}$$
$$\displaystyle\dot{\phi}^{c}_{0}\delta\phi^{d}_{\mathbf{k}}+\frac{k^{2}}{a^{2}}%
\delta\phi^{a}_{\mathbf{k}}+U^{;a}_{\;\;;b}\delta\phi^{b}_{\mathbf{k}}=\frac{1%
}{a^{3}}\frac{D}{dt}\left(\frac{a^{3}}{H}\dot{\phi}^{a}_{0}\dot{\phi}^{b}_{0}%
\right)h_{bc}\delta\phi^{c}_{\mathbf{k}}~{},$$
(2.13)
where $DX^{a}=dX^{a}+\Gamma^{a}_{bc}X^{b}d\phi^{c}_{0}$ is analogous to the directional derivative in curved spacetime and $\Gamma^{a}_{bc}=\frac{1}{2}h^{ad}(\partial_{b}h_{cd}+\partial_{c}h_{db}-%
\partial_{d}h_{bc})$ is the Christoffel symbol for the curved field space. It is easy to show that $\frac{D}{dt}=\dot{\phi}^{a}_{0}\nabla_{a}$. Note that the equations of motion for perturbations have already been transformed into those for spatial Fourier modes. The equations of motion of the scalar fields can be regarded as modified geodesic equations in the curved field space. The first term in (2.12) is just the ordinary geodesic equation while the second and the third term represent modifications from cosmic expansion and the scalar field potential, respectively.
Correspondingly, the field perturbation equations (2.13) can be regarded as geodesic deviation. Their equations of motion are also modified by cosmic expansion and the potential.
With all the effects taken into account, the trajectory of the fields traces neither the geodesics in the curved field space nor the bottom of the valley of the potential as postulated in [17]. Note that since generally the trajectory take turns during inflation, we also expect that there will be effects due to the turning.
3 Slow-Roll Inflation and Curvature Perturbations
In this paper, we mainly focus on the parameter regime where $\xi>0$ and fix the self coupling
at a typical value $\lambda=0.01$ from phenomenology.
We will also briefly discuss the situation when $\xi<0$ at the end.
\@fb@secFB
3.1 Features of the Potential
Here we give two examples for different combinations of $\xi$ and $M$ in Figure 1. The potential (2.7) is invariant under $\chi\rightarrow-\chi$. One can calculate the effective mass of the Higgs field, $m^{2}_{\chi}$, by taking derivatives of the potential. The dominant contribution in small $\chi$ regime comes from a term proportional to $\xi$, $-3\xi M^{2}\exp(-\sqrt{2/3}\psi/M_{p})$. For positive $\xi$, Higgs field obtains a negative $m^{2}_{\chi}$ around the origin where its amplitude will grow exponentially. In large $\chi$ regime, $m^{2}_{\chi}$ is dominated by a term proportional to $\chi^{2}$, $3\lambda(1+3\xi^{2}M^{2}/\lambda M^{2}_{p})\exp(-2\sqrt{2/3}\psi/M_{p})\chi^{2}$, whose coefficient is always positive. These properties imply the existence of a local minimum on the potential for a given $\psi$ which corresponds to the valleys in Figure 1. Thus, independent of the initial position of $\chi$, with a large $\xi$, the Higgs field will quickly fall into one of the valleys and evolve around the local minimum. If $\xi$ takes a small value, i.e. $m^{2}_{\chi}$ is small, $\chi$ direction will become flatter. In this case, if the initial conditions start from a large $\chi$ value, it is possible for Higgs field to slowly roll down the potential wall which is similar to the
situation discussed in [23]. As for $\psi$ direction, it is always flat in large $\psi$ regime so that it has similar behavior to the scalaron in the $R^{2}$ inflation. As we shall see later, there is a turning in the trajectory which can affect the sound speed of the curvature perturbations during inflationary phase. The angular velocity at this turning is not large in the parameter regime we consider here, though. After the end of inflation, the fields will oscillate around the global minimum of the potential at $(\chi,\psi)=(0,0)$ where reheating is expected to happen. According to the recent work [15], the particle production during preheating will be violent due to the appearance of non-minimal coupling between Higgs and gravity.
\@fb@secFB
3.2 Slow-Roll Inflation
As mentioned in previous sections, the evolution trajectory of two scalar fields are affected by the curved nature of the field space, the potential shape and the expansion of the universe. Thus, it would be more convenient to discuss the features of this trajectory by defining unit vectors $T^{a}$ and $N^{a}$ [24] as
$$\displaystyle T^{a}\equiv\frac{\dot{\phi}^{a}_{0}}{\dot{\phi}_{0}}$$
$$\displaystyle=\frac{1}{\sqrt{\dot{\psi}^{2}+e^{-\sqrt{\frac{2}{3}}\frac{\psi}{%
M_{p}}}\dot{\chi}^{2}}}(\dot{\psi},\dot{\chi}),$$
(3.1)
$$\displaystyle\dot{\theta}N^{a}$$
$$\displaystyle\equiv-\frac{DT^{a}}{dt}~{},$$
(3.2)
which are tangent and normal to the trajectory, respectively. Here we denote $\dot{\phi}^{2}_{0}\equiv h_{ab}\dot{\phi}^{a}_{0}\dot{\phi}^{b}_{0}$ and $\dot{\theta}$ is the angular velocity describing the turning in the trajectory which, according to the normalization condition, is given by
$$\displaystyle\dot{\theta}^{2}$$
$$\displaystyle=h_{ab}\frac{DT^{a}}{dt}\frac{DT^{b}}{dt}$$
(3.3)
$$\displaystyle=e^{\sqrt{\frac{2}{3}}\frac{\psi}{M_{p}}}\frac{\left(\frac{%
\partial U}{\partial\chi}\dot{\psi}-e^{-\sqrt{\frac{2}{3}}\frac{\psi}{M_{p}}}%
\frac{\partial U}{\partial\psi}\dot{\chi}\right)^{2}}{\left(\dot{\psi}^{2}+e^{%
-\sqrt{\frac{2}{3}}\frac{\psi}{M_{p}}}\dot{\chi}^{2}\right)^{2}}~{},$$
(3.4)
so that $N^{a}$ is explicitly given by
$$\displaystyle N^{a}=\frac{e^{\sqrt{\frac{1}{6}}\frac{\psi}{M_{p}}}}{\left(\dot%
{\psi}^{2}+e^{-\sqrt{\frac{2}{3}}\frac{\psi}{M_{p}}}\dot{\chi}^{2}\right)^{1/2%
}}(-e^{-\sqrt{\frac{2}{3}}\frac{\psi}{M_{p}}}\dot{\chi},\dot{\psi})~{}.$$
(3.5)
We define the slow-roll parameters analogous to the single-field case as
$$\displaystyle\epsilon$$
$$\displaystyle\equiv-\frac{\dot{H}}{H^{2}}=\frac{\dot{\phi}^{2}_{0}}{2M^{2}_{p}%
H^{2}}~{},$$
(3.6)
$$\displaystyle\eta^{a}$$
$$\displaystyle\equiv-\frac{1}{H\dot{\phi}_{0}}\frac{D\dot{\phi}^{a}_{0}}{dt}~{}.$$
(3.7)
Note that $\eta^{a}$ is no longer a scalar but a vector which means that one needs two different $\eta$s to describe the evolution of these two different directions. Using the unit vectors, one can easily obtain an $\eta$ for each direction as
$$\displaystyle\eta^{a}$$
$$\displaystyle=\eta_{||}T^{a}+\eta_{\perp}N^{a}~{},$$
(3.8)
$$\displaystyle\eta_{||}$$
$$\displaystyle\equiv-\frac{\ddot{\phi}_{0}}{H\dot{\phi}_{0}}~{},$$
(3.9)
$$\displaystyle\eta_{\perp}$$
$$\displaystyle\equiv\frac{U_{N}}{\dot{\phi}_{0}H}~{},$$
(3.10)
where $U_{N}\equiv N^{a}U_{,a}$. Slow-roll inflation requires that $\epsilon\ll 1$ and $\eta_{||}\ll 1$. Note that the slow-roll requirement does not impose any constraint on $\eta_{\perp}$ which means that it can be large. Then the angular velocity $\dot{\theta}$ can also be expressed in terms of the slow-roll parameter as
$$\displaystyle\dot{\theta}=H\eta_{\perp}~{}.$$
(3.11)
Since $\eta_{\perp}$ can be large, we may expect $\dot{\theta}$ to be large as well. However, this does not spoil the validity of the effective field theory used below [24] as long as the adiabatic condition, $|\ddot{\theta}/\dot{\theta}^{2}|\ll M_{\text{eff}}$, is satisfied.
We now consider perturbations in this formalism. In flat gauge, the comoving curvature perturbation and the isocurvature perturbation are defined as [24]
$$\displaystyle\mathcal{R}$$
$$\displaystyle\equiv-\frac{H}{\dot{\phi}_{0}}\delta\phi^{a}T_{a}~{},$$
(3.12)
$$\displaystyle\mathcal{F}$$
$$\displaystyle\equiv N_{a}\delta\phi^{a}~{}.$$
(3.13)
Expanding the perturbed action to second order, we find
$$\displaystyle S_{2}=\frac{1}{2}\int d^{4}x~{}a^{3}\left[\frac{\dot{\phi}^{2}_{%
0}}{H^{2}}\dot{\mathcal{R}}^{2}-\frac{\dot{\phi}^{2}_{0}}{H^{2}}\frac{(\nabla%
\mathcal{R})^{2}}{a^{2}}+\dot{\mathcal{F}}^{2}-\frac{(\nabla\mathcal{F})^{2}}{%
a^{2}}-M^{2}_{\text{eff}}\mathcal{F}^{2}-4\dot{\theta}\frac{\dot{\phi}_{0}}{H}%
\dot{\mathcal{R}}\mathcal{F}\right]$$
(3.14)
from which it is clear that the curvature perturbations evolve along the light (massless) direction while the isocurvature modes have an effective mass $M^{2}_{\text{eff}}=U_{\text{NN}}+M^{2}_{p}\epsilon H^{2}R_{ab}h^{ab}-\dot{%
\theta}^{2}$ where $U_{NN}\equiv N^{a}N^{b}\nabla_{a}\nabla_{b}U$. The explicit form of $U_{NN}$ and $M^{2}_{\text{eff}}$ is given by
$$\displaystyle\begin{split}\displaystyle U_{NN}=\frac{1}{\dot{\psi}^{2}+e^{-%
\sqrt{\frac{2}{3}}\frac{\psi}{M_{p}}}\dot{\chi}^{2}}&\displaystyle\left(e^{-%
\sqrt{\frac{2}{3}}\frac{\psi}{M_{p}}}\dot{\chi}^{2}\frac{\partial^{2}U}{%
\partial\psi^{2}}+e^{\sqrt{\frac{2}{3}}\frac{\psi}{M_{p}}}\dot{\psi}^{2}\frac{%
\partial^{2}U}{\partial\chi^{2}}\right.\\
&\displaystyle\left.-2\dot{\psi}\dot{\chi}\frac{\partial^{2}U}{\partial\psi%
\partial\chi}-\frac{1}{\sqrt{6}}\dot{\psi}^{2}\frac{\partial U}{\partial\psi}-%
\sqrt{\frac{2}{3}}\dot{\psi}\dot{\chi}\frac{\partial U}{\partial\chi}\right)~{%
},\end{split}$$
(3.15)
$$\displaystyle\begin{split}\displaystyle M^{2}_{\text{eff}}=\frac{1}{\dot{\psi}%
^{2}+e^{-\sqrt{\frac{2}{3}}\frac{\psi}{M_{p}}}\dot{\chi}^{2}}&\displaystyle%
\left(e^{-\sqrt{\frac{2}{3}}\frac{\psi}{M_{p}}}\dot{\chi}^{2}\frac{\partial^{2%
}U}{\partial\psi^{2}}+e^{\sqrt{\frac{2}{3}}\frac{\psi}{M_{p}}}\dot{\psi}^{2}%
\frac{\partial^{2}U}{\partial\chi^{2}}-2\dot{\psi}\dot{\chi}\frac{\partial^{2}%
U}{\partial\psi\partial\chi}-\frac{1}{\sqrt{6}}\dot{\psi}^{2}\frac{\partial U}%
{\partial\psi}\right.\\
\displaystyle\left.-\sqrt{\frac{2}{3}}\dot{\psi}\dot{\chi}\frac{\partial U}{%
\partial\chi}\right)&\displaystyle-\frac{\dot{\psi}^{2}+e^{-\sqrt{\frac{2}{3}}%
\frac{\psi}{M_{p}}}\dot{\chi}^{2}}{6M^{2}_{p}}-e^{\sqrt{\frac{2}{3}}\frac{\psi%
}{M_{p}}}\frac{\left(\frac{\partial U}{\partial\chi}\dot{\psi}-e^{-\sqrt{\frac%
{2}{3}}\frac{\psi}{M_{p}}}\frac{\partial U}{\partial\psi}\dot{\chi}\right)^{2}%
}{\left(\dot{\psi}^{2}+e^{-\sqrt{\frac{2}{3}}\frac{\psi}{M_{p}}}\dot{\chi}^{2}%
\right)^{2}}~{},\end{split}$$
(3.16)
respectively. Thus, light modes and massive modes are separated. Integrating out the high energy degrees of freedom as in [24], the massive modes, $\mathcal{F}$, are completely determined by the massless modes, $\mathcal{R}$, so that one gets its effective action to quadratic order,
$$\displaystyle\mathcal{F}$$
$$\displaystyle=-\frac{\dot{\phi}_{0}}{H}\frac{2\dot{\theta}\dot{\mathcal{R}}}{k%
^{2}/a^{2}+M^{2}_{\text{eff}}}~{},$$
(3.17)
$$\displaystyle S_{\text{eff}}=\frac{1}{2}\int$$
$$\displaystyle d^{4}x~{}a^{3}\frac{\dot{\phi}^{2}_{0}}{H^{2}}\left[\frac{\dot{%
\mathcal{R}}^{2}}{c^{2}_{s}(k)}-\frac{k^{2}\mathcal{R}^{2}}{a^{2}}\right]~{}.$$
(3.18)
The appearance of the turning gives corrections to the sound speed, $c^{-2}_{s}(k)=1+4\dot{\theta}^{2}/(k^{2}/a^{2}+M^{2}_{\text{eff}})$ which is exact unity in single-field models. Therefore, the effective action obtained for curvature perturbations is that of a single-field theory with a modified sound speed. When $\dot{\theta}^{2}$ is close to $U_{NN}$, $c_{s}^{-2}\gg 1$, the effect of the turn in the trajectory becomes significant, so that the sound speed is largely modified. However, in the region of the parameter space we consider, the modification is not significant.
In this action, only the adiabatic mode appears but it does not mean that the heavy mode has no influence on the evolution of the adiabatic mode. Both light and heavy modes have high energy and low energy contributions. Integrating out the high energy part to get the low energy effective theory does not mean decoupling between light and heavy modes. As long as a turning exists, adiabatic and isocurvature modes couple with each other
and the isocurvature mode is forced to oscillate coherently with the light field at low frequency [24]. In the slow-roll regime, $\dot{\theta}$ is automatically small and slowly changing in time, so that approximately we can quantize the quadratic action considering the sound speed as a constant close to unity. As a result, the power spectrum is just
$$\displaystyle\Delta^{2}_{\mathcal{R}}=\frac{k^{3}}{2\pi^{2}}\mathcal{P}_{%
\mathcal{R}}(k)~{},$$
(3.19)
$$\displaystyle\mathcal{P}_{\mathcal{R}}(k)=\frac{H^{2}}{4c_{s}k^{3}}(1$$
$$\displaystyle+c^{2}_{s}k^{2}\tau^{2})\frac{2H^{2}}{\dot{\phi}^{2}_{0}}%
\xrightarrow[\text{large scale}]{}\frac{H^{2}}{4c_{s}k^{3}}\frac{1}{\epsilon}~%
{},$$
(3.20)
which gives the scalar index and the scalar-to-tensor ratio
$$\displaystyle n_{\text{s}}-1\equiv$$
$$\displaystyle\frac{d\ln\Delta^{2}_{\mathcal{R}}}{d\ln k}\approx 2\eta_{||}-4\epsilon$$
(3.21)
$$\displaystyle r\approx 16\epsilon c_{s}~{}.$$
(3.22)
These results are just like those in single-field models with modification from the non-trivial sound speed. However, as mentioned already, large modification is not expected because in the slow-roll regime as well as the region of the parameter space we consider, the sound speed does not deviate from unity too much.
\@fb@secFB
3.3 Predictions for observations
As we can see above, the dynamics as well as power spectrum are determined by three parameters, Higgs self-coupling $\lambda$, non-minimal coupling $\xi$ and scalaron mass $M$.
Fixing $\lambda=0.01$ and the amplitude of curvature perturbation at the pivot scale to be $2\times 10^{-9}$, we choose several groups of $\xi$ and $M$ to calculate $n_{\text{s}}$ and $r$. All the results are completely degenerate (Shown in Figure 2) with those of the $R^{2}$ inflation.
This should not be surprising because this two-field model is built from the $R^{2}$ inflation and the Higgs inflation both of which give predictions staying right at the center of the famous $n_{s}-r$ plot from Planck’s observational data in 2015 [7]. Also, due to the presence of mass hierarchy and considering slow-roll inflation, the effective theory of this model reduces to a single-field model with slightly modified sound speed as one can see above. Therefore, we should not expect this model to give predictions which largely deviate from those of the Higgs inflation or the $R^{2}$ inflation in this level.
4 To the Effective $R^{2}$ Inflation
\@fb@secFB
4.1 Relation between $\xi$ and $M$
The degeneracy phenomenon implies that with a fixed $\lambda=0.01$, the amplitude of curvature perturbations, $\mathcal{P}_{\mathcal{R}}(k)\simeq 2\times 10^{-9}$, gives a strong constraint on $\xi$ and $M$.
Varying $\xi$ from $0.1$ to $4000$, one obtains the relation depicted in Figure 3. For small enough $\xi$, $M$ remains almost constant at around $10^{-5}M_{p}$ which coincides with the case of the standard $R^{2}$ inflation. This situation holds until $\xi$ reaches $1000$ where $M$ starts to grow rapidly to compensate the change of $\xi$. In this logarithmic plot (Figure 3), the relationship between $M$ and $\xi$ can be approximately separated into two branches. One is $\xi\lesssim 1000$ where we can just take the scalaron mass $M$ to be the same as in the standard $R^{2}$ inflation. The other is $\xi\gtrsim 1000$ where the value of the scalaron mass in the $R^{2}$ inflation is no longer valid since the non-minimal coupling is so large that it modifies the model significantly. In order to maintain the amplitude of curvature perturbations, $M$ must take a much larger value.
So far we have qualitative understanding of this relation. We present more precise explanation in the following.
\@fb@secFB
4.2 Explanation
To explain this, we firstly have a look at the potential (2.7). In slow-roll regime, the Friedmann equation approximately gives
$$\displaystyle H^{2}\approx\frac{1}{3M^{2}_{p}}U(\psi,\chi)\approx\frac{1}{4}M^%
{2}\left(1-e^{-\sqrt{\frac{2}{3}}\frac{\psi}{M_{p}}}\right)^{2}$$
(4.1)
in large $\psi$ regime with small $\xi$ while
$$\displaystyle H^{2}\approx\frac{1}{3M^{2}_{p}}U(\psi,\chi)\approx\frac{1}{4}M^%
{2}\left(1-\frac{1}{M^{2}_{p}}\xi\chi^{2}e^{-\sqrt{\frac{2}{3}}\frac{\psi}{M_{%
p}}}\right)^{2}$$
(4.2)
in large $\psi$ regime with large $\xi$. In the case of (4.1), the second term in the parenthesis is negligible compared with unity so that Hubble parameter is just a constant completely determined by $M$. However, in the case of (4.2), the second term in the parenthesis is not negligible compared with unity which means that the second factor in (4.2) could possibly be much smaller than unity for large enough $\xi$. In order to preserve the amplitude of curvature perturbations which is determined by Hubble parameter and its derivatives as mentioned above, we need a larger value of $M$ to ”protect” the Hubble parameter from being too small.
Intuitively, the scalaron mass $M$ mainly controls the height of the hill in the middle of the potential and while the non-minimal coupling $\xi$ mainly controls the depth and position of valleys on both sides of the hill. For a given amplitude of curvature perturbations, we require the inflaton to slowly roll down a trajectory whose height is around a certain value during inflation. If $M$ is not too large that means that the height of the central hill is small, the inflaton is allowed to roll along the central region of the potential, e.g. the left panel in Figure 1. On the contrary, if $M$ is too large, one then needs a larger value of $\xi$ which would generate a deep valley on each side of the hill so that the inflaton can leave the too high hill top to roll along a trajectory which is of proper height to generate small enough curvature perturbations.
One may see this relation more clearly without any conformal transformation by considering the relation between this two-field model and the $R^{2}$ inflation directly in the Jordan frame. From the action in this frame (2.1), we find that the non-minimal coupling can be partially regarded as an extra contribution to the
kinetic term for Higgs field $\chi$ proportional to $\xi$ since there are second derivatives in Ricci scalar so that we can realize this by using integration by parts. Then, for large $\xi$ and $\psi$ regime, the original kinetic term in (2.1) is negligible. Dropping the kinetic term for $\chi$, the original action becomes
$$\displaystyle S_{\text{J}}\approx\tilde{S_{\text{J}}}=\int d^{4}x\sqrt{-\hat{g%
}}\left[\frac{M_{p}^{2}}{2}\hat{R}+\frac{1}{2}\xi\chi^{2}\hat{R}+\frac{M_{p}^{%
2}}{12M^{2}}\hat{R}^{2}-\frac{\lambda}{4}\chi^{4}\right]~{}.$$
(4.3)
As a result, the equation of motion for $\chi$ just becomes a constraint on $\chi$ [22] and $R$ as $\chi^{2}=\xi\hat{R}/\lambda$ which simplifies the action as
$$\displaystyle\tilde{S_{\text{J}}}$$
$$\displaystyle=\int d^{4}x\sqrt{-\hat{g}}\left[\frac{M_{p}^{2}}{2}\hat{R}+\left%
(\frac{M_{p}^{2}}{12M^{2}}+\frac{\xi^{2}}{4\lambda}\right)\hat{R}^{2}\right]$$
(4.4)
$$\displaystyle=\int d^{4}x\sqrt{-\hat{g}}\left[\frac{M_{p}^{2}}{2}\hat{R}+\frac%
{M_{p}^{2}}{12\tilde{M}^{2}}\hat{R}^{2}\right]$$
(4.5)
where
$$\displaystyle\tilde{M}^{2}\equiv\frac{M^{2}}{1+3\xi^{2}\frac{M^{2}}{\lambda M^%
{2}_{p}}}$$
(4.6)
is the effective mass squared of the scalaron, which should take $\tilde{M}=1.3\times 10^{-5}M_{p}$ [30, 31] to reproduce the observed amplitude of curvature perturbation.
From (4.6) we can understand two characteristic regimes of Figure 3. When the second term in the denominator is much smaller than unity, namely, $\xi\ll\sqrt{\frac{\lambda}{3}}\frac{M_{p}}{M}\equiv\xi_{c}\cong 4.3\times 10^{3}$, the scalaron mass should simply take the original value of $R^{2}$ model. As $\xi$ increases, approaching the above critical value, $M^{2}$ also starts to increase according to
$$\displaystyle M^{2}=\frac{\tilde{M}^{2}}{1-\frac{3\xi^{2}\tilde{M}^{2}}{%
\lambda M^{2}_{p}}}~{},$$
(4.7)
reaching infinity at $\xi=\xi_{c}$. This is in perfect agreement with Figure 3. However, note that in the simplified case we do not have modification of sound speed. The effective scalaron mass can be regarded as the modification of potential that is different from the change of sound speed although they may both produce similar features on the power spectrum.
As is well-known, only with either one of the two ingredients, $R^{2}$ or Higgs, it is enough to achieve successful inflation model with proper parameter value which is favored by the observation we have so far. Since the model we consider is the combination of these two single-field models, it goes back to either of them in some limits of the parameters.
One can easily see from the Lagrangian that if we take $\lambda\rightarrow 0$ and $\xi\rightarrow 0$ in the model or just simply set $\chi=0$, what we get is just the $R^{2}$ inflation with only one parameter $M$, the mass of the scalaron. The other limit is the Higgs inflation where we take $M\rightarrow\infty$. Note that we cannot just take $\psi=0$ which is analogous to what we did above. The reason is that the new degree of freedom in $\psi$ comes from the quadratic term of $\hat{R}$ in (2.4). Only when $\hat{R}^{2}$ term vanishes, this ”new” scalar field is completely determined by Higgs field, i.e. it is no longer a new degree of freedom. Thus, we go back to the Higgs inflation in this case with two parameters $\xi$ and $\lambda$.
5 Conclusion and Outlook
In this paper, we have analyzed a two-field inflation model consisting of the $R^{2}$ term and the Higgs field in detail. This model can easily go back to the two single-field models, the $R^{2}$ inflation [1] and the Higgs
inflation [8, 9, 10].
We have considered the parameter space where $\lambda=0.01$ and $\xi>0$. In the presence of mass hierarchy and considering slow-roll regime, one can integrate out the high energy part and obtain an effective single-field model with a slightly modified sound speed where we can easily calculate the power spectrum of curvature perturbations. The modification of sound speed comes from the presence of turning in the inflation trajectory, but in our case it turned out to be negligibly small.
For the amplitude of curvature perturbations to coincide with observation, we find that the predictions of this model are just the same as in the $R^{2}$ inflation or the Higgs inflation. Fixing the amplitude, $\mathcal{P}_{\mathcal{R}}(k)$, we find a relation between the scalaron mass $M$ and the non-minimal coupling $\xi$ which helps us to notice the relation between this two-field model and the $R^{2}$ inflation directly in the Jordan frame. We can effectively regard this model in the parameter space considered as $R^{2}$ inflation with an effective scalaron mass which naturally explains the existence of a special relation between the two free parameters.
For typical values of the self coupling parameters of the standard Higgs field at high energy, this model
gives essentially the same predictions as the $R^{2}$ inflation and the original Higgs inflation as far as the power spectrum is concerned. These two models, however, have quite different reheating mechanisms
[1, 28, 25, 26, 27]
with a much higher reheating temperature for the latter model with a possible violent behaviors due
to the non-minimal coupling [15]. Since our model smoothly connects the
two limits, the number of $e$-folds, $N_{\ast}$, of
the pivot scale of CMB observation is also expected to shift from the value corresponding to
the pure Higgs model to that of $R^{2}$ model, which leads to an observational
consequence [29]. This shift, however, is degenerate to the expansion
history or the amount of entropy production after reheating which may be measured by the
direct observation of high frequency tensor perturbations [32].
For smaller values of the self coupling than the case of the standard Higgs field with smaller
$\xi$, we may realize a situation both fields are in the slow roll regime and
acquire non-negligible quantum fluctuations so that
the isocurvature mode may also play an important role. Furthermore,
for $\xi<0$, we expect our model would have similar behaviors as in [23] where the two-field model can generate large fluctuations on small scale.
Though we considered the model with only one scalar field, our results can be straightforwardly generelaized to an arbitrary number of mutually interacting scalar fields sufficiently strongly coupled to the Ricci scalar.
Acknowledgements
We thank the M. Sasaki, T. Suyama, Y. Watanabe, Y. Wang, Y. P. Wu, S. Pi, Y. Zhang, H. Lee for useful discussion. MH was supported by the Global Science Graduate Course (GSGC) program of the University of Tokyo. AS acknowledges RESCEU hospitality as a visiting professor. He was also partially supported by the RFBR grant 17-02-01008. The work of JY was supported by JSPS KAKENHI, Grant-in-Aid for Scientific Research 15H02082 and Grant-in-Aid for Scientific Research on Innovative Areas 15H05888.
References
[1]
A. A. Starobinsky,
Phys. Lett. 91B, 99 (1980).
doi:10.1016/0370-2693(80)90670-X
[2]
K. Sato,
Mon. Not. Roy. Astron. Soc. 195, 467 (1981).
[3]
A. H. Guth,
Phys. Rev. D 23, 347 (1981).
doi:10.1103/PhysRevD.23.347
[4]
A. D. Linde,
Phys. Lett. 108B, 389 (1982).
doi:10.1016/0370-2693(82)91219-9
[5]
A. Albrecht and P. J. Steinhardt,
Phys. Rev. Lett. 48, 1220 (1982).
doi:10.1103/PhysRevLett.48.1220
[6]
For a review of single-field inflation, see e.g. K. Sato and J. Yokoyama,
Int. J. Mod. Phys. D 24, no. 11, 1530025 (2015).
doi:10.1142/S0218271815300256
[7]
P. A. R. Ade et al. [Planck Collaboration],
Astron. Astrophys. 594, A20 (2016)
doi:10.1051/0004-6361/201525898
[arXiv:1502.02114 [astro-ph.CO]].
[8]
J. L. Cervantes-Cota and H. Dehnen,
Nucl. Phys. B 442, 391 (1995)
doi:10.1016/0550-3213(95)00128-X
[astro-ph/9505069].
[9]
F. L. Bezrukov and M. Shaposhnikov,
Phys. Lett. B 659, 703 (2008)
doi:10.1016/j.physletb.2007.11.072
[arXiv:0710.3755 [hep-th]].
[10]
A. O. Barvinsky, A. Y. Kamenshchik and A. A. Starobinsky,
JCAP 0811, 021 (2008)
doi:10.1088/1475-7516/2008/11/021
[arXiv:0809.2104 [hep-ph]].
[11]
K. Kamada, T. Kobayashi, T. Takahashi, M. Yamaguchi and J. Yokoyama,
Phys. Rev. D 86, 023504 (2012)
doi:10.1103/PhysRevD.86.023504
[arXiv:1203.4059 [hep-ph]].
[12]
H. Kodama and M. Sasaki,
Prog. Theor. Phys. Suppl. 78, 1 (1984).
doi:10.1143/PTPS.78.1
[13]
X. Chen and Y. Wang,
Phys. Rev. D 81, 063511 (2010)
doi:10.1103/PhysRevD.81.063511
[arXiv:0909.0496 [astro-ph.CO]].
[14]
D. Polarski and A. A. Starobinsky,
Phys. Rev. D 50, 6123 (1994)
doi:10.1103/PhysRevD.50.6123
[astro-ph/9404061].
[15]
Y. Ema, R. Jinno, K. Mukaida and K. Nakayama,
JCAP 1702, no. 02, 045 (2017)
doi:10.1088/1475-7516/2017/02/045
[arXiv:1609.05209 [hep-ph]].
[16]
Y. Ema,
Phys. Lett. B 770, 403 (2017)
doi:10.1016/j.physletb.2017.04.060
[arXiv:1701.07665 [hep-ph]].
[17]
Y. C. Wang and T. Wang,
Phys. Rev. D 96, no. 12, 123506 (2017)
doi:10.1103/PhysRevD.96.123506
[arXiv:1701.06636 [gr-qc]].
[18]
K. Maeda,
Phys. Rev. D 39, 3159 (1989).
doi:10.1103/PhysRevD.39.3159
[19]
K. Maeda,
Phys. Rev. D 37, 858 (1988).
doi:10.1103/PhysRevD.37.858
[20]
M. Sasaki and E. D. Stewart,
Prog. Theor. Phys. 95, 71 (1996)
doi:10.1143/PTP.95.71
[astro-ph/9507001].
[21]
A. Achucarro, J. O. Gong, S. Hardeman, G. A. Palma and S. P. Patil,
JCAP 1101, 030 (2011)
doi:10.1088/1475-7516/2011/01/030
[arXiv:1010.3693 [hep-ph]].
[22]
A. Kehagias, A. Moradinezhad Dizgah and A. Riotto,
Phys. Rev. D 89, no. 4, 043527 (2014)
doi:10.1103/PhysRevD.89.043527
[arXiv:1312.1155 [hep-th]].
[23]
S. Pi, Y. l. Zhang, Q. G. Huang and M. Sasaki,
arXiv:1712.09896 [astro-ph.CO].
[24]
A. Achucarro, V. Atal, S. Cespedes, J. O. Gong, G. A. Palma and S. P. Patil,
Phys. Rev. D 86, 121301 (2012)
doi:10.1103/PhysRevD.86.121301
[arXiv:1205.0710 [hep-th]].
[25]
M. B. Mijic, M. S. Morris and W. M. Suen,
Phys. Rev. D 34, 2934 (1986).
doi:10.1103/PhysRevD.34.2934
[26]
D. S. Gorbunov and A. G. Panin,
Phys. Lett. B 700, 157 (2011)
doi:10.1016/j.physletb.2011.04.067
[arXiv:1009.2448 [hep-ph]].
[27]
E. V. Arbuzova, A. D. Dolgov and L. Reverberi,
JCAP 1202, 049 (2012)
doi:10.1088/1475-7516/2012/02/049
[arXiv:1112.4995 [gr-qc]].
[28]
A. A. Starobinsky, ”Nonsingular model of the Universe with
the quantum-gravitational de Sitter stage and its observational
consequences”, Proc. of the Second Seminar ”Quantum Theory of gravity”
(Moscow, 13-15 Oct. 1981), INR Press, Moscow, 1982, pp. 58-72; Quantum
Gravity, eds. M A. Markov and P. C. West, Plenum Publ. Co., New York,
1984, pp. 103-128.
[29]
J. Martin, C. Ringeval and V. Vennin,
Phys. Rev. Lett. 114, no. 8, 081303 (2015)
doi:10.1103/PhysRevLett.114.081303
[arXiv:1410.7958 [astro-ph.CO]].
[30]
A. A. Starobinsky,
Sov. Astron. Lett. 9, 302 (1983).
[31]
T. Faulkner, M. Tegmark, E. F. Bunn and Y. Mao,
Phys. Rev. D 76, 063505 (2007)
doi:10.1103/PhysRevD.76.063505
[astro-ph/0612569].
[32]
K. Nakayama, S. Saito, Y. Suwa and J. Yokoyama,
JCAP 0806, 020 (2008)
doi:10.1088/1475-7516/2008/06/020
[arXiv:0804.1827 [astro-ph]]. |
A Practical Introduction to Side-Channel Extraction of Deep Neural Network Parameters
Raphaël Joud, Pierre-Alain Moëllic, Simon Pontié
CEA Tech, Centre CMP, Equipe Commune CEA Tech - Mines Saint-Etienne, F-13541 Gardanne, France
Univ. Grenoble Alpes, CEA, Leti, F-38000 Grenoble, France
{name}.{surname}@cea.fr
&Jean-Baptiste Rigaud
Mines Saint-Etienne, CEA, Leti, Centre CMP,
F-13541 Gardanne France
rigaud@emse.fr
Abstract
Model extraction is a major threat for embedded deep neural network models that leverages an extended attack surface. Indeed, by physically accessing a device, an adversary may exploit side-channel leakages to extract critical information of a model (i.e., its architecture or internal parameters). Different adversarial objectives are possible including a fidelity-based scenario where the architecture and parameters are precisely extracted (model cloning). We focus this work on software implementation of deep neural networks embedded in a high-end 32-bit microcontroller (Cortex-M7) and expose several challenges related to fidelity-based parameters extraction through side-channel analysis, from the basic multiplication operation to the feed-forward connection through the layers. To precisely extract the value of parameters represented in the single-precision floating point IEEE-754 standard, we propose an iterative process that is evaluated with both simulations and traces from a Cortex-M7 target.
To our knowledge, this work is the first to target such an high-end 32-bit platform. Importantly, we raise and discuss the remaining challenges for the complete extraction of a deep neural network model, more particularly the critical case of biases.
Keywords Side-Channel Analysis $\cdot$
Confidentiality $\cdot$
Machine Learning $\cdot$
Neural Network
1 Introduction
Deep Neural Network (DNN) models are widely used in many domains with outstanding performances in several complex tasks. Therefore, an important trend in modern Machine Learning (ML) is a large-scale deployment of models in a wide variety of hardware platforms from FPGA to 32-bit microcontroller. However, major concerns related to their security are regularly highlighted with milestones works focused on availability, integrity, confidentiality and privacy threats. Even if adversarial examples are the flagship of ML security, confidentiality and privacy threats are becoming leading topics with mainly training data leakage and model extraction, the latest being the core subject of this work.
Model extraction.
The valuable aspects of a DNN model gather
its architecture and internal parameters finely tuned to the task it is dedicated to. These
carefully crafted parameters represent an asset for model owners and generally must remain secret. Jagielski et al. introduce an essential distinction between the objectives of an attacker that aims at extracting the parameters of a target model [1], by defining a clear difference between fidelity and accuracy:
•
Fidelity measures how well extracted model predictions match those from the victim model. In that context, an adversary aims to precisely extract model’s characteristics in order to obtain a clone model. In such a scenario, the extraction precision is important. Additionally to model theft, the adversary may aim to enhance his level of control over the system in order to shift from a black-box to a white-box context and design more powerful attacks against the integrity, confidentiality or availability of the model.
•
Accuracy aims at performing well over the underlying learning task of the original model: the attacker’s objective is to steal the performance of the model and, effortlessly, reach equal or even superior performance. In such a case, a high degree of precision is not compulsory.
Attack surface.
The large-scale deployment of DNN models raises many security issues. Most of the studied attacks target a model as an abstraction,
exploiting theoretical flaws. However, implementing a model to a physically accessible device open doors toward a new attack surface taking advantage of physical threats [2] [3], like side-channel (SCA) or fault injection analysis (FIA). This work is focused on fidelity-oriented attack targeting model confidentiality using SCA techniques.
Structure of the paper.
In Section 2, we first provide basic deep learning backgrounds that introduce most of our formalism. Related works are presented in Section 3, followed by an explanation of our positioning and contributions (Section 4). Details on our experimental setups and comments on our implementations are presented in Section 5, before a description of the threat model setting, discussed in Section 6. As an introduction to all our experiments, we expose the main challenges related to fidelity-based parameter extraction and describe our overall methodology in Section 7. Then, in Section 8, we detail our extraction method, our experiments and results with a progressive focus on: (1) one multiplication operation, (2) one neuron, (3) sign extraction, (4) several neurons and (5) successive layers. As future works (Section 9), we discuss the critical case of bias and the scaling up of our approach on state-of-the-art models. Fig. 1 illustrate the structure of the experimental sections with respect to the basic structural elements of a model. Finally, we conclude with possible mitigations.
2 Background
2.1 Neural Networks
2.1.1 Formalism
This work is about supervised DNN models. Input-output pairs $(x,y)\in\mathcal{X}\times\mathcal{Y}$ depend on the underlying task. A neural network model $\mathcal{M}_{W}:\mathcal{X}\rightarrow\mathcal{Y}$, with parameters $W$, predicts an input $x\in\mathcal{X}$ to an output $\mathcal{M}_{W}(x)\in\mathcal{Y}$ (e.g., a label for classification task). $W$ are optimized during the training phase in order to minimize a loss function that evaluates the quality of a prediction compared to the ground-truth $y$. Note, that a model $\mathcal{M}$, seen as an abstract algorithm, is distinguished from its physical implementations $M^{*}$, for example embedded models in microcontroller platforms.
From a pure functional point of view, the embedded models rely on the same abstraction but differ in terms of implementation along with potential optimization processes (e.g., quantization, pruning) to reach hardware requirements (e.g., memory constaints).
2.1.2 Perceptron
is the basic functional element of a neural network. The perceptron (also called neuron in the paper) first processes a weighted sum of the input with its trainable parameters $w$ (also called weights) and $b$ (called bias), then non-linearly maps the output thanks to an activation function (noted $\sigma$):
$$a(x)=\sigma\Big{(}\sum_{j=0}^{n-1}{w_{j}}x_{j}+b\Big{)}$$
(1)
where $x=(x_{0},...,x_{j},x_{n-1})\in\mathbb{R}^{n}$ is the input, $w_{j}$ the weights, $b$ the bias, $\sigma$ is activation function and $a$ the perceptron output. The historical perceptron used the sign function as $\sigma$ but others are available, as detailed hereafter.
2.1.3 MultiLayer Perceptrons (MLP)
are deep neural networks composed of many perceptrons stacked vertically, called a layer, and multiple layers stacked horizontally.
A neuron from layer $l$ gets information form all neurons belonging to the previous layer $l-1$. Therefore, MLP are also called feedforward fully-connected neural networks (i.e, information goes straight from input layer to output one). For a MLP, Equation 1 can be generalized as:
$$a^{l}_{j}(x)=\sigma\Big{(}\sum_{i\in(l-1)}{w_{i,j}}a^{l-1}_{i}+b_{j}\Big{)}$$
(2)
where $w_{i,j}$ is the weight that connects the neuron $j$ of the layer $l$ and the neuron $i$ of the previous layer ($l-1$), $b_{j}$ is the bias of the neuron $j$ of the layer $l$ and $a^{l-1}_{i}$ and $a^{l}_{j}$ are the output of neuron $i$ of layer $(l-1)$ and neuron $j$ of layer $l$.
2.1.4 Activation functions
inject non-linearity through the layers. Typical functions maps the output of a neuron into a well-defined space like $[0,+\infty]$, $[-1,+1]$ or $[0,1]$.
The Rectified Linear Unit function (hereafter, ReLU) is the most popular function because of its simplicity and constant-gradient property.
ReLU is piece-wise linear and defined as $ReLU(x)=max(0,x)$.
We focus our work on ReLU but other activations are possible: $tanh$ , $sigmoid$ or $softmax$ that is typically used at the end of classification models to normalize output to a probability distribution.
2.2 IEEE-754 Standard for Floating-Point Arithmetic
We study single-precision floating-point values on a 32-bit microcontroller. IEEE-754 standard details floating-point representation and arithmetic. Floating value are composed of three parts: Sign, Exponent and Mantissa as in Eq. 3 for a 32-bit single-precision floating-point value, $a$:
$$\displaystyle a$$
$$\displaystyle=(-1)^{b_{31}}\times 2^{(b_{30}...b_{23})_{2}-127}\times\Big{(}1.b_{22}...b_{0}\Big{)}_{2}$$
(3)
$$\displaystyle=(-1)^{S_{a}}\times 2^{E_{a}-127}\times\Big{(}1+2^{-23}\times M_{a}\Big{)}$$
This allows to represent values from almost $10^{-38}$ to $10^{+38}$ and considers specific case like infinity or Not a Number (NaN) values which are not considered here. We emphasize on the usual case when the exponent value belongs to $[\![1;254]\!]$. In this case, the final floating-point value $a$ is as in Eq. 3 where $S_{a}$, $E_{a}$ and $M_{a}$ correspond respectively to the sign, exponent and mantissa values. With this representation, result of the multiplication operation $c=a\times b$ with $b$ another single floating-point value, leads to the sign ($S_{c}$), exponent ($E_{c}$) and mantissa ($M_{c}$) detailed in Eq. 4.
Note that these do not necessarily correspond the very final representation of $c$: depending on the value of $M_{c}$, some realignment can be performed affecting both $M_{c}$ and $E_{c}$. However, it appears clearly that $M_{a}\times M_{b}$ have less impact on $M_{c}$ value than $M_{a}+M_{b}$.
$$\displaystyle S_{c}$$
$$\displaystyle=S_{a}\oplus S_{b}$$
(4)
$$\displaystyle E_{c}$$
$$\displaystyle=E_{a}+E_{b}-127$$
$$\displaystyle M_{c}$$
$$\displaystyle=M_{a}+M_{b}+2^{-23}\times M_{a}\times M_{b}$$
3 Related Work
Table 1 presents works that are – to the best of our knowledge – references for the topic of model extraction. These works are distinguished through the adversary’s objective (recover the architecture or recover the parameters) and the attack surface (API-based attacks or side-channel-based approaches). In this section, we detail works related to our scope. Interested readers may refer to surveys with a wider panorama such as [4] or [5]111More particularly, cache-based attacks that are out of our scope..
3.1 API-based attacks
These approaches exploit input/output pairs and information about the target model.
Carlini et al. consider the extraction of parameters as a cryptanalytic problem [6] and demonstrate significant improvements from [1]. The threat model sets an adversary that knows the architecture of the target model but not the internal parameters. The attack is only focused on ReLU-based multi-layer perceptron (MLP) models with one (scalar) output. The basic principle of this attack exploits the fact that the second derivative of ReLU is null everywhere except at a critical point, i.e. at the boundary of the negative and the positive input space of ReLU. By forcing exactly one neuron at this critical state thanks to chosen inputs and binary search, absolute values of weight matrix can be reconstructed progressively. Then, the sign is obtained thanks to small variations on the input and by checking activation output.
Experimental results (state-of-the-art) show a complete extraction of a 100,000 parameters MLP (one hidden layer) with $2^{21.5}$ queries with a worst-case extraction error of $2^{-25}$. Although the attack is an important step forward, limitations rely on its high complexity for deeper models and its strict dependence to ReLU.
3.2 Timing Analysis
In [8], Gongye et al. exploit, on a x86 processor, extra CPU cycles that significantly appear for IEEE-754 multiplication or addition with subnormal values. They precisely recover a 4-layer MLP models (weights and bias). However, a potential simple countermeasure against this attack is to enable flush-to-zero mode which turns subnormal values into zeros.
Maji et al. also demonstrate a timing-based parameter recovery that mainly rely on ReLU and the multiplication operation [9] with floating point,
fixed point, and binary models deployed on three platforms without FPU (ATmega-328P, ARM Cortex-M0+, RISC-V RV32IM). Countermeasures encompass adapted floating-point representation and a constant-time ReLU implementation. However, they highlight the fact that even with constant-time implementations, correlation power analysis (CPA) may be efficient and demonstrate a CPA (referencing to [11]) on only one multiplication.
3.3 SCA-based extractions
[11] from Batina et al. , is a milestone work
that covers the extraction of model’s architecture, activation function and parameters with SCA. Two platforms are mentioned, Atmel ATmega328P (opened) and SAM3X8E ARM Cortex-M3 for which floating-point operations are performed without FPU.
Activation functions are characterized with a timing analysis that enables a clear distinction between ReLU, sigmoid, tanh and softmax and relies on the strong assumption that an adversary is capable of measuring precisely execution delay of each activation functions of the targeted model during inference.
The main contribution, for our work, is related to the parameter extraction method that is mainly demonstrated on the 8-bit ATmega328P. Bias extraction is not taken into account nor mentioned. The method is focused on a low-precision recovery of the IEEE-754 float32 weights. Correlation Electromagnetic Analysis is used to identify the Hamming Weight ($HW$) of multiplication result (STD instructions to the 8-bit registers). The weight values are set in a realistic range $[-N,+N]$ with a precision $p=10^{-2}$ (therefore, $2N/p$ possible values). They extract the three bytes of the mantissa (three 8-bit registers) and the byte including the sign and the exponent222Due to IEEE-754 encoding, second byte of an encoded value contains the least significant bit of the exponent and the 7 most significant bits of mantissa.
There is no mention of an adaptation of this technique when dealing with the 32-bit Cortex-M3. Since desynchronization is strong (software multiplication and non-constant time activation function), the EM traces are resynchronized each time according to the target neuron. Note that, because the scope of [11] also encompass timing-based characterization and structure extraction, the scaling up from one weight to a complete deep model extraction and the related issues are not detailed.
Finally, model’s topology is extracted during the weight extraction procedure: new correlation scores are used to detect layer boundaries, i.e. distinguish if currently targeted neuron belongs to the same layer as previously attacked neurons or to the next one.
Presented methods are confronted to a MLP trained on MNIST dataset and a 8-bit convolutional neural network (CNN) trained on CIFAR-10333The specific features of CNN compared to MLP that should impact the leakage exploitation are not discussed in [11].. Original and recovered models have an accuracy difference of 0.01% and 0.36% respectively, with an average weight error of 0.0025 for the MLP. Implementations are not available and the compilation level is not mentioned.
4 Scope and Contributions
Our scope is a fidelity-based extraction of parameters of a MLP model embedded for inference purpose in an AI-suitable 32-bit microcontroller thanks to correlation-based SCA, such as CPA or CEMA.
Our principal reference is the work from Batina et al. [11] (and to certain extend [6] as a state-of-the-art fidelity-based extraction approach) and we position our contributions as follow:
•
Contrary to [11] (precision of $10^{-2}$), we set in a fidelity scenario and aim at studying how SCA can precisely extract parameter values.
•
Our claim is that the problem of parameter extraction raises several challenges, hardly mentioned in the literature, that we progressively describe. A wrong assumption may reduce this problem to a naive series of attacks targeting independent multiplications (that are actually not independent).
•
From the basic operation (multiplication) to an overall model, we propose and discuss methods to extract the complete value of 32-bit floating point weights. Extraction error can reach IEEE-754 encoding error level.
•
We do not claim to be able to fully recover all the parameters of a software embedded MLP model: we show that extraction of a secret weight absolute value from multiplication operation is necessary but not sufficient to generalize to the extraction of a complete MLP model. We discuss open issues preventing this generalization such as the extraction of bias values.
•
We highlight the choice of our target, based on a ARM-Cortex M7, i.e. a high-end device particularly adapted to deep learning applications (STM32H7). To the best of our knowledge, such a target does not appear in the literature despite its DNN convenient attributes (e.g., FPU, memory capacity). Electromagnetic (EM) acquisitions have been made with an unopened chip which corresponds to a more restrictive attack context compared to literature.
•
To foster further experiments and help the hardware security community to take on this topic, our traces and implementations are publicly available444https://gitlab.emse.fr/securityml/SCANN-ex.git.
5 Experimental setup
5.1 Target device and setup
Our experimental platform is a ARM Cortex-M7 based STM32H7 board. This high-end board provides large memories (2 MBytes of flash memory and 1 MByte of RAM) allowing to embed state-of-the-art models (e.g., 8-bit quantized MobileNet for image classification task). A 25 MHz quartz has been melted as part of the HSE oscillator to have more stable clock. CPU is running at 25 MHz as well, as its clock is directly derived from the melted quartz.
EM emanations coming from the chip are measured with a probe from Langer (EMV-Technik LF-U 2,5 with a frequency range going from 100 kHz to 50 MHz) connected to a 200 MHz amplifier (Fento HVA-200M-40-F) with a 40 dB gain, as shown in Fig. 2. Acquisitions are collected and saved thanks to a Lecroy oscilloscope (4 GHz WaveRunner 640Zi).
To reduce noise and ease leakage exploitation, all traces acquired experimentally from Cortex M7 are averaged over 50 program executions.
5.2 Inference program
Because of the scope, objective and methodology of this work, we need to perfectly master the programs under analysis to properly understand the leakage properties and their potential exploitation. Therefore, instead of attacking black-box off-the-shelf inference libraries,
we implement our own C programs for every experiments mentioned in this paper
and compile them with O0 gcc optimization level to ensure each multiplication is followed by STR instruction saving result in SRAM. This point is discussed as further works in Section 10.
As in [9], some approaches exploit timing inconsistency to recover model information. In this work, we consider implementations protected from such kind of attacks as model inferences are performed in a timing constant manner. We claim that these choices represent more real-world applications, as for the selection of an high-end AI-suitable board:
•
We use the floating-point unit (FPU) module that performs floating-point calculations in a single cycle rather than passing through C compiler library. When available, usage of such hardware module is preferred to its software counterpart as it speeds up program execution and relieve CPU.
•
ReLU function has been implemented in a timing constant way as in Listing 1. It has been confirmed by checking on thousands of execution that its delay standard deviation is lower than one clock cycle.
6 Threat model
Adversary’s goal.
Considered adversary aims at reverse-engineering an embedded MLP model as closely as possible by cloning the targeted model with a fidelity-oriented approach. This objective implies that parameters values resulting from target model training phase have to be estimated.
Adversary’s knowledge.
The attack context corresponds to a gray-box setting since the adversary knows several information about the target system: (1) the model architecture, (2) the used activation function is ReLU, (3) model parameters are stored as single-precision float following the IEEE-754 standard.
With an appropriate expertise in Deep Learning, the attacker may also carry out upstream analyses more particularly on the typical distribution of the weigths (ranges, normalization…) he aims at extracting.
Adversary’s capacity.
The adversary is able to acquire side-channel information (in our case, EM emanations with an appropriate probe), leaking from the system embedding the targeted DNN model. The collected traces only results from the usual inference and the attacker does not alter the program execution.
We assume a classical linear leakage model: the leakage captured is linearly dependent of the $HW$ of the processed secret value (e.g., a floating-point multiplication between the secret $w$ and an input coming from the previous layer). Typically, a gaussian noise encompasses the intrinsic and acquisition noise.
The adversary can feed the model with crafted inputs, without any limitation (nor normalization), allowing to control the distribution of the inputs according to the chosen leakage model. However, these chosen inputs belong to the usual values according to the IEEE-754 standard. Contrary to API-based attacks, the attacker does not need to access the outputs of the model.
To simplify the scope of this introductory work, we set in a worst case scenario according to defender. Attacker is considered able to access a clone device and have enough knowledge and expertise to take benefit of his own implementations to estimate the temporal windows in which he will perform his analysis. We discuss that point in Section 7 and 9.
7 Challenges and overall methodology
7.1 Critical challenges related to SCA-based parameter extraction
Impact of ReLU.
The assumption stating that inputs of targeted operation are controlled by the adversary is partially correct: if we focus on a single hidden layer, the inputs are the outputs of the previous one after passing through ReLU. Therefore, the input range is necessarily restricted to non-negative values.
Fully-connected model. MLP parameters are not shared555Contrary to Convolutional Neural Network models.: the activated output of a neuron is connected to all neurons of the next layer. Then, an input is involved in as many multiplications as neurons in the considered layer. As such, when performing a CEMA at the layer-level, several hypothesis would stand out from the analysis and would likely be correct as they would correspond to the leakage of each neuron of the layer. However, these hypothesis would not stand out at the same time if neurons outputs are computed sequentially. Therefore, knowledge of the order of the neurons is compulsory to correctly associate CEMA results to neurons. That point is closely related to the threat model we defined in Section 6 and the profiling ability of the attacker is also discussed in Section 9.
Error propagation problem.
Because of the feedforward functioning of a MLP, extraction techniques must be designed as well: the correct extraction of parameters related to a layer cannot be achieved without fully recovering the previous parameters. A strong estimation error in the recovery of a weight (and therefore in the estimation of the neuron output) will impact the extraction of remaining neuron weights. The impact of this error will spread to the weight extraction of all neurons belonging to forthcoming layers as illustrated in Fig. 1.
Temporal profiling. In [11], side-channel patterns could be visually recognized on 8-bit microcontroller on which most of the results have been demonstrated. In our context, SPA is hardly feasible (e.g., see Fig. 3) and the localization of all the relevant parts of the traces is a challenging issue that we consider as out the scope of this work. As mentioned in our threat model (Section 6), we set in a worst case scenario where the attacker is able to perform a temporal profiling on a clone device to have an estimation of the parts of the traces to target since he has several secrets to recover spread all over the traces. This estimation can be more or less precise to enable attacks at neuron or layer-level.
Bias values.
Knowledge of the bias value is compulsory to compute the entire neuron output. This parameter is not involved in multiplications with the inputs but added to the accumulated sum between neuron inputs and its weights. Thus, leakage of bias and weight must be exploited differently. Bias extraction is treated in the API-based attack [6] and the timing attack from [9] but not mentioned in [11]. In this work, we clearly states that we keep the extraction of bias values as a future work but discuss this challenging point in Section 9.1.
7.2 Our methodology
Our methodology starts with analyzing the most basic operation of a model – i.e. a multiplication – then, to widen our scope, to a neuron, one layer, then several layers as illustrated in Fig.1. Corresponding steps are evaluated with both simulated and real traces.
Dealing with an entire model means to recover parameters layer by layer, following the (feedforward) network flow: extractions of a layer $l$ being used to infer the inputs of the layer $l+1$.
Since our main objective is a practical fidelity-based extraction, we aims at crafting an efficient extraction method, faster than a brute-force CEMA on $2^{32}$ hypothesis, that enables a progressive precision. With this introduction work, in addition to expose challenges that suffer analysis in the literature, we assess the precision degree SCA can reach.
8 Extraction method and experiments
8.1 Targeting multiplication operation
We first focus on the multiplication $c=w\times x$, between two IEEE-754 single-precsion floating-point operands: a secret weight ($w$) and a known input ($x$). We remind that we use hardware operations thanks to the FPU.
8.1.1 Our approach
is composed of multiple CEMA to extract the absolute value $|w|$. Importantly, the sign bit is not considered yet and is extracted later on (Section 8.3).
With fidelity-oriented extraction as objective, our method has no fixed accuracy objective and avoids exhaustive analysis over $2^{32}$ hypothesis. It allows to see how accurate SCA-based extraction can perform.
It relies on two successive steps. First one tends to recover as much information as possible in a single attack by targeting most significant bits from a variable encoded with IEEE-754 standard. The second step is made to correct possible approximations from the previous one and enhance extraction accuracy by refining the granularity of tested hypothesis.
In this step, no focus is made on specific bits, entire variable with all 32-bit varying are considered. Fig. LABEL:xtr_process_flowchart describes these two steps. The attack relies on different parameters:
•
$d_{0}$: size of the initial interval that is centered on the value $C$. Thus, the tested hypothesis belong to $[C-d_{0}/2,C+d_{0}/2]$.
•
$\lambda_{1}>\lambda_{2}$: two shrinking factors that narrow the interval of analysis ($\lambda_{1}$ for the first iteration of step 2, $\lambda_{2}$ for successive step 2 iterations).
•
$m$: number of times the step 2 is repeated.
•
$N$: number of kept hypothesis (after CEMA) at each extraction step.
Step 1 of the extraction process is as follows:
1.
First, we generate an exhaustive set of hypothesis with all possible $2^{16}$ combinations of the 8 bits of exponent and the 8 most significant bits of mantissa (remaining bits are set to 0) and filter out unlikely values (in a DNN context) by keeping hypothesis in $[C-d_{0}/2,C+d_{0}/2]$. Kept hypothesis are not linearly distributed in this interval.
2.
We compute the HW of the targeted intermediate values: here, the result of the products between inputs and weight hypothesis.
3.
We perform a CEMA between EM traces and our HW hypothesis and keep the $N$ best ones according to the absolute values of correlation scores.
Step 2 is processed in an iterative way and depends on the previous extraction that could be the output of Step 1 or from the previous Step 2 iteration:
1.
For each best hypothesis $\hat{w}_{i}$ kept from the previous step ($i\in\llbracket 0;N-1\rrbracket$), a set of assumptions is linearly sampled around $\hat{w}_{i}$ with an narrower interval of size $d_{1}=d_{0}/\lambda_{1}$ (if the previous step was step 1) or $d_{i+1}=d_{i}/\lambda_{2}$ (otherwise).
2.
As in step 1, we compute the HW of the intermediate values and perform a CEMA to select the $N$ best hypothesis among the $N$ considered hypothesis sets (so that we always keep $N$ hypothesis at each iteration) according to absolute value of the correlation scores.
Fig. 4 shows the two steps of this extraction process ($w=0.793281$, $d_{0}=5$ and $C=2.5$) for 3,000 real traces on a STM32H7, obtained from the $\texttt{PRGM}_{2}$ experiment described hereafter. The second step is iterated three times and we progressively reach a high correlation score.
8.1.2 Validation on simulated traces
We first confirm our approach on simulated traces by computing the success rate of our extraction with respect to several extraction error ($\epsilon_{rr}$) thresholds. We randomly generate 5,000 positive secret values $w$ and for each of them, we craft 1,000 3-dimensional traces using random inputs $x$. At the middle sample, the trace value is the Hamming Weight of the multiplication: $HW(x\times w)$. A random uniform variable is used for the other samples. An additional gaussian noise ($\mu=0$, $\sigma$) is applied on the entire trace. We set $N=5$, $d_{0}=5$, $C=2.5$, $m=3$, $\lambda_{1}=100$ and $\lambda_{2}=50$. Results according to the noise level are presented in Tab. 2. We reach a significant success rate over 90% for the extraction process until a recovering error of $10^{-6}$.
8.1.3 Experiments on Cortex-M7
This extraction method is also confronted to real traces obtained from our target board. For these experiments, the secret values are positive and hard-written in the code and input values are sent from a python script through UART interface, 150,000 traces have been acquired for each of them, then averaged to 3,000 traces. Two programs have been implemented: $\texttt{PRGM}_{1}$ performs a single multiplication and $\texttt{PRGM}_{2}$ performs two multiplications with distinct secret values and inputs (corresponding EM activity is depicted in Fig. 5).
Both being compiled with O0 gcc optimization level, this implies that each multiplication is followed by a STR instruction saving the multiplication result in SRAM as in Listing 2. One source of leakages exploited to recover secret values is these store instructions. Inference EM activity to be analyzed is framed by a trigger added by hand at assembler level.
For this experiment, our extraction method allows to recover the secret values with high extraction level as presented in Tab. 3.
8.2 Extracting parameters of a perceptron
8.2.1 Neuron computation implementation
After extracting secret value from an isolated multiplication and studying success-rate of such attack, we scale up to a single neuron computation as described in Eq. 1. The most important difference is that the output of a neuron is the result of an accumulation of several multiplications.
This accumulation is processed through two successive FPU instructions (fmul then fadd, as it is the case in our experiments with Listing 3) or a dedicated multiplication-addition instruction.
That leads to two new challenges in the extraction of the secret values compared to our first experiments on isolated multiplications: (1) hypothesis have to be made on accumulated values, (2) the attacker needs to know the order in which multiplication are computed (i.e., how the accumulation evolves).
8.2.2 Extraction of neuron weights
We assume that the attacker knows the computation order. The first weight $w_{0}$ can be extracted as done before by exploiting the direct result of $w_{0}\times x_{0}$. Then, the second weight $w_{1}$ can also be extracted with the same approach by targeting $w_{0}\times x_{0}+w_{1}\times x_{1}$ because $w_{0}$ was recovered before.
This process can be applied again for each weight value as long as all previous ones have been correctly extracted. Actually, that point is a critical one since the extraction quality of currently targeted weight strongly depends on the extraction accuracy of previously extracted weights.
8.2.3 Experiments on Cortex-M7
We apply that method on 2,000 averaged traces that capture the inference of a 4-input neuron. As presented in Table 4, we reach a very precise extraction of the four weights.
8.3 Targeting the sign
8.3.1 Problem statement
As seen before, for a ReLU-MLP model, a neuron belonging to an hidden layer is fed with non-negative input values. An obvious but important observation is that, for $w\times x=c$, if the secret value $w$ is multiplied with positive value $x\geq 0$ then $sign(c)=sign(w)$. Therefore, in such a context, CEMA is not able to distinguish sign by leveraging the input-weight product.
8.3.2 Extracting the sign at the neuron-level
A way to overcome this issue is to set the sign extraction problem at the neuron-level, i.e. to build hypothesis on sign changes throughout the overall multiplication-accumulation process.
Let consider the accumulation of two successive multiplications: $acc=w_{0}\times x_{0}+w_{1}\times x_{1}$ with inputs $x_{0},x_{1}\geq 0$. $acc$ variations would change whether $sign(w_{0})\neq sign(w_{1})$ or $sign(w_{0})=sign(w_{1})$. Based on that, by focusing on variation of $|acc|$ value, it is possible to find out if $w_{0}$ and $w_{1}$ have an opposed sign or not.
Thus, weight sign estimation can be done progressively, by checking if the sign associated to the weight currently extracted is similar or opposed to the sign of the previous weight. However, since the sign extraction is processed relatively to the sign of $w_{0}$, an additional verification has to be done to confirm which of the current extracted signs or the opposite is correct. This can be done thanks to ReLU output that matches with only one hypothesis.
8.3.3 Validation on simulated traces
As in Section 8.1, we craft simulated 50-dimensional traces for a $m$-input neuron with $m$ randomly picked in $\llbracket 2;8\rrbracket$. We generate 5,000 neurons with $m$ signed weights, no bias and fed by 3,000 positive inputs sets (i.e., 25M of traces). The generation process is similar to the previous experiment apart from the leakage placement which depends on $m$. Thus, leakages correspond to $m$ product accumulations and one for the $ReLU$ output. We uniformly place these $m+1$ leakage samples in the traces with random uniform values for the other samples. To characterize the principle of the method, we set in a low-noise simulation ($\sigma^{2}=0.5$). We reach the following results:
•
78.8% of neurons have been extracted with all signs correctly assigned.
•
For 91.6% of the weights, the sign is correctly assigned. Table 5 details the extraction success rates for these weights (consistent with Table 2).
8.3.4 Experiments on Cortex-M7
We use 5,000 averaged traces capturing the inference of one neuron with signed weights. With Table 6, we observe that the sign inversion and the relative value have been correctly affected.
In addition to our previous experiences, our approach performs well at the neuron-level. We progressively scale-up in the next section at a layer-level.
8.4 Targeting one layer
Previous structure has inputs involved in only one multiplication with weights. However, neural network interconnections between layers implies that an input value is passed to each neuron of the layer
and thus is involved in several multiplications with different weights. If neurons are computed sequentially, this means that CEMA would likely bring out several hypothesis that would match weights of different neurons that leak at different moments.
In this context, to associate the extracted values to a specific neuron, we assume that neurons computation is made sequentially from top to bottom of the layer.
To ensure an already extracted value is not associated again to another neuron, leaking time sample of tested hypothesis are filtered. Consider only leaking sample greater than the one from last extracted value prevents this.
8.4.1 Experiments on Cortex-M7
Two experiments have been made:
1.
2-neuron layer with 3 inputs each: the 6 weights are positives and 3,000 traces have been captured by feeding the layer with random positive inputs (as for an hidden layer). The six weights have been recovered with an averaged estimation error $\epsilon_{rr}=2.68e^{-6}$ (worst: $5.18e^{-6}$, best: $4.48e^{-8}$).
2.
5-neuron layer with 4 inputs each: the 20 weights encompass positive and negative values and 5,000 traces have been acquired by feeding the layer with random positive or negative values (as for an input layer). We reach a similar extraction error $\epsilon_{rr}=1.03e^{-6}$ (worst: $3.10e^{-6}$, best: $1.55e^{-7}$). All weight signs have been correctly guessed.
8.5 Targeting few layers
DNN are characterized by layers stacked horizontally. Proposed method is able to extract weights from one layer and is supposed to be applied to each of them one after the other, by progressively reconstructing intermediate layer outputs.
8.5.1 Experiments on Cortex-M7
To verify this principle, we craft a MLP with 5 hidden layers with respectively 5, 4, 3, 2 and 3 neurons. The 64 corresponding weights are positive and the model is fed with 4-dimensional positive inputs. Every weights have been recovered with an estimation error lower than $10^{-6}$ ($SR=95.31\%$ for $\epsilon_{rr}<10^{-7}$, best $\epsilon_{rr}=7.63e^{-10}$, worst $\epsilon_{rr}=6.67e^{-6}$).
Note that sign is not considered in this experiments because the tested model has been crafted and is not functional (i.e., not the result of a training process). Such none functional models are likely to present too many dead neurons and even dead layers because of the accumulated ReLU effect. The scaling-up to a fully functional state-of-the-art model with signed weights and biases is planned for future works and discussed in the next section.
9 Future Works
9.1 What about neuron bias?
So far, biases have not been considered even though these values may significantly impact neuron outputs and also the way a neuron is implemented: the weighted sum between weights and inputs could be initialized to $0$ or directly to the bias. In the latter case, our extraction method cannot be directly applied and needs an initial and challenging bias extraction based on IEEE-754 addition.
To better explain this challenge, lets consider that the accumulation is well initialised to 0 (i.e, bias is added after the weighted sum). Using simulated traces, we perform our extraction method to recover the weights and the bias by focusing on the final accumulation $\sum_{j}(w_{i,j}\times x_{j})+b$.
5,000 neurons with $m$ secret weights $w_{0..m}$ ($m$ is randomly picked in $\llbracket 2;8\rrbracket$) and one secret bias $b$ have been generated. 5,000 simulated traces have been crafted for each neuron with random positive inputs. Success rates (SR) are presented in Table 7. These SR only concern weight and bias for which sign has correctly been recovered. This corresponds to $93.25\%$ over 24956 attacked weights and $92.14\%$ over 5000 bias. While SR related to weight extraction remains consistent with previous simulations, SR corresponding to bias extraction significantly drops (e.g., $SR=35.8$ for $10^{-3}$).
A possible explanation relies on the IEEE-754 addition that requires a strict exponent realignment contrary to multiplication. If $a\gg b$ then $a+b=a$ because $b$ value disappears in front of $a$. In our context, as inputs $x$ (controlled by the attacker that aims at covering as much as possible the float32 range) are defined randomly, then multiplied by weights, it is likely that $\sum_{j=0}^{n}{w_{j}}x_{j}\gg b$. Thus, secret information related to bias could be hardly recovered by exploiting our EM traces. Therefore, we need to develop a different strategy (including a coherent selection of the inputs) to exploit potential IEEE-754 addition leakages.
9.2 Targeting state-of-the-art functional models
Further experiments will encompass compressed embedded models thanks to deployment libraries (e.g., TFLite, NNoM) with a focus on Convolutional Neural Network (CNN) models. Indeed, for memory constrains, deep embedded models in 32-bit microcontrollers are usually stored with parameters quantized in 8-bit integers with training-aware or post-training quantization methods. For the most straightforward quantization and embedding approaches, this quantization should simplify the extraction process with only $2^{8}$ hypothesis for each weight and bias as well as an additional extraction of a scaling factor that enables the mapping from 8-bit to 32-bit values. Regarding CNN, these models also rely on multiplication-accumulation operations (and the same activation principle), but the fact that parameters are shared across the inputs should interestingly impact the way leakages could be exploited for a practical extraction.
10 Conclusion
Side-channel analysis is a well-known, powerful, mean to extract information from an embedded system. However,
with this work, we clearly question the practicability of a complete parameters extraction with SCA when facing state-of-the-art models and real-world platforms. By demonstrating promising results on a high-end 32-bit microcontroller on a high fidelity-based extraction scenario, we do not claim this challenge as impracticable but we aim at inciting further (open) works focused on the exposed challenges as well as bridging different approaches with combined API and SCA-based methods.
An additional outcome from our experiments concerns defenses. Classical hiding countermeasures, already demonstrated in other context (e.g., cryptographic modules), should be relevant (as also mentioned in [11]). More precisely, randomizing multiplication and/or accumulation order (including the bias) should significantly impact an adversary. An efficient complementary defense could be to randomly add fake or neutral operations at a neuron or layer-level. We keep as future works, the proper evaluation of such state-of-the-art protections in a model extraction context.
Acknowledgements
This work is supported by (CEA-Leti) the European project InSecTT (ECSEL JU 876038)666www.insectt.eu, InSecTT: ECSEL Joint Undertaking (JU) under grant agreement No 876038. The JU receives support from the European Union’s Horizon 2020 research and innovation program and Austria, Sweden, Spain, Italy, France, Portugal, Ireland, Finland, Slovenia, Poland, Netherlands, Turkey. The document reflects only the author’s view and the Commission is not responsible for any use that may be made of the information it contains.
and by the French ANR in the framework of the Investissements d’avenir program (ANR-10-AIRT-05, irtnanoelec); and (Mines Saint-Etienne) by the French program ANR PICTURE (AAPG2020). This work benefited from the French Jean Zay supercomputer with the AI dynamic access program.
References
[1]
M. Jagielski, N. Carlini, D. Berthelot, A. Kurakin, and N. Papernot.
High accuracy and high fidelity extraction of neural networks.
In 29th $\{$USENIX$\}$ Security Symposium ($\{$USENIX$\}$
Security 20), pages 1345–1362, 2020.
[2]
Mathieu Dumont, Pierre-Alain Moëllic, Raphael Viera, Jean-Max Dutertre, and
Rémi Bernhard.
An overview of laser injection against embedded neural network
models.
In 2021 IEEE 7th World Forum on Internet of Things (WF-IoT),
pages 616–621. IEEE, 2021.
[3]
Raphaël Joud, Pierre-Alain Moëllic, Rémi Bernhard, and
Jean-Baptiste Rigaud.
A review of confidentiality threats against embedded neural network
models.
In 2021 IEEE 7th World Forum on Internet of Things (WF-IoT).
IEEE, 2021.
[4]
Maria Méndez Real and Ruben Salvador.
Physical side-channel attacks on embedded neural networks: A survey.
Applied Sciences, 11(15):6790, 2021.
[5]
Hervé Chabanne, Jean-Luc Danger, Linda Guiga, and Ulrich Kühne.
Side channel attacks for architecture extraction of neural networks.
CAAI Transactions on Intelligence Technology, 6(1):3–16, 2021.
[6]
Nicholas Carlini, Matthew Jagielski, and Ilya Mironov.
Cryptanalytic extraction of neural network models.
In Daniele Micciancio and Thomas Ristenpart, editors, Advances
in Cryptology – CRYPTO 2020, pages 189–218, Cham, 2020. Springer
International Publishing.
[7]
Seong Joon Oh, Bernt Schiele, and Mario Fritz.
Towards reverse-engineering black-box neural networks.
In Explainable AI: Interpreting, Explaining and Visualizing Deep
Learning, pages 121–144. Springer, 2019.
[8]
Cheng Gongye, Yunsi Fei, and Thomas Wahl.
Reverse-Engineering Deep Neural Networks Using
Floating-Point Timing Side-Channels.
In 2020 57th ACM/IEEE Design Automation Conference
(DAC), pages 1–6, July 2020.
ISSN: 0738-100X.
[9]
Saurav Maji, Utsav Banerjee, and Anantha P Chandrakasan.
Leaky nets: Recovering embedded neural network models and inputs
through simple power and timing side-channels–attacks and defenses.
IEEE Internet of Things Journal, 2021.
[10]
Y. Xiang, Z. Chen, et al.
Open DNN Box by Power Side-Channel Attack.
IEEE Transactions on Circuits and Systems II: Express Briefs,
67(11):2717–2721, 2020.
[11]
Lejla Batina, Shivam Bhasin, Dirmanto Jap, and Stjepan Picek.
{CSI} {NN}: Reverse Engineering of Neural Network
Architectures Through Electromagnetic Side Channel.
In 28th $\{$USENIX$\}$ Security Symposium ($\{$USENIX$\}$
Security 19), pages 515–532, 2019.
[12]
W. Hua, Z. Zhang, and G. E. Suh.
Reverse Engineering Convolutional Neural Networks Through
Side-channel Information Leaks.
In 2018 55th ACM/ESDA/IEEE Design Automation
Conference (DAC), pages 1–6, June 2018.
[13]
A. Dubey, R. Cammarota, and A. Aysu.
MaskedNet: The First Hardware Inference Engine Aiming
Power Side-Channel Protection.
In 2020 IEEE International Symposium on Hardware
Oriented Security and Trust (HOST), pages 197–208, December 2020.
[14]
H. Yu, H. Ma, K. Yang, Y. Zhao, and Y. Jin.
DeepEM: Deep Neural Networks Model Recovery through em
Side-Channel Information Leakage.
In 2020 IEEE International Symposium on Hardware Oriented
Security and Trust (HOST), pages 209–218. IEEE, 2020.
[15]
Jakub Breier, Dirmanto Jap, Xiaolu Hou, Shivam Bhasin, and Yang Liu.
Sniff: reverse engineering of neural networks with fault attacks.
IEEE Transactions on Reliability, 2021. |
Jet substructure shedding light on heavy Majorana neutrinos at the LHC
Arindam Das,
b
Partha Konar,
c
and Arun Thalapillil
arindam@kias.re.kr
konar@prl.res.in
thalapillil@iiserpune.ac.in
School of Physics, KIAS, Seoul 130-722, KoreaTheoretical Physics Group, Physical Research Laboratory, Ahmedabad-380009, IndiaIndian Institute of Science Education and Research,
Homi Bhabha Rd, Pashan, Pune 411 008, India
Abstract
The existence of tiny neutrino masses and flavor mixings can be explained naturally in various seesaw models, many of which typically having additional Majorana type SM gauge singlet right handed neutrinos ($N$). If they are at around the electroweak scale and furnished with sizeable mixings with light active neutrinos, they can be produced at high energy colliders, such as the Large Hadron Collider (LHC). A characteristic signature would be same sign lepton pairs, violating lepton number, together with light jets – $pp\to N\ell^{\pm},\;N\to\ell^{\pm}W^{\mp},\;W^{\mp}\to jj$. We propose a new search strategy utilising jet substructure techniques, observing that for a heavy right handed neutrino mass $M_{N}$ much above $M_{W^{\pm}}$, the two jets coming out of the boosted $W^{\pm}$ may be interpreted as a single fat-jet ($J$). Hence, the distinguishing signal topology will be $\ell^{\pm}\ell^{\pm}J$. Performing a comprehensive study of the different signal regions along with complete background analysis, in tandem with detector level simulations, we compute statistical significance limits. We find that heavy neutrinos can be explored effectively for mass ranges $300$ GeV $\leq M_{N}\leq 800$ GeV and different light-heavy neutrino mixing $|V_{\mu N}|^{2}$. At the 13 TeV LHC with 3000 $\mathrm{fb}^{-1}$ integrated luminosity one can competently explore mixing angles much below present LHC limits, and moreover exceed bounds from electroweak precision data.
Keywords:Large Hadron Collider, Seesaw neutrino mass, Jet substructure.
1 Introduction
The experimental evidence for neutrino oscillations Neut1 ; Neut2 ; Neut3 ; Neut4 ; Neut5 ; Neut6 and lepton flavor mixings, from the various experiments, motivate extensions of the SM incorporating non-zero neutrino masses and mixings.
After the pioneering realization of the unique $d=5$ operator Weinberg:1979sa within the SM with $\Delta L=2$ lepton number violation $(L=\rm{Lepton~{}number})$, it was realized that the Seesaw mechanism seesaw0 ; seesaw1 ; seesaw2 ; seesaw3 ; seesaw4 ; seesaw5 ; seesaw6 could be the simplest idea to explain the smallness of the neutrino masses and flavor mixings. In many of these models, SM is extended by gauge singlet, Majorana type, heavy right handed neutrinos (RHNs). After electroweak (EW) symmetry breaking, the light Majorana neutrino masses are generated by, for instance, the so called type-I seesaw mechanism.
Through the seesaw mechanism, the flavor eigenstates of the SM light neutrino mix with the mass eigenstates of the light neutrinos and RHNs. The SM singlet RHNs ($N$) interact with the SM gauge bosons through lepton mixing. Such Majorana type RHNs, if at the EW scale, can be produced at the Large Hadron Collider (LHC) with a distinguishing signature – Same Sign Di-Leptons (SSDL) and di-jets. In this channel the heavy RHNs decay into a $W^{\pm}$ and a lepton. In cases where the RHNs are sufficiently massive, very often the gauge bosons are significantly boosted, resulting in collimated energy deposits in the hadronic calorimeter. With a suitable jet algorithm, these collimated hadron four momenta may be reconstructed as a fat-jet $(J)$. Fat-jets retain information of their origins and have several distinct properties that may be leveraged for tagging and signal discrimination. The resulting signal of interest therefore becomes SSDL + fat-jet. In this paper we consider searches for RHNs with masses $M_{N}\geq 300$ GeV, which is sufficient to produce boosted jets. It is important to search for such relatively small-mass RHNs at colliders, since from a very general theoretical viewpoint, a small $M_{N}$ may be considered technically natural Fujikawa:2004jy ; deGouvea:2005er . This is because in the limit $M_{N}\rightarrow 0$ one regains $U(1)_{\text{\tiny{B-L}}}$ as a global symmetry of the Lagrangian. Different experiments such as ATLAS Aad:2015xaa and CMS CMS8:2016olu ; Khachatryan:2015gha have already searched for RHNs in the SSDL + dijets channel, assuming non-boosted $W^{\pm}$.
At the 8 TeV LHC, with $20.3$ fb${}^{-1}$ luminosity and $95\%$ confidence limit (C. L.), ATLAS Aad:2015xaa has probed mixings for muon flavor down to a $|V_{\mu N}|^{2}$ of $3.5\times 10^{-3}$, for $M_{N}=100$ GeV. The limits further goes down to $2.9\times 10^{-3}$ for $M_{N}=110$ GeV and then monotonically weakens with mass, up to $M_{N}=500$ GeV. At $M_{N}=500$ GeV the limits are $|V_{\mu N}|^{2}=4\times 10^{-1}$. The limits are nearly two orders of magnitude weaker in the case of electron flavor mixings $|V_{eN}|^{2}$ at the $95\%$ C. L.
CMS has also studied the SSDL plus dijet signal and obtain the exclusion limits for $|V_{eN}|^{2}$ CMS8:2016olu and $|V_{\mu N}|^{2}$ Khachatryan:2015gha . Both studies are performed at the 8 TeV LHC with $19.7$ fb${}^{-1}$ luminosity at $95\%$ C. L. The limits for the mixed $e^{\pm}\mu^{\pm}+jj$ final state was also considered in CMS8:2016olu . CMS observed upper limits for $|V_{eN}|^{2}$ at $1.2\times 10^{-4}$ for $M_{N}=40$ GeV, $2\times 10^{-2}$ for $M_{N}=85$ GeV, $8\times 10^{-3}$ for $M_{N}=130$ GeV and $1.2\times 10^{-2}$ for $M_{N}=200$ GeV. Thus, the $|V_{eN}|^{2}$ limits were found to again weaken with $M_{N}$. Alternatively, RHNs may be excluded as large as $M_{N}=480$ GeV, assuming the mixing is unity. The limits on $|V_{\mu N}|^{2}$ from the SSDL + dijet final state with $\mu$ flavor is probed down to $2\times 10^{-5}$ for $M_{N}=40$ GeV, $4.5\times 10^{-3}$ for $M_{N}=90$ GeV, $1.75\times 10^{-3}$ for $M_{N}=125$ GeV and $7\times 10^{-3}$ for $M_{N}=175$ GeV with $|V_{\mu N}|^{2}$ again weakening subsequently with $M_{N}$. For $M_{N}=500$ GeV the limit is $|V_{\mu N}|^{2}=0.6$.
In this paper we leverage boosted $W^{\pm}$ production from massive RHN, and its subsequent decay into a fat-jet in association with $\mu^{\pm}\mu^{\pm}$ pairs. The $P_{T}$ of the $W^{\pm}$ scale as $P_{T}^{W}\sim(M_{N}^{2}-M_{W}^{2})/M_{N}$ and the separation between the hadronic decay products of $W^{\pm}$ scale as $\sim M_{W}/P_{T}^{W}$. Therefore, a natural region of focus may be the intermediate to heavy RHN mass range, say $M_{N}\geq 300$ GeV. In this mass range, the only other competent limit that exists comes from indirect EW precision data (EWPD). The EWPD limit is around $|V_{\mu N}|^{2}=0.009$ deBlas:2013gla ; delAguila:2008pw ; Akhmedov:2013hec . The mixing limits may also be obtained from the Higgs data BhupalDev:2012zg ; Das:2017zjc ; Das:2017rsu for $10$ GeV $\leq M_{N}\leq 200$ GeV.
For simplicity and clarity, we consider only the $\mu$ flavor for the SSDL. Moreover, $\mu$ detection efficiencies are better, compared to electrons and tau leptons. We place limits on $|V_{\mu N}|^{2}$ at the 13 TeV LHC, with 3000 fb ${}^{-1}$ luminosity, in the $300$ GeV $\leq M_{N}\leq 800$ GeV mass range. A representative diagram for the parton level production and decay of RHN, leading to final states of interest, is shown in figure 1.
Search strategies utilising boosted and collimated objects have proven to be spectacularly successful in searches at the LHC. The seminal ideas Seymour:1993mx ; Butterworth:2007ke ; Brooijmans:1077731 ; Butterworth:2008iy have burgeoned into many sophisticated methods that enable tagging jets arising from the decay of boosted heavy particles, improving searches for new topologies, investigating jet properties and mitigating underlying events and pile-up (please see Adams:2015hiv and references therein for a review of some these techniques).
In the context of sterile neutrinos and related models there have been a few studies that have, in a broader sense, leveraged the effectiveness of collimated objects in the signal topology Izaguirre:2015pga ; Antusch:2016ejd ; Mitra:2016kov ; Dube:2017jgo ; Cox:2017eme . Nevertheless, surprisingly, there have not been any investigations in the SSDL+fat-jet channel, in the RHN collider search context. We utilise for the first time, jet substructure techniques to augment RHN searches, in the $l^{\pm}l^{\pm}J$ channel corresponding to figure 1.
The paper is organized as follows. In section 2, we discuss the prototypical model along with the RHN production cross sections at the 13 TeV LHC. We also calculate the decay widths and the corresponding branching ratios there. In section 3, we briefly describe the fat-jet technique for W-tagging.
Sections 4 and 5 are dedicated to the setup, collider analysis, discussion of kinematic distributions, and presentation of the salient results and limits. We conclude in section 6.
2 Model and heavy Majorana neutrinos at the LHC
In the simplest model of seesaw, we only introduce SM gauge-singlet Majorana RHNs $N_{R}^{\beta}$ (where $\beta$ is a flavor index). $N_{R}^{\beta}$ would couple with the SM lepton doublet $\ell_{L}^{\alpha}$ and the Higgs doublet $H$.
The relevant part of the Lagrangian density is
$$\displaystyle\mathcal{L}\supset-Y_{D}^{\alpha\beta}\overline{\ell_{L}^{\alpha}%
}HN_{R}^{\beta}-\frac{1}{2}M_{N}^{\alpha\beta}\overline{N_{R}^{\alpha C}}N_{R}%
^{\beta}+H.c..$$
(1)
After EW symmetry breaking by a vacuum expectation value (VEV) $H=\begin{pmatrix}\frac{v}{\sqrt{2}}~{}0\end{pmatrix}^{T}$, we obtain the Dirac mass matrix $M_{D}=\frac{Y_{D}v}{\sqrt{2}}$. Using these Dirac and Majorana mass matrices, we can write the full neutrino mass matrix as
$$\displaystyle M_{\nu}=\begin{pmatrix}0&&M_{D}\\
M_{D}^{T}&&M_{N}\end{pmatrix}.$$
(2)
Diagonalizing this matrix, we obtain the well-known seesaw formula for the light Majorana neutrinos
$$\displaystyle m_{\nu}\simeq-M_{D}M_{N}^{-1}M_{D}^{T}.$$
(3)
With $M_{N}\sim 100$ GeV, we require $Y_{D}\sim 10^{-6}$ for $m_{\nu}\sim 0.1$ eV. However, in the general parameterization for the seesaw formula Casas:2001sr , $Y_{D}$ can be large and sizable, which is the case we are going to consider in this paper. An interesting class of models have mass matrices $M_{D}$ and $M_{N}$ with specific textures, enforced by some symmetries Pilaftsis:2003gt ; Kersten:2007vk ; Xing:2009in ; He:2009ua ; Ibarra:2010xw ; Deppisch:2010fr ; Dev:2013oxa , so that a large light-heavy neutrino mixing occurs even at a low scale, satisfying the neutrino oscillation data.
If these RHNs reside at the electroweak scale, then they can be produced in high energy colliders such as the LHC Keung:1983uu ; Datta:1992qw ; Datta:1993nm ; AguilarSaavedra:2009ik ; Chen:2011hc ; Das:2012ze ; Das:2014jxa ; Das:2015toa ; Dev:2015pga ; Das:2016akd ; Gluza:2016qqv ; delAguila:2008hw ; delAguila:2007qnc ; AguilarSaavedra:2012gf ; Nayak:2015zka ; Nayak:2013dza ; AguilarSaavedra:2012fu ; LalAwasthi:2011aa ; Fong:2011xh ; Dias:2011sq ; Ibarra:2011xn ; ILC1 ; ILC2 ; type-I1 ; type-I2 ; Batell:2015aha ; Leonardi:2015qna ; p-photon ; konar1 ; konar2 ; Dutta:1994wz ; Haba:2009sd ; Matsumoto:2010zg ; Mondal:2012jv ; Helo:2013esa ; Hessler:2014ssa ; Deppisch:2015qwa ; Arganda:2015ija ; Dib:2015oka . Searches for Majorana RHNs can be performed via the ‘smoking-gun’ tri-lepton, as well as, SSDL+dijet signals. The rates will generally be suppressed by the square of light-heavy mixing $|V_{\ell N}|^{2}\simeq|M_{D}M_{N}^{-1}|^{2}$.
A comprehensive, general study111The study uses data from neutrino oscillation experiments Neut1 ; Neut2 ; Neut3 ; Neut4 ; Neut5 ; Neut6 , bounds from Lepton Flavor Violation (LFV) Adam:2011ch ; Aubert:2009ag ; OLeary:2010hau , Large Electron-Positron (LEP) Achard:2001qv ; delAguila:2008pw ; Akhmedov:2013hec experiments using the non-unitarity effects Antusch:2006vwa ; Abada:2007ux applying the Casas- Ibarra conjecture Casas:2001sr ; Ibarra:2010xw ; Ibarra:2011xn ; Dinh:2012bp .
of $|V_{\ell N}|^{2}$ and associated parameters is given in Das:2017nvm .
Bounds may be placed on the light-heavy mixing angles using results from different experiments, as in Aad:2015xaa ; CMS8:2016olu ; Khachatryan:2015gha ; delAguila:2008pw ; Akhmedov:2013hec ; deBlas:2013gla ; BhupalDev:2012zg ; KamLAND-Zen:2016pfg ; Das:2017zjc ; Das:2017rsu ; Achard:2001qv ; Rasmussen:2016njh , considering degenerate Majorana RHNs.
Through the seesaw mechanism, a flavor eigenstate ($\nu$) of the SM neutrino may be expressed in terms of the mass eigenstates of the light ($\nu_{m}$) and heavy ($N_{m}$) Majorana neutrinos as
$$\displaystyle\nu\simeq\mathcal{N}\nu_{m}+\mathcal{R}N_{m}\;.$$
(4)
Here
$$\displaystyle\mathcal{R}=M_{D}M_{N}^{-1}~{}~{},~{}~{}~{}\mathcal{N}=\big{(}1-%
\frac{1}{2}\epsilon\big{)}U_{\rm{PMNS}}\;,$$
(5)
with $\epsilon=\mathcal{R^{\ast}}\mathcal{R}^{T}$ and $U_{\rm{PMNS}}$ Pontecorvo:1957qd ; Maki:1962mu the usual neutrino mixing matrix. In terms of mass eigenstates, the charged current interactions for the heavy neutrinos is then given by
$$\displaystyle\mathcal{L}_{CC}=-\frac{g}{\sqrt{2}}W_{\mu}\bar{e}\gamma^{\mu}P_{%
L}\Big{(}\mathcal{N}\nu_{m}+\mathcal{R}N_{m}\Big{)}+h.c.,$$
(6)
where $e$ denotes three generations of charged leptons, in vector form, and
$P_{L}=\frac{1}{2}(1-\gamma_{5})$. Similarly, the neutral current interactiona are given by
$$\displaystyle\mathcal{L}_{NC}$$
$$\displaystyle=$$
$$\displaystyle-\frac{g}{2c_{w}}Z_{\mu}\Big{[}\overline{\nu_{m}}\gamma^{\mu}P_{L%
}({\cal N}^{\dagger}{\cal N})\nu_{m}+\overline{N_{m}}\gamma^{\mu}P_{L}({\cal R%
}^{\dagger}{\cal R})N_{m}$$
(7)
$$\displaystyle+$$
$$\displaystyle\Big{\{}\overline{\nu_{m}}\gamma^{\mu}P_{L}({\cal N}^{\dagger}{%
\cal R})N_{m}+h.c.\Big{\}}\Big{]},$$
where $c_{w}=\cos\theta_{w}$ with $\theta_{w}$ being the weak mixing angle.
At the LHC, the heavy neutrinos can be produced through charged current interactions, via the $s$-channel exchange of W bosons. The main production process at the parton level is $u\bar{d}\rightarrow\mu^{+}N$ (and $\bar{u}d\rightarrow\mu^{-}N$). The differential cross section is found to be
$$\displaystyle\frac{d\hat{\sigma}_{LHC}}{d\cos\theta}$$
$$\displaystyle=$$
$$\displaystyle(3.89\times 10^{8}\;{\rm pb})\times\frac{\beta}{32\pi\hat{s}}%
\frac{\hat{s}+M_{N}^{2}}{\hat{s}}\Big{(}\frac{1}{2}\Big{)}^{2}3\Big{(}\frac{1}%
{3}\Big{)}^{2}\frac{g^{4}}{4}$$
(8)
$$\displaystyle\frac{({\hat{s}}^{2}-M_{N}^{4})(2+\beta\cos^{2}\theta)}{({\hat{s}%
}-M_{W}^{2})^{2}+M_{W}^{2}\Gamma_{W}^{2}},$$
where $\sqrt{\hat{s}}$ is the center-of-mass energy of the colliding partons, $M_{N}$ the mass of $N$, and $\beta=({\hat{s}}-M_{N}^{2})/({\hat{s}}+M_{N}^{2})$. The total production cross section at the LHC is thus given by
$$\displaystyle\sigma_{LHC}$$
$$\displaystyle=$$
$$\displaystyle\int d\sqrt{\hat{s}}\int d\cos\theta\int^{1}_{{\hat{s}}/E_{CMS}^{%
2}}dx\frac{\sqrt{4{\hat{s}}}}{xE_{CMS}^{2}}f_{u}(x,Q)f_{\bar{d}}\left(\frac{%
\hat{s}}{xE_{CMS}},Q\right)\frac{d\hat{\sigma}_{LHC}}{d\cos\theta}$$
(9)
$$\displaystyle+$$
$$\displaystyle(u\to{\bar{u}},{\bar{d}}\to d)\;.$$
We take $E_{CMS}=13$ TeV, for the center-of-mass energy of the LHC. In the numerical analysis, we further employ CTEQ5M Pumplin:2002vw for the $u$-quark ($f_{u}$) and ${\bar{d}}$-quark ($f_{\bar{d}}$) parton distribution functions, with a factorization scale $Q=\sqrt{\hat{s}}$. The total cross section thus computed, as a function of $M_{N}$, is depicted in figure 2 (Left pane), normalized by $|V_{\mu N}|^{2}$. Hence, the resultant cross sections shown in figure 2 correspond to maximum values for a fixed $M_{N}$.
The main decay modes of the heavy neutrino are
$N\to\ell W$, $\nu_{\ell}Z$, $\nu_{\ell}h$.
The corresponding partial decay widths Das:2012ze ; Das:2014jxa ; Das:2015toa ; Das:2016hof ; Das:2017pvt are given by
$$\displaystyle\Gamma(N\rightarrow\ell W)$$
$$\displaystyle=$$
$$\displaystyle\frac{g^{2}|V_{\ell N}|^{2}}{64\pi}\frac{(M_{N}^{2}-M_{W}^{2})^{2%
}(M_{N}^{2}+2M_{W}^{2})}{M_{N}^{3}M_{W}^{2}},$$
$$\displaystyle\Gamma(N\rightarrow\nu_{\ell}Z)$$
$$\displaystyle=$$
$$\displaystyle\frac{g^{2}|V_{\ell N}|^{2}}{128\pi c_{w}^{2}}\frac{(M_{N}^{2}-M_%
{Z}^{2})^{2}(M_{N}^{2}+2M_{Z}^{2})}{M_{N}^{3}M_{Z}^{2}},$$
$$\displaystyle\Gamma(N\rightarrow\nu_{\ell}h)$$
$$\displaystyle=$$
$$\displaystyle\frac{|V_{\ell N}|^{2}(M_{N}^{2}-M_{h}^{2})^{2}}{32\pi M_{N}}%
\left(\frac{1}{v}\right)^{2}.$$
(10)
Note that the decay width of heavy neutrinos into $W^{\pm}$ is about twice as large as that into $Z^{0}$, owing to the two degrees of freedom.
We plot the branching ratios $BR_{i}\left(\equiv{\Gamma_{i}}/{\Gamma_{\rm total}}\right)$ of the various decay modes $\left(\Gamma_{i}\right)$ in figure 2 (Right pane). Note that for larger values of $M_{N}$, the branching ratios are related as
$$\displaystyle BR\left(N\rightarrow\ell W\right):BR\left(N\rightarrow\nu Z%
\right):BR\left(N\rightarrow\nu H\right)\simeq 2:1:1.$$
(11)
As mentioned earlier, in our analysis we will consider Majorana RHNs having mass in the range $300$ GeV $\leq M_{N}\leq 800$ GeV. In this mass range, the $W^{\pm}$ boson from the leading decay mode $N\rightarrow\ell W$ (see figure 2), will be boosted. These boosted $W^{\pm}$ can decay hadronically to produce a fat-jet, with the characteristic final state $\mu^{\pm}\mu^{\pm}J$.
3 Fat-jets and Jet Substructure for $W$-like jet tagging
As we have emphasized, in scenarios where the right-handed neutrino is very heavy, the hadronically decaying daughter $W^{\pm}$ will typically have a large boost. This causes the jets from the $W^{\pm}$ to be very collimated and one would detect them as a single jet – a ‘fat-jet’ ($J$). The boosted topology and its associated substructure is extremely powerful in reducing backgrounds, mitigating underlying event contamination and in event tagging Adams:2015hiv . In our context, the jet substructure analysis primarily appears as a means to efficiently tag the hadronically decaying boosted-$W^{\pm}$. Our strategy will be to leverage two variables – N-subjettiness and jet-mass – to achieve efficient W-tagging in the $\mu^{\pm}\mu^{\pm}J$ final state.
N-subjettiness Thaler:2010tr ; Thaler:2011gf is an inclusive jet shape variable defined as
$$\tau_{N}=\frac{1}{\mathcal{N}_{0}}\sum\limits_{i}p_{i,T}\min\left\{\Delta R_{i%
1},\Delta R_{i2},\cdots,\Delta R_{iN}\right\}.$$
(12)
The normalization is defined as $\mathcal{N}_{0}=\sum\limits_{i}p_{i,T}R$. $i$ runs over the constituent particles in the jet. $p_{i,T}$ are transverse momenta of the constituent particles, $\Delta R_{i\alpha}=\sqrt{(\Delta\eta)^{2}_{i\alpha}+(\Delta\phi)^{2}_{i\alpha}}$ is the $\eta-\phi$ distance between a candidate $\alpha$-subjet and a constituent particle $i$ and $R$ is the jet radius. $\tau_{N}$ tries to quantify if the original jet consists of N daughter subjets. A low value of $\tau_{N}$ suggests that the original jet consists of $N$ or fewer daughter subjets. Thus, information from $\tau_{N}$ may potentially be used to identify an object that has an N-prong hadronic decay. In fact, it has been shown that a better discriminant to tag an N-subjet object is to consider ratios $\tau_{N}/\tau_{N-1}$ Thaler:2010tr ; Thaler:2011gf . For W-tagging, the $W^{\pm}$ yields two subjets that are collimated, and hence the variable of interest is $\tau_{21}^{J}=\tau_{2}/\tau_{1}$. The mass of the fat-jet ($M_{J}$), after suitable jet grooming, is another variable that can help in distinguishing signal events from background. At each iteration in a sequential recombination jet algorithm, in the E-scheme, the mother proto jet four-momentum is the vector sum of the daughter proto jet four-momenta. In this fashion, the jet algorithm at the end of the iteration provides a $P_{T}^{J}$ for the full fat-jet. $M_{J}^{2}$ is computed as the invariant mass square of the fat-jet four momentum ($P_{J}^{2}$).
To reconstruct the candidate fat-jet, Delphes 3.3.2 deFavereau:2013fsa hadron calorimeter outputs are clustered using FastJet 3.1.3 hep-ph/0512210 ; Cacciari:2011ma . The $\tau_{21}^{J}$ is computed with the aid of the N-subjettiness extension, available as part of the FastJet-contrib hep-ph/0512210 ; Cacciari:2011ma . Following Khachatryan:2014vla for W-tagging, we will choose for the jet clustering algorithm Cambridge-Achen Dokshitzer:1997in ; Wobisch:1998wt with a jet-cone radius $R=0.8$. We will in addition require specific cuts on $\tau_{21}^{J}$ and $M_{J}$ for efficient W-tagging, as we shall discuss in section 5.
4 Analysis setup and Simulation
In preparation for our exploration of the SSDL+fat-jet channel at 13 TeV LHC, along with establishing the setup in terms of signal RHN model files and a jet substructure analysis strategy, we must also consider the relevant backgrounds carefully. Towards this end we will perform detailed background simulations and study the prospects of our proposed channel, in terms of statistical significance.
Consider the production of heavy RHN, through an off-shell $W^{\pm}$. This in a further decay produces relatively clean, same sign di-muon pair $\mu^{\pm}\mu^{\pm}$ final states, in association with a boosted $W^{\pm}$. Our primary objective is to unmask these $W^{\pm}$ from other hadronic backgrounds. This is efficiently achieved by utilizing jet substructure to W-tag the fat-jet originating from $W^{\pm}$. Of course, one expects from our previous discussions that the fat-jet and jet substructure techniques become significant when $W^{\pm}$ bosons are generated with sufficient boost. Hence, as mentioned earlier, our primary region of interest is when $M_{N}\gg M_{W}$. Notably, these are also the ranges where conventional searches at colliders fail to probe the mixing parameters very effectively. Corresponding to the signal production channel depicted in figure 1, we will consider
$$\displaystyle pp$$
$$\displaystyle\rightarrow\ell_{1}^{+}N,\;\;N\rightarrow\ell_{2}^{+}W^{-},\;\;W^%
{-}\rightarrow J$$
$$\displaystyle pp$$
$$\displaystyle\rightarrow\ell_{1}^{-}\overline{N},\;\;\overline{N}\rightarrow%
\ell_{2}^{-}W^{+},\;\;W^{+}\rightarrow J.$$
(13)
As mentioned before, for concreteness we assume the light-heavy mixing is non-zero only for the muon flavor in a simplified model. The muons also provide cleaner lepton signals. Hence, all leptons we consider in this study will be muons. It is straightforward to extend the analysis if more lepton flavors are allowed.
Backgrounds for our SSDL+fat-jet channel can originate from electroweak gauge boson decays along with a fat-jet; the latter for instance produced from a W boson decaying to $J$. Additionally, some of the QCD jets can also mimic $J$. Hence, one is required to simulate all such processes accompanied by hard jet(s) at the parton level, and then match them with shower jet events.
Dominant contributions come from same-sign $W^{\pm}$ pair production in association with jets – $W^{\pm}W^{\pm}+jets$. Here, $W^{\pm}$ would decay leptonically. One of these jets has the possibility to resemble a $W^{\pm}$-like fat-jet. Another significantly large contribution comes from $WZ$ production, where both vector bosons decay leptonically. Subsequently, one of the charged lepton is missed in the detector, giving an SSDL signature. An additional fat-jet like component can come either from a radiated jet or an associated $W^{\pm}$ boson decaying hadronically.
We implement the parton level event generation using MadGraph5-aMC@NLO MG ; MG5 and signal model files are generated with FeynRules Christensen:2008py ; Alloul:2013bka . CTEQ6L Pumplin:2002vw is adopted for the parton distribution functions (PDF) and the factorization scale $\mu_{F}$ is set to the default MadGraph option. The showering, fragmentation and hadronization of the generated events were performed with PYTHIA6.4 Sjostrand:2006za .
The matching is done using the MLM schemeHoche:2006ph ; based on a shower-kT algorithm with pT-ordered showers. For SM backgrounds, the matching scale QCUT is set between 20 and 30 GeV. The showered events are passed through Delphes 3.3.2 deFavereau:2013fsa for detector level simulations with the default CMS card. The jets and associated substructure variables are constructed as described in section 3.
To establish specific features and kinematic characteristics related to our RHN signal and backgrounds, we start by focusing on signal identification. Our prototypical signal is Same Flavour ($\mu^{\pm}$ ) SSDL, in association with a fat-jet. We adopt the following selection criteria
•
Muons $\mu^{\pm}$ are identified with a minimum transverse momentum $p^{\mu}_{T}>10$ GeV and rapidity range $|\eta^{\mu}|<2.4$, with a maximum efficiency of $95\%$. Efficiency decreases for $p^{\mu}_{T}$ above $1\,\mathrm{TeV}$.
•
Only events with reconstructed di-muons having same sign are selected for further analysis.
•
Hard jets having at least $p_{T}^{j}>10\,\mathrm{GeV}$ and $|\eta^{j}|<2.4$ are identified.
•
Candidate fat-jets are to be identified, following the criteria in section 3 (an $R=0.8$, CA jet with $|\eta^{J}|<2.4$).
•
We identify the hardest fat-jet with the $W^{\pm}$ candidate jet ($J$), and this is required to have $p^{J}_{T}>100$ GeV.
The above basic selection criteria are like primary level cuts required for effective signal identification. The last requirement is to ensure robust fat-jet properties. As argued earlier, features of the boosted fat-jet are rather more prominent for large $M_{N}$; showing up emphatically for 300 GeV and above. In the next section we introduce some additional event criteria and then illustrate various results by considering several signal benchmark points.
5 Results and Discussion
In the previous section, basic selection criteria were set. We are now in a position to identify specific features and kinematic characteristics that can further differentiate RHN events from SM backgrounds. To highlight the differences, we focus on four key characteristic distributions, considering backgrounds along with three signal benchmark points (based on $M_{N}=300,\;500$ and $700$ GeV).
Figure 3 illustrates the normalized differential distribution of events as a function of missing transverse momentum, after the application of the basic selection cuts. Missing transverse momentum (MET) is calculated from the contributions of isolated electrons, muons, photons and jets along with unclustered deposits. Our signal of interest from RHN involves no missing particle at the detector and is thus expected to have low MET. The only MET contributions may be from the mismeasurement of hard jets. On the contrary, in almost all relevant background processes, leptons originate from $W^{\pm}$ along with a neutrino. The neutrinos are not detected and contribute to a large MET. Distribution of one prototypical signal region with all dominant backgrounds is shown in the plot. It clearly shows the larger MET contribution for the backgrounds. Inset shows the same distribution for three benchmark signal points, $M_{N}=300,\;500$ and $700$ GeV. Distributions are very mildly sensitive to $M_{N}$, since heavier masses contribute to harder boosted jets and the jet-energy mis-measurements have a $P_{T}$ dependence.
Figure 4 presents the normalized differential distributions for the fat-jet transverse momentum in a similar way. Here, minimum $P_{T}^{J}$ of 100 GeV has already been imposed. As we discussed in section 3, $P_{T}^{J}$ is the vector sum of all constituent four momenta in $J$. Signal distribution is noticeably harder compared to background distributions, which fall faster. Comparison of different signal distributions is also quite interesting. As expected, heavier $M_{N}$ produces harder $J$ candidates.
Imposing a minimum $P_{T}^{J}$ selection brings out marked differences in the distribution of $M_{J}$ and $\tau_{21}^{J}$, between signal and backgrounds events. These jet shape variables can be very powerful in further containing the backgrounds.
Two fat-jet invariant mass peaks are evident from the figure 5. Second peak at around 80 GeV reflects the jet mass of $W$ like fat-jet $J$. This peak is absent for those backgrounds where fat-jet is faked by QCD jets. Only the triple gauge boson background, where fat-jet can originate from hadronic decay of one of the W’s, provide some contamination to signal. The signal plots in the inset are also quite instructive, showing the significant $W$ like fat-jet contributions for higher $M_{N}$. The small spurious peak is due to events where some four-momenta from the hadronically decaying boosted-$W^{\pm}$ is missed in the jet clustering. This spurious peak around $20\,\mathrm{GeV}$ may be reduced by imposing a larger $P_{T}^{J}$. This would of course cut down the signal as well, and we find $P_{T}^{J}\gtrsim 100\,\mathrm{GeV}$ to provide the most optimal signal significance.
Another excellent discriminant to tag a hadronic two-pronged object is $\tau_{21}^{J}=\tau_{2}/\tau_{1}$, as we discussed in section 3. Corresponding distributions are shown in figure 6. $\tau_{21}^{J}$ for $W^{\pm}$-like fat-jets peak around small values and this is clearly visible in figure 6. It becomes more prominent for larger $M_{N}$, as the inset figure shows, due to the $J$ being more boosted.
It is important to reemphasize that the choice of a higher, minimum $P_{T}^{J}$ effectively selects purer, $W^{\pm}$-like fat-jet events, but probably at the cost of some signal. This is essentially reflected in the larger event fractions in the higher (lower) peaks for $m^{J}$ ($\tau_{21}^{J}$). This would result in a sharper peak and background reductions. We find $P_{T}^{J}>150\,\mathrm{GeV}$ to be optimal for selecting events, while maintaining good signal significance, as mentioned.
We list below our final event selection criteria motivated by the kinematic distributions.
•
Leading muon should have $p_{T}(\mu_{1})>20$ GeV and the next hardest muon must have $p_{T}(\mu_{2})>15$ GeV.
•
Minimum invariant mass for the same sign muon pair must satisfy $m_{\mu\mu}>50$ GeV. This is easily satisfied for the signal events, and can control backgrounds with non-prompt muon pairs.
•
Lacking any missing particles for our signal, require $E_{T}^{\rm{miss}}<35$ GeV. This can control background events with large MET contributions.
•
The hardest, reconstructed fat-jet must have $p_{T}^{J}>150$ GeV.
•
We also demand the invariant mass of the hardest, reconstructed fat-jet to satisfy $M_{J}>50$ GeV. In principle one may use a mass window around the $W^{\pm}$ mass, but we find that a simple lower bound suffices.
•
The N-subjettiness ratio corresponding to the reconstructed fat-jet must satisfy $\tau_{21}^{J}<0.5$.
With these we are able to achieve very significant background elimination, relative to the signal.
Now we present our results. The effects of the different cuts, as we have motivated, are summarized in Table 1 in the form of a cut-flow. Three reference RNH benchmark points are presented with masses $300$ GeV, $500$ GeV and $700$ GeV. It is quite clear, in reference to the different distributions shown earlier, that the choice of these cuts are extremely efficient in controlling the large SM backgrounds. This enables the RHN signal to be probed to a significant mass range, or alternatively to smaller mixing angles, at the LHC.
The statistical significance ($\cal{S}$) of the observed signal events ($S$) over the total SM background events ($B$) has been calculated using
$$\displaystyle{\cal S}$$
$$\displaystyle=$$
$$\displaystyle S/\sqrt{B}\text{ %
for $5\sigma$ significance,}$$
(14)
$$\displaystyle{\cal S}$$
$$\displaystyle=$$
$$\displaystyle\sqrt{2\times\left[(S+B)\ln(1+\frac{S}{B})-S\right]}\text{ %
for $2\sigma$ and $3\sigma$ significance.}$$
(15)
figure 7 displays the significance contours in the $(M_{N},|V_{\mu N}|^{2})$ plane. These contours reflect the extensive capability of RHN searches augmented by jet substructure techniques. One obtains interesting limits all the way from $M_{N}=300$ GeV with $|V_{\mu N}|^{2}=2.6\times 10^{-4}$ to $M_{N}=800$ GeV with $|V_{\mu N}|^{2}=9.6\times 10^{-4}$.
It is instructive to compare our projected collider limits with existing LHC limits, as well as indirect EWPD bounds. This is shown in figure 8. There are currently no limits above $M_{N}=500$ GeV, from any experiment. From ATLAS searches Aad:2015xaa at $\sqrt{s}=8\,\mathrm{TeV}$, the blue solid line shows the limits for the $e^{\pm}e^{\pm}jj$ channel and the brown solid line for the $\mu^{\pm}\mu^{\pm}jj$ channel. The orange dashed line shows the limits for $e^{\pm}e^{\pm}jj$ and the green dot-dashed line shows limits for $\mu^{\pm}\mu^{\pm}jj$, both from CMS CMS8:2016olu ; Khachatryan:2015gha . The light gray solid line stands for the EWPD limit for $\mu$ flavor mixings Achard:2001qv ; delAguila:2008pw ; Akhmedov:2013hec .
Note that our event selection criteria is optimized to the lower side of the heavy neutrino mass and we have applied the same cuts universally for the full signal mass range. Now, we have already observed in the inset plots of Figs. 3-6, that distributions change with $M_{N}$. Hence, there is sufficient room left to improve our results for higher $M_{N}$, by focused optimizations at each mass point. Instead of fine tuning the analysis, our main aim here was to demonstrate the efficacy and usefulness of jet substructure analysis in the general RHN collider search context self-ongoing .
6 Conclusions
Seesaw models can naturally incorporate the existence of tiny neutrino masses and flavor mixings through simple extensions of the SM, many of which have Majorana RHNs. Such RHNs, if they exist at the TeV scale, can be produced and detected at the LHC. Searches have been performed for these states in the dilepton+dijet channel. Here we propose for the first time a search in the dilepton+fat-jet channel, leveraging jet substructure methods which can very significantly increase the LHC reach for these RHN states. In our case we considered the mass region $M_{N}\geq 300$ GeV. We used the unique kinematic characteristics of a fat-jet – such as $P_{T}^{J}$, $M^{J}$ and $\tau_{21}^{J}$– to W-tag it and it was found that this helps greatly in discriminating the RHN SSDL+fat-jet signal from backgrounds. Exclusion limits are obtained by computing signal significance and the bounds we obtain are many orders of magnitude stronger than current LHC limits.
Acknowledgements.We thank Akanksha Bhardwaj for independent validation of this work and also facilitating JaxoDraw Binosi:2003yf ; Binosi:2008ig generated diagram.
References
(1)
K. Abe et. al. [T2K Collaboration] Phys. Rev. Lett. 107, 041801 (2011).
(2)
P. Adamson et al. [MINOS Collaboration], Phys. Rev. Lett. 107, 181802 (2011).
(3)
Y. Abe et al. [DOUBLE-CHOOZ Collaboration], Phys. Rev. Lett. 108, 131801 (2012).
(4)
F. P. An et al. [DAYA-BAY Collaboration], Phys. Rev. Lett. 108, 171803 (2012).
(5)
J. K. Ahn et al. [RENO Collaboration], Phys. Rev. Lett. 108, 191802 (2012).
(6)
C. Patrignani et al. [Particle Data Group],
“Review of Particle Physics,”
Chin. Phys. C 40, no. 10, 100001 (2016).
doi:10.1088/1674-1137/40/10/100001
(7)
S. Weinberg,
“Baryon and Lepton Nonconserving Processes,”
Phys. Rev. Lett. 43, 1566 (1979).
doi:10.1103/PhysRevLett.43.1566
(8)
P. Minkowski,
“$\mu\to e\gamma$ at a Rate of One Out of $10^{9}$ Muon Decays?,”
Phys. Lett. 67B, 421 (1977).
doi:10.1016/0370-2693(77)90435-X
(9)
T. Yanagida, “Horizontal Symmetry and Masses of Neutrinos,” Prog. Theor. Phys. 64, 1103 (1980).
(10)
J. Schechter and J. W. F. Valle, “Neutrino Masses in SU(2) $\otimes$ U(1) Theories,” Phys. Rev. D 22, 2227 (1980).
(11)
T. Yanagida, in Proceedings of the Work- shop on the Unified Theory and the Baryon Number in the Universe (O. Sawada and A. Sugamoto, eds.), KEK, Tsukuba, Japan, 1979, p. 95.
(12)
M. Gell-Mann, P. Ramond, and R. Slansky, Supergravity (P. van Nieuwenhuizen et al. eds.), North Holland, Amsterdam, 1979, p. 315.
(13)
S. L. Glashow, The future of elementary particle physics, in Proceedings of the 1979 Carg‘ese Summer Institute on Quarks and Leptons (M. Levy et al. eds.), Plenum Press, New York, 1980, p. 687.
(14)
R. N. Mohapatra and G. Senjanovic, “Neutrino Mass and Spontaneous Parity Violation,” Phys. Rev. Lett. 44, 912 (1980).
(15)
K. Fujikawa,
Prog. Theor. Phys. 113 (2005) 1065
doi:10.1143/PTP.113.1065
[hep-ph/0407331].
(16)
A. de Gouvea,
Phys. Rev. D 72 (2005) 033005
doi:10.1103/PhysRevD.72.033005
[hep-ph/0501039].
(17)
G. Aad et al. [ATLAS Collaboration],
“Search for heavy Majorana neutrinos with the ATLAS detector in pp collisions at $\sqrt{s}=8$ TeV,”
JHEP 1507, 162 (2015)
doi:10.1007/JHEP07(2015)162
[arXiv:1506.06020 [hep-ex]].
(18)
V. Khachatryan et. al. [CMS Collaboration],
“Search for heavy Majorana neutrinos in e${}^{\pm}$e${}^{\pm}$+jets and e${}^{\pm}\mu^{\pm}$+jets events in proton-proton collisions at $\sqrt{s}=8$ TeV,”
JHEP 1604, 169 (2016)
doi: 10.1007/JHEP04 (2016)169
[arxiv:1603.02248 [hep-ex]].
(19)
V. Khachatryan et al. [CMS Collaboration],
“Search for heavy Majorana neutrinos in $\mu^{\pm}\mu^{\pm}+$ jets events in proton-proton collisions at $\sqrt{s}$ = 8 TeV,”
Phys. Lett. B 748, 144 (2015)
doi:10.1016/j.physletb.2015.06.070
[arXiv:1501.05566 [hep-ex]].
(20)
F. del Aguila, J. de Blas and M. Perez-Victoria,
“Effects of new leptons in Electroweak Precision Data,”
Phys. Rev. D 78, 013010 (2008)
doi:10.1103/PhysRevD.78.013010
[arXiv:0803.4008 [hep-ph]].
(21)
J. de Blas,
“Electroweak limits on physics beyond the Standard Model,”
EPJ Web Conf. 60, 19008 (2013)
doi:10.1051/epjconf/20136019008
[arXiv:1307.6173 [hep-ph]].
(22)
A. Gando et al. [KamLAND-Zen Collaboration],
“Search for Majorana Neutrinos near the Inverted Mass Hierarchy Region with KamLAND-Zen,”
Phys. Rev. Lett. 117, no. 8, 082503 (2016)
Addendum: [Phys. Rev. Lett. 117, no. 10, 109903 (2016)]
doi:10.1103/PhysRevLett.117.109903, 10.1103/PhysRevLett.117.082503
[arXiv:1605.02889 [hep-ex]] .
(23)
E. Akhmedov, A. Kartavtsev, M. Lindner, L. Michaels and J. Smirnov,
“Improving Electro-Weak Fits with TeV-scale Sterile Neutrinos,”
JHEP 1305, 081 (2013)
doi:10.1007/JHEP05(2013)081
[arXiv:1302.1872 [hep-ph]].
(24)
P. S. Bhupal Dev, R. Franceschini and R. N. Mohapatra,
“Bounds on TeV Seesaw Models from LHC Higgs Data,”
Phys. Rev. D 86, 093010 (2012)
doi:10.1103/PhysRevD.86.093010
[arXiv:1207.2756 [hep-ph]].
(25)
A. Das, P. S. B. Dev and C. S. Kim,
“Constraining Sterile Neutrinos from Precision Higgs Data,”
Phys. Rev. D 95, no. 11, 115013 (2017)
doi:10.1103/PhysRevD.95.115013
[arXiv:1704.00880 [hep-ph]].
(26)
A. Das, Y. Gao and T. Kamon,
“Heavy Neutrino Search via the Higgs boson at the LHC,”
arXiv:1704.00881 [hep-ph].
(27)
M. H. Seymour, Searches for new particles using cone and cluster jet
algorithms: A Comparative study, Z.Phys. C62 (1994) 127–138.
(28)
J. Butterworth, J. R. Ellis, and A. Raklev, Reconstructing sparticle mass
spectra using hadronic decays, JHEP 0705 (2007) 033,
[hep-ph/0702150].
(29)
G. Brooijmans, High pt hadronic top quark identification, Tech. Rep.
ATL-PHYS-CONF-2008-008. ATL-COM-PHYS-2008-001, CERN, Geneva, Jan, 2008.
(30)
J. M. Butterworth, A. R. Davison, M. Rubin, and G. P. Salam, Jet
substructure as a new Higgs search channel at the LHC, Phys.Rev.Lett. 100 (2008) 242001,
[arXiv:0802.2470].
(31)
D. Adams et al.,
“Towards an Understanding of the Correlations in Jet Substructure,”
Eur. Phys. J. C 75 (2015) no.9, 409
doi:10.1140/epjc/s10052-015-3587-2
[arXiv:1504.00679 [hep-ph]].
(32)
E. Izaguirre and B. Shuve,
“Multilepton and Lepton Jet Probes of Sub-Weak-Scale Right-Handed Neutrinos,”
Phys. Rev. D 91 (2015) no.9, 093010
doi:10.1103/PhysRevD.91.093010
[arXiv:1504.02470 [hep-ph]].
(33)
S. Antusch, E. Cazzato and O. Fischer,
“Sterile neutrino searches at future $e^{-}e^{+}$, $pp$, and $e^{-}p$ colliders,”
arXiv:1612.02728 [hep-ph].
(34)
M. Mitra, R. Ruiz, D. J. Scott and M. Spannowsky,
“Neutrino Jets from High-Mass $W_{R}$ Gauge Bosons in TeV-Scale Left-Right Symmetric Models,”
Phys. Rev. D 94, no. 9, 095016 (2016)
doi:10.1103/PhysRevD.94.095016
[arXiv:1607.03504 [hep-ph]].
(35)
S. Dube, D. Gadkari and A. M. Thalapillil,
“Lepton-Jets and Low-Mass Sterile Neutrinos at Hadron Colliders,”
Phys. Rev. D 96 (2017) no.5, 055031
doi:10.1103/PhysRevD.96.055031
[arXiv:1707.00008 [hep-ph]].
(36)
P. Cox, C. Han and T. T. Yanagida,
“LHC Search for Right-handed Neutrinos in $Z^{\prime}$ Models,”
arXiv:1707.04532 [hep-ph].
(37)
J. A. Casas and A. Ibarra,
“Oscillating neutrinos and muon$--->e$, gamma,”
Nucl. Phys. B 618, 171 (2001)
doi:10.1016/S0550-3213(01)00475-8
[hep-ph/0103065].
(38)
W. Y. Keung and G. Senjanovic,
“Majorana Neutrinos and the Production of the Right-handed Charged Gauge Boson,”
Phys. Rev. Lett. 50, 1427 (1983).
doi:10.1103/PhysRevLett.50.1427
(39)
A. Datta, M. Guchait and D. P. Roy,
“Prospect of heavy right-handed neutrino search at SSC / CERN LHC energies,”
Phys. Rev. D 47, 961 (1993)
doi:10.1103/PhysRevD.47.961
[hep-ph/9208228].
(40)
A. Datta, M. Guchait and A. Pilaftsis,
“Probing lepton number violation via majorana neutrinos at hadron supercolliders,”
Phys. Rev. D 50, 3195 (1994)
doi:10.1103/PhysRevD.50.3195
[hep-ph/9311257].
(41)
J. A. Aguilar-Saavedra,
“Heavy lepton pair production at LHC: Model discrimination with multi-lepton signals,”
Nucl. Phys. B 828, 289 (2010)
doi:10.1016/j.nuclphysb.2009.11.021
[arXiv:0905.2221 [hep-ph]].
(42)
C. Y. Chen and P. S. B. Dev,
“Multi-Lepton Collider Signatures of Heavy Dirac and Majorana Neutrinos,”
Phys. Rev. D 85, 093018 (2012)
doi:10.1103/PhysRevD.85.093018
[arXiv:1112.6419 [hep-ph]].
(43)
A. Das and N. Okada,
“Inverse seesaw neutrino signatures at the LHC and ILC,”
Phys. Rev. D 88, 113001 (2013)
doi:10.1103/PhysRevD.88.113001
[arXiv:1207.3734 [hep-ph]].
(44)
A. Das, P. S. Bhupal Dev and N. Okada,
“Direct bounds on electroweak scale pseudo-Dirac neutrinos from $\sqrt{s}=8$ TeV LHC data,”
Phys. Lett. B 735, 364 (2014)
doi:10.1016/j.physletb.2014.06.058
[arXiv:1405.0177 [hep-ph]].
(45)
A. Das and N. Okada,
“Improved bounds on the heavy neutrino productions at the LHC,”
Phys. Rev. D 93, no. 3, 033003 (2016)
doi:10.1103/PhysRevD.93.033003
[arXiv:1510.04790 [hep-ph]].
(46)
P. S. Bhupal Dev and R. N. Mohapatra,
“Unified explanation of the $eejj$, diboson and dijet resonances at the LHC,”
Phys. Rev. Lett. 115, no. 18, 181803 (2015)
doi:10.1103/PhysRevLett.115.181803
[arXiv:1508.02277 [hep-ph]].
(47)
A. Das, N. Nagata and N. Okada,
“Testing the 2-TeV Resonance with Trileptons,”
arXiv:1601.05079 [hep-ph].
(48)
J. Gluza, T. Jelinski and R. Szafron,
“Lepton Number Violation and ‘Diracness’ of massive neutrinos composed of Majorana states,”
arXiv:1604.01388 [hep-ph].
(49)
F. del Aguila and J. A. Aguilar-Saavedra,
“Electroweak scale seesaw and heavy Dirac neutrino signals at LHC,”
Phys. Lett. B 672, 158 (2009)
doi:10.1016/j.physletb.2009.01.010
[arXiv:0809.2096 [hep-ph]].
(50)
F. del Aguila, J. A. Aguilar-Saavedra and R. Pittau,
“Heavy neutrino signals at large hadron colliders,”
JHEP 0710, 047 (2007)
doi:10.1088/1126-6708/2007/10/047
[hep-ph/0703261].
(51)
J. A. Aguilar-Saavedra and F. R. Joaquim,
“Measuring heavy neutrino couplings at the LHC,”
Phys. Rev. D 86, 073005 (2012)
doi:10.1103/PhysRevD.86.073005
[arXiv:1207.4193 [hep-ph]].
(52)
B. P. Nayak and M. K. Parida,
“Dilepton events with displaced vertices, double beta decay, and resonant leptogenesis with Type-II seesaw dominance, TeV scale $Z^{\prime}$ and heavy neutrinos,”
arXiv:1509.06192 [hep-ph].
(53)
B. P. Nayak and M. K. Parida,
“New mechanism for Type-II seesaw dominance in SO(10) with low-mass $Z^{\prime}$, RH neutrinos, and verifiable LFV, LNV and proton decay,”
Eur. Phys. J. C 75, 183 (2015)
doi:10.1140/epjc/s10052-015-3385-x
[arXiv:1312.3185 [hep-ph]].
(54)
J. A. Aguilar-Saavedra, F. Deppisch, O. Kittel and J. W. F. Valle,
“Flavour in heavy neutrino searches at the LHC,”
Phys. Rev. D 85, 091301 (2012)
doi:10.1103/PhysRevD.85.091301
[arXiv:1203.5998 [hep-ph]].
(55)
R. Lal Awasthi and M. K. Parida,
“Inverse Seesaw Mechanism in Nonsupersymmetric SO(10), Proton Lifetime, Nonunitarity Effects, and a Low-mass $Z^{\prime}$ Boson,”
Phys. Rev. D 86, 093004 (2012)
doi:10.1103/PhysRevD.86.093004
[arXiv:1112.1826 [hep-ph]].
(56)
C. S. Fong, R. N. Mohapatra and I. Sung,
“Majorana Neutrinos from Inverse Seesaw in Warped Extra Dimension,”
Phys. Lett. B 704, 171 (2011)
doi:10.1016/j.physletb.2011.08.069
[arXiv:1107.4086 [hep-ph]].
(57)
A. G. Dias, C. A. de S.Pires and P. S. R. da Silva,
“How the Inverse See-Saw Mechanism Can Reveal Itself Natural, Canonical and Independent of the Right-Handed Neutrino Mass,”
Phys. Rev. D 84, 053011 (2011)
doi:10.1103/PhysRevD.84.053011
[arXiv:1107.0739 [hep-ph]].
(58)
A. Ibarra, E. Molinaro and S. T. Petcov,
“Low Energy Signatures of the TeV Scale See-Saw Mechanism,”
Phys. Rev. D 84, 013005 (2011)
doi:10.1103/PhysRevD.84.013005
[arXiv:1103.6217 [hep-ph]].
(59)
T. Saito et al., “Extra dimensions and Seesaw Neutrinos at the International Linear Collider,” Phys. Rev. D 82, 093004 (2010) [arXiv:1008.2257 [hep-ph]]
(60)
S. Banerjee, P. S. B. Dev, A. Ibarra, T. Mandal and M. Mitra,“Prospects of Heavy Neutrino Searches at Future Lepton Colliders,” arXiv:1503.05491 [hep-ph].
(61)
F. del Aguila, J. A. Aguilar-Saavedra and R. Pittau, “Heavy neutrino signals at large hadron colliders,” JHEP 0710, 047 (2007) [hep-ph/0703261].
(62)
F. del Aguila and J. A. Aguilar-Saavedra, “Distinguishing seesaw models at LHC with multi-lepton signals,” Nucl. Phys. B 813, 22 (2009) [arXiv:0808.2468 [hep-ph]].
(63)
B. Batell and M. McCullough,
“Neutrino Masses from Neutral Top Partners,”
Phys. Rev. D 92, no. 7, 073018 (2015)
doi:10.1103/PhysRevD.92.073018
[arXiv:1504.04016 [hep-ph]].
(64)
R. Leonardi, L. Alunni, F. Romeo, L. Fano and O. Panella,
“Hunting for heavy composite Majorana neutrinos at the LHC,”
arXiv:1510.07988 [hep-ph].
(65)
P. S. B. Dev, A. Pilaftsis, U. K. Yang, Phys. Rev. Lett. 112 (2014) 081801, arXiv: 1308.2209 [hep-ph].
(66)
G. Bambhaniya, S. Khan, P. Konar and T. Mondal, “Constraints on a seesaw model leading to quasidegenerate neutrinos and signatures at the LHC,” Phys. Rev. D 91, no. 9, 095007 (2015)
[arXiv:1411.6866 [hep-ph]].
(67)
G. Bambhaniya, S. Goswami, S. Khan, P. Konar and T. Mondal, “Looking for hints of a reconstructible seesaw model at the Large Hadron Collider,” Phys. Rev. D 91, 075007 (2015)
[arXiv:1410.5687 [hep-ph]].
(68)
G. Dutta and A. S. Joshipura,
“PseudoDirac neutrinos in seesaw model,”
Phys. Rev. D 51, 3838 (1995)
doi:10.1103/PhysRevD.51.3838
[hep-ph/9405291].
(69)
N. Haba, S. Matsumoto and K. Yoshioka,
“Observable Seesaw and its Collider Signatures,”
Phys. Lett. B 677, 291 (2009)
doi:10.1016/j.physletb.2009.05.042
[arXiv:0901.4596 [hep-ph]].
(70)
S. Matsumoto, T. Nabeshima and K. Yoshioka,
“Seesaw Neutrino Signals at the Large Hadron Collider,”
JHEP 1006, 058 (2010)
doi:10.1007/JHEP06(2010)058
[arXiv:1004.3852 [hep-ph]].
(71)
S. Mondal, S. Biswas, P. Ghosh and S. Roy,
“Exploring novel correlations in trilepton channels at the LHC for the minimal supersymmetric inverse seesaw model,”
JHEP 1205, 134 (2012)
doi:10.1007/JHEP05(2012)134
[arXiv:1201.1556 [hep-ph]].
(72)
J. C. Helo, M. Hirsch and S. Kovalenko,
“Heavy neutrino searches at the LHC with displaced vertices,”
Phys. Rev. D 89, 073005 (2014)
doi:10.1103/PhysRevD.89.073005
[arXiv:1312.2900 [hep-ph]].
(73)
A. G. Hessler, A. Ibarra, E. Molinaro and S. Vogl,
“Impact of the Higgs boson on the production of exotic particles at the LHC,”
Phys. Rev. D 91, no. 11, 115004 (2015)
doi:10.1103/PhysRevD.91.115004
[arXiv:1408.0983 [hep-ph]].
(74)
F. F. Deppisch, P. S. Bhupal Dev and A. Pilaftsis,
“Neutrinos and Collider Physics,”
New J. Phys. 17, no. 7, 075019 (2015)
doi:10.1088/1367-2630/17/7/075019
[arXiv:1502.06541 [hep-ph]].
(75)
E. Arganda, M. J. Herrero, X. Marcano and C. Weiland,
“Exotic $\mu\tau jj$ events from heavy ISS neutrinos at the LHC,”
Phys. Lett. B 752, 46 (2016)
doi:10.1016/j.physletb.2015.11.013
[arXiv:1508.05074 [hep-ph]].
(76)
C. O. Dib and C. S. Kim,
“Discovering sterile Neutrinos ligther than $M_{W}$ at the LHC,”
Phys. Rev. D 92, no. 9, 093009 (2015)
doi:10.1103/PhysRevD.92.093009
[arXiv:1509.05981 [hep-ph]].
(77)
A. Das and N. Okada,
“Bounds on heavy Majorana neutrinos in type-I seesaw and implications for collider searches,”
arXiv:1702.04668 [hep-ph].
(78)
J. Adam et al. [MEG Collaboration],
“New limit on the lepton-flavour violating decay $\mu^{+}\to e^{+}\gamma$,”
Phys. Rev. Lett. 107, 171801 (2011)
doi:10.1103/PhysRevLett.107.171801
[arXiv:1107.5547 [hep-ex]].
(79)
B. Aubert et al. [BaBar Collaboration],
“Searches for Lepton Flavor Violation in the Decays tau$+---->$ e$+-$ gamma and tau$+---->$ mu$+-$ gamma,”
Phys. Rev. Lett. 104, 021802 (2010)
doi:10.1103/PhysRevLett.104.021802
[arXiv:0908.2381 [hep-ex]].
(80)
B. O’Leary et al. [SuperB Collaboration],
“SuperB Progress Reports – Physics,”
arXiv:1008.1541 [hep-ex].
(81)
P. Achard et al. [L3 Collaboration],
“Search for heavy isosinglet neutrino in $e^{+}e^{-}$ annihilation at LEP,”
Phys. Lett. B 517, 67 (2001)
doi:10.1016/S0370-2693(01)00993-5
[hep-ex/0107014].
(82)
S. Antusch, C. Biggio, E. Fernandez-Martinez, M. B. Gavela and J. Lopez-Pavon,
“Unitarity of the Leptonic Mixing Matrix,”
JHEP 0610, 084 (2006)
doi:10.1088/1126-6708/2006/10/084
[hep-ph/0607020].
(83)
A. Abada, C. Biggio, F. Bonnet, M. B. Gavela and T. Hambye,
“Low energy effects of neutrino masses,”
JHEP 0712, 061 (2007)
doi:10.1088/1126-6708/2007/12/061
[arXiv:0707.4058 [hep-ph]].
(84)
N. Okada, Y. Orikasa and T. Yamada,
“Minimal Flavor Violation in the Minimal $U(1)_{B-L}$ Model and Resonant Leptogenesis,”
Phys. Rev. D 86, 076003 (2012)
doi:10.1103/PhysRevD.86.076003
[arXiv:1207.1510 [hep-ph]].
(85)
A. Ibarra, E. Molinaro and S. T. Petcov,
“TeV Scale See-Saw Mechanisms of Neutrino Mass Generation, the Majorana Nature of the Heavy Singlet Neutrinos and $(\beta\beta)_{0\nu}$-Decay,”
JHEP 1009, 108 (2010)
doi:10.1007/JHEP09(2010)108
[arXiv:1007.2378 [hep-ph]].
(86)
D. N. Dinh, A. Ibarra, E. Molinaro and S. T. Petcov,
“The $\mu-e$ Conversion in Nuclei, $\mu\to e\gamma,\mu\to 3e$ Decays and TeV Scale See-Saw Scenarios of Neutrino Mass Generation,”
JHEP 1208, 125 (2012)
Erratum: [JHEP 1309, 023 (2013)]
doi:10.1007/JHEP09(2013)023, 10.1007/JHEP08(2012)125
[arXiv:1205.4671 [hep-ph]].
(87)
R. W. Rasmussen and W. Winter,
“Perspectives for tests of neutrino mass generation at the GeV scale: Experimental reach versus theoretical predictions,”
Phys. Rev. D 94, no. 7, 073004 (2016)
doi:10.1103/PhysRevD.94.073004
[arXiv:1607.07880 [hep-ph]].
(88)
B. Pontecorvo,
“Inverse beta processes and nonconservation of lepton charge,”
Sov. Phys. JETP 7, 172 (1958)
[Zh. Eksp. Teor. Fiz. 34, 247 (1957)].
(89)
Z. Maki, M. Nakagawa and S. Sakata,
“Remarks on the unified model of elementary particles,”
Prog. Theor. Phys. 28, 870 (1962).
doi:10.1143/PTP.28.870
(90)
A. Pilaftsis and T. E. J. Underwood,
“Resonant leptogenesis,”
Nucl. Phys. B 692, 303 (2004)
doi:10.1016/j.nuclphysb.2004.05.029
[hep-ph/0309342].
(91)
J. Kersten and A. Y. Smirnov,
“Right-Handed Neutrinos at CERN LHC and the Mechanism of Neutrino Mass Generation,”
Phys. Rev. D 76, 073005 (2007)
doi:10.1103/PhysRevD.76.073005
[arXiv:0705.3221 [hep-ph]].
(92)
Z. z. Xing,
“Naturalness and Testability of TeV Seesaw Mechanisms,”
Prog. Theor. Phys. Suppl. 180, 112 (2009)
doi:10.1143/PTPS.180.112
[arXiv:0905.3903 [hep-ph]].
(93)
X. G. He, S. Oh, J. Tandean and C. C. Wen,
“Large Mixing of Light and Heavy Neutrinos in Seesaw Models and the LHC,”
Phys. Rev. D 80, 073012 (2009)
doi:10.1103/PhysRevD.80.073012
[arXiv:0907.1607 [hep-ph]].
(94)
F. F. Deppisch and A. Pilaftsis,
“Lepton Flavour Violation and theta(13) in Minimal Resonant Leptogenesis,”
Phys. Rev. D 83, 076007 (2011)
doi:10.1103/PhysRevD.83.076007
[arXiv:1012.1834 [hep-ph]].
(95)
C. H. Lee, P. S. Bhupal Dev and R. N. Mohapatra,
“Natural TeV-scale left-right seesaw mechanism for neutrinos and experimental tests,”
Phys. Rev. D 88, no. 9, 093010 (2013)
doi:10.1103/PhysRevD.88.093010
[arXiv:1309.0774 [hep-ph]].
(96)
J. Pumplin, D. R. Stump, J. Huston, H. L. Lai, P. M. Nadolsky and W. K. Tung,
“New generation of parton distributions with uncertainties from global QCD analysis,”
JHEP 0207, 012 (2002)
doi:10.1088/1126-6708/2002/07/012
[hep-ph/0201195].
(97)
A. Das, P. Konar and S. Majhi,
“Production of Heavy neutrino in next-to-leading order QCD at the LHC and beyond,”
JHEP 1606, 019 (2016)
doi:10.1007/JHEP06(2016)019
[arXiv:1604.00608 [hep-ph]].
(98)
A. Das,
“Pair production of heavy neutrinos in next-to-leading order QCD at the hadron colliders in the inverse seesaw framework,”
arXiv:1701.04946 [hep-ph].
(99)
J. Thaler and K. Van Tilburg,
“Identifying Boosted Objects with N-subjettiness,”
JHEP 1103, 015 (2011)
doi:10.1007/JHEP03(2011)015
[arXiv:1011.2268 [hep-ph]].
(100)
J. Thaler and K. Van Tilburg,
“Maximizing Boosted Top Identification by Minimizing N-subjettiness,”
JHEP 1202, 093 (2012)
doi:10.1007/JHEP02(2012)093
[arXiv:1108.2701 [hep-ph]].
(101)
J. de Favereau et al. [DELPHES 3 Collaboration],
“DELPHES 3, A modular framework for fast simulation of a generic collider experiment,”
JHEP 1402 (2014) 057
doi:10.1007/JHEP02(2014)057
[arXiv:1307.6346 [hep-ex]].
(102)
M. Cacciari, G. P. Salam and G. Soyez,
“FastJet user manual,”
Eur. Phys. J. C 72 (2012) 1896
[arXiv:1111.6097 [hep-ph]].
(103)
M. Cacciari and G. P. Salam,
“Dispelling the $N^{3}$ myth for the $k_{t}$ jet-finder,”
Phys. Lett. B 641 (2006) 57
[hep-ph/0512210].
(104)
V. Khachatryan et al. [CMS Collaboration],
“Identification techniques for highly boosted W bosons that decay into hadrons,”
JHEP 1412, 017 (2014)
doi:10.1007/JHEP12(2014)017
[arXiv:1410.4227 [hep-ex]].
(105)
Y. L. Dokshitzer, G. D. Leder, S. Moretti, and B. R. Webber, Better Jet
Clustering Algorithms, JHEP 08 (1997) 001,
[hep-ph/9707323].
(106)
M. Wobisch and T. Wengler, Hadronization corrections to jet cross
sections in deep- inelastic scattering,
hep-ph/9907280.
(107)
J. Alwall, M. Herquet, F. Maltoni, O. Mattelaer and T. Stelzer, “MadGraph 5 : Going Beyond,” JHEP 1106, 128 (2011)[arXiv:1106.0522 [hep-ph]].
(108)
J. Alwall, M. Herquet, F. Maltoni, O. Mattelaer, T. Stelzer, J. High Energy Phys. 1106 (2011) 128, arXiv: 1106.0522 [hep-ph]
(109)
N. D. Christensen and C. Duhr,
“FeynRules - Feynman rules made easy,”
Comput. Phys. Commun. 180, 1614 (2009)
doi:10.1016/j.cpc.2009.02.018
[arXiv:0806.4194 [hep-ph]].
(110)
A. Alloul, N. D. Christensen, C. Degrande, C. Duhr and B. Fuks,
“FeynRules 2.0 - A complete toolbox for tree-level phenomenology,”
Comput. Phys. Commun. 185, 2250 (2014)
doi:10.1016/j.cpc.2014.04.012
[arXiv:1310.1921 [hep-ph]].
(111)
J. Pumplin, D. R. Stump, J. Huston, H. L. Lai, P. M. Nadolsky and W. K. Tung,
“New generation of parton distributions with uncertainties from global QCD analysis,”
JHEP 0207, 012 (2002)
doi:10.1088/1126-6708/2002/07/012
[hep-ph/0201195].
(112)
T. Sjostrand, S. Mrenna and P. Z. Skands,
“PYTHIA 6.4 Physics and Manual,”
JHEP 0605, 026 (2006)
doi:10.1088/1126-6708/2006/05/026
[hep-ph/0603175].
(113)
S. Hoeche, F. Krauss, N. Lavesson, L. Lonnblad, M. Mangano, A. Schalicke and S. Schumann,
“Matching parton showers and matrix elements,”
doi:10.5170/CERN-2005-014.288
hep-ph/0602031.
(114)
M. Cacciari, G. P. Salam and G. Soyez,
“The Anti-k(t) jet clustering algorithm,”
JHEP 0804, 063 (2008)
doi:10.1088/1126-6708/2008/04/063
[arXiv:0802.1189 [hep-ph]].
(115)
Akanksha Bhardwaj, Arindam Das, Partha Konar, Arun Thalapillil, in preparation.
(116)
D. Binosi and L. Theussl,
“JaxoDraw: A Graphical user interface for drawing Feynman diagrams,”
Comput. Phys. Commun. 161, 76 (2004)
doi:10.1016/j.cpc.2004.05.001
[hep-ph/0309015].
(117)
D. Binosi, J. Collins, C. Kaufhold and L. Theussl,
“JaxoDraw: A Graphical user interface for drawing Feynman diagrams. Version 2.0 release notes,”
Comput. Phys. Commun. 180, 1709 (2009)
doi:10.1016/j.cpc.2009.02.020
[arXiv:0811.4113 [hep-ph]]. |
EPHOU-20-011
KEK-TH-2265
WU-HEP-20-10
*[50pt]
Spontaneous CP violation and symplectic modular symmetry
in Calabi-Yau compactifications
Keiya Ishiguro${}^{a}$***
E-mail address: keyspire@ruri.waseda.jp
,
Tatsuo Kobayashi${}^{b}$†††
E-mail address: kobayashi@particle.sci.hokudai.ac.jp
and Hajime Otsuka${}^{c}$‡‡‡
E-mail address: hotsuka@post.kek.jp
${}^{a}$Department of Physics, Waseda University, Tokyo 169-8555, Japan
${}^{b}$Department of Physics, Hokkaido University, Sapporo 060-0810, Japan
${}^{c}$KEK Theory Center, Institute of Particle and Nuclear Studies, KEK,
1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan
Abstract
We explore the geometrical origin of CP and the spontaneous CP violation in Calabi-Yau compactifications. We find that the CP symmetry is identified with an outer automorphism of the symplectic modular group in the large complex structure regime of Calabi-Yau threefolds, thereby enlarging the symplectic modular group to their semidirect product group.
The spontaneous CP violation is realized by the introduction of fluxes, whose effective action is invariant under CP as well as the discrete $\mathbb{Z}_{2}$ symmetry or $\mathbb{Z}_{4}$ R-symmetry.
We explicitly demonstrate the spontaneous CP violation on a specific Calabi-Yau threefold.
Contents
1 Introduction
2 CP and symplectic modular symmetries
3 CP-invariant flux compactifications and discrete symmetry
4 CP-conserving and -breaking vacua
4.1 CP-conserving vacua in generic CY
4.2 One modulus case: $\mathbb{CP}_{11111}[5]$
4.3 Two moduli case: $\mathbb{CP}_{11222}[8]$
4.4 CP-breaking vacua on $\mathbb{CP}_{11169}[18]$
5 Conclusions
1 Introduction
The origin of CP violation is one of important issues to study in particle physics.
Indeed, it has been studied extensively in various scenarios.
CP violation may originate from an underlying theory such as superstring theory.
As discussed in Refs. [1, 2, 3],
one can embed the four-dimensional (4D) CP symmetry into
proper Lorentz symmetry in higher dimensional theory, e.g.,
ten-dimensional (10D) proper Lorentz symmetry in superstring theory
on six-dimensional (6D) Calabi-Yau (CY) manifolds.
That is, as the 10D proper Lorentz transformation,
one performs simultaneously the 4D spacetime orientation and orientation reversing transformation of CY threefolds.
The latter 6D transformation can be achieved by $z_{i}\rightarrow{-z^{\ast}_{i}}$ with $z_{i}$, $i=1,2,3$, being the complex coordinates, corresponding to the negative determinant in the transformation of the 6D CY manifolds.
(See also for CP symmetry on 6D orbifold compactifications [4].)
The 6D transformation reverses the sign of the volume form as well as the Kähler form of CY threefolds.
To clarify the nature of CP symmetry, let us consider Majorana-Weyl spinor representation of 10D Lorentz group $SO(1,9)$ which decomposes as ${\bf 16}=({\bf 2},{\bf 4})\oplus({\bf 2^{\prime}},{\bf\bar{4}})$ under
the 4D and the 6D Lorentz groups
$SO(1,3)\times SO(6)$.
Here, ${\bf 2}$ and ${\bf 2^{\prime}}$ denote the left- and right-handed spinor representations of $SL(2,\mathbb{C})$,
and ${\bf 4}$ and ${\bf\bar{4}}$ represent the positive- and negative-chirality spinor representations of $SO(6)\simeq SU(4)$,
respectively.
Then, simultaneous transformations of 4D parity and 6D orientation reversing exchange
$({\bf 2},{\bf 4})$ with $({\bf 2^{\prime}},{\bf\bar{4}})$.
For example, in the context of standard embedding
in the heterotic string theory, fundamental and
anti-fundamental representations of $E_{6}$, ${\bf 27}$ and ${\bf\overline{27}}$, correspond to ${\bf 4}$ and ${\bf\bar{4}}$,
respectively.
Hence, they are exchanged under such simultaneous transformations.
Such a CP transformation is identified with an outer
automorphism transformation of gauge symmetries [5] and
each of 4D and 6D Lorentz symmetries [6].
In Type II D-brane models,
massless fermions localize on D-branes wrapping certain cycles of the CY threefold.
When we consider the orientation reversing transformation on the internal cycles wrapped by D-branes,
one can also embed the 4D CP into the higher-dimensional Lorentz transformation on the worldvolume of D-branes on which CP exchanges
left-handed massless matters with their complex conjugates.
The torus compactification as well as toroidal orbifold compactifications
is one of simplest compactifications.
The 2D torus has two independent one-cycles.
We have the degree of freedom to change bases of these cycles.
That is the $SL(2,\mathbb{Z})$ transformation, which is called as
the modular symmetry.
The modular group is the geometrical symmetry, and also
transforms zero-modes corresponding to quarks and leptons,
that is, the flavor symmetry.
(See for the modular symmetry in magnetized D-brane models [7, 8, 9] and the classification of
its subgroup [10].)111
Recently, the modular flavor symmetries are studied extensively from
the phenomenological viewpoint [11].
When there are 4D models having non-trivial flavor symmetries,
the 4D CP symmetry can be enlarged to a generalized CP symmetry,
which includes outer automorphisms of flavor symmetries [12, 13, 14].
Such a generalized CP symmetry is also extended to
the modular symmetry on the torus compactification, where
CP symmetry is identified with the outer automorphism of $SL(2,\mathbb{Z})$ [15, 16].
The size and shape of a compact space is described by moduli parameters,
which are vacuum expectation values of moduli fields.
Their vacuum expectation values can be stabilized at potential minima of moduli fields.
The complex structure moduli also transform under the CP symmetry,
$z_{i}\rightarrow{-z^{\ast}_{i}}$.
Thus, the CP symmetry can be spontaneously broken
through the moduli stabilization.
For example, the spontaneous CP-violation was studied by assuming
non-perturbative moduli potentials [17, 18, 19, 20].
The three-form flux background is of controllable ways to stabilize
the complex structure moduli as well as the axio-dilaton [21].
Indeed, in Ref. [22], spontaneous CP-violation was studied
within the framework of Type IIB superstring theory on
toroidal orbifold compactifications with three-form flux background.
The CP-invariant superpotential, which is induced by three-form fluxes,
is either even or odd polynomial functions of the complex structure moduli
and the axio-dilaton.
The potential minima were examined, but
the spontaneous CP-violation can not be realized in
the torus compactifications.
Our purpose in this paper is to reveal the geometrical origin of CP and search for the spontaneous CP violation
in CY compactifications, with an emphasis on
the effective action of the complex structure moduli.
As discussed in Ref. [1], the orientation reversing isometry of CY threefolds
induces anti-holomorphic transformations for the complex structure,
which restrict the form of the low-energy effective action.
We find that CP symmetry is identified with the outer automorphism of the symplectic modular group of generic CY
threefolds in the large complex structure regime.
That is a natural extension of the discussion in toroidal backgrounds, where the
CP symmetry is
also identified with the outer automorphism of $SL(2,\mathbb{Z})$ modular group on the
2D torus.
Furthermore, we discuss the spontaneous CP violation in the CY flux compactifications.
Flux compactifications in the string theory have a potential to break CP spontaneously.
In a similar discussion with the toroidal background [22], the flux-induced superpotential is restricted to be odd or even polynomials with respect to the moduli fields, having the discrete $\mathbb{Z}_{2}$ symmetry or $\mathbb{Z}_{4}$ R-symmetry
from the field theoretical point of view.
It turns out that the CP-conserving vacua exist in the large complex structure regime of generic CY threefolds.
The spontaneous CP violation occurs in some class of CY threefolds whose prepotentials have a structure
different from the toroidal one.
This paper is organized as follows.
In Sec. 2, we show a relationship between the CP symmetry and the symplectic modular symmetry.
Sec. 3 is devoted to the construction of the flux-induced effective action in a CP-invariant way.
Concrete CY flux compactifications are analyzed in Sec. 4, in which the CP-breaking and -conserving
vacua are demonstrated.
Finally, we conclude our results in Sec. 5.
2 CP and symplectic modular symmetries
In this section, we focus on the complex structure moduli of CY threefolds ${\cal M}$,
whose effective action is described by the Kähler potential in the reduced Planck mass unit $M_{\rm Pl}=1$,
$$\displaystyle K_{\rm cs}$$
$$\displaystyle=-\ln\biggl{[}-i\int_{{\cal M}}\Omega\wedge\bar{\Omega}\biggl{]},$$
(1)
where $\Omega$ denotes a holomorphic three-form of CY threefolds determining the complex structure
of ${\cal M}$.
We examine the CP invariance of the holomorphic three-form $\Omega$ as well as its
relation to the symplectic modular symmetry on ${\cal M}$.
Recalling that $i\Omega\wedge\bar{\Omega}$ is represented by the volume form of CY threefolds $dV$, namely $i\Omega\wedge\bar{\Omega}=||\Omega||^{2}dV$ with $||\Omega||^{2}$ being a scalar,
the orientation reversing transformation changes the sign of the volume form leading
$i\Omega\wedge\bar{\Omega}\rightarrow-i\Omega\wedge\bar{\Omega}$.
It results in the transformation of $\Omega$ under
the orientation reversing222Although the following discussion also holds for the anti-holomorphic involution $\Omega\rightarrow\bar{\Omega}$ corresponding to $z_{i}\rightarrow z_{i}^{\ast}$ in the local patch,
we adopt the orientation reversing in
Eq. (2) without loss of generality.,
$$\displaystyle\Omega\rightarrow{-\bar{\Omega}}.$$
(2)
Note that in the local coordinates of CY threefolds $\{z_{i}\}$, the holomorphic three-form is given by
$\Omega=dz_{1}\wedge dz_{2}\wedge dz_{3}$. Hence, the orientation reversing transformation
$z_{i}\rightarrow{-z^{\ast}_{i}}$ gives rise to Eq. (2).
Since CY threefolds are described by the special geometry333For more details about the special geometry,
we refer Refs. [23, 24].,
the holomorphic three-form is expanded in the symplectic basis.
When we denote $(A^{I},B_{I})$ ($I,J=0,1,\cdots,h^{2,1}({\cal M})$) a canonical homology basis for $H_{3}({\cal M},\mathbb{Z})$, the dual cohomology basis ($\alpha_{I},\beta^{I}$) is defined such that
$$\displaystyle\int_{A^{J}}\alpha_{I}=-\int_{B_{I}}\beta^{J}=\int_{{\cal M}}%
\alpha_{I}\wedge\beta^{J}=\delta^{J}_{I},\quad\int_{{\cal M}}\alpha_{I}\wedge%
\alpha_{J}=\int_{{\cal M}}\beta^{I}\wedge\beta^{J}=0.$$
(3)
In terms of the real three-form basis ($\alpha_{I},\beta^{I}$), the holomorphic three-form can be expanded as
$$\displaystyle\Omega$$
$$\displaystyle=X^{I}\alpha_{I}-{\cal F}_{I}\beta^{I},$$
(4)
where ${\cal F}_{I}\equiv\partial_{I}{\cal F}$ is a holomorphic function of the prepotential ${\cal F}=\frac{1}{2}X^{I}{\cal F}_{I}$ as a function of homogeneous coordinates $X^{I}$ on the moduli space. Note that we have rescaling degrees of freedom on $X^{I}$.
Let us analyze the CP transformation of $\Omega$ in the symplectic basis in more detail.
The transformation of $\Omega$ in Eq. (2)
is satisfied when
$$\displaystyle X^{0}\alpha_{0}\rightarrow{-\bar{X}^{0}\alpha_{0}},\quad X^{i}%
\alpha_{i}\rightarrow{-\bar{X}^{i}\alpha_{i}},\quad{\cal F}_{0}\beta^{0}%
\rightarrow{-\bar{{\cal F}}_{0}\beta^{0}},\quad{\cal F}_{i}\beta^{i}%
\rightarrow{-\bar{{\cal F}}_{i}\beta^{i}}.$$
(5)
We define the coordinates $u^{i}=X^{i}/X^{0}$, $i=1,2,\cdots,h^{2,1}$.
The orientation reversing transformation requires that the complex structure moduli should transform $u^{i}\rightarrow{\pm\bar{u}^{i}}$.
In the following discussion, we adopt $u\rightarrow-\bar{u}^{i}$, restricting ourselves to the ${\rm Im}u^{i}>0$ plane444
That is a generalization of the upper half plane of the complex plane, realized in
the $SL(2,\mathbb{Z})$ moduli space of the torus. and focus on
the large complex structure regime.
Note that the orientation reversing transformation in the 6D internal space corresponds
to
$$\displaystyle\int_{{\cal M}}\alpha_{I}\wedge\beta^{J}=\delta^{J}_{I}%
\rightarrow-\int_{{\cal M}}\alpha_{I}\wedge\beta^{J}=-\delta^{J}_{I}.$$
(6)
The Kähler potential is CP-invariant because $\Omega\wedge\bar{\Omega}$ transforms into $-\Omega\wedge\bar{\Omega}$.
Then, we arrive at the CP transformations of $\{X^{I},{\cal F}_{I}\}$ and three-form basis:
$$\displaystyle X^{0}$$
$$\displaystyle\rightarrow{\pm\bar{X}^{0}},\quad X^{i}\rightarrow{\mp\bar{X}^{i}%
},\quad{\cal F}_{0}\rightarrow{\mp\bar{{\cal F}}_{0}},\quad{\cal F}_{i}%
\rightarrow{\pm\bar{{\cal F}}_{i}},$$
$$\displaystyle\alpha_{0}$$
$$\displaystyle\rightarrow{\mp\alpha_{0}},\quad\alpha_{i}\rightarrow{\pm\alpha_{%
i}},\quad\beta^{0}\rightarrow{\pm\beta^{0}},\quad\beta^{i}\rightarrow{\mp\beta%
^{i}},$$
(7)
where the double-sign corresponds.
However, it is difficult to achieve these CP transformations for a generic form of the prepotential.
In the large complex structure regime, the prepotential ${\cal F}(X)=(X^{0})^{2}F(u)$ is indeed expanded as555We still call $F(u)$ as the prepotential throughout this paper.
$$\displaystyle F(u)=\frac{1}{3!}\kappa_{ijk}u^{i}u^{j}u^{k}+\frac{1}{2!}\kappa_%
{ij}u^{i}u^{j}+\kappa_{i}u^{i}+\frac{1}{2}\kappa_{0},$$
(8)
up to geometrical corrections [25].
Here, the coefficients $\kappa_{ijk}$, $\kappa_{ij}$, $\kappa_{i}$ $\kappa_{0}$ are the topological quantities determined by the CY data,
i.e.
$$\displaystyle\kappa_{ijk}$$
$$\displaystyle=\int_{\tilde{M}}J_{i}\wedge J_{j}\wedge J_{k},\quad\kappa_{ij}=%
\frac{1}{2}\int_{\tilde{M}}J_{i}\wedge J_{j}^{2},\quad\kappa_{i}=-\frac{1}{24}%
\int_{\tilde{M}}c_{2}(\tilde{M})\wedge J_{i},\quad\kappa_{0}=-\frac{\zeta(3)%
\chi({\tilde{M}})}{(2\pi i)^{3}}.$$
(9)
These classical topological quantities are calculated on the mirror CY threefold $\tilde{M}$, where
the $(1,1)$-forms are denoted by $J_{i}$, and the second Chern class and the Euler characteristic of ${\tilde{M}}$
are represented by $c_{2}(\tilde{M})$ and $\chi({\tilde{M}})$, respectively.666Note that the quadratic and linear terms of the prepotential, $\kappa_{ij}$ and $\kappa_{i}$, are only determined modulo integers, where we adopt the convention in Ref. [26].
To be invariant under the CP transformation (7) taking into account $u^{i}\rightarrow-\bar{u}^{i}$,
the prepotential is restricted to be a cubic type, i.e.
$$\displaystyle F_{\rm cubic}=\frac{1}{3!}\kappa_{ijk}u^{i}u^{j}u^{k}.$$
(10)
The linear term in the prepotential is also allowed under the CP transformation, but we focus on a strict large complex structure regime of generic CY threefolds in
the following analysis.
Before analyzing the CP-invariant effective potential,
we discuss the relation between the CP symmetry and the symplectic modular symmetry of CY threefolds, described by the special geometry.
Given the holomorphic three-form $\Omega$ expanded on the basis of the symplectic basis $(\alpha_{I},\beta^{I})$ in $H^{3}({\cal M},\mathbb{Z})$ as in Eq. (4),
the symplectic basis transforms under the symplectic modular symmetry as
$$\displaystyle\begin{pmatrix}\alpha\\
\beta\\
\end{pmatrix}\rightarrow\begin{pmatrix}a&b\\
c&d\\
\end{pmatrix}\begin{pmatrix}\alpha\\
\beta\\
\end{pmatrix},$$
(11)
with
$$\displaystyle\begin{pmatrix}a&b\\
c&d\\
\end{pmatrix}\in Sp(2h^{2,1}+2;\mathbb{Z}).$$
(12)
That corresponds to the group of $(2h^{2,1}+2)\times(2h^{2,1}+2)$ matrices preserving the symplectic matrix:
$$\displaystyle\Sigma=\begin{pmatrix}0&\bf{1}\\
-\bf{1}&0\\
\end{pmatrix}.$$
(13)
Under the symplectic transformations, the so-called
period integrals of the holomorphic three-form $\Omega$ on the three-cycles $\{A^{I},B_{I}\}$ transform as
$$\displaystyle\Pi=\begin{pmatrix}\int_{A}\Omega\\
\int_{B}\Omega\\
\end{pmatrix}=\begin{pmatrix}\vec{X}\\
\vec{{\cal F}}\\
\end{pmatrix}\rightarrow\begin{pmatrix}d&c\\
b&a\\
\end{pmatrix}\begin{pmatrix}\vec{X}\\
\vec{{\cal F}}\\
\end{pmatrix},$$
(14)
with $\vec{X}=\{X^{I}\}$ and $\vec{{\cal F}}=\{{\cal F}_{I}\}$.
These transformations are reflected by the fact that holomorphic three-form itself is
invariant under the symplectic transformation.
Let us introduce the period matrix
$$\displaystyle{\cal F}_{IJ}=\frac{\partial^{2}}{\partial u^{I}\partial u^{J}}{%
\cal F}=\partial_{I}{\cal F}_{J},$$
(15)
which transforms under the symplectic transformation as
$$\displaystyle{\cal F}_{IJ}\rightarrow\left((a{\cal F}+b)(c{\cal F}+d)^{-1}%
\right)_{IJ},$$
(16)
analogous to the period matrices of Riemann surfaces with genus $g$.
Correspondingly, when we take the gauge $X^{0}=1$, the complex structure
moduli form a vector-valued modular form of $Sp(2h^{2,1}+2;\mathbb{Z})$,
$$\displaystyle u\rightarrow(c{\cal F}+d)u,$$
(17)
with $u^{0}=X^{0}=1$.
We recall that the Kähler potential
$$\displaystyle K_{\rm cs}=-\ln\biggl{[}-i\Pi^{\dagger}\cdot\Sigma\cdot\Pi\biggl%
{]},$$
(18)
is indeed invariant under the symplectic transformation of the period integrals (14).
Such a symplectic modular symmetry arises from the fact that
the CY moduli space is described by the special geometry.
On the toroidal background $T^{2n}$, the geometrical space group is the $SL(2n,\mathbb{Z})$
modular group.
For example, the CP transformation on $T^{2}$ is identified with the outer automorphism
of $SL(2,\mathbb{Z})$ [15, 16].
It is interesting to ask whether CP is embedded into the symplectic modular transformation on CY threefolds.
The CP transformation we discussed corresponds to the
following transformation for the period integrals,
$$\displaystyle\Pi=\begin{pmatrix}X^{0}\\
X^{i}\\
{\cal F}_{0}\\
{\cal F}_{i}\\
\end{pmatrix}\rightarrow\pm\begin{pmatrix}X^{0}\\
-\bar{X}^{i}\\
-\bar{{\cal F}}_{0}\\
\bar{{\cal F}}_{i}\\
\end{pmatrix}=\pm\begin{pmatrix}1&0&0&0\\
0&-\bf{1}&0&0\\
0&0&-1&0\\
0&0&0&\bf{1}\\
\end{pmatrix}\bar{\Pi}.$$
(19)
We find that the matrix
$$\displaystyle{\cal CP}=\pm\begin{pmatrix}1&0&0&0\\
0&-\bf{1}&0&0\\
0&0&-1&0\\
0&0&0&\bf{1}\\
\end{pmatrix}\notin Sp(2h^{2,1}+2;\mathbb{Z}),$$
(20)
is not the element of $Sp(2h^{2,1}+2;\mathbb{Z})$,
due to the property ${\cal CP}^{T}\cdot\Sigma\cdot{\cal CP}=\Sigma^{T}\neq\Sigma$.
We have two options to define the CP transformation as in Eq. (20), but
both of them share the common properties.
Hence, we consider the case with the positive sign in Eq. (20) without loss
of generality.
Under CP and symplectic modular transformation $\gamma$, we obtain
$$\displaystyle\Pi\xrightarrow{\rm CP}{\cal CP}\bar{\Pi}\xrightarrow{\gamma}{%
\cal CP}\cdot\gamma\bar{\Pi}\xrightarrow{{\rm CP}^{-1}}{\cal CP}\cdot\gamma%
\cdot{\cal CP}^{-1}\Pi,$$
(21)
namely
$$\displaystyle{\cal CP}\cdot\gamma\cdot{\cal CP}^{-1}=\left(\begin{array}[]{cc}%
\hat{\sigma}^{3}&0\\
0&-\hat{\sigma}^{3}\end{array}\right)\left(\begin{array}[]{cc}d&c\\
b&a\end{array}\right)\left(\begin{array}[]{cc}\hat{\sigma}^{3}&0\\
0&-\hat{\sigma}^{3}\end{array}\right)=\begin{pmatrix}\hat{\sigma}^{3}d\hat{%
\sigma}^{3}&-\hat{\sigma}^{3}c\hat{\sigma}^{3}\\
-\hat{\sigma}^{3}b\hat{\sigma}^{3}&\hat{\sigma}^{3}a\hat{\sigma}^{3}\\
\end{pmatrix}=\begin{pmatrix}d&-c\\
-b&a\\
\end{pmatrix},$$
(22)
with
$$\displaystyle\hat{\sigma}^{3}\equiv\begin{pmatrix}1&0\\
0&-\bf{1}\\
\end{pmatrix}.$$
(23)
In this way, we can define the outer automorphism ${\cal Q}$:
$$\displaystyle\gamma=\begin{pmatrix}d&c\\
b&a\\
\end{pmatrix}\rightarrow{\cal Q}(\gamma)\equiv{\cal CP}\cdot\gamma\cdot{\cal CP%
}^{-1}=\begin{pmatrix}d&-c\\
-b&a\\
\end{pmatrix},$$
(24)
satisfying
$$\displaystyle{\cal Q}(\gamma_{1}){\cal Q}(\gamma_{2})={\cal CP}\cdot\gamma_{1}%
\cdot{\cal CP}^{-1}{\cal CP}\cdot\gamma_{2}\cdot{\cal CP}^{-1}={\cal Q}(\gamma%
_{1}\gamma_{2}),$$
(25)
and no group element $\hat{\gamma}\in Sp(2h^{2,1}+2,\mathbb{Z})$ exists such that ${\cal Q}(\gamma)=\hat{\gamma}^{-1}\gamma\hat{\gamma}$.
In this respect, we argue that CP has a geometrical origin in the CY moduli space,
namely the outer automorphism of the $Sp(2h^{2,1}+2,\mathbb{Z})$ modular group.
The whole group is described by the semidirect product group $Sp(2h^{2,1}+2,\mathbb{Z})\rtimes{\cal CP}$, since there exists a group homomorphism from ${\cal CP}$ to the automorphism group of $Sp(2h^{2,1}+2,\mathbb{Z})$ and $Sp(2h^{2,1}+2,\mathbb{Z})$ is a normal subgroup of $Sp(2h^{2,1}+2,\mathbb{Z})\rtimes{\cal CP}$.
That is a natural extension of the $SL(2,\mathbb{Z})\simeq Sp(2,\mathbb{Z})$ modular
group known in the $T^{2}$ toroidal background [15, 16],
where the whole group is given by the âgeneralized modular group”
$GL(2,\mathbb{Z})\simeq SL(2,\mathbb{Z})\rtimes{\cal CP}$.777We can treat an element of $Sp(2h^{2,1}+2,\mathbb{Z})\rtimes{\cal CP}$ as a pair $(g,h)$ where $g\in Sp(2h^{2,1}+2,\mathbb{Z})$, $h\in{\cal CP}\simeq\mathbb{Z}_{2}$, and a product between two pairs is represented as $(g_{1},h_{1})(g_{2},h_{2})=(g_{1}h_{1}^{-1}g_{2}h_{1},h_{1}h_{2})$ with respect to the outer automorphism. Nevertheless, in the toroidal case, it is enough to consider a set of elements $(g,h)(h\neq 1)$ as a negative determinant part of $GL(2,\mathbb{Z})$, so we simply use $GL(2,\mathbb{Z})$ instead of $SL(2,\mathbb{Z})\rtimes{\cal CP}$ thanks to the isomorphism. However, because of the explicit form of $\hat{\sigma}^{3}$, we cannot think $Sp(2h^{2,1}+2,\mathbb{Z})\rtimes{\cal CP}$ as the same with the toroidal case for odd $h^{2,1}$, where ${\cal CP}$ has a positive determinant. For even $h^{2,1}$, ${\cal CP}$ has a negative determinant then it could be similar to the toroidal case that we can distinguish new elements of the whole group $Sp(2h^{2,1}+2,\mathbb{Z})\rtimes{\cal CP}$ that are not in original $Sp(2h^{2,1}+2,\mathbb{Z})$ by their signs of determinants.
3 CP-invariant flux compactifications and discrete symmetry
In this section, we introduce the fluxes into the previous effective action
to determine the size of CP-violation, namely the vacuum expectation values of moduli fields.
We mainly focus on the effective action of Type IIB flux compactifications on CY orientifolds ${\cal M}$, but it is straightforward to extend our analysis to T-dual Type IIA flux
compactifications as well as the heterotic string theory in the large volume/complex structure and weak coupling regime.
In the following, we first derive the CP-invariant flux compactifications and next discuss the
discrete $\mathbb{Z}_{2}$ or $\mathbb{Z}_{4}$ symmetry appearing in the effective action.
In addition to the effective action of the complex structure moduli shown in Sec. 2,
the moduli effective action includes the axio-dilaton and the Kähler moduli,
$$\displaystyle K$$
$$\displaystyle=-\ln(i(\bar{S}-S))-\ln\biggl{[}-i\int_{{\cal M}}\Omega\wedge\bar%
{\Omega}\biggl{]}-2\ln{\cal V},$$
$$\displaystyle W$$
$$\displaystyle=\int_{{\cal M}}G_{3}\wedge\Omega,$$
(26)
where ${\cal V}$ represents for the CY volume in the Einstein frame, depending on the Kähler moduli of ${\cal M}$.
Here, we introduce the flux-induced superpotential [21] generated by
a linear combination of Ramond-Ramond (RR) three-form flux $F_{3}$ and Neveu-Schwarz (NS) three-form flux $H_{3}$,
namely $G_{3}=F_{3}-SH_{3}$ as a function of the axio-dilaton $S$.
Since the CY volume is invariant under the orientation reversing transformation,
kinetic terms of the Kähler moduli as well as the complex structure moduli are CP-invariant quantities.
This is because both of them originate from the Einstein-Hilbert
action in 10D supergravity action, which is invariant under the 10D proper Lorentz transformation.
Since the axio-dilaton $S$ is a 4D CP-odd field,
$$\displaystyle S\rightarrow{-\bar{S}},$$
(27)
the kinetic term of $S$ is also invariant under the CP transformation.
However, the three-form fluxes generically break CP as well as the 6D Lorentz symmetry.
Hence, not all the flux quanta are allowed in the CP-invariant effective action.
We classify the possible pattern of
CP-invariant three-form fluxes inserted on three-cycles of CY threefolds.
The CP invariance of the 4D effective action requires $W\rightarrow{e^{i\gamma}\bar{W}}$
for the superpotential, where $\gamma$ denotes a complex phase.
Taking into account the CP transformation of $\Omega$ in Eq. (2),
the three-form flux $G_{3}$ transforms into
$$\displaystyle G_{3}$$
$$\displaystyle\rightarrow{-e^{i\gamma}\bar{G}_{3}},$$
(28)
from which the integral flux quanta require $\gamma=0$ or $\pi$.
Recalling that the axio-dilaton $S$ is a 4D CP-odd field,
RR and NS three-form fluxes should transform as
$$\displaystyle\left\{\begin{array}[]{l}\gamma=0,\quad F_{3}\rightarrow{-F_{3}},%
\quad H_{3}\rightarrow{H_{3}}\\
\gamma=\pi,\quad F_{3}\rightarrow{F_{3}},\quad H_{3}\rightarrow{-H_{3}}\end{%
array}\right..$$
(29)
Hence, the expansion of the three-form fluxes on the symplectic basis is categorized into two classes:
•
$\gamma=0$
$$\displaystyle F_{3}$$
$$\displaystyle=f^{0}\alpha_{0}+f_{i}\beta^{i},$$
$$\displaystyle H_{3}$$
$$\displaystyle=h^{i}\alpha_{i}+h_{0}\beta^{0}.$$
(30)
•
$\gamma=\pi$
$$\displaystyle F_{3}$$
$$\displaystyle=f^{i}\alpha_{i}+f_{0}\beta^{0},$$
$$\displaystyle H_{3}$$
$$\displaystyle=h^{0}\alpha_{0}+h_{i}\beta^{i}.$$
(31)
As a result, we obtain two classes of 4D CP-invariant effective action in the large complex structure
regime of CY threefolds.
In both classes,
the Kähler potential is described by
$$\displaystyle K_{\rm cs}$$
$$\displaystyle=-\ln\biggl{[}-i\int_{{\cal M}}\Omega\wedge\bar{\Omega}\biggl{]}=%
-\ln\biggl{[}-i(\bar{u}^{I}F_{I}-u^{I}\bar{F}_{I})\biggl{]}$$
$$\displaystyle=-\ln\biggl{[}\frac{i}{3!}\kappa_{ijk}(u^{i}-\bar{u}^{i})(u^{j}-%
\bar{u}^{j})(u^{k}-\bar{u}^{k})\biggl{]},$$
(32)
where we employ Eq. (10)888Here and in what follows, we adopt the
cubic-type prepotential under the gauge $X^{0}=1$, and the dimension of the complex structure moduli space is given by $h^{2,1}_{-}$. with $i=1,2,\cdots,h^{2,1}_{-}$.
However, for the superpotential we have two options:
•
$\gamma=0$
$$\displaystyle W$$
$$\displaystyle=-f_{i}u^{i}-f^{0}(2F-u^{i}\partial_{i}F)-S\left(-h_{0}-h^{i}%
\partial_{i}F\right)$$
$$\displaystyle=-f_{i}u^{i}-f^{0}\left(-\frac{1}{6}\kappa_{ijk}u^{i}u^{j}u^{k}%
\right)+h_{0}S+h^{i}S\left(\frac{1}{2}\kappa_{ijk}u^{j}u^{k}\right).$$
(33)
•
$\gamma=\pi$
$$\displaystyle W$$
$$\displaystyle=-f_{0}-f^{i}\partial_{i}F-S\biggl{[}-h_{i}u^{i}-h^{0}(2F-u^{i}%
\partial_{i}F)\biggl{]}$$
$$\displaystyle=-f_{0}-f^{i}\left(\frac{1}{2}\kappa_{ijk}u^{j}u^{k}\right)+h_{i}%
u^{i}S+h^{0}S\left(-\frac{1}{6}\kappa_{ijk}u^{i}u^{j}u^{k}\right).$$
(34)
Hence, the CP-invariant superpotential is restricted to be either odd
or even polynomials with respect to the moduli fields.
Note that these fluxes induce the D3-brane charge:
$$\displaystyle N_{\rm flux}=\int H_{3}\wedge F_{3}=\left\{\begin{array}[]{l}-f^%
{0}h_{0}+\sum_{i}f_{i}h^{i},\quad(\gamma=0)\\
f_{0}h^{0}-\sum_{i}f^{i}h_{i},\quad(\gamma=\pi)\end{array}\right.,$$
(35)
which should be cancelled by mobile D3-branes and orientifold
contributions.
Finally, we comment on the (accidental) symmetry which arises in the CP-invariant effective action
from the field theoretical point of view.
The superpotential of even degree with respect to the moduli fields in Eq. (34)
is invariant under the discrete $\mathbb{Z}_{2}$ symmetry (not related to the discrete R-symmetry)
$$\displaystyle u^{i}\rightarrow-u^{i},\quad S\rightarrow-S.$$
(36)
For the superpotential with the odd degree of moduli fields,
one can assign the R-charge 2 for all the moduli fields due to an existence of
linear terms. However, the existence of cubic terms breaks the continuous R-symmetry
and it results in the discrete $\mathbb{Z}_{4}$ R-symmetry in the above superpotential.
Note that in the superpotential of both odd and even degrees, the Kähler potential is also invariant under
the discrete $\mathbb{Z}_{2}$ and $\mathbb{Z}_{4}$ symmetries, taking into account transformations of
the axio-dilaton and the complex structure moduli at the same time.
Let us consider the generic flux-induced superpotential having both odd and even degrees of polynomials with
respect to the moduli fields:
$$\displaystyle W=-f_{i}u^{i}-f^{0}(2F-u^{i}\partial_{i}F)-S\left(-h_{0}-h^{i}%
\partial_{i}F\right)-f_{0}-f^{i}\partial_{i}F-S\biggl{[}-h_{i}u^{i}-h^{0}(2F-u%
^{i}\partial_{i}F)\biggl{]}.$$
(37)
When we impose the discrete $\mathbb{Z}_{2}$ or $\mathbb{Z}_{4}$ symmetry for moduli fields
in the supersymmetric effective action,
the generic flux-induced superpotential (37) in the large complex structure
regime has a CP-invariance
categorized by two classes, i.e. Eqs. (33) and (34).
Hence, the necessary and sufficient condition to posses CP in the moduli effective action
is the existence of discrete $\mathbb{Z}_{2}$ or $\mathbb{Z}_{4}$ symmetry and the supersymmetry (SUSY).
Recalling that the spontaneous CP violation is realized by the non-zero vacuum expectation values
of axionic fields ${\rm Re}u^{i}$ and ${\rm Re}S$,
the discrete $\mathbb{Z}_{2}$ or $\mathbb{Z}_{4}$ symmetry is spontaneously broken at the CP-breaking vacua.
4 CP-conserving and -breaking vacua
We are ready to analyze the vacuum structure of CP-invariant
effective action.
We first discuss the existence of CP-conserving vacua in
the large complex structure regime of generic CY threefolds in Sec. 4.1.
Next, we deal with concrete CY threefolds in the large complex structure regime to clarify whether the
spontaneous CP violation is realized or not.
It turns out that for the prepotential having a similar structure of factorizable tori in Secs. 4.2 and 4.3,
it is difficult to achieve the spontaneous CP violation.
When the structure of prepotential is deviated from the toroidal one,
one can find the CP-breaking vacua as discussed in detail in Sec. 4.4,
where we choose a specific CY such as the degree 18 hypersurface in a weighted projective space $\mathbb{CP}_{11169}$.
4.1 CP-conserving vacua in generic CY
Before analyzing the CP-breaking minimum of the
scalar potential, we investigate the CP-conserving minima satisfying
the SUSY condition,
$$\displaystyle D_{I}W=\left(\partial_{I}+K_{I}\right)W=0,$$
(38)
where $I=\{S,u^{i}\}$ runs over the axio-dilaton $S$ and all the complex structure moduli $u^{i}$,
and $K_{I}=\partial_{I}K$.
Since the effective action in the large volume regime possesses the no-scale structure for the
Kähler moduli, we focus on the dynamics of the axio-dilaton and the complex structure moduli
in the following analysis.
The superpotential and its covariant derivatives transform under the CP transformations $\{S\rightarrow-\bar{S},u^{i}\rightarrow-\bar{u}^{i}\}$
as
•
$\gamma=0$
$$\displaystyle W$$
$$\displaystyle\rightarrow\overline{W},\quad D_{S}W\rightarrow\overline{D_{S}W},%
\quad D_{i}W\rightarrow{\overline{D_{i}W}},$$
(39)
•
$\gamma=\pi$
$$\displaystyle W$$
$$\displaystyle\rightarrow-\overline{W},\quad D_{S}W\rightarrow-\overline{D_{S}W%
},\quad D_{i}W\rightarrow-{\overline{D_{i}W}},$$
(40)
where we use $K_{\bar{i}}=-K_{i}$ and $K_{i}$ is a CP-even function due to the axionic
shift symmetries.
Hence, for both $\gamma=0$ and $\pi$ cases, the SUSY minimum $D_{I}W=0$ has a symmetry,
$\langle{\rm Re}S\rangle\rightarrow{-\langle{\rm Re}S\rangle}$ and
$\langle{\rm Re}u^{i}\rangle\rightarrow{-\langle{\rm Re}u^{i}\rangle}$.
It suggests that CP-invariant moduli effective action may have the CP-conserving
vacua in generic CY flux compactifications.
Indeed, among the $h^{2,1}_{-}+1$ number of SUSY conditions $D_{I}W=0$,
the CP-conserving vacuum ${\rm Re}S={\rm Re}u^{i}=0$ is a solution for half of them, namely ${\rm Re}(D_{I}W)=0$ and ${\rm Im}(D_{I}W)=0$ for $\gamma=\pi$ and $\gamma=0$, respectively.
By introducing
$$\displaystyle{\cal X}^{i}\equiv\kappa_{ijk}{\rm Re}u^{j}{\rm Im}u^{k},\quad{%
\cal Y}^{i}\equiv\kappa_{ijk}{\rm Re}u^{j}{\rm Re}u^{k},\quad{\cal Z}^{i}%
\equiv\kappa_{ijk}{\rm Im}u^{j}{\rm Im}u^{k},$$
(41)
one can explicitly check that ${\rm Re}S={\rm Re}u^{i}=0$ lead to
•
$\gamma=0$
$$\displaystyle{\rm Im}(D_{S}W)$$
$$\displaystyle=h^{i}{\cal X}^{i}+{\rm Im}K_{S}{\rm Re}W=0,$$
$$\displaystyle{\rm Im}(D_{i}W)$$
$$\displaystyle=f^{0}{\cal X}^{i}+{\rm Re}S\kappa_{ijk}h^{j}{\rm Im}u^{k}+{\rm Im%
}S\kappa_{ijk}h^{j}{\rm Re}u^{k}+{\rm Im}K_{i}{\rm Re}W=0,$$
(42)
with
$$\displaystyle{\rm Re}W=-f_{i}{\rm Re}u^{i}+\frac{f^{0}}{6}\left({\cal Y}^{i}-3%
{\cal X}^{i}\right){\rm Re}u^{i}+h_{0}{\rm Re}S+\frac{h^{i}}{2}{\rm Re}S\left(%
{\cal Y}^{i}-{\cal Z}^{i}\right)-{\rm Im}Sh^{i}{\cal X}^{i}.$$
(43)
•
$\gamma=\pi$
$$\displaystyle{\rm Re}(D_{S}W)$$
$$\displaystyle=h_{i}{\rm Re}u^{i}-\frac{h^{0}}{6}\left({\cal Y}^{i}-3{\cal Z}^{%
i}\right){\rm Re}u^{i}-{\rm Im}K_{S}{\rm Im}W=0,$$
$$\displaystyle{\rm Re}(D_{i}W)$$
$$\displaystyle=-\kappa_{ijk}f^{j}{\rm Re}u^{k}+{\rm Re}S\left(h_{i}-\frac{h^{0}%
}{2}{\cal Y}^{i}+\frac{h^{0}}{2}{\cal Z}^{i}\right)+h^{0}{\rm Im}S{\cal X}^{i}%
-{\rm Im}K_{i}{\rm Im}W=0,$$
(44)
with
$$\displaystyle{\rm Im}W=-f^{i}{\cal X}^{i}+{\rm Re}S\left(h_{i}-\frac{h^{0}}{2}%
{\cal Y}^{i}+\frac{h^{0}}{6}{\cal Z}^{i}\right){\rm Im}u^{i}+{\rm Im}S\left(h_%
{i}-\frac{h^{0}}{6}{\cal Y}^{i}+\frac{h^{0}}{2}{\cal Z}^{i}\right){\rm Re}u^{i}.$$
(45)
Note that ${\rm Im}K_{I}$ are functions of imaginary parts of moduli fields.
In this way, half of the SUSY conditions $D_{I}W=0$ are satisfied by ${\rm Re}S={\rm Re}u^{i}=0$, and the imaginary parts are determined by the remaining SUSY conditions for both $\gamma=0$ and $\gamma=\pi$ cases.
In the following, we explicitly check an existence of CP-conserving vacua and
search for the CP-breaking vacua on concrete CY threefolds.
4.2 One modulus case: $\mathbb{CP}_{11111}[5]$
We begin with the CY threefold with a single modulus, especially
the mirror dual of the quintic $\mathbb{CP}_{11111}[5]$ in Ref. [27], defined by
the degree 5 hypersurface in a projective space $\mathbb{CP}_{11111}$.
To realize the CP-invariant moduli potential, we restrict ourselves
to the large complex structure regime ${\rm Im}U>1$,
where we denote the single complex structure modulus by $U$.
Given the triple intersection number $\kappa_{UUU}=5$,
the Kähler potential is given by
$$\displaystyle K=-\ln(i(\bar{S}-S))-\ln\biggl{[}\frac{5i}{6}(U-\bar{U})^{3}%
\biggl{]},$$
(46)
and the superpotential is categorized by two classes:
$$\displaystyle W=\left\{\begin{array}[]{c}-f_{U}U+\frac{5f^{0}}{6}U^{3}+h_{0}S+%
\frac{5h^{U}}{2}SU^{2}\quad(\gamma=0)\\
-f_{0}-\frac{f^{U}}{2}U^{2}+h_{U}US-\frac{5h^{0}}{6}SU^{3}\quad(\gamma=\pi)%
\end{array}\right.,$$
(47)
with $\{f^{0},f_{0},f_{U},f^{U},h_{0},h^{0},h^{U},h_{U}\}$ being flux quanta.
By solving SUSY conditions $D_{I}W=0$ for $S$ and $U$, we find the CP-conserving minima:
•
$\gamma=0$
$$\displaystyle{\rm Re}U={\rm Re}S=0,\quad{\rm Im}U=\sqrt{\frac{2}{5}}\left(-%
\frac{3f_{U}h_{0}}{f^{0}h^{U}}\right)^{1/4},\quad{\rm Im}S=\sqrt{\frac{2}{5}}%
\left(-\frac{f^{0}}{3h_{0}}\right)^{1/4}\left(\frac{f_{U}}{h^{U}}\right)^{3/4},$$
(48)
•
$\gamma=\pi$
$$\displaystyle{\rm Re}U={\rm Re}S=0,\quad{\rm Im}U=\sqrt{2}\left(-\frac{3f_{0}h%
_{U}}{5f^{U}h^{0}}\right)^{1/4},\quad{\rm Im}S=\sqrt{\frac{1}{2}}\left(\frac{3%
f_{0}}{5h^{0}}\right)^{1/4}\left(-\frac{f^{U}}{h_{U}}\right)^{3/4},$$
(49)
and four classes of degenerate CP-conserving and -breaking minima:
$$\displaystyle{\rm(i)}\,|U|^{2}=\frac{6f_{U}}{5f^{0}}=-\frac{2h_{0}}{5h^{U}},%
\quad S=-\frac{f^{0}}{3h^{U}}\bar{U},$$
$$\displaystyle{\rm(ii)}\,|U|^{2}=-\frac{2f_{U}}{5f^{0}}=\frac{6h_{0}}{5h^{U}},%
\quad|S|^{2}=\left(\frac{f^{0}}{h^{U}}\right)^{2}|U|^{2},\quad{\rm Im}S=\frac{%
5(f^{0})^{2}({\rm Im}U)^{3}}{8f_{U}h^{U}+15f^{0}h^{U}({\rm Im}U)^{2}},$$
(50)
for $\gamma=0$ and
$$\displaystyle{\rm(iii)}\,|U|^{2}=\frac{6h_{U}}{5h^{0}},\quad S=-\frac{f^{U}}{2%
h_{U}}U,\quad f_{0}=-\frac{3f^{U}h_{U}}{5h^{0}},$$
$$\displaystyle{\rm(iv)}\,|U|^{2}=-\frac{2h_{U}}{5h^{0}},\quad|S|^{2}=-\frac{(f^%
{U})^{2}}{10h^{0}h_{U}},\quad{\rm Im}S=\frac{5f^{U}h^{0}({\rm Im}U)^{3}}{4(h_{%
U})^{2}-30h^{0}h_{U}({\rm Re}U)^{2}},\quad f_{0}=-\frac{f^{U}h_{U}}{15h^{0}},$$
(51)
for $\gamma=\pi$, respectively.
Hence, we cannot realize the spontaneous CP-violation in the large complex structure regime.
This is because the prepotential has a structure similar to one of the toroidal background
with the overall complex structure modulus, where it was pointed out in Ref. [22] that
the spontaneous CP violation is difficult to achieve.
This argument also holds for other one-parameter CY threefolds in the large complex structure regime by
changing the value of triple intersection number, for instance
the mirror dual of $\mathbb{CP}_{11112}[6]$, $\mathbb{CP}_{11114}[8]$ and
$\mathbb{CP}_{11125}[10]$ defined on a single polynomial in an ambient weighted projective space.
Interestingly, for a particular choice of fluxes, CP is embedded into $SL(2,\mathbb{Z})_{S}$ duality group
of the axio-dilaton and/or the modular symmetry of the complex structure moduli.
For instance, the vacuum expectation value of the axio-dilaton in the solution (i)
is given by
$$\displaystyle|S|^{2}=\frac{2f^{0}f_{U}}{15(h^{U})^{2}}=1$$
(52)
by setting $f^{0}f_{U}=15(h^{U})^{2}/2$.
The $S$-transformation of the $SL(2,\mathbb{Z})_{S}$ duality group at the vacuum (52)
$$\displaystyle S\rightarrow-1/S=-\bar{S}$$
(53)
corresponds to the CP transformation in Eq. (27).
In this respect, CP is unified into the duality group for a particular choice of fluxes.
4.3 Two moduli case: $\mathbb{CP}_{11222}[8]$
The next example is the CY threefold with two complex structure moduli
labelled by $u^{1}$ and $u^{2}$.
In particular, we deal with the mirror dual of CY threefold defined by
the degree 8 hypersurface in a weighted projective space $\mathbb{CP}_{11222}$ studied
in Refs. [28, 29],
where the triple intersection numbers are specified by
$$\displaystyle\kappa_{111}=8,\qquad\kappa_{112}=4,$$
(54)
and otherwise 0.
By restricting ourselves to the large complex structure regime $\{{\rm Im}u^{1},{\rm Im}u^{2}>1\}$,
the Kähler potential is given by
$$\displaystyle K$$
$$\displaystyle=-\ln(i(\bar{S}-S))-\ln\biggl{[}\frac{i}{6}\left(8(u^{1}-\bar{u}^%
{1})^{3}+12(u^{1}-\bar{u}^{1})^{2}(u^{2}-\bar{u}^{2})\right)\biggl{]},$$
(55)
and the superpotential is categorized by two classes:
•
$\gamma=0$
$$\displaystyle W$$
$$\displaystyle=-f_{1}u^{1}-f_{2}u^{2}+\frac{f^{0}}{6}(u^{1})^{2}\left(8u^{1}+12%
u^{2}\right)+h_{0}S+\frac{h^{1}}{2}S\left(8(u^{1})^{2}+8u^{1}u^{2}\right)+%
\frac{h^{2}}{2}S\left(8u^{1}u^{2}\right),$$
(56)
•
$\gamma=\pi$
$$\displaystyle W$$
$$\displaystyle=-f_{0}-\frac{f^{1}}{2}\left(8(u^{1})^{2}+8u^{1}u^{2}\right)-%
\frac{f^{2}}{2}\left(8u^{1}u^{2}\right)+(h_{1}u^{1}+h_{2}u^{2})S-\frac{h^{0}}{%
6}S\left(8(u^{1})^{3}+12(u^{1})^{2}u^{2}\right),$$
(57)
where we denote the flux quanta $\{f^{0},f_{0},f_{1,2},f^{1,2},h_{0},h^{0},h^{1,2},h_{1,2}\}$.
By solving SUSY conditions $D_{I}W=0$ with $I=S,u^{1},u^{2}$, we find CP-conserving
solutions:
•
$\gamma=0$
$$\displaystyle{\rm Re}u^{1}$$
$$\displaystyle={\rm Re}u^{2}={\rm Re}S=0,$$
$$\displaystyle{\rm Im}u^{1}$$
$$\displaystyle=2^{-3/4}\left(\frac{3f_{2}h_{0}}{f^{0}(2h^{2}-h^{1})}\right)^{1/%
4},\quad{\rm Im}u^{2}=-\frac{h_{0}}{2f^{0}}\frac{{\rm Im}S}{({\rm Im}u^{1})^{2%
}}-\frac{2}{3}{\rm Im}u^{1},$$
$$\displaystyle{\rm Im}S$$
$$\displaystyle=2^{-1}\left(\frac{f^{0}(2f_{2}-3f_{1})}{h_{0}(h^{1}+h^{2})}%
\right)^{1/2}\left(\frac{f_{2}h_{0}}{6f^{0}(2h^{2}-h^{1})}\right)^{1/4}.$$
(58)
•
$\gamma=\pi$
$$\displaystyle{\rm Re}u^{1}$$
$$\displaystyle={\rm Re}u^{2}={\rm Re}S=0,$$
$$\displaystyle{\rm Im}u^{1}$$
$$\displaystyle=2^{-3/4}\left(\frac{3f_{0}h_{2}}{h^{0}(2f^{2}-f^{1})}\right)^{1/%
4},\quad{\rm Im}u^{2}=\frac{-3h_{1}+2h_{2}}{12(f^{1}+f^{2})}{\rm Im}S-\frac{2}%
{3}{\rm Im}u^{1},$$
$$\displaystyle{\rm Im}S$$
$$\displaystyle=2\left(\frac{f^{1}+f^{2}}{h^{0}h_{2}(2h_{2}-3h_{1})}\right)^{1/2%
}\left(6f_{0}h^{0}h_{2}(2f^{2}-f^{1})\right)^{1/4}.$$
(59)
On the other hand, we rely on the numerical search to find the CP-breaking
vacua in both $\gamma=0$ and $\pi$ cases.999Specifically, we used “FindRoot” of Mathematica (v12.0) to solve SUSY conditions $D_{I}W=0$ with randomly generated fluxes and initial values of moduli fields.
Numerical search for the randomly generated $2\times 10^{7}$ dataset of fluxes within $-30\leq\{f^{0},f_{0},f_{1,2},f^{1,2},h_{0},h^{0},h^{1,2},h_{1,2}\}\leq 30$ leading to $0\leq N_{\rm flux}\leq 150$ allows only $5.8\times 10^{3}$ and 87 stable CP-conserving vacua for $\gamma=0$ and $\gamma=\pi$, respectively.
The reason why the CP-breaking vacuum is absent is that the Kähler
potential and the superpotential have a similar structure with the toroidal
one due to the torus-type prepotential.
Indeed, when we redefine the modulus field as $12u^{2}=-8u^{1}+u$,
the prepotential is given by
$$\displaystyle F(u)$$
$$\displaystyle=\frac{1}{6}\left(8(u_{1})^{3}+12(u_{1})^{2}u_{2}\right)=\frac{1}%
{6}(u_{1})^{2}u.$$
(60)
Because of the moduli redefinition, the prepotential has a similar structure to the factorizable $T^{6}$ torus by identifying the two complex structure moduli with the identical one.
4.4 CP-breaking vacua on $\mathbb{CP}_{11169}[18]$
Finally, we analyze the different two-parameter CY threefold, especially
the mirror dual of the degree 18 hypersurface in a weighted projective space $\mathbb{CP}_{11169}$
studied in Ref. [30], where the complex structure moduli are labelled by $u^{1}$ and $u^{2}$.
We can also consider the original CY threefold as follows.101010For more details, see, e.g. Ref. [31].
Taking into account a $G=\mathbb{Z}_{6}\times\mathbb{Z}_{18}$ discrete symmetry of this CY,
the complex structure moduli space parametrized by $u^{1}$ and $u^{2}$ is invariant under this action.
Other non-invariant complex structure moduli can
be fixed at the fixed points under $G$, thanks
to the three-form fluxes along the $G$-invariant three-forms.
Hence, the period integrals in the mirror dual of the CY
are the same with the original two-parameter CY threefold.
To realize the CP-invariant moduli potential, we further restrict ourselves
to the large complex structure regime $\{{\rm Im}u^{1},{\rm Im}u^{2}>1\}$.
Given the non-vanishing triple intersection numbers $\kappa_{111}=9$, $\kappa_{112}=3$ and $\kappa_{122}=1$,
the Kähler potential is given by
$$\displaystyle K=-\ln(i(\bar{S}-S))-\ln\biggl{[}\frac{i}{6}\left(9(u^{1}-\bar{u%
}^{1})^{3}+9(u^{1}-\bar{u}^{1})^{2}(u^{2}-\bar{u}^{2})+3(u^{1}-\bar{u}^{1})(u^%
{2}-\bar{u}^{2})^{2}\right)\biggl{]},$$
(61)
and the superpotential is categorized by two classes:
$$\displaystyle W=\left\{\begin{array}[]{l}-f_{1}u^{1}-f_{2}u^{2}+\frac{f^{0}}{6%
}\left(9(u^{1})^{3}+9(u^{1})^{2}u^{2}+3u^{1}(u^{2})^{2}\right)\\
+h_{0}S+\frac{h^{1}}{2}S\left(9(u^{1})^{2}+6u^{1}u^{2}+(u^{2})^{2}\right)+%
\frac{h^{2}}{2}S\left(3(u^{1})^{2}+2u^{1}u^{2}\right)\quad(\gamma=0)\\
\\
-f_{0}-\frac{f^{1}}{2}\left(9(u^{1})^{2}+6u^{1}u^{2}+(u^{2})^{2}\right)-\frac{%
f^{2}}{2}\left(3(u^{1})^{2}+2u^{1}u^{2}\right)\\
+(h_{1}u^{1}+h_{2}u^{2})S-\frac{h^{0}}{6}S\left(9(u^{1})^{3}+9(u^{1})^{2}u^{2}%
+3u^{1}(u^{2})^{2}\right)\quad(\gamma=\pi)\end{array}\right.,$$
(62)
where we denote the flux quanta $\{f^{0},f_{0},f_{1,2},f^{1,2},h_{0},h^{0},h^{1,2},h_{1,2}\}$. These integral flux quanta are constrained by the tadpole cancellation condition [32]
$$\displaystyle 0\leq N_{\rm flux}\leq 138.$$
(63)
By solving SUSY conditions $D_{I}W=0$ with $I=S,u^{1},u^{2}$, we find that CP-conserving solutions are realized to
satisfy
•
$\gamma=0$
$$\displaystyle{\rm Re}u^{1}$$
$$\displaystyle={\rm Re}u^{2}={\rm Re}S=0,$$
$$\displaystyle\frac{h_{0}}{f^{0}}$$
$$\displaystyle=-\frac{{\rm Im}u^{1}(3({\rm Im}u^{1})^{2}+3{\rm Im}u^{1}{\rm Im}%
u^{2}+({\rm Im}u^{2})^{2})}{2{\rm Im}S},$$
$$\displaystyle\frac{h^{1}}{{\rm Im}u^{1}}$$
$$\displaystyle=-\frac{3f_{2}({\rm Im}u^{1}+{\rm Im}u^{2})(3{\rm Im}u^{1}+{\rm Im%
}u^{2})-f_{1}(3({\rm Im}u^{1})^{2}+6{\rm Im}u^{1}{\rm Im}u^{2}+2({\rm Im}u^{2}%
)^{2})}{{\rm Im}u^{2}(3{\rm Im}u^{1}+{\rm Im}u^{2})(3({\rm Im}u^{1})^{2}+3{\rm
Im%
}u^{1}{\rm Im}u^{2}+({\rm Im}u^{2})^{2}){\rm Im}S},$$
$$\displaystyle h^{2}$$
$$\displaystyle=-\frac{3f_{1}({\rm Im}u^{1})^{2}({\rm Im}u^{1}+{\rm Im}u^{2})-f_%
{2}(9({\rm Im}u^{1})^{3}+9({\rm Im}u^{1})^{2}{\rm Im}u^{2}+3{\rm Im}u^{1}({\rm
Im%
}u^{2})^{2}+({\rm Im}u^{2})^{3})}{{\rm Im}u^{1}{\rm Im}u^{2}(3({\rm Im}u^{1})^%
{2}+3{\rm Im}u^{1}{\rm Im}u^{2}+({\rm Im}u^{2})^{2}){\rm Im}S},$$
(64)
•
$\gamma=\pi$
$$\displaystyle{\rm Re}u^{1}$$
$$\displaystyle={\rm Re}u^{2}={\rm Re}S=0,$$
$$\displaystyle\frac{f_{0}}{h^{0}}$$
$$\displaystyle=\frac{{\rm Im}u^{1}{\rm Im}S\left(3({\rm Im}u^{1})^{2}+3{\rm Im}%
u^{1}{\rm Im}u^{2}+({\rm Im}u^{2})^{2}\right)}{2},$$
$$\displaystyle\frac{f^{1}}{{\rm Im}u^{1}}$$
$$\displaystyle=-\frac{3f^{2}{\rm Im}u^{1}({\rm Im}u^{1}+{\rm Im}u^{2})(3{\rm Im%
}u^{1}+{\rm Im}u^{2})+2h_{1}{\rm Im}S(3({\rm Im}u^{1})^{2}+3{\rm Im}u^{1}{\rm
Im%
}u^{2}+({\rm Im}u^{2})^{2})}{(3{\rm Im}u^{1}+{\rm Im}u^{2})(9({\rm Im}u^{1})^{%
3}+9({\rm Im}u^{1})^{2}{\rm Im}u^{2}+3{\rm Im}u^{1}({\rm Im}u^{2})^{2}+({\rm Im%
}u^{2})^{3})},$$
$$\displaystyle\frac{h_{2}}{{\rm Im}u^{1}}$$
$$\displaystyle=\frac{-f^{2}{\rm Im}u^{2}(3({\rm Im}u^{1})^{2}+{\rm Im}u^{1}{\rm
Im%
}u^{2}+({\rm Im}u^{2})^{2})+3h_{1}{\rm Im}u^{1}({\rm Im}u^{1}+{\rm Im}u^{2}){%
\rm Im}S}{(9({\rm Im}u^{1})^{3}+9({\rm Im}u^{1})^{2}{\rm Im}u^{2}+3{\rm Im}u^{%
1}({\rm Im}u^{2})^{2}+({\rm Im}u^{2})^{3}){\rm Im}S}.$$
(65)
On the other hand, we numerically searched111111We used the same numerical method in Sec. 4.3, but the result could be highly dependent of initial values.
Hence, this non-existence of CP-breaking vacua for the even polynomial case could not be a general statement but may capture some tendency. for the CP-breaking vacua
under the set of randomly generated fluxes satisfying the
tadpole cancellation condition (63).
For the even polynomial case, one cannot find the CP-breaking vacua
for $4\times 10^{7}$ dataset of fluxes within $-30\leq\{f_{0},f^{1,2},h^{0},h_{1,2}\}\leq 30$, whereas there exist CP-breaking
vacua in the superpotential with odd degrees.
Indeed, the numerical search under the randomly generated $2\times 10^{7}$ dataset of fluxes
within $-30\leq\{f^{0},f_{1,2},h_{0},h^{1,2}\}\leq 30$ leads to 559 stable CP-breaking vacua in the large complex structure regime ${\rm Im}u^{1}>1$.
For the benchmark dataset of fluxes:
$$\displaystyle(f^{0},f_{1},f_{2},h_{0},h^{1},h^{2})=(-1,25,-23,22,1,-3),$$
(66)
leading to $N_{\rm flux}=116$,
the vacuum expectation values of moduli fields are evaluated as
$$\displaystyle\langle{\rm Re}u^{1}\rangle\simeq 1.86,\quad\langle{\rm Re}u^{2}%
\rangle\simeq-2.70,\quad\langle{\rm Re}S\rangle\simeq 4.32,$$
$$\displaystyle\langle{\rm Im}u^{1}\rangle\simeq 3.01,\quad\langle{\rm Im}u^{2}%
\rangle\simeq 4.15,\quad\langle{\rm Im}S\rangle\simeq 4.81,$$
(67)
showing that CP is spontaneously broken due to the nonvanishing values of
axionic fields.
Note that masses squared of the moduli fields are positive as shown in the descendent order,
$$\displaystyle{\cal V}^{-2}(9.72,5.90,4.73,2.38,5.29\times 10^{-1},6.60\times 1%
0^{-2}).$$
(68)
Hence, CP is spontaneously broken in this class of flux compactification.
The realization of spontaneous CP violation depends on the structure of
the prepotential and the functional form of the superpotential.
The reason why the CP-breaking vacua are absent in the choice of the
superpotential with even degrees is unclear owing to the fact
that the SUSY conditions are non-linear functions of the axionic fields as in
Eqs. (42) and (44).
The profound understanding of the origin of CP-breaking vacua will be reached
by studying other CY flux compactifications, which will be investigated
in future work.
5 Conclusions
We have revealed the geometrical origin of CP embedded into the 10D proper Lorentz
transformation with an emphasis on the complex structure of compact
6D spaces, in particular the CY threefolds.
We find that the anti-holomorphic involution of the complex structure is regarded
as the anti-holomorphic involution of period integrals on CY threefolds
with the large complex structure.
Consequently, the anti-holomorphic involution of the period integrals corresponds to
the outer automorphism of $Sp(2h^{2,1}+2,\mathbb{Z})$ symplectic modular group,
rather than the element of $Sp(2h^{2,1}+2,\mathbb{Z})$ in the complex structure moduli space.
The moduli group is then enlarged into $Sp(2h^{2,1}+2,\mathbb{Z})\rtimes{\cal CP}$.
That is a natural extension of the known toroidal cases, where the CP symmetry is regarded
as the outer automorphism of the $SL(2,\mathbb{Z})$ modular group [15, 16].
The CP violation is strongly correlated with the dynamics of the moduli fields,
whose vacuum expectation values determine the size of CP violation.
Our approach to realize the spontaneous CP violation is based on the flux compactifications
in the string theory.
For concreteness, we deal with Type IIB flux compactifications, where
we turn on three-form fluxes on CY three-cycles.
Imposing CP invariance on the moduli effective action requires the restricted choices of RR and NSNS flux quanta
in the large complex structure regime of CY threefolds.
It results in the flux-induced superpotential consisting of either odd or even
polynomials with respect to the moduli fields.
These CP-invariant superpotential possesses the discrete $\mathbb{Z}_{2}$ symmetry
or $\mathbb{Z}_{4}$ R-symmetry from the field theoretical point of view.
We analyzed the vacuum structure of CP-invariant effective action.
It turned out that CP-conserving vacua appear in generic CY flux compactifications
in the large complex structure regime.
To check the existence of CP-breaking vacua, we work with some concrete CY threefolds.
It indicates that the prepotential having the toroidal structure does not lead to the spontaneous CP violation.
For a particular choice of fluxes, CP is embedded into the duality symmetry of the axio-dilaton and/or
the modular symmetry of the complex structure moduli.
When the structure of the prepotential differs from the toroidal one,
the spontaneous CP violation
can be achieved on the CY, illustrated on $\mathbb{CP}_{11169}[18]$.
It is interesting to examine more examples to reveal the nature of spontaneous CP
violation in CY compactifications.
In this paper, we examined the geometrical origin of CP and its violation in the complex structure moduli space of CY threefolds, but
phenomenologically it is required to consider the CP violation in the matter sector.
The CP and the flavor symmetries can be unified in the common group as discussed in
toroidal orbifolds [33].
We will report the analysis of CP and the flavor
on curved CY manifolds in a separate paper.
Note added
After finishing this work, Ref. [34] appeared, where symplectic modular symmetries were studied
from the phenomenological viewpoints.
Acknowledgements
T. K. was supported in part by MEXT KAKENHI Grant Number JP19H04605. H. O. was
supported in part by JSPS KAKENHI Grant Numbers JP19J00664 and JP20K14477.
References
[1]
A. Strominger and E. Witten,
Commun. Math. Phys. 101 (1985), 341.
[2]
M. Dine, R. G. Leigh and D. A. MacIntire,
Phys. Rev. Lett. 69 (1992) 2030
[hep-th/9205011].
[3]
K. w. Choi, D. B. Kaplan and A. E. Nelson,
Nucl. Phys. B 391 (1993) 515
[hep-ph/9205202].
[4]
T. Kobayashi and C. S. Lim,
Phys. Lett. B 343, 122-127 (1995)
[arXiv:hep-th/9410023 [hep-th]].
[5]
W. Grimus and M. N. Rebelo,
Phys. Rept. 281 (1997), 239-308
[arXiv:hep-ph/9506272 [hep-ph]].
[6]
I. L. Buchbinder, D. M. Gitman and A. L. Shelepin,
Int. J. Theor. Phys. 41 (2002), 753-790
[arXiv:hep-th/0010035 [hep-th]].
[7]
T. Kobayashi, S. Nagamoto, S. Takada, S. Tamba and T. H. Tatsuishi,
Phys. Rev. D 97, no.11, 116002 (2018)
[arXiv:1804.06644 [hep-th]].
[8]
S. Kikuchi, T. Kobayashi, S. Takada, T. H. Tatsuishi and H. Uchida,
[arXiv:2005.12642 [hep-th]].
[9]
S. Kikuchi, T. Kobayashi, H. Otsuka, S. Takada and H. Uchida,
[arXiv:2007.06188 [hep-th]].
[10]
T. Kobayashi and H. Otsuka,
Phys. Rev. D 101 (2020) no.10, 106017
[arXiv:2001.07972 [hep-th]].
[11]
F. Feruglio,
arXiv:1706.08749 [hep-ph];
T. Kobayashi, K. Tanaka and T. H. Tatsuishi,
Phys. Rev. D 98, no. 1, 016004 (2018)
[arXiv:1803.10391 [hep-ph]];
J. T. Penedo and S. T. Petcov,
Nucl. Phys. B 939, 292 (2019)
[arXiv:1806.11040 [hep-ph]];
J. C. Criado and F. Feruglio,
SciPost Phys. 5, no. 5, 042 (2018)
[arXiv:1807.01125 [hep-ph]];
T. Kobayashi, N. Omoto, Y. Shimizu, K. Takagi, M. Tanimoto and T. H. Tatsuishi,
JHEP 1811, 196 (2018)
[arXiv:1808.03012 [hep-ph]];
P. P. Novichkov, J. T. Penedo, S. T. Petcov and A. V. Titov,
JHEP 1904, 005 (2019)
[arXiv:1811.04933 [hep-ph]];
JHEP 1904, 174 (2019)
[arXiv:1812.02158 [hep-ph]];
F. J. de Anda, S. F. King and E. Perdomo,
arXiv:1812.05620 [hep-ph];
H. Okada and M. Tanimoto,
Phys. Lett. B 791, 54 (2019)
[arXiv:1812.09677 [hep-ph]];
T. Kobayashi, Y. Shimizu, K. Takagi, M. Tanimoto, T. H. Tatsuishi and H. Uchida,
Phys. Lett. B 794, 114 (2019)
[arXiv:1812.11072 [hep-ph]];
P. P. Novichkov, S. T. Petcov and M. Tanimoto,
Phys. Lett. B 793, 247 (2019)
[arXiv:1812.11289 [hep-ph]].
[12]
F. Feruglio, C. Hagedorn and R. Ziegler,
JHEP 1307 (2013) 027
[arXiv:1211.5560 [hep-ph]].
[13]
M. Holthausen, M. Lindner and M. A. Schmidt,
JHEP 1304 (2013) 122
[arXiv:1211.6953 [hep-ph]].
[14]
M. C. Chen, M. Fallbacher, K. T. Mahanthappa, M. Ratz and A. Trautner,
Nucl. Phys. B 883 (2014) 267
[arXiv:1402.0507 [hep-ph]].
[15]
H. P. Nilles, M. Ratz, A. Trautner and P. K. S. Vaudrevange,
Phys. Lett. B 786 (2018) 283
[arXiv:1808.07060 [hep-th]].
[16]
P. P. Novichkov, J. T. Penedo, S. T. Petcov and A. V. Titov,
JHEP 07 (2019), 165
[arXiv:1905.11970 [hep-ph]].
[17]
T. Dent,
Phys. Rev. D 64, 056005 (2001)
[hep-ph/0105285].
[18]
S. Khalil, O. Lebedev and S. Morris,
Phys. Rev. D 65, 115014 (2002)
[hep-th/0110063].
[19]
J. Giedt,
Mod. Phys. Lett. A 17, 1465 (2002)
[hep-ph/0204017].
[20]
T. Kobayashi, Y. Shimizu, K. Takagi, M. Tanimoto, T. H. Tatsuishi and H. Uchida,
Phys. Rev. D 101, no.5, 055046 (2020)
[arXiv:1910.11553 [hep-ph]].
[21]
S. Gukov, C. Vafa and E. Witten,
Nucl. Phys. B 584, 69 (2000)
Erratum: [Nucl. Phys. B 608, 477 (2001)]
[hep-th/9906070].
[22]
T. Kobayashi and H. Otsuka,
Phys. Rev. D 102 (2020) no.2, 026004
[arXiv:2004.04518 [hep-th]].
[23]
A. Strominger,
Commun. Math. Phys. 133 (1990), 163-180.
[24]
P. Candelas and X. de la Ossa,
Nucl. Phys. B 355 (1991), 455-481.
[25]
S. Hosono, A. Klemm, S. Theisen and S. T. Yau,
AMS/IP Stud. Adv. Math. 1 (1996), 545-606
[arXiv:hep-th/9406055 [hep-th]].
[26]
P. Mayr,
JHEP 01 (2001), 018
[arXiv:hep-th/0010223 [hep-th]].
[27]
P. Candelas, X. C. De La Ossa, P. S. Green and L. Parkes,
AMS/IP Stud. Adv. Math. 9 (1998), 31-95.
[28]
P. Berglund, P. Candelas, X. De La Ossa, A. Font, T. Hubsch, D. Jancic and F. Quevedo,
Nucl. Phys. B 419 (1994), 352-403
[arXiv:hep-th/9308005 [hep-th]].
[29]
P. Candelas, X. De La Ossa, A. Font, S. H. Katz and D. R. Morrison,
AMS/IP Stud. Adv. Math. 1 (1996), 483-543
[arXiv:hep-th/9308083 [hep-th]].
[30]
P. Candelas, A. Font, S. H. Katz and D. R. Morrison,
Nucl. Phys. B 429 (1994), 626-674
[arXiv:hep-th/9403187 [hep-th]].
[31]
A. Giryavets, S. Kachru, P. K. Tripathy and S. P. Trivedi,
JHEP 04 (2004), 003
[arXiv:hep-th/0312104 [hep-th]].
[32]
J. Louis, M. Rummel, R. Valandro and A. Westphal,
JHEP 10 (2012), 163
[arXiv:1208.3208 [hep-th]].
[33]
A. Baur, H. P. Nilles, A. Trautner and P. K. S. Vaudrevange,
Phys. Lett. B 795 (2019) 7
[arXiv:1901.03251 [hep-th]].
[34]
G. J. Ding, F. Feruglio and X. G. Liu,
[arXiv:2010.07952 [hep-th]]. |
The Filamentary Large Scale Structure around the $z=2.16$ Radio Galaxy PKS 1138-262
Steve Croft11affiliation: Institute of Geophysics and Planetary Physics, Lawrence Livermore National Laboratory L-413, 7000 East Avenue, Livermore, CA 94550 , Jaron Kurk22affiliation: INAF, Osservatorio Astrofisico di Arcetri, Largo Enrico Fermi 5, 50125, Firenze, Italy , Wil van Breugel11affiliation: Institute of Geophysics and Planetary Physics, Lawrence Livermore National Laboratory L-413, 7000 East Avenue, Livermore, CA 94550 , S. A. Stanford11affiliation: Institute of Geophysics and Planetary Physics, Lawrence Livermore National Laboratory L-413, 7000 East Avenue, Livermore, CA 94550 33affiliation: Department of Physics, University of California at Davis, 1 Shields Avenue, Davis, CA 95616 , Wim de Vries11affiliation: Institute of Geophysics and Planetary Physics, Lawrence Livermore National Laboratory L-413, 7000 East Avenue, Livermore, CA 94550 , Laura Pentericci44affiliation: Dipartimento di Fisica, Università degli Studi Roma Tre, Italy and Huub Röttgering55affiliation: Leiden Observatory, Niels Bohrweg 2, NL-2333 CA Leiden, The Netherlands
Abstract
PKS 1138-262 is a massive radio galaxy at $z=2.16$ surrounded by
overdensities of Ly$\alpha$ emitters, H$\alpha$ emitters, EROs and X-ray
emitters. Numerous lines of evidence exist that it is located in a
forming cluster. We report on Keck spectroscopy of candidate members of
this protocluster, including nine of the 18 X-ray sources detected by
Pentericci et al. (2002) in this field. Two of these X-ray sources (not counting
PKS 1138-262 itself) were previously confirmed to be members of the
protocluster; we have discovered that an additional two (both AGN) are
members of a filamentary structure, at least $3.5$ Mpc in projection,
aligned with the radio jet axis, the 150 kpc-sized emission-line halo,
and the extended X-ray emission around the radio galaxy. Three of the
nine X-ray sources observed are lower redshift AGN, and three are
M-dwarf stars.
galaxies: active — galaxies: individual (PKS 1138-262 (catalog ))
1 Introduction
PKS 1138-262 is a massive forming radio galaxy at $z=2.156$ (Pentericci et al., 1998), which is surrounded by overdensities of Ly$\alpha$ emitters
(Pentericci et al., 2000), H$\alpha$ emitters (Kurk et al., 2004b), EROs
(Kurk et al., 2004a) and X-ray emitters (Pentericci et al., 2002), several of which
are spectroscopically confirmed to be close to the radio galaxy
redshift. These overdensities appear to be spatially aligned with each
other, and with the radio axis of PKS 1138-262 (§ 5).
At least half a dozen high redshift radio galaxies, with $2<z<5.2$,
are now known to be located in such overdense regions (protoclusters; Venemans et al., 2002). Candidate members of such protoclusters may be found using color selection, line emission in a narrow band targeted at the redshift of interest, or X-ray emission – as noted above, when applied to the field of PKS 1138-262, these methods all revealed overdensities. Protoclusters have also been found using Lyman Break techniques (e. g. , Steidel et al., 1998) and deep millimeter and sub-mm observations (e. g. Ivison et al., 2000; Smail et al., 2003), and confirmed with spectroscopic follow-up. Due to the atmospheric cutoff, $z\sim 2$ is the lowest redshift at which the Ly$\alpha$ selection technique will work using ground-based optical instruments, but the technique has been successfully used to detect clumps and filamentary large scale structure even at $z\sim 6$ (e. g. , Ouchi et al., 2005, and references therein).
Recent simulations suggest that the colors of massive galaxies in the local Universe can only be explained if AGN feedback quenches star formation in the host galaxy (Springel et al., 2005). The link between supermassive black hole and galaxy formation (Haehnelt & Kauffmann, 2000) may depend critically on AGN feedback, via coupling of the mechanical power (via winds or jets) of AGN to the baryonic component of forming galaxies (Rawlings, 2003). The mergers, galaxy harassment, and other process invoked in AGN triggering, are common in clusters, and become increasingly important as redshift increases (Moore et al., 1998). High redshift clusters are dynamic regions at the nodes of the Large Scale Structure, where huge amounts of gravitational energy involved in structure formation may be dissipated. In order to understand the link between AGN, starburst and ULIRG galaxies (e. g. Nagar et al., 2003), and the life-cycle of a typical AGN, the study of AGN in clusters and protoclusters is important. Studies of the AGN / galaxy populations, kinematics, and substructures in protoclusters provide insight into how galaxies, AGN and clusters of galaxies form and evolve.
PKS 1138-262 is one of the first, brightest, and lowest redshift radio galaxy protoclusters discovered
using Ly$\alpha$ narrow-band selection techniques. By studying PKS 1138-262, we examine an important link between protoclusters at higher redshift, and their more evolved, virialised counterparts seen at $z\sim 1$ (Ford et al., 2004), and can help constrain models of structure formation and cosmology. This, and the many X-ray emitters with unknown redshifts, motivated
us to obtain further spectra of candidate members of this protocluster using slitmasks at Keck.
The previous spectroscopically confirmed overdensities led us to suspect that even more of these targets, in addition to some of the candidate Ly$\alpha$ emitters which did not yet have redshifts, would turn out to be members of the protocluster. If a large fraction of the X-ray sources turned out to be at the cluster redshift, this would suggest that the AGN activity in this cluster is unusually high, and if a large fraction of the Ly$\alpha$ sources turn out to be protocluster members, this would tend to suggest that star formation is also enhanced.
We assume an $\Omega_{m}=0.27,\Omega_{\Lambda}=0.73,H_{0}=71~{}{{\rm km~{}s}^{-1}~{}{\rm Mpc%
}^{-1}}$ cosmology (Spergel et al., 2003). In comparing with the analyses of Kurk et al. (2000) and Pentericci et al. (2000) note that these authors use an $\Omega_{m}=1.0,\Omega_{\Lambda}=0.0,H_{0}=50~{}{{\rm km~{}s}^{-1}~{}{\rm Mpc}^%
{-1}}$ cosmology. However, due to a calculation error, the comoving volume in the cosmology of these papers should have been 2740 Mpc${}^{3}$ rather than the 3830 Mpc${}^{3}$ quoted in Pentericci et al. (2000). In considering the comoving volume in which the objects are located, note also that, as discussed by Kurk et al. (2004a), the latest Ly$\alpha$-candidate catalog is based on a re-reduction of the imaging data of Kurk et al. (2000). The area used for detection by Kurk et al. (2004a) was 38.90 arcmin${}^{2}$ (not, in fact, 43.6 arcmin${}^{2}$ as stated by those authors), 10% larger than the 35.4 arcmin${}^{2}$ used by Kurk et al. (2000) and Pentericci et al. (2000).
In our $\Lambda$CDM cosmology (similar to the cosmology of Kurk et al. 2004a), the comoving area of the field used for target selection is 98 Mpc${}^{2}$. This yields a comoving volume of 4249 Mpc${}^{3}$ probed by the spectroscopic observations (in the redshift range of 2.139 – 2.170), fortuitously rather close to the 3830 Mpc${}^{3}$ of Pentericci et al. (2000). Using the redshift range of 2.110 – 2.164 probed by the narrow-band imaging (Kurk et al., 2000) yields a comoving volume of 7405 Mpc${}^{3}$. The conclusions of Kurk et al. (2004a) are essentially unchanged, i. e. , we can estimate the strength of the overdensity to be a factor $2\pm 1$, since the comoving volume density of candidate Ly$\alpha$ emitters is 3.1 times smaller than the factor six overdensity of galaxies at $z=3.09$ discovered by Steidel et al. (2000) – see Kurk et al. (2004a) for more details. The conclusions of Pentericci et al. (2002) are also unaffected by our analysis as here the comparison to other fields is made on the basis of an areal overdensity. As noted by Pentericci et al., the strength of the overdensity (around 50% when compared to the Chandra Deep Fields) is similar to those found in galaxy clusters such as 3C 295 and RX J0030 (Cappi et al., 2001), and may, at least in part, be due to an excess of AGN in this field.
2 Keck observations
Our observations were made on 2004 January 19 – 20, using the Low
Resolution Imaging Spectrometer (LRIS; Oke et al., 1995) on Keck I, with
the D680 dichroic and a slit width of 1.5″. On the blue side, a
400 line / mm grism, blazed at 3400 Å was employed, giving 1.09
Å / pixel and spectral resolution 8.1 Å. On the red side, a 400
line / mm grating, blazed at 8500 Å was used, giving 1.86 Å /
pixel and spectral resolution 7.3 Å. This setup gives spectral coverage of $\sim 3150-9400$ Å, although this varies somewhat from slitlet to slitlet. Two slitmasks were observed for
9000 s each. Seeing was $0.8$″ during both sets of observations.
Our primary targets were randomly selected from those X-ray emitters of
Pentericci et al. (2002) and candidate Ly$\alpha$ emitters of Kurk et al. (2004a) which did not yet have spectroscopically confirmed
redshifts. Kurk et al. (2004a) select LEG (Ly$\alpha$-emitting galaxy) candidates from a 38.90 arcmin${}^{2}$ field on the basis of excess narrow versus broad band flux, from which they calculate a continuum subtracted narrow band flux, $F=F_{NB}-F_{B}$ (where $F_{NB}$ is the flux measured in a narrow band filter with central wavelength 3814 Å and FWHM 65 Å, and $F_{B}$ is the flux measured in Bessel $B$-band). They also calculate a corresponding rest frame equivalent width, $EW_{0}$ (assuming that the narrow band excess is due to Ly$\alpha$ emission at $z=2.16$). Their values of $F$ and $EW_{0}$ for our targets are shown in Table 1.
Our
slitmasks were designed by attempting to maximise the number of primary
targets per mask. Gaps were then filled with secondary targets: five
objects confirmed as members of the protocluster by
Pentericci et al. (2000). In total we observed 31 targets
(Table 1). The slitmasks were designed with slitlets ranging in length from $\sim 15-60$″, in all cases adequate for good sky subtraction.
The data were reduced in the standard manner using bogus111http://astron.berkeley.edu/$∼$dan/homepage/bogus.html
in iraf, and spectra extracted in a 1.3″-wide aperture.
3 Properties of the spectroscopic targets
Table 1 summarises the results for all of our spectroscopic
targets. Four targets were undetected and three showed continuum but no
clear emission lines. Five showed an obvious single emission line which
we were unable to unambiguously identify; since continuum is seen
blueward of the line in these cases it is likely to be [O ii]
3727. Assuming this to be the case, we assign redshifts in
Table 1 but mark these instances with a question mark.
In total we obtained secure redshifts for nineteen new targets which we classify
on the basis of the broadness of spectral lines; [O iii] / H$\beta$,
[O ii] / [O iii] and other relevant line ratios (where these
could be measured) and / or X-ray characteristics,
as discussed below. Spectral lines were fit with Gaussian profiles using splot in iraf. For L968 (X3), where broad and narrow components are present, deblending was performed by fitting with multiple Gaussians. Measured parameters for the emission lines in those objects believed to be members of the PKS 1138-262 protocluster are listed in Table 2. The observed-frame equivalent width of the fitted profile, $EW_{\lambda}$, was used to compute the rest-frame equivalent width, $EW_{0}=EW_{\lambda}\times(\lambda_{rest}/\lambda_{obs})$, which is shown in the table. In some cases our measured equivalent widths differ from the estimates of Kurk et al. (2004a); this is because it is difficult to get an accurate measure of the line flux and the faint continuum from filters which are relatively broad compared to the spectral features of interest. All lines discussed below are in emission unless otherwise noted.
L54
Previously confirmed at $z=2.143$ by Pentericci et al. (2000). We obtain $z=2.145$. Our spectrum is shown in Fig. 4.
L522
Previously confirmed at $z=2.166$ by Pentericci et al. We obtain $z=2.161$ (see Fig. 4). The Ly$\alpha$ emission is extended (Fig. 2) over 2.7″ (23 kpc) across the slit (PA = 82°), due to the Ly$\alpha$ halo which encompasses this object and the radio galaxy (Kurk, 2003). Indeed, in the narrow-band image, the halo is seen to be several hundred kiloparsecs in extent – this is not uncommon for such objects (e. g. , van Ojik et al., 1996), and is proposed to be due to the infall of primordial gas and galaxy “building blocks” such as Lyman Break Galaxies (Reuland et al., 2003).
L675
Previously confirmed at $z=2.163$ by Pentericci et al. (we measure $z=2.162$; see Fig. 4), and also detect N v 1240.
L778 (X16)
A new spectroscopic confirmation at $z=2.149$, showing broad Ly$\alpha$ (rest-frame deconvolved FWHM 890 km s${}^{-1}$), N v 1240, C iv 1549, He ii 1640, C iii] 1909, [C ii] 2326 and Mg ii 2798 (Fig. 4).
L891
Listed as a $z=2.147$ Ly$\alpha$ emitter by Pentericci et al., this object was observed in both of our masks, and shows continuum blueward of the supposed Ly$\alpha$ line at 3828 Å, as well as an emission line at 7176 Å. We were unable to identify a plausible combination of emission lines arising from a single object at any redshift which could give rise to this line combination. We note, however, that in the 2D spectra (Fig. 3), the “Ly$\alpha$” appears offset slightly (around 0.7″) from the continuum emission. This may be a LEG at the cluster redshift (we measure $z=2.148$), while the continuum and 7176 Å emission line are continuum and [O ii] 3727 from a foreground object at $z=0.925$ (which also shows some evidence of a continuum break in around the right place for Ca H + K).
L968 (X3)
Previously confirmed at 2.183 (with caution that Ly$\alpha$ appears self-absorbed). We obtain $z=2.162$, and see only slight self-absorption blueward of Ly$\alpha$ (Fig. 4). This AGN shows broad and narrow Ly$\alpha$, broad and narrow C iv 1549, and broad N v. The C iv emission was fit with two Gaussian components, and the broad and narrow Ly$\alpha$ and N v were fit with three Gaussian components.
X2, X11 & X17
H$\alpha$ through H$\zeta$ at $z=0$ are seen, along with molecular absorption bands. Comparison with the spectroscopic standards of Kirkpatrick, Henry, & McCarthy (1991) shows these three objects to be M-dwarf stars.
X5
Faint but significant Ly$\alpha$ emission (rest-frame deconvolved FWHM 400 km s${}^{-1}$); a new spectroscopic confirmation at $z=2.162$ (Fig. 4). This object is also listed as an ERO (ERO 226; $I-K=5.0$) by Kurk et al. (2004a). Although it is difficult to say, based on only the observed Ly$\alpha$ line, whether this object is a starburst or AGN, the soft X-ray to optical flux ratio (Pentericci et al., 2002), in addition to an inferred soft X-ray luminosity ($4\times 10^{43}$ erg s${}^{-1}$) much too high for a starburst, provide strong evidence for the presence of an AGN.
X18
This $z=0.436$ Seyfert 2 shows strong [O ii] 3727, [O iii] 4959, 5007, [Ne iii] 3869, [Ne v] 3426 and H$\alpha$ lines. It is the nucleus of a galaxy undergoing a merger as seen on the VLT image.
4 Discussion
All of the nine X-ray selected targets were spectroscopically detected
and classified. Six of these were AGN, three of which are members of
the protocluster $z\sim 2.2$ (including one reconfirmation). Two of
these were also LEG candidates, showing again the efficacy of the narrow-band imaging
selection method. Of the three lower-redshift AGN, one is at $z=1.117$ (a member of the $z=1.16$ spike discussed below).
The three remaining X-ray targets are M-dwarf stars. This is an
unusually large fraction. Among the 42 objects with spectroscopic
classifications in the Chandra Deep Field South catalog of Szokoly et al. (2004) only one X-ray
source brighter than $10^{-15}$ erg s${}^{-1}$ cm${}^{-2}$ is an M-dwarf. So the X-ray
overdensity in the PKS 1138-262 field is at least in part due to an excess of
M-dwarfs in the foreground. However, with five out of 18 X-ray sources
(including PKS 1138-262) now confirmed to be AGN this still suggests a higher
AGN fraction in the PKS 1138-262 protocluster than in local clusters
(see Pentericci et al., 2002) and in the fields of other HzRGs (Overzier et al., 2005).
Of the 24 targets selected on the basis of their Ly$\alpha$ excess, five were
at the redshift of the
protocluster (including four previously confirmed), seven were undetected or showed only faint continuum, and the remainder are most likely
foreground galaxies. Four of these show a single emission line in the range
7927 – 8235 Å, which, if interpreted as [O ii] 3727, places them at
$z=1.16\pm 0.05$. The remaining five are at $z<1$.
The apparent overdensity at $z=1.16$, which also includes L479 and
X4 in addition to the four single-emission-line objects,
makes it all the more necessary to obtain high quality spectra
of more targets in this field to determine whether they belong to the system at $z=1.2$, the
PKS 1138-262 protocluster at $z=2.2$, or are unassociated with either.
For example, the starburst galaxy L479 at $z=1.171$ lies at the
center of a 20″-long string of six EROs (Kurk et al., 2004a), suggesting that at least
some of the EROs may be physically associated with the $z=1.16$
overdensity. Apparently, at this redshift, starburst galaxies
with strong [O ii] emission are also picked up
by the Ly$\alpha$ selection process (at quite low significance) due to
bluening of their continuum by young stars.
The pure LEG-selected targets in our multislit observations did not
reveal additional PKS 1138-262 protocluster members (with the exception of
L778, which is also an X-ray emitter). This is unlike the
high success rate (up to 70%)
of spectroscopically confirmed $z\sim 2.2$ LEGs reported by Pentericci et al. (2000). This is probably because objects with
large $EW$ (estimated from narrow-band photometry) are easiest to detect spectroscopically.
However, these also tend to be
the faintest objects in B-band (by definition) and large $EW$ combined with low expected Ly$\alpha$ flux creates a signal-to-noise
problem. Indeed, we see that our four non-detections are those listed
by Kurk et al. (2004a) as having large $EW$, and comparatively low
Ly$\alpha$ flux (Table 1).
To dispel any concern that our observations were less sensitive than
those of Pentericci et al. (2000), note that the five of their confirmed $z\sim 2.2$ LEGs which we observed are reconfirmed by our observations
(with the possible exception of L891, as noted above — which does in
any case show a line which could plausibly be Ly$\alpha$). In all cases
except L968 (see discussion above), values of $1+z$ obtained are in
agreement at the $\sim 0.2\%$ level. It seems therefore that our
success rate in confirming Ly$\alpha$ emitters is lower simply due to our
targets being those which are intrinsically fainter and / or closer to
the cutoff of the selection criteria. We note that the Ly$\alpha$ selection
technique has shown great success in other fields, particularly at
higher redshift, and it is perhaps not surprising that our success rate
is somewhat lower here due to redshifted Ly$\alpha$ being towards blue
optical wavelengths. Here spectroscopic efficiency begins to drop, and
the measurement of broad-band fluxes below the Ly$\alpha$ line or Lyman limit as an additional
selection criterion is impossible due to the atmospheric cutoff.
5 Conclusions
There is still a good deal of evidence that PKS 1138-262 is located in a
forming cluster; this conclusion is bolstered by our confirmation of
two new objects at the same redshift as the radio galaxy. That one of
these is also an ERO lends credence to the hypothesis that at least
some of the ERO overdensity may be attributed to the protocluster.
However, as noted by Steidel et al. (2000), the narrow-band (Equivalent
Width) selection technique is effective but also difficult to
quantify. Sensitive, wide spectral wavelength coverage is essential,
the more so because there is a non-trivial chance that overdense fields
at high redshifts maybe confused by other overdense fields in the
foreground (van Haarlem, Frenk, & White, 1997); another example of such a superposition of unrelated overdensities is seen by Francis et al. (2004). Curiously, Francis et al. also detect an object showing two emission lines which they are unable to indentify, similar to L891 as discussed above.
The confirmation that around one third of the X-ray sources in this field are associated with the protocluster ties in well with the conclusions of Pentericci et al. (2002) that there is a 50% excess of X-ray sources in the field of PKS 1138-262 compared to the Chandra Deep Fields. The high AGN fraction in PKS 1138-262 suggests that the AGN were probably triggered at around the same time, presumably by the ongoing formation of the
protocluster. This supports models where AGN feedback is an important component of the early phases of galaxy and cluster formation.
Intriguingly, our two new spectroscopic confirmations
lie along the line of an apparent overdensity of
spectroscopically-confirmed Ly$\alpha$ emitters (Fig. 1), in
the same direction as the overdensity of X-ray emitters, radio axis of
the central galaxy, extended X-ray emission around PKS 1138-262, and the
general distribution of H$\alpha$ emitters in the cluster (Pentericci et al., 2002).
This is not entirely surprising if what we are seeing is a filament of
the Large Scale Structure associated with this forming galaxy cluster.
The data presented herein were obtained at the W. M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California and the National Aeronautics and Space Administration. The Observatory was made possible by the generous financial support of the W. M. Keck Foundation. Work was performed under the auspices of the U. S. Department of Energy, National Nuclear Security Administration by the University of California, Lawrence Livermore National Laboratory under contract No. W-7405-Eng-48.
References
Best (2000)
Best, P. N. 2000, MNRAS, 317, 720
Cappi et al. (2001)
Cappi, M., et al. 2001, ApJ, 548, 624
Carilli et al. (1998)
Carilli, C. L., Harris, D. E., Pentericci, L., Rottergering, H. J. A., Miley, G. K., & Bremer, M. N. 1998, ApJ, 494, L143
Chambers, Miley, & van Breugel (1987)
Chambers, K. C., Miley, G. K., & van Breugel, W. 1987, Nature, 329, 604
Ford et al. (2004)
Ford, H., et al. 2004, in Penetrating Bars through Masks of Cosmic Dust: The Hubble Tuning Fork Strikes a New Note, ed. D. L. Block, et al. (New York: Springer); astro-ph/0408165
Francis et al. (2004)
Francis, P. J.,
Palunas, P., Teplitz, H. I., Williger, G. M., & Woodgate, B. E. 2004,
ApJ, 614, 75
Haehnelt & Kauffmann (2000)
Haehnelt, M. G., & Kauffmann, G. 2000, MNRAS, 318, L35
Hill & Lilly (1991)
Hill, G. J. & Lilly, S. J. 1991, ApJ, 367, 1
Ivison et al. (2000)
Ivison, R. J., Dunlop, J. S., Smail, I., Dey, A., Liu, M. C., & Graham, J. R. 2000, ApJ, 542, 27
Kirkpatrick, Henry, & McCarthy (1991)
Kirkpatrick, J. D., Henry, T. J., & McCarthy, D. W. 1991, ApJS, 77, 417
Kurk et al. (2000)
Kurk, J. D., et al. 2000, A&A, 358, L1
Kurk et al. (2004a)
Kurk, J. D., Pentericci,
L., Röttgering, H. J. A., & Miley, G. K. 2004, A&A, 428, 793
Kurk et al. (2004b)
Kurk, J. D., Pentericci,
L., Overzier, R. A., Röttgering, H. J. A., & Miley, G. K. 2004,
A&A, 428, 817
Kurk (2003)
Kurk, J. D. 2003, PhD Thesis, Leiden Observatory
Lacey & Cole (1993)
Lacey, C. & Cole, S. 1993, MNRAS, 262, 627
Moore et al. (1998)
Moore, B., Lake, G., & Katz, N. 1998, ApJ, 495, 139
Nagar et al. (2003)
Nagar, N. M., Wilson, A. S., Falcke, H., Veilleux, S., & Maiolino, R. 2003, A&A, 409, 115
Oke et al. (1995)
Oke, J. B., et al. 1995, PASP, 107, 375
Ouchi et al. (2005)
Ouchi, M., et al. 2005, ApJ, 620, L1
Overzier et al. (2005)
Overzier, R. A., Harris, D. E., Carilli, C. L., Pentericci, L., Röttgering, H. J. A.,
& Miley, G. K. 2005, A&A, 433, 87
Pentericci et al. (1998)
Pentericci, L., Röttgering, H. J. A., Miley, G. K., Spinrad, H., McCarthy, P. J., van
Breugel, W. J. M., & Macchetto, F. 1998, ApJ, 504, 139
Pentericci et al. (2000)
Pentericci, L., et al. 2000, A&A, 361, L25
Pentericci et al. (2002)
Pentericci, L., Kurk, J. D., Carilli, C. L., Harris, D. E., Miley, G. K., & Röttgering, H. J. A. 2002, A&A, 396, 109
Rawlings (2003)
Rawlings, S. 2003, New Astronomy Review, 47, 397
Reuland et al. (2003)
Reuland, M., et al. 2003, ApJ, 592, 755
Simpson & Rawlings (2002)
Simpson, C. & Rawlings, S. 2002, MNRAS, 334, 511
Smail et al. (2003)
Smail, I., Scharf, C. A., Ivison, R. J., Stevens, J. A., Bower, R. G., & Dunlop, J. S. 2003, ApJ, 599, 86
Spergel et al. (2003)
Spergel, D. N., et al. 2003, ApJS, 148, 175
Springel et al. (2005)
Springel, V., Di Matteo, T., & Hernquist, L. 2005, ApJ, 620, L79
Steidel et al. (1998)
Steidel, C. C., Adelberger, K. L., Dickinson, M., Giavalisco, M., Pettini, M., & Kellogg, M. 1998, ApJ, 492, 428
Steidel et al. (2000)
Steidel, C. C., Adelberger, K. L., Shapley, A. E., Pettini, M., Dickinson, M., & Giavalisco, M. 2000, ApJ, 532, 170
Szokoly et al. (2004)
Szokoly, G. P., et al.
2004, ApJS, 155, 271
Venemans et al. (2002)
Venemans, B. P., et al. 2002, ApJ, 569, L11
van Haarlem, Frenk, & White (1997)
van Haarlem, M. P., Frenk, C. S., & White, S. D. M. 1997, MNRAS, 287, 817
van Ojik et al. (1996)
van Ojik, R., Roettgering, H. J. A., Carilli, C. L., Miley, G. K., Bremer, M. N., & Macchetto, F. 1996, A&A, 313, 25 |
On the edge-reconstruction number of a tree
K. Asciak111kevin.j.asciak@um.edu.mt and J. Lauri222josef.lauri@um.edu.mt
Department of Mathematics
University of Malta
Malta
W. Myrvold333wendym@cs.uvic.ca
Dept. of Computer Science
University of Victoria
Victoria, B.C.
Canada V8N 6K3
V. Pannone444virgilio.pannone@unifi.it
Dipartment of Mathematics and Informatics
University of Florence
Italy.
Abstract
The edge-reconstruction number $\mbox{ern}(G)$ of a graph $G$ is equal to the minimum number of edge-deleted subgraphs $G-e$ of $G$ which are sufficient to determine $G$ up to isomorphsim. Building upon the work of Molina and using results from computer searches by Rivshin and more recent ones which we carried out, we show that, apart from three known exceptions, all bicentroidal trees have edge-reconstruction number equal to 2. We also exhibit the known trees having edge-reconstruction number equal to 3 and we conjecture that the three infinite families of unicentroidal trees which we have found to have edge-reconstruction number equal to 3 are the only ones.
1 Introduction
Trees have often been the test-bed for various graph theoretic conjectures, not least being the Reconstruction Conjecture. Kelly’s proof that trees are reconstructible [7] was the first substantial reconstructibility proof. This result was later improved by various authors who showed that trees can be reconstructed using only their endvertex- or peripheral-vertex- or cutvertex-deleted subgraphs [6, 2, 10].
A vertex-deleted subgraph $G-v$ of $G$ is called a card of $G$; the collection of cards of $G$ is called the deck of $G$, denoted by ${\cal{D}}(G)$. Our main focus in this paper will be on the analogously defined edge-cards of $G$ which are the edge-deleted subgraphs $G-e$ of $G$; the collection of edge-cards of $G$ is called the edge-deck of $G$ and is denoted by ${\cal{ED}}(G)$.
In [5], Harary and Plantholt introduced the notion of reconstruction numbers. The reconstruction number $\mbox{rn}(G)$ of a graph $G$ is defined to be the least number of vertex-deleted subgraphs of $G$ which alone reconstruct $G$ uniquely (up to isomorphism). The class reconstruction number ${\cal C}\mbox{rn}(G)$ is defined as follows. Let $\cal C$ be a class of graphs closed under isomorphism. Then the class reconstruction number of a graph $G$ in $\cal C$ is the minimum number of vertex-deleted subgraphs of $G$ which, together with the information that $G$ is in $\cal C$, reconstruct $G$ uniquely.
It is clear that the reconstruction number of a graph is always at least 3 and that ${\cal C}\mbox{rn}(G)\leq\mbox{rn}(G)$. In fact, the class reconstruction number can even be 1, for example, when $\cal C$ is the class of regular graphs.
The edge-reconstruction number $\mbox{ern}(G)$ of a graph $G$ and the class edge-reconstruction number ${\cal C}\mbox{ern}(G)$ for a graph $G$ in $\cal C$ are analogously defined.
In [3], Harary and Lauri tackled the reconstruction number of a tree. Let $\cal T$ be the class of trees. In their paper, Harary and Lauri tried to show that ${\cal T}\mbox{ern}(T)\leq 2$. Although they managed to achieve this in many of the cases they considered, in some cases they had to settle for the upper bound of 3. So, what was accomplished in [3] was to show that ${\cal T}\mbox{rn}(T)\leq 3$ and to make plausible their conjecture that, in fact, ${\cal T}\mbox{rn}(T)\leq 2$ for all trees $T$. Myrvold [14] soon improved the first result by showing that
$rn(T)\leq 3$. The conjecture ${\cal T}\mbox{rn}(T)\leq 2$, however, still stood. A significant step forward was recently taken by Welhan [16] who proved that the class reconstruction number of trees is at most 2 for trees without vertices of degree 2.
The situation for the edge-reconstruction numbers of trees is less clear, somewhat surprisingly compared with what happens in the Reconstruction Problem where edge-reconstruction is easier than vertex-reconstruction. Although Harary and Lauri conjectured that ${\cal T}\mbox{rn}(T)\leq 2$ for all trees $T$, they presented in [3] a few trees with class edge-reconstruction number ${\cal T}\mbox{ern}$ equal to 3 even though their class (vertex) reconstruction number was equal to 2. In [13], Molina started to tackle the edge-reconstruction number of trees. In summary, these are Molina’s main results.
1.
Let $T$ be a unicentroidal tree with at least four edges, then $\mbox{ern}(T)\leq 3$.
2.
Let $T$ be bicentroidal with centroidal vertices $a$ and $b$, and let $G$ and $H$ be the two components of $T-ab$ with $a$ in $G$ and $b$ in $H$. Then
(a)
If one of the centroidal vertices has degree equal to two, then $\mbox{ern}(T)\leq 3$.
(b)
If both centroidal vertices have degree at least three and if $G$ or $H$ has an irreplaceable endvertex (defined below), then $\mbox{ern}(T)=2$.
(c)
If both centroidal vertices have degree at least three and if either $G$ or $H$ has no irreplaceable endvertex, then $\mbox{ern}(T)\leq 3$.
In this paper we shall improve the above results on bicentroidal trees by showing that $\mbox{ern}(T)=2$ when the degrees of the centroidal vertices are 2 and even when both $G$ and $H$ have no irreplaceable vertices, giving our main result is that all bicentroidal trees, with only three exceptions, have ern equal to 2. We shall also prove some results on unicentroidal trees and, based on these results and empirical evidence which we shall present, we give a conjecture stating which infinite classes of unicentroidal trees have ern equal to 3.
One final definition: suppose we are considering $\mbox{rn}(G)$ or $\mbox{ern}(G)$ and suppose that a graph $H\not\simeq G$ has in its deck (edge-deck) the cards (edge-cards) $G-v_{1},\ldots,G-v_{k}$ ($G-e_{1},\ldots,G-e_{k}$) we then says that $H$ is a blocker for these cards (edge-cards) or that $H$ blocks these cards (edge-cards).
2 Main techniques
We shall here present the main techniques and supporting results used in this paper. Many of these were first used or proved in [3]. While all work on the reconstruction of trees prior to [3] depended on the centre of a tree, in [3] the centroid was used instead. Since then, all investigations of reconstruction numbers of trees depended heavily on centroids. Non-pseudosimilarity and irreplaceabilty of endvertices were also very important techniques first used in the proofs in [3]. These ideas will be explained below. We shall also present a new technique and a result which will be used for the first time in this paper.
2.1 The centre and the centroid of a tree, rooted trees and branches
The diameter $\mbox{diam}(G)$ of a connected graph $G$ is the length of a longest path in $G$. The eccentricity of a vertex $v$ in $G$ is the longest distance from $v$ to any other vertex in the graph. The centre of $G$ is the set of vertices with minimum eccentricity. It is well-known that if $G$ is a tree then the centre is either one vertex or two adjacent vertices.
We now turn our attention to the centroid. Define the weight of a vertex $v$ of a tree $T$, denoted by $wt(v)$, to be the number of vertices in a largest component of $T-v$. For example all endvertices in a $n$-vertex tree have weight $n-1$. The centroid of a tree $T$ is the set of all vertices with minimum weight denoted by $wt(T)$. A centroidal vertex is a vertex in the centroid. It is well-known that the centroid of a tree consists of either one vertex or two adjacent vertices. A tree with one centroidal vertex is called unicentroidal while a tree with two centroidal vertices is called
bicentroidal. In the latter case, the edge joining the centroidal vertices is called the centroidal edge. When $T$ is bicentroidal with centroidal edge $e$, the two components of $T-e$ are also said to be centroidal components.
The following simple observation will be very useful. The second part, especially, tells us that for a graph $T$ which we know to be a tree, if it is bicentroidal, then one can determine from an edge-deleted subgraph $T-e$ of $T$ alone, whether or not $e$ is the centroidal edge of $T$ and also, if $e$ is the centroidal edge, the isomorphism type of the two centroidal components.
Observation 2.1
Let $T$ be a tree of order $n$ and let $v$ be a vertex of $T$. Then $wt(v)\leq\frac{n}{2}$ if and only if $v$ is in the centroid of $T$. Also, $T$ is bicentroidal with centroidal vertices $a$ and $b$ if and only if $T-ab$ has two components $G,H$ each of order $\frac{\left|V(T)\right|}{2}$.
Notation. In the rest of the paper, $a$ and $b$ will denote the centroidal vertices of a bicentroidal tree with centroidal components $G$ and $H$ such that $a$ is in $G$ and $b$ is in $H$.
A rooted tree is a tree which has one identified vertex. Let $P$ be a path in a tree and let $v$ be an internal vertex on $P$. The branch at $v$ relative to $P$ is the subtree, rooted at $v$, induced by all those vertices connected to $v$ by a path not containing other vertices of $P$.
A vertex of degree 1 is said to be an endvertex. A cutvertex in a tree which is adjacent to only one vertex of degree greater than 1 is said to be an end-cutvertex. An edge incident to an endvertex is called an end-edge.
2.2 Pseudosimilar vertices, irreplaceable edges and conjugate pairs of trees
Most of the works which we mentioned and which deal with reconstruction numbers of trees of some sort make heavy use of the impossibility of endvertices being pseudosimilar in a tree and of the fact that only a few very special type of trees have the property that any end-edge can be exchanged with another giving us a tree isomorphic to the one which we started with. Since we shall be using these results even in this paper we shall explain them and their general use in this section. We shall also prove another result in this vein which we shall be needing, namely a result about a pair of trees such that any one can be obtained from the other by exchanging some end-edges in a particular way
Let $u$ and $v$ be two vertices in a graph $K$ such that an automorphism of $K$ maps $u$ into $v$. Then $u$ and $v$ are said to be similar in $K$. Now suppose that $u$ and $v$ are such that $K-u$ is isomorphic to $K-v$; we call such a pair of vertices removal-similar. If $u$ and $v$ are removal-similar in $K$ but not similar, then $u$ and $v$ are said to be pseudosimilar vertices in $K$. The following results say that endvertices and end-cutvertices in a tree cannot be pseudosimilar.
Theorem 2.1
(Harary and Palmer) [4] (i) Any two removal-similar endvertices in a tree are similar.
(Kirkpatrick, Klawe and Corneil) [8] (ii) Any two removal-similar end-cutvertices in a tree are similar.
Since we shall be expanding on this and the subsequent result in this paper it is interesting to see one way in which these two results have been extended by Krasikov in [9]. Let $T$ be a tree and $a,b\in V(T)$, and let $A,B$ be two rooted trees. Then $T_{a,b}(A,B)$ denotes the tree obtained by identifying the root of $A$ with $a$ and the root of $B$ with $b$. Krasikov proved the following.
Theorem 2.2
If $A$ and $B$ are two non-isomorphic rooted trees and
$$T_{a,b}(A,B)\simeq T_{a,b}(B,A)$$
then $a$ and $b$ are similar in $T$.
Clearly, if we take $A$ to be the tree on two vertices and $B$ a single vertex, then this result gives that endvertices cannot be pseudosimilar in a tree.
Now let $e=xv$ be an end-edge of $T$ with $\deg(v)=1$. Let $y\not=x$ be another vertex of $T$ and let $T^{\prime}=T-e+e^{\prime}$, where $e^{\prime}=yv$. If $T^{\prime}$ is isomorphic to $T$, then $e$ is called a replaceable end-edge. If there is no such vertex $y$ then $e$ is called an irreplaceable end-edge. Let $S_{1}$ and $S_{2}$ be the graphs shown in Figure 1. A tree which is isomorphic either to a path $P_{k}$ on $k$ vertices or to one of $S_{1}$ or $S_{2}$ is said to be a pseudopath.
The following theorem was proved in [3] and was also profitably used in [13].
Theorem 2.3
Any tree which is not a pseudopath has an irreplaceable end-edge.
The use of non-pseudosimilarity of endvertices and irreplaceable edges are important techniques
which are used in these two broad scenarios in this paper. First of all, suppose that we have two trees $G,H$ and we know that the tree $T$ to be reconstructed is obtained by joining together with a new edge an endvertex $a$ of $G$ to another endvertex $b$ of $H$ (we do not know which vertices are $a$ and $b$). Suppose, however, that we know the isomorphism types of both $G^{\prime}=G-a$ and
$H^{\prime}=H-b$. Then, since endvertices in a tree cannot be pseudosimilar, we can pick any endvertex $x$ in $G$ such that $G-x\simeq G^{\prime}$ and similarly any endvertex $y$ in $H$ such that $H-y\simeq H^{\prime}$, and join the two vertices $x$ and $y$ giving the reconstruction of $T$ which is unique up to isomorphism.
The second scenario is basically this. Suppose that we know again that the tree $T$ to be reconstructed is obtained by joining vertex $a$ in $G$ to vertex $b$ in $H$ ($a,b$ need not be endvertices now). We are also given the tree $T^{\prime}$ which is composed of $G$ joined correctly to $H^{\prime}$, where $H^{\prime}$ is $H$ less an endvertex and we can identify the edge $ab$ in $T^{\prime}$. We therefore know from $T^{\prime}$ the components $G$ and $H^{\prime}$ and how they are connected. We just need to be able to put back the missing endvertex in $H^{\prime}$. In order to have unique reconstruction up to isomorphism, non-pseudosimilarity of the missing endvertex is not enough here. We now require that the missing vertex be irreplaceable in $H$.
In this paper we shall also need a notion which is in some way an extension of the idea of replaceable endvertices. Instead of asking that exchanging an end-edge in a tree gives us the same tree, we ask that a pair of trees are related by a particular exchange of end-edges. This is quite a natural occurrence when considering reconstruction of trees. First we need a technical definition which, however, will find its natural place in our reconstruction results later in Theorem 3.2.
Suppose $G$ and $H$ are two non-isomorphic trees. Let $a,b$ be endvertices of $G$ and $H$, respectively. Suppose also that:
1.
$G-a+e_{1}\simeq H$ for some new end-edge $e_{1}$ added to $G-a$;
2.
$G+aa^{\prime}-e_{2}\simeq H$ for some new endvertex $a^{\prime}$ added to $G$ and some end-edge $e_{2}$ of $G$;
3.
$H-b+e_{3}\simeq G$ for some new end-edge $e_{3}$ added to $H-b$;
4.
$H+bb^{\prime}-e_{4}\simeq G$ for some new endvertex $b^{\prime}$ added to $H$ and some end-edge $e_{4}$ of $H$.
Then $G$ and $H$ are said to be a conjugate pair of trees.
The theorem we shall need is the following.
Theorem 2.4
Let $G$ and $H$ be a conjugate pair of trees as in the definition. Then $G$ and $H$ must be trees as shown in Figure 2.
Proof. Let $c$ be the neighbour of $a$ in $G$ and $d$ the neighbour of $b$ in $H$. We shall consider two cases:
Case 1: At least one of the trees on the left-hand-side of equations (1)–(4) in the definition of conjugate pairs has a different centre from the original tree on the right-hand-side.
We shall suppose that the change of centre occurs in Equation 1 of the definition. The arguments for Equation 2 are similar, and those for Equations 3 and 4 follow by symmetry. Therefore we have that the centre of $G-a+e_{1}$ is not the same as in $G$. We shall consider the case when the centre of $G$ has one vertex. The bicentral case is similar.
The condition implies that $a$ is on the unique longest path of $G$. Call this path $P$. Let $x$ be the vertex at the other end of $P$. It also follows that $\deg(c)=2$. If $z$ is the other neighbour of $c$ on $P$, then the branch at $z$ relative to $P$ must be at most an edge; the branch at the next vertex $z^{\prime}$ cannot have a vertex distant more than 2 from $z^{\prime}$, and so on. Let the branches relative to $P$ at the internal vertices of $P$ be $B_{1},B_{2},\ldots,B_{k}$ in that order starting from the branch at $z$ as shown in Figure 3. Since, by definition, $G$ and $H$ are not isomorphic, the branches cannot all be trivial (consisting of only the root vertex).
Also, from Equation 1 it follows that $\mbox{diam}(H)\leq\mbox{diam}(G)$ and, from Equation 2, that $\mbox{diam}(G)\leq\mbox{diam}(H)$. Therefore $G$ and $H$ have the same diameter. Therefore the end-edge $e_{1}$ in Equation 1 of the definition is $xx^{\prime}$ for some new vertex $x^{\prime}$.
Now, consider $H$ given as $G-a+xx^{\prime}$ as depicted in Figure 4. Which would be the vertex $b$ in $H$ which satisfies Equation 4? Recall that $H+bb^{\prime}-e_{4}$, being isomorphic to $G$, would have to have a unique path of maximum length and the branches of the internal vertices of this path would have to be $B_{1},B_{2},\ldots,B_{k}$ in that order.
Therefore $b$ cannot be $x^{\prime}$ nor any endvertex in any of $B_{1},B_{2},\ldots,B_{k}$. Therefore $b$ must be the vertex $c$ and $d$ must be the vertex $z$ in Figure 4.
But, in order to satisfy Equation 3, the branch $B_{1}$ must be a single edge and the end-edge $e_{3}$ must be as shown in Figure 5.
But then, comparing $G$ as in Figure 5 with Figure 3 shows that all the branches are single edges and $G$ is as in Figure 2. This finishes Case 1.
Case 2: Every tree on the left-hand-side of equations (1)–(4) in the definition of conjugate pairs has the same centre as that in the original tree on the right-hand-side.
We shall prove that this leads to a contradiction, therefore only Case 1 can hold.
We shall only consider the unicentral case. The bicentral case can be treated similarly. First we need to define exactly what we mean by a central branch of a central tree $T$ with central vertex $v$. Let $A$ be a component of $T-v$ and let $u$ be the neighbour of $v$ in $A$. Let $B=A+uv$. Then $B$ will be called a central branch of $T$. Clearly, the number of central branches of $T$ is equal to $\deg(v)$.
Consider first Equation 1: $G-a+e_{1}\simeq H$. Let $B_{1}$ be the central branch of $G$ containing the edge $ac$. We now have two sub-cases.
Case 2.1: The edge $e_{1}$ is incident to a vertex in $B_{1}$.
Therefore $G$ and $H$ have exactly the same collection of branches except that $H$ has the branch $B_{1}^{\prime}\simeq B_{1}-a+e_{1}$ instead of $B_{1}$. So, the only way of obtaning $G$ back from $H$ in the way stipulated by Equations 3 and 4 is by changing $B_{1}^{\prime}$ to $B_{1}$ and, similarly, the only way of going from $G$ to $H$, according to Equations 1 and 2, is by changing $B_{1}$ to $B_{1}^{\prime}$. Therefore Equations 1 to 4 hold for the trees $B_{1}$ and $B^{\prime}_{1}$, that is, they form a conjugate pair of trees. Applying induction on the number of vertices gives us that $B_{1}$ and $B^{\prime}_{1}$ are as specified by the theorem, that is, as in Figure 2. Therefore $G$ is the tree $B_{1}$ with extra branches joined to $v$ (which is an endvertex in $B_{1}$). But then, $G$ cannot satisfy Equations 1 to 4, that is, it cannot be a member of a conjugate pair of trees.
Case 2.2: The edge $e_{1}$ is not incident to a vertex in $B_{1}$.
Let $B_{2}$ be the central branch of $G$ containing $e_{1}$. Therefore the branches of $G$ and $H$ are identical except that $H$ has $B_{1}^{\prime}=B_{1}-c$ instead of $B_{1}$ and $B_{2}^{\prime}=B_{2}+e_{1}$ instead of $B_{2}$.
Now, the endvertex $a$ of $G$ which is in $B_{1}$ is also involved in Equation 2: $G+aa^{\prime}-e_{2}\simeq H$. Let us consider where the edge $e_{1}$ can come from so that $H$ is isomorphic to both $G-a+e_{1}$ and $G+aa^{\prime}-e_{2}$. We point out that we need to obtain the same collection of branches for $H$ (with $B_{1}^{\prime}$ and $B_{2}^{\prime}$ instead of $B_{1}$ and $B_{2}$, respectively) and that we cannot do this by moving the centre. That is, we can only make modifications to the existing central branches.
The only way this can happen is if $e_{2}$ comes from some third central branch $B_{3}$. Now consider the orders of $B_{1},B_{2},B_{3}$. Let these orders be $r,s,t$, respectively. Then, a moment’s consideration shows that we must have that $r=p+1,s=p$ and $t=p+2$, for some $p$.
Therefore $G$ and $H$ are as shown in Figure 6.
But also, $H$ is isomorphic to the tree shown in Figure 7. So we get, for example, by considering orders, that $B_{3}\simeq B_{1}+aa^{\prime}$ and $B_{2}\simeq B_{1}-a$. Switching over from $G$ to $H$ and from $H$ to $G$ using Equations 1 to 4 involves exchanges endvertices between these three branches (or three branches in $G$ or $H$ isomorphic to them).
So, when we are considering $G$ in Equations 1 and 2, the vertex equivalent to $a$ would be in that branch which has order $p+1$, the new edge $e_{1}$ would be attached to the branch of order $p$, and $e_{2}$ would be removed from the branch of order $p+2$. Similarly, if we are considering $H$, for $b,e_{3}$ and $e_{4}$ in Equations 3 and 4. But we are always permuting between the same (up to isomorphism) three branches which become isomorphic to $B_{1},B_{2},B_{3}$ in $G$ and $B_{1}^{\prime},B_{2}^{\prime},B_{3}$ in $H$. But this would force the two trees in Figure 8 to be isomorphic, therefore $G$ and $H$ would be isomorphic, a contradiction which completes our proof.
Note that it is the fact that $G$ and $H$ are not isomorphic which forces conjugate pairs to be as described in the theorem and which gives us our final contradiction. If $G$ and $H$ are allowed to be isomorphic then, for example, two trees both isomorphic to the one shown in Figure 9 do satisfy Equations 1 to 4. Note, in this example, the three central branches as described in Case 2.2 of the above proof.
2.3 Recognising trees and Molina’s Lemma
There is a simple but very useful result proved by Molina in [13] which often allows us to identify a graph as a tree from two given edge-cards. We reproduce its short proof for completeness’ sake.
Lemma 2.1
Let $G$ be a graph with edges $e_{1}$ and $e_{2}$. Suppose that the edge-card $G-e_{1}$ has two components which are trees of orders $p_{1}$ and $p_{2}$ while the edge-card $G-e_{2}$ has another two components which are trees of orders $q_{1}$ and $q_{2}$. If $\{p_{1},p_{2}\}\not=\{q_{1},q_{2}\}$, then $G$ is a tree.
Proof. Suppose $G$ is not a tree.
Without loss of generality we assume that $e_{1}$ joins two vertices in the same component of $G-e_{1}$; call this component $H$, that is, $H+e_{1}$ contains a cycle. Therefore to obtain the second
edge-card with two trees as components an edge must be removed from $H+e_{1}$. But this contradicts that $\{p_{1},p_{2}\}\not=\{q_{1},q_{2}\}$.
2.4 Some special types of tree
A special type of tree denoted by $S_{p,q,r}$ is a unicentroidal tree similar to a star (that is, the tree on $n$ vertices, $n-1$ of which are endvertices) which consists of three paths on $p$, $q$ and $r$ edges, respectively, emerging from the centroidal vertex. Some examples are shown in Figure 10. Note that the pseudopaths $S_{1}$ and $S_{2}$ defined above are $S_{1,1,2}$ and $S_{1,2,3}$, respectively.
A caterpillar is a tree such that the removal of all of its endvertices results in a path.
This path is called the spine of the caterpillar. A caterpillar whose spine is the
path $v_{1}v_{2}\ldots v_{s}$ and such that the vertex $v_{i}$ is adjacent to $a_{i}$ endvertices will be denoted by
$C(a_{1},...,a_{s})$. Two examples are shown in Figure 11. Finally, a path on $n$ vertices is denoted by $P_{n}$.
3 Bicentroidal trees
3.1 The centroidal component $G$ is not a pseudopath and $\deg(b)\geq 3$
Theorem 3.1
Let $T$ be a bicentroidal tree with bicentroidal edge $ab$, bicentroidal components $G,H$, $a\in V(G)$, $b\in V(H)$. Suppose $\deg(b)\geq 3$ and $G$ is not a pseudopath.
Then $\mbox{ern}(T)=2$.
Proof. Since $T$ is bicentroidal and $ab$ is the centroidal edge then the two components $G$ and $H$ of the card $T-ab$ have the same number of vertices, namely $\frac{\left|V(T)\right|}{2}$.
Let $f$ be an irreplaceable end-edge of $G$ (such an $f$ exists since $G$ is not a pseudopath). We claim that $T$ is reconstructible from $T-ab$ and $T-f$.
By Lemma 2.1 we can recognise from $T-ab$ and $T-f$ that the graph to be reconstructed is a tree.
By Observation 2.1 one can therefore recognise from the edge-card $T-ab$ that the edge $ab$ is the centroidal edge and also that $G$ and $H$ are the centroidal components of $T$.
Now, we would like to show
that the centroidal edge is recognisable in the edge-card $T-f$. There is surely an edge $e$ such that $(T-f)-e$ has non-trivial components $G-f$ and $H$, but we can definitely say that $e$ is the edge $ab$ only if:
(i)
there is only one edge $e$ such that the non-trivial components of $(T-f)-e$ are
isomorphic to $H$ and some $T-f$;
and
(ii)
there is no edge $e^{\prime}$ such that the non-trivial component of $(T-f)-e^{\prime}$ is isomorphic to
$G$ and some $H-f$.
If both (i) and (ii) hold then we can distinguish the centroidal edge in $T-f$ and we can reconstruct uniquely by putting $f$ back into $G-f$, since $f$ is an irreplaceable end-edge (note that this proof also works if the end-edge $f$ happens to be adjacent to the centroidal edge).
But cases (i) and (ii) can fail to occur only if the degree of the centroidal vertex $b$ is two. Since $\deg(b)>2$ it follows that $T$ is reconstructible from $T-ab$ and $T-f$.
We shall come back to what happens when $\deg(b)=2$ but $G$ is still not a pseudopath in Lemma 5.2 after having obtained some more results and discussed some special cases.
3.2 Both $\deg(a)$ and $\deg(b)$ equal 2 and none of $G$ or $H$ is a pseudopath
Theorem 3.2
Let $T$ be a bicentroidal tree with bicentroidal edge $ab$. Let $\deg(a)=\deg(b)=2$ and suppose that none of the two centroidal components $G$, $H$ is a pseudopath. Then $\mbox{ern}(T)=2$.
Proof. Recall that $b\in V(H)$; let $d\in V(H)$ be the other neighbour of $b$. We shall first try to show that $T$ is reconstructible from $T-ab$ and $T-bd$ and we shall see where this can go wrong.
As before, from the two given edge-cards we can recognise that $T$ is a tree and that $G,H$ are its centroidal components. Consider $T-bd$. If we can definitely tell that the larger component of $T-bd$ is $G$ plus some edge then we would only need to decide which is the extra end-edge in the larger component. But, since endvertices cannot be pseudosimilar, we can choose any endvertex whose deletion gives $G$. We therefore know, up to isomorphism, which of the vertices of $G$ is incident to the centroidal edge. Now we would need to do the same with $H$.
Let $H^{\prime}$ be the smaller component of $T-bd$. Recall that we know the component $H$. We look for any endvertex $d^{\prime}$ such that $H-d^{\prime}\simeq H^{\prime}$. Again, by non-pseudosimilarity of endvertices, any such choice is equivalent to $d$ up to isomorphism. So we also know the vertex of $H$ which is incident to the centroidal edge, hence $T$ can be uniquely reconstructed.
This proof fails if we cannot tell whether the larger component is $G$ plus an end-edge or $H$ plus an end-edge. This ambiguity can only happen if $G\not\simeq H$ and $G+ab-\alpha\simeq H$ for some end-edge $\alpha$ of $G$ and $H-b+\beta\simeq G$ for some new end-edge $\beta$.
Therefore let us assume that this is the case and let us proceed to reconstruct, this time from $T-ab$ and $T-ac$, where $c$ is the other neighbour of $a$ in $G$.
Reconstruction will proceed as above unless we cannot tell whether the larger component of $T-ac$ is $G$ plus an end-edge or $H$ plus an end-edge. But this ambiguity can only happen if $H+bd-\gamma\simeq G$ for some end-edge $\gamma$ of $H$ and $G-a+\delta\simeq H$ for some new end-edge $\delta$.
But this means that $G$ and $H$ are conjugate pairs and, by Theorem 2.4, $T$ is therefore as shown in Figure 12. But then $T$ is reconstructible from $T-ab$ and $T-e$, where $e$ is as shown in Figure 12. Therefore $ern(T)=2$.
4 Edge-reconstruction number 3: three infinite families
Molina, in [13] had stated that $\mbox{ern}(P_{n})=3$ if $T$ is a path with four or more edges.
We shall show that his statement is correct provided that $n$, the number of vertices, is odd, that is, $P_{n}$ is unicentroidal.
In the following theorem we shall show that $\mbox{ern}(P_{n})=2$ when $n$ is even. We shall also show that $\mbox{ern}(P_{n})=3$ when $n$ is odd. To do this second part we need to show that, for each pair of cards in the edge-deck ${\cal{ED}}(P_{n})$, there exists a graph $H\not\simeq P_{n}$ which has the same pair of edge-cards in its edge-deck, that is, $H$ is a blocker for that particular pair of edge-cards.
Theorem 4.1
If $n$ is even then $\mbox{ern}(P_{n})=2$ while if $n$ is odd then $\mbox{ern}(P_{n})=3$
Proof.
Consider the graph $P_{n}$, $n$ even. Let $e_{1}$ be the central edge of $P_{n}$ and $e_{2}$ any of the two edges adjacent to $e_{1}$. We claim that the two edge-cards $C_{1}=P_{n}-e_{1}=P_{\frac{n}{2}}\cup P_{\frac{n}{2}}$ and $C_{2}=P_{n}-e_{2}=P_{\frac{n}{2}+1}\cup P_{\frac{n}{2}-1}$ reconstruct $P_{n}$.
By Molina’s Lemma the graph to be reconstructed must be a tree.
Consider the missing edge of $P_{n}-e_{1}$. This edge can be made incident to (i) two endvertices of $P_{n}-e_{1}$; or (ii) two vertices of degree two in $P_{n}-e_{1}$; or (iii) one endvertex and one vertex of degree two. Case (i) gives $P_{n}$, and Case (ii) is impossible because no other edge-card of the resulting tree can be equal to the union of two paths. Therefore we need only consider Case (iii).
Let $w$ be the vertex of degree three incident to $e_{1}$ after this edge is put back into $P_{n}-e_{1}$. Then the second edge-card $C_{2}$ must be obtained by removing one of the other two edges incident to $w$. But this will always give a component $P_{k}$ with $k>\frac{n}{2}+1$, which is a contradiction. This proves our claim.
We now consider the odd path $P_{n}$ for $n=2s+1$ . When two edge-cards are obtained by deleting the two edges incident to the central vertex, then a blocker would consist of the cycle $C_{s}$ union the path $P_{s+1}$. The only exception is $P_{5}$ whose blocker in this case is the union of $C_{3}$ and $P_{2}$. We therefore consider any other pair of deleted edges. Let the edges of $P_{n}$ be ordered as
$$e_{1},e_{2},\ldots,e_{n-1}.$$
Suppose we are given the two cards $P_{n}-e_{i}$ and $P_{n}-e_{j}$, where $i\leq j$. (We can assume, by symmetry, that $j\leq s$. Also, we may assume that we do not have $i=j=s$, corresponding to $i=(n-1)/2$ and $j=(n+1)/2$ since we have already observed that the blocker then is $C_{s}\cup P_{s+1}$.) The blocker will then consist of $S_{p,q,r}$ where $p=i$, $q=j$ and $r=2s-i-j$.
Therefore $\mbox{ern}(P_{n})>2$. But Molina has shown that for any tree $T$ on at least four edges $\mbox{ern}(T)\leq 3$, therefore $\mbox{ern}(P_{n})=3$ when $n$ is odd.
We now show that a class of caterpillars also has $\mbox{ern}=3$.
Theorem 4.2
The caterpillars $C(2,0,\ldots,0,2)$ of even diameter greater than 3 have edge-reconstruction number equal to 3.
Proof.
Let $C$ = $C(2,0,\ldots,0,2)$ have even diameter $d>3$.
Let $(v_{0},\dots,v_{d})$ be a longest path of $C$.
By the the first result of Molina, $\mbox{ern}(C)\leq 3$, so we only have to prove that $\mbox{ern}(C)>2$. Thus,
we have to prove that for every pair of edge-cards $A$ and $B$ of
$C$ ($A$ and $B$ might be isomorphic), there is a blocker, that is, a graph $X$, non-isomorphic to $C$, having
two edge-cards isomorphic to edge-cards $A$ and $B$, respectively.
Let $F_{i}$ be the forest obtained by deleting edge $v_{i-1}v_{i}$,
$i=1,...,d$. Note that, because of symmetry, we need only consider
$F_{1},\ldots,F_{d/2}$. For $d>5$ we argue as follows:
•
If the pair $F_{1}$, $F_{1}$ is chosen, we construct the graph $X$
by adding to $F_{1}$ edge $v_{0}v_{d-1}$. From $X$, we obtain $F_{1}$,
by deleting, of course, $v_{0}v_{d-1}$, and also by deleting $v_{d-1}v_{d}$.
Note that $X$ is a tree.
•
If the pair $F_{1}$, $F_{i}$ is chosen, $2\leq i\leq d/2$, we
construct graph $X$ by adding to $F_{1}$ the edge $v_{0}v_{d-i-1}$.
From $X$, we obtain $F_{i}$
by deleting $v_{d-i}v_{d-i+1}$. Also in these cases, $X$ is a tree.
•
If the pair $F_{j}$, $F_{i}$ is chosen, $j=2,...,(d/2)-1$,
$j\leq i\leq d/2$, we construct $X$ by adding to $F_{j}$
the edge $v_{j-1}v_{d-i}$. From $X$ we obtain $F_{i}$ by deleting
$v_{d-i}v_{d-i+1}$. Again,
$X$ is a tree.
•
If the pair $F_{d/2}$, $F_{d/2}$ is chosen, construct $X$
by adding to $F_{d/2}$ the edge $v_{(d/2)+1}v_{d}$. From $X$ we obtain
$F_{d/2}$ by deleting the edge $v_{d-1}v_{d}$. In this last case, $X$
is not a tree, and it can be seen that there is no tree, non-isomorphic
to $C$, having $F_{d/2}$ as two of its edge-cards.
For $d=4$, we have the caterpillar $C(2,0,2)$ which we have already noted that it
has $\mbox{ern}=3$. For completeness’ sake we give the same analysis as for $d>5$ above.
•
If the pair $F_{1}$, $F_{1}$ is chosen, we construct $X$
by adding to $F_{1}$ the edge $v_{0}v_{3}$. From $X$, we obtain $F_{1}$,
by deleting, of course, $v_{0}v_{3}$, and also by deleting $v_{3}v_{4}$.
In this case $X$ is a tree.
•
If the pair $F_{1}$, $F_{2}$ is chosen, we construct $X$ by adding
to $F_{1}$ again the edge $v_{0}v_{3}$. From $X$, we obtain
$F_{2}$ by deleting $v_{2}v_{3}$.
The graph $X$ is a tree in this case too.
•
If the pair $F_{2}$, $F_{2}$ is chosen, we construct $X$ by adding
to $F_{2}$ the edge $v_{0}x$, where $x$ is the other edge of degree
1 in the same connected component as $v_{0}$. From $X$, we obtain
$F_{2}$ both by deleting $v_{0}x$ , and by deleting $v_{1}x$. In
this case $X$ is a not a forest.
[Comment. The caterpillars $C(2,0,...,0,2)$ of odd diameter
$d$ all have $\mbox{ern}=2$. Indeed, it can be directly verified that (with the same
notation as before) the pair $F_{(d-1)/2}$, $F_{(d+1)/2}$ is a pair
of edge-cards which are not in the edge-deck of any other graph not isomorphic to $C(2,0,...,0,2)$. This observation and also Rivshin’s computer search show that $C(2,0,0,0,0,2)$, which is not covered by our previous results, does indeed have $\mbox{ern}=2$.]
Finally, we note that the infinite family of trees $T_{k}$ ($k\geq 2$) shown in Figure 14 also has $\mbox{ern}=3$. (Note that, when $k=2$, $T_{k}$ is the caterpillar $C(2,0,2)$ and, when $k=3$, $T_{k}$ is the graph $G_{1}$ shown in Figure 13(a).) Since there are only two types of edges up to isomorphism in $T_{k}$, it is easy to verify that $\mbox{ern}(T_{k})=3$. For example, if $e_{1}$ and $e_{2}$ are two edges of $T_{k}$ incident to the central vertex then $T_{k}-e_{1}$ and $T_{k}-e_{2}$ are isomorphic. The blocker having two copies of these graphs in its edge-deck is $T_{k-1}\cup R$, where $R$ is a triangle. Therefore these two subgraphs do not reconstruct $T_{k}$.
5 Empirical Evidence
Empirical evidence, which was provided to us by David Rivshin [15], showed that out of more than a billion graphs on at most eleven vertices and at least four edges, only seventeen trees have edge-reconstruction number equal to 3. Four of these trees are paths of odd order which we have already considered in the previous section. Other trees are the graphs $S_{2,2,2}$, $S_{3,3,3}$ which were already noticed by Harary and Lauri [3]. Nine other trees are the caterpillars $C(2,2)$, $C(2,0,2)$, $C(1,0,1,0,1)$, $C(2,1,2)$, $C(2,0{{}^{3}},2)$, $C(2,3,2)$, $C(2,1,1,2)$, $C(1,0,1,0,1,0,1)$ and $C(2,0{{}^{5}},2)$, while the remaining two trees are $G_{1}$ and $G_{2}$ shown in Figure 13.
One can notice that only three out of the seventeen trees are bicentroidal namely the two caterpillars $C(2,2)$ and $C(2,1,1,2)$, and the graph named $G_{2}$ in Figure 13. These trees do not contradict Theorems 3.1 and 3.2 since, in all three cases, the centroidal components are both pseudopaths. These small examples show that the condition that not both centroidal components are a pseudopath is required for ern to be equal to 2. Therefore the only bicentroidal trees $T$ for which we have not determined their ern because they are not covered by Theorem 3.1 or Theorem 3.2 are: (i) those with both centroidal components equal to pseudopaths; or (ii) those with one centroidal component being a pseudopath and the centroidal vertex in the other component having degree 2. In the next section, we shall return to the arbitrarily large instances of these two cases, that is, when the pseudopaths involved are paths. But for the smaller cases we now present our computer search which not only covers these cases but also gives empirical evidence for our later conjecture on unicentroidal trees.
Rivshin’s computer analysis considered all graphs and went up to order 11. Here, by considering only trees we extend the analysis up to order 23 (24 may be done soon, 25 and 26 are probably feasible). Because we think that it has independent interest, we shall briefly describe how this search was carried out.
5.1 The computer search
The program geng distributed with Brendan McKay’s program
nauty [12] was used to generate the trees on up to 20 vertices
and nauty was used for isomorphism testing.
For trees on 21-24 vertices, Li and Ruskey’s program [11] was used
because it generates the trees much faster.
The approach applied to determine the edge reconstruction
number of each tree $T$ works as follows:
For each way to select two different edges $e_{1}$ and $e_{2}$ of $T$,
first create the set $S_{1}$ of all graphs having a card $T-e_{1}$
by adding one edge back to $T-e_{1}$ in all possible ways.
The number of ways to add back an edge is $n(n-1)/2-(n-2)$.
Similarly, determine the set $S_{2}$ of graphs having a card isomorphic
to $T-e_{2}$.
These graphs are put into their canonical forms using nauty
(two isomorphic graphs have the same canonical form).
Next find the intersection $S$ of $S_{1}$ and $S_{2}$ which is equal
to the set of all graphs having both cards. If the two cards
are not isomorphic to each other and $S$
only contains one graph then return the message that
$\mbox{ern}(T)$ is equal to two. If the two cards are isomorphic to
each other, then remove from $S$ any graphs having only
one card isomorphic to $T-e_{1}$.
If $|S|$ is equal to one after removing these graphs then
return the message that $\mbox{ern}(T)$ is equal to two.
If all pairs of edges are tested without determining
that $ern(T)$ is equal to two, return the message
that $\mbox{ern}(T)$ is equal to three.
The results of this computer search match the results that came from David Rivshin’s data
for up to 11 vertices. This search also enabled us to discover the infinite family of trees $T_{k}$ with $\mbox{ern}=3$ which we described above. We also found the graph $G_{15}$ on fifteen vertices (shown in Figure 15) which does not fall within any known infinite class but which also has $\mbox{ern}=3$.
5.2 Bicentroidal trees: the remaining cases
Let us now take stock of the situation for bicentroidal trees in the light of the results we have presented. We can summarise the situation as follows. If both centroidal components are not pseudopaths, then $\mbox{ern}(T)=2$. If one component is $S_{1}$ then $\mbox{ern}(T)=2$ except when the other component is also $S_{1}$ and $T$ is the caterpillar $C(2,1,1,2)$, in which case $\mbox{ern}(T)=3$. If both components are $S_{2}$ then $\mbox{ern}(T)=2$. The case when only one centroidal component is $S_{2}$ is covered by our computer search which confirms that, in this case too, all these trees have $\mbox{ern}=2$.
Now, if one of the two centroidal components is $P_{k}$, for $k\leq 5$, then $\mbox{ern}(T)=2$ except when both components are $P_{3}$ and therefore $T$ is the caterpillar $C(2,2)$, and when the two components are $P_{5}$ and therefore $T$ is the graph $G_{2}$ of Figure 13. We shall therefore consider next the case when both components are $P_{k}$ for $k>5$.
First a bit of notation: Let $T$ be a bicentroidal tree with centroidal edge $ab$ and such that the two components of $T-ab$ are both isomorphic to $P_{k}$, the path on $k$ vertices. Let the two paths starting from $a$, but not counting $a$, have $p$ and $q$ vertices, and similarly for $b$, let the lengths be $r$ and $s$. Then we say that $T$ is of type $T(p,q;r,s)$. We shall only give a sketch of the proof of the following lemma.
Lemma 5.1
Let $T$ be a bicentroidal tree with both centroidal components equal to $P_{k},k>5$. Then $\mbox{ern}(T)=2$.
Proof. As usual, let the centroidal edge be $ab$. It is clear that $T-ab$ cannot be one of two edge-cards giving reconstruction of $T$. Therefore let us first consider the case when we delete a non-centroidal edge $ax$ incident to $a$ and an edge $uv$ where $u$ is the endvertex on the same path from $a$ passing through $x$. Again, Lemma 2.1 gives that $T$ is a tree. The isolated vertex $u$ must be joined to the rest of $T-uv$ in such a way that the resulting tree has an edge-card isomorphic to $T-ax$. Obviously, one way this can happen is if $u$ is joined to $v$, and this gives $T$. But there are two “wrong” ways in which $u$ can be joined to $T-uv$ such that the edge-card isomorphic to $T-ax$ can be obtained. Firstly, (i) $u$ can be joined to an endvertex $w$ of $T-uv$ different from $v$; or (ii) $u$ is joined to vertex $x$ of $T-uv$ .
In case (i) let, for example, $w$ be an endvertex of the other path $P_{k}$ giving $T^{\prime}$. Then $T^{\prime}-ay$, where $y$ is the remaining neighbour of $a$, will be the edge-card isomorphic to $T-ax$; but this can happen only if $T$ is of the form $T(p,p;p,p)$. In case (ii), $T^{\prime}-ax$ will be the edge-card isomorphic to $T-ax$; and this can happen only if $T$ is of the form $T(p,p;r,s)$, since $T$ is not the caterpillar $C(2,2)$ nor the graph $G_{2}$ of Figure 13.
So now we need to consider separately the case when $T$ is of the form $T(p,q;r,s)$ with $q=p$. Let $a$ be the centroidal vertex joining the two paths $P_{k}$, as above let $x$ be a neighbour of $a$ different from the other centroidal vertex, and let $x^{\prime}$ be the other neighbour of $x$; $x^{\prime}$ exists since $T$ is not the graph $G_{2}$ in Figure 13.
We shall, in this case, use the edge cards $T-ax$ and $T-xx^{\prime}$. Checking all the possibilities one finds that, again since $T$ is not the graph $G_{2}$, the only way to join the two components of $T-xx^{\prime}$ in such a way that the resulting tree has an edge-card isomorphic to $T-ax$ is by joining the vertices $x$ and $x^{\prime}$ in $T-xx^{\prime}$.
[Comment. We think that $T$ can be reconstructed in all cases from the edge-cards $T-ax$ and $T-xx^{\prime}$, as in the last paragraph of the above proof. However, reducing the problem first to the case when $q=p$ reduces the number of ways one can join the two components of $T-xx^{\prime}$. This makes checking the proof shorter and easier.]
Therefore the only remaining case of an infinite class of bicentroidal tree whose ern is not known is when one of the centroidal components is the path $P_{k}$ and the other component is not a path or the graphs $S_{1}$, $S_{2}$, that is, not a pseudopath. We can now easily deal with this case in our final result which therefore neatly complements our first recosntruction result, Theorem 3.1.
Lemma 5.2
Let $T$ be a bicentroidal tree one of whose centroidal components is the path $P_{k}$ while the other component is not a pseudopath. Then $\mbox{ern}(T)=2$.
Proof. As usual, let the two centroidal components of $T$ be $G$ and $H$ containing, respectively, the centroidal vertices $a$ and $b$. Suppose $H$ is the path $P_{k}$. Therefore, since $G$ is not a pseudopath, we may assume that $\deg(b)=2$, otherwise we know that $\mbox{ern}(T)=2$, by Theorem 3.1. Let $d$ be the other neighbour of $b$, therefore $d\in V(H)$. We shall first try to reconstruct $T$ from $T-ab$ and $T-bd$.
As usual, we can determine from these two edge-cards that $T$ must be a tree, by Lemma 2.1, and we therefore know the two centroidal components of $T$.
We now consider $T-bd$. The two usual situations can arise: (1) the large component of $T-bd$ is isomorphic to $G$ plus an edge and the smaller component is isomorphic to $P_{k}$ less an edge; or (2) vice-versa, with the roles of $G$ and $P_{k}$ reversed. Suppose the first but not the second case holds. Therefore we need to find, in the large component $K$ of $T-bd$, an end-edge $e$ such that $K-e$ is isomorphic to $G$. Reconstruction would then proceed by joining the endvertex incident to $e$ with an endvertex of the other component of $T-bd$. The edge $ab$ is surely one such edge $e$. But even if there is another edge $vw$ with $\deg(v)=1$ such that $K-vw\simeq G$, then $w$ and $b$ would be similar in $K$. Therefore, joining the vertex $w$ or the vertex $b$ to an endvertex of the other component would give isomorphic reconstructions. This shows that $T$ is reconstructible from $T-ab$ and $T-bd$.
We can now suppose that case (2) holds. This means that $G+ab-\alpha\simeq P_{k}$ for some end-edge $\alpha$ of $G$.
So, since $G$ is not $P_{k}$, it must be the path $P_{2k-1}$ with an extra end-edge incident to one of its interior vertices. Therefore $T$ is the path $P_{2k-1}$ plus an end-edge incident to some interior vertex.
But in this case it is easy to shown that $\mbox{ern}(T)=2$. Let the vertices of the path $P_{2k-1}$
be, in order, $v_{1},v_{2},\ldots,v_{2k-1}$. Let the extra end-edge be $v_{i}x$ where $i$ is not equal to 1 or $2k-1$. Since $T$ is bicentroidal, $v_{i}$ is not the central vertex $v_{k}$ of $P_{2k-1}$. Also, we may assume, without loss of generality, that $i<k$. But then it is easily checked that $T$ is reconstructible from the edge-cards $T-v_{i}x$ and $T-v_{2i-1}v_{2i}$.
This final result and the previous comments gives the main result of this paper.
Theorem 5.1
Every bicentroidal tree except $C(2,2)$, $C(2,1,1,2)$ and the graph $G_{2}$ shown in Figure 13(b) has edge-reconstruction number equal to 2.
6 Final comments
We have managed to fill in the gaps in our knowledge of the edge-reconstruction number of bicentroidal trees. The computer search described above also leads us to make this conjecture for unicentroidal trees.
Conjecture 6.1
The only infinite classes of trees which have $\mbox{ern}=3$ are the paths on an odd number of vertices, the caterpillars $C(2,0,\ldots,0,2)$ of even diameter, and the family of trees $T_{k}$ depicted in Figure 14.
Proving this conjecture might not be easy. The difficulty of determining which unicentroidal trees have ern equal to 2 or 3 when the (vertex) reconstruction number of trees is known is again evidence for the phenomenon, commented upon in [1], when determining the edge-reconstruction number of a class of graphs is sometimes more difficult than determining the (vertex) reconstruction number.
Acknowledgement
We are again grateful to David Rivshin whose data and computer programs helped us when we could not see the trees for the wood when considering the exceptional cases amongst smaller trees.
References
[1]
K. Asciak, M.A. Francalanza, J. Lauri, and W. Myrvold.
A survey of some open questions in reconstruction numbers.
Ars Combin., 97:443–456, 2010.
[2]
J.A. Bondy.
On Kelly’s congrunce theorem for trees.
Proc. Cambridge Philos. Soc., 65:387–397, 1969.
[3]
F. Harary and J. Lauri.
On the class-reconstruction number of trees.
Quart. J. Math. Oxford (2), 39:47–60, 1988.
[4]
F. Harary and E. M. Palmer.
On similar points of a graph.
J. Math. Mech., 15:623–630, 1966.
[5]
F. Harary and M. Plantholt.
The graph reconstruction number.
J. Graph Theory, 9(4):451–454, 1985.
[6]
Frank Harary and Ed Palmer.
The reconstruction of a tree from its maximal subtrees.
Canad. J. Math., 18:803–810, 1966.
[7]
P. J. Kelly.
A congruence theorem for trees.
Pacific J. Math., 7:961–968, 1957.
[8]
D. G. Kirkpatrick, M. M. Klawe, and D. G. Corneil.
On pseudosimilarity in trees.
J. Combin. Theory Ser. B, 34(3):323–339, 1983.
[9]
I. Krasikov.
Interchanging branches and similarity in a tree.
Graphs Combin., 7(2):165–175, 1991.
[10]
J. Lauri.
Proof of Harary’s conjecture on the reconstruction of trees.
Discrete Math., 43(1):79–90, 1983.
[11]
G. Li and F. Ruskey.
The advantages of forward thinking in generating rooted and free
trees.
In 10th annual ACM-SIAM Symposium on Discrete
Alogorithms (SODA), 1999.
[12]
B. D. McKay.
nauty user’s guide (version 1.5).
Technical report, TR-CS-90-02. Australian National University,
Computer Science Department, http://cs.anu.edu.au/people/bdn/nauty/,
1990.
[13]
R. Molina.
The edge reconstruction number of a tree.
Vishwa International Journal of Graph Theory, 2(2):117–130,
1993.
[14]
W. J. Myrvold.
The ally-reconstruction number of a tree with five or more vertices
is three.
J. Graph Theory, 14:149–166, 1990.
[15]
D. Rivshin.
Personal communication.
[16]
M. Welhan.
Reconstructing trees from two cards.
J. Graph Theory, 63(3):243–257, 2010. |
Holographic model of exciton condensation in double monolayer Dirac semimetal
A. Pikalov${}^{1,2}$
arseniy.pikalov@phystech.edu
${}^{1}$Moscow Institute of Physics and Technology, Dolgoprudny 141700,
Russia
${}^{2}$Institute for Theoretical and Experimental Physics, Moscow,
Russia
(January 11, 2021)
Abstract
In this paper we consider holographic model of exciton condensation in double monolayer Dirac semimetal. Excitons is a bound states of an electron and a hole. Being Bose particles, excitons can form a Bose-Einstein condensate. We study formation of two types of condensates. In first case both the electron and the hole forming the exciton are in the same layer (intralayer condensate), in the second case the electron and the hole are in different layers (interlayer condensate). We study how the condensates depend on the distance between layers and the mass of the quasiparticles in presence of a strong magnetic field. In order to take into account possible strong Coulomb interaction between electrons we use holographic appoach. The holographic model consists of two $D5$ branes embedded into anti de Sitter space. The condensates are described by geometric configuration of the branes. We show that the distance between layers at which interlayer condensate disappears decreases with quasiparticle mass.
Introduction.
An exciton is a bound state of an electron and a hole.
Excitons have been studied in condensed matter literature for a long time.
One of interesting questions concerning excitons is whether they can form Bose-Einstein condensate.
Exciton condenstation might be easier to achieve in case we have electrons and holes in different layers of a double layer two dimensional structure.
An insulator between the layer prevents electron and holes from annihilation thus increasing exciton lifetime.
The exciton condensation in double layer systems in magnetic field has been extensively discussed in condensed matter literature (see for example excitons1 ; exingraphene ; skyrm ).
In case the electron quasiparticles can be described as massless (gapless) Dirac fermions, exciton condensation is similar to the spontaneous chiral symmetry breaking in Quantum Chromodynamics.
The condensate breaks the chiral symmetry of massless fermions creating an energy gap in the spectrum.
From this point of view the chiral symmetry of graphene was discussed in Semenoff_chiral .
This analogy allows to test some basic notions of Quantum Chromodynamics in condensed matter systems.
Exciton condensate was observed in AlAs/GaAs heterostructures (double quantum well) in strong magnetic field hallex .
Later the spontaneous coherence of excitons was observed in cold atom gas excitongas . The coherence was interpreted as an sign of condensate formation. Another signature of the exciton condensation is Coulomb drag phenomenon coulomb_drag , that is the current in one layer creates voltage in the other one.
In this paper we discuss physics of the systems, consisting of
two layers of Dirac semimetal. We consider only zero temperature case.
There is strong magnetic field perpendicular to the layers.
The electrons and holes in the layers have quasirelativistic dispersion law $\epsilon(p)\sim\sqrt{m^{2}+p^{2}}$. For instance, such dispersion law can be created in graphene by applying appropriate deformation graphene_gap . The layers are separated by a thin dialectic preventing electrons from tunneling between layers. There are two possible types of excitons in the system. If the electron and the hole are both from the same layer, the exciton is intralayer, otherwise it is interlayer. If $\psi_{1}$ and $\psi_{2}$ stand for Dirac fermion operators in the first and second layers, intralayer condensate corresponds to the average $\langle\bar{\psi_{1}}\psi_{1}\rangle$, while interlayer condensate corresponds to $\langle\bar{\psi_{1}}\psi_{2}\rangle$. Formation of such condensates is similar to the chiral symmetry breaking in Quantum Chromodynamics Semenoff_chiral . However, there are important differences because we consider a system in two spatial dimensions instead of three.
Coulomb interaction between electrons might be strong strong .
Thus perturbation theory can be not working.
In order to describe exciton condensation in strong coupling regime we use holographic approach holqm . This model was introduce in paper Grignani2014 for zero temperature case and generalized to the case of finite temperature in Grignani2016 . However, only the case of Dirac quasiparticles with zero masses ($m=0$) was considered.
We generalize this discussion to the case $m\neq 0$. However, we consider only the zero temperature case and electrically neutral layers.
It was found in Grignani2014 that even infinitely small charge imbalance between the layers destroys exciton condensation in the strong coupling regime.
Therefore it is interesting to check if the fermion mass can lead to the similar effect.
Other holographic models of exciton condensation in bilayer systems can be found in Semenoff_ex ; Semenoff_d7 ; excitontrans
More generally, holographic chiral symmetry breaking is discussed in phase2 ; phase5 ; phase6 . Condensed matter aspects of exciton condensation in bilayer systems are described in room_excitons ; su4 ; roomtemp without holography.
The rest of the paper is organized as follows. First, we introduce the holographic set up and derive the basic equations. Second, we describe the possible types of solutions and their effective energies. By comparing the values of energy we derive the phase diagram of the system. The values of the condensates depends on the quasiparticle mass and the separation between the layers.
The model.
In this note we consider $D3/D5$ model of exciton condensation in a double monolayer system.
The model consists of large number $N$ of $D3$ branes that create $AdS_{5}\times S^{5}$ geometry. Here $AdS_{5}$ stands for a five-dimensional anti de Sitter space while $S^{5}$ is a five dimensional sphere.
The two layers of Dirac semimetal are modeled by two $D5$ branes embedded into this geometry. We treat them in probe approximation that is we do not consider the $D5$ branes back-reaction on the geometry.
$AdS_{5}$ geometry is dual to the $\mathcal{N}=4$ super Yang-Mills (SYM) theory. Each of the $D5$ branes supports massless Dirac fermions and connected brane configuration gives the fermions mass Bergman . The $\mathcal{N}=4$ SYM leads to the electron interaction energy proportional to $1/r$ and does not take into account screening.
Standard metric of
$AdS_{5}\times S^{5}$
$$\displaystyle ds^{2}=\frac{d\rho^{2}}{\rho^{2}}+\rho^{2}\left(-dt^{2}+dx^{2}+%
dy^{2}+dz^{2}\right)+\\
\displaystyle+d\psi^{2}+\sin^{2}\psi\,d\Omega_{2}^{2}+\cos^{2}\psi\,d\hat{%
\Omega}_{2}^{2}.$$
(1)
can be written in different coordinate system. Namely we can
introduce variables $\rho$ and $l$ as
$$r=\rho\sin\psi;\quad l=\rho\cos\psi.$$
(2)
In this coordinates the metric is
$$\displaystyle ds^{2}=\frac{1}{r^{2}+l^{2}}\left(dr^{2}+r^{2}d\Omega_{2}+dl^{2}%
+l^{2}d\hat{\Omega}_{2}\right)+\\
\displaystyle(r^{2}+l^{2})(-dt^{2}+dx^{2}+dy^{2}+dz^{2}).$$
(3)
This metric is more convenient when we are dealing with the case of finite mass.
The $D5$ branes are stretched along the $r$, $t$, $x$, $y$ directions and also are wrapped around the $\Omega_{2}$ sphere. The variable $\psi$ controls the radius of the sphere while the $z$ is the separation between the two $D5$ branes. Also there is magnetic field $b$ in $xy$ plane and gauge field $a_{0}$. All this field depend on radial variable $r$.
The Dirac-Born-Infeld (DBI) action for the D5 branes is
$$S=\mathcal{N}_{5}F=\mathcal{N}_{5}\int dr\,r^{2}\sqrt{1+\frac{b^{2}}{\rho^{4}}%
}\sqrt{1+l^{\prime 2}+\rho^{4}z^{\prime 2}}.$$
(4)
The variable $\rho^{2}=r^{2}+l(r)^{2}$. Here $\mathcal{N}_{5}$ is some normalization constant that includes the volume of the system. We do not need its exact value. Also we set $2\pi\alpha^{\prime}=1$. We will refer to the $F$ as free energy of the system (or more precisely free energy density).
The lagrangian does not depend on $z$ and corresponding canonical momentum is constant
$$f=\frac{\partial\mathcal{L}}{\partial z^{\prime}}=r^{2}\sqrt{1+\frac{b^{2}}{%
\rho^{4}}}\frac{\rho^{4}z^{\prime}}{\sqrt{1+l^{\prime 2}+\rho^{4}z^{\prime 2}}}.$$
(5)
We can express derivative as
$$z^{\prime}=\frac{f\sqrt{1+l^{\prime 2}}}{\rho^{2}\sqrt{r^{4}(b^{2}+\rho^{4})+-%
f^{2}}}.$$
(6)
The equation of motion for the $l$ is
$$\frac{\rho^{2}l^{\prime\prime}}{1+l^{\prime 2}}+\frac{2l^{\prime}r\left(f^{2}+%
b^{2}r^{2}l^{2}+r^{2}\rho^{6}\right)+2l(b^{2}r^{4}-f^{2})}{r^{4}(b^{2}+\rho^{4%
})-f^{2}}=0.$$
(7)
At large distances $r$ we have the following asymptotics for the solutions
$$l=m+\frac{c}{r}+\dots.$$
(8)
$$z=\frac{L}{2}-\frac{f}{5r^{5}}+\dots.$$
(9)
Here for the moment we consider the constants $m$, $c$, $L$, $f$ as free parameters that characterize the solution. We have a system of two differential equation of second order, therefore the solution is fully fixed by four constants. However, the requirement that the solution must be regular at small values of $r$ restricts two of them.
The variation of the free energy with respect to the parameters of the solution
$$\delta F=-q\delta\mu+f\delta L/2-c\delta m.$$
(10)
By the standard holographic dictionary we have the following interpretation of the variables in the solution:
$m$ is proportional effective mass of the electron in a layer;
$c$ is proportional intralayer exciton condensate;
$L$ is equal distance between layers;
$f$ is proportional interlayer exciton condensate.
We will not need the values of the proportionality coefficients.
Connection between $m$ and the mass is explained in Bergman ; kiritsis .
The values of $L$ and $m$ are fixed by the physical properties of the system.
We examine solutions with different types of asymptotics but the same values of $m$ and $L$. The solution with the lowest energy corresponds to thermodynamically stable state.
In principle, there are four options
1.
Brane separation $z$ is constant, $l=0$.Therefore $f=c=m=0$. It is massless solution without condensates.
2.
Brane separation is constant, but $l\neq 0$. There is mass and intralayer condensate.
3.
Branes annihilate, $l=0$. There is interlayer condensate but fermions are massless.
4.
Branes annihilate and $l\neq 0$. Both interlayer and intralayer condensates are nonzero.
We are going to focus on massive case, therefore we are intersted in second and fourth cases.
The solutions.
For numerical computations we use units in which the applied magnetic field $b=1$. First we consider type two solution. In this case branes do not interact with each other and in fact we deal with one brane solutions. Electrons from one layer do not form bound states with holes from the other layer.
The equation for $l(r)$ simplifies to
$$\frac{\rho^{2}l^{\prime\prime}}{1+l^{\prime 2}}+\frac{2l^{\prime}\left(b^{2}l^%
{2}+\rho^{6}\right)+2lb^{2}r}{r(b^{2}+\rho^{4})}=0.$$
(11)
Asymptotic of the solution for the small values of $r$ is (in units $b=1$)
$$l(r)=l_{0}-\frac{r^{2}}{3l_{0}\left(l_{0}^{4}+1\right)}+\frac{81l_{0}^{8}+45l_%
{0}^{4}-4}{270l_{0}^{3}\left(l_{0}^{4}+1\right)^{3}}r^{4}+\dots$$
(12)
The solution is fully specified by the value $l(0)=l_{0}$, the derivative $l^{\prime}(0)=0$ due to regularity of the solution.
For the large distances we have
$$l(r)=m+\frac{c}{r}-\frac{m}{6r^{4}}+\frac{c^{3}-2c}{10r^{5}}+\frac{m^{3}}{5r^{%
6}}+\frac{4cm^{2}}{7r^{7}}+\dots.$$
(13)
Full solution can be found only numerically.
Finally we need regularized version of free energy. Regularization is performed by subtraction of free energy of the brane in zero magnetic field with $f=c=0$.
$$F=\int_{0}^{\infty}dr\,\left[r^{2}\sqrt{1+\frac{1}{\rho^{4}}}\sqrt{1+l^{\prime
2%
}}-r^{2}\right].$$
(14)
For computation we introduce some large cut-off $r_{1}$. In the region $r<r_{1}$ integration is performed numerically, for large $r$ we use asymptotic expansion for $l$. Contribution of $r>r_{1}$ is
$$F_{1}=\frac{c^{2}}{2r_{1}}-\frac{m^{2}}{3r_{1}^{3}}-\frac{2mc}{3r_{1}^{4}}+%
\frac{3c^{4}-14c^{2}+12m^{4}}{40r_{1}^{5}}+\frac{6m^{3}c}{5r_{1}^{6}}+\dots$$
(15)
As a result, we can obtain free energy of the system as a function of mass. This can be performed only numerically. It does not depend on the distance between the layers. When mass is small enough, condensate $c$ is of order of unity and decreases with $m$.
Now we turn to the solution of type four when both $c$ and $f$ are nonzero.
Asymptotics are
$$\displaystyle l(r)=m+\frac{c}{r}-\frac{m}{6r^{4}}+\frac{c^{3}-2c}{10r^{5}}+%
\dots;\\
\displaystyle z(r)=\frac{L}{2}-\frac{f}{5r^{5}}+\frac{2fm^{2}}{7r^{7}}+\dots\\
$$
(16)
The solution at small $r$ starts at certain point $r_{0}$ defined by the constant $f$ and the initial value of $l$. At the point where two branes connect we must have $z^{\prime}(r_{0})=\infty$ therefore
$$f^{2}=r_{0}^{4}(1+(r_{0}^{2}+l_{0}^{2})^{2}).$$
It is more convenient to specify the solution by the values $r_{0}$ and $l_{0}$ and calculate corresponding parameter $f$.
Value of $l^{\prime}(r_{0})$ can be derived from the regularity of the solution at the point $r_{0}$:
$$l^{\prime}(r_{0})=\frac{l_{0}r_{0}(r_{0}^{2}+l_{0}^{2})^{2}}{r_{0}^{2}(1+(r_{0%
}^{2}+l_{0}^{2})^{2})+l_{0}^{2}+(r_{0}^{2}+l_{0}^{2})^{3}}.$$
(17)
Second derivative can also be calculated from the equation of motion, but the expression is very clumsy.
We need to choose the parameters $l_{0}$ and $r_{0}$ in such a way, that the solution has the correct value of mass $m$ and layer separation $L$. Such solution exists only if separation is small enough. Then for a given layer configuration we can calculate the free energy in cases with and without interlayer condensate. Note that there are at least two different solutions for a given pair $m$, $L$.
We need to choose the solution with minimal energy.
Comparison of free energies yields phase diagram in plane $m$, $L$. We find that for large enough separation $L>L_{c}$ interlayer condensate disappears. Critical layer separation decreases with mass. The results are summarized in Fig. 1. Above the yellow line there is no solution with interlayer condensate and above the blue (lower) line phase with interlayer condensate is energetically disfavored. As the mass increases, the two lines become closer.
Intralayer condensate is always nonzero in case $m\neq 0$, but in region of small masses is quite small and approximately proportional to the value of mass. If branes are not connected the value of condensate for $m<0.4$ is $c\approx 0.4-0.5$.
Conclustion.
In conclusion, we have studied the holographic model of double monolayer Dirac semimetal. In this model we calculated the energy of different brane configurations. By comparing the energies we obtained the phase diagram of the system. and showed that the critical distance between layers at which interlayer condensate disappears, decreases with mass of the quaisiparticles.
This results cannot be checked directly against experiment because we have not identified the parameters of holographic model in terms of physical parameters of the system. However, the model has some methodological value enabling us to access the properties of the system in strong coupling regime.
The holographic model confirms that exciton condensate exists for the finite fermion mass even for the strong coupling case.
This result is in qualitative agrement with the results of strong . Further investigation of the holographic condensate properties we postpone to future work.
The author is grateful to Alexander Gorsky for suggesting the problem and numerous discussions. The work of the author was supported by Basis Foundation fellowship and RFBR grant 19-02-00214.
References
(1)
O. L. Berman1, R. Ya. Kezerashvili1, Yu. E. Lozovik,
Nanotechnology 21, 134019 (2010).
(2)
C. H. Zhang, Y. N. Joglekar,
Phys. Rev. B 77, 233405 (2008).
(3)
K. Moon, H. Mori, K. Yang, et. al.,
Phys. Rev. B 51, 5138 (1994).
(4)
G. W. Semenoff,
Phys. Scr. 146 014016 (2012).
(5)
L. V. Butov, A. I. Filin,
Phys. Rev. B 58, 1980 (1998).
(6)
A. A. High, J. R. Leonard, A. T. Hammack, et.al.,
Nature 483, 584 (2012).
(7)
D. Nandi, A. D. K. Finck, J. P. Eisenstein, et.al.,
Nature 488, 481 (2012).
(8)
G. Cocco, E. Cadelano, L. Colombo,
Phys. Rev. B 81, 241412(R) (2010).
(9)
B. Debnath, Y. Barlas, D. Wickramaratne, et. al.,
Phys. Rev. B 96, 174504 (2017).
(10)
S. A. Hartnoll, A. Lucas, S. Sachdev,
Holographic quantum matter (The MIT press, 2018).
(11)
G. Grignani, N. Kim, A. Marini, G.W. Semenoff,
JHEP12(2014)091.
(12)
G. Grignani, A. Marini, A. Pigna, G.W. Semenoff,
JHEP06(2016)141.
(13)
G. Grignani, N. Kim , A. Marini, et.al.,
Phys. Lett. B 750, 22 (2015).
(14)
G. Grignani, N. Kim, G. W. Semenoff,
Phys. Lett. B 722, 360 (2013).
(15)
E. Gubankova, M. Cubrovic, J. Zaanen
Phys. Rev. D 92, 086004 (2015).
(16)
Veselin G. Filev, Matthias Ihl, and Dimitrios Zoakos,
JHEP07(2014)043.
(17)
N. Evans, A. Gebauer, K. Kim, M. Magou,
Phys. Lett. B 698, 91 (2011).
(18)
N. Evans and K. Kim,
Phys. Lett. B 728, 658 (2014).
(19)
Z. Wang, D. A. Rhodes, K. Watanabe, et.al.,
Nature 574, 76 (2019).
(20)
Z. F. Ezawa, K. Hasebe,
Phys. Rev. B 65, 075311 (2002).
(21)
H. Min, R. Bistritzer, J. Su, A. H. MacDonald,
Phys. Rev. B 78, 121401 (2008).
(22)
O. Bergman, S. Seki, J. Sonnenschein,
JHEP12(2007)037.
(23)
R. Casero, E. Kiritsisa, A. Paredes,
Nucl. Phys. B 787, 98 (2007). |
A NEW PHASE IN THE BILAYERS OF SEMICONDUCTORS IN QUANTUM HALL EFFECT
KESHAV N. SHRIVASTAVA
School of Physics, University of Hyderabad,
Hyderabad 500046, India
We find that a bilayer of semiconductors emits a new Goldstone
quasiparticle when Landau levels in the layers are half filled.
The emission of the new quasiparticle is associated with the
divergence in the energy of the system characteristic of a phase
transition.
keshav@mailaps.org Fax. +91-40-3010145
1. Introduction.
Recently, Spielman et al${}^{1,2}$ have found a new type of
Goldstone boson in quantum Hall effect in bilayers. The
Goldstone boson is accompanied with a phase transition or broken
symmetry. In the vacuum state if there is a broken symmetry,
then a massless particle is emitted. Spielman et al have
discovered such a quasiparticle. However, it remains to be
proved that there is a phase transition associated with the new
quasiparticle. Usually the Hall resistivity is quantized in
units of $h/ie^{2}$ where $i$ is an integer. It has been reported
that $i$ need not be an integer. In particular, it can be equal
to 1/3 or some other fraction. We have found${}^{3}$ that the
experimentally observed fractions can be accurately predicted by
a special spin-orbit interaction based on a pseudoscalar which
describes a $\vec{v}.\vec{s}$ type interaction, where $\vec{v}$
is the velocity of the electron and $\vec{s}$ the spin. Thus
there is a new spin-dependent force on the electron. This
spin-orbit interaction is of the order of $v/c$ whereas the
usual spin-orbit interaction in atoms is of the order of
$v^{2}/c^{2}$. It has been found experimentally that the masses of
some of the quasiparticles are equal. This equality of masses
has been understood by using the particle-hole symmetry.${}^{4}$
Further details of this calculation have been described in
detail${}^{5}$. The effective charge and the high Landau levels are
well understood by our calculation${}^{6}$.
In the present paper, we show that at half filled Landau level,
the energy of the system diverges so that there is a phase
transition. This phase transition is of second order and hence
it is accompanied by a new type of Goldstone boson. In the
present case, boson is emitted when $\sqrt{2}$ times the
magnetic length is an integer multiple of the distance between
the two layers of the semiconductors.
2. Theory.
We show that when a flux is moving between two walls at a
distance $d$ apart, a sharp peak in the conductance is predicted
when magnetic length, $t$, is an integer multiple of $d$,
otherwise a broad peak occurs. Using an earlier work, we show
that there is a phase transition when both layers are having
half-filled Landau levels. This phase is accompanied by a
Goldstone type boson. When the magnetic field is applied in the
$z$ direction and the electric field along $y$ direction, Hall
voltage develops along the $x$ direction. The Hall resistivity
is linearly proportional to the applied magnetic field and can
be used to determine the carrier concentration, $n$ as,
$$\rho_{H}={B\over nec}$$
(1)
where $B$ is the magnetic induction and $e$ is the charge of the
electron with $c$ as the velocity of light. In 1980 von Klitzing
et al${}^{5}$ discovered that there is a plateau at integer
multiples of $h/ie^{2}$ ($i$=integer), so that
$$\rho_{xy}={h\over ie^{2}}.$$
(2)
Equating (1) and (2) we find that
$${B\over nec}={h\over ie^{2}}\,\,.$$
(3)
We define the concentration as the number of electrons per unit
area, $n=n_{o}/A$ so that the above relation becomes
$$BA={n_{o}\over i}(hc/e)$$
(4)
which shows that flux is quantized with unit charge $e$. The
magnetic field may be measured in terms of a distance such that
the area $A=t^{2}$ above, may be taken as the square of the
magnetic length, $t$ so that (4) may be written as,
$$Bt^{2}={1\over i}(hc/e)$$
(5)
where we have used $n_{o}=1$ as the number of electrons per unit
area. For $i=1$,
$$Bt^{2}=hc/e$$
(6)
and
$$t=\left({hc\over eB}\right)^{1/2}$$
(7)
which is called the magnetic length. In the case of a two layer
system with distance $d$ between layers, the following theorems
are predicted. For
$$t=id$$
(8)
sharp line in the conductance versus voltage is predicted where
magnetic length is perfectly matched with the distance between
semicodncutor layers. When $t<<d$ or $t>>d$, the sharp line
disappears and a broad line is then predicted. Thus for a sharp
line in the characteristic conductance,
$$Bi^{2}d^{2}={hc\over e}$$
(9)
otherwise a broad line is predicted. Since the integer $i$ has
come from the area, the square of this integer is generated. The
fraction of conductivity, $\sigma=dI/dV$, in the sharp line is
$$\sigma_{o}=\sigma_{t}{\sin^{2}(\pi d/t)\over(\pi d/t)^{2}}\,\,,$$
(10)
where $\sigma_{o}$ is contained in the sharp line and $\sigma_{t}$
is the total integrated value of the conductivity. This spectrum
of $dI/dV$ as a function of $V$ is equivalent to the motion of
vortex waves with $t$ as the wave length. Since the condition
(9) gives a Goldstone mode, there must be a phase transition
associated with zero of an order parameter or the divergence of
a physical quantity, such as susceptibility of a ferromagnet. We
will show that the energy of the system diverges at half filled
Landau level. Since the fractional charges predicted by us${}^{3}$
are exactly the same as those experimentally observed by
Eisenstein and Störmer${}^{7}$ and others,${}^{8-11}$ we make use of
this model to understand the divergence. All of the aspects of
our model${}^{3}$ are in agreement with the experimental data${}^{7-11}$.
We have reported that a special type of spin-orbit interaction
predicts the fractions of charges correctly. This interaction is
of first order in $v/c$ whereas the ordinary spin-orbit coupling
of an electron in an atom is of second order, varying as
$v^{2}/c^{2}$. The spin dependence in the interaction is introduced
by the product $\vec{s}.\hat{n}$ where $\hat{n}$ is a unit
vector in the direction of the radius vector $\vec{r}$. From the
vectors $\vec{s}.\vec{n}$ and $\vec{v}$, a true scalar of the
form $\hat{n}\times\vec{v}.\vec{s}$ may be constructed. The new
spin-orbit coupling operator is of the form,
$$V_{sl}=-\phi(r)\hat{n}\times\vec{v}.\vec{s}$$
(11)
where $\phi(r)$ is a function of $\vec{r}$ because
$\vec{l}=\vec{r}\times\vec{p}$. This interaction may be written
as a spin-orbit interaction,
$$V_{sl}=-f(r)\vec{l}.\vec{s}$$
(12)
where $f(r)=\hbar\phi(r)/rm$. It splits the levels with orbital
angular momentum $l$ into two levels, $j=l\pm 1/2$. Since,
$$\displaystyle-\vec{l}.\vec{s}$$
$$\displaystyle=$$
$$\displaystyle-{1\over 2}l\qquad\mbox{for}\qquad j=l+{1\over 2}$$
(13)
$$\displaystyle=$$
$$\displaystyle+{1\over 2}(l+1)\qquad\mbox{for}\qquad j=l-{1\over 2}$$
the energy difference between the two states is
$$\displaystyle\Delta E$$
$$\displaystyle=$$
$$\displaystyle E_{(l-{1\over 2})}-E_{(l+{1\over 2})}$$
(14)
$$\displaystyle=$$
$$\displaystyle f(r)(l+{1\over 2})$$
As $l\to\infty$, one of the levels with $j=l+{1\over 2}$ in (13)
goes to $-\infty$ and the other with $j=l-{1\over 2}$ goes to
$+\infty$ and the energy different in (14) always goes to
$\infty$. Therefore, there is a phase transition as
$l\to\infty$. Therefore the Goldstone mode is accompanied by a
phase transition. Next, we show that at this phase transition
the Landau level is half filled. We obtain the effective
fractional charge from the charge in the Bohr magneton or from
the cyclotron frequency. Usually $l=0$, for the conduction
electrons but we have to allow the finite values of $l$ to
obtain the required result. We consider the spin as well as the
orbital motion so that,
$$g_{j}\vec{j}=g_{s}\vec{s}+g_{l}\vec{l}={1\over 2}(g_{l}+g_{s})\vec{j}+{1\over 2%
}(g_{l}-g_{s})(\vec{l}-\vec{s})\,\,\,.$$
(15)
We consider the bound electrons which have finite $l$. Upon
substituting $s={1\over 2}$, the above expression gives,
$$g_{j}=g_{l}\pm{g_{s}-g_{l}\over 2l+1}$$
(16)
which for $j=l\pm 1/2$ and $g_{s}=2$, $g_{l}=1$ gives,
$$g_{\pm}=1\pm{1\over 2l+1}\,\,\,.$$
(17)
The cyclotron frequency is defined in terms of the magnetic
field as,
$$\omega={eB\over mc}\,\,\,.$$
(18)
Corresponding to this frequency, the voltage along $y$ direction
is
$$\hbar\omega=eV_{y}\,\,\,\,.$$
(19)
From the above two relations
$${e^{2}B\over 2\pi mc}={e^{2}\over h}V_{y}$$
(20)
which describes the current in the $x$ direction so that the
resistivity is
$$\rho_{xy}={h\over e^{2}}$$
(21)
which is the same as (2) for $i=1$. We include (17) in (18)
so that the current becomes,
$$I_{x}={1\over 2}g{e^{2}B\over 2\pi mc}={1\over 2}{ge^{2}V_{y}\over h}\,\,\,\,.$$
(22)
For $l=0$, $g=2$ and,
$$I_{x}={e^{2}\over h}V_{y}$$
(23)
so that we define the effective charge as,
$$\nu_{\pm}={1\over 2}g_{\pm}\,\,\,.$$
(24)
For $l=0,{1\over 2}g_{+}=1$ and ${1\over 2}g_{-}=0$, for $l=1,{1\over 2}g_{+}={2\over 3}$ and ${1\over 2}g_{-}={1\over 3}$ which
determines the fractional charge of a quasiparticle. From (17)
and (24) we show below the predicted effective charge,
$$l$$
0
1
2
3
4
5
6
$$\infty$$
$$\nu_{-}={1\over 2}g_{-}$$
0
1/3
2/5
3/7
4/9
5/11
6/13
1/2
$$\nu_{+}={1\over 2}g_{+}$$
1
2/3
3/5
4/7
5/9
6/11
7/13
1/2
which occurs in two series such that for a given value
of $l$ the sum of the two values is always equal to unity,
$$\nu_{+}+\nu_{-}=1$$
(25)
because of the Kramers conjugate pairs, one has spin + and the
other -. At $l=\infty$, $g_{\pm}=1$ for both the series and hence
$\nu_{\pm}={1\over 2}$ which corresponds to half filled Landau
level. Thus we find that at $l=\infty$, the Landau level is half
filled, and as we have seen below eq.(14), there is a divergence
and hence phase transition. The modified values of the magnetic
length is
$$t^{*}=\left({hc\over\nu eB}\right)^{1/2}=id$$
(26)
for the Goldstone mode which for $\nu={1\over 2}$ is at,
$$\left({2hc\over eB}\right)^{1/2}=id\,\,\,.$$
(27)
Thus the divergence in energy at $l=\infty$, $\nu={1\over 2}$
which describes a phase transition, is accompanied by a Goldstone
boson. We have obtained three different features, (a) a
Goldstone boson, (b) divergence in energy and (c) the correct
fractional charge by using the $\vec{v}.\vec{s}$ type
interaction given by (11) alone. We can write the velocity in
the form of a current as $\vec{J}=n_{e}e\vec{v}$ where $n_{e}$ is
the concentration of electrons so that the relevent interaction becomes,
$${\cal H}^{\prime}=-{\phi(r)\over n_{e}e}\hat{n}\times\vec{J}.\vec{s}$$
(28)
which describes all of the essential features of the
quantum Hall effect correctly.
3. Comparison with the experimental data.
Usually the Goldstone boson in solids may be a soft phonon
associated with a lattice distortion. The zero-phonon lines are
common in solids. The Mössbauer lines are also zero-phonon
lines. However, in the present problem of semiconductor
heterostructures, there are no phonons and the problem is
electromagnetic at low temperatures and hence these are new type
of Goldstone bosons of intensity
$$I\propto\exp(-d^{2}/t^{2})\,\,\,.$$
(29)
The predicted values of the effective fractional charge given
above eq.(25) are exactly the same as those experimentally
observed by Eisenstein and Störmer.${}^{7}$ Not only the numerical
values but also the grouping into two groups predicted here is
in agreement with that observed. Similarly, the predicted values
are in agreement with the experimentally observed values in many
other measurements. When $\nu$ is an observable fractional
charge, $n\nu$ ($n=$ integer) also becomes observable. Since
$l=\infty$ limit gives $\nu=1/2$, the values $n/2$ become
observable. This prediction is also in accord with the
experimental data of Yeh et al.${}^{11}$ The divergence in energy
at $\nu=1/2$ shows that there is a phase transition at these
values in the semiconductor bilayer to a new state of matter.
The symmetry breaking at this point is accompanied with a
Goldstone boson. This prediction of emission of a boson is in
agreement with the experimental observation of a Goldstone boson
at zero voltage${}^{1-2}$.
We have predicted that there is a Goldstone boson at $\nu=1/2$
when the condition (27) is satisfied in the bilayer
semiconductor. This mode is associated with a divergence in
energy at $l\to\infty$ as determined from (14). The predicted
fractional charges are in agreement with the experimental data
in several works${}^{7-11}$. The predicted Goldstone boson at the
phase transition is also in agreement with that found
experimentally. A Josephson type flux flow has recently been
considered${}^{12,13}$ in the semiconductor bilayers which is not
in contradiction with our work because we can convert the flux to current. Several advanced theories of the
quantum Hall effect have been reviewed from which we find that
our theory is consistent with those of others${}^{14}$. The fractions
given by Fig. 18 of Störmer${}^{15}$ are exactly the same as those
predicted by us${}^{3}$. There is no doubt in our interpretation of the
Goldstone boson observed by measuring the conductance in the data
of Spielman et al${}^{1}$. The conductance is analogous to the
Doppler brodened line calculated by Dicke${}^{16}$. However,
in the case of Dicke’s line there are ordinary distances whereas
in the present case there is the magnetic length. The interaction
(11) involves the vector product of the velocity with the unit vector
in the direction of the radius vector which generates the rotations
or ”magnetic rotons” which are often observed in the experiments. Thus
the Goldstone boson, divergence in energy at $l=1/2$ required
for a singularity in the energy, effective fractional charges and
rotations, predicted by our calculations are all in good agreement
with the data.
4. Bose condensation or phase transition.
The equation (24) gives the effective fractional charge. The
series with + sign gives $\nu_{+}=1/2$ as $l\to\infty$. Similarly,
$\nu_{-}=1/2$ occurs also at $l=\infty$. We call these states
$A^{(+)}$ and $B^{(-)}$, respectively. The energy of the state
$A^{(+)}$ from (13) is $-(1/2)l$ and that of $B^{(-)}$ is
$+{1\over 2}(l+1)$. When the quasiparticles from these states are
added to form a mixed state, the energy is
$[-{1\over 2}l+{1\over 2}(l+1)]f(r)={1\over 2}f(r)$, i.e., we
consider $\epsilon_{1}$ and $\epsilon_{2}$ as the single particle
energies for the two states and form a two particle state with
energy, $\epsilon_{1}+\epsilon_{2}$. The Bose distribution for the
mixed state is,
$$n={1\over e^{(\epsilon_{1}+\epsilon_{2}-\mu)/k_{B}T}-1}$$
(30)
where $\mu$ is the chemical potential of the mixed state arising
from the interactions between the $A^{(+)}$ and $B^{(-)}$ states.
So far we have not introduced the wave vector space and hence
$\epsilon_{1}$ and $\epsilon_{2}$ are not associated with any wave
vectors but $\epsilon_{1}$ has spin + and $\epsilon_{2}$ has spin -.
Therefore $\epsilon_{1}+\epsilon_{2}$ actually is similar to a Cooper
pair with infinite value of $l$. The two particle energy is,
$$\epsilon_{1}+\epsilon_{2}={\hbar^{2}k^{2}_{1}\over 2m}+{\hbar^{2}k^{2}_{2}%
\over 2m}+{1\over 2}f(r)+\hbar\omega_{c}(n_{1}+{1\over 2})+\hbar\omega_{c}(n_{%
2}+{1\over 2})$$
where the last two terms are the result of Landau levels when the
Landau number in state $\epsilon_{1}$ need not be equal to that in
$\epsilon_{2}$ state. The minimum energy is obtained at $k_{1}=k_{2}=0$
and $n_{1}=n_{2}=0$,
$$2\epsilon_{o}=(\epsilon_{1}+\epsilon_{2})_{min}={1\over 2}f(r)+\hbar\omega_{c}%
\,\,\,.$$
(31)
The number of particles in the ground state is
$$n_{o}={1\over e^{(2\epsilon_{o}-\mu)/k_{B}T}-1}\,\,\,.$$
(32)
If these particles are confined in a 2-dimensional plane, the
average number of particles is given by,
$$N=n_{o}+{A\over(2\pi)^{2}}\int{kdk\over e^{(\epsilon_{1k}+(\epsilon_{2k}-\mu)/%
k_{B}T}-1}\,\,\,.$$
(33)
We devide this expression by the area $A$ to obtain the number of
particles per unit area as,
$${N\over A}={n_{o}\over A}+{1\over(2\pi)^{2}}\int{kdk\over e^{(\epsilon_{1k}+%
\epsilon_{2k}-\mu)/k_{B}T}-1}$$
(34)
which diverges for $\epsilon_{1k}+\epsilon_{2k}=\mu$. This
divergence in the areal density is associated with the Bose
condensation. We write $(\epsilon_{1k}+\epsilon_{2k}-\mu)/k_{B}T=x$
and $(\hbar^{2}k^{2}_{1}+\hbar^{2}k_{2}^{2})/2m\simeq ak^{2}$ so that
$$k_{B}Tx+\mu=ak^{2}\,\,\,.$$
(35)
Differentiating this expression,
$$kdk={k_{B}T\over 2a}dx$$
(36)
which substituted in the above gives the number of quasiparticles
per unit area as,
$${N\over A}={n_{o}\over A}+{(k_{B}T)/2a\over(2\pi)^{2}}\int{dx\over e^{x}-1}\,%
\,\,.$$
(37)
This means that there is a divergence in $N/A$ when $x=0$ and
away from the divergence, $N/A$ depends linearly on temperature.
The particles in the states $A^{(+)}$ and $B^{(-)}$ may Bose
condense in the two particle states and the resulting liquid
exhibits superflow and hence can tunnel between layers just like
a Josephson tunneling. As we have pointed out this Bose condensed
state is made from pairs of quasiparticles with one electron spin
up and the other spin down but not $s$ wave because of large $l$. Therefore, the pairs
can tunnel through the layers of GaAs/AlGaAs as in Josephson
effect. However, in the Josephson effect in superconductors, the
pairs are usually in the $s$ wave, $l=0$ and in some materials a
$d$-wave has been found with $l=2$. In the present problem
$\nu={1over2}$ is achieved with $l\to\infty$. Hence, the present
state has $l\to\infty$ which is never the case in
superconductors. The $l\to\infty$ state with spin singlet state
is a boson but in superconductors, the Cooper pairs are made from
fermions. The eq.(24) gives the effective fractional charge. The
series with + sign gives $\nu_{+}=1/2$ at $l\to\infty$. Similarly
$\nu_{-}=1/2$ occurs alos at $l=\infty$. We call these states as
$A^{+}$ and $B^{-}$, respectively. The energy of the state $A^{+}$ from
(13) is $-(1/2)l$ and that of $B^{-}$ is ${1\over 2}(l+1)$. The
Bose-Einstein distribution for the $A^{+}$ state is,
$$n={1\over e^{(\epsilon_{A}-\mu)/k_{B}T}-1}$$
(38)
where the single particle energy in the state $A^{+}$ is,
$$\epsilon_{A}={\hbar^{2}k^{2}\over 2m}-{1\over 2}lf(r)+\hbar\omega_{c}(n+{1%
\over 2}).$$
(39)
At $\nu={1\over 2}$, $\l\to\infty$ and hence $A^{+}$ state energy
diverges so that there is a phase transition when single-particle
$A^{+}$ and $B^{-}$ states are formed. This phase transition is
associated with a Goldstone mode and not with Bose condensation.
The paired state with $l\to\infty$, with one particle spin up and
the other down is a Bose condensed state. Therefore, the two
particle state, one spin up and the other down, with $l=\infty$
for both, gives a Bose condensed state which is not a phase
transition. The single particle state at $l=\infty$ does not Bose
condense but gives the phase transition and hence the Goldstone
mode. The two particle state gives the Bose condensed state. Thus
two results emerge. Firstly, the two particle state gives Bose
condensation and in this case the Bose condensate itself has to
be its own Goldstone mode. Secondly, the single particle
$\nu=1/2$ does not Bose condense but describes a phase transition
and hence a Goldestone mode emerges.
We have compared our theory with a number of data and found that
in all cases the theory is in accord with the experimental data${}^{17}$.
We find that there is a new change in the magnetic moment of the electron${}^{18}$ in the quantum Hall effect. We find that the rate of sweep of the magnetic field plays an important role particularly in NMR${}^{19}$ and even if NMR is not being taken. The theory of the composite fermions (CFs)in found to be internally inconsistent${}^{20}$ so there are no CFs in the real material. The polarization of the electrons as a function of temperature exhibits ordinary two-level type behavior and shows that fluxex are not attached to the electrons${}^{21}$.
5. Conclusions
The interaction with dot product of velocity with spin, given by
(11), exerts a new force on the electron. The energies arising
from this interaction show that there is a divergence and hence
a phase transition so that there is a new type of Goldstone
boson. We predict that the quasiparticles exhibit effective
fractional charge. The predicted Goldstone boson and the
fractional charges are in agreement with the experimental observations.
There is a Bose condensation of a two component fluid at $\nu=1/2$.
References
1.
I.B. Spielman, J.P. Eisenstein, L.N. Pfeiffer and K.W.
West, Phys. Rev. Lett. 84, 5808 (2000).
2.
I.B. Spielman, J.P. Eisenstein, L.N. Pfeiffer and K.W.
West, Cond-mat/0012094.
3.
K.N. Shrivastava, Phys. Lett. A113, 435 (1986);
115, 459(E) (1986).
4.
K.N. Shrivastava, Mod. Phys. Lett. 13, 1087 (1999).
5.
K.N. Shrivastava, Superconductivity: Elementary Topics,
World Scientific, Singapore (2000).
6.
K.N. Shrivastava, Mod. Phys. Lett. 14, 1009(2001).
7.
J.P. Eisenstein and H.L. Störmer, Science 248, 1510 (1990).
8.
R.R. Du, H.L. Störmer, D.C. Tsui, L.N. Pfeiffer and K.W.
West, Phys. Rev. Lett. 70, 2944 (1993).
9.
W. Pan, J.S. Xia, V. Shavarts, D.E. Adams, H.L. Störmer,
D.C. Tsui, L.N. Pfeiffer, K.W. Baldwin and K.W. West,
Phys. Rev. Lett. 83, 3530 (1999).
10.
J.P. Eisenstein, M.P. Lilly, K.B. Cooper, L.N. Pfeiffer
and K.W. West, Physica E6, 29 (2000).
11.
A.S. Yeh, H.L. Störmer, D.C. Tsui, L.N. Pfeiffer, K.W.
Baldwin and K.W. West, Phys. Rev. Lett. 62, 592
(1999).
12.
M.M. Fogler and F. Wilczek, Phys. Rev. Lett. 86, 1833
(2001).
13.
M.M. Fogler and F. Wilczek, cond-mat/0007403.
14.
M. Stone, ed., Quantum Hall effect, World Scientific,
Singapore 1992.
15.
H. L. Störmer, Rev. Mod. Phys. 71, 875 (1999).
16.
R. H. Dicke, Phys. Rev. 89, 472 (1953).
17.
K. N. Shrivastava, CERN SCAN-0103007.
18.
K. N. Shrivastava, cond-mat/0104004.
19.
K. N. Shrivastava, cond-mat/0104577.
20.
K. N. Shrivastava, cond-mat/0105559.
21.
K. N. Shrivastava, cond-mat/0106160. |
Can Planets Exist in the Habitable Zone of 55 Cancri?
Suman Satyal11affiliation: Department of Physics, University of Texas at Arlington, Box 19059,
Arlington, TX 76019, USA
suman.satyal@uta.edu
Manfred Cuntz11affiliation: Department of Physics, University of Texas at Arlington, Box 19059,
Arlington, TX 76019, USA
cuntz@uta.edu
Abstract
The aim of our study is to explore the possible existence of Earth-mass planets in the
habitable zone of 55 Cancri, an effort pursued based on detailed orbital stability simulations.
This star is known to possess (at least) five planets with masses ranging between super-Earth
and Jupiter-type. Additionally, according to observational constraints, there is a space without
planets between $\sim$0.8 au and $\sim$5.7 au, noting that the inner part of this gap largely
coincides with 55 Cnc’s habitable zone — a sincere motivation for the search of potentially
habitable planets. It has previously been argued that terrestrial habitable planets are able
to exist in the 55 Cnc system, including a planet at $\sim$1.5 au. We explore this possibility
through employing sets of orbital integrations and assuming an integration time of 50 Myr.
We found that the possibility of Earth-mass planets in the system’s habitable zone strongly
depends on the adopted system parameters, notably the eccentricity of 55 Cnc-f, which is
controversial as both a high value ($e\sim 0.32$) and a low value ($e\sim 0.08$) have
previously been deduced. In case that the low value is adopted (together with other updates
for the system parameters), the more plausible and most recent value, Earth-mass planets would
be able to exist in the gap between 1.0 au and 2.0 au, thus implying the possibility of habitable
system planets. Thus, 55 Cnc should be considered a favorable target for future
habitable planet search missions.
\draft\Received
$\langle$reception date$\rangle$
\Accepted$\langle$acception date$\rangle$
\Published$\langle$publication date$\rangle$
\KeyWords
astrobiology — celestial mechanics — methods: numerical —
planetary systems — stars: individual (55 Cnc)
1 Introduction
A prevalent topic at the crossroad between astrophysics and astrobiology is given by the
study of stellar habitable zones (HZ). Following the landmark paper by Kasting et al. (1993), which focused on
the extent of HZs for main-sequence stars based on simplified atmospheric climate models two and a
half decades ago, there has been an explosion of literature on that topic. Immense progress has been
made especially in the study of planetary atmospheres, see, e.g., Kopparapu et al. (2013); Kasting et al. (2015); Ramirez & Kaltenegger (2018), among many
other studies. Recently, a catalog of HZ exoplanet candidates based on results of the Kepler mission
has been published as well by Kane et al. (2016). Clearly, the progress made in investigating HZ also goes
hand-in-hand with ongoing discoveries of extrasolar planets around different types of stars111For
updated information, please visit http://exoplanet.eu, including G and K-type stars.
However, it is well-acknowledged that the location of a planet inside the HZ is by itself insufficient
for ensuring habitability, as sets of other conditions need to be fulfilled, including (but not limited to)
the mass and size of that planets, geodynamic aspects, and environmental forcings mostly associated with
the parent star; see, e.g., Lammer et al. (2009), and subsequent work.
Regarding astrobiology, orange dwarfs, i.e., late-type G and early-type K stars have previously
received a heightened amount of attention (e.g., Heller & Armstrong, 2014; Cuntz & Guinan, 2016). These types of stars are
considered particularly suitable for hosting planets with exolife (potentially
even advanced forms of life) due to numerous supportive features including the frequency of those stars,
the relatively large size of their HZs (if compared to M-type stars)222This aspect has recently
been challenged as observations and theoretical work show that the occurrence rate of low-mass planets
increases with smaller orbital distances from the star, thus favoring the population of HZs of very
low-mass stars; see Mulders et al. (2018) and Ogihara et al. (2018). However, this finding does not reduce the
privileged situation of orange dwarfs relative to other types of stars as they are advantageous
to hotter stars, but not disadvantageous to cooler stars such as M dwarfs (although those
may host an increased number of low-mass planets in their HZs), noting that the latter are often
considered subideal for hosting habitable planets owing to their energetic radiative environments.
their relatively long life time on the main-sequence (i.e., 15 Gyr to 30 Gyr, compared to $\sim$10 Gyr for
stars akin to the Sun), and the modest amounts of magnetic-dynamo-generated X-ray-UV emissions
(except young stars) resulting in not-so-much destructive planetary atmospheric environments; see,
e.g., Cuntz & Guinan (2016) and references therein.
In this work, we focus on 55 Cancri (55 Cnc, $\rho^{1}$ Cnc), a G8 V star (Gonzalez, 1998), with
an effective temperature and luminosity lower than those of the Sun (e.g., Fischer & Valenti, 2005; Ligi et al., 2016). Moreover,
55 Cnc is considerably older than the Sun with an estimated age between 7.4 Gyr and 8.7 Gyr (Mamajek & Hillenbrand, 2008).
It has a mass of 0.96 $M_{\odot}$ (Ligi et al., 2016).
Additionally, 55 Cnc is known to host five planets, discovered between 2008 and 2011 by Fischer et al. (2008),
Dawson et al. (2008), and Winn et al. (2011) based on the radial velocity method. All of these planets have masses
significantly larger than Earth. However, based on previous theoretical studies, see, e.g.,
von Bloh et al. (2003), Rivera & Haghighipour (2007), Raymond et al. (2008), and Smith & Lissauer (2009), the general possibility of Earth-mass
planets in that system is implied.
The focus of this study is to explore whether 55 Cnc would be able to host Earth-mass planets,
especially in its HZ. This topic is of interest to both astrophysics and astrobiology, and
is expected to be relevant to future planetary search missions. Thus, we will employ detailed
orbital stability simulations for Earth-mass test planets assuming adequate initial conditions (ICs).
In this respect, we consider previous data for this system, as well as updated findings by
Bourrier et al. (2018). Regarding those studies there is a pivotal difference in the eccentricity
of 55 Cnc-f, which is of critical importance for the outcome of our simulations.
Our paper is structured as follows. In Section 2, we convey our theoretical approach, including
information on the system set up as well as the theoretical simulations. Our results and discussion
are given in Section 3. Comments on previous studies are given in Section 4, whereas Section 5
states our summary and conclusions.
2 Theoretical Approach
2.1 System Set-Up
The star 55 Cnc is host to (at least) five planets, discovered between 2008 and 2011, with masses ranging from super-Earth to Jupiter-type; see Table 1 for details and references333In this study, we will assume minimum masses for the five 55 Cnc system planets. If higher masses were assumed, the domain of orbital stability for hypothetical Earth-mass planets would be further reduced..
The location of the planets in the 55 Cnc system, as identified, exhibits a relatively large gap between 55 Cnc-f and 55 Cnc-d. Therefore, to numerically test if any additional planet could remain in a stable orbit in this region, we inject Earth-mass planets with different ICs and perform dynamical analyses. The integrations are carried out to determine the nearest stable semi-major axis exterior to 55 Cnc-f, which is at $\sim$0.77 au. Injected planets are set at initially circular and coplanar orbits
at distances of $a_{\rm pl}=a_{0}$.
Next, the planetary initial inclination ($i_{\rm pl}=i_{0}$) and initial eccentricity ($e_{\rm pl}=e_{0}$) are gradually increased to explore different dynamical settings. However, for the inclination no values above 20${}^{\circ}$
are considered as no significant dynamical changes were found to occur in the region-of-interest. Compact systems like 55 Cnc are typically identified as having nearly coplanar orbits. For example, the planets of the Kepler-62 and TRAPPIST-1 systems have inclinations varying by less than 1${}^{\circ}$ (Borucki et al., 2013; Gillon et al., 2016). However, some of these data may have been biased by the applied observation method.
The initial orbital parameters, i.e., semi-major axis ($a_{0}$), eccentricity ($e_{0}$), inclination ($i_{0}$),
argument of periapsis ($\omega_{0}$), ascending node ($\Omega_{0}$), and mean anomaly ($MA_{0}$) of the five planets are obtained from
the original discovery papers and follow-up papers, if applicable; see Fischer et al. (2008),
Dawson et al. (2008), Winn et al. (2011), Endl et al. (2012), and Ligi et al. (2016). These parameters are given in Table 1. We also considered a different set of orbital parameters recently reported by Bourrier et al. (2018); they differ significantly from the previous works (see Table 2). We will present and discuss the outcome of simulations from both sets of data. However, we will not consider the observational uncertainties in these parameters as part of our simulation; the latter are based on the planets’ best-fit values. The orbital parameters, i.e., $\omega$, $\Omega$, and $MA$, not reported in the respective articles are set to zero.
Technically, 55 Cancri constitutes a wide binary system. 55 Cancri A, the focus of our study, is accompanied by a red dwarf, 55 Cancri B, with a mass of
0.13 $M_{\odot}$. Due to the large separation of the secondary companion, given as $\sim$1065 au (Duquennoy & Mayor, 1991; Eggenberger et al., 2004), there is no need to consider its gravitational effects on the planets for our integrations444Methods for the calculation of HZs for 55 Cancri-type binaries have been given by, e.g., Eggl et al. (2013) and Cuntz (2014, 2015)..
2.2 Numerical Simulations
The numerical calculations are performed using the orbital integration package mercury (Chambers, 1997, 1999) where the multi-body system is set in an astro-centric coordinate system. The program calculates the orbital evolution of the planets as they orbit in the gravitational field given by the central star and the five system planets. Among different N-body algorithms within the program, the Hybrid Symplectic/Bulirsch-Stoer Integrator has been chosen because of its fastness and its ability of computing close encounters. In the mercury setup files, the time step is fixed at $\epsilon=10^{-3}$ year/step in order to minimize the error accumulation. The 7-body system (5 planets as observed, 1 Earth-mass test planet, and the star) is simulated for up to 50 Myr. The data are sampled every 1000 years for short-term simulations and every 10,000 years for long-term simulations. Additionally, we monitor the
changes in the system’s total energy and angular momentum; both of them consistently remain close to zero. Other forces as, e.g., tides are not considered as part of the simulations.
For the initial orbital parameters of the test planets, the semi-major axis is assumed to take values between 0.0 au and 6.0 au with a step size of 0.01 au. First, we consider coplanar and circular orbits. Thereafter, the test planets are integrated for non-coplanar orbits by varying the orbital inclination from 0${}^{\circ}$ to 20${}^{\circ}$ with a step size of 0.5${}^{\circ}$, and the orbital eccentricity from 0.0 to 0.4 with a step size of 0.01. Hence, a total of 24,000 ICs are simulated in the $a_{\rm pl}$ - $i_{\rm pl}$ phase space, and 20,000 ICs are simulated in the $a_{\rm pl}$ - $e_{\rm pl}$ phase space. The mass of the test planets is always set to 1 $M_{\oplus}$; furthermore, the other orbital parameters (i.e., argument of periapsis, ascending node, and mean anomaly) are set to zero in all models. The relevant parameters, including those for the test planets
are given in Table Can Planets Exist in the Habitable Zone of 55 Cancri? for earlier work and Table Can Planets Exist in the Habitable Zone of 55 Cancri? based on work by Bourrier et al. (2018); see discussion below.
The orbits of the Earth-mass test planets were considered stable if they survived the total simulation time and their maximum eccentricities remained close to their initial values, without approaching $\sim$1. In the cases where the test planets collided with one of the system planets or the star, or if they were ejected from the system, the orbits were considered unstable. For each simulation, this information can be extracted via mercury output files for further analyses.
3 Results and Discussion
3.1 Stability Analysis Based on Earlier Work
In the following, we report on numerical results based on data given in Table 1, which encompasses results published between 2008 and 2016.
Generally, an important aspect pertains to the comparison of the domain of planetary orbital stability to the stellar HZ.
The HZ of 55 Cnc is calculated using the formalism given by Kopparapu et al. (2013, 2014). They have specified a general habitable zone (GHZ) as well as an optimistic habitable zone (OHZ); the latter is given by the recent Venus / early Mars limits, previously coined by Kasting et al. (1993). Based on the luminosity and stellar effective temperature of 55 Cnc, given as $L=0.589~{}L_{\odot}$ and $T_{\rm eff}=5165$ K, respectively, see Ligi et al. (2016), the GHZ extends from $\sim$0.78 au to $\sim$1.37 au, and the OHZ extends from $\sim$0.59 au to $\sim$1.43 au. Hence, our analysis is concentrated at distances akin to the GHZ and OHZ, as well as at distances close to $\sim$1.51 au previously advocated by Cuntz (2012). Regions akin to those distances will be selected to inject hypothetical Earth-mass planets in order to explore their orbital stability.
A preliminary stability map for the injected planet based on $e_{\rm max}$ while using a relatively short simulation time of 10 kyr is given in Fig. 1a. The magenta vertical lines represent the location of the five system planets. The color bar represents the $e_{\rm max}$ values; here green means 0.0 and white means 1.0. Other intermediate colors change from dark green to dark blue to light blue as the eccentricity values change from 0.2 to 0.8. The phase-space map of $i_{\rm pl}$ versus $a_{\rm pl}$ exhibits a stability region with the $e_{\rm max}$ values remaining close to zero; it extends from 1.6 au to 4.0 au. However, this stability region considerably shifts to the right from 1.6 au to 2.0 au when the simulation time is increased to 100 kyr (Fig. 1b). The unstable regions between 1.0 au and 2.0 au become more prominent for 1 Myr and 10 Myr simulation times (Fig. 1c, d). A smaller phase space (1.0 au to 2.6 au) is considered for longer simulation times to focus on the planetary behaviors in the stellar HZ and to make our simulations computationally less expensive. A few resonance structures can be seen at 1.6 au and 1.8 au where the eccentricity values deviate the least from the initial values $e_{0}$; however, we expect them to disappear for simulation times larger than 10 Myr. In the region of phase space between 1.2 au and 1.6 au, the eccentricity rises to $\sim$1 for almost all ICs. Hence, this makes it impossible for any additional planet to remain in a dynamically stable orbit in the HZ; i.e., neither the GHZ nor the OHZ.
Figure 2 shows the $e_{\rm max}$ map for the injected planet with 1 $M_{\oplus}$, which is generated from the maximum eccentricities attained by the orbits during simulation times of 100 kyr and 10 Myr. Four of the interior planets (Cnc-b, Cnc-c, Cnc-d, and Cnc-f) are shown by colored circles at their respective locations. The maps indicate an unstable region between 1.0 au and 2.0 au followed by a stable region between 2.0 au and 4.0 au. The map shows significant changes in the $e_{\rm max}$ values in the regions below 2.0 au, which partially coincides with the stellar HZ. The stability regions decrease as the simulation time is increased from 100 kyr to 10 Myr as also observed in Fig. 1. Also, similar to the Fig. 1d, only a small portion of the phase space, located between 0.8 au and 2 au, is simulated for 10 Myr. This region is expected to shrink further for simulation times in excess of 10 Myr. However, the regions of stability around 3 au are expected to remain, and may thus be able to host additional planets.
Figure 3 (top panel), conveys the evolution of the semi-major axes $a_{\rm pl}$ for Earth-mass test planets originally placed between 1.0 au and 2.0 au. All test planets in this phase space, including those originally set in the stellar HZ, become unstable by either colliding with the one of the system planets or by being ejected from the system. The amplitude of the semi-major axis variation is rather small for some cases, like $a_{\rm pl}$ = 1.6 au and $a_{\rm pl}$ = 2 au; nevertheless, the planetary orbits become chaotic after 10 Myr when $e_{\rm pl}$ exceeds 0.5. This behavior eventually leads to instability. The $a_{\rm pl}$ time series is complemented by the bottom panel depicting the evolution of $e_{\rm pl}$ for the same planetary orbits. Note that the amplitude of $e_{\rm pl}$ oscillations varies significantly between 0.0 and 1.0 for all cases, which indicates that the orbits will eventually become unstable. The test planets set at distances below 1.5 au also became orbitally unstable during a relatively short simulation time due to collisions with the system planets. Survival times and $e_{\rm max}$ values for all test planets set between 1.5 au and 1.6 au are given in Table 3.
Figure 4 depicts the orbits of the five 55 Cancri system planets, based on data from earlier work, as well as the added Earth-mass planets originally set at 1.50 au, 1.55 au, 1.57 au, and 1.60 au, respectively; see Table Can Planets Exist in the Habitable Zone of 55 Cancri? for information on the data of the system planets as used.
The variations exhibited by the semi-major axis of the test planet during the simulations are relatively large. They are observed to scatter by as much as 2 au, and are displayed in red in all four panels. The system planets (depicted in magenta, cyan, blue, green, and black for 55 Cnc-b, 55 Cnc-c, 55 Cnc-d, 55 Cnc-e, and 55 Cnc-f, respectively) are found to exhibit periodic orbits with small variations in their semi-major axis values. The bottom two panels, i.e., for (a) $a_{0}$ = 1.57 au and (b) $a_{0}$ = 1.60 au, omit the data for 55 Cnc-d to better display the smaller inner orbits. The Earth-mass planet reaches instability within 2.5 Myr for all four $a_{0}$ values. Thus, based on our results as demonstrated via phase-space maps and time series analyses (see Figs. 1 to 4), we conclude that — for the assumed data set — no Earth-mass planets are able to exist at distances below $\sim$2.2 au, a domain that also encompasses the stellar HZ. Simulation times beyond 10 Myr are expected to somewhat further reduce the zone of planetary orbital stability.
3.2 Stability Analysis Based on the Data by Bourrier et al. (2018)
The choices of phase space and map resolution are similar to those considered in Section 3.1. However, the ICs used to simulate the planetary orbits are now based on the best-fit values reported by Bourrier et al. (2018), see Table 2. The primary difference between their work and the previous work is given by the eccentricity of Cnc 55-f, which was updated from 0.32 to 0.08. This lower value for the eccentricity results in a spatially increased dynamically stable region as given in Figs. 5 and 6, see discussion below.
Figures 5a and 5b show the stability region between 1 au and 2 au, which also largely coincides with the stellar HZ; see the $e_{\rm pl}$ - $a_{\rm pl}$ phase space. The locations of the five system planets are shown as color-filled circles. The $e_{\rm max}$ values (color coded) remain relatively low as shown in Fig. 5a for the added Earth-mass planets. In the HZ, the $e_{\rm max}$ is less than 0.1 for relatively low initial eccentricities ($e_{0}$), thus indicating stable orbits. The stability region extends up to $\sim$4 au for $e_{0}$ less than 0.2. The unstable mean motion resonances (MMRs) due to the interactions with Cnc 55-f and Cnc 55-d reveal themselves in the phase space at around 1.65 au, 1.9 au, 2.0 au, 2.5 au, and 3.0 au. Despite these MMRs, the injected Earth-mass planets are expected to maintain stable orbits in the inner regions ranging from 1.0 au to 1.6 au.
In order to explore the time series evolution of the semi-major axis and the eccentricity of the added Earth-mass planet, we chose 33 different ICs in the stable phase space region as given in Fig. 5b. Integrating fewer ICs is computationally less expensive, thus allowing integrations for longer orbital periods.
Therefore, the initial semi-major axis of the Earth-mass planet $a_{0}$ was varied from 1.50 au to 1.60 au with a step size of 0.01, and the initial eccentricity $e_{0}$ was varied from 0.0 to 0.5 with a step size of 0.1. Each IC was simulated for 50 Myr. These time series plots are shown in Fig. 6. Figure 6a depicts the evolution of $a_{\rm pl}$ and $e_{\rm pl}$ with $e_{0}$ set at 0.0. To avoid overcrowding, only 6 and 3 points are plotted in the $a_{\rm pl}$ and $e_{\rm pl}$ panels, respectively. The amplitude of the $a_{\rm pl}$ variation remains low for all the ICs for the total time of integration. However, the $e_{\rm pl}$ variation is large, i.e., up to 0.3 for some values of $a_{0}$. Furthermore, the orbits are periodic for up to 50 Myr. Similarly, when $e_{0}$ was set to 0.1 and 0.2 (see Figs. 6b and 6c), the time series for $a_{\rm pl}$ and $e_{\rm pl}$ exhibit stable periodic orbits even though the amplitude of eccentricity oscillations varied by as much as 0.3. For $e_{0}$ higher than 0.2, the system displayed unstable orbits within 50 Myr of simulation time.
The actual survival times of Earth-mass planets for various ICs and different system parameters, as given in Table 1 (earlier data) and Table 2 (data by Bourrier et al. (2018)), are given in Table 3. These simulations pertain to circular orbits, i.e., $e_{0}=0.0$. Here $a_{0}$ is set between 1.50 au and 1.60 au with a step size of 0.01 au. For each combination of $a_{0}$ and $e_{0}$, the maximum eccentricity is recorded for the total time of simulation. The results show that Earth-mass planets with $a_{0}$ between 1.50 au and 1.60 au based on the data of Table 1 did not survive (except for $a_{0}=1.60$ au). However, when the updated data by Bourrier et al. (2018) are used,
the Earth-mass planets survived for (at least) a 50 Myr timespan for all ICs. The outcomes for the test planets, including their survival times, have directly been taken from the mercury output files. This latter result is attributable to the updated value of the eccentricity for Cnc 55-f, now identified as close to circular.
4 Comments on Previous Studies
There had been previous studies on the possibility of Earth-mass planets around 55 Cnc, especially planets located in 55 Cnc’s HZ, including work based on detailed orbital stability simulations. For example, Raymond et al. (2008) identified a large stable zone extending from 0.9 to 3.8 au at planetary eccentricities below 0.4. This zone of stability encompasses a large fraction of 55 Cnc’s OHZ, determined to extend from $\sim$0.59 to $\sim$1.43 au (see Sect. 3). Raymond et al. (2008) argued that the zone of stability may in fact contain 2 to 3 (albeit unobserved) additional planets. In fact, based on the system data given by Bourrier et al. (2018), we are largely able to confirm the previous results by Raymond et al. (2008).
This statement is based on a set of orbital integrations for Earth-mass test planets where we studied the time series evolution of their semi-major axis and eccentricity.
However, there is a bigger picture here. As it turns out, the eccentricity of 55 Cnc-f is of pivotal importance
for the outcome of the simulations. Previously, based on work by Endl et al. (2012) and Ligi et al. (2016), that eccentricity
was determined as 0.32. With this value of $e_{\rm pl}$ the planet’s apocenter is at 1.03 au, thus located
within the previously identified domain for stability for possibly habitable terrestrial planets. Hence, the
stability regime is pushed considerably further outward (i.e., as far as $\sim$2.2 au), which is beyond the
outer limit of the stellar HZ. Thus, it is found that possible Earth-mass planets originally placed between
1.0 au and 4.0 au (a region also encompassing most of 55 Cnc’s HZ) are unable to survive their total simulation
time of 10 Myr, a result impressively demonstrated in phase space and through eccentricity analysis.
The recent study by Bourrier et al. (2018) yields 0.08 as eccentricity for 55 Cnc-f, and possible Earth-mass planets
in the system’s inner region are found to be orbitally stable for simulation times of 50 Myr. Although it is beyond
the scope of this study to decide on the most realistic observational data for the planetary system of 55 Cnc,
the close-to-circular value for 55 Cnc-f’s eccentricity appears to be most plausible. Previously, it has been
shown that planetary eccentricities derived from radial velocity measurements can be seriously overestimated
(Shen & Turner, 2008; Zakamska et al., 2011). Additionally, both 55 Cnc-e (and 55 Cnc-f) are very unlikely to have high eccentricities
because the eccentricities of close-in planets are expected to be damped due to tidal dissipation (Bolmont et al., 2013).
The recent work by Bourrier et al. (2018) is highly comprehensive as it studies both the thermal and orbital properties
of the system planets in great detail; it also considers the impact of the stellar magnetic cycle on the results.
In a separate approach Cuntz (2012) argued in favor of the existence of a possibly habitable planet at $\sim$1.5 au. This opinion was based on an application of the Titius-Bode rule (TBR) to the system. Despite some positive outcomes regarding the TBR, including work by Bovaird & Lineweaver (2013) who applied a generalized TBR to numerous exoplanetary systems (including those identified by Kepler), the applicability of the TBR continues to remain controversial555Previously, the nature of the TBR has been associated with planetary orbital stability, planet formation scenarios, or sheer numerology; see, e.g., Lynch (2003) and Cuntz (2012) for references and background information. For example, Hayes & Tremaine (1998) concluded that
“[…] the significance of [the TBR] is simply that stable planetary systems tend to be regularly spaced […]”.. In the view of our study,
a planet-as-conjectured at $\sim$1.5 au is identified to be possible in consideration of the relatively small
orbital eccentricity of 55 Cnc-f.
5 Summary and Conclusions
The aim of this study is to explore the possible existence of Earth-mass planets
around 55 Cnc, an orange dwarf of spectral type G8 V, with an age somewhat larger
than the Sun. Our main focus concerns the 55 Cnc’s HZ, which assuming optimistic limits
extends between approximately 0.59 au to and 1.43 au. 55 Cnc is homestead of (at least) five planets
with masses ranging between super-Earth and Jupiter-type. The planet closest to the stellar HZ
is 55 Cnc-f, with a semi-major axis of 0.77 au. Furthermore, based on observational constraints,
there is a gap without planets between $\sim$0.8 au and $\sim$5.7 au. Previously, it has been argued,
see, e.g., Raymond et al. (2008), that terrestrial planets may be able to exist within that gap,
including planets located in 55 Cnc’s HZ.
Our study is based on detailed orbital stability simulations for hypothetical Earth-mass planets assuming
integration times of up to 50 Myr. These planets are originally set between 0.0 au and 6.0 au.
We explore detailed phase-space maps and detailed time series analyses with focus on the evolution of
the eccentricity of the hypothetical Earth-mass planets. As expected, those planets interact
gravitationally with the five system planets as well as the center star. It is found that
the possibility for the existence of Earth-mass planets in the system, especially its HZ, strongly
depends on the adopted system parameters, notably the eccentricity of 55 Cnc-f. Note that previously
both a high value ($e\sim 0.32$) and a low value ($e\sim 0.08$) have previously been deduced.
If the low value is adopted, the more plausible and most recent value, as given by Bourrier et al. (2018),
Earth-mass planets would be able to exist in the gap between 1.0 au and 2.0 au, thus implying the
possibility of habitable system planets. We pursued test simulations for various parts of
the stellar HZ to verify this result. If the high value for 55 Cnc-f’s eccentricity is adopted,
typically, the Earth-mass test planets are found to collide with the star or the system planets,
or being ejected from the system. This result is particularly evident for the region beyond 1.50 au,
which is most affected by 55 Cnc-f.
In conclusion, 55 Cnc should continue to be considered a favorable target for future habitable planet search missions.
Dynamical studies by Satyal et al. (2017) for other systems have shown that Earth-mass planets can remain in stable
orbits for at least 1 Gyr, with a very small $a_{\rm pl}$ and $e_{\rm pl}$ deviations from their initial values.
Regarding the GJ 832 system, the planetary stability region was prominent and extended to a larger phase space,
i.e., up to 0.5 au, between the inner and outer planets (Wittenmyer et al., 2014). Moreover, Hinse et al. (2015) predicted
a possible third planet in the Kepler-47 circumbinary system by using the MEGNO based on long-term stability
simulations, which allowed identifying quasi-periodic planetary orbits. In case of 55 Cnc, Earth-mass planets
in stable orbits are found to be possible in its HZ (and in various regions outside of that domain as well),
which is consistent with findings by Raymond et al. (2008) and others.
{ack}
This work has been supported by the Department of Physics, University
of Texas at Arlington. We wish to thank the anonymous referee for their
comments which helped to improve the quality of our manuscript.
References
Bolmont et al. (2013)
Bolmont, E., Selsis, F., Raymond, S. N., et al. 2013, \aap, 556, A17
Borucki et al. (2013)
Borucki, W. J., Agol, E., Fressin, F., et al. 2013, Science, 340, 587
Bourrier et al. (2018)
Bourrier, V., Dumusque, X., Dorn, C., Henry, G. W., et al. 2018, A&A, 619, A1
Bovaird & Lineweaver (2013)
Bovaird, T., & Lineweaver, C. H. 2013, \mnras, 435, 1126
Chambers (1999)
Chambers, J. E. 1999, MNRAS, 304, 793
Chambers (1997)
Chambers, J. E., & Migliorini, F. 1997, AAS/DPS Abstracts, \baas, 29, 1024
Cuntz (2012)
Cuntz, M. 2012, \pasj, 64, 73
Cuntz (2014)
Cuntz, M. 2014, \apj, 780, 14
Cuntz (2015)
Cuntz, M. 2015, \apj, 798, 101
Cuntz & Guinan (2016)
Cuntz, M., & Guinan, E. F. 2016, \apj, 827, 79
Dawson et al. (2008)
Dawson, R. I., & Fabrycky, D. C. 2008, \apj, 722, 937
Duquennoy & Mayor (1991)
Duquennoy, A., & Mayor, M. 1991, \aap, 248, 485
Eggenberger et al. (2004)
Eggenberger, A., Udry, S., & Mayor, M. 2004, \aap, 417, 353
Eggl et al. (2013)
Eggl, S., Haghighipour, N., & Pilat-Lohinger, E. 2013, \apj, 764, 130
Endl et al. (2012)
Endl, M., Robertson, P., Cochran, W. D., et al. 2012, \apj, 759, 19
Fischer & Valenti (2005)
Fischer, D. A., & Valenti, J. 2005, \apj, 622, 1102
Fischer et al. (2008)
Fischer, D. A., Marcy, G. W., Butler, R. P., et al. 2008, \apj, 675, 790
Gillon et al. (2016)
Gillon, M., Jehin, E., Lederer, S. M., et al. 2016, \nat, 533, 221
Gonzalez (1998)
Gonzalez, G. 1998, \aap, 334, 221
Hayes & Tremaine (1998)
Hayes, W., & Tremaine, S. 1998, Icarus, 135, 549
Heller & Armstrong (2014)
Heller, R., & Armstrong, J. 2014, Astrobiology, 14, 50
Hinse et al. (2015)
Hinse, T. C., Haghighipour, N., Kostov, V. B., & Goździewski, K. 2015 \apj, 799, 88
Kane et al. (2016)
Kane, S. R., Hill, M. L., Kasting, J. F., et al. 2016, \apj, 830, 1
Kasting et al. (1993)
Kasting, J. F., Whitmore, D. P., & Reynolds, R. T. 1993, Icarus, 101, 108
Kasting et al. (2015)
Kasting, J. F., Chen, H., & Kopparapu, R. K. 2015, \apjl, 813, L3
Kopparapu et al. (2013)
Kopparapu, R. K., Ramirez, R., Kasting, J. F., et al. 2013,
\apj, 765, 131; Erratum 770, 82
Kopparapu et al. (2014)
Kopparapu, R. K., Ramirez, R. M., SchottelKotte, J., Kasting, J. F.,
Domagal-Goldman, S., & Eymet, V. 2014, \apj, 787, L29
Lammer et al. (2009)
Lammer, H., Bredehöft, J. H., Coustenis, A., et al. 2009,
A&A Rev. 17, 181
Ligi et al. (2016)
Ligi, R., Creevey, O., Mourard, D., et al. 2016, A&A, 586, A94
Lynch (2003)
Lynch, P. 2003, \mnras, 341, 1174
Mamajek & Hillenbrand (2008)
Mamajek, E. E., & Hillenbrand, L. A. 2008, \apj, 687, 1264
Mulders et al. (2018)
Mulders, G. J., Pascucci, I., Apai, D., & Ciesla, F. J. 2018, \aj, 156, 24
Ogihara et al. (2018)
Ogihara, M., Kokubo, E., Suzuki, T. K., & Morbidelli, A. 2018, \aap, A63
Ramirez & Kaltenegger (2018)
Ramirez, R. M., & Kaltenegger, L. 2018, \apj, 858, 72
Raymond et al. (2008)
Raymond, S. N., Barnes, R., & Gorelick, N. 2008, \apj, 689, 478
Rivera & Haghighipour (2007)
Rivera, E., & Haghighipour, N. 2007, \mnras, 374, 599
Satyal et al. (2017)
Satyal, S., Griffith, J., & Musielak, Z. E. 2017, \apj, 845,
Shen & Turner (2008)
Shen, Y., & Turner, E. L. 2008, \apj, 685, 553
Smith & Lissauer (2009)
Smith, A. W., & Lissauer, J. J. 2009, Icarus, 201, 381
von Bloh et al. (2003)
von Bloh, W., Cuntz, M., Franck, S., & Bounama, C. 2003, Astrobiology, 3, 681
Winn et al. (2011)
Winn, J. N., Matthews, J. M., Dawson, R. I., et al. 2011, \apj, 737, L18
Wittenmyer et al. (2014)
Wittenmyer, R. A., Tuomi, M., Butler, R. P., et al. 2014, \apj, 791, 114
Zakamska et al. (2011)
Zakamska, N. L., Pan, M., & Ford, E. B. 2011, \mnras, 410, 1895
=========================================================== |
Testing models of triggered star formation: theory and observation
Thomas J. Haworth
Thomas Haworth University of Exeter, 22email: haworth@astro.ex.ac.uk
Tim J. Harries and David M. Acreman
Thomas Haworth University of Exeter, 22email: haworth@astro.ex.ac.uk
Abstract
One of the main reasons that triggered star formation is contentious is the failure to accurately link the observations with models in a detailed, quantitative, way. It is therefore critical to continuously test and improve the model details and methods with which comparisons to observations are made. We use a Monte Carlo radiation transport and hydrodynamics code torus to show that the diffuse radiation field has a significant impact on the outcome of radiatively driven implosion (RDI) models. We also calculate SEDs and synthetic images from the models to test observational diagnostics that are used to determine bright rimmed cloud conditions and search for signs of RDI.
We have investigated the impact of polychromatic and diffuse field radiation on radiatively driven implosion (RDI) models using the Monte Carlo radiation transport and hydrodynamics code torus H00 ; 2012MNRAS.420..562H . The details of the code implementation, model parameters and results are given in 2012MNRAS.420..562H . We ran three types of RDI calculation. One with a monochromatic radiation field, one with a polychromatic radiation field and one that is both polychromatic and includes the diffuse radiation field. The addition of polychromatic radiation to the calculation does not significantly alter the outcome of the model. However, including the diffuse field can lead to significantly different evolution of the cloud, altering the morphology and increasing the maximum accumulated density after 200 kyr up to about a factor of 10.
Using these RDI models from 2012MNRAS.420..562H we calculated synthetic images and SEDs to test observational diagnostics of bright rimmed clouds (BRCs) in HHD12 . We calculated the neutral cloud properties by fitting the cloud SED as a greybody to determine the dust temperature, which can then be used to calculate the cloud mass following H83 . The temperature and electron density in the ionized boundary layer (IBL) and the cloud mass loss rate were calculated using simulated VLA 20 cm continuum images, an example of which is given in Figure 1, and the standard techniques of LL94 ; LL97 . We have also tested the use of forbidden line ratios from long slit spectroscopy to determine the IBL conditions and found that they are a viable tool, giving a direct and more accurate measure of the IBL temperatures compared to the radio method which assumes a canonical value of 10${}^{4}$ K.
Using the inferred cloud and IBL conditions we calculated the cloud support and IBL pressures to determine whether or not the clouds are being compressed. We find that this pressure comparison diagnostic is a reasonable indicator of whether or not the IBL is driving into the cloud. The accuracy of the techniques was investigated by comparing the derived conditions and behaviours with those known from the model grid. For example, we have demonstrated that as the beam size increases the IBL conditions are increasingly underestimated in the radio diagnostic because the IBL flux is contaminated by the neutral cloud and HII region. We also found that the contribution to the SED from warm dust causes a slight overestimation of the dominant cloud temperature by $1-2$ K, which leads to an overestimation of the mass by up to a factor of 35%. Furthermore, use of a constant mass conversion factor $C_{\nu}$ in the mass calculations of H83 for BRCs of different class is found to introduce errors up to a factor 3.6. This comparison of the known conditions in simulations with those inferred through observational diagnostics of synthetic data means that more reliable conclusions can be drawn from studies of real BRCs.
Bibliography
(1)
T.J. Harries, MNRAS 315, 722 (2000).
DOI 10.1046/j.1365-8711.2000.03505.x
(2)
T.J. Haworth, T.J. Harries, MNRAS420, 562 (2012).
DOI 10.1111/j.1365-2966.2011.20062.x
(3)
T.J. Haworth, T.J. Harries, D.M. Acreman, ArXiv e-prints:1205.6993
(2012)
(4)
R.H. Hildebrand, QJRAS 24, 267 (1983)
(5)
B. Lefloch, B. Lazareff, A&A 289, 559 (1994)
(6)
B. Lefloch, B. Lazareff, A. Castets, A&A 324, 249 (1997) |
Two Globally Convergent Adaptive Speed Observers for Mechanical Systems
[
Jose.Romero-Velazquez@lirmm.fr
[
ortega@lss.supelec.fr
Laboratoire d’Informatique, de Robotique et de Microélectronique de Montpellier, 161 Rue Ada, 34090 Montpellier, France
Laboratoire des Signaux et Systémes, Supélec, Plateau du Moulon, 91192 Gif-sur-Yvette, France
Abstract
A globally exponentially stable speed observer for mechanical systems was recently reported in the literature, under the assumptions of known (or no) Coulomb friction and no disturbances. In this note we
propose and adaptive version of this observer, which is robust vis–à–vis constant disturbances. Moreover, we propose a new globally convergent speed observer that, besides rejecting the disturbances,
estimates some unknown friction coefficients for a class of mechanical systems that contains several practical examples.
M
††thanks: [
footnoteinfo]Corresponding author J.G. Romero Tel. +33 4 67 14 95 68 .
LIRMM]Jose Guadalupe Romero* , LSS]Romeo Ortega,
echanical systems, adaptive observers, robustness.
1 Introduction
The design of speed observers for mechanical systems is a problem of great practical importance that has attracted the attention of researchers for over $25$ years—the reader is referred to the recent books [1, 3] for an exhaustive list of references. The first globally exponentially convergent speed observer for general, simple mechanical systems was recently reported in [2], where the Immersion and Invariance (I&I) techniques developed in [1] were used. Although the observer of [2] considers the case of systems with non–holonomic constraints, it relies on the assumptions of no friction and the absence of disturbances. In [13] this speed observer was redesigned to accommodate the presence of known Coulomb friction and it was used to design a uniformly globally exponentially stable tracking controller using only position feedback for mechanical systems (without non–holonomic constraints).
In this paper we propose two new robust velocity observers for mechanical systems (without non–holonomic constraints). First, we add an adaptation stage to the observer of [2] to reject constant input disturbances. Second, for mechanical systems with zero Riemann symbols (ZRS) [11, 14, 16], we propose a new adaptive speed observer that, besides rejecting the disturbances, estimates some of the unknown friction coefficients. It should be noted that in this case there are products of unmeasurable states and unknown parameters, a situation for which very few results are available in the observer design literature—even for the case of linear systems. A similar transformation has been presented in [4, 5], where a coordinate transformation that removes the quadratic terms in velocity is found to solve challenging position–feedback tracking problems for surface ships and mobile robots.
The paper is organized as follows. The two adaptive observation problems addressed in the paper are presented in Section 2. The standing assumptions and a preliminary lemma are given in Section 3. In Section 4 we present an adaptive observer for systems with unknown friction and disturbances. In Section 5 the I&I observer of [13] is redesigned to accommodate the possible presence of disturbances and known friction. Some physical examples are given in Section 6. The paper is wrapped–up with some future research in Section 7.
Notation. To avoid cluttering the notation, throughout the paper $\kappa$ and $\alpha$ are generic positive constants. $I_{n}$ is the $n\times n$ identity matrix and $0_{n\times s}$ is an
$n\times s$ matrix of zeros, $0_{n}$ is an $n$–dimensional column vector of zeros. Given $a_{i}\in\mathbb{R},\;i\in\bar{n}:=\{1,\dots,n\}$, we denote with $\mbox{col}(a_{i})$ the $n$–dimensional column vector with
elements $a_{i}$. For any matrix $A\in\mathbb{R}^{n\times n}$, $(A)_{i}\in\mathbb{R}^{n}$ denotes the $i$–th column, $(A)^{i}$ the $i$–th row and $(A)_{ij}$ the $ij$–th element. That is, with $e_{i}\in\mathbb{R}^{n},\;i\in\bar{n}$, the Euclidean basis vectors, $(A)_{i}:=Ae_{i}$, $(A)^{i}:=e_{i}^{\top}A$ and $(A)_{ij}:=e_{i}^{\top}Ae_{j}$. For $x\in\mathbb{R}^{n}$, $S\in\mathbb{R}^{n\times n}$, $S=S^{\top}>0$, we denote the Euclidean norm $|x|^{2}:=x^{\top}x$, and the weighted–norm $\|x\|^{2}_{S}:=x^{\top}Sx$. Given a function $f:\mathbb{R}^{n}\to\mathbb{R}$ we define the differential operators
$$\nabla f:=\left(\frac{\displaystyle\partial f}{\displaystyle\partial x}\right)%
^{\top},\;\nabla_{x_{i}}f:=\left(\frac{\displaystyle\partial f}{\displaystyle%
\partial x_{i}}\right)^{\top},$$
where $x_{i}\in\mathbb{R}^{p}$ is an element of the vector $x$. For a mapping $g:\mathbb{R}^{n}\to\mathbb{R}^{m}$, its Jacobian matrix is defined as
$$\nabla g:=\left[\begin{array}[]{cc}(\nabla g_{1})^{\top}\\
\vdots\\
(\nabla g_{m})^{\top}\end{array}\right],$$
where $g_{i}:\mathbb{R}^{n}\to\mathbb{R}$ is the $i$-th element of $g$.
2 FORMULATION OF TWO ROBUST SPEED OBSERVATION PROBLEMS
In the paper we consider $n$–degrees of freedom, perturbed, simple, mechanical systems described in port–Hamiltonian (pH) form [15] by
$$\left[\begin{array}[]{c}\dot{q}\\
\dot{\mathbf{p}}\end{array}\right]=\left[\begin{array}[]{cc}0&I_{n}\\
-I_{n}&-\mathbf{\mathfrak{R}}\end{array}\right]\nabla{H(q,\mathbf{p})}+\left[%
\begin{array}[]{c}0\\
G(q)\end{array}\right]u+\left[\begin{array}[]{c}0\\
d\end{array}\right]$$
(1)
with total energy function $H:\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}$
$$H(q,\mathbf{p})=\frac{1}{2}\mathbf{p}^{\top}M^{-1}(q)\mathbf{p}+V(q),$$
(2)
where $q,\mathbf{p}\in\mathbb{R}^{n}$ are the generalized positions and momenta, respectively, $u\in\mathbb{R}^{m}$ is the control input, $G:\mathbb{R}^{n}\to\mathbb{R}^{n\times m}$ is the input matrix, the inertia
matrix $M:\mathbb{R}^{n}\to\mathbb{R}^{n\times n}$ verifies $M(q)=M^{\top}(q)>0$ and $V:\mathbb{R}^{n}\to\mathbb{R}$ is the potential energy function. As customary in the observer literature, it is assumed that
the control signal $u(t)$ is such that trajectories exist for all $t\geq 0$.
The system is subject to two different perturbations.
-
Unknown constant disturbances $d=\mbox{col}(d_{i})\in\mathbb{R}^{n}$.
-
Coulomb friction captured by
$$\mathbf{\mathfrak{R}}=\mbox{diag}\{r_{1},r_{2},.,r_{n}\}\in\mathbb{R}^{n\times
n},$$
(3)
with unknown $r_{i}\geq 0,\;i\in\bar{n}$.
The problem is to design a globally convergent robust adaptive observer for the momenta $\mathbf{p}$. The main contributions of the paper are the following.
(i)
For systems with ZRS design a new observer that is globally convergent in spite of the presence of the disturbance $d$ and some unknown friction coefficients $r_{i}$.
(ii)
If the friction is known robustify the observer of [2] to reject the disturbance $d$.
Remark 2.1.
The qualifier “some" in item (i) is essential because, as will become clear later, except for the case of constant inertia matrix, we will not be able to consider the presence of unknown friction in all generalized
coordinates.
Remark 2.2.
Notice that the objective is to observe only the momenta (equivalently, the velocity) not to ensure consistent estimation of the parameters $d$ and $r:=\mbox{col}(r_{i})$. As is well–known in the identification literature, a
necessary condition for parameter convergence is that the signals satisfy a persistency of excitation condition [10]. In this respect, we notice that the system (1), (2), which can be
represented in the state space form
$$\displaystyle\dot{q}$$
$$\displaystyle=v$$
$$\displaystyle\dot{v}$$
$$\displaystyle=U(q,v)-\mathbf{\mathfrak{R}}v+G(q)u+d$$
$$\displaystyle\dot{r}$$
$$\displaystyle=0$$
$$\displaystyle\dot{d}$$
$$\displaystyle=0,$$
for some $U:\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}^{n}$, with measurement $q$ does not satisfy the observability rank condition [7] at zero velocity, hampering the observation of the parameters $r$.111The authors thank Prof.
Witold Respondek for this insightful remark.
Remark 2.3.
See Remark 1 in [12] for a physical interpretation of the disturbances $d$ that, we underscore, enter at the level of the momenta.
3 A Suitable pH Representation
As shown in [16], the change of coordinates
$$(q,p)\mapsto(q,T^{\top}(q)\mathbf{p}),\hskip 8.535827pt$$
where $T:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n\times n}$ is a full rank factorization of the inverse inertia matrix, that is,
$$M^{-1}(q)=T(q)T^{\top}(q),$$
(4)
transforms (1) into
$$\displaystyle\left[\begin{array}[]{c}\dot{q}\\
\dot{p}\end{array}\right]$$
$$\displaystyle=\left[\begin{array}[]{cc}0&T(q)\\
-T^{\top}(q)&J(q,p)-R(q)\end{array}\right]\nabla W(q,p)$$
$$\displaystyle+\left[\begin{array}[]{cc}0\\
T^{\top}(q)G(q)\end{array}\right]u+\left[\begin{array}[]{cc}0\\
T^{\top}(q)d\end{array}\right],$$
(5)
with new Hamiltonian $W:\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}$
$$W(q,p)=\frac{1}{2}|p|^{2}+V(q),$$
the $jk$ element of the skew–symmetric matrix $J:\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}^{n\times n}$ given by
$${J}_{jk}(q,p)=-p^{\top}[(T)_{j},(T)_{k}],$$
(6)
with $[\cdot,\cdot]$ the standard Lie bracket [14] and the transformed friction matrix
$$R(q):=T^{\top}(q)\mathbf{\mathfrak{R}}T(q)\geq 0.$$
(7)
Remark 3.1.
One possible choice of the factorization (4) is the Cholesky factorization [6, 8]. But, as will become clear below, other choices may prove more suitable for the solution of the problem.
4 ROBUST OBSERVER FOR A CLASS SYSTEMS WITH FRICTION AND DISTURBANCES
In this section we solve the problem of robust observation of momenta in the presence of disturbances and some unknown friction coefficients for a class of mechanical systems.
4.1 Assumption on $M(q)$ and a preliminary lemma
Assumption 4.1.
$M^{-1}(q)$ admits a factorization (4) with a factor $T(q)$ verifying
$$\Big{[}\Big{(}T(q)\Big{)}_{i},\Big{(}T(q)\Big{)}_{j}\Big{]}=0,\;i,j\in\bar{n}.$$
(8)
Instrumental for the developments of this paper is the following result, whose proof may be found in [16].
Lemma 4.1.
The following statements are equivalent:
(i)
$M(q)$ satisfies Assumption 4.1.
(ii)
The Riemann symbols of $M(q)$ are all zero.222See equations (6) and (7) of [16] for the definition of these symbols.
(iii)
There exists a mapping $Q:\mathbb{R}^{n}\to\mathbb{R}^{n}$ such that
$$\nabla Q(q)=T^{-1}(q).$$
(9)
Remark 4.1.
Mechanical systems verifying Assumption 4.1 have been extensively studied in analytical mechanics and have a deep geometric significance—stemming from Theorem 2.36 in [11]. They belong to the
class of systems that are partially linearizing via change of coordinates studied in [16]—see that paper for some additional references.
4.2 Assumptions on friction
To design our robust adaptive observer, besides Assumption 4.1, a restriction on the friction coefficients is imposed. Namely, we assume that there are $s$, with $s\leq n$, unknown coefficient and decompose the friction matrix $\mathbf{\mathfrak{R}}$ (3) as
$$\mathbf{\mathfrak{R}}=\mathbf{\mathfrak{R}_{k}}+\mathbf{\mathfrak{R}_{u}}$$
where $\mathbf{\mathfrak{R}_{k}},\mathbf{\mathfrak{R}_{u}}$ are $n\times n$ diagonal matrices containing the known and the unknown friction coefficients respectively. As a working example consider the case $n=3$ and $s=2$ with
$$\mathbf{\mathfrak{R}_{k}}=\mbox{diag}\{0,r_{2},0\},\;\mathbf{\mathfrak{R}_{u}}%
=\mbox{diag}\{r_{1},0,r_{3}\}$$
Similarly, with an obvious definition, we decompose the transformed friction matrix (7) into
$$R(q)=R_{k}(q)+R_{u}(q).$$
To streamline the presentation all friction coefficients are grouped in a vector $r=\mbox{col}(r_{i})\in\mathbb{R}^{n}$ with the unknown and known coefficients in vectors $r_{u}\in\mathbb{R}^{s}$ and $r_{k}\in\mathbb{R}^{n-s}$, respectively. Thus, for our working example we have
$$r=\mbox{col}(r_{1},r_{2},r_{3}),\;r_{k}=r_{2},\;r_{u}=\mbox{col}(r_{1},r_{3}).$$
We also define a set of integers $\kappa\subset\bar{n}$ that contains the indices of the unknown coefficients of $r$, which in the example is $\kappa=\{1,3\}$.
Finally, we define a matrix $C\in\mathbb{R}^{n\times s}$ such that
$$C^{\top}r=r_{u}.$$
(10)
Clearly, the matrix $C$ verifies:
•
$\mbox{rank}\{C\}=s$.
•
For $j\in\kappa$, $(C)_{j}=e_{\kappa_{j}}$ .
In our example
$$C=\left[\begin{array}[]{ccc}1&0\\
0&0\\
0&1\end{array}\right].$$
The following assumption and lemma are instrumental for our future developments.
Assumption 4.2.
The $i$–th row of factor $T(q)$ is independent of $q$ for $i\in\kappa$.333Consequently, the unknown friction coefficients are located in these rows.
Lemma 4.2.
Under Assumption 4.2, there exists constant matrices $Y_{j}\in\mathbb{R}^{n\times s},\;j\in\bar{n},$ such that, for all vectors $z=\mbox{col}(z_{i})\in\mathbb{R}^{n}$ we have
$$R_{u}(q)z=(\sum_{j=1}^{n}Y_{j}z_{j})r_{u}.$$
(11)
Proof 4.1.
From (10) it follows that
$$\mathfrak{R}_{u}=\sum_{i=1}^{n}e_{i}e_{i}^{\top}(e_{i}^{\top}Cr_{u}).$$
(12)
Using the definition of $R_{u}(q)$ we get
$$\displaystyle R_{u}(q)z$$
$$\displaystyle=$$
$$\displaystyle T^{\top}(q)\left[\sum_{i=1}^{n}e_{i}e_{i}^{\top}(e_{i}^{\top}Cr_%
{u})\right]T(q)z,$$
(13)
$$\displaystyle=$$
$$\displaystyle\sum_{i=1}^{n}T^{\top}(q)e_{i}e_{i}^{\top}T(q)ze_{i}^{\top}Cr_{u}$$
$$\displaystyle=$$
$$\displaystyle\sum_{i=1}^{n}\left[\sum_{j=1}^{n}T^{\top}(q)e_{i}e_{i}^{\top}T(q%
)\right]e_{j}z_{j}e_{i}^{\top}Cr_{u}.$$
Hence, (11) follows swapping the sums and defining
$$\displaystyle Y_{j}$$
$$\displaystyle:=$$
$$\displaystyle\sum_{i=1}^{n}T^{\top}(q)e_{i}e_{i}^{\top}T(q)e_{j}e_{i}^{\top}C$$
(14)
$$\displaystyle=$$
$$\displaystyle\sum L_{i}e_{j}e_{i}^{\top}C,\;j\in\bar{n}$$
with matrices $L_{i}$ defined as
$$L_{i}:=T^{\top}(q)e_{i}e_{i}^{\top}T(q)\hskip 2.845276pt,\;i\in\bar{n}$$
(15)
It only remains to prove that the matrices $Y_{j}$ and $L_{i}$ are constant. Towards this end we refer to (14) and notice that, in view of Assumption 4.2, the term $e_{i}^{\top}T(q)$ is constant for $i\in\kappa$ while the term $e_{i}^{\top}C$ is an $1\times s$ zero vector for $i\notin\kappa$. Completing the proof.
Remark 4.2.
As indicated in Remark 2.1, except for the case when $M$ (and, consequently, the factor $T$) are constant, to satisfy Assumption 4.2 we have to assume that some of the elements of $r$ are known. See Section 6 for some physical examples.
4.3 First robust momenta observer
{prop}
Consider the system (1), (2) where the inertia matrix $M(q)$ and the friction matrix $\mathbf{\mathfrak{R}}$ verify Assumptions 4.1 and 4.2. The $2n+s$ dimensional I$\&$I adaptive momenta observer
$$\displaystyle\dot{p}_{I}$$
$$\displaystyle=-T^{\top}(q)[\nabla V-G(q)u-\hat{d}]$$
$$\displaystyle-(\sum_{i=1}^{n}Y_{i}\hat{p}_{i})\hat{r}_{u}-[\lambda Q(q)+R_{k}(%
q)]\hat{p}$$
$$\displaystyle\dot{r}_{u_{I}}$$
$$\displaystyle=(\sum_{i=1}^{n}Y^{\top}_{i}\hat{p}_{i})(\dot{p}_{I}+\lambda\hat{%
p})$$
$$\displaystyle\dot{d}_{I}$$
$$\displaystyle=T(q)\hat{p}$$
$$\displaystyle\hat{p}$$
$$\displaystyle=p_{I}+{\lambda}Q(q)$$
$$\displaystyle\hat{r}_{u}$$
$$\displaystyle=r_{u_{I}}+{1\over 2\lambda}(\sum_{i=1}^{s}\hat{p}^{\top}L_{i}%
\hat{p})e_{i}$$
$$\displaystyle\hat{d}$$
$$\displaystyle=d_{I}+q$$
$$\displaystyle\hat{\mathbf{p}}$$
$$\displaystyle=T^{-\top}(q)\hat{p}$$
with the constant $n\times n$ matrices $L_{i}$ given by (15), $Q(q)$ given in (9), $Y_{i}\in\mathbb{R}^{n\times s},\;i\in\bar{n}$ given in (14) and $\lambda>0$ a free parameter, ensures boundedness of all signals and
$$\lim_{t\rightarrow\infty}[\hat{\mathbf{p}}(t)-\mathbf{p}(t)]=0.$$
(16)
for all initial conditions $(q(0),\mathbf{p}(0))\in\mathbb{R}^{n}\times\mathbb{R}^{n}.$
Proof 4.2.
Let the observation and parameter estimation errors be defined as
$$\displaystyle\tilde{p}$$
$$\displaystyle=\hat{p}-p$$
$$\displaystyle\tilde{r}_{u}$$
$$\displaystyle=\hat{r}_{u}-r_{u}$$
(17)
$$\displaystyle\tilde{d}$$
$$\displaystyle=\hat{d}-d.$$
(18)
Following the I&I adaptive observer procedure [1] we propose to generate the estimates as the sum of a proportional and an integral term, that is,
$$\displaystyle\hat{p}$$
$$\displaystyle=p_{I}+p_{P}(q)$$
$$\displaystyle\hat{r}_{u}$$
$$\displaystyle=r_{u_{I}}+r_{u_{P}}(\hat{p})$$
(19)
$$\displaystyle\hat{d}$$
$$\displaystyle=d_{I}+d_{P}(q),$$
(20)
where the mappings
$$\displaystyle p_{P}$$
$$\displaystyle:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$$
$$\displaystyle r_{u_{P}}$$
$$\displaystyle:\mathbb{R}^{n}\rightarrow\mathbb{R}^{s}$$
$$\displaystyle d_{P}$$
$$\displaystyle:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n},$$
and the observer states $p_{I},d_{I}\in\mathbb{R}^{n}$ and $r_{u_{I}}\in\mathbb{R}^{s}$ will be defined below.444The reason for the particular selection of the arguments of the proportional terms will become clear below.
First, we study the dynamic behavior of $\tilde{p}$ and compute
$$\displaystyle\dot{\tilde{p}}$$
$$\displaystyle=\dot{p}_{I}+\nabla{p_{P}}T(q)p+T^{\top}(q)[\nabla V-G(q)u]$$
$$\displaystyle+[R_{k}(q)+R_{u}(q)](\hat{p}-\tilde{p})-T^{\top}(q)(\hat{d}-%
\tilde{d}),$$
where we have invoked Assumption 4.1 that ensures—via (6) and (8)—that $J(q,p)=0$, and used (4.2) to obtain the terms in the second row. Invoking (11) we can write
$$R_{u}(q)\hat{p}=(\sum_{i=1}^{n}Y_{i}\hat{p}_{i})r_{u}.$$
Hence, proposing
$$\displaystyle\dot{p}_{I}$$
$$\displaystyle=-\nabla{p_{P}}{T}(q)\hat{p}-T^{\top}(q)[\nabla V-G(q)u]$$
$$\displaystyle-\Big{(}\sum_{i=1}^{n}Y_{i}\hat{p}_{i}\Big{)}\hat{r}_{u}-R_{k}(q)%
\hat{p}+T^{\top}(q)\hat{d},$$
(21)
yields
$$\displaystyle\dot{\tilde{p}}$$
$$\displaystyle=-[R(q)+\nabla_{q}p_{P}T(q)]\tilde{p}-(\sum_{i=1}^{n}Y_{i}\hat{p}%
_{i})\tilde{r}_{u}+T^{\top}(q)\tilde{d}$$
$$\displaystyle=-[R(q)+\lambda I_{n}]\tilde{p}-(\sum_{i=1}^{n}Y_{i}\hat{p}_{i})%
\tilde{r}_{u}+T^{\top}(q)\tilde{d},$$
(22)
where to obtain the second equations we have selected
$$p_{P}(q)=\lambda Q(q),$$
with $Q(q)$ given in (9).
Now, the time derivative of $\tilde{r}_{u}$ is given as
$$\displaystyle\dot{\tilde{r}}_{u}$$
$$\displaystyle=\dot{r}_{u_{I}}+\nabla r_{u_{P}}\dot{\hat{p}}$$
$$\displaystyle=\dot{r}_{u_{I}}+\nabla r_{u_{P}}[\dot{p}_{I}+\nabla p_{P}T(q)p]$$
$$\displaystyle=\dot{r}_{u_{I}}+\nabla r_{u_{P}}[\dot{p}_{I}+\lambda(\hat{p}-%
\tilde{p})].$$
Hence, choosing
$$\dot{r}_{u_{I}}=-\nabla r_{u_{P}}(\dot{p}_{I}+\lambda\hat{p}),$$
yields
$$\dot{\tilde{r}}_{u}=-\lambda\nabla r_{u_{P}}\tilde{p}.$$
(23)
Finally, the time derivative of $\tilde{d}$ is given as
$$\displaystyle\dot{\tilde{d}}$$
$$\displaystyle=\dot{d}_{I}+\nabla d_{P}T(q)p$$
$$\displaystyle=\dot{d}_{I}+\nabla d_{P}T(q)(\hat{p}-\tilde{p}).$$
Hence, choosing
$$\dot{d}_{I}=-\nabla d_{P}T(q)\hat{p},$$
yields
$$\dot{\tilde{d}}=-\nabla d_{P}T(q)\tilde{p}.$$
(24)
We will now analyze the stability of the error model (22), (23) and (24) with the aid of the proper Lyapunov function candidate
$$V(\tilde{p},\tilde{d},\tilde{r}_{u})={1\over 2}(|\tilde{p}|^{2}+|\tilde{d}|^{2%
}+|\tilde{r}_{u}|^{2}).$$
(25)
Taking its time-derivative we obtain
$$\displaystyle\dot{V}$$
$$\displaystyle=-\tilde{p}^{\top}\Big{[}R(q)+\lambda I_{n}\Big{]}\tilde{p}-%
\tilde{p}^{\top}\big{[}(\sum_{i=1}^{n}Y_{i}\hat{p}_{i})\tilde{r}_{u}-T^{\top}(%
q)\tilde{d}\big{]}$$
$$\displaystyle-\big{[}\lambda\tilde{r}_{u}^{\top}\nabla r_{u_{P}}+\tilde{d}^{%
\top}\nabla d_{P}T(q)\big{]}\tilde{p}.$$
(26)
Clearly, if the mappings $r_{u_{P}}(\hat{p})$ and $d_{P}(q)$ solve the partial differential equations (PDEs)
$$\displaystyle\nabla r_{u_{P}}$$
$$\displaystyle=-{1\over\lambda}(\sum_{i=1}^{n}Y^{\top}_{i}\hat{p}_{i})$$
(27)
$$\displaystyle\nabla d_{P}$$
$$\displaystyle=I_{n},$$
one gets
$$\dot{V}=-\tilde{p}^{\top}[R(q)+\lambda I_{n}]\tilde{p}\leq-\lambda|\tilde{p}|^%
{2}.$$
(28)
From (25), (28) we conclude that $\tilde{p}$ $\in$ $\mathcal{L}_{2}$ $\cap$ $\mathcal{L}_{\infty}$ and $\tilde{d},\tilde{r}_{u}\in\mathcal{L}_{\infty}$. Doing some standard signal chasing it is
straightforward to prove from here that (16) holds.
Motivated by the conclusion above let us now study the PDEs (27). The second one has the trivial solution $d_{P}=q$. Regarding the first one, it is clear that the elements of the mapping
$r_{u_{P}}(\hat{p})$ must be of the quadratic form
$$(r_{u_{P}}(\hat{p}))_{i}={1\over 2{\lambda}}\hat{p}^{\top}L_{i}\hat{p},\;i\in%
\bar{s},$$
with constant, symmetric matrices $L_{i}\in\mathbb{R}^{n\times n}$. Replacing the expression above in the PDE (27) yields
$$\displaystyle\left[\begin{array}[]{c}\hat{p}^{\top}L_{1}\\
\vdots\\
\hat{p}^{\top}L_{s}\end{array}\right]=-\sum_{i=1}^{n}Y^{\top}_{i}\hat{p}_{i}.$$
That lead us to the solution
$$\left[\begin{array}[]{ccc}L_{1}^{\top}e_{j}&\ldots&L_{s}^{\top}e_{j}\end{array%
}\right]=-Y_{j},\;j\in\bar{n}.$$
(29)
It only remains to show that the resulting matrices $L_{i}$ are symmetric. From (29) we get
$$\displaystyle-L_{j}^{\top}$$
$$\displaystyle=$$
$$\displaystyle-\left[\begin{array}[]{cccc}L_{j}^{\top}e_{1}&L_{j}^{\top}e_{2}&%
\ldots&L_{j}^{\top}e_{n}\end{array}\right]$$
$$\displaystyle=$$
$$\displaystyle\left[\begin{array}[]{cccc}Y_{1}e_{j}&Y_{2}e_{j}&\ldots&Y_{n}e_{j%
}\end{array}\right]$$
Clearly, the matrix $L_{j}$ is symmetric if and only if
$$e_{i}^{\top}Y_{k}=e_{k}^{\top}Y_{i},\;\forall i,k\in\bar{n},\;i\neq k.$$
This fact can be easily verified using (14)
$$\displaystyle e_{k}^{\top}Y_{i}$$
$$\displaystyle=$$
$$\displaystyle e_{k}^{\top}\sum_{j=1}^{n}T^{\top}(q)e_{j}e_{j}^{\top}T(q)e_{i}e%
_{j}^{\top}C$$
$$\displaystyle=$$
$$\displaystyle e_{i}^{\top}\sum_{j=1}^{n}T^{\top}(q)e_{j}e_{j}^{\top}T(q)e_{k}e%
_{j}^{\top}C$$
$$\displaystyle=$$
$$\displaystyle e_{i}^{\top}Y_{k}.$$
Replacing all the derivations above in $\dot{p}_{I}$, $\dot{d}_{I}$ and $\dot{r}_{u_{I}}$ gives the equations given in the proposition completing the proof.
Remark 4.3.
If Assumption 4.1 is not imposed a term $J(q,p)p$ appears in the error equation (22) and (23). Even though this term is quadratic in the unknown state $p$, the properties of
$J(q,p)$ can be used to handle this term in the first error equations—this is done in the second observer in the next section. However, there is no obvious way to create a suitable error term for the second
error equation styming the relaxation of Assumption 4.1.
Remark 4.4.
If the matrices $Y_{i}$ are not constant the first PDE in (27) does not admit a solution, hence Assumption 4.2 is required. It can be shown that making $r_{u_{P}}$ function of $q$ does not
solve the problem, because a quadratic function of $\hat{p}$ will appear in $\dot{V}$.
Remark 4.5.
As shown in Proposition 6 of [16] the dynamics of mechanical systems satisfying Assumption 4.1 expressed in the coordinates $(Q,p)$ take the form
$$\displaystyle\dot{Q}$$
$$\displaystyle=p$$
$$\displaystyle\dot{p}$$
$$\displaystyle=-\tilde{R}(Q)p-\tilde{T}^{\top}(Q)[\nabla\tilde{V}(Q)-\tilde{G}(%
Q)u+d],$$
(30)
where $\tilde{(\cdot)}(Q):=(\cdot)(Q^{I}(Q))$, with $Q^{I}:\mathbb{R}^{n}\to\mathbb{R}^{n}$ a left inverse of $Q(q)$, that is, $Q(Q^{I}(z))=z$ for all $z\in\mathbb{R}^{n}$. Although the construction of an observer for
(30) when the friction is known is straightforward, the case of unknown friction is far from trivial. Applying the I$\&$I procedure used in Proposition 4.3 leads, for the definition of
$d_{P}(q)$, to a PDE of the form
$$\nabla S(Q)=\tilde{T}(Q),$$
whose solution is not obvious.
5 ROBUST OBSERVER FOR GENERAL PERTURBED MECHANICAL SYSTEMS WITH KNOWN FRICTION
In this section we redesign the I&I speed observer of [13], see also [2], to ensure its global convergence in spite of the presence of the unknown disturbances $d$ and known friction forces in all coordinates.
{prop}
Consider the system (1), (2) with known friction matrix $\mathfrak{R}$. There exist smooth mappings
$$\displaystyle\mathbf{A}$$
$$\displaystyle:\mathbb{R}^{4n}\times\mathbb{R}_{\geq 0}\times\mathbb{R}^{n}%
\times\mathbb{R}^{n}\to\mathbb{R}^{4n+1}$$
$$\displaystyle\mathbf{B}$$
$$\displaystyle:\mathbb{R}^{4n}\times\mathbb{R}_{\geq 0}\times\mathbb{R}^{n}%
\rightarrow\mathbb{R}^{n}$$
such that the interconnection of (1), (2) with
$$\displaystyle\dot{\mathrm{X}}$$
$$\displaystyle=\mathbf{A}(\mathrm{X},q,u)$$
(31)
$$\displaystyle\hat{\mathbf{p}}$$
$$\displaystyle=\mathbf{B}(\mathrm{X},q),$$
where $\mathrm{X}\in\mathbb{R}^{4n}\times\mathbb{R}_{\geq 0},\;\hat{\mathbf{p}}\in%
\mathbb{R}^{n}$, ensures (16) holds for all initial conditions
$$(q(0),\mathbf{p}(0),\mathrm{X}(0))\in\mathbb{R}^{n}\times\mathbb{R}^{n}\times%
\mathbb{R}^{4n}\times\mathbb{R}_{\geq 0}.$$
This implies that, in spite of the presence of the unknown disturbances $d$, (31) is a globally convergent momenta observer for the mechanical system with friction (1), (2).
Proof 5.1.
The construction of the observer follows very closely the one reported in [13] with the only difference of the inclusion of an adaptation law for the unknown disturbance parameters $d$. However, for the sake of completeness, a detailed derivation of all the steps is given.
Define the estimation errors
$$\displaystyle\tilde{p}$$
$$\displaystyle=\hat{p}-p$$
(32)
$$\displaystyle\tilde{d}$$
$$\displaystyle=\hat{d}-d.$$
Following the I&I adaptive observer procedure [1] we propose to generate the estimates as
$$\displaystyle\hat{p}$$
$$\displaystyle:=$$
$$\displaystyle p_{I}+p_{P}(q,\textbf{{q|}},\textbf{{{|p}}})$$
(33)
$$\displaystyle\hat{d}$$
$$\displaystyle:=$$
$$\displaystyle{d_{I}+d_{P}(q,r)}$$
where the mappings $p_{P}:\mathbb{R}^{n}\times\mathbb{R}^{n}\times\mathbb{R}^{n}\rightarrow\mathbb%
{R}^{n}$ and $d_{P}\in\mathbb{R}^{n}$, and the observer states $d_{I}\in\mathbb{R}^{n}$, $p_{I}\in\mathbb{R}^{n}$ and $r\in\mathbb{R}$ are defined such that
(16) holds.
We, therefore, study the dynamic behavior of $\tilde{p}$ and compute
$$\displaystyle\dot{\tilde{p}}$$
$$\displaystyle=\dot{p_{I}}+\nabla_{q}{p_{P}}{\dot{q}}+\nabla_{\textbf{{q|}}}{p_%
{P}}\dot{\textbf{{q|}}}+\nabla_{\textbf{{{|p}}}}{p_{P}}{\dot{\textbf{{{|p}}}}}-$$
$$\displaystyle-J(q,p)p+T^{\top}(q)[\nabla V-G(q)u]+R(q)p-T^{\top}(q)d.$$
In [2] it has been shown that the mapping $J(q,p)$ defined in (6) verifies the following properties:
(P.i)
$J(q,p)$ is linear in the second argument, that is
$$J(q,\alpha_{1}p+\alpha_{2}\bar{p})=\alpha_{1}{J}(q,p)+\alpha_{2}{J}(q,\bar{p})$$
for all $q$, $p$, $\bar{p}$ $\in\mathbb{R}^{n}$, and $\alpha_{1}$, $\alpha_{2}$ $\in\mathbb{R}$.
(P.ii)
There exists a mapping ${\bar{J}}:\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}^{n\times n}$ satisfying
$${J}(q,p)\bar{p}={\bar{J}}(q,\bar{p})p.$$
Hence, proposing
$$\displaystyle\dot{p_{I}}$$
$$\displaystyle:=-\nabla_{\textbf{{q|}}}{p_{P}}\dot{\textbf{{q|}}}-\nabla_{%
\textbf{{{|p}}}}{p_{P}}{\dot{\textbf{{{|p}}}}}+J(q,\hat{p})\hat{p}-R(q)\hat{p}-$$
$$\displaystyle-T(q)^{\top}(q)\nabla V+v-\nabla_{q}{p_{P}}T(q)\hat{p}+T^{\top}(q%
)\hat{d},$$
together with Properties (P.i) and (P.ii) yields
$$\dot{\tilde{p}}=[J(q,p)+{\bar{J}}(q,\hat{p})-R(q)-\nabla_{q}p_{P}T(q)]\tilde{p%
}+T^{\top}(q)\tilde{d}.$$
(35)
It is clear that if the mapping $p_{P}$ solves the PDE
$$\nabla_{q}p_{P}=[\psi{I_{n}}+{\bar{J}}(q,\hat{p})]T^{-1}(q),$$
with $\psi>0$ a design constant, the $\tilde{p}$–dynamics reduces to
$$\dot{\tilde{p}}=[J(q,p)-\psi{I_{n}}-R(q)]\tilde{p}+T^{\top}(q)\tilde{d}.$$
Recalling that $J(q,p)$ is skew–symmetric and $R(q)\geq 0$ the unperturbed part of the error dynamics above, i.e. when $\tilde{d}=0$, is exponentially stable.
Similarly to [13], to avoid the solution of the PDE, the dynamic scaling technique is used. Towards this end, define the mapping
$${\mathcal{H}}(q,\hat{p}):=[\psi{I_{n}}+{\bar{J}}(q,\hat{p})]T^{-1}(q).$$
(36)
and define $p_{P}$ as
$$p_{P}(q,\textbf{{q|}},\textbf{{{|p}}}):={\mathcal{H}}({\textbf{{q|}},\textbf{{%
{|p}}}})q.$$
(37)
The choice above yields $\nabla_{q}p_{P}={\mathcal{H}}({\textbf{{q|}},\textbf{{{|p}}}})$, which may be written as
$$\nabla_{q}p_{P}={\mathcal{H}}(q,\hat{p})-[{\mathcal{H}}(q,\hat{p})-{\mathcal{H%
}}({\textbf{{q|}},\textbf{{{|p}}}})].$$
(38)
Now, since the term in brackets in (38) is equal to zero if $\textbf{{{|p}}}=\hat{p}$ and $\textbf{{q|}}=q$, there exist mappings
$${\Delta}_{q},{\Delta}_{p}:\mathbb{R}^{n}\times\mathbb{R}^{n}\times\mathbb{R}^{%
n}\to\mathbb{R}^{n\times n}$$
verifying
$${\Delta}_{q}(q,\textbf{{{|p}}},0)=0,\quad{\Delta}_{p}(q,\textbf{{{|p}}},0)=0,$$
(39)
and such that
$${\mathcal{H}}(q,\hat{p})-{\mathcal{H}}({\textbf{{q|}},\textbf{{{|p}}}})={%
\Delta}_{q}(q,\textbf{{q|}},e_{q})+{\Delta}_{p}(q,\textbf{{{|p}}},e_{p}),$$
(40)
where
$$e_{q}:=\textbf{{q|}}-q,\quad e_{p}:=\textbf{{{|p}}}-\hat{p}.$$
(41)
Substituting (36), (38) and (40) in (35), yields
$$\displaystyle\dot{\tilde{p}}$$
$$\displaystyle=[J(q,p)-\psi{I_{n}}-R]\tilde{p}$$
$$\displaystyle+\Big{(}{\Delta}_{q}(q,\textbf{{q|}},e_{q})+{\Delta}_{p}(q,%
\textbf{{{|p}}},e_{p})\Big{)}T(q)\tilde{p}+T^{\top}(q)\tilde{d}.$$
The mappings ${\Delta}_{q}$, ${\Delta}_{p}$ play the role of disturbances that are dominated with a dynamic scaling and a proper choice of the observer dynamics. For, define the dynamically scaled
off–the–manifold coordinate
$$\eta=\frac{\displaystyle 1}{\displaystyle r}\tilde{p},$$
(42)
where $r$ is a scaling factor to be defined. The dynamic behavior of $\eta$ is given by
$$\dot{\eta}=({J}-R-\psi{I}){\eta}+({\Delta_{q}}+{\Delta_{p}})T{\eta}+\frac{1}{r%
}T^{\top}\tilde{d}-\frac{\dot{r}}{r}{\eta},$$
(43)
where, for brevity, the arguments of the mappings are omitted.
Mimicking [2] select the dynamics of q|, |p as
$$\displaystyle{\dot{\textbf{{q|}}}}$$
$$\displaystyle=T(q)\hat{p}-\psi_{1}{e_{q}}$$
(44)
$$\displaystyle\dot{\textbf{{{|p}}}}$$
$$\displaystyle=-T^{\top}(q)\nabla V+v+J(q,\hat{p})\hat{p}-R\hat{p}$$
$$\displaystyle-\psi_{2}{e_{p}}+T^{\top}(q)\hat{d}$$
where $\psi_{1},\psi_{2}$ are some positive functions of the state defined later. Using (44), together with (41), we get
$$\displaystyle\dot{e}_{q}$$
$$\displaystyle=T(q)\eta r-\psi_{1}{e_{q}}$$
(45)
$$\displaystyle\dot{e}_{p}$$
$$\displaystyle=\nabla_{q}p_{P}T(q)\eta r-\psi_{2}{e_{p}}.$$
Moreover, select the dynamics of $r$ as
$$\dot{r}=-{\psi\over 4}(r-1)+{r\over\psi}(\|{\Delta_{p}T}\|^{2}+\|{\Delta_{q}T}%
\|^{2}),\;r(0)\geq 1,$$
(46)
with $\|\cdot\|$ the matrix induced $2$–norm. At this point we make the important observation that the set
$$\{r\in\mathbb{R}:r\geq 1\}$$
is invariant for the dynamics (46). Hence, $r(t)\geq 1,\;\forall t\geq 0$.
On other hand, taking the time-derivative of $\tilde{d}$, we get
$$\displaystyle\dot{\tilde{d}}$$
$$\displaystyle=\dot{d}_{I}+\nabla_{q}d_{P}T(q)p+\nabla_{r}d_{P}\dot{r}$$
$$\displaystyle=\dot{d}_{I}+\nabla_{q}d_{P}(q,r)T(q)(\hat{p}-r\eta)+\nabla_{r}d_%
{P}\dot{r}$$
and choosing
$$\dot{d}_{I}=-\nabla_{q}d_{P}T(q)\hat{p}-\nabla_{r}d_{P}\dot{r},$$
(47)
the $\tilde{d}$–dynamics take the form
$$\dot{\tilde{d}}=-r\nabla_{q}d_{P}T(q)\eta$$
(48)
We now analyze the error system (42), (45), (46), (48)—with the coordinate $\tilde{r}=(r-1)$. For, define the proper Lyapunov function candidate.
$$V(\eta,e_{q},e_{p},\tilde{r},z_{a}):={1\over 2}\left(|\eta|^{2}+|e_{q}|^{2}+|e%
_{p}|^{2}+\tilde{r}^{2}+|\tilde{d}|^{2}\right).$$
(49)
Taking its time-derivative we obtain
$$\displaystyle\dot{V}$$
$$\displaystyle\leq-\left({\psi\over 4}-1\right)|\eta|^{2}-\left(\psi_{1}-{1%
\over 2}r^{2}\|T(q)\|^{2}\right)|e_{q}|^{2}$$
$$\displaystyle-\left(\psi_{2}-{1\over 2}r^{2}\|\nabla_{q}p_{P}\|^{2}\|T(q)\|^{2%
}\right)|e_{p}|^{2}+\tilde{r}\dot{r}+$$
$$\displaystyle+\tilde{d}^{\top}\left(\frac{1}{r}-r\nabla_{q}d_{P}\right)T(q)\eta$$
Clearly, if we set
$${\psi=4(1+\psi_{3})},\;\psi_{1}={1\over 2}r^{2}\|T(q)\|^{2}+\psi_{4}$$
(50)
and
$$\psi_{2}={1\over 2}r^{2}\|\nabla_{q}p_{P}\|^{2}\|T(q)\|^{2}+\psi_{5},$$
where $\psi_{3},\psi_{4},\psi_{5}$ are positive functions of the state defined below, one gets
$$\displaystyle\dot{V}$$
$$\displaystyle\leq-\psi_{3}|\eta|^{2}-\psi_{4}|e_{q}|^{2}-\psi_{5}|e_{p}|^{2}+%
\tilde{r}\dot{r}$$
$$\displaystyle+\tilde{d}^{\top}\left(\frac{1}{r}-r\nabla_{q}d_{P}(q,r)\right)T(%
q)\eta.$$
To eliminate the cross term appearing in the last term of the right hand side above we select
$$d_{P}(q,r)={1\over r^{2}}q,$$
(51)
which clearly solves the PDE
$${1\over r}-r\nabla_{q}d_{P}(q,r)=0.$$
It only remains to study the term $\tilde{r}\dot{r}$, which is given by
$$\tilde{r}\dot{r}=-{\psi\over 4}\tilde{r}^{2}+\tilde{r}{r\over\psi}(\|{\Delta_{%
p}T}\|^{2}+\|{\Delta_{q}T}\|^{2}).$$
Now, (39) ensures the existence of mappings $\bar{\Delta}_{q}$, $\bar{\Delta}_{p}:\mathbb{R}^{n}\times\mathbb{R}^{n}\times\mathbb{R}^{n}\to%
\mathbb{R}^{n\times n}$ such that
$$\displaystyle\|\Delta_{q}(q,\textbf{{{|p}}},e_{q})\|$$
$$\displaystyle\leq\|\bar{\Delta}_{q}(q,\textbf{{{|p}}},e_{q})\|\;|e_{q}|$$
$$\displaystyle\|\Delta_{p}(q,\textbf{{{|p}}},e_{p})\|$$
$$\displaystyle\leq\|\bar{\Delta}_{p}(q,\textbf{{{|p}}},e_{p})\|\;|e_{p}|.$$
Hence
$$\|{\Delta_{p}T}\|^{2}+\|{\Delta_{q}T}\|^{2}\leq\|T\|^{2}|(\|\bar{\Delta}_{p}\|%
^{2}|e_{p}|^{2}+|\bar{\Delta}_{q}\|^{2}|e_{q}|^{2}).$$
Setting
$$\displaystyle\psi_{3}$$
$$\displaystyle=\kappa$$
$$\displaystyle\psi_{4}$$
$$\displaystyle={r\tilde{r}\over 4(1+\psi_{3})}\|T\|^{2}\|\bar{\Delta}_{q}\|^{2}+\kappa$$
$$\displaystyle\psi_{5}$$
$$\displaystyle={r\tilde{r}\over 4(1+\psi_{3})}\|T\|^{2}\|\bar{\Delta}_{p}\|^{2}%
+\kappa,$$
one gets
$$\dot{V}\leq-\kappa(|\eta|^{2}+|e_{q}|^{2}+|e_{p}|^{2}+\tilde{r}^{2}]$$
(52)
for some positive constant $\kappa$.
The proof is completed invoking the arguments of [13], selecting the observer state as
$${\mathrm{X}}:=(\textbf{{q|}},\textbf{{{|p}}},p_{I},d_{I},r-1),$$
and defining $\mathbf{A}(\mathrm{X},q,u)$ from (LABEL:dot_xi), (44), (46) and (47) and setting $\mathbf{B}(\mathrm{X},q)$ via
(33).
Remark 5.1.
It is clear from the proof that the key step to reject the disturbances is to make the proportional term of the parameter estimator, $d_{P}$ a function of the dynamic scaling factor $r$, see (51).
6 PHYSICAL EXAMPLES
In this section we present three physical mechanical systems that satisfy the conditions of Proposition 4.3. Consequently, robust adaptive speed observation is possible for them.
6.1 Constant inertia matrix
In the case of constant inertia matrix Assumption 4.1 is trivially satisfied, because the factor $T$ can be taken to be constant. Assumption 4.2 is also satisfied with $\dim(q)=n$ and $C=I_{n}$, hence all friction coefficients can be identified.
Given any constant factor $T$, the vector field $Q(q)$ that solves (9) is given by
$$Q(q)=T^{-1}q.$$
Finally, from (14) and (15) we get
$$\displaystyle Y_{j}$$
$$\displaystyle=\sum_{i=1}^{n}T^{\top}(q)e_{i}e_{i}^{\top}T(q)e_{j}e_{i}^{\top}%
\;j\in\bar{n}$$
$$\displaystyle L_{i}$$
$$\displaystyle=[(T)^{i}]^{\top}(T)^{i},\;i\in\bar{n}.$$
6.2 Planar redundant manipulator with one elastic degree of freedom
This is a 4-dof underactuated mechanical system depicted in 1.
The inverse inertia matrix is given by
$$M^{-1}(q)=\left[\begin{array}[]{cccc}\hfil 1\over I&\hfil-1\over I&0&0\\
*&{1\over a_{2}^{2}}+{1\over I}&-{1\over m\ell}S_{12}&{1\over m\ell}C_{12}\\
*&*&{1\over a_{3}^{2}}&0\\
*&*&*&{1\over a_{3}^{2}}\end{array}\right],$$
where we defined
$$\displaystyle S_{12}$$
$$\displaystyle:=\sin(q_{1}+q_{2}),\;C_{12}:=\cos(q_{1}+q_{2})$$
$$\displaystyle a_{2}$$
$$\displaystyle:={{\sqrt{Mm}\ell}\over\sqrt{m+M}},\;a_{3}:=\sqrt{M+m},$$
and the definition of all constants may be found in [16]. The Cholesky factorization is given as
$$T(q)=\left[\begin{array}[]{cccc}\hfil 1\over\sqrt{I}&0&0&0\\
\hfil-1\over\sqrt{I}&{1\over a_{2}}&0&0\\
0&-\sqrt{M\"{\i}\textquestiondown\textonehalf\over m}{1\over a_{3}}S_{12}&%
\hfil 1\over a_{3}&0\\
0&\sqrt{M\"{\i}\textquestiondown\textonehalf\over m}{1\over a_{3}}C_{12}&0&%
\hfil 1\over a_{3}\end{array}\right],$$
and the vector field $Q(q)$ that solves (9) is
$$Q(q)=\left[\begin{array}[]{c}\sqrt{I}q_{1}\\
a_{2}(q_{1}+q_{2})\\
-a_{2}\sqrt{M\"{\i}\textquestiondown\textonehalf\over m}C_{12}+a_{3}q_{3}\\
-a_{2}\sqrt{M\"{\i}\textquestiondown\textonehalf\over m}S_{12}+a_{3}q_{4}\end{%
array}\right].$$
A matrix $C$ that satisfies Assumption 4.2 is
$$C=\left[\begin{array}[]{cc}I_{2}\\
0_{2\times 2}\end{array}\right].$$
Hence, we can consider as unknown the frictions in the elastic coordinate $r_{1}$ and the revolute joint $r_{2}$.
Finally,
$$\displaystyle Y_{1}^{\top}$$
$$\displaystyle=\left[\begin{array}[]{cccc}\hfil 1\over I&0&0&0\\
{1\over I}&-{a_{2}\over\sqrt{I}}&0&0\end{array}\right]$$
$$\displaystyle Y_{2}^{\top}$$
$$\displaystyle=\left[\begin{array}[]{cccc}0&0&0&0\\
-{a_{2}\over\sqrt{I}}&a_{2}^{2}&0&0\end{array}\right]$$
$$\displaystyle L_{1}$$
$$\displaystyle=\left[\begin{array}[]{ccc}\left[\begin{array}[]{cc}{1\over I}&0%
\\
0&0\end{array}\right]&0_{2\times 2}\\
0_{2\times 2}&0_{2\times 2}\end{array}\right]$$
$$\displaystyle L_{2}$$
$$\displaystyle=\left[\begin{array}[]{ccc}\left[\begin{array}[]{cc}{1\over I}&-{%
a_{2}\over\sqrt{I}}\\
-{a_{2}\over\sqrt{I}}&a_{2}^{2}\end{array}\right]&0_{2\times 2}\\
0_{2\times 2}&0_{2\times 2}\end{array}\right]$$
6.3 2D-Spider crane gantry cart
This is a 3-dof underactuated mechanical system depicted in Fig. 2.
The inertia matrix is
$$M(q)=\left[\begin{array}[]{ccc}m_{r}+m&0&{mL_{3}}C_{3}\\
*&m_{r}+m&mL_{3}S_{3}\\
*&*&mL_{3}^{2}\end{array}\right],$$
with inverse
$$M^{-1}(q)=\left[\begin{array}[]{ccc}\hfil{m_{r}+mC_{3}^{2}}\over{(m_{r}+m)m_{r%
}}&\hfil{mC_{3}S_{3}}\over{(m_{r}+m)m_{r}}&\hfil-{C_{3}}\over{L_{3}m_{r}}\\
*&{{m_{r}+m-mC_{3}^{2}}\over{(m_{r}+m)m_{r}}}&-{{S_{3}}\over{m_{r}L_{3}}}\\
*&*&{{m_{r}+m}\over{m_{r}L_{3}^{2}m}}\end{array}\right]$$
where, to simplify the notation, we have defined
$$S_{3}:=\sin(q_{3}),C_{3}:=\cos(q_{3}),$$
and the definition of all constants may be found in [9]. An upper triangular factorization of $M^{-1}(q)$ is given as
$$T(q)=\left[\begin{array}[]{ccc}a&0&-bC_{3}\\
0&a&-bS_{3}\\
0&0&c\end{array}\right],$$
(53)
where we defined the constants
$$a:={{1}\over{\sqrt{m_{r}+m}}},b:={{1}\over{cL_{3}m_{r}}},c:=\sqrt{{{m_{r}+m}%
\over{mL_{3}^{2}m_{r}}}}.$$
We can check that the columns of $T(q)$ satisfy (8) and thus the system verifies Assumption 1. Taking the inverse of $T(q)$ we get
$$T^{-1}(q)=\left[\begin{array}[]{ccc}\hfil{1}\over{a}&0&aL_{3}mC_{3}\\
0&\hfil{1}\over{a}&aL_{3}mS_{3}\\
0&0&\hfil{1}\over{c}\end{array}\right],$$
From the equation above it is clear that a mapping $Q(q)$ that solves (9) is
$$Q(q)=\left[\begin{array}[]{c}{1\over a}q_{1}+aL_{3}mS_{3}\\
{1\over a}q_{2}-aL_{3}mC_{3}\\
{1\over c}q_{3}\\
\end{array}\right].$$
From the definition of $T(q)$ in (53) it is clear that
$$C^{\top}=\left[\begin{array}[]{ccc}0&0&1\end{array}\right]$$
satisfies Assumption 4.2. Hence, we can consider as unknown the friction parameter $r_{3}$.
Finally, from (14) and (15) we get
$$\displaystyle Y_{1}^{\top}$$
$$\displaystyle=\left[\begin{array}[]{ccc}0&0&c^{2}\\
\end{array}\right],$$
$$\displaystyle L_{1}$$
$$\displaystyle=\left[\begin{array}[]{ccc}0&0&0\\
0&0&0\\
0&0&c^{2}\\
\end{array}\right].$$
We show several simulations of the proposed observer. The system has the two forces shown in Fig. 2 as control inputs, that is, $u=\mbox{col}(F_{x},F_{y})$ and constant input matrix $G$ of the form
$$G=\left[\begin{array}[]{cc}1&0\\
0&1\\
0&0\end{array}\right].$$
The parameters are taken as $m_{r}=0.5$ $kg$ for ring mass, $m=1$ $kg$ for payload mass and $L_{3}=0.5$ $m$ for the cable length. We fix the control inputs as $F_{x}=1.535\cos(t)$ and $F_{y}=7.67\sin(t)$. The disturbances are taken as $d=\mbox{col}(0.1,0.2,0.2)$ and the friction coefficients $r=\mbox{col}(0,0,0.5)$, with $r_{3}$ being unknown.
The transient behavior of the error signals $\tilde{p},\tilde{r}_{3}$ and $\tilde{d}$ with the tuning parameter $\lambda=0.8$ and different initial conditions of $d_{I},r_{3_{I}}$ and $P_{I}$ are shown in Fig.3 and Fig.4. As seen from the figures, besides the convergence to zero of the momenta estimate predicted by the theory, we also observe that the estimated parameters converge to their true value—assessing the fact that the signals chosen for the simulation are persistently exciting.
To evaluate the effect of the tuning parameter $\lambda$ on the transient behavior we also show in Fig.5 and Fig.6 the transient behavior of the error signals for different values of $\lambda$.
Finally to illustrate the robustness of the adaptive observer, we carried out a simulation considering that the input disturbance $d_{1}$ is subject to step changes. The trajectories of $d_{1}(t)$ and its estimation, depicted in Fig. 7, clearly illustrate the tracking capability of the proposed observer.
Remark 6.1.
It is interesting to note that the standard (lower triangular) Cholesky factorization of $M^{-1}(q)$ does not satisfy (8).
7 Future Research
The design of the observer in Proposition 4.3 requires the explicit solution of the PDE (9). This requirement restricts the practical applicability of the approach. Indeed, this PDE
has no free parameters and its explicit solution may be even impossible. As indicated in [16] this is the case of the classical cart–pendulum example.
Current research is under way to extend the realm of application of the observer in Proposition 4.3. In particular, it is possible to consider the following generalization of the Lyapunov function
candidate (25)
$$|\tilde{p}|^{2}+\tilde{r}_{u}^{\top}P^{-1}\tilde{r}_{u}+|\tilde{d}|^{2},$$
with $P>0$ a constant matrix. It is straightforward to show that the second PDE in (27)—that cancels the cross term appearing in the derivative of the new Lyapunov function—becomes
$$\nabla r_{u_{P}}=-{1\over\lambda}(\sum_{i=1}^{n}PY_{i}^{\top}\hat{p}_{i}),$$
where we underscore the presence of the matrix $P$ in front of $Y_{i}$. There are inertia matrices where the corresponding $Y_{i}$ are not constant but there exists positive definite $P$ that will make
$PY_{i}^{\top}$ constant—hence relaxing Assumption 4.2. We are currently investigating whether there exist physical systems for which such property holds.
Another, quite challenging, task is the extension of Proposition 4.3 to systems that do not have ZRS. One possibility is to look into the next class of systems partially linearizable via coordinate changes characterized in [16].
{ack}
This work was supported by the Ministry of Education and Science of Russian Federation (Project 14.Z50.31.0031).
References
[1]
A. Astolfi, D. Karagiannis and R. Ortega. Nonlinear and Adaptive Control Design with Applications. Springer-Verlag; London, 2007
[2]
A. Astolfi, R. Ortega and A. Venkatraman. A globally exponentially convergent immersion and invariance speed observer for mechanical systems with non-holonomic constraints. Automatica, 46(1);182–189, 2010.
[3]
G. Besançon (Ed.). Nonlinear Observers and Applications. Springer-Verlag, 2007.
[4]
K.D. Do, Z.P. Jiang and J. Pan. A global output–feedback controller for simultaneous tracking and stabilization of unicycle–type mobile robots. IEEE Transactions on Robotics and Automation, 20(3); 589–594, 2004.
[5]
K.D. Do and J. Pan. Underactuated ships follow smooth paths with integral actions and without velocity measurements for feedback: theory and experiments. IEEE Transactions on Control Systems Technology , 14(2); 308–322, 2006.
[6]
R. A. Horn and C. R. Johnson. Matrix Analysis. Cambridge University Press, 1990.
[7]
A. Isidori. Nonlinear Control Systems. Springer -Verlag, 3rd edition, 1995.
[8]
A. Jain and G. Rodriguez. Diagonalized lagrangian robot dynamics. Proc. IEEE Int. Conf. Robot. & Autom, 11(4); 571–583, 1995.
[9]
F. Kazi, R. Banavar, P. Mullhaupt and D. Bonvin. Stabilization of a 2d-spidercrane mechanism using interconnection and damping assignment passivity–based control. Proceedings of the 17th IFAC World Congress, 17(1); 3155–3160, 2008.
[10]
L.Ljung. Systems Identification: Theory for the User. Prentice Hall, 1987.
[11]
H. Nijmeijer and A. J. van der Schaft. Nonlinear Dynamical Control Systems. Springer-Verlag, London, 1990.
[12]
J. G. Romero, A. Donaire and R. Ortega. Robust energy shaping control of mechanical systems. Systems & Control Letters, 62(9); 770–780, 2013.
[13]
J. G. Romero, I. Sarras and R. Ortega. A globally exponentially stable tracking controller for mechanical systems using position feedback. American Control Conference , pages 4969–4974, 2013. (To be published in IEEE Trans. Automatic Control)
[14]
M. Spivak. A Comprehensive Introduction to Differential Geometry. Perish, Inc, 3rd edition, 1999.
[15]
A. J. van der Schaft. $L_{2}$-Gain and Passivity Techniques in Nonlinear Control. Springer, London, 1999.
[16]
A. Venkatraman, R. Ortega, I. Sarras and A. van der Schaft. Speed observation and position feedback stabilization of partially linearizable mechanical systems. IEEE Transactions on Automatic Control , 55(5); 1059–1074, 2010. |
Quantization of the Hall conductivity well beyond the
adiabatic limit in pulsed magnetic fields
V. T. Dolgopolov and A. A. Shashkin
Institute of Solid State Physics, Chernogolovka, Moscow
District 142432, Russia
J. M. Broto
H. Rakoto
and S. Askenazy
SNCMP INSA 135 avenue de Rangueil 31077 Toulouse cedex, 4,
France
Abstract
We measure the Hall conductivity, $\sigma_{xy}$, on a Corbino
geometry sample of a high-mobility AlGaAs/GaAs heterostructure in a
pulsed magnetic field. At a bath temperature about 80 mK, we observe
well expressed plateaux in $\sigma_{xy}$ at integer filling factors.
In the pulsed magnetic field, the Laughlin condition of the phase
coherence of the electron wave functions is strongly violated and,
hence, is not crucial for $\sigma_{xy}$ quantization.
pacs: PACS numbers: 72.20 My, 73.40 Kp
On recognizing the crucial role of the edge channels in
two-dimensional (2D) electron transport in a quantizing magnetic
field [1, 2], it became pretty clear that the
quantization of the Hall resistance, $R_{xy}$, in Hall bar samples,
which corresponds to the quantum Hall effect [3], is not
directly connected with that of the Hall conductivity, $\sigma_{xy}$.
Even if the longitudinal resistance, $R_{xx}$, is negligible, the
measured resistance tensor cannot be converted into the conductivity
one: the net Hall current is a sum of the bulk and edge currents
while the conductivities $\sigma_{xx}$ and $\sigma_{xy}$ are related
to the bulk of the 2D system. Therefore, the conductivity tensor and
the accuracy of $\sigma_{xy}$ quantization should be investigated
independently using the Corbino geometry which allows separation of
the bulk contribution to the measured current. Such an arrangement
was described in the Laughlin [4] and Widom-Clark
[5] gedanken experiments. A (Hall) charge transfer below the
Fermi level between the coasts of a Corbino sample is induced by
magnetic field sweep and thus the shunting effect of the edge
currents is completely excluded. The concept of Ref. [4]
based on gauge invariance leads to the conclusion that at integer
filling factor the conductivity $\sigma_{xy}$ will be quantized if
the magnetic field, $B$, is changed adiabatically so as to keep the
phase coherence of the wave functions on the sample size. The
quantization of $\sigma_{xy}$ follows from the fact that an integer
number of electrons is transferred between the ring edges if the
magnetic flux changes by one quantum. It is clear that the phase
coherence should be the case at field sweep rates when the magnetic
flux change, $\Delta\Phi=\tau L^{2}{\rm d}B/{\rm d}t$, in a sample with
size $L$ within the settling time, $\tau$, of the wave function phase
is small compared to the flux quantum, $h/e$:
$${\rm d}B/{\rm d}t<2\pi\Omega_{c}B(l/L)^{4},$$
(1)
where $\Omega_{c}$ is the cyclotron frequency, $l$ is the magnetic
length, and the phase settling time is estimated as the ratio of the
sample size and the phase velocity of an electron,
$\tau=L^{2}/l^{2}\Omega_{c}$.
Doubts about the correctness of the gauge invariance approach were
expressed in Ref. [6] and were thought to be supported by
results of the microwave studies, e.g., of Ref. [7]. In
fact, those studies as well as edge magnetoplasmon [8] and
related [9] experiments are not free of edge current
contribution so that they do not yield the pure $\sigma_{xy}$ and
cannot be an argument against the approach of Ref. [4]. The
value $\sigma_{xy}$ can be measured in the arrangement of the above
gedanken experiments which was employed in the work of
Refs. [10, 13, 14]. Plateaux with the quantized values of
$\sigma_{xy}$ were indeed observed in the quasistatic measurements of
Ref. [10], even though the quantization accuracy was about 1%.
Here, we study the charge transfer in a Corbino geometry sample
subjected to a pulsed magnetic field with sweep rate up to $5\times 10^{2}$ T/s at low temperatures. The conductivity $\sigma_{xy}$ has
been found to be quantized at integer filling factor. This result is
very similar to the data obtained in quasistatically changing
magnetic fields, although at such high sweep rates of the pulsed
magnetic field, the phase coherence of the electron wave functions is
strongly broken. So, the condition of adiabaticity is sufficient but
not necessary for $\sigma_{xy}$ to be quantized.
The samples are Corbino disks fabricated from two wafers of
AlGaAs/GaAs heterostructures containing a 2D electron gas with
mobility $1.2\times 10^{6}$ and $4\times 10^{5}$ cm${}^{2}$/Vs at 4.2 K and
density $3.6\times 10^{11}$ and $3.2\times 10^{11}$ cm${}^{-2}$,
respectively. Each sample has a circular gate covering a part of the
sample area so that the gated region of the 2D electron system is
separated from the contacts by guarding rings, see Fig. Quantization of the Hall conductivity well beyond the
adiabatic limit in pulsed magnetic fields. The
radii of the Corbino ring are $r_{1}$ and $r_{2}$, and the circular gate
is restricted by radii $r_{1g}$ and $r_{2g}$; two sets of the radii
are employed: (i) $r_{1}=0.2$ mm, $r_{2}=0.5$ mm, $r_{1g}=0.305$ mm, and
$r_{2g}=0.39$ mm and (ii) $r_{1}=1.013$ mm, $r_{2}=1.119$ mm,
$r_{1g}=1.025$ mm, and $r_{2g}=1.108$ mm. The sample is placed into
the mixing chamber of a dilution refrigerator with a base temperature
of 80 mK. In each pulse, the magnetic field sweeps up to 34 T (or
\egtrm
FIG. 1.
Schematic view of the sample and measurement circuit. The magnetic
field pulse is shown in the inset.
lower) with rising and falling times of about 50 ms and 1 s,
respectively (inset to Fig. Quantization of the Hall conductivity well beyond the
adiabatic limit in pulsed magnetic fields). The azimuthal electric field
induced by magnetic field sweep gives rise to an electric current
only in the radial direction if $\sigma_{xx}\rightarrow 0$. In the
experiment we study the charge, $Q$, brought out of the gated region,
which is equal to the difference between the charge exiting and
entering the gated region [10]
$$Q=\pi(r_{2g}^{2}-r_{1g}^{2})\sigma_{xy}\Delta B.$$
(2)
This charge induces the voltage, $V=Q/C$, across a sufficiently large
capacitance, $C$, connected in parallel to the gate, which is
measured using a preamplifier and a digitizer. The capacitance $C$
allows one to restrict the induced voltage in order to avoid the
breakdown of the dissipationless quantum Hall state [11]. The
equilibrium ($B=0$) electron density in the gated region can be
changed by using a gate bias, $V_{g}$. All data we show in the paper
refer to the gate voltage $V_{g}=0$; we have checked that for gate
voltages between 0 and $-80$ mV (the threshold voltage $V_{th}\approx-0.3$ V), the results discussed below are not sensitive to $V_{g}$. In
the experiment with quasistatically changing magnetic fields (with
sweep rates in the range $(1-5)\times 10^{-3}$ T/s), we measure the
charge $Q$ using an electrometer.
Typical experimental traces of the voltage induced on the sample in a
pulsed magnetic field in the vicinity of filling factor $\nu=1$ and
$\nu=2$ are shown in Fig. Quantization of the Hall conductivity well beyond the
adiabatic limit in pulsed magnetic fields for up and down sweeps. When
sweeping the magnetic field up, at small $\sigma_{xx}$ the voltage
rises linearly with $B$, in accordance with Eq. (2), until it
drops above a certain value of the magnetic field thereby signaling
the breakdown of
\egtrm
FIG. 2.
Experimental traces of the induced voltage on one of the samples in a
pulsed magnetic field for $C=10.4$ and 32 nF at filling factor
$\nu=2$ and $\nu=1$. The expected slopes $V/\Delta B$ are shown by
dashed lines. The sweep direction is indicated by
arrows.
the dissipationless quantum Hall state. On changing
the sweep direction, the voltage polarity reverses so that the up and
down traces form a hysteresis loop. The asymmetry between its top and
bottom parts is caused by larger overheating of the sample in
sweeping the field up, which leads to a more pronounced narrowing of
the quantum plateaux. A change in the background signal below and
above the hysteresis loop originates from chemical potential
oscillations [10]. Similar dependences $V(B)$ are observed also
at higher integer $\nu\leq 6$ ($\nu\leq 10$) for up (down) sweeps;
below we discuss the two lowest filling factors at which the observed
structures occupy the widest magnetic field intervals.
It is important that the slope in the linear interval of the
dependence $V(B)$ is in excellent agreement with the calculated one
using Eq. (2) with the quantized
\egtrm
FIG. 3.
Charge brought out of the 2D electron system as a function of $B$ for
filling factor $\nu=1$ as obtained from the data of Fig. Quantization of the Hall conductivity well beyond the
adiabatic limit in pulsed magnetic fields.
The arrows indicate the direction of magnetic field sweep. The
numerical derivative ${\rm d}Q/{\rm d}B$ is displayed in the
inset.
value $\sigma_{xy}=\nu e^{2}/h$, see Fig. Quantization of the Hall conductivity well beyond the
adiabatic limit in pulsed magnetic fields. As the magnetic field is increased within
the linear interval of $V(B)$, electrons are brought into the 2D
system, and the electron density increases, in accordance with
Eq. (2), by $\Delta n_{s}=(\nu e/h)\Delta B$ (where
$\nu=1,2...$). As a result of the aligned change of magnetic field
and electron density, the filling factor remains approximately
constant: it is about 10% larger (smaller) than the integer $\nu$
for up (down) sweep of the magnetic field (Fig. Quantization of the Hall conductivity well beyond the
adiabatic limit in pulsed magnetic fields). Thus, the
observed dependences $V(B)$ yield well expressed plateaux in
$\sigma_{xy}$ as a function of filling factor.
Figure Quantization of the Hall conductivity well beyond the
adiabatic limit in pulsed magnetic fields shows the corresponding dependence $Q(B)$ in a pulsed
magnetic field. As seen from the figure, within the whole hysteresis
loop, the behaviour of the charge $Q$ brought out of the 2D electron
system is independent of shunting capacitance $C$. This implies that
the observed linear $B$ dependence of the charge $Q$ is limited by a
capacitance discharge that is controlled by the dependence of
$\sigma_{xx}$ on magnetic field [16]. The derivative ${\rm d}Q/{\rm d}B$ yields plateaux in $\sigma_{xy}$ as a function of
magnetic field (inset to Fig. Quantization of the Hall conductivity well beyond the
adiabatic limit in pulsed magnetic fields).
Typical curves of the voltage induced by the charge $Q$ in the
quasistatic measurement are displayed in Fig. Quantization of the Hall conductivity well beyond the
adiabatic limit in pulsed magnetic fields. In addition
to the curves obtained by sweeping the magnetic field up and down all
way through the hysteresis loop, two more traces correspond to
reversal of the sweep direction within hysteresis loop. The expected
linear behaviour of $V$ against $B$ is shown for comparison by dashed
lines. As seen from Fig. Quantization of the Hall conductivity well beyond the
adiabatic limit in pulsed magnetic fields, the intervals of the upper and
lower curves, in which $V(B)$ is linear, are narrower as compared to
the pulsed field data of Fig. Quantization of the Hall conductivity well beyond the
adiabatic limit in pulsed magnetic fields.
\egtrm
FIG. 4.
The induced voltage on another sample in a quasistatically changing
magnetic field as indicated by arrows; $C=30$ nF. Also shown by
dashed lines is the expected slope $V/\Delta B$.
This is undoubtedly caused by
leakage currents emerging because of non-zero dissipative
conductivity $\sigma_{xx}$ whose influence is suppressed in pulsed
measurements. As a result, the accuracy of $\sigma_{xy}$ quantization
for pulsed magnetic fields turns out to be the same or even higher
than in quasistatic measurement.
One can easily see that the above mentioned adiabatic limit of the
inequality (1) is not fulfilled in our experiments. This
limit corresponds to the magnetic field sweep rate $\sim 10^{-4}$ T/s
if $L=0.5$ mm, which is already an order of magnitude lower than
sweep rates in the quasistatic measurement. Moreover, the
conductivity $\sigma_{xy}$ is still found to be quantized even at
much higher sweep rates of the pulsed magnetic field, at least six
orders of magnitude beyond the estimated adiabatic limit of the
expression (1). This finding unambiguously shows that the
condition of the phase coherence of the electron wave functions is
not crucial for $\sigma_{xy}$ quantization.
Apparently, our line of reasoning holds if the temperature-dependent
dephasing time, $\tau_{\phi}(T)$, is much larger than the phase
settling time $\tau$. The former can be evaluated from the balance
condition for thermal electron excitation to the upper quantum level
and relaxation of the excited electrons
$$n_{0}\tau_{\phi}^{-1}=n_{0}\exp(-\Delta/2k_{B}T)\tau_{\text{exc}}^{-1},$$
(3)
where $n_{0}$ is the quantum level degeneracy, $\Delta$ is the level
splitting, and $\tau_{\text{exc}}$ is the lifetime of an excited
electron. As known from optical studies (see, e.g., Ref. [15]),
the lifetime $\tau_{\text{exc}}$ exceeds $\hbar\Delta^{-1}$ for both
spin and cyclotron splittings. Hence, we obtain $\tau_{\phi}>\hbar\Delta^{-1}\exp(\Delta/2k_{B}T)\gg\tau$ in our experiment.
From the expression (1) it follows that for our highest sweep
rates, the phase coherence of the wave functions is broken on the
length $\sim 10$ $\mu$m, which is still much larger than the magnetic
length. In other words, $\sigma_{xy}$ is found to be quantized when
the adiabatic limit is not the case for the whole sample but still
holds on macroscopic distances. Whether there exists a maximum sweep
rate of the magnetic field for $\sigma_{xy}$ to be quantized remains
to be seen.
In summary, we have measured the Hall conductivity in the arrangement
of Laughlin’s gedanken experiment in pulsed magnetic fields. Well
expressed plateaux in $\sigma_{xy}$ have been observed at integer
filling factors, which is similar to the data obtained in quasistatic
measurements. Although in the pulsed magnetic field, the phase
coherence of the electron wave functions is strongly broken, the
$\sigma_{xy}$ quantization is still the case. Therefore, the
gauge-invariance-based argumentation [4] is sufficient but
not necessary for $\sigma_{xy}$ quantization.
We gratefully acknowledge discussions with A. Gold, S. V. Iordanskii,
V. B. Shikin, S. Ulloa, and A. Wixforth. This work was supported by
RFBR Grants 00-02-17294 and 01-02-16424 and the Programme
”Nanostructures” from the Russian Ministry of Sciences. V.T.D.
acknowledges hospitality and support of Paul Sabatier University
during his stay in Toulouse as well as financial support of the BMBF
via a Max Planck research award.
References
[1]
B. I. Halperin, Phys. Rev. B 25, 2185 (1982).
[2]
M. Büttiker, Phys. Rev. B 38, 9375 (1988).
[3]
For a review, see The Quantum Hall Effect,
edited by R. E. Prange and S. M. Girvin (Springer Verlag, Berlin,
1987).
[4]
R. B. Laughlin, Phys. Rev. B 23, 5632
(1981).
[5]
A. Widom and T. D. Clark, J. Phys. D 15,
L181 (1982).
[6]
J. Riess, Europhys. Lett. 12, 253 (1990).
[7]
F. Kuchar, R. Meisels, G. Weimann, and W. Schlapp,
Phys. Rev. B 33, 2965 (1986).
[8]
S. A. Govorkov, M. I. Reznikov, A. P. Senichkin, and
V. I. Talyanskii, JETP Lett. 44, 487 (1986).
[9]
E. Yahel, D. Orgad, A. Palevski, and H. Shtrikman,
Phys. Rev. Lett. 76, 2149 (1996).
[10]
V. T. Dolgopolov, A. A. Shashkin, N. B. Zhitenev, S. I.
Dorozhkin, and K. von Klitzing, Phys. Rev. B 46, 12560
(1992); V. T. Dolgopolov, A. A. Shashkin, G. V. Kravchenko, S. I.
Dorozhkin, and K. von Klitzing, Phys. Rev. B 48, 8480
(1993).
[11]
Similar experiments on quasi-2D organic crystals in
Corbino geometry in pulsed magnetic fields were performed without a
shunting capacitance [12]. As a result, only a
breakdown-caused limiting behaviour was observed with no sign of
quantization.
[12]
M. M. Honold, N. Harrison, J. Singleton, M.-S. Nam,
S. J. Blundell, C. H. Mielke, M. V. Kartsovnik, and N. D. Kushch,
Phys. Rev. B 59, R10417 (1999).
[13]
J. P. Watts, A. Usher, A. J. Matthews, M. Zhu, M.
Elliott, W. G. Herrenden-Harker, P. R. Morris, M. Y. Simmons, and
D. A. Ritchie, Phys. Rev. Lett. 81, 4220 (1998).
[14]
S. A. J. Wiegers, J. G. S. Lok, M. Jeuken, U.
Zeitler, J. C. Maan, and M. Henini, Phys. Rev. B 59, 7323
(1999).
[15]
I. V. Kukushkin and V. B. Timofeev, Adv. Phys. 45, 147 (1996).
[16]
In quasistatically changing magnetic fields, a
universal breakdown curve $V(B)$ was observed [10]. The
different behaviour of the 2D electron system in the breakdown
regime, found here for pulsed magnetic fields is likely to be due to
higher electron temperatures achieved in a field pulse. |
Renormalized entanglement entropy
Marika Taylor and William Woodhead
Mathematical Sciences and STAG Research Centre, University of Southampton,
Highfield, Southampton, SO17 1BJ, UK.
E-mail
m.m.taylor@soton.ac.uk; w.r.woodhead@soton.ac.uk
Abstract:
We develop a renormalization method for holographic entanglement entropy based on area renormalization of entangling surfaces.
The renormalized entanglement entropy is derived for entangling surfaces in asymptotically locally anti-de Sitter spacetimes in general dimensions and
for entangling surfaces in four dimensional holographic renormalization group flows. The renormalized entanglement entropy for disk regions in $AdS_{4}$
spacetimes agrees precisely with the holographically renormalized action for $AdS_{4}$ with spherical slicing and hence with the F quantity,
in accordance with the Casini-Huerta-Myers map. We present a generic class of holographic RG flows associated with deformations by operators of dimension
$3/2<\Delta<5/2$ for which the F quantity increases along the RG flow, hence violating the strong version of the F theorem. We conclude by explaining
how the renormalized entanglement entropy can be derived directly from the renormalized partition function using the replica trick i.e. our renormalization method
for the entanglement entropy is inherited directly from that of the partition function. We show explicitly how the entanglement entropy
counterterms can be derived from the standard holographic renormalization counterterms for asymptotically locally anti-de Sitter spacetimes.
1 Introduction
In recent years there has been considerable interest in entanglement entropy and its holographic implementation, following the proposal of
[1] that entanglement entropy can be computed from the area of a bulk minimal surface
homologous to a boundary entangling region. This proposal was proved for spherical entangling regions in conformal field theories in [2] and
arguments supporting the Ryu-Takayanagi prescription based on generalized entropy were given in [3]. Entanglement entropy has by now been computed in a wide range of holographic systems, see the review [4]. General properties of holographic entanglement entropy are reviewed in
[5].
Entanglement entropy is a UV divergent quantity, with the leading UV divergences scaling with the area of the boundary of the entangling region. For a quantum field theory in D spatial dimensions, the boundary of the entangling region is $(D-1)$-dimensional and thus $S\approx\Lambda^{D-1}{\cal A}_{D-1}$
where $\Lambda$ is the UV cutoff and ${\cal A}_{D-1}$ is the area of the boundary of the entangling region.
If one is interested in the entanglement entropy of a discrete system, in which there is a natural UV cutoff set by, for example, the lattice scale, then it may be natural to work with this “bare” entanglement entropy. If however one is interested in entanglement entropy in a quantum field theory context, then it natural to explore whether and how entanglement entropy can be renormalized.
Finite terms in the entanglement entropy are used in a number of contexts. Firstly, they arise as order parameters for phase transitions, see the pioneering works [6, 7]. Finite terms in the entanglement entropy for disk regions in three dimensional conformal field theories are also related by conformal transformations [2] to the free energy on a three sphere, which is the quantity appearing in the proposed F theorem [8].
As we will review in section 2, in previous works the finite terms in the entanglement entropy have been isolated using differentiation of the entanglement entropy with respect to geometric parameters characterizing the entangling region. Such procedures can be implemented in a simple way, both holographically and in field theory calculations, but they have several disadvantages. The differentiation prescriptions depend on the specific geometry of the entangling region, and thus it is hard to implement such renormalization in situations where the shape of the entangling region is itself being varied. Renormalization by differentiation is furthermore not directly related to the renormalization procedures used for other quantum field theory quantities. Thus, in particular, it is hard to understand issues such as the scheme dependence of the finite answer.
In this paper we will develop a systematic renormalization procedure for entanglement entropy. We begin by setting up holographic renormalization for the Ryu-Takayanagi entanglement entropy functional. Since the entanglement entropy is described by the area of a minimal surface homologous to the boundary entangling region, the UV divergences of the entanglement entropy are in direct correspondence with the area divergences of this minimal surface. Following the holographic renormalization methods of [9, 10, 11] one can identify covariant counterterms on the conformal boundary of the minimal surface which renormalize the area of the minimal surface.
In section 3 we derive the renormalized Ryu-Takayanagi functional for static entangling surfaces in AdS spacetimes. Assuming flat spatial slices of the background for the dual quantum field theory (i.e. a Poincaré representation of $AdS_{D+2}$) the renormalized functional takes the form
$$\displaystyle S_{\rm ren}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{4G_{D+2}}\int_{\Sigma}\mathrm{d}^{D}\sigma^{\alpha}\,\sqrt{\gamma}$$
$$\displaystyle\qquad-\frac{1}{4G_{D+2}}\int_{\partial\Sigma}\mathrm{d}^{D-1}x\sqrt{\tilde{\gamma}}\left(\frac{1}{D-1}+\frac{1}{2(D-1)^{2}(D-3)}{\cal K}^{2}\cdots\right),$$
Here $\Sigma$ is the entangling surface,with induced metric $\gamma$, and $\partial\Sigma$ is its boundary, with induced metric $\tilde{\gamma}$. The extrinsic curvature ${\cal K}$ refers to the extrinsic curvature of $\partial\Sigma$ embedded into a spatial slice of the boundary of the bulk manifold.
The first counterterm becomes logarithmic for $D=1$. Only the first
counterterm given above is needed for gravity in four bulk dimensions ($D=2$). The second counterterm becomes logarithmic at $D=3$ and is needed in the form given above for $D>3$. Additional counterterms involving higher order curvature invariants are needed for $D\geq 5$. The counterterms for entangling surfaces in general asymptotically locally $AdS$ spacetimes can be found in section 5.
We then show that the renormalized entanglement entropy for a disk region in a three dimensional conformal field theory dual to $AdS_{4}$ is in precise agreement with the holographically renormalized Euclidean action for $AdS_{4}$ with spherical slicing, i.e. the CHM map [2] holds at the level of renormalized quantities.
In section 4 we consider holographic RG flows in four bulk dimensions which respect Poincaré invariance of the dual theory. For flows driven by a single scalar we compute the renormalized Ryu-Tayakanagi functional, expressing the counterterms in terms of the superpotential associated with the flow.
We then use the renormalized entanglement entropy to explore the change in the F quantity along RG flows. In particular, we consider a disk entangling region and calculate the change the renormalized entanglement entropy (and hence F quantity) perturbatively in the source of the relevant deformation, $\phi_{(0)}$. For operators of dimension
$3/2<\Delta_{+}<3$ we find that
$$\delta S_{\rm ren}=\frac{\pi}{16(2\Delta_{+}-5)G_{4}}\phi_{(0)}^{2}R^{2(3-\Delta_{+})}+{\cal O}\left(\phi_{(0)}^{3}\right),$$
(2)
where $R$ is the radius of the disk entangling region while $\delta S_{\rm ren}=0$ for exactly marginal operators. This quantity is clearly negative for $\Delta_{+}<5/2$ which, since $\delta S_{\rm ren}=-\delta F$, corresponds to an increase in the F quantity. We should note however that the corresponding deformations on the three sphere are inhomogeneous and do not therefore correspond to RG flows which respect the $SO(4)$ invariance.
Direct calculation of the F quantity for $SO(4)$ invariant RG flows on $S^{3}$ driven by such operators also gives an increase in the F quantity to quadratic order in the source, see the companion paper [12]. It would be interesting to understand whether such flows are unphysical or if the strong version of the proposed F theorem is indeed violated.
In section 5 we show that the holographically renormalized entanglement entropy can be obtained from the holographically renormalized action. Using the replica trick,
the entropy associated with a density matrix $\rho$ is expressed as
$$S=-n\partial_{n}\left[\log Z(n)-n\log Z(1)\right]_{n=1}$$
(3)
where $Z(n)={\rm Tr}(\rho^{n})$ and $Z(1)={\rm Tr}(\rho)$ is the usual partition function. If we are interested in the entropy of a thermal state, then $Z(n)$ is constructed by extending the period of the thermal circle by a factor of $n$. In the case of entanglement entropy, $Z(n)$ is constructed by extending the period of the circle around the boundary of the entangling region by a factor of $n$, where implicitly $n$ is an integer. Assuming that the resulting expression is analytic in $n$, one can obtain the entropy by analytically continuing to $n=1$.
Holographically $Z(n)$ can be computed in terms of the onshell Euclidean action [3] as
$$S=n\partial_{n}\left[I(n)-nI(1)\right]_{n=1}.$$
(4)
Here $I(1)$ represents the onshell Euclidean action for the bulk geometry while $I(n)$ represents the onshell Euclidean action for the replica bulk geometry.
For a thermal state, the bulk geometry associated with $Z(1)$ is a black hole and the replica is constructed by extending the period of the thermal circle by a factor of $n$.
For the entanglement entropy, the bulk geometry associated with $Z(1)$ corresponds to the usual bulk dual of the given state in the field theory and
the replica is constructed by extending the period of the circle around the entangling region boundary by a factor of $n$. Following the same logic as in Lewkowycz-Maldacena
[3], the expression (4) localises on the minimal surface corresponding to the extension of the boundary of the entangling region into the bulk. However, the entangling surface itself has area divergences, unlike the black hole setup analysed in detail in [3].
In section 5 we show that the renormalized entanglement entropy can be expressed in terms of the renormalized onshell action i.e.
$$S_{\rm ren}=n\partial_{n}\left[I_{\rm ren}(n)-nI_{\rm ren}(1)\right]_{n=1}.$$
(5)
In particular, using the standard counterterms for asymptotically locally $AdS$ spacetimes [11], together with results on the curvature invariants of the replica space
[13, 14], one obtains exactly the same $S_{\rm ren}$ as computed directly via area renormalization. Thus, the renormalization scheme for the entanglement entropy is inherited directly from the renormalization scheme used for the partition function.
This result provides evidence for the applicability of the replica trick in the holographic context. Note that the derivation of the entanglement entropy functional from the Euclidean action functional requires only the local geometry of the replica; any potential anomalies in the replica symmetry do not affect the derivation. The holographic renormalization counterterms for higher derivative gravity theories such as Gauss-Bonnet also imply counterterms for the entanglement entropy, as we discuss at the end of section 5.
The plan of this paper is as follows. In section 2 we review the renormalization of entanglement entropy by differentiation. In section 3 we setup area renormalization for entangling surfaces in $AdS$ spacetimes, and show that the renormalized entanglement entropy for disk regions in $AdS_{4}$ indeed agrees with the F quantity. In section 4 we consider entanglement entropy for RG flows while in section 5 we show how the renormalized entanglement entropy can be obtained from the renormalized action via the replica trick. We conclude in section 6.
2 Renormalization by differentiation
In previous works, the finite terms in the entanglement entropy have been isolated by differentiation of the entanglement entropy. In the case of a strip of width $R$, UV divergent contributions to the entanglement entropy in a local quantum field theory are necessarily independent of $R$ and therefore
$$S_{R}=R\frac{\partial S}{\partial R}$$
(6)
is finite. This expression has been used in a number of earlier works, including [15, 16, 17].
For a spherical entangling region, the radius of the sphere controls the local curvature of the boundary of the entangling region and therefore it is no longer true that UV divergences are independent of the scale of the entangling region. In [18] it was noted that the following quantity
$$F(R)=-S(R)+R\frac{\partial S}{\partial R}$$
(7)
is manifestly finite in any 3d field theory which has a UV fixed point. (Analogous expressions for general dimensions were given in [18].) In particular, for a three-dimensional CFT the regulated entanglement entropy for a disc entangling region is
$$S_{\rm reg}=\frac{a_{-1}R}{\delta}+a_{0}$$
(8)
where $\delta\ll 1$ is the UV cutoff and $(a_{0},a_{-1})$ are constants. Then by construction
$$F(R)=-a_{0}.$$
(9)
For theories with a holographic dual one can show (see section 3) that
$$S_{\rm reg}=\frac{\pi}{2G_{4}}\left(\frac{R}{\delta}-1\right)$$
(10)
and therefore
$$F(R)=\frac{\pi}{2G_{4}}.$$
(11)
The normalization of (7) is chosen so that the latter indeed agrees with the F quantity.
The renormalized entanglement entropy defined by (7) has both positive and negative features. On the positive side, there is evidence that $F(R)$ behaves monotonically as a function of $R$ in free field theory and holographic examples [19, 20]. Also by construction
$$\frac{\partial F}{\partial R}=R\frac{\partial^{2}S}{\partial R^{2}}$$
(12)
and strong subadditivity of the entanglement entropy implies that in any Poincaré invariant field theory $\partial^{2}S/\partial R^{2}\leq 0$ [21], so $F(R)$ is a non-increasing function of the radius $R$.
Let us suppose we deform a conformal field theory by an operator ${\cal O}_{\Delta}$ of dimension $\Delta<3$:
$$I_{CFT}\rightarrow I_{CFT}+\int d^{3}x\sqrt{h}\lambda{\cal O}_{\Delta}.$$
(13)
The dimension of $\lambda$ is then $(3-\Delta)$; the coupling provides another dimensionful scale and it is no longer the case that (8) are the only divergences. There are in general additional divergences which are analytic in the deformation parameter $\lambda$ and hence for a disk region the change in the entanglement entropy under the relevant deformation is
$$\delta S_{\rm reg}=a_{5-2\Delta}\frac{\lambda^{2}R}{\delta^{\Delta-5/2}}+a_{8-3\Delta}\frac{\lambda^{3}R}{\delta^{\Delta-8/3}}+\cdots$$
(14)
where the coefficients $a_{m}$ are dimensionless. Hence for $\Delta>5/2$ the relevant deformation generates additional UV divergences in the entanglement entropy; additional divergences arise for $\Delta>3-1/n$. The form of this expression follows from conformal perturbation theory; in particular the term linear in $\lambda$ vanishes, while all divergences scale extensively with the length of the boundary of the entangling region. By construction $F(R)$ is finite for all such deformations although it is not a priori clear that $F(R)$ agrees with the F quantity.
On the negative side, there is evidence that $F(R)$ is not stationary at a UV fixed point [22]. Consider perturbations of a two-dimensional CFT by a slightly relevant operator of dimension $2-\delta_{\Delta}$. Then Zamoldchikov’s c-function behaves as
$$c(g)=c_{UV}-g^{2}\delta_{\Delta}+{\cal O}(g^{3})$$
(15)
where $g$ is the renormalised coupling. For a theory with several coupling constants
$$\frac{\partial c}{\partial g^{i}}=G_{ij}\beta^{j}$$
(16)
where $G_{ij}$ is the Zamalodchikov metric and $\beta^{j}=\mu\frac{\partial g^{j}}{\partial\mu}$ are the beta functions. Then non-singularity of the Zamalodchikov metric guarantees the stationarity of the $c$ function in two dimensions. In [22] it was shown that the proposed $F(R)$ is not stationary in this sense at the UV fixed point in free massive scalar field theory examples.
Another drawback of the definition of the renormalized entanglement entropy (7) is that the definition is only applicable to disk entangling regions, or to regions which are characterized by one overall scale. This drawback is not an issue for applications to the F theorem, for which only disk regions are needed, but prevents using (7) to explore the general shape dependence of entanglement entropy.
The renormalization that we propose in this paper by contrast is inherited directly from the renormalization of the partition function, making scheme dependence and the relation to the F quantity manifest, and is applicable to any shape entangling region. Moreover, our renormalization is applicable in theories which are not conformal in the UV.
3 Renormalized entanglement entropy in anti-de Sitter
In this section we will define the renormalized area of (static) entangling surfaces in anti-de Sitter. We parameterise the $AdS_{d+1}$ metric as
$$\mathrm{d}s^{2}=\frac{\mathrm{d}\rho^{2}}{4\rho^{2}}+\frac{1}{\rho}\eta_{\mu\nu}\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}$$
(17)
where $\rho\to 0$ corresponds to the conformal boundary and $\eta_{\mu\nu}$ is the Minkowski metric.
The Ryu-Takayanagi function for the entanglement entropy is the area functional for a codimension two surface:
$$S=\frac{1}{4G_{d+1}}\int_{\Sigma}\mathrm{d}^{D}\sigma^{\alpha}\,\sqrt{\gamma}$$
(18)
where $G_{d+1}$ denotes the Newton constant (with the number of spatial dimensions in the field theory being $D=(d-1)$) and $\gamma$ is the determinant of the induced metric on the surface. Throughout this section we work in a static setup, in which the entangling surface is independent of time. To find the bulk minimal surface $\Sigma$, we solve the equations of motion following from (18), subject to boundary conditions which define the entangling region in the dual field theory. In particular, as shown in Figure 1, the minimal surface $\Sigma$ has a conformal boundary $\partial\Sigma$ as $\rho\rightarrow 0$ which is conformal to the boundary $\partial A$ of the entangling region $A$ in the dual field theory.
When one evaluates the onshell value of the functional (18), it has area divergences which may conveniently be regulated by setting $\rho=\epsilon$, see Figure 2. Let us denote the bulk manifold as ${\cal M}$ and the regulated
conformal boundary at $\rho=\epsilon$ as $\partial{\cal M}_{\epsilon}$.
Since the entangling surface itself is asymptotically locally hyperbolic, the regulated functional (18) diverges as
$$S_{\rm reg}\sim\frac{{\cal A}_{\partial A}}{\epsilon^{\frac{d}{2}-1}}+\cdots$$
(19)
where ${\cal A}_{\partial A}$ is the area of the $(d-2)$-dimensional boundary of the entangling region $\partial A$.
Following the principles of [9, 10, 11]
we can now define a renormalized functional $S_{\rm ren}$ as
$$S_{\rm ren}={\cal L}_{\epsilon\rightarrow 0}\left(S_{\rm reg}+S_{ct}\right)$$
(20)
where the counterterm action $S_{ct}$ is defined in terms of covariant properties of the boundary of the minimal surface and of the cutoff surface. Let the induced metric on the cutoff surface be $h_{\mu\nu}$ and the metric on the boundary of the minimal surface be $\tilde{\gamma}_{ab}$. Let us further denote the Ricci scalar of the boundary of the minimal surface as ${\cal R}$, with the corresponding Ricci tensor being ${\cal R}_{ab}$. Similarly we denote
the extrinsic curvature of the minimal surface embedded into the cutoff surface as ${\cal K}_{ab}$ with trace ${\cal K}$ . Then counterterms must be expressible as
$$S_{\rm ct}=\int_{\partial\Sigma}\mathrm{d}^{D-1}x\sqrt{\tilde{\gamma}}{\cal L}\left({\cal K},{\cal R},{\cal R}_{ab}{\cal R}^{ab},{\cal K}_{ab}{\cal K}^{ab},\cdots\right),$$
(21)
i.e. as a functional of extrinsic and intrinsic curvature invariants. In our setup there are three extrinsic curvatures arising from the following three different embeddings: the embedding of $\Sigma_{\varepsilon}$ in $\mathcal{M}_{\varepsilon}$, the embedding of $\partial\Sigma_{\varepsilon}$ in $\Sigma_{\varepsilon}$, and the embedding of $\partial\Sigma_{\varepsilon}$ in $\partial\mathcal{M}_{\varepsilon}$. We should emphasise that it is the final one which is relevant for the counterterms, as the first two are not intrinsic to the regulated boundary.
There is a further restriction on the allowed counterterms. The entanglement entropy of region $A$ is the same as the entanglement entropy of the complementary region $B$. If we require that the renormalized entanglement entropy satisfies the same property, then the counterterms should only depend on even powers of the extrinsic curvature ${\cal K}$, since the extrinsic curvature of $A$ is minus the extrinsic curvature of the complementary region $B$.
Finally, we should note that the intrinsic and extrinsic curvature are related by Gauss-Codazzi relations. Throughout this section we will be interested in the case in which the background for the dual field theory is flat, in which case
$${\cal R}={\cal K}^{2}-{\cal K}_{ab}{\cal K}^{ab},$$
(22)
with analogous Gauss-Codazzi relations holding between higher order scalar invariants of the intrinsic and extrinsic curvature.
3.1 Explicit computation of counterterms
Let us now express the area functional as
$$S=\frac{1}{4G_{d+1}}\int\mathrm{d}\rho\int\,\mathrm{d}^{D-1}\sigma^{a}\,\sqrt{\gamma}$$
(23)
where $\gamma_{\alpha\beta}=g_{mn}\partial_{\alpha}x^{m}\partial_{\beta}x^{n}$, $g_{mn}$ is the metric on the $AdS$ target space and $x^{m}(x^{\alpha})$ defines the embedding in terms of the worldvolume coordinates $\sigma^{\alpha}$. We have implicitly fixed a static gauge, in which the time coordinate $t$ is constant and $\rho$ is one of the worldvolume coordinates, i.e. $\sigma^{\alpha}=\{\rho,\sigma^{a}\}$. The spatial coordinates $x^{i}$ are then functions of $\rho$ and $\sigma^{a}$ and $D$ represents the number of spatial directions in the boundary theory.
In such a gauge the induced metric on the minimal surface is
$$\displaystyle\gamma_{\rho\rho}$$
$$\displaystyle=\frac{1}{4\rho^{2}}+\frac{1}{\rho}x^{i}_{,\rho}x^{i}_{,\rho}$$
(24)
$$\displaystyle\gamma_{\rho a}$$
$$\displaystyle=\frac{1}{\rho}x^{i}_{,\rho}x^{i}_{,a}\qquad\gamma_{ab}=\frac{1}{\rho}x^{i}_{,a}x^{i}_{,b}$$
where we denote $x^{i}_{,\rho}=\partial_{\rho}x^{i}$ and $x^{i}_{,a}=\partial_{\sigma^{a}}x^{i}$. One can often (but not always) further gauge fix, setting $x^{a}=\sigma^{a}$ and $x^{D}\equiv y(\rho,x^{a})$, so that
$$\displaystyle\gamma_{\rho\rho}$$
$$\displaystyle=\frac{1}{4\rho^{2}}+\frac{1}{\rho}y_{,\rho}y_{,\rho}$$
(25)
$$\displaystyle\gamma_{\rho a}$$
$$\displaystyle=\frac{1}{\rho}y_{,\rho}y_{,a}\qquad\gamma_{ab}=\frac{1}{\rho}(\delta_{ab}+y_{,a}y_{,b}),$$
reflecting the fact that a codimension one spatial minimal surface has only one transverse direction. Note however that such gauge fixing cannot be used to describe minimal surfaces with cusps, but in this paper we restrict to the case of surfaces without cusps.
The gauge fixed minimal surface action is given by
$$\displaystyle S$$
$$\displaystyle=\frac{1}{4G_{d+1}}\int\mathrm{d}\rho\int\mathrm{d}^{D-1}x\,\left(\frac{1}{4\rho^{D+1}}(1+y_{,a}y_{,a}+4\rho y_{,\rho}^{2})\right)^{1/2}$$
$$\displaystyle=\frac{1}{4G_{d+1}}\int\mathrm{d}\rho\int\mathrm{d}^{D-1}x\,\frac{1}{2\rho^{(D+1)/2}}m(\rho,x^{a})$$
(26)
where we have introduced the shorthand $m(\rho,x^{a})=\sqrt{1+y_{,a}y_{,a}+4\rho y_{,\rho}^{2}}$.
The regulated action is then of the form
$$\displaystyle S_{\rm reg}$$
$$\displaystyle=\frac{1}{4G_{d+1}}\int_{\epsilon}\mathrm{d}\rho\int\mathrm{d}^{D-1}x\,\left(\frac{1}{4\rho^{D+1}}(1+y_{,a}y_{,a}+4\rho y_{,\rho}^{2})\right)^{1/2}$$
(27)
$$\displaystyle=\frac{1}{4G_{d+1}}\int_{\partial\Sigma}\mathrm{d}^{D-1}x\sum\epsilon^{-k}(a_{k}(x)+\log\epsilon b_{k}(x))+\cdots$$
where the explicit powers arising in the divergences and their coefficients $(a_{k}(x),b_{k}(x))$ are determined by analysing solutions to the minimal surface equations with the required boundary conditions asymptotically near the conformal boundary.
Note that the action does not depend explicitly on $y$ and the minimal surface equation is:
$$\displaystyle 0$$
$$\displaystyle=\partial_{a}\left(\frac{y_{,a}}{m\,\rho^{3/2}}\right)+\partial_{\rho}\left(\frac{y_{,\rho}}{m\,\rho^{1/2}}\right),$$
(28)
which should be solved near $\rho=0$ subject to the boundary condition
$${\cal L}_{\rho\rightarrow 0}\left(y(\rho,x^{a})\right)=y_{(0)}(x^{a}),$$
(29)
where $y_{(0)}(x^{a})$ specifies the entangling region in the dual geometry.
We wish to solve this equation iteratively for $y(\rho,x^{a})$ as a series expansion in $\rho$. We consider the following Taylor series expansions for $y(\rho,x^{a})$:
$$\displaystyle y(\rho,x^{a})$$
$$\displaystyle=y_{(0)}(x)+y_{(\beta_{1})}(x)\rho^{\beta_{1}}+y_{(\beta_{2})}(x)\rho^{\beta_{2}}+\ldots$$
(30)
where we assume that $0<\beta_{1}<\beta_{2}<\ldots$. To solve the PDE we insert these expansions into the minimal surface equation and set $\rho=0$. We then fix $\beta_{1}$ and $y_{(\beta_{1})}$ to solve the resulting equation such that $y_{(0)}$ remains unconstrained. We then differentiate the minimal surface equation with respect to $\rho$ and repeat to find $\beta_{2}$.
After substituting the expansions into the minimal surface equation, one finds that the leading order terms are $\rho^{0}$ and $\rho^{\beta_{1}-1}$. To leave $y_{(0)}$ unconstrained we must therefore set $\beta_{1}=1$ and deduce that:
$$y_{(1)}(x)=2\sqrt{1+y_{(0),a}y_{(0),a}}\partial_{a}\left(\frac{\partial_{a}y_{(0)}}{\sqrt{1+y_{(0),a}y_{(0),a}}}\right).$$
(31)
To find higher terms in the asymptotic expansion we can use radial derivatives of the minimal surface equations. Before carrying out this procedure, let us consider the regulated onshell action (27) and determine the leading divergences, which are
$$S_{\rm reg}=\frac{1}{4(D-1)G_{d+1}\epsilon^{\frac{1}{2}(D-1)}}\int_{\partial\Sigma}\mathrm{d}^{D-1}x\sqrt{1+y_{(0),a}y_{(0),a}}+{\cal O}\left(\epsilon^{\frac{3-D}{2}}\right)$$
(32)
for $D>1$. As anticipated above, this divergence scales with the area ${\cal A}_{\partial A}$ of the boundary of the entangling region
$${\cal A}_{\partial A}=\int_{\partial\Sigma}\mathrm{d}^{D-1}x\sqrt{1+y_{(0),a}y_{(0),a}}.$$
(33)
The case of $D=1$, corresponding to a dual two-dimensional conformal field theory, is degenerate. The divergence is logarithmic:
$$S_{\rm reg}=\frac{1}{8G_{3}}\Sigma_{k}y_{k}\log\epsilon$$
(34)
with $y_{k}$ being the endpoints of the intervals defining the entangling region. The required counterterm action is therefore
$$S_{\rm ct}=-\frac{1}{8G_{3}}\Sigma_{k}y_{k}\log\left(\frac{\epsilon}{\mu}\right),$$
(35)
where $\mu$ is an arbitrary renormalization scale.
3.2 Entangling surfaces in $AdS_{4}$
For minimal surfaces in $AdS_{4}$ the only divergence in the onshell functional is
$$S_{\rm reg}=\frac{1}{4G_{4}}\int_{\partial\Sigma}\mathrm{d}x\left(\frac{1}{\epsilon^{\frac{1}{2}}}\sqrt{1+y_{(0),x}y_{(0),x}}\right),$$
(36)
where the entangling region in the boundary is defined by a curve $y_{(0)}(x)$ in two dimensional space. Noting that the induced line element on the boundary of the entangling surface is
$$\gamma^{h}_{xx}=\frac{1}{\epsilon}(1+y_{,x}y_{,x})$$
(37)
the divergence is manifestly removed by the covariant counterterm
$$S_{\rm ct}=-\frac{1}{4G_{4}}\int_{\partial\Sigma}\mathrm{d}x\sqrt{\tilde{\gamma}},$$
(38)
where $\tilde{\gamma}$ is the determinant of the induced metric on $\partial\Sigma$.
This is the only possible divergent counterterm but the following counterterm is finite:
$$S_{\rm ct}=\frac{a_{s}}{4}\int_{\partial\Sigma}\mathrm{d}x\sqrt{\tilde{\gamma}}{\cal K}$$
(39)
where ${\cal K}$ denotes the trace of the extrinsic curvature of the boundary of the minimal surface embedded into the regulated cutoff surface. For a curve $y(x,\epsilon)$ embedded into the cutoff surface
$$ds^{2}=\frac{1}{\epsilon}\left(-dt^{2}+dx^{2}+dy^{2}\right)$$
(40)
the trace of the extrinsic curvature is
$${\cal K}=\epsilon^{\frac{1}{2}}\frac{y_{,xx}}{(1+y_{,x}^{2})^{\frac{3}{2}}}$$
(41)
and thus
$$\sqrt{\gamma^{h}}{\cal K}=\frac{y_{,xx}}{(1+y_{,x}^{2})}=\frac{y_{(0),xx}}{(1+y_{(0),x}^{2})}+{\cal O}(\epsilon),$$
(42)
which is indeed finite.
Thus the complete renormalized action for the minimal surface is
$$S_{\rm ren}=\frac{1}{4G_{4}}\int_{\Sigma}d^{2}\sigma\sqrt{\gamma}+\frac{1}{4G_{4}}\int_{\partial\Sigma}dx\sqrt{\tilde{\gamma}}\left(a_{s}{\cal K}-1\right).$$
(43)
Note that terms depending on higher powers of the extrinsic curvature cannot contribute in the limit $\epsilon\rightarrow 0$. The finite counterterm is however not
consistent with the requirement that the renormalized entropy for any region is equal to that of its complement, and we must therefore set $a_{s}=0$.
As an example, let us evaluate the renormalized action for a disk entangling region, of radius $R$. The exact solution for the minimal surface is conveniently expressed
in terms of the following coordinates
$$ds^{2}=\frac{d\rho^{2}}{4\rho^{2}}+\frac{1}{\rho}\left(-dt^{2}+dr^{2}+r^{2}d\phi^{2}\right)$$
(44)
as the circularly symmetric surface at constant time:
$$r^{2}+\rho=R^{2}.$$
(45)
The renormalized action for this surface is then
$$S_{\rm ren}=-\frac{\pi}{2G_{4}}.$$
(46)
Note that this is independent of the choice of the radius $R$.
Implicitly our Newton constant has been fixed to be dimensionless, as we chose the anti-de Sitter metric to have unit radius, absorbing the curvature radius into the overall prefactor of the bulk action.
To reinsert the AdS radius we need only rescale the bulk metric by $\ell$ and the covariant counterterm by a further $\ell$. The result of these insertions is to simply rescale the results for the entanglement entropy by $\ell^{2}$:
$$\displaystyle S_{\rm ren}$$
$$\displaystyle=-\frac{\pi\ell^{2}}{2G_{N}},$$
(47)
where the Newton constant $G_{N}$ now has the standard dimensions.
Since the dual field theory is conformal there is no other scale apart from $R$ and therefore $S_{\rm ren}$, which is dimensionless, cannot depend explicitly on $R$.
Next we consider an entangling surface of two infinitely long parallel lines with separation $R$. We will regulate the lines to have length $L$ and by symmetry we may choose these lines to lie in the $x$ direction and to be located at $y=\pm\frac{R}{2}$.
The minimal surface can be characterized by worldsheet coordinates $(\rho,x)$ and by symmetry the transverse coordinate $y$ depends only on $\rho$. The surface equations can be solved to obtain
$$y(\rho)=\pm\left(-\frac{R}{2}+\frac{\rho^{3/2}}{3\rho_{0}}{}_{2}F_{1}\left(\frac{1}{2},\frac{3}{4};\frac{7}{4},\frac{\rho^{2}}{\rho_{0}^{2}}\right)\right).$$
(48)
We can also rewrite this hypergeometric function in terms of the incomplete beta function $B_{z}(a,b)$ using the identity
$${}_{2}F_{1}(a,b;1+b;z)=bz^{-b}B_{z}(b,1-a).$$
(49)
The surface has a turning point at $\rho_{0}$, where by symmetry $y(\rho_{0})=0$, and hence
$$y(\rho_{0})=0\implies\rho_{0}=\frac{9\Gamma(5/4)^{2}}{4\pi\Gamma(7/4)^{2}}R^{2}.$$
(50)
The regularised holographic entanglement entropy is then given by
$$S_{\rm reg}=\frac{L}{8G_{4}}\int_{\varepsilon}^{\rho_{0}}\mathrm{d}\rho\frac{\rho_{0}}{\sqrt{\rho^{3}(\rho_{0}^{2}-\rho^{2})}},$$
(51)
This integral is elliptic and can be calculated analytically using
$$\displaystyle\int\frac{1}{\sqrt{w^{3}(a^{2}-w^{2})}}\mathrm{d}w$$
$$\displaystyle=$$
$$\displaystyle 2\frac{2}{a^{2}\sqrt{w}}\sqrt{a^{2}-w^{2}}$$
$$\displaystyle+\frac{2}{\sqrt{a^{3}}}\left(F\left(\left.\sin^{-1}\left(\sqrt{\frac{w}{a}}\right)\right|-1\right)-E\left(\left.\sin^{-1}\left(\sqrt{\frac{w}{a}}\right)\right|-1\right)\right)$$
where $F(\phi|k^{2})$ and $E(\phi|k^{2})$ are the incomplete elliptic integrals of the first and second kind respectively.
The renormalized holographic entanglement entropy is:
$$S_{\rm ren}=-\frac{\sqrt{2}\pi^{2}\Gamma(7/4)}{3G_{4}\Gamma(1/4)^{2}\Gamma(5/4)}\frac{L}{R}$$
(53)
Note that in the above calculation we have implicitly assumed that $L\gg R$ and that there are no contributions from the lines $x=\pm L/2$, $-R/2\leq y\leq R/2$. To take the limit of $L\rightarrow\infty$ we can calculate the renormalized entropy density
$$s_{\rm ren}={\cal L}_{L\rightarrow\infty}\left(\frac{S_{\rm ren}}{L}\right)=-\frac{\sqrt{2}\pi^{2}\Gamma(7/4)}{3RG_{4}\Gamma(1/4)^{2}\Gamma(5/4)}.$$
(54)
Finally let us consider the half plane entangling region with a boundary at $y=0$; again we regulate the $x$ direction to have length $L$. The bulk minimal surface has worldsheet coordinates $(\rho,x)$ and by symmetry $y=0$ over the surface. The regularised holographic entanglement entropy is
$$S_{\rm reg}=\frac{L}{8G_{4}}\int^{\infty}_{\epsilon}\frac{d\rho}{\rho^{\frac{3}{2}}}=\frac{L}{4G_{4}\epsilon^{\frac{1}{2}}}$$
(55)
and this term exactly cancels the counterterm giving
$$S_{\rm ren}=0$$
(56)
which was to be expected since there is no other scale in the problem but $L$ and the dual theory is conformal.
The calculation of the renormalized area of a two-dimensional minimal surface in four bulk dimensions has arisen in other contexts, including Wilson loops. In particular,
anomalies were discussed in [10] while
the counterterm involving the regulated length of the boundary of the surface was discussed in the context of Wilson loops in [23]; the counterterm was derived
by requiring a well-defined variational principle. The relation of holographic renormalization to variational principles for minimal surfaces was discussed in detail in
[24].
Minimal surfaces in hyperbolic spaces were also analysed in [25]: generalizing [10], it was noted that submanifold observables have conformal anomalies for specific codimensions. In particular, the results of [25] imply that codimension two minimal surfaces in odd bulk dimensions have logarithmic divergences in their regulated volumes. This is consistent with our $D=1$ result above, and the $D=3$ result we will give below.
According to the results of [25] the renormalized area of a codimension two minimal surface in an even dimensional hyperbolic space should be a conformal invariant. This is not however apparent from the above results: the renormalized entropy of the half plane was found to be zero (56), while the renormalized entropy of the disk is finite and negative. Yet, as is well known, one can find a conformal bijective map between the disk and the half plane and therefore these entangling regions are conformally equivalent. We will explore this issue further in the next section.
The renormalized entropy for the strip entangling region is negative. This is unsurprising: in [15, 16] the entanglement entropy for free scalars and fermions was calculated for strip entangling regions and it was found that the entanglement entropy contains finite terms of the form
$$S_{\rm finite}=-k\frac{L}{R}$$
(57)
where again $L$ is the regulated length of the strip, $R$ is its width (with $L/R\gg 1$) and $k$ is a positive constant, which takes the value of $k=0.039$ for a real scalar and $k=0.072$ for a Dirac fermion.
3.2.1 Relation to F theorem
More generally, we should be unsurprised about finding negative values for the renormalized entanglement entropy.
The conjectured F-theorem in three dimensions is the following. For a three-dimensional CFT we define the F quantity in terms of the (renormalized) partition function of the theory on a three sphere [8], i.e.
$$F=-\ln Z_{S^{3}}$$
(58)
and then the F theorem states that $F_{UV}\geq F_{IR}$. More precisely, in [8] it was conjectured that $F$ is positive in a unitary CFT, that it decreases along any RG flow and that it is stationary at fixed points. Support for the conjecture can be found in [8, 26, 27] and many subsequent works.
In odd spacetime dimensions
the finite terms in the entanglement entropy of a spherical region
$$S_{\rm finite}=(-)^{\frac{1}{2}(d-1)}2\pi a_{d}$$
(59)
are conjectured to satisfy the relation $(a_{d})_{UV}\geq(a_{d})_{IR}$ for any RG flows between fixed points [28].
Indeed it has been shown that the sphere partition function and the sphere entanglement entropy are proportional
using the CHM map [2], thus establishing a connection between the F theorem and monotonous running of the finite part of the disk entanglement entropy. In three
dimensions
$$F=-2\pi a_{3}$$
(60)
and hence positivity of F is equivalent to negativity of the finite parts of the entanglement entropy.
To understand the relation between (58) and (59) it is useful to recall the arguments of CHM [2] in more detail. Let us parameterise the flat three-dimensional metric in as
$$ds^{2}=-dt^{2}+dr^{2}+r^{2}d\phi^{2}$$
(61)
Now consider the following change of coordinates
$$\displaystyle t$$
$$\displaystyle=$$
$$\displaystyle R\frac{\cos\theta\sinh\tau/R}{(1+\cos\theta\cosh\tau/R)};$$
(62)
$$\displaystyle r$$
$$\displaystyle=$$
$$\displaystyle R\frac{\sin\theta}{(1+\cos\theta\cosh\tau/R)};$$
so that the metric becomes
$$ds^{2}=\Omega^{2}\left(-\cos^{2}\theta d\tau^{2}+R^{2}(d\theta^{2}+\sin^{2}\theta d\phi^{2})\right)$$
(63)
with conformal factor
$$\Omega=(1+\cos\theta\cosh\tau/R)^{-1}.$$
(64)
One can clearly absorb the $R$ dependence as an overall factor by introducing $\tilde{\tau}=\tau/R$, so that the metric is conformal to the static patch of de Sitter space. Since $0\leq\theta<\pi/2$ the new coordinates cover $0<r<R$, i.e. the disk of radius $R$ in the original flat coordinates, with $\theta\rightarrow\pi/2$ (the cosmological horizon) corresponding to $r=R$. The limits $\tau\rightarrow\pm\infty$ correspond to $t\rightarrow\pm R$ and therefore the new coordinates cover the causal development of the disk $r\leq R$ from $t=0$.
Modular transformations inside the causal development act as time translations in de Sitter space, and therefore the state in the de Sitter geometry is thermal with $\beta=2\pi R$.
One can then identify the entanglement entropy for the disc of radius $R$ in flat space with the thermodynamic entropy of the thermal state in de Sitter space, which in turn is given by
$$S_{\rm deSitter}=-W$$
(65)
where $W=-\ln Z$ is the free energy of the partition function $Z$. This relation is the origin of the above statement that the disc entanglement entropy is related to the partition function on the sphere, since the analytic continuation of de Sitter is the three-dimensional sphere.
The corresponding Euclidean transformations begin from the metric
$$ds^{2}=dt_{E}^{2}+dr^{2}+r^{2}d\phi^{2}$$
(66)
with the transformations being
$$\displaystyle t_{E}$$
$$\displaystyle=$$
$$\displaystyle R\frac{\cos\theta\sin\tau_{E}/R}{(1+\cos\theta\cos\tau_{E}/R)};$$
(67)
$$\displaystyle r$$
$$\displaystyle=$$
$$\displaystyle R\frac{\sin\theta}{(1+\cos\theta\cos\tau_{E}/R)};$$
so that the metric becomes
$$ds^{2}=\Omega^{2}\left(\cos^{2}\theta d\tau_{E}^{2}+R^{2}(d\theta^{2}+\sin^{2}\theta d\phi^{2})\right)$$
(68)
with conformal factor
$$\Omega=(1+\cos\theta\cos\tau_{E}/R)^{-1}.$$
(69)
In the transformed coordinates the Euclidean time $\tau_{E}$ is periodic with period $2\pi R$ for the sphere to be regular and $0\leq\theta<\pi/2$.
Implicitly the finite parts of the partition function on the $S^{3}$ are computed by renormalization; the CHM map thus relates the (renormalized) F quantity to the
the corresponding renormalized entanglement entropy i.e.
$$F=-S_{\rm ren}$$
(70)
with $F$ being positive and decreasing along an RG flow. For the disk entangling region we thus find holographically that
$$F=\frac{\pi}{2G_{4}}$$
(71)
which is indeed positive.
Let us now review the evaluation of the partition function on $S^{3}$ for a conformal field theory with a holographic dual described by Einstein gravity. The renormalized partition function is then calculated by evaluating the renormalized Euclidean action [11]:
$$I=-\frac{1}{16\pi G_{4}}\int d^{4}x\sqrt{g}(R_{g}+6)+\frac{1}{8\pi G_{4}}\int d^{3}x\sqrt{h}(1-\frac{R}{4}),$$
(72)
where $R_{g}$ is the bulk Ricci scalar and $R$ is the Ricci scalar for the boundary metric $h$.
For the $AdS_{4}$ geometry with spherical slicing
$$ds^{2}=d\rho^{2}+\sinh^{2}\rho d\Omega_{3}^{2}$$
(73)
the renormalized onshell action is then
$$I=\frac{\pi}{2G_{4}}.$$
(74)
Comparing (74) with (71), the values indeed agree.
Note that there is no ambiguity in the holographically renormalized action (72): there are no candidate covariant finite counterterms. We will explain further in section 5 how the renormalization schemes for the bulk action and for the entanglement entropy are related.
3.3 Renormalization for AdS in general dimensions
In this section we describe the holographic renormalization of the entanglement entropy for AdS in general dimensions, noting the generic forms of possible counterterms, anomalies, and finding the first two counterterms.
We begin by establishing the notational conventions we will use in this section. We will take our bulk manifold $\mathcal{M}$ to be $AdS_{D+2}$ and will work exclusively in coordinates in which the metric takes the form
$$\mathrm{d}s^{2}_{\mathcal{M}}=g_{mn}\mathrm{d}x^{m}\mathrm{d}x^{n}=\frac{\mathrm{d}\rho^{2}}{4\rho^{2}}-\frac{1}{\rho}\mathrm{d}t^{2}+\frac{1}{\rho}\delta_{ij}\mathrm{d}x^{i}\mathrm{d}x^{j}$$
(75)
where $i,j=1,\ldots,D$ are the boundary spatial directions. The entangling surface $\Sigma$ is a codimension 2 surface of $\mathcal{M}$ satisfying the appropriate boundary conditions. We choose the coordinates on $\Sigma$ to be $(\rho,x^{a})$ where $a=1,\ldots,D-1$. The embedding of $\Sigma$ in $\mathcal{M}$ is then given by:
$$X^{m}=(\rho,t,x^{1},\ldots,x^{D-1},y(\rho,x^{a}))$$
(76)
where $t$ is a constant. This is an appropriate gauge whenever the boundary entangling region specified by $y(0,x^{a})$ is smooth.
We regulate the bulk as $\mathcal{M}_{\varepsilon}$ by restricting $\rho\geq\varepsilon>0$, and similarly define the regulated entangling surface $\Sigma_{\varepsilon}$ by the same restriction. The surface $\Sigma_{\varepsilon}$ is a constant time hypersurface of $\mathcal{M}_{\varepsilon}$.
The metric $\gamma_{\alpha\beta}$ on $\Sigma_{\varepsilon}$ is given by
$$\mathrm{d}s^{2}_{\Sigma_{\varepsilon}}=\left(\frac{1}{4\rho^{2}}+\frac{1}{\rho}y_{,\rho}^{2}\right)\mathrm{d}\rho^{2}+\frac{2}{\rho}y_{,\rho}y_{,a}\mathrm{d}x^{a}+\frac{1}{\rho}(\delta_{ab}+y_{,a}y_{,b})\mathrm{d}x^{a}\mathrm{d}x^{b}.$$
(77)
In this gauge the regulated bare entanglement entropy is given by
$$S_{\rm reg}=\frac{1}{4G_{D+2}}\int_{\partial\Sigma_{\varepsilon}}\mathrm{d}^{D-1}\int_{\varepsilon}^{\rho_{0}}\mathrm{d}\rho\frac{1}{2\rho^{(D+1)/2}}\sqrt{1+4\rho y_{,\rho}^{2}+y_{,a}^{2}}$$
(78)
where summation is implicit for the $a,b,\ldots$ indices.
From the action one can find the equation of motion for $y(\rho,x^{a})$, as in the previous section. Expanding the solution near the conformal boundary one finds:
$$y(\rho,x^{a})=y^{(0)}+y^{(1)}\rho+O(\rho^{2});\qquad y^{(1)}=\frac{1}{2(D-1)}\left(y^{(0)}_{,aa}-\frac{y^{(0)}_{,a}y^{(0)}_{,ab}y^{(0)}_{,b}}{1+{y^{(0)}_{,c}}^{2}}\right).$$
(79)
Note that this result agrees with the $D=2$ case we considered above.
Inserting the asymptotic expansion into the regulated functional yields for $AdS_{5}$ ($D=3$)
$$S_{\rm reg}=\frac{1}{4G_{5}}\int_{\partial\Sigma_{\varepsilon}}\mathrm{d}^{2}x(1+{y^{(0)}_{,c}}^{2})^{1/2}\left(\frac{1}{2\varepsilon}-\frac{y^{(0)}_{,a}y^{(1)}_{,a}+2{y^{(1)}}^{2}}{2(1+{y^{(0)}_{,b}}^{2})}\ln\varepsilon+\ldots\right)$$
(80)
where the ellipses denote finite terms. Similarly for $D>3$
$$S_{\rm reg}=\frac{1}{4G_{D+2}}\int_{\partial\Sigma_{\varepsilon}}\mathrm{d}^{D-1}x(1+{y^{(0)}_{,c}}^{2})^{1/2}\left(\frac{\varepsilon^{-\frac{(D-1)}{2}}}{D-1}+\frac{\varepsilon^{-\frac{(D-3)}{2}}}{D-3}\frac{y^{(0)}_{,a}y^{(1)}_{,a}+2{y^{(1)}}^{2}}{1+{y^{(0)}_{,b}}^{2}}+\ldots\right)$$
(81)
where the ellipses denote subleading divergences and terms that are finite as $\varepsilon\to 0$.
Our task is now to find counterterms which are integrals of covariant quantities defined on $\partial\Sigma_{\varepsilon}$, i.e. scalars constructed from the intrinsic and extrinsic curvature tensors. The induced metric on $\partial\Sigma_{\varepsilon}$, $\tilde{\gamma}_{ab}$ is given by
$$\mathrm{d}s^{2}_{\tilde{\gamma}}=\frac{1}{\varepsilon}(\delta_{ab}+y_{,a}y_{,b})\mathrm{d}x^{a}\mathrm{d}x^{b}$$
(82)
which has determinant
$$\tilde{\gamma}=\det(\tilde{\gamma}_{ab})=\varepsilon^{-\frac{D-1}{2}}(1+y_{,a}^{2})$$
(83)
by Sylvester’s determinant theorem.
Using the asymptotic expansion we can expand the volume form to first subleading order in $\varepsilon$ as:
$$\sqrt{\tilde{\gamma}}=\varepsilon^{-\frac{D-1}{2}}(1+{y^{(0)}_{,c}}^{2})^{1/2}\left(1+\varepsilon\frac{y^{(0)}_{,b}y^{(0)}_{,b}}{1+{y^{(0)}_{,c}}^{2}}+\ldots\right).$$
(84)
On dimensional grounds we can show that all curvature scalars will be at least $O(\varepsilon^{1/2})$ and so we can uniquely identify the leading divergence in $S_{\rm ren}$ as coming from the area divergence, as expected. Our first counterterm is therefore
$$S_{ct,1}=-\frac{1}{4G_{D+2}}\frac{1}{D-1}\int_{\partial\Sigma_{\varepsilon}}\mathrm{d}^{D-1}x\sqrt{\tilde{\gamma}}$$
(85)
which is again consistent with our previously found $AdS_{4}$ ($D=2$) result.
We now need to find the counterterms for the subleading divergences. Let us consider first the case of $D=3$.
Using integration by parts we can rewrite:
$$\int_{\partial\Sigma_{\varepsilon}}\mathrm{d}^{D-1}x\frac{y_{,A}^{(0)}y_{,A}^{(1)}}{(1+{y^{(0)}_{,C}}^{2})^{1/2}}=-\int_{\partial\Sigma_{\varepsilon}}\mathrm{d}^{D-1}x\frac{{y^{(1)}}^{2}}{(1+{y^{(0)}_{,C}}^{2})^{1/2}}$$
(86)
and hence for $D=3$
$$S_{\rm reg}+S_{ct,1}=-\frac{1}{8G_{5}}\int_{\partial\Sigma_{\varepsilon}}\mathrm{d}^{2}x\sqrt{\tilde{\gamma}}\frac{{y^{(1)}}^{2}}{1+{y_{,C}^{(0)}}^{2}}\ln\varepsilon+\ldots$$
(87)
To rewrite this term covariantly we note that the metric on a constant time hypersurface of the regulated boundary is given by
$$\mathrm{d}s^{2}_{D}=\tilde{g}_{ij}\mathrm{d}x^{i}\mathrm{d}x^{j}=\frac{1}{\varepsilon}\delta_{ij}\mathrm{d}x^{i}\mathrm{d}x^{j}$$
(88)
and the embedding of $\partial\Sigma_{\varepsilon}$ is given by $x^{D}\equiv y(\varepsilon,x^{a})$. The unit normal covector is then given by
$$n_{\flat}=\varepsilon^{-\frac{1}{2}}(1+y_{,c}^{2})^{-\frac{1}{2}}(y_{,a}\mathrm{d}x^{a}-\mathrm{d}x^{D})$$
(89)
From this we define the induced metric $\tilde{\gamma}_{ij}$, and the extrinsic curvature ${\cal K}_{ij}$ by
$$\tilde{\gamma}_{ij}=\tilde{g}_{ij}-n_{i}n_{j}\qquad{\cal K}_{ij}=\tilde{\gamma}^{k}_{i}\nabla_{k}n_{j}$$
(90)
where $\nabla_{k}$ is the covariant derivative with respect to $\tilde{g}_{ij}$. The trace of the extrinsic curvature is then given by
$${\cal K}=\frac{\varepsilon^{\frac{1}{2}}}{(1+y_{,c}^{2})^{\frac{1}{2}}}\left(y_{,aa}-\frac{y_{,a}y_{,ab}y_{,b}}{1+y_{,e}^{2}}\right)=2(D-1)\frac{\varepsilon^{\frac{1}{2}}y^{(1)}}{(1+{y^{(0)}_{,c}}^{2})^{\frac{1}{2}}}+\ldots$$
(91)
By contrast, the Ricci scalar has a qualitatively different structure:
$${\cal R}=\frac{2\varepsilon}{(1+y_{,e}^{2})}\left(y^{2}_{,aa}-y_{,ab}y_{,ab}-y_{,a}y_{,b}\frac{y_{,ab}y_{,cc}-y_{,ac}y_{,bc}}{1+y_{,d}^{2}}\right)$$
(92)
Comparing with (87) we can see that the required logarithmic counterterm is hence written in terms of the extrinsic curvature as
$$S_{ct,2}=\frac{1}{64G_{5}}\int_{\partial\Sigma_{\varepsilon}}\mathrm{d}^{2}x\sqrt{\tilde{\gamma}}{\cal K}^{2}\ln\frac{\varepsilon}{\mu},$$
(93)
with $\mu$ a cutoff scale.
Similarly for $D>3$ we can show that
$$S_{EE}+S_{ct,1}=\frac{1}{4G_{D+2}}\frac{2}{(D-3)}\int_{\partial\Sigma_{\varepsilon}}\mathrm{d}^{D-1}x\sqrt{\tilde{\gamma}}\frac{\varepsilon{y^{(1)}}^{2}}{1+{y_{,C}^{(0)}}^{2}}+\ldots$$
(94)
At this order the only possible intrinsic curvature term would be ${\cal R}$, the Ricci scalar on $\partial\Sigma_{\varepsilon}$ but from (92) this does not have the right structure to be the correct counterterm. Using (91) we can show that the required counterterm is
$$S_{ct,2}=-\frac{1}{8G_{D+2}}\frac{1}{(D-1)^{2}(D-3)}\int_{\partial\Sigma_{\varepsilon}}\mathrm{d}^{D-1}x\sqrt{\tilde{\gamma}}{\cal K}^{2}.$$
(95)
Note that other extrinsic curvature invariants again either do not have the correct $\varepsilon$ structure or the correct ${y^{(1)}}^{2}$ behaviour to arise as possible counterterms.
In the $D=2$ analysis we found that ${\cal K}$ would be a finite counterterm, but it is excluded by the requirement that the renormalised entanglement entropy of the complementary region is equal to that of the original region. For $D>2$ we can further note that
$$\int_{\partial\Sigma_{\varepsilon}}\mathrm{d}^{D-1}x\sqrt{\tilde{\gamma}}{\cal K}^{D-1}$$
(96)
are finite counterterms. The complementarity requirement rules out such counterterms for even $D$ (corresponding to field theories in odd spacetime dimensions).
For odd $D$ (corresponding to field theories in even spacetime dimensions), these counterterms are consistent with the requirement that the entropy of the complement is the same as that of the original region. Indeed we already saw that such a term arises in $D=3$: it is automatically included in (93) in the $\mu$ dependent part.
In addition, higher dimensions allows the possibility of other finite counterterms constructed from curvature invariants such as
$$\int_{\partial\Sigma_{\varepsilon}}\mathrm{d}^{D-1}x\sqrt{\tilde{\gamma}}({\cal K}_{ab}{\cal K}^{ab})^{(D-1)/2};\quad\int_{\partial\Sigma_{\varepsilon}}\mathrm{d}^{D-1}x\sqrt{\tilde{\gamma}}\tilde{R}^{(D-1)/2}$$
(97)
which are both valid for odd $D$, so that they are analytic. These counterterms are however not linearly independent of each other, due to the Gauss-Codazzi relations.
In general there will always be finite counterterms possible in even spacetime dimensions and the number of such terms will increase with $D$ implying there are an increasing number of scheme dependent terms. We will understand in section 5 how these finite counterterms relate to the scheme dependence of the partition function.
Thus, to summarise the results in this section, the renormalized entanglement entropy for static surfaces in $AdS_{D+2}$ is
$$\displaystyle S_{\rm ren}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{4G_{D+2}}\int_{\Sigma}\mathrm{d}^{D}\sigma^{\alpha}\,\sqrt{\gamma}$$
$$\displaystyle\qquad-\frac{1}{4G_{D+2}}\int_{\partial\Sigma}\mathrm{d}^{D-1}x\sqrt{\tilde{\gamma}}\left(\frac{1}{D-1}+\frac{1}{2(D-1)^{2}(D-3)}{\cal K}^{2}\cdots\right),$$
where $D$ represents the number of spatial dimensions in the dual field theory. The first counterterm is logarithmic for $D=1$. Only the first
counterterm given above is needed for gravity in four bulk dimensions ($D=2$). The second counterterm is logarithmic at $D=3$ and is needed in the form given above for $D>3$.
Additional counterterms involving higher order curvature invariants are needed for $D\geq 5$; the additional counterterms are associated with logarithmic divergences (i.e. conformal anomalies) in odd dimensions.
The analysis in this section assumed a Poincaré parameterisation of $AdS_{D+2}$, i.e. we assumed a flat background metric for the dual field theory. We will generalize these results in section 5. Note that although the renormalized
entanglement entropy can be covariantized as shown in section 5 the complete holographic dictionary would also need to take into account real time issues [29] for non-static setups.
4 Entanglement entropy for holographic RG flows
A holographic RG flow (for a field theory in a flat background) can be described by a domain wall geometry
$$\mathrm{d}s^{2}=dw^{2}+e^{2{\cal A}(w)}dx^{\mu}dx_{\mu}$$
(99)
where the warp factor ${\cal A}(w)$ is linear in $w$ at a fixed point. The geometry satisfies the equations of motion derived from Einstein gravity
coupled to scalar fields $\phi^{A}$, and the scalar fields have corresponding radial profiles $\phi^{A}(w)$. In what follows we will consider the case of a single scalar field
with the bulk action being
$$I=\frac{1}{16\pi G_{4}}\int\mathrm{d}^{4}x\,\sqrt{-g}\left(R_{g}-\frac{1}{2}{(\partial\phi)}^{2}+V(\phi)\right),$$
(100)
with $V(\phi)$ being the scalar potential. The generalisation to multiple scalar fields would be straightforward.
We restrict to UV conformal theories, so that
the scalar potential $V(\phi)$ can be expanded as a power series in $\phi$ near the boundary:
$$V(\phi)=6-\sum_{n=1}^{\infty}\frac{\lambda_{(2n)}}{(2n)!}\phi^{2n}.$$
(101)
The mass $M$ of the scalar is then given by $M^{2}=\lambda_{(2)}$, so the scalar field is dual to a dimension $\Delta$ operator in the boundary CFT where $M^{2}=\Delta(\Delta-3)$. In what follows we will denote
$$\Delta_{+}=\frac{3}{2}+\frac{1}{2}\sqrt{9+4M^{2}}.$$
(102)
For $-9/4<M^{2}<-5/4$, two quantizations are possible with the operator dimension corresponding to the second quantization being
$$\Delta_{-}=\frac{3}{2}-\frac{1}{2}\sqrt{9+4M^{2}}.$$
(103)
The equations of motion are
$$\displaystyle\ddot{\cal A}=-\frac{1}{4}(\dot{\phi})^{2}$$
(104)
$$\displaystyle\ddot{\phi}+3\dot{\cal A}\dot{\phi}=-\frac{dV}{d\phi}$$
where a dot denotes a derivative with respect to $w$. It is well-known, see [30], that these equations are always equivalent to first order equations
$$\dot{\cal A}=W\qquad\dot{\phi}=-4\frac{dW}{d{\phi}}$$
(105)
where the superpotential $W(\phi)$ is given by
$$V=-2\left(4\left(\frac{dW}{d\phi}\right)^{2}-3W^{2}\right),$$
(106)
with
$$W=1+\frac{1}{8}(3-\Delta_{+})\phi^{2}+\cdots$$
(107)
Note that the superpotential is not unique at higher orders in the scalar field: different choices are associated with different RG flows and in a supersymmetric theory only one choice will be supersymmetric.
For flat sliced domain walls corresponding to holographic RG flows, the appropriate counterterm for the bulk action can be expressed in terms of the superpotential as
$$I_{\rm ct}=-\frac{1}{4\pi G_{4}}\int_{{\cal M}_{\epsilon}}\mathrm{d}^{3}x\sqrt{-h}W.$$
(108)
To match with conventions in earlier sections, it is convenient to express the asymptotically AdS${}_{4}$ domain wall spacetime in the coordinates
$$\mathrm{d}s^{2}=\frac{\mathrm{d}\rho^{2}}{4\rho^{2}}+\frac{1}{\rho}e^{A(\rho)}\eta_{\mu\nu}\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}$$
(109)
where $\rho\to 0$ corresponds to the conformal boundary, $\eta_{\mu\nu}$ is the flat metric, with coordinates $(t,x,y)$. Near the conformal boundary
$$e^{A(\rho)}=1+\ldots,$$
(110)
where the subleading terms depend on the form of the scalar potential. In these coordinates the Einstein and scalar equations become
$$\displaystyle A^{\prime\prime}+\frac{1}{\rho}A^{\prime}=-\frac{1}{4}(\phi^{\prime})^{2};$$
(111)
$$\displaystyle 4\rho^{2}\phi^{\prime\prime}+2(3\rho A^{\prime}-1)\rho\phi^{\prime}=-\frac{dV}{d\phi}.$$
These equations can also be rewritten in terms of the superpotential as
$$-\rho A^{\prime}=\tilde{W}\qquad\rho\phi^{\prime}=2\frac{d\tilde{W}}{d\phi}$$
(112)
where $\tilde{W}=W-1$.
4.1 Renormalization of entanglement entropy
Consider a codimension two minimal spacelike surface probing the domain wall spacetime.
The entanglement entropy functional is
$$S=\frac{1}{4G_{4}}\int\mathrm{d}\rho\,\mathrm{d}x\,\sqrt{\gamma}$$
(113)
where $\gamma_{\mu\nu}=g_{mn}\partial_{\mu}X^{m}(\rho,x)\partial_{\nu}X^{n}(\rho,x)$, $g_{mn}$ is the metric on the full target space, and $X^{m}(\rho,x)$ is the embedding. We will again work in static gauge where the embedding is given by $X^{m}(\rho,x)=(\rho,t,x,y(\rho,x))$ and $t$ is a constant. Therefore
$$\displaystyle\gamma_{\rho\rho}$$
$$\displaystyle=\frac{1}{4\rho^{2}}+\frac{e^{A(\rho)}}{\rho}y_{\rho}^{2}$$
(114)
$$\displaystyle\gamma_{\rho x}$$
$$\displaystyle=y_{\rho}y_{x}\frac{e^{A(\rho)}}{\rho}$$
$$\displaystyle\gamma_{xx}$$
$$\displaystyle=\frac{1}{\rho}e^{A(\rho)}+\frac{e^{A(\rho)}}{\rho}y_{x}^{2}$$
where we denote $y_{\rho}=\partial_{\rho}y$ and $y_{x}=\partial_{x}y$. The entanglement entropy is thus given by
$$\displaystyle S$$
$$\displaystyle=\frac{1}{4G_{4}}\int\mathrm{d}\rho\,\mathrm{d}x\,\frac{e^{A(\rho)/2}}{2\rho^{3/2}}\sqrt{1+y_{x}^{2}+4\rho e^{A}y_{\rho}^{2}}$$
(115)
$$\displaystyle=\frac{1}{4G_{4}}\int\mathrm{d}\rho\,\mathrm{d}x\,\frac{e^{A/2}}{2\rho^{3/2}}m(\rho,x)$$
where we have introduced the shorthand $m(\rho,x)=\sqrt{1+y_{x}^{2}+4\rho e^{A}y_{\rho}^{2}}$.
The minimal surface equation is:
$$\displaystyle 0$$
$$\displaystyle=(1+4\rho e^{A}y_{\rho}^{2})y_{xx}+\rho e^{A}(1+y_{x}^{2})y_{\rho\rho}-5\rho e^{A}y_{\rho}y_{x}y_{\rho x}$$
(116)
$$\displaystyle+\frac{1}{2}e^{A}y_{\rho}(\rho A^{\prime}(3+3y_{x}^{2}+8\rho e^{A}y_{\rho}^{2})-1-y_{x}^{2}-8\rho e^{A}y_{\rho}^{2}).$$
We now solve this equation iteratively for $y(\rho,x)$ as a series expansion in $\rho$. We assume the following Taylor series expansions for $A(\rho)$ and $y(\rho,x)$:
$$\displaystyle e^{A(\rho)}$$
$$\displaystyle=1+A_{(\alpha)}\rho^{\alpha}+\ldots$$
(117)
$$\displaystyle y(\rho,x)$$
$$\displaystyle=y_{(0)}(x)+y_{(\beta_{1})}(x)\rho^{\beta_{1}}+y_{(\beta_{2})}(x)\rho^{\beta_{2}}+\ldots$$
where we assume that $\alpha>0$ and $0<\beta_{1}<\beta_{2}<\ldots$. To solve the PDE we insert these expansions into equation (4.1) and set $\rho=0$. We then fix $\beta_{1}$ and $y_{(\beta_{1})}$ to solve the resulting equation to leave $y_{(0)}$ unconstrained and differentiate equation (4.1) with respect to $\rho$ and repeat to find $\beta_{2}$.
After substituting the expansions into the minimal surface equation, one finds that the leading order behaviour is a term constant in $\rho$ and a term scaling as $\rho^{\beta_{1}-1}$. To leave $y_{(0)}$ unconstrained we must set $\beta_{1}=1$ (as before) and deduce that:
$$y_{(1)}(x)=\frac{2y_{(0)xx}}{1+({y_{(0)x}})^{2}}.$$
(118)
Next we substitute the expansions into the $\rho$ derivative of equation (4.1). In all cases the lowest power involving $\beta_{2}$ is $\rho^{\beta_{2}-2}$ and we choose $\beta_{2}$ so as to cancel the leading order divergence involving $\alpha$.
In the case that $\alpha<1$ the leading order divergence involving $\alpha$ goes as $\rho^{\alpha-1}$ which requires $\beta_{2}=1+\alpha$ and the following value of $y_{(1+\alpha)}$ to cancel the divergence:
$$y_{(1+\alpha)}=-\frac{A_{(\alpha)}(3\alpha-1)y_{(1)}}{2\alpha^{2}+\alpha-1}$$
(119)
Note that the denominator here vanishes when $\alpha=\frac{1}{2}$ (and when $\alpha=-1$ which is excluded by the boundary conditions) and this case needs to be treated separately.
In the case $\alpha>1$ the leading order term involving $\alpha$ is not divergent and we can set $\beta_{2}=2$ with
$$y_{(2)}=\frac{4y_{(1)}^{3}+6y_{(1)}y_{(0)x}y_{(1)x}-4y_{(1)}^{2}y_{(0)xx}-y_{(1)xx}}{1+({y_{(0)x}})^{2}}$$
(120)
In the case where $\alpha=1$ these two results overlap to give $\beta_{2}=2$ and
$$y_{(2)}=\frac{4y_{(1)}^{3}+6y_{(1)}y_{(0)x}-4y_{(1)}^{2}y_{(0)xx}-y_{(1)xx}}{1+({y_{(0)x}})^{2}}-A_{(1)}y_{(1)}.$$
(121)
One can similarly analyse the asymptotic expansions to higher order but this will not be needed in calculating the regularised entanglement entropy.
Let us now turn to the regularisation of the entanglement entropy functional. Using the series expansion for $y(\rho,x)$ the
small $\rho$ behaviour of the action is
$$S=\frac{1}{4G_{4}}\int\mathrm{d}x\,\mathrm{d}\rho\frac{e^{A/2}}{2\rho^{3/2}}\sqrt{1+({y_{(0)x}})^{2}}\left(1+\frac{1}{2}\rho B(\rho,x)+\ldots\right).$$
(122)
where $B(\rho,x)$ is a function which is constant in $\rho$ to leading order. The full expression for $B(\rho,x)$ is given by
$$\rho B(\rho,x)=\frac{y_{x}^{2}+4\rho e^{A}y_{\rho}^{2}-({y_{(0)x}})^{2}}{1+({y_{(0)x}})^{2}}$$
(123)
where it is understood that the series expansions for $e^{A}$ and $y$ are inserted above. It is clear that
$$\int\mathrm{d}x\,\int_{\varepsilon}\mathrm{d}\rho\frac{e^{A/2}}{4}\sqrt{1+({y_{(0)x}})^{2}}\rho^{-1/2}B\sim\varepsilon^{1/2}+\ldots$$
(124)
which vanishes as the cutoff is removed.
Hence to find the regularised action we only need to expand the function $e^{A(\rho)/2}$ and keep terms which are powers of $\rho^{1/2}$ or lower:
$$S_{\rm reg}=\frac{1}{4G_{4}}\int\mathrm{d}x\,\sqrt{1+({y_{(0)x}})^{2}}\int\mathrm{d}\rho\frac{e^{A/2}}{2\rho^{3/2}}$$
(125)
The latter radial integral depends only on the background and not on the specific embedding.
The first counterterm we require is needed in all cases independently of $\alpha$: this is the volume divergence associated with the asymptotically AdS background. The necessary counterterm here is as before
$$S_{\rm ct}=-\frac{1}{4G_{4}}\int_{\partial\Sigma}\mathrm{d}x\,\sqrt{\tilde{\gamma}}.$$
(126)
The remaining divergent terms depend explicitly on $A_{(\alpha)}$ and $\alpha$. These terms can only be non-trivial if there is a non-trivial matter content in the bulk and consequentially the counterterms must be functions of the scalar fields on the $\rho=\varepsilon$ slice pulled back on to the minimal surface.
Solving the field equations (111) to leading orders in $\rho$ implies that
$$\phi=\phi_{(0)}\rho^{\frac{1}{2}(3-\Delta_{+})}+\cdots$$
(127)
and for the warp factor:
$$\alpha=3-\Delta_{+};\qquad A_{(\alpha)}=-\frac{1}{8}\phi_{(0)}^{2}.$$
(128)
Subleading divergences in the entanglement entropy are only present when $\Delta_{+}>5/2$. The regulated onshell action up to the first subleading divergence is:
$$S_{\rm reg}=\frac{1}{4G_{4}}\int_{\partial\Sigma_{\epsilon}}\mathrm{d}x\,\sqrt{\tilde{\gamma}}\left(1+\frac{3-\Delta_{+}}{8(5-2\Delta_{+})}\phi_{(0)}^{2}\varepsilon^{3-\Delta_{+}}+\ldots\right)$$
(129)
and to leading order we also know that on the $\rho=\varepsilon$ hypersurface $\phi=\phi_{(0)}\varepsilon^{(3-\Delta_{+})/2}$ so it is simple to write the $n=1$ divergence in a covariant form so that the corresponding counterterm can then be read off:
$$S_{\rm ct}=-\frac{1}{4G_{4}}\int_{\partial\Sigma}\mathrm{d}x\,\sqrt{\tilde{\gamma}}\frac{3-\Delta_{+}}{8(5-2\Delta_{+})}\phi^{2}.$$
(130)
At $\Delta_{+}=5/2$ the divergence becomes logarithmic and is associated with a conformal anomaly; we will discuss such anomalies further below.
Given a superpotential for the RG flow one can find an exact expression for the counterterms to all orders as follows. We have argued that the counterterms can be written covariantly as
$$S_{\rm ct}=-\frac{1}{4G_{4}}\int_{\partial\Sigma}\mathrm{d}x\,\sqrt{\tilde{\gamma}}Y(\phi)$$
(131)
where $Y(\phi)$ is analytic in the scalar field. (Here we exclude conformal anomalies, which we will discuss below.)
By construction the counterterm is chosen to cancel divergences and hence
$$\int_{\epsilon}d\rho\frac{e^{\frac{A}{2}}}{2\rho^{\frac{3}{2}}}=\frac{e^{\frac{A}{2}}}{\epsilon^{\frac{1}{2}}}Y(\phi),$$
(132)
where implicitly the latter is evaluated at $\rho=\epsilon$. Differentiating this expression with respect to the radius we then obtain
$$A^{\prime}Y+2\frac{dY}{d\phi}\phi^{\prime}-\frac{Y}{\rho}=-\frac{1}{\rho}.$$
(133)
One can then substitute in the superpotential to get
$$(1+\tilde{W})Y-4\frac{dY}{d\phi}\frac{d\tilde{W}}{d\phi}=1,$$
(134)
i.e. an expression for $Y(\phi)$ in terms of the superpotential $\tilde{W}(\phi)$ with no explicit radial dependence. The superpotential $\tilde{W}(\phi)$ can be expressed as
$$\tilde{W}(\phi)=\sum_{n\geq 2}w_{n}\phi^{n}\qquad w_{2}=\frac{1}{8}(3-\Delta_{+})$$
(135)
and correspondingly
$$Y(\phi)=1+\sum_{n\geq 2}y_{n}\phi^{n}$$
(136)
with
$$y_{2}=\frac{(3-\Delta_{+})}{8(5-2\Delta_{+})};\qquad y_{3}=\frac{1+24y_{2}}{(8-3\Delta_{+})}w_{3},$$
(137)
and so on. Here the cubic counterterm is required for $\Delta_{+}>8/3$, and there is a corresponding
logarithmic divergence at $\Delta_{+}=8/3$ which is cubic in the scalar field.
For a free scalar in the bulk $w_{n}=0$ for $n>2$, but the expansion of $Y(\phi)$ does not terminate at $n=2$:
$$Y(\phi)=e^{\frac{1}{4}\phi^{2}}\sum_{m\geq 0}\frac{(-1)^{m}}{4^{m}m!}\frac{\phi^{2m}}{(2m\Delta_{+}-2m+1)}.$$
(138)
However, one should implicitly only retain terms from this series which contribute to divergences. The order $m$ term is required for
$$\Delta_{+}>3-\frac{1}{2m}.$$
(139)
The associated divergence becomes logarithmic at $\Delta_{+}=3-1/2m$: the coefficient at order $m$ in (138) becomes ill-defined, corresponding to the breakdown of the assumed form of the counterterms. Note that logarithmic terms appear in the asymptotic expansion of the scalar field $\phi$ for half integer conformal dimensions but these are not related to conformal anomalies in the entanglement entropy.
In the $m=1$ case the regulated onshell action has a logarithmic divergence when $\Delta_{+}=5/2$
$$S_{\rm reg}=\frac{1}{4G_{4}}\int\mathrm{d}x\,\sqrt{\tilde{\gamma}}\left(1-\frac{1}{4}A_{3-\Delta_{+}}\varepsilon^{1/2}\log\varepsilon\right)+\ldots$$
(140)
Here
$A_{1/2}=-\frac{1}{8}\phi_{(0)}^{2}$ and $\phi=\phi_{(0)}\varepsilon^{1/4}+\ldots$, so we can write this divergence as
$$S_{\rm reg}=\frac{1}{4G_{4}}\int\mathrm{d}x\,\sqrt{\tilde{\gamma}}\left(1+\frac{1}{32}\phi^{2}\log\varepsilon\right)+\ldots.$$
(141)
The corresponding logarithmic counterterm is then simply
$$S_{ct}=-\frac{1}{4G_{4}}\int\mathrm{d}x\,\sqrt{\tilde{\gamma}}\frac{1}{32}\phi^{2}\log\varepsilon.$$
(142)
This result is consistent with that of [31] who found a logarithmic divergence, in their notation, given by
$$\delta S=\frac{\mathcal{A}}{8G_{N}}(d-2)\lambda^{2}h_{0}\log(\varepsilon/\varepsilon_{IR})$$
(143)
which matches our expression under the substitutions $4G_{N}=1$, $\mathcal{A}=\int\mathrm{d}x\,\sqrt{\tilde{\gamma}}$, $d=3$, $h_{0}=\frac{1}{8}$, $\lambda=\phi$ and the relabelling of the cut-off $\varepsilon\to\varepsilon^{1/2}$. This relabelling of the cut off is necessary as theirs is imposed on a $z=\varepsilon$ surface where $\rho=z^{2}$.
Thus, to summarise the results of this section, the required counterterms are
$$S=-\frac{1}{4G_{4}}\int_{\partial\Sigma}\mathrm{d}x\sqrt{\tilde{\gamma}}\left(1+\frac{3-\Delta_{+}}{8(5-2\Delta_{+})}\phi^{2}+\cdots\right),$$
(144)
where the ellipses denote terms involving higher powers of the scalar field. The counterterm quadratic in scalar fields is necessary for $\Delta_{+}>5/2$ and is logarithmic at $\Delta_{+}=5/2$. More generally, new logarithmic divergences involving $n$ powers of
the scalar field arise at
$$\Delta_{+}=3-\frac{1}{n}$$
(145)
and an additional counterterm involving $n$ powers of the scalar field is switched on for $\Delta_{+}>3-1/n$. The counterterms can be expressed compactly in terms
of an analytic function of the scalar field $Y(\phi)$
$$S=-\frac{1}{4G_{4}}\int_{\partial\Sigma}\mathrm{d}x\sqrt{\tilde{\gamma}}Y(\phi),$$
(146)
where $Y(\phi)$ is defined in terms of the superpotential for the flow by (134). We should emphasise that both expressions (144) and (146) are applicable to entangling surfaces in holographic RG flows with flat slicings. For entangling surfaces in generic Einstein-scalar backgrounds there could be additional counterterms dependent on gradients of the scalar field.
4.2 Entanglement entropy change under relevant perturbation
In this section we will calculate the change in the renormalized entanglement entropy of a disk entangling region under a small relevant perturbation of the CFT, i.e. we work perturbatively in $\phi_{(0)}$, the source of the relevant operator. As in [32, 33] it is convenient to express
the change in the bare entanglement entropy as
$$\delta S=\frac{1}{8G_{4}}\int\mathrm{d}^{2}x\sqrt{\gamma}T^{mn}_{\rm min}\delta g_{mn}$$
(147)
where $\gamma$ is the metric on the unperturbed minimal surface, $T^{mn}_{\rm min}$ is the energy momentum tensor for the minimal surface
$$T^{mn}_{\rm min}=\gamma^{\alpha\beta}\partial_{\alpha}X^{m}\partial_{\beta}X^{n}$$
(148)
and $\delta g_{mn}$ is the change in the (Einstein) metric induced by the relevant deformation. The latter can always be parameterised as
$$ds^{2}=\frac{d\rho^{2}}{4\rho^{2}}(1+\delta f(\rho))+\frac{1}{\rho}(1+\delta h(\rho))dx^{\mu}dx^{\mu},$$
(149)
and we can furthermore use the gauge freedom to fix $\delta f(\rho)=0$. The latter gauge choice was implicit in our earlier parameterisation of domain wall geometries.
One can then show that the change in the regulated (bare) entanglement entropy for a disk is
$$\delta S_{\rm reg}=\frac{\pi R}{4G_{4}}\int_{\epsilon}^{R^{2}}\frac{d\rho}{\rho^{\frac{3}{2}}}(1+\frac{\rho}{R^{2}})\delta h(\rho).$$
(150)
Note that this expression holds for any small perturbation of the metric which preserves Poincaré invariance of the dual field theory.
Working perturbatively in the scalar field amplitude, and taking into account Poincaré invariance, the most general solution possible for the scalar field
is
$$\phi=\phi_{(0)}\rho^{\frac{1}{2}(3-\Delta_{+})}+\phi_{(\Delta_{+})}\rho^{\frac{1}{2}\Delta_{+}}$$
(151)
(where we assume that $\Delta_{+}\neq 3/2$) and $\phi_{(\Delta_{+})}$ is the normalizable mode of the scalar field. Correspondingly the warp factor is given by
$$\delta h=-\frac{1}{8}\left(\phi_{(0)}^{2}\rho^{(3-\Delta_{+})}+\frac{8\Delta_{+}}{9}(3-\Delta_{+})\phi_{(0)}\phi_{(\Delta_{+})}\rho^{\frac{3}{2}}+\phi_{(\Delta_{+})}^{2}\rho^{\Delta_{+}}\right)$$
(152)
Since we are working perturbatively in the scalar field, we need only retain counterterms which are quadratic in the scalars. In the case of a single scalar field this implies that the only contributing counterterms are those given in (144).
At $\Delta_{+}=5/2$, the change in the entanglement entropy involves a logarithmic divergence, and thus the renormalized entanglement entropy will be renormalization scheme dependent.
The change in the renormalized entanglement entropy is hence (for $\Delta_{+}\neq 5/2$)
$$\displaystyle\delta S_{\rm ren}$$
$$\displaystyle=$$
$$\displaystyle\frac{\pi}{16(2\Delta_{+}-5)G_{4}}\phi_{(0)}^{2}R^{2(3-\Delta_{+})}$$
$$\displaystyle+\frac{\pi}{36G_{4}}\Delta_{+}(\Delta_{+}-3)\phi_{(0)}\phi_{(\Delta_{+})}R^{3}+\frac{\pi}{16(2\Delta_{+}-1)G_{4}}\phi_{(\Delta_{+})}^{2}R^{2\Delta_{+}}.$$
Working to quadratic order in the scalar field one cannot impose regularity in the bulk as $\rho\rightarrow\infty$ as both modes are unbounded.
On dimensional grounds, however,
$$\phi_{(\Delta)}\propto\phi_{(0)}^{\frac{\Delta_{+}}{(3-\Delta_{+})}}$$
(154)
for $\frac{3}{2}<\Delta_{+}<3$. Hence $\phi_{(\Delta_{+})}\sim\phi_{(0)}^{\delta}$ with $\delta>1$, and the normalizable mode is subleading in powers of the non-normalizable mode, as we will see in the full solution given in the next section.
Therefore
$$\delta S_{\rm ren}=\frac{\pi}{16(2\Delta_{+}-5)G_{4}}\phi_{(0)}^{2}R^{2(3-\Delta_{+})}+\cdots$$
(155)
where ellipses denote terms which are of higher order in the source. This quantity is positive for $\Delta_{+}>5/2$ but negative for relevant deformations with $3/2<\Delta_{+}<5/2$. Recalling that the F quantity is proportional to minus the renormalized entanglement entropy the change in the F quantity is positive for relevant deformations with $3/2<\Delta_{+}<5/2$. A related result was obtained in [34], although the sign of the quantity was not explicitly identified in that work.
For operators of dimension $\Delta_{-}<3/2$, the non-normalizable mode $\phi_{(0)}$ is not the operator source: the correct source is obtained from a Legendre transformation of the onshell action [35]. Such a Legendre transformation cannot be carried out without working to higher orders in the non-normalizable mode $\phi_{(0)}$ and thus we cannot obtain the entanglement entropy for this case without knowledge of the higher order solution.
Now let us consider the special case of $\Delta=3$, i.e. marginal operators. In this case the warp factor is unchanged by the non-normalizable mode of the scalar field $\phi_{(0)}$, i.e. integrating the equations of motion we obtain
$$\delta h=-\frac{1}{8}\phi_{(3)}^{3}\rho^{3}.$$
(156)
Since the non-normalizable mode does not affect the metric, there are no new divergences and no counterterms depending on the scalar field. At $\Delta=3$ there are also no
possible finite counterterms since the finite counterterm
$$S_{\rm ct}=-\frac{1}{4G_{4}}\int dx\sqrt{\gamma^{h}}\left({\cal K}\phi^{2}\right)$$
(157)
does not respect the complementarity requirement.
The renormalized entanglement entropy for a marginal deformation is thus
$$\delta S_{\rm ren}=\frac{\pi}{80G_{4}}\phi_{(3)}^{2}R^{6}.$$
In the vacuum of the marginally deformed conformal field theory $\phi_{(3)}=0$ and the change in the renormalised entanglement entropy is therefore zero for the vacuum of the deformed conformal field theory. Note that the change in the renormalized entanglement entropy is hence implicitly not analytic in the operator dimension as
$\Delta\rightarrow 3$; this is however permissible, since the spectrum of operators is discrete.
We can also compute the change in the quantity $F(R)$ defined in section 2:
$$\delta F(R)=-\delta S_{\rm reg}(R)+R\frac{\partial S_{\rm reg}(R)}{\partial R}$$
(158)
For $\Delta<3$
$$\delta F(R)=-\frac{\pi\phi_{(0)}^{2}}{16G_{4}}R^{6-2\Delta}$$
(159)
which is negative for all relevant deformations. This does not agree numerically with $\delta S_{\rm ren}$, but it is the latter which is by construction related to the renormalized F quantity by the CHM map.
For $\Delta=3$, the change in the regulated entanglement entropy is zero, as the metric is unchanged, and therefore
$$\delta F(R)=0.$$
(160)
Note that implicitly the change in $\delta F(R)$ is therefore also non-analytic at $\Delta\rightarrow 3$.
It may seem surprising that the F quantity decreases along RG flows generated by operators of dimensions $3/2<\Delta_{+}<5/2$.
The results discussed above actually follow directly from the subadditivity property of the (regularised) entanglement entropy: recall that the latter implies that $\partial^{2}S_{\rm reg}/\partial R^{2}\leq 0$. Our analysis implies that the counterterms scale with the size of the entangling region, i.e. $S_{\rm ct}\propto R$. Therefore subadditivity implies
$$\frac{\partial^{2}S_{\rm ren}}{\partial R^{2}}=\frac{\partial^{2}S_{\rm reg}}{\partial R^{2}}\leq 0.$$
(161)
However, on dimensional grounds, when we work to quadratic order in the source $S_{\rm ren}$ must take the form
$$S_{\rm ren}=-\frac{\pi}{2G_{4}}+a_{2(3-\Delta_{+})}\phi_{(0)}^{2}R^{2(3-\Delta_{+})}+\cdots$$
(162)
for $\Delta_{+}>3/2$ where $a_{2(3-\Delta_{+})}$ is a dimensionless constant. Here we use the explicit form for the
leading term, which is independent of $R$. Differentiating twice with respect to $R$ then gives
$$\frac{\partial^{2}S_{\rm ren}}{\partial R^{2}}=2(3-\Delta_{+})(5-2\Delta_{+})a_{2(3-\Delta_{+})}\phi_{(0)}^{2}R^{2(2-\Delta_{+})}+\cdots$$
(163)
This is negative semi-definite (as required by strong subadditivity) provided that
$$(3-\Delta_{+})(5-2\Delta_{+})a_{2(3-\Delta_{+})}\leq 0,$$
(164)
i.e. provided that $a_{2(3-\Delta_{+})}\geq 0$ for $\Delta_{+}\geq 5/2$ and $a_{2(3-\Delta_{+})}\leq 0$ for $\Delta_{+}\leq 5/2$, as we found above.
A related result is found by
directly computing the change in the free energy to quadratic order in the source for holographic RG flows on a sphere driven by the same operators. Deformations of the theory on the sphere
$$I_{\rm CFT}\rightarrow I_{\rm CFT}+\int_{S^{3}}d^{3}\Omega\;\psi_{(0)}{\cal O}_{\Delta_{+}},$$
(165)
where the source $\psi_{(0)}$ is independent of the spherical coordinates and $d\Omega$ is the measure on the $S^{3}$,
may be described holographically by spherical sliced domain walls. Again working to quadratic order in the source, the change in the free energy is positive for operators of dimensions $3/2<\Delta_{+}<5/2$ [12].
Note that such deformations are not equivalent to conformal transformations of the holographic RG flows considered here, which are dual to deformations of the theory on flat space:
$$I_{\rm CFT}\rightarrow I_{\rm CFT}+\int_{R^{3}}d^{3}x\phi_{(0)}{\cal O}_{\Delta_{+}}.$$
(166)
To understand this point further, it is useful to recall the relationship between spherical and Poincaré coordinates for anti-de Sitter. The former can be described in terms of the following embedding into $R^{1,4}$:
$$X^{0}=\cosh w\qquad X^{1}+iX^{2}=\sinh w\cos\theta e^{i\tau_{E}}\qquad X^{3}+iX^{4}=\sinh w\sin\theta e^{i\phi}$$
(167)
so that
$$ds^{2}=dw^{2}+\sinh^{2}w\left(d\theta^{2}+\cos^{2}\theta d\tau_{E}^{2}+\sin^{2}\theta d\phi^{2}\right).$$
(168)
Poincaré coordinates can be obtained by setting
$$\displaystyle X^{0}+X^{1}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{\rho^{1/2}}\qquad X^{0}-X^{1}=\left(\rho^{1/2}+\frac{1}{\rho^{1/2}}(t_{E}^{2}+x^{2}+y^{2})\right)$$
(169)
$$\displaystyle X^{2}$$
$$\displaystyle=$$
$$\displaystyle\frac{t}{\rho^{1/2}}\qquad X^{3}=\frac{x}{\rho^{1/2}}\qquad X^{4}=\frac{y}{\rho^{1/2}},$$
resulting in
$$ds^{2}=\frac{d\rho^{2}}{4\rho^{2}}+\frac{1}{\rho}\left(dt_{E}^{2}+dx^{2}+dy^{2}\right).$$
(170)
From these relations it is clear that the radial coordinate in spherical slicings, $w$, depends on both $\rho$ and $|x|\equiv(t_{E}^{2}+x^{2}+y^{2})^{\frac{1}{2}}$. Conversely
the Poincaré radial coordinate $\rho$ depends on $(w,\theta,\tau_{E})$. Therefore flows which depend only on $w$ or $\rho$, respectively, are not equivalent to each
other: a flow which depends only on $w$ will depend on the Poincaré norm $|x|$ as well as $\rho$.
From the field theory perspective, the theories on the $S^{3}$ and on $R^{3}$ are related by the conformal transformation described earlier, with the relevant conformal factor being
given by (69). While the original conformal field theory is of course unaffected by this conformal factor, mapping (166) to the sphere results in
$$\phi_{(0)}\rightarrow\Omega^{\Delta_{+}-3}(\theta,\tau_{E})\phi_{(0)},$$
(171)
i.e. the transformed source is not homogeneous over the $S^{3}$, and therefore the deformations on $S^{3}$ and $R^{3}$ by homogeneous sources are not conformally equivalent. Thus,
while the change in the renormalized entanglement entropy is indeed related to a change in the free energy on the $S^{3}$, the latter is the change under a deformation which breaks the $SO(4)$ invariance.
4.3 Top down RG flow
Let us now consider entanglement entropy in holographic RG flows which have top down embeddings. We will discuss the following single scalar example, taken from [36]. Let the potential be
$$V(\phi)=6\cosh\left(\frac{\phi}{\sqrt{3}}\right)$$
(172)
which arises in a consistent truncation of ${\cal N}=8$ gauged supergravity, which in turn is a consistent truncation of M theory compactified on $S^{7}$.
The RG flow equations can be used to construct analytic domain wall solutions in which the metric is conveniently expressed as
$$ds^{2}=\frac{(1+\nu r+\sqrt{1+2\nu r+r^{2}})}{2r^{2}\sqrt{1-r^{2}}(1+2\nu r+r^{2})}dr^{2}+\frac{\sqrt{1-r^{2}}}{2r^{2}}(1+\nu r+\sqrt{1+2\nu r+r^{2}})dx^{\mu}dx_{\mu}$$
(173)
and the scalar field profile is
$$\phi=\sqrt{3}\tanh^{-1}(r).$$
(174)
The parameter $\nu\geq-1$ is arbitrary with $\nu=-1$ corresponding to a supersymmetric domain wall of the supergravity theory. Here $r\rightarrow 0$ corresponds to the conformal boundary. Note that in all cases the metric has a singularity at $r=1$; this singularity is null in the supersymmetric case and timelike in all other cases but the singularity is good according to the standard criteria.
The scalar mass associated with the potential is $M^{2}=-2$, which corresponds to the cases of $\Delta_{-}=1$ and $\Delta_{+}=2$, i.e. the mass is such that both quantisations are possible and mixed boundary conditions can be considered.
We can reintroduce the scalar field amplitude as a parameter by letting
$$r=c\tilde{r};\qquad x^{\mu}=c\tilde{x}^{\mu}$$
(175)
so that
$$\displaystyle ds^{2}$$
$$\displaystyle=$$
$$\displaystyle\frac{(1+\nu c\tilde{r}+\sqrt{1+2\nu c\tilde{r}+c^{2}\tilde{r}^{2}})}{2\tilde{r}^{2}\sqrt{1-c^{2}\tilde{r}^{2}}(1+2\nu c\tilde{r}+c^{2}\tilde{r}^{2})}d\tilde{r}^{2}+\frac{\sqrt{1-c^{2}\tilde{r}^{2}}}{2\tilde{r}^{2}}(1+\nu c\tilde{r}+\sqrt{1+2\nu c\tilde{r}+c^{2}\tilde{r}^{2}})d\tilde{x}^{\mu}d\tilde{x}_{\mu}$$
$$\displaystyle\phi$$
$$\displaystyle=$$
$$\displaystyle\sqrt{3}\tanh^{-1}(c\tilde{r}).$$
(176)
We can then change coordinates for $c\tilde{r}\ll 1$ as
$$\tilde{r}^{2}=\rho+\nu c\rho^{\frac{3}{2}}+\cdots$$
(177)
to obtain
$$\displaystyle ds^{2}$$
$$\displaystyle=$$
$$\displaystyle\frac{d\rho^{2}}{4\rho^{2}}+\frac{1}{\rho}(1-\frac{3}{8}c^{2}\rho+\cdots)d\tilde{x}^{\mu}d\tilde{x}_{\mu};$$
(178)
$$\displaystyle\phi$$
$$\displaystyle=$$
$$\displaystyle\sqrt{3}c\left(\rho^{\frac{1}{2}}+\frac{1}{2}\nu c\rho+\cdots\right)$$
from which we can read off that
$$\phi_{(0)}=\sqrt{3}c;\qquad\phi_{(\Delta_{+})}\equiv\phi_{(1)}=\frac{1}{2}\sqrt{3}\nu c^{2},$$
(179)
i.e. the normalizable mode is of order the non-normalizable mode squared. (This had to be true on dimensional grounds in a solution which depends on only one dimensionful parameter, $c$.)
Thus substituting into (4.2) we obtain
$$\delta S_{\rm ren}=-\frac{\pi}{48G_{4}}\phi_{(0)}^{2}R^{4}+{\cal O}\left(\phi_{(0)}^{3}\right),$$
(180)
in agreement with (155) in the case of $\Delta_{+}=2$.
The result (180) can be interpreted as follows. There are only two physical scales in the field theory: the source for the operator deformation $c$ and the size of the entangling region $R$. When $cR\ll 1$, the entangling surface is small and does not penetrate far into the bulk. The region probed by the entangling surface is well-described by the asymptotic Fefferman-Graham expansion (178), and therefore one can use the results of the previous section to compute the entanglement entropy. Note that the result does not depend on the parameter $\nu$, i.e. it is same for supersymmetric and non-supersymmetric RG flows.
Now consider increasing the radius of the entangling surface at fixed source. On dimensional grounds $\delta S_{\rm ren}$ is a function of $\phi_{(0)}R$. Since
$\partial^{2}S_{\rm ren}/\partial R^{2}\leq 0$, $\partial\delta S_{\rm ren}/\partial R$ must decrease monotonically with the radius $R$ and $\delta S_{\rm ren}$ must be negative for all $R$.
5 Renormalization via the replica trick
In the previous sections we have described a renormalization procedure for entanglement entropy which is based on the holographic realisation of entanglement entropy in terms of minimal
surfaces. It is difficult to translate this procedure directly into a field theoretic definition of renormalization, since the Ryu-Takayanagi functional itself does not follow directly from field theory.
A conceptual derivation of the Ryu-Takayanagi functional has been obtained by Lewkowycz-Maldacena [3] via the replica trick.
The entropy associated with a density matrix $\rho$ is expressed as
$$S=-n\partial_{n}\left[\log Z(n)-n\log Z(1)\right]_{n=1}$$
(181)
where $Z(n)={\rm Tr}(\rho^{n})$ and $Z(1)={\rm Tr}(\rho)$ is the usual partition function. If we are interested in the entropy of a thermal state, then $Z(n)$ is constructed by extending the period of the thermal circle by a factor of $n$. In the case of entanglement entropy, $Z(n)$ is constructed by extending the period of the circle around the boundary of the entangling region by a factor of $n$, where implicitly $n$ is an integer. Assuming that the resulting expression is analytic in $n$, one can obtain the entropy by analytically continuing to $n=1$.
Holographically $Z(n)$ can be computed in terms of the Euclidean actions:
$$S=n\partial_{n}\left[I(n)-nI(1)\right]_{n=1}.$$
(182)
Here $I(1)$ represents the onshell Euclidean action for the bulk geometry while $I(n)$ represents the onshell Euclidean action for the replica bulk geometry.
For a thermal state, the bulk geometry associated with $Z(1)$ is a black hole and the replica is constructed by extending the period of the thermal circle by a factor of $n$.
It was shown by Lewkowycz-Maldacena [3] that for a bulk theory described by Einstein gravity (181) then localises on the horizon of the black hole, i.e.
$$S=\frac{A}{4G_{d+1}}.$$
(183)
In particular, the volume divergences of the onshell actions (associated with UV divergences in the field theory) by construction cancel, since the replica geometry asymptotically
matches $n$ copies of the original geometry.
For the entanglement entropy, the bulk geometry associated with $Z(1)$ corresponds to the usual bulk dual of the given state in the field theory.
The replica is constructed by extending the period of the circle around the entangling region boundary by a factor of $n$. Following the same logic as in Lewkowycz-Maldacena,
the expression (182) localises on the minimal surface corresponding to the extension of the boundary of the entangling region into the bulk (see the discussions in [37]). However, unlike the black
hole case, the volume divergences of the bulk actions in (182) do not cancel, as the entangling surface itself has area divergences.
We can formally write down a renormalized entanglement entropy as
$$S_{\rm ren}=n\partial_{n}\left[I_{\rm ren}(n)-nI_{\rm ren}(1)\right]_{n=1}$$
(184)
where the quantities appearing on the right hand side are the renormalized bulk actions. Equivalently,
$$S_{\rm ct}=n\partial_{n}\left[I_{\rm ct}(n)-nI_{\rm ct}(1)\right]_{n=1}$$
(185)
Let us first focus on the specific case of entangling surfaces in $AdS_{4}$, for which the usual counterterms for the onshell action are [11]
$$I_{\rm ct}(1)=\frac{1}{4\pi G_{4}}\int_{\partial{\cal M}}d^{3}x\sqrt{h}\left(-\frac{1}{2}K+1+\frac{1}{4}R\right).$$
(186)
Here we define the bulk geometry to be ${\cal M}$ and its boundary to be $\partial{\cal M}$, and $K$ denotes the trace of the extrinsic curvature of $\partial{\cal M}$ embedded into
${\cal M}$. (The first term is the usual Gibbons-Hawking term.)
Since the replica geometry is also asymptotically locally $AdS_{4}$, the counterterms are
$$I_{\rm ct}(n)=\frac{1}{4\pi G_{4}}\int_{\partial{\cal M}_{n}}d^{3}x\sqrt{h_{n}}\left(-\frac{1}{2}K_{n}+1+\frac{1}{4}R_{n}\right).$$
(187)
where $h_{n}$ is the boundary metric for the replica geometry and $K_{n}$ and $R_{n}$ are the associated extrinsic curvature and Ricci scalar, respectively.
Now the replica geometry by construction matches the original
geometry except at the fixed point set of $\partial_{\tau}$, where $\tau$ is the circle around the boundary of the entangling region and its extension into the bulk.
At this fixed point set the metric and the extrinsic curvature of the replica match the original metric, but the intrinsic curvature invariants of the replica
receive contributions from the conical singularity. In the case of interest $R=0$ but in the replica geometry due to the conical singularity
$$\int d^{3}x\sqrt{h_{n}}R_{n}=4\pi(1-n)\int_{\partial\Sigma}dx\sqrt{\tilde{\gamma}}$$
(188)
and hence we find that
$$S_{\rm ct}=-\frac{1}{4G_{4}}\int_{\partial\Sigma}dx\sqrt{\tilde{\gamma}},$$
(189)
which matches the counterterm obtained by our explicit calculations in section 3.
For the case of entangling surfaces in holographic RG flows the counterterms to quadratic order in the scalar field are [11]
$$I_{\rm ct}(1)=\frac{1}{4\pi G_{4}}\int_{\partial{\cal M}}d^{3}x\sqrt{h}\left(1+\frac{1}{16}(3-\Delta_{+})\phi^{2}+\frac{1}{4}R+\frac{\Delta_{+}-3}{32(2\Delta_{+}-5)}R\phi^{2}\right),$$
(190)
where we drop the Gibbons-Hawking term as it does not contribute to the entanglement entropy counterterms, and we
also neglect terms involving derivatives of the scalar field, i.e. we restrict to homogeneous scalar field configurations.
Following the same steps as above, we can then show that
$$S_{\rm ct}=-\frac{1}{4G_{4}}\int_{\partial\Sigma}dx\sqrt{\tilde{\gamma}}\left(1+\frac{\Delta_{+}-3}{8(2\Delta_{+}-5)}\phi^{2}\right)$$
(191)
which is again in agreement with our explicit results of section 4.
Let us now move to general dimensions. For an asymptotically locally $AdS_{D+2}$ spacetime the counterterms are [11]
$$\displaystyle I_{\rm ct}(1)$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{16\pi G_{D+2}}\int_{\partial{\cal M}}d^{D+1}x\sqrt{h}\left(2D+\frac{1}{(D-1)}R\right.$$
$$\displaystyle\qquad\qquad\left.+\frac{1}{(D-3)(D-1)^{2}}\left(R_{ab}R^{ab}-\frac{D+1}{4D}R^{2}\right)+\cdots\right).$$
This expression should be understood as containing only the appropriate divergent terms in any given dimension; moreover, for odd $D$ there are logarithmic
counterterms. In particular, for $D=3$ the third counterterm is replaced by the logarithmic counterterm
$$\frac{1}{16\pi G_{5}}\int_{\partial{\cal M}_{\varepsilon}}d^{4}x\sqrt{h}\frac{1}{8}\left(R_{ab}R^{ab}-\frac{1}{3}R^{2}\right)\ln\varepsilon.$$
(193)
In the replica geometry, the contributions to the curvature from the conical singularity are given by [13]
$$\displaystyle R_{n}$$
$$\displaystyle=$$
$$\displaystyle R+4\pi(1-n)\delta_{\partial\Sigma}+{\cal O}(1-n)^{2};$$
(194)
$$\displaystyle R_{nab}$$
$$\displaystyle=$$
$$\displaystyle R_{ab}+2\pi(1-n)n_{a}n_{b}\delta_{\partial\Sigma}+{\cal O}(1-n)^{2},$$
where $\delta_{\partial\Sigma}$ is a delta function localised on the entangling surface. Here $n_{a}^{k}$ with $k=1,2$ represent orthonormal vectors to the entangling surface and
$$n_{a}n_{b}=\sum_{k}n^{k}_{a}n^{k}_{b}.$$
(195)
Following the same steps as above, we can immediately read off the leading counterterm for the entanglement entropy as
$$S_{\rm ct,1}=-\frac{1}{4(D-1)G_{D+2}}\int_{\partial\Sigma}d^{D-1}x\sqrt{\tilde{\gamma}},$$
(196)
in agreement with our earlier result.
For the higher order counterterms, one can use the following expressions [14]
$$\displaystyle\int_{\partial{\cal M}_{n}}d^{D+1}x\sqrt{h_{n}}R_{n}^{2}$$
$$\displaystyle=$$
$$\displaystyle n\int_{\partial{\cal M}}d^{D+1}x\sqrt{h}R^{2}+8\pi(1-n)\int_{\partial\Sigma}d^{D-1}x\sqrt{\tilde{\gamma}}R$$
(197)
$$\displaystyle\int_{\partial{\cal M}_{n}}d^{D+1}x\sqrt{h_{n}}R_{nab}R_{n}^{ab}$$
$$\displaystyle=$$
$$\displaystyle n\int_{\partial{\cal M}}d^{D+1}x\sqrt{h}R_{ab}R^{ab}+4\pi(1-n)\int_{\partial\Sigma}d^{D-1}x\sqrt{\tilde{\gamma}}(R_{ii}-\frac{1}{2}k^{2}),$$
where implicitly we work to leading order in $(1-n)$ and we define
$$k^{2}=\sum_{k}({\cal K}^{k})^{2}$$
(198)
with $R_{ii}$ corresponding to invariant projections of the Ricci tensor onto the subspace orthogonal to $\partial\Sigma$, see [13].
In section 3, we analysed the entanglement entropy counterterms assuming that the entangling surface is static and that the curvature of the boundary metric is zero.
In such a case $R_{ii}=R=0$ and the extrinsic curvature in the time direction is zero. Thus the second counterterm becomes
$$S_{\rm ct,2}=-\frac{1}{8(D-1)^{2}(D-3)G_{D+2}}\int_{\partial\Sigma}d^{D-1}x\sqrt{\tilde{\gamma}}{\cal K}^{2},$$
(199)
where ${\cal K}$ refers to the trace of the extrinsic curvature of the surface embedded into a constant time hypersurface. Similarly in $D=3$ the logarithmic counterterm is
$$S_{\rm ct,2}=\frac{1}{64G_{5}}\int_{\partial\Sigma}d^{3}x\sqrt{\tilde{\gamma}}{\cal K}^{2}\ln\varepsilon,$$
(200)
which is in agreement with the expression obtained in [13] for the anomaly in the entanglement entropy for 4d CFTs with a holographic dual. (See [13] for the conformal
anomaly in a general 4d conformal field theory in which $a\neq c$.)
One can now immediately generalize the entanglement entropy counterterms to the case of a general embedding into a curved boundary metric obtaining
$$\displaystyle S_{\rm ct}$$
$$\displaystyle=$$
$$\displaystyle-\frac{1}{4(D-1)G_{D+2}}\int_{\partial\Sigma}d^{D-1}x\sqrt{\tilde{\gamma}}$$
$$\displaystyle\qquad-\frac{1}{4(D-1)^{2}(D-3)G_{D+2}}\int_{\partial\Sigma}d^{D-1}x\sqrt{\tilde{\gamma}}\left(R_{ii}-\frac{1}{2}k^{2}-\frac{D+1}{2D}R\right),$$
where one can use the Gauss-Codazzi relations to write $R_{ii}$ and $R$ in terms of intrinsic and extrinsic curvatures of $\partial\Sigma$.
5.1 Higher derivative generalizations
Using the replica trick, we can derive the renormalized entanglement entropy functional from any higher derivative gravity for which the renormalized bulk action is known. Let
us consider the particular example of Gauss-Bonnet gravity, with bulk action
$$I=-\frac{1}{16\pi G_{D+2}}\int_{{\cal M}}d^{D+2}x\sqrt{g}\left[R_{g}+D(D+1)+\lambda\left(R_{mnpq}R^{mnpq}-4R_{mn}R^{mn}+R_{g}^{2}\right)\right]$$
(202)
where $\lambda$ is the Gauss-Bonnet coupling.
One can derive the entanglement entropy functional by the replica trick used above, see [14, 38], using the bulk versions of (197) together with the additional relation
$$\displaystyle\int_{{\cal M}_{n}}d^{D+2}x\sqrt{g}R_{mnpq}R^{mnpq}$$
$$\displaystyle=$$
$$\displaystyle n\int_{{\cal M}_{n}}d^{D+2}x\sqrt{g}R_{mnpq}R^{mnpq}$$
$$\displaystyle\qquad+8\pi(1-n)\int_{\Sigma}d^{D}y\sqrt{\gamma}\left(R_{ijij}-{\rm Tr}(k^{2})\right),$$
where we neglect terms of higher order in $(n-1)$ and $R_{ijij}$ denotes the projection of the Riemann tensor in the directions orthogonal to the entangling surface. Also
$${\rm Tr}(k^{2})=\sum_{k=1}^{2}{\cal K}^{k}_{ab}{\cal K}^{kab}.$$
(204)
Thus the entanglement entropy functional consists of the usual Ryu-Takayanagi term plus additional terms
$$\displaystyle S=\frac{1}{4G_{D+2}}\int_{\Sigma}d^{D}y\sqrt{\gamma}+\frac{\lambda}{G_{D+2}}\int_{\Sigma}d^{D}y\sqrt{\gamma}\left(R_{ijij}-{\rm Tr}(k^{2})-2R_{ii}+k^{2}+R\right),$$
(205)
where implicitly all terms can be written in terms of extrinsic and intrinsic curvatures on the entangling surface.
As shown in [14], in five bulk dimensions the latter term can be simplified using the Gauss-Codazzi relations to give
$$S=\frac{1}{4G_{5}}\int_{\Sigma}d^{3}y\sqrt{\gamma}\left(1+2\lambda\hat{R}\right),$$
(206)
with $\hat{R}$ the intrinsic curvature of the entangling surface.
Now the bulk equations of motion admit as $AdS_{5}$ as a solution, but the radius of the $AdS_{5}$ depends on the Gauss-Bonnet coupling, i.e. the $AdS_{5}$ metric is
$$ds^{2}=l^{2}(\lambda)\left(\frac{d\rho^{2}}{4\rho^{2}}+\frac{1}{\rho}dx\cdot dx\right)$$
(207)
where the radius is given by
$$l^{4}(\lambda)-l^{2}(\lambda)+2\lambda=0.$$
(208)
One can then straightforwardly show that the leading order counterterm for the entanglement entropy is given by
$$S_{\rm ct}=-\frac{1}{8G_{5}}\int_{\partial\Sigma}d^{2}x\sqrt{\tilde{\gamma}}\left(l(\lambda)-12\frac{\lambda}{l(\lambda)}\right),$$
(209)
where we use the fact that the entangling surface is asymptotically locally hyperbolic and thus $\hat{R}=-6//l(\lambda)^{2}+\cdots$. There is also a subleading logarithmic divergence associated with the conformal anomaly; this is known from the work of [13].
Now the leading order counterterm for the entanglement entropy is inherited from the subleading counterterm for the bulk action, i.e. the counterterm
$$I_{ct}=a_{2}\int d^{4}x\sqrt{h}R.$$
(210)
This counterterm is not known to all orders in $\lambda$, although it was derived perturbatively in $\lambda$ in [39, 40, 41]. The relation with entanglement entropy immediately gives the coefficient of this counterterm to be
$$a_{2}=\frac{1}{32G_{5}}\left(l(\lambda)-12\frac{\lambda}{l(\lambda)}\right),$$
(211)
i.e. the entanglement entropy counterterms provide a quick method of deriving or checking counterterms in the bulk action involving the curvature.
5.2 Domain walls
In this section we show how the counterterms for asymptotically locally $AdS$ solutions of a theory with a single scalar imply the entanglement entropy counterterms discussed
in section 4. The bulk Euclidean action is
$$I=-\frac{1}{16\pi G_{4}}\int_{{\cal M}}d^{4}x\sqrt{g}\left(R_{g}-\frac{1}{2}(\partial\phi)^{2}+V(\phi)\right).$$
(212)
In general the counterterms for asymptotically locally $AdS$ solutions of this action can be expressed in the form
$$I_{ct}=\frac{1}{16\pi G_{4}}\int_{\partial{\cal M}}d^{3}x\sqrt{h}\left({\cal W}(\phi)+{\cal Y}(\phi)R+\cdots\right),$$
(213)
where ${\cal W}(\phi)$ and ${\cal Y}(\phi)$ are analytic functions of the scalar field $\phi$. Here the ellipses denote terms which depend on gradients of the scalar field; as in the discussions above, such terms are not relevant when using the replica trick to derive the entanglement entropy counterterms. In the above expression we assume generic values of the dual operator dimension such that there are no conformal anomalies; for specific values of the operator dimension there will however be conformal anomalies.
For a flat domain wall solution, characterized by a given superpotential $W(\phi)$, the only contributing counterterm is ${\cal W}(\phi)=4W(\phi)$, since in this case $R=0$.
To use the replica trick we need to know how the counterterms for the bulk action depend on the curvature of the boundary metric i.e. we cannot restrict to flat sliced domain walls: the counterterm for the entanglement entropy follows from the term involving the Ricci scalar above, i.e.
$$S_{\rm ct}=-\frac{1}{4G_{4}}\int_{\partial\Sigma}dx\sqrt{\tilde{\gamma}}{\cal Y}(\phi).$$
(214)
We can understand the specific form of ${\cal Y}(\phi)$ for entanglement entropy in a flat sliced domain wall as follows.
We begin with solutions of the equation of motion correspondins to domain walls with homogeneous slicing, i.e.
the metric is
$$ds^{2}=dw^{2}+e^{2{\cal A}(w)}d\Omega_{3}^{2}$$
(215)
and the scalar field profile is $\phi(w)$. We let the Ricci scalar of the slicing be $\hat{r}$ where, for example, $\hat{r}=6$ for unit radius spherical slices. The equations of motion are then
$$\displaystyle\ddot{\phi}+3\dot{\phi}\dot{\cal A}$$
$$\displaystyle=$$
$$\displaystyle-V^{\prime}(\phi);$$
(216)
$$\displaystyle-\frac{\hat{r}}{6}e^{-2{\cal A}}-\frac{1}{4}\dot{\phi}^{2}$$
$$\displaystyle=$$
$$\displaystyle\ddot{\cal A}.$$
These equations are identical to those discussed in section 4, apart from the curvature contribution to the second equation.
Now let us work in the limit that $\hat{r}\ll 1$. For $\hat{r}=0$, the equations admit the first order form discussed in section 4, in terms of the superpotential $W(\phi)$. For $\hat{r}\ll 1$, the equations of motion are solved to order $\hat{r}^{2}$ by
$$\dot{\cal A}=W\qquad\dot{\phi}=-4\frac{dW}{d\phi}+\hat{r}f(\phi)$$
(217)
provided that
$$\displaystyle 3Wf(\phi)-4\frac{d}{d\phi}\left(f(\phi)\frac{dW}{d\phi}\right)$$
$$\displaystyle=$$
$$\displaystyle 0;$$
(218)
$$\displaystyle f(\phi)\frac{dW}{d\phi}=\frac{1}{6}e^{-2{\cal A}}.$$
The regulated onshell action (including the Gibbons-Hawking term) thus becomes
$$\displaystyle I_{\rm reg}$$
$$\displaystyle=$$
$$\displaystyle-\int^{R}dw\int d\Omega_{3}\left(e^{\cal A}\hat{r}+{\cal O}(\hat{r}^{2})\right)-\frac{1}{4\pi G_{4}}\int d\Omega_{3}\left[e^{3\cal A}W\right]_{R},$$
$$\displaystyle=$$
$$\displaystyle-\frac{1}{16\pi G_{4}}\int^{R}dw\int d\Omega_{3}\left(e^{\cal A}\hat{r}+{\cal O}(\hat{r}^{2})\right)-\frac{1}{4\pi G_{4}}\int_{\partial M}d^{3}x\sqrt{h}W,$$
where we have used the field equations to linear order in $\hat{r}$ and in the second line we write the boundary term in covariant form. The bulk term can be expressed
as a covariant boundary term
$$-\frac{1}{16\pi G_{4}}\int_{\partial M}d^{3}x\sqrt{h}RY(\phi)$$
(220)
provided that
$$\frac{d}{dw}\left(\sqrt{h}RY(\phi)\right)=e^{\cal A}\hat{r}.$$
(221)
However,
$$\frac{d}{dw}\left(\sqrt{h}RY\right)=\hat{r}\frac{d}{dw}\left(e^{\cal A}Y\right)=e^{\cal A}\hat{r}\left(WY-4\frac{dW}{d\phi}\frac{dY}{d\phi}\right),$$
(222)
where we drop terms of higher order in $\hat{r}$ and use the field equations. Therefore the required counterterms are
$$I_{\rm ct}=\frac{1}{16\pi G_{4}}\int_{\partial M}d^{3}x\sqrt{h}\left(4W+RY\right),$$
(223)
with
$$\left(WY-4\frac{dW}{d\phi}\frac{dY}{d\phi}\right)=1,$$
(224)
as we found in section 4. Note that terms of higher order in $\hat{r}$ would not contribute to the counterterms, as they do not give rise to divergent terms.
We calculated the curvature term in (223) by working with a homogeneous domain wall. To use the replica trick, we need to consider a replica space in which
the curvature of the boundary is given by (194), in the limit that $n\rightarrow 1$, i.e. it is not homogeneous, but (223) is covariant and still applies.
Note that the slices of the domain wall are flat, up to conical singularity terms which are proportional to $(n-1)$, and hence $R$ is small as $n\rightarrow 1$. It is therefore
indeed true to leading order in $(n-1)$ that the replica geometry is still governed by the superpotential $W$. Following the same steps as earlier in this section, we can then
immediately read off the counterterm action for the entanglement entropy as
$$S_{\rm ct}=-\frac{1}{4G_{4}}\int_{\partial\Sigma}dx\sqrt{\tilde{\gamma}}Y(\phi),$$
(225)
as we found in section 4.
It is important to note that this expression holds specifically for flat domain wall geometries associated with a superpotential $W$. A generic curved domain wall geometry is not governed by a single real superpotential (see [42, 43]) and the analysis above would need to be generalized for such cases.
6 Conclusions
In this paper we have shown how the holographic entanglement entropy may be renormalized using appropriately covariant boundary counterterms. This renormalization procedure is inherited directly from the renormalization of the partition function, using the replica trick.
We analysed renormalization for entangling surfaces in asymptotically locally AdS spacetimes in any dimension and in flat sliced holographic RG flows in four bulk dimensions. We also showed that the renormalization procedure can be extended to higher derivative theories such as Gauss-Bonnet. It would be straightforward to generalize our results to include entangling surfaces with cusps and to non-conformal holographic setups using [44]. It would be interesting to explore real-time holography in the context of entanglement entropy, using the techniques of [29] for the HRT functional [45].
While it is difficult to relate the area renormalization of the holographic entanglement entropy functional directly to field theory renormalization, the replica trick expresses our renormalised entanglement entropy in terms of renormalized partition functions, i.e.
$$S_{\rm ren}=-n\partial_{n}\left[\log Z_{\rm ren}(n)-n\log Z_{\rm ren}(1)\right]_{n=1}.$$
(226)
This expression can be directly implemented in a field theoretical calculation: having fixed a renormalization scheme for the partition function, the partition function on the replica space (which has the same UV divergence structure) will inherit a renormalization scheme and thus $S_{\rm ren}$ will be determined. This assumes that the replica trick is applicable but in practice most explicit calculations of entanglement entropy in field theory do in any case make use of the replica trick. Computations of the renormalized entanglement entropy in free field theory examples will be presented elsewhere.
There has been considerable interest recently in supersymmetric renormalization schemes for field theories on curved spaces and, in particular, in analysing how much supersymmetry is required for the partition function to be uniquely defined [46]. It would be interesting to understand the role of supersymmetry in our analysis.
In section 4 we showed that the renormalized entanglement entropy of a disk decreases under deformations of a conformal field theory by operators of dimension $3/2<\Delta<5/2$. Under the CHM map, this corresponds to an increase in the F quantity when one makes corresponding deformations of the theory on a three sphere; note however that these deformations do not preserve the symmetry of the $S^{3}$. In the companion paper [12] we find analogous results for flows which are homogeneous on the three sphere. It would be interesting to understand whether these examples indeed disprove the strong version of the F theorem, or whether the flows under consideration are unphysical.
Acknowledgments
We would like to thank Kostas Skenderis and Balt van Rees for useful comments and discussions.
This work was supported by the Science and Technology Facilities Council (Consolidated Grant “Exploring the Limits of the Standard Model and Beyond”).
We thank the Simons Center for partial support during the completion of this work. This project has received funding from the European Union’s Horizon 2020
research and innovation programme under the Marie Sklodowska-Curie grant
agreement No 690575.
References
[1]
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy
from AdS/CFT, Phys.Rev.Lett. 96 (2006) 181602,
[hep-th/0603001].
[2]
H. Casini, M. Huerta, and R. C. Myers, Towards a derivation of
holographic entanglement entropy, JHEP 05 (2011) 036,
[arXiv:1102.0440].
[3]
A. Lewkowycz and J. Maldacena, Generalized gravitational entropy, JHEP 08 (2013) 090, [arXiv:1304.4926].
[4]
T. Takayanagi, Entanglement Entropy from a Holographic Viewpoint, Class.Quant.Grav. 29 (2012) 153001,
[arXiv:1204.2450].
[5]
M. Headrick, General properties of holographic entanglement entropy,
JHEP 03 (2014) 085,
[arXiv:1312.6717].
[6]
I. R. Klebanov, D. Kutasov, and A. Murugan, Entanglement as a probe of
confinement, Nucl.Phys. B796 (2008) 274–293,
[arXiv:0709.2140].
[7]
L. Huijse, S. Sachdev, and B. Swingle, Hidden Fermi surfaces in
compressible states of gauge-gravity duality, Phys.Rev. B85
(2012) 035121, [arXiv:1112.0573].
[8]
D. L. Jafferis, I. R. Klebanov, S. S. Pufu, and B. R. Safdi, Towards the
F-Theorem: N=2 Field Theories on the Three-Sphere, JHEP 06
(2011) 102, [arXiv:1103.1181].
[9]
M. Henningson and K. Skenderis, The Holographic Weyl anomaly, JHEP 07 (1998) 023,
[hep-th/9806087].
[10]
M. Henningson and K. Skenderis, Weyl anomaly for Wilson surfaces, JHEP 06 (1999) 012,
[hep-th/9905163].
[11]
S. de Haro, K. Skenderis, and S. N. Solodukhin, Holographic
Reconstruction of Spacetime and Renormalization in the AdS/CFT
Correspondence, Communications in Mathematical Physics 217
595–622, [hep-th/0002230].
[12]
M. Taylor and W. Woodhead, The Holographic F Theorem,
arXiv:1604.xxxx.
[13]
S. N. Solodukhin, Entanglement entropy, conformal invariance and
extrinsic geometry, Phys. Lett. B665 (2008) 305–309,
[arXiv:0802.3117].
[14]
D. V. Fursaev, A. Patrushev, and S. N. Solodukhin, Distributional
Geometry of Squashed Cones, Phys. Rev. D88 (2013), no. 4
044054, [arXiv:1306.4000].
[15]
H. Casini and M. Huerta, Entanglement and alpha entropies for a massive
scalar field in two dimensions, J. Stat. Mech. 0512 (2005)
P12012, [cond-mat/0511014].
[16]
H. Casini and M. Huerta, Universal terms for the entanglement entropy in
2+1 dimensions, Nucl. Phys. B764 (2007) 183–201,
[hep-th/0606256].
[17]
P. Calabrese and J. Cardy, Entanglement entropy and conformal field
theory, J. Phys. A42 (2009) 504005,
[arXiv:0905.4013].
[18]
H. Liu and M. Mezei, A Refinement of entanglement entropy and the number
of degrees of freedom, JHEP 04 (2013) 162,
[arXiv:1202.2070].
[19]
I. R. Klebanov, T. Nishioka, S. S. Pufu, and B. R. Safdi, On Shape
Dependence and RG Flow of Entanglement Entropy, JHEP 07 (2012)
001, [arXiv:1204.4160].
[20]
H. Liu and M. Mezei, Probing renormalization group flows using
entanglement entropy, JHEP 01 (2014) 098,
[arXiv:1309.6935].
[21]
H. Casini and M. Huerta, On the RG running of the entanglement entropy of
a circle, Phys. Rev. D85 (2012) 125016,
[arXiv:1202.5650].
[22]
I. R. Klebanov, T. Nishioka, S. S. Pufu, and B. R. Safdi, Is Renormalized
Entanglement Entropy Stationary at RG Fixed Points?, JHEP 10
(2012) 058, [arXiv:1207.3360].
[23]
N. Drukker, D. J. Gross, and H. Ooguri, Wilson loops and minimal
surfaces, Phys. Rev. D60 (1999) 125006,
[hep-th/9904191].
[24]
I. Papadimitriou, Holographic renormalization as a canonical
transformation, JHEP 11 (2010) 014,
[arXiv:1007.4592].
[25]
C. R. Graham and E. Witten, Conformal anomaly of submanifold observables
in AdS / CFT correspondence, Nucl. Phys. B546 (1999) 52–64,
[hep-th/9901021].
[26]
I. R. Klebanov, S. S. Pufu, and B. R. Safdi, F-Theorem without
Supersymmetry, JHEP 10 (2011) 038,
[arXiv:1105.4598].
[27]
I. R. Klebanov, S. S. Pufu, S. Sachdev, and B. R. Safdi, Entanglement
Entropy of 3-d Conformal Gauge Theories with Many Flavors, JHEP 05 (2012) 036, [arXiv:1112.5342].
[28]
R. C. Myers and A. Sinha, Seeing a c-theorem with holography, Phys. Rev. D82 (2010) 046006,
[arXiv:1006.1263].
[29]
K. Skenderis and B. C. van Rees, Real-time gauge/gravity duality:
prescription, renormalization and examples, JHEP 2009 085–085,
[0812.2909].
[30]
D. Z. Freedman, C. Nunez, M. Schnabl, and K. Skenderis, Fake supergravity
and domain wall stability, Phys. Rev. D69 (2004) 104027,
[hep-th/0312055].
[31]
P. A. R. Jones and M. Taylor, Entanglement entropy and differential
entropy for massive flavors, JHEP 1508 (2015) 014, [arXiv:1505.0769].
[32]
H.-C. Chang and A. Karch, Entanglement Entropy for Probe Branes, JHEP 1401 (2014) 180, [arXiv:1307.5325].
[33]
A. Karch and C. F. Uhlemann, Generalized gravitational entropy of probe
branes: flavor entanglement holographically, JHEP 2014 17,
[arXiv:1402.4497].
[34]
T. Nishioka, Relevant perturbation of entanglement entropy and
stationarity, Physical Review D 90 045006,
[arXiv:1405.3650].
[35]
I. R. Klebanov and E. Witten, AdS / CFT correspondence and symmetry
breaking, Nucl. Phys. B556 (1999) 89–114,
[hep-th/9905104].
[36]
I. Papadimitriou, Non-Supersymmetric Membrane Flows from Fake
Supergravity and Multi-Trace Deformations, JHEP 02 (2007) 008,
[hep-th/0606038].
[37]
P. A. R. Jones and M. Taylor,
Entanglement entropy in top-down models,
arXiv:1602.04825 [hep-th].
[38]
A. Bhattacharyya, A. Kaviraj, and A. Sinha, Entanglement entropy in
higher derivative holography, JHEP 08 (2013) 012,
[arXiv:1305.6694].
[39]
J. T. Liu and W. A. Sabra, Hamilton-Jacobi Counterterms for
Einstein-Gauss-Bonnet Gravity, Class. Quant. Grav. 27 (2010)
175014, [arXiv:0807.1256].
[40]
S. Cremonini, J. T. Liu, and P. Szepietowski, Higher Derivative
Corrections to R-charged Black Holes: Boundary Counterterms and the
Mass-Charge Relation, JHEP 03 (2010) 042,
[arXiv:0910.5159].
[41]
V. Jahnke, A. S. Misobuchi, and D. Trancanelli, Holographic
renormalization and anisotropic black branes in higher curvature gravity,
JHEP 01 (2015) 122,
[arXiv:1411.5964].
[42]
K. Skenderis and P. K. Townsend, Hidden supersymmetry of domain walls and
cosmologies, Phys. Rev. Lett. 96 (2006) 191301,
[hep-th/0602260].
[43]
I. Papadimitriou, Holographic Renormalization of general dilaton-axion
gravity, JHEP 08 (2011) 119,
[arXiv:1106.4826].
[44]
I. Kanitscheider, K. Skenderis, and M. Taylor, Precision holography for
non-conformal branes, JHEP 09 (2008) 094,
[arXiv:0807.3324].
[45]
V. E. Hubeny, M. Rangamani, and T. Takayanagi, A Covariant holographic
entanglement entropy proposal, JHEP 07 (2007) 062,
[arXiv:0705.0016].
[46]
J. Gomis, Z. Komargodski, P.-S. Hsin, A. Schwimmer, N. Seiberg, and S. Theisen,
Anomalies, Conformal Manifolds, and Spheres,
arXiv:1509.0851. |
Spatio-Temporal Random Partition Models
Garritt L. Page
Brigham Young University, Provo, USA
BCAM - Basque Center of Applied Mathematics, Bilbao, Spain
and
Fernando A. Quintana
Pontificia Universidad Católica de Chile, Santiago, Chile
and
David B. Dahl
Brigham Young University, Provo, USA.
Partially supported by grant FONDECYT 1180034 and by Iniciativa Científica Milenio - Minecon Núcleo Milenio MiDaS
Abstract
The number of scientific fields that regularly collect data that are spatio-temporal continues to grow. An intuitive feature of this type of data is that measurements taken on experimental units near each other in time and space tend to be similar. As such, many methods developed to accommodate spatio-temporal dependent structures attempt to borrow strength among units close in space and time, which constitutes an implicit space-time grouping. We develop a class of dependent random partition models that explicitly models this spatio-temporal clustering by way of a dependent random partition model. We first detail how temporal dependence is incorporated so that partitions evolve gently over time. Then conditional and marginal properties of the joint model are derived. We then demonstrate how space can be integrated. Computation strategies are detailed and we illustrate the methodology through simulations and an application.
\externaldocument
[]SupplementaryMaterial1
Keywords: correlated partitions, spatio-temporal clustering, hierarchical Bayes modeling, Bayesian nonparametrics, time dependent partitions.
1 Introduction
We introduce a method to directly model spatio-temporal dependence in a
sequence of random partitions. Our approach is motivated
by the practical problem of modeling a prior distribution for a sequence of random partitions
that exhibit substantial overlap over time, and where cluster formation may also
be spatially influenced. Traditionally, dependencies in random partitions
(i.e., the clustering of units) have been obtained as a by-product of dependent
random measures in Bayesian nonparametric (BNP). We will argue, however, that when
partitions are the inferential objects of principal interest, then the
partition should be modeled directly rather than relying on induced random
partition models such as those originating from temporal, or spatio-temporal dependent BNP models.
But first, we review the literature on dependent BNP methods.
BNP methods that incorporate time include Caron et al. (2007), Nieto-Barajas et al. (2012), Antoniano-Villalobos and Walker (2016),
Gutiérrez et al. (2016), Jo et al. (2017) and Caron et al. (2017). Those accommodating space include Gelfand et al. (2005), Griffin and Steel (2006), Duan et al. (2007), Petrone et al. (2009), and Gelfand et al. (2010). The BNP literature is more sparse for combined space-time methods, with Kottas et al. (2008) being the first to construct a spatio-temporal BNP model for areal data by adding an AR(1)-like temporal transition structure to the spatial Dirichlet process of Gelfand et al. (2005). Zhang et al. (2016) consider a model for functional magnetic resonance imaging data and model temporal dependence in the error term and spatial dependence through a hierarchical Dirichlet process mixture model on voxel-specific coefficients (whose clustering induces spatial dependence in the partition). Savitsky (2016) apply a spatio-temporal BNP model to the American Community Survey with varying spatial resolution. Cassese et al. (2019) construct a space-time species sampling model that permits the identification of disease outbreaks.
A common aspect of all these methods is that temporal, spatial, or spatio-temporal dependence is accommodated in
the sequence of random measures by way of the atoms or weights of the stick-breaking representation
(Sethuraman, 1994). The induced random partitions, however, exhibit only weak dependence even when a
sequence of random probability measures is highly correlated. To illustrate this point, we conducted a small Monte
Carlo simulation where a sequence of partitions were generated with 10 time points and 20 units using the method of
Caron et al. (2017). To measure similarity of partitions at different time points, we use the lagged adjusted rand index (ARI).
Figure 1 shows these values averaged over 10,000 Monte Carlo samples.
Notice that as $\alpha$ increases, the partitions from time period $t$ to $t+1$ only become slightly more similar,
such that the dependance between partitions is, at best, only weak. Further, the dependence is not temporally intuitive
as it does not decay as a function of lag. This behavior is not unique to Caron et al. (2017)’s approach,
as Wade et al. (2014) noticed the same type of behavior when using a linear dependent Dirichlet process mixture model. In fact, all BNP methods that model a sequence of random probability measures will induce a random partition model with similar weak-correlation behavior. This behavior is analogous to trying to induce dependence among random variables from distributions with correlated parameters. There is no guarantee that correlated parameters would produce strong correlations among the random variables themselves.
Paci and Finazzi (2018)’s motivation is more similar to ours as their principal interest is spatially referenced partitions over time. However, their approach is based on a mixture of experts model whose weights depend on space and time. As such, their method retains the same properties as the BNP methods.
Our approach is to consider the sequence of partitions indexed by time as the object of principal interest and propose a method that models it directly. This will provide more control over how “smoothly” partitions evolve over time. Perhaps the work closest to ours (in the sense of explicitly modeling a sequence of partitions) can be found in Zanini et al. (2019). Their modeling approach for temporally-referenced sequence of partitions differs from ours in that they do not focus on smooth evolution of spatial partitions over time.
The rest of the article is organized as follows. Section 2 details our approach to modeling partitions temporally and spatially. In Section 2 we also provide a few theoretical results and some computational strategies. In Section 3 we detail a number of simulation studies that illustrate the method and highlight its utility. Then we consider a PM${}_{10}$ data set that is publicly available. Section 4 contains some concluding remarks.
2 Joint Model for a Sequence of Partitions
We begin with some notation. Let $i=1,\ldots,m$ denote the $m$ experimental units at time $t$ for $t=1,\ldots,T$. Let $\rho_{t}=\{S_{1t},\ldots,S_{k_{t}t}\}$ denote a partition of the $m$ experimental units at time $t=1,\ldots,T$ into $k_{t}$ clusters. An alternative partition notation is based on $m$ cluster labels at time $t$ denoted by $\bm{c}_{t}=\{c_{1t},\ldots,c_{mt}\}$ where $c_{it}=j$ implies that $i\in S_{jt}$. Notice the one-to-one correspondence between $\rho_{t}$ and $\bm{c}_{t}$. Finally, any quantity with a “$\star$” superscript will be cluster-specific. For example, we will use $\mu^{\star}_{jt}$ to denote the mean of cluster $j$ at time $t$ so that $\mu_{it}=\mu^{\star}_{jt}$ if $c_{it}=j$.
2.1 Temporal Modeling for Sequences of Partitions
We first describe our approach to correlating partitions over time and subsequently, in the next subsection, detail the inclusion of space. Introducing temporal dependence in a collection of partitions requires formulating a joint probability model for $\{\rho_{1},\ldots,\rho_{T}\}$. Generically, we will denote this joint model with $\text{Pr}(\rho_{t},\ldots,\rho_{T})$. Temporal dependence among the $\rho_{t}$’s implies that the cluster configurations found in $\rho_{t-1},\rho_{t-2},\ldots,\rho_{1}$ could impact the cluster configuration in $\rho_{t}$. However, we assume that the probability model for the sequence of partitions has a Markovian structure. That is,
the conditional distribution of $\rho_{t}$ given $\rho_{t-1},\rho_{t-2},\ldots,\rho_{1}$
only depends on $\rho_{t-1}$. Thus, we
construct $\text{Pr}(\rho_{t},\ldots,\rho_{T})$ as
$$\displaystyle\text{Pr}(\rho_{1},\ldots,\rho_{T})=\text{Pr}(\rho_{T}|\rho_{T-1}%
)\text{Pr}(\rho_{T-1}|\rho_{T-2})\cdots\text{Pr}(\rho_{2}|\rho_{1})\text{Pr}(%
\rho_{1}).$$
(1)
Here $\text{Pr}(\rho_{1})$ is an exchangeable partition probability function (EPPF) that describes how the $m$ experimental units at time period 1 are grouped into $k_{1}$ distinct groups with frequencies $n_{11},\ldots,n_{1k_{1}}$. One characteristic of an exchangeable EPPF that will prove useful in what follows is sample size consistency (or what De Blasi et al. (2015) refer to as the addition rule). This property dictates that
marginalizing the last of $m+1$ elements leads to the same model as if we only had $m$ elements. A commonly encountered EPPF is that induced by a Dirichlet process (DP). This particular EPPF is sometimes referred to as a Chinese restaurant process (CRP) and corresponds to a special case from the family of product partition models (PPM).
For more details see De Blasi et al. (2015). Because we employ the CRP-type EPPF in what follows, we provide its form here
$$\displaystyle\text{Pr}(\rho|M)=\frac{M^{k}}{\prod_{i=1}^{n}(M+i-1)}\prod_{i=1}%
^{k}(|S_{i}|-1)!,$$
(2)
where $k$ is the number of clusters in $\rho$ and $M$ is a concentration parameter controlling the number of clusters. We will denote this random partition distribution as $CRP(M)$.
Although conceptually straightforward, (1) is silent regarding how $\rho_{t-1}$ influences the form of $\rho_{t}$. To make this explicit, we introduce an auxiliary variable that guides how similar $\rho_{t}$ is to $\rho_{t-1}$. Now, if two partitions are highly correlated, then the cluster configurations between them will change very little and as a result only a few of the $m$ experimental units will change cluster assignment. Conversely, two partitions that exhibit low correlation will likely be comprised of very different cluster configurations. The auxiliary variable we introduce identifies which of the experimental units at time $t-1$ will be considered for possible cluster reallocation at time $t$. Specifically, let $\gamma_{it}$ denote the following
$$\displaystyle\gamma_{it}=\left\{\begin{array}[]{c l}1&\mbox{if unit $i$ is {%
\it not} reallocated when moving from time $t-1$ to $t$}\\
0&\mbox{otherwise}.\end{array}\right.$$
(3)
By construction we set $\gamma_{i1}=0$ for all $i$ (i.e., all experimental units are allocated to clusters during the
first time period). We then assume that $\gamma_{it}\stackrel{{\scriptstyle ind}}{{\sim}}Ber(\alpha_{t})$. Note that each of the
$\alpha_{t}\in[0,1]$ acts as a temporal dependence parameter. Specifically, we will interpret $\alpha_{t}=1$ as
implying that $\rho_{t}=\rho_{t-1}$ with probability 1. Conversely, when $\alpha_{t}=0$, then $\rho_{t}$ is independent
of $\rho_{t-1}$. For notational
convenience we introduce $\bm{\gamma}_{t}=(\gamma_{1t},\gamma_{2t},\ldots,\gamma_{mt})$ which is an
$m$-tuple comprised of zeros and ones. The augmented joint model changes (1) to
$$\displaystyle\text{Pr}(\bm{\gamma}_{1},\rho_{1},\ldots,\bm{\gamma}_{T},\rho_{T%
})=\text{Pr}(\rho_{T}|\bm{\gamma}_{T},\rho_{T-1})\text{Pr}(\bm{\gamma}_{T})%
\times\\
\displaystyle\text{Pr}(\rho_{T-1}|\bm{\gamma}_{T-1},\rho_{T-2})\text{Pr}(\bm{%
\gamma}_{T-1})\cdots\text{Pr}(\rho_{2}|\bm{\gamma}_{2},\rho_{1})\text{Pr}(\bm{%
\gamma}_{2})\text{Pr}(\rho_{1}).$$
(4)
In Section LABEL:toy.example of the online Supplementary Material, we provide a toy example that illustrates how our construction produces intuitive conditional partition distributions. In addition to exhibiting intuitive behavior conditionally, it would be appealing if marginally each of the $\rho_{t}$ follow the parent EPPF (i.e., the probability model assumed for $\rho_{1}$), so that
the joint probability model for partitions would become stationary. The following proposition establishes this result which is a consequence of the fact that conditioning on $\bm{\gamma}_{t}$ provides a “reduced” EPPF.
Proposition 2.1.
Let $\rho_{1}\sim EPPF$ and $\bm{\gamma}_{1}=0$. If a joint model for $\rho_{1}\ldots,\rho_{T}$ is constructed as described above by introducing $\bm{\gamma}_{t}$ for $t=2,\ldots,T$,
then we have that marginally $\rho_{1},\ldots,\rho_{T}$ are identically distributed with law coming from the EPPF used to model $\rho_{1}$. Specifically, letting $\rho_{-t}=(\rho_{1},\ldots,\rho_{t-1},\rho_{t+1},\ldots,\rho_{T})$ and $\bm{\gamma}=(\gamma_{1},\ldots,\gamma_{T})$, we have that for all $\lambda\in P$,
$$\displaystyle\textup{Pr}(\rho_{t}=\lambda)=\sum_{\rho_{-t}\in P^{\otimes}}\sum%
_{\bm{\gamma}\in\Gamma^{\otimes}}\textup{Pr}(\bm{\gamma}_{1},\rho_{1},\ldots,%
\rho_{t}=\lambda,\ldots,\bm{\gamma}_{T},\rho_{T})=\textup{Pr}(\rho_{1}=\lambda),$$
where $P^{\otimes}=P\times P\times\ldots\times P$, $P$ a collection of all partitions of $m$ units and $\Gamma^{\otimes}=\Gamma\times\Gamma\times\ldots\times\Gamma$, $\Gamma$ a collection of all possible binary vectors of size $m$.
Proof.
See the Appendix.
∎
In what follows we will use $tRPM(\bm{\alpha},M)$ to denote our temporal random partition model (4) parameterized by $\alpha_{1},\ldots,\alpha_{T}$ and EPPF (2).
We briefly mention that introducing $\gamma_{it}$ is similar in spirit to the approach taken by Caron et al. (2007, 2017).
However, they use $\bm{\gamma}_{t}$ to identify a partial partition at time $t$ that informs how all the observational
units will be reallocated at time $t+1$. While this difference may seem benign at first glance, it has drastic ramifications on
the type of dependence that exists among the actual sequence of partitions. To see this, similar to what was done in
the simulation described in the Introduction, we generate 10,000 sequences of partitions based on our construction an
provide the average lagged ARI values in Figure 2. Notice now that the similarity of the partitions behaves in an
intuitive way as a function of lag. Mainly, that as lags increase the similarity between partitions decreases. Further,
$\alpha$ has a clear impact on the dependence between partitions with large $\alpha$ values resulting in strong dependence.
Observe also that the range of ARI values achieved by this construction can be substantially higher than what was
described earlier in the discussion leading to Figure 1.
2.2 Spatio-Temporal Model for a Sequence of Partitions
Before studying how our joint partition model can be employed in Bayesian modeling, we next describe our approach to incorporating space in the partition model. One possible way of adding a spatial component in the joint model would be to make the auxiliary variables $\bm{\gamma}_{it}$ spatially referenced. However, sample size consistency would be lost and as a result the marginal property in Proposition 2.1 would not hold. An alternative approach that we adopt is to include spatial information directly in the EPPF. If the spatially referenced EPPF employed preserves sample size consistency, then Proposition 2.1 still holds. To this end, we consider the spatial product partition model (sPPM) developed in Page and Quintana (2016). As a way of introducing the sPPM, let $\bm{s}_{i}$ denote the spatial coordinates of the $i$th item (note that these coordinates do not change over time) and let $\bm{s}^{\star}_{jt}$ be the subset of spatial coordinates that belong to the $j$th cluster at time $t$. Then we express the EPPF of the $t$th partition with the following product form
$$\displaystyle\text{Pr}(\rho_{t}|\nu_{0},M)\propto\prod_{j=1}^{k_{t}}c(S_{jt}|M%
)g(\bm{s}^{\star}_{jt}|\nu_{0}).$$
(5)
Here $c(\cdot|M)\geq 0$ is called the cohesion and is a set function that produces cluster weights a priori. We consider the cohesion $c(S_{jt}|M)=M\times(|S_{jt}|-1)!$ as it has connections with the CRP making this version of the sPPM a type of spatially re-weighted CRP. The similarity function $g(\cdot|\nu_{0})$ is a set function parametrized by $\nu_{0}$ that measures the compactness of the spatial coordinates in $\bm{s}^{\star}_{jt}$ producing higher values if the spatial coordinates in $\bm{s}^{\star}_{jt}$ are
less alike. Not all similarity functions preserve sample size consistency so to ensure this, after standardizing spatial locations, we employ
$$\displaystyle g(\bm{s}^{\star}_{jt}|\nu_{0})=\int\prod_{i\in S_{jt}}N(\bm{s}_{%
i}|\bm{m},\bm{V})NIW(\bm{m},\bm{V}|\bm{0},1,\nu_{0},\bm{I})d\bm{m}d\bm{V},$$
(6)
where $N(\cdot|\bm{m},\bm{V})$ denotes a bivariate normal density and $NIW(\cdot,\cdot|\bm{0},1,\nu_{0},\bm{I})$ a normal-inverse-Wishart density with mean $\bm{0}$, scale equal to 1, inverse scale matrix equal to $\bm{I}$, and $\nu_{0}$ being the user-supplied degrees of freedom. Note that larger values of $\nu_{0}$ increase spatial influence on partition probabilities. For more details on why this formulation preserves sample size consistency, see Müller et al. (2011) and Quintana et al. (2018). For more information regarding the impact of $\nu_{0}$ on product form of the partition model, see Page and Quintana (2016, 2018). We will denote the random partition distribution defined in (5) and (6) using $sPPM(\nu_{0},M)$.
We mention briefly that it would be very straightforward to build a partition model based on space and time by extending the sPPM so similarity function $g$ is a function of both space and time. Although this ensures that partitions will be influenced by space and time, the desire for partitions to evolve over time would be lost. In this setting, each spatial location by time point combination would be treated as an observational unit and would create clusters that transect time, which is something we wanted to avoid in our formulation.
We will use $stRPM(\bm{\alpha},\nu_{0},M)$ to denote our spatio-temporal random partition model (4) parameterized by $\alpha_{1},\ldots,\alpha_{T}$ and EPPF detailed in (5) and (6).
2.3 Hierarchical Data Model
Once a partition model is specified, there is tremendous flexibility regarding how to model space/time (global or cluster-specific) at different levels of a hierarchical model (at the data level or process level or both). Since we are interested to see how including space/time in the partition model impacts clustering and model fits, in the simulations of the next section, we consider a hierarchical model where space/time only appears in the partition model. In particular, using cluster label notation, we will employ the following hierarchical model
$$\displaystyle Y_{it}|\bm{\mu}^{\star}_{t},\bm{\sigma}_{t}^{2\star},\bm{c}_{t}$$
$$\displaystyle\stackrel{{\scriptstyle ind}}{{\sim}}N(\mu^{\star}_{c_{it}t},%
\sigma_{c_{it}t}^{2\star}),\ i=1,\ldots,m\ \mbox{and}\ t=1,\ldots,T,$$
(7)
$$\displaystyle(\mu_{jt}^{\star},\sigma^{\star}_{jt})|\theta_{t},\tau^{2}$$
$$\displaystyle\stackrel{{\scriptstyle ind}}{{\sim}}N(\theta_{t},\tau^{2})\times
UN%
(0,A_{\sigma}),\ j=1,\dots,k_{t},$$
$$\displaystyle(\theta_{t},\tau)$$
$$\displaystyle\stackrel{{\scriptstyle iid}}{{\sim}}N(\phi_{0},\lambda^{2})%
\times UN(0,A_{\tau}),\ t=1,\ldots,T,$$
$$\displaystyle\{\bm{c}_{t},\ldots,\bm{c}_{T}\}$$
$$\displaystyle\sim\mbox{{\it joint RPM}},$$
where $Y_{it}$ denotes the response measured on the $i$th unit at time $t$, joint RPM denotes some joint random partition model, and $UN$ denotes a uniform distribution.
The remaining assumptions (e.g., independence across clusters and exchangeability within each cluster) are commonly employed. Notice that in this model three entities are in some sense “competing” when determining cluster membership, namely: a) time, b) space, and c) response. This competition, however, is carried out in a probabilistic and coherent fashion.
2.4 Computation
As the posterior distribution implied by the model in (7) is not available in closed form, we build an algorithm
that permits sampling from it. The construction of $\text{Pr}(\rho_{1},\ldots,\rho_{T})$ naturally leads one to consider a Gibbs
sampler. In the Gibbs sampler, $\bm{\gamma}_{t}$ will need to be updated in addition to $\rho_{t}$ (by way of $\bm{c}_{t}$). But
the Markovian assumption reduces some of the cost as we only need to consider $\rho_{t-1}$ and $\rho_{t+1}$ when
updating $\rho_{t}$. Even though each update of $\rho_{t}$ and $\bm{\gamma}_{t}$ for $t=1,\ldots,T$ needs to be checked
for compatibility (i.e. proposed moves do not violate the prior construction), it is fairly straightforward to adapt standard
algorithms, e.g. Algorithm 8 of Neal (2000), with care to make sure that only experimental units with
$\gamma_{it}=0$ are considered when updating cluster labels at time $t$. In what follows we assume that the
joint RPM in (7) is the $stRPM(\bm{\alpha},\nu_{0},M)$ described earlier.
The MCMC algorithm we employ depends on deriving the complete conditionals for $\rho_{t}$ and $\gamma_{t}$. Before describing them, we introduce some needed notation. Let $N_{0t}=\sum_{j=1}^{m}I[\gamma_{jt}=0]$ denote the number of units to be reallocated when moving from time $t-1$ to $t$ (note that $N_{0t}\sim{\rm Bin}(m,1-\alpha_{t})$) and denote with $\rho_{t}^{-N_{0t}}$ the “reduced” partition that remains after removing the $N_{0t}$ units that are to be reallocated at time $t$ as indicated by $\bm{\gamma}_{t}$. A key result needed to derive the full conditionals of $\gamma_{it}$ and $c_{it}$ is provided in the following proposition.
Proposition 2.2.
Based on the construction of a joint probability model as described in Section 2.1 and $\rho_{1}\sim EPPF$, then we have
$$\displaystyle\textup{Pr}(\rho_{t}|\bm{\gamma}_{t},\rho_{t-1})=\left\{\begin{%
array}[]{l l}\textup{Pr}(\rho_{t})/\textup{Pr}(\rho^{-N_{0t}}_{t})&\rho_{t}%
\asymp\rho_{t-1}\\
0&otherwise,\end{array}\right.$$
(8)
where $\rho_{t}\asymp\rho_{t-1}$ indicates that $\rho_{t}$ is compatible with $\rho_{t-1}$.
Proof.
See the Appendix.
∎
When updating $\gamma_{it}$ in a Gibbs sampler, one can think of removing $\gamma_{it}$ from $\bm{\gamma}_{t}$, and then reinsert it as either a 0 or 1. To this end, let $N^{(-i)}_{0t}=\sum_{j\neq i}I[\gamma_{it}=0]$ denote the case when $\gamma_{it}$ is reinserted as a 1 and $N^{(+i)}_{0t}=N^{(-i)}_{0t}+1$ denote the case when $\gamma_{it}$ is reinserted as a 0. Now, the full conditional for $\gamma_{it}=1$, denoted by $\text{Pr}(\gamma_{it}=1|-)$, is
$$\displaystyle\text{Pr}(\gamma_{it}=1|-)$$
$$\displaystyle\propto\text{Pr}(\rho_{t}|\bm{\gamma}_{t},\rho_{t-1})\text{Pr}(%
\bm{\gamma}_{t})\text{I}[\rho_{t}\asymp\rho_{t-1}],$$
$$\displaystyle\propto\frac{\text{Pr}(\rho_{t})}{\text{Pr}(\rho^{-N^{(+i)}_{0t}}%
_{t})}\alpha_{t}^{\gamma_{it}}\text{I}[\rho_{t}\asymp\rho_{t-1}].$$
Here $\text{I}[\cdot]$ denotes an indicator function. The resulting normalized full conditional for $\gamma_{it}$ is
$$\displaystyle\text{Pr}(\gamma_{it}=1|-)=\displaystyle\frac{\alpha_{t}\text{Pr}%
(\rho^{-N^{(-i)}_{0t}}_{t})}{\alpha_{t}\text{Pr}(\rho^{-N^{(-i)}_{0t}}_{t})+(1%
-\alpha_{t})\text{Pr}(\rho^{-N^{(+i)}_{0t}}_{t})}\text{I}[\rho_{t}\asymp\rho_{%
t-1}].$$
(9)
For a given EPPF that has a closed form (e.g., CRP), it is straightforward to compute $\text{Pr}(\rho^{-N^{(-i)}_{0t}}_{t})$ and $\text{Pr}(\rho^{-N^{(+i)}_{0t}}_{t})$. If, however, the EPPF does not have a closed form (e.g., sPPM), then note that (9) can be re-expressed as
$$\displaystyle\text{Pr}(\gamma_{it}=1|-)=\displaystyle\frac{\alpha_{t}\text{I}[%
\rho_{t}\asymp\rho_{t-1}]}{\alpha_{t}+(1-\alpha_{t})\text{Pr}(\rho^{-N^{(+i)}_%
{0t}}_{t})/\text{Pr}(\rho^{-N^{(-i)}_{0t}}_{t})}.$$
(10)
The quantity $\text{Pr}(\rho^{-N^{(+i)}_{0t}}_{t})/\text{Pr}(\rho^{-N^{(-i)}_{0t}}_{t})$ is a commonly encountered expression in MCMC methods employed in random partition modeling. See for example Neal’s Algorithm 8 (Neal, 2000). Those same methods can be employed to calculate the desired probabilities.
The full conditional for $c_{it}=h$ corresponding to $\gamma_{it}=0$ is the following
$$\displaystyle\text{Pr}(c_{it}=h|-)$$
$$\displaystyle\propto N(Y_{it}|\mu^{\star}_{c_{it}=h,t},\sigma^{2\star}_{c_{it}%
=h,t})\text{Pr}(c_{1t},\ldots,c_{it}=h,\ldots,c_{mt})\text{I}[\rho_{t}\asymp%
\rho_{t-1}].$$
The case that unit $it$ forms a new cluster must also be considered so that
$$\displaystyle\text{Pr}(c_{it}=h|-)\propto\left\{\begin{array}[]{cl}N(Y_{it}|%
\mu^{\star}_{c_{it}=h,t},\sigma^{2\star}_{c_{it}=h,t})\text{Pr}(c_{it}=h)\text%
{I}[\rho_{t}\asymp\rho_{t-1}]&\mbox{$h=1,\ldots,k_{t}^{-i}$, }\\
N(Y_{it}|\mu^{\star}_{new_{h},t},\sigma^{2\star}_{new_{h},t})\text{Pr}(c_{it}=%
h)\text{I}[\rho_{t}\asymp\rho_{t-1}]&\mbox{$h=k_{t}^{-i}+1$,}\end{array}\right.$$
where $\text{Pr}(c_{it}=h)=\text{Pr}(c_{1t},\ldots,c_{it}=h,\ldots,c_{mt})$, $\mu^{\star}_{new_{h},t}$ and $\sigma^{2\star}_{new_{h},t}$ are auxiliary parameters drawn from the prior as in Neal (2000)’s Algorithm 8 (with one auxiliary parameter) and $k_{t}^{-i}$ are the number of clusters at time $t$ when the $i$th unit has been removed. Details of computation procedures associated with the sPPM can be found in Page and Quintana (2016). Given $\rho_{t}$ and $\bm{\gamma}_{t}$, the full conditionals of the remaining parameters in model (7) follow standard techniques. A sample can be drawn from the posterior distribution implied by model (7) by iterating through the complete conditionals for $\bm{\gamma}_{t}$ and $\rho_{t}$ and those of other model parameters.
3 Simulation Studies
In this section we describe three simulation studies that explore the performance of our proposal. The first simulation study is focused on the temporal dependence among estimated partitions, the second on the temporal dependence that the joint partition model induces among the $\bm{Y}_{i}=(Y_{i1},\ldots,Y_{iT})$, and the third on the impact that including space in the partition model has on model fit.
3.1 Simulation 1: Temporal Dependence in Estimated Partitions
The purpose of the first simulation is to study the accuracy of partition estimates (i.e., $\hat{\rho}_{t}$) and how much they change over time. As such, in this study we do not consider spatial clustering. We do however, explore accuracy in estimating $\mu_{it}$ and $\alpha_{t}$. To this end, we considered model (7) as a data generating mechanism to create one hundred datasets with fifty observations at five time points. For the joint RPM in model (7) we used $tRPM(\bm{\alpha})$ with $\alpha_{t}=\alpha$ for all $t$ and generate synthetic datasets under $\alpha\in\{0,0.1,0.25,0.5,0.75,0.9,0.999\}$. For all $i$ and $t$, we set $\sigma_{c_{it}t}^{2\star}=\sigma^{2}=1$, $\tau^{2}=25$, and $\theta_{t}=0$.
To each synthetic data set we fit model (7) using the MCMC algorithm detailed in Section 2.4 by collecting 10,000 iterates and discarding the first 5,000 as burn-in and thinning by 5 (resulting in 1,000 MCMC samples) after setting $A_{\sigma}=5$ and $A_{\tau}=10$. All partition point estimates were estimated using the method developed in the salso R package (Dahl 2019) with the Binder loss function (Binder 1978, Lau and Green 2007). To measure similarity between partitions, we employed the adjusted Rand index (Rand 1971; Hubert and Arabie 1985) and we used WAIC (Gelman et al. 2014) to measure model fit.
Table 1 displays the lagged 1 and 4 adjusted Rand index (ARI) as a function of $\alpha$. As expected, for both lags the ARI increases as $\alpha$ increases. Also as expected lagged 4 ARI increases less as a function of $\alpha$ compared to the lagged 1 ARI. Note that on average the lagged 1 ARI for $\alpha\in\{0.1,0.25\}$ is smaller than that for $\alpha=0$. This is because the variability associated with lagged 1 ARI when $\alpha=0$ is much larger than when $\alpha>0$ producing a few lagged ARI values that are large. The median of the lagged ARI values increase as a function of $\alpha$ monotonically.
To study the ability to recover $\mu_{it}$ and $\alpha$, 95% credible intervals for each were computed and coverage was estimated. Results are provided in Table 1. Notice that coverage for $\alpha$ is low when the true $\alpha$ is at or near the boundary (e.g., $\alpha\in\{0,0.9999\})$ which is to be expected. The coverage associated with $\mu_{it}$ is close to the nominal rate regardless of the value of $\alpha$. Therefore, temporal dependence in the partition model does not adversely impact the ability to estimate individual means.
Lastly, to compare model fit when using $tRPM(\alpha,M)$ as the RPM in model (7) relative to $\rho_{t}\stackrel{{\scriptstyle iid}}{{\sim}}CRP(M)$, we calculated the WAIC for each data set when fitting model (7) under both RPMs. Results are provided in Table 1 where each entry is an average WAIC value over all 100 datasets. Notice that, when the independent partitions were used to generate data (i.e., $\alpha=0$), modeling partitions independently produces slightly better model fit as would be expected. But even if relatively weak temporal dependence exists among the sequence of partitions, there are gains in modeling the sequence of partitions with $tRPM(\alpha,M)$, with gains becoming substantial as $\alpha$ increases.
The upshot from this simulation study is that lagged partition estimates when employing $tRPM(\alpha,M)$ display intuitive behavior in that similarity between partition estimates decreases as lag increases. In addition, employing the $tRPM(\alpha,M)$ partition model does not negatively impact parameter estimation and produces improved model fits when dependence is present in the sequence of partitions and a minimal cost in model fit when it is not.
3.2 Simulation 2: Induced Correlation at the Response Level
A potential benefit of developing a joint model for partitions is the ability to accommodate temporal dependence that may
exist between $Y_{it}$ and $Y_{it+1}$. To study this, we conducted a small Monte Carlo simulation study that is comprised
of sampling repeatedly from the $tRPM(\alpha,M)$ using the computational approach of Section 2.4.
Once the partition is generated, the temporal dependence among the $\bm{Y}_{i}$ depends on specific model choices for
$\mu_{jt}^{\star}$. Here we use $\mu^{\star}_{jt}\sim N(\phi_{1}\mu_{jt-1}^{\star},\tau^{2}(1-\phi_{1}^{2}))$ for $t>2$, $j=1,\ldots,k_{t}$,
and $|\phi_{1}|\leq 1$. For $t=1$ we use $\mu^{\star}_{j1}\sim N(0,\tau^{2})$ and if $k_{t+1}>k_{t}$ new $\mu^{\star}_{jt+1}$ values are drawn from $N(0,\tau^{2})$.
Now setting $m=25$, $T=10$, $\tau=10$, and $\sigma=1$, 100 data sets were generated for $\phi_{1}\in\{0,0.25,0.5,075,0.9,1\}$. For each data set generated, the lagged auto-correlations among $\bm{Y}_{i}$ were computed for $i=1,\ldots,m$. The results found in Figure 3 are the lagged auto-correlations averaged over the $m$ units for $\alpha\in\{0,0.25,0.5,0.75,0.9\}$.
As can be seen in Figure 3, when partitions are independent (i.e., $\alpha=0$), no correlation propagates to the data level. The same can be said if atoms are $iid$ (i.e., $\phi_{1}=0$). As the temporal dependance among $\mu_{jt}^{\star}$ increases (i.e., $\phi_{1}$ increases), there is stronger temporal dependence among $Y_{i1},\ldots,Y_{iT}$. Notice further that this dependence persists longer in time as $\alpha$ increases as one would expect.
3.3 Simulation 3: Dependence in Estimated Partitions
We now discuss our final simulation study, where we investigated the performance of our procedure when both space and time are considered. To do so, we created synthetic data sets that contain spatio-temporal structure. Each employs a $15\times 15$ regular grid with spatial locations coming from the unit interval. In addition, either 5 or 10 time points were considered resulting in 1,125 or 2,250 total observations. Response values were generated in two ways. The first employs a Gaussian process with a separable spatio-temporal exponential covariance function. We set the spatial scale to 0.3, the temporal scale to 2 and the sill to 1.75 (see Padoan and Bevilacqua 2015 for more details). Note that no “true” partition exists for this data generating mechanism. However, we study it to explore performance of our method when spatial structure exists among observations but was not induced through partitioning. The second method of generating response values essentially employs model (7) as a data generating mechanism. Spatio-temporal partitions were generated using (6) together with conditional cluster label probabilities of Müller et al. (2011, pg. 265) and setting $\alpha_{t}=\alpha$ for all $t$ with $\alpha\in\{0,0.5,0.9\}$ (note that for $\alpha=0$ no temporal dependence exists among partitions). In the similarity function (6) we considered $\nu_{0}\in\{2,20\}$ where $\nu_{0}=2$ corresponds to light weight on spatial proximity and $\nu_{0}=20$ moderate weight. Finally, we set $\tau^{2}=1$ and $\sigma^{2\star}_{c_{it}t}=\sigma^{2}=0.04$ for all $i$ and $t$ resulting in smaller with-in cluster variability relative to between-cluster variability.
To determine the impact that each component of our spatio-temporal partition model has on model fit, we fit the hierarchical
model (7) to each synthetic data set using a variety of random partition models which are listed below.
As a competitor, we consider a linear dependent Dirichlet process (MacEachern, 2000; De Iorio et al., 2009), indexing the
random probability measure through the mean function of the atoms by space and time. To ensure sufficient flexibility, B-spline basis functions for both spatial
coordinates were employed. The details of each model considered are
Model 1: $(\rho_{1},\ldots,\rho_{T})\sim stRPM(\bm{\alpha},\nu_{0},M)$
Model 2: $\rho_{t}\stackrel{{\scriptstyle iid}}{{\sim}}sPPM(\nu_{0},M)$ for $t=1,\ldots,T$.
Model 3: $(\rho_{1},\ldots,\rho_{T})\sim tRPM(\bm{\alpha},M)$
Model 4: $\rho_{t}\stackrel{{\scriptstyle iid}}{{\sim}}CRP(M)$ for $t=1,\ldots,T$.
Model 5: linear dependent Dirichlet process mixture model (DDPM).
Additionally, for each model that employs the sPPM, we considered both $\nu_{0}=2$ (models 1a, 2a) and $\nu_{0}=20$ (models 1b, 2b). For each data generating scenario, 100 data sets were created and each of the models listed was fit by collecting 1,000 MCMC samples after discarding the first 5,000 as burn-in and thinning by 5 after setting $A_{\sigma}=1$ and $A_{\tau}=2$. Model fits were compared using WAIC. Results can be found in Figures 4 and 5.
The primary purpose of Figure 4 is to compare model fit from the spatio-temporal partition model we develop to that from the linear DDPM (model 5). It appears that all methods are competitive to the linear DDPM, which is particularly true with 10 time points. Thus, our dependent partition model accommodates temporal dependence more efficiently relative to the linear DDPM under this data generating scenario. Note that regardless of the number of time points, model 1b ($stRPM(\alpha,\nu_{0},M)$ with $\nu_{0}=20$) appears to perform best. Surprisingly, $tRPM(\alpha,M)$ (model 4) is quite competitive, particularly with 10 time points. The conclusion here is that employing $stRPM(\alpha,\nu_{0},M)$ to model partitions appears to accommodate spatio-temporal dependence even if there is no underlying partition structure.
From Figure 5 we see that when partitions are generated independently, there is very little lost by employing the dependent joint model in terms of model fit (see top left panel for model 3 and 4). However, as spatial and/or temporal structure is introduced in the partition model, there are clear gains in terms of model fit when employing $tRPM(\bm{\alpha},M)$ and/or $stRPM(\bm{\alpha},\nu_{0},M)$. From this simulation it seems that employing the $tRPM(\alpha,M)$ regardless of the strength of temporal dependence among partitions is reasonable as there is minimal cost in terms of model fit even when partitions are generated independently. Finally, it appears that $stRPM(\bm{\alpha},\nu_{0},M)$ performed best.
3.4 Application
In this section we apply our method to a real-world data set coming from the field of environmental science. A second application in educational measurement is provided in Section LABEL:SIMCE.application of the online Supplementary Material. As mentioned previously, once a partition model is specified there is quite a bit of flexibility regarding how (or if) temporal dependence is incorporated in other parts of a hierarchical model. To illustrate this, we incorporate temporal dependence in three places of the hierarchical model we construct.
As part of preliminary exploratory data analysis (not shown), we examined serial dependence for each experimental (monitoring station), and concluded that they all exhibited a particular type of
temporal dependence. Because of this, we introduce a unit-specific temporal dependence parameter $|\eta_{1i}|\leq 1$ and model observations from a single unit over time ($Y_{1i},\ldots,Y_{iT}$) with an AR(1) structure. In addition, motivated by a desire for parsimony, we employed a Laplace prior for $\eta_{1i}$. Finally, to permit the temporal dependence in the partition model to propagate through the hierarchical model, we model $\theta_{t}$ with an AR(1) structure. The full hierarchical model is detailed in (11).
$$\displaystyle Y_{it}|Y_{it-1},\bm{\mu}^{\star}_{t},\bm{\sigma}^{2\star}_{t},%
\bm{\eta},\bm{c}_{t}$$
$$\displaystyle\stackrel{{\scriptstyle ind}}{{\sim}}N(\mu^{\star}_{c_{it}t}+\eta%
_{1i}Y_{it-1},\sigma_{c_{it}t}^{2\star}(1-\eta_{1i}^{2})),$$
(11)
$$\displaystyle Y_{i1}$$
$$\displaystyle\stackrel{{\scriptstyle ind}}{{\sim}}N(\mu^{\star}_{c_{i1}1},%
\sigma_{c_{i1}1}^{2\star}),$$
$$\displaystyle\xi_{i}=\mbox{Logit}(0.5(\eta_{1i}+1))$$
$$\displaystyle\stackrel{{\scriptstyle iid}}{{\sim}}Laplace(a,b),$$
$$\displaystyle(\mu_{jt}^{\star},\sigma^{\star}_{jt})$$
$$\displaystyle\stackrel{{\scriptstyle ind}}{{\sim}}N(\theta_{t},\tau^{2})\times
UN%
(0,A_{\sigma}),$$
$$\displaystyle\theta_{t}|\theta_{t-1}$$
$$\displaystyle\stackrel{{\scriptstyle ind}}{{\sim}}N(\phi_{0}+\phi_{1}\theta_{t%
-1},\lambda^{2}(1-\phi_{1}^{2})),$$
$$\displaystyle(\theta_{1},\tau)$$
$$\displaystyle\sim N(\phi_{0},\lambda^{2})\times UN(0,A_{\tau}),$$
$$\displaystyle(\phi_{0},\phi_{1},\lambda)$$
$$\displaystyle\sim N(0,s^{2})\times UN(-1,1)\times UN(0,A_{\lambda}),$$
$$\displaystyle\{\bm{c}_{t},\ldots,\bm{c}_{T}\}$$
$$\displaystyle\sim stRPM(\bm{\alpha},\nu_{0},M),\ \mbox{with $\alpha_{t}%
\stackrel{{\scriptstyle iid}}{{\sim}}Beta(a_{\alpha},b_{\alpha})$},$$
where all Roman letters correspond to parameters that are user supplied. Notice that there are a number of special cases embedded in our hierarchical model. For example, $\eta_{i1}=0$ for all $i$ results in conditionally independent observations. Further, $\phi_{1}=0$ results in independent atoms and $\alpha_{t}=0$ for all $t$ in independent partitions over time. Note that model (7) used in the simulation studies is a special case of (11) ($\phi_{1}=0$ and $\eta_{i1}=0$ for all $i$). $A_{\sigma}$ may influence partition formation. If this value is selected to be too large, then all observational units could plausibly be allocated to one cluster. If it is too small then many spurious clusters could potentially be formed. Therefore, this parameter must be selected thoughtfully. Our approach is to set $A_{\sigma}$ to about half the sample standard deviation computed using all observations.
3.5 Rural Background PM${}_{10}$ Data Application
The rural background PM${}_{10}$ data is taken from the European air quality database.
These data are comprised of the daily measurements of particulate matter with a diameter less than 10 $\mu$m
from rural background stations in Germany and are publicly available in the gstat package (Gräler et al. 2016)
found on CRAN in R (R Core Team 2018). We focus on average monthly PM${}_{10}$ measures from the year 2005.
Of the 69 stations, 9 were removed because of missing values.
We fit the hierarchical model (11) to these data and consider all the possible special cases (i.e., $\eta_{1i}=0$ or not, $\phi_{1}=0$ or not, $\alpha_{t}=0$ or not, with and without space). This resulted in 16 total models that were fit by collecting 1,000 MCMC iterates after discarding the first 10,000 as burn-in and thinning by 10. The prior values employed were $A_{\sigma}=A_{\tau}=5$, $s^{2}=100$, $a=0$, $b=1$, $a_{\alpha}=b_{\alpha}=1$, and $\nu_{0}=5$. The WAIC and log pseudo marginal likelihood (LPML) for each model are presented in Table 2.
Notice that among all the model fits, employing a variant of $tRPM(\bm{\alpha},M)$ (i.e., rows with “Yes” in the “In Partition” column) generally improves model fit. The best performing model in terms of WAIC and LPML includes spatio-temporal dependence in the partition model, temporal dependence among the atoms, and temporal dependence in the likelihood. To see how the different models impact how partitions evolve over time, we provide Figure 6. This figure displays the lagged ARI values for each of the 16 models. Notice that when partitions are modeled independently (first or third rows of Figure 6) then partitions evolve over time quite erratically in the sense that the cluster configuration can change dramatically from one time point to the next. However, when employing $tRPM(\bm{\alpha},M)$ (second row of Figure 6) the partitions seemed to evolve much more “smoothly” as there is less drastic changes in cluster configuration. Finally, it appears that employing the $stRPM(\bm{\alpha},\nu_{0},M)$ (fourth row of Figure 6) not only produces partitions that evolve “smoothly” over time, but the temporal dependence seems to decay quicker than when employing $tRPM(\bm{\alpha},M)$ only. In fact the model that produces the best model fit metrics (right most plot of the bottom row) seems to produce partitions that change quite gently over time as desired.
4 Conclusions
We developed a joint probability model for a sequence of partitions that explicitly considers temporal dependence among the partitions. Further we showed that our methodology is capable of accommodating partitions that evolve slowly over time in that the adjusted Rand index between estimated partitions decays as the lag in time increases. Further, we showed that in the absence of temporal dependence between partitions, the cost in terms of model fit is minimal.
Even though our main focus is constructing a dependent probability model for a sequence of random partitions, our method, when coupled with a simple hierarchical model, could provide an alternative approach to general space-time modeling that completely avoids inverting matrices. This could result in computation gains compared to employing computationally intense non-separable covariance functions. In addition, assumptions associated with stationarity and/or isotropy can be avoided.
The predictive nature of the spatio-temporal prior on a sequence of random partitions we have presented has a
(first-order) Markovian structure. Various extensions can be considered, such as adding higher order dependence
across time or dependence in baseline covariates. All of these cases would build on our constructive
definition, as extra refinements of the basic idea of carrying smooth transitions on time and space. The Markovian
approach can also be used for predictive inference, although that was not our main motivation for the models
implemented here, and therefore we have not explored this avenue.
Appendix A Proof of Proposition 2.1
Proof.
For clarity, here we introduce notation that highlights the dependence of partitions on sample size. For example, $\rho_{t,m}=(S_{1,t},\ldots,S_{k_{t}(m),t})$ and $[m]=\{1,\ldots,m\}$.
By assumption $\text{Pr}(\rho_{1,m})$ is specified by means of an EPPF which we now construct. Denote $\mathbb{N}^{*}=\cup_{k=0}^{\infty}\mathbb{N}^{k}$, and identify any
$\bm{n}=(n_{1},\ldots,n_{k})\in\mathbb{N}^{*}$ with the infinite sequence
$(n_{1},\ldots,n_{k},0,0,\ldots)$. Given $\bm{n}\in\mathbb{N}^{*}$, let
$k(\bm{n})$ denote the number of non-zero entries in $\bm{n}$ and denote by
$\bm{n}^{j+}$ the result of incrementing $\bm{n}$’s $j$th component (i.e., $n_{j}$) by 1, with $1\leq j\leq k(\bm{n})+1$. An EPPF is then any function
$r:\mathbb{N}^{*}\longrightarrow[0,1]$ that is symmetric in its arguments and
where
$$r(1)=1\qquad\mbox{and}\qquad r(\bm{n})=\sum_{j=1}^{k(\bm{n})+1}r(\bm{n}^{j+})%
\qquad\mbox{for all $\bm{n}\in\mathbb{N}^{*}$.}$$
(12)
Condition (12) implies that a EPPF is sample
size consistent, i.e., marginalizing the $(n+1)$st element leads to the
model for $n$ elements. The EPPF also implies exchangeability of
configurations in the sense that a EPPF is invariant under permutations of
the elements that keep the cluster sizes unaltered. We also note that any
valid EPPF defines a predictive rule of the form
$$r_{j}(\bm{n})=\frac{r(\bm{n}^{j+})}{r(\bm{n})},\qquad\mbox{for $1\leq j\leq k(%
\bm{n})+1$,}$$
(13)
where it is assumed that $r(\bm{n})>0$ and $r_{j}(\bm{n})$ represents the probability of a new element joining the
$j$th already existing cluster, for $1\leq j\leq k(\bm{n})$, or starting a
new one (the $k(\bm{n})+1$). The one-step rule (13) can also be
extended to predictions of two or more elements by simply iterating the
one-step rule as many times as needed. Now, given an EPPF $r$, we have that
$$\text{Pr}(\rho_{1,m}=(S_{1,1},\ldots,S_{k_{1}(m),1}))=r(n_{1,1},\ldots,n_{k_{1%
}(m),1}).$$
(14)
To prove the result, it suffices to show that it holds for $\rho_{2,m}$ and then by induction the result holds generally. Denote by
$[\Gamma]=\{i\in\{1,\ldots,m\}:\,\gamma_{i2}=0\}$ the (random) set of elements
removed from $\rho_{1,m}$. Then, $\rho_{1,m}^{-N_{02}}$ is a partition of the
elements of $[m]-[\Gamma]$ (where as before $N_{02}=\sum_{j=1}^{m}I[\gamma_{j2}=0]$). By exchangeability and the fact that an EPPF is
sample size consistent, we have that for any partition $S^{-}_{1},\ldots,S^{-}_{k([m]-[\Gamma])}$ of $[m]-[\Gamma]$:
$$\displaystyle\text{Pr}(\rho_{2,m}^{-N_{02}}=(S^{-}_{1},\ldots,S^{-}_{k([m]-[%
\Gamma])})\mid[\Gamma])$$
$$\displaystyle=\text{Pr}(\rho_{1,m}^{-N_{02}}=(S^{-}_{1},\ldots,S^{-}_{k([m]-[%
\Gamma])})\mid[\Gamma])$$
$$\displaystyle=r(|S^{-}_{1}|,\ldots,|S^{-}_{k([m]-[\Gamma])}|),$$
where $|S_{j}|$ is the number of elements in $S_{j}$. In addition, and again by exchangeability and
sample size consistency, the predictive rule starting
from $[m]-[\Gamma]$ (or from any subset of $[m]$ for that matter) depends only on the sizes
of the subsets in that partition. Thus, conditioning on all reallocation configurations and
initial partition after subject removal we have:
$$\displaystyle\begin{split}\displaystyle\text{Pr}(\rho_{2,m}=(S_{1},\ldots,S_{k%
}))&\displaystyle=\sum_{[\Gamma]}\sum_{\rho_{2,m}^{-N_{02}}}\text{Pr}(\rho_{2,%
m}=(S_{1},\ldots,S_{k})\mid[\Gamma],\,\rho_{2,m}^{-N_{02}})\times\\
&\displaystyle\qquad\qquad\qquad\text{Pr}(\rho_{2,m}^{-N_{02}}\mid[\Gamma])%
\text{Pr}([\Gamma]),\end{split}$$
$$\displaystyle\begin{split}&\displaystyle=\sum_{[\Gamma]}\sum_{\rho_{1,m}^{-N_{%
02}}}\text{Pr}(\rho_{1,m}=(S_{1},\ldots,S_{k})\mid[\Gamma],\,\rho_{1,m}^{-N_{0%
2}})\times\\
&\displaystyle\qquad\qquad\qquad\text{Pr}(\rho_{1,m}^{-N_{02}}\mid[\Gamma])%
\text{Pr}([\Gamma]),\end{split}$$
$$\displaystyle=\text{Pr}(\rho_{1,m}=(S_{1},\ldots,S_{k})),$$
where the second to last equality follows from the constructive description given earlier
and the properties of the EPPF. The result then follows.
∎
Appendix B Proof of Proposition 2.2
Proof.
Let $P_{C_{t}}=\{\rho_{t}\in P:\rho_{t}\asymp\rho_{t-1}\}$ denote the collection of all partitions of the elements of $[m]$ at time $t$ that are compatible with $\rho_{t-1}$ based on $\bm{\gamma}_{t}$. Then by construction, $\text{Pr}(\rho_{t}|\bm{\gamma}_{t},\rho_{t-1})$ is a random partition distribution whose support is $P_{C_{t}}$ so that
$$\displaystyle\text{Pr}(\rho_{t}=\lambda|\bm{\gamma}_{t},\rho_{t-1})=%
\displaystyle\frac{\text{Pr}(\rho_{t}=\lambda)I[\lambda\in P_{C_{t}}]}{\sum_{%
\lambda}\text{Pr}(\rho_{t}=\lambda)I[\lambda\in P_{C_{t}}]}.$$
It only remains to show that $\sum_{\lambda\in P_{C_{t}}}\text{Pr}(\rho_{t}=\lambda)=\text{Pr}(\rho_{t}^{-N_%
{0t}})$ which is more easily seen employing cluster label notation. Let $c_{\gamma_{t}}=\{c_{it}:\gamma_{it=0}\}$. By iteratively invoking the sample size consistency property we have that
$$\displaystyle\text{Pr}(\rho_{t}^{-N_{0t}})$$
$$\displaystyle=\sum_{c_{\gamma_{t}}}\text{Pr}(\rho_{t}=\{c_{1t},\ldots,c_{mt}\})$$
$$\displaystyle=\sum_{\lambda\in P_{C_{t}}}\text{Pr}(\rho_{t}=\lambda),$$
where the last equality holds since summing over $c_{\gamma_{t}}$ is based only on cluster labels that are not fixed from time point $t-1$ to $t$ which results in summing over all possible compatible partitions (i.e., $\lambda\in P_{C_{t}}$).
∎
SUPPLEMENTARY MATERIAL
Title:
Supplementary Material. This file contains details of our model applied to an additional application in the field of education.
R-package for the $stRPM$ routine:
An R-package titled drpm contains code used to fit model described in (11).
References
Antoniano-Villalobos and Walker (2016)
Antoniano-Villalobos, I. and Walker, S. G. (2016), “A nonparametric
model for stationary time series,” J. Time Series Anal., 37,
126–142.
Binder (1978)
Binder, D. A. (1978), “Bayesian Cluster Analysis,” Biometrika,
65, 31–38.
Caron et al. (2007)
Caron, F., Davy, M., and Doucet, A. (2007), “Generalized Polya Urn for
Time-varying Dirichlet Process Mixtures,” in Proceedings of the
Twenty-Third Conference on Uncertainty in Artificial Intelligence,
Arlington, Virginia, United States: AUAI Press, UAI’07, pp. 33–40.
Caron et al. (2017)
Caron, F., Neiswanger, W., Wood, F., Doucet, A., and Davy, M. (2017),
“Generalized Pólya Urn for Time-Varying Pitman-Yor Processes,”
Journal of Machine Learning Research, 18, 1–32.
Cassese et al. (2019)
Cassese, A., Zhu, W., Guindani, M., and Vannucci, M. (2019), “A
Bayesian Nonparametric Spiked Process Prior for Dynamic Model Selection,”
Bayesian Analysis, advance publication.
Dahl (2019)
Dahl, D. B. (2019), salso: Sequentially-Allocated Latent Structure
Optimization, r package version 0.1.10.
De Blasi et al. (2015)
De Blasi, P., Favaro, S., Lijoi, A., Mena, R. H., Prünster, I., and
Ruggiero, M. (2015), “Are Gibbs-Type Priors the Most Natural
Generalization of the Dirichlet Process?” IEEE Transactions on
Pattern Analysis and Machine Intelligence, 37, 212–229.
De Iorio et al. (2009)
De Iorio, M., Johnson, W., Müller, P., and Rosner, G. (2009),
“Bayesian Nonparametric Nonproportional Hazards Survival Modeling,”
Biometrics, 65, 762–771.
Duan et al. (2007)
Duan, J. A., Guindani, M., and Gelfand, A. E. (2007), “Generalized
spatial Dirichlet process models,” Biometrika, 94, 809–825.
Gelfand et al. (2010)
Gelfand, A. E., Diggle, P. J., Fuentes, M., and Guttorp, P. (eds.) (2010),
Handbook of Spatial Statistics, Boca Raton: Chapman and Hall/CRC,
1st ed.
Gelfand et al. (2005)
Gelfand, A. E., Kottas, A., and MacEachern, S. N. (2005), “Bayesian
Nonparametric Spatial Modeling With Dirichlet Process Mixing,”
Journal of the American Statistical Association, 100, 1021–1035.
Gelman et al. (2014)
Gelman, A., Hwang, J., and Vehtari, A. (2014), “Understanding
predictive information criteria for Bayesian models,” Statistics and
Computing, 24, 997–1016.
Gräler et al. (2016)
Gräler, B., Pebesma, E., and Heuvelink, G. (2016), “Spatio-Temporal
Interpolation using gstat,” The R Journal, 8, 204–218.
Griffin and Steel (2006)
Griffin, J. E. and Steel, M. F. J. (2006), “Order-Based Dependent
Dirichlet Processes,” Journal of the American Statistical
Association, 101, 179–194.
Gutiérrez et al. (2016)
Gutiérrez, L., Mena, R. H., and Ruggiero, M. (2016), “A time
dependent Bayesian nonparametric model for air quality analysis,”
Computational Statistics & Data Analysis, 95, 161 – 175.
Hubert and Arabie (1985)
Hubert, L. and Arabie, P. (1985), “Comparing Partitions,”
Journal of Classification, 2, 193–218.
Jo et al. (2017)
Jo, S., Lee, J., Müller, P., Quintana, F. A., and Trippa, L. (2017),
“Dependent Species Sampling Models for Spatial Density Estimation,”
Bayesian Analysis, 12, 379–406.
Kottas et al. (2008)
Kottas, A., Duan, J. A., and Gelfand, A. E. (2008), “Modeling Disease
Incidence Data with Spatial and Spatio-Temporal Dirichlet Process Mixtures,”
Biometrical Journal, 50, 29–42.
Lau and Green (2007)
Lau, J. W. and Green, P. J. (2007), “Bayesian Model-Based Clustering
Procedures,” Journal of Computational and Graphical Statistics, 16,
526–558.
MacEachern (2000)
MacEachern, S. N. (2000), “Dependent Dirichlet processes,” Tech. rep.,
Ohio State University.
Müller et al. (2011)
Müller, P., Quintana, F., and Rosner, G. L. (2011), “A Product
Partition Model With Regression on Covariates,” Journal of
Computational and Graphical Statistics, 20, 260–277.
Neal (2000)
Neal, R. M. (2000), “Markov Chain Sampling Methods for Dirichlet
Process Mixture Models,” Journal of Computational and Graphical
Statistics, 9, 249–265.
Nieto-Barajas et al. (2012)
Nieto-Barajas, L. E., Muller, P., Ji, Y., Lu, Y., and Mills, G. B. (2012),
“A Time-Series DDP for Functional Proteomics Profiles,”
Biometrics, 68, 859–868.
Paci and Finazzi (2018)
Paci, L. and Finazzi, F. (2018), “Dynamic model-based clustering for
spatio-temporal data,” Statistics & Computing, 28, 359 – 374.
Padoan and Bevilacqua (2015)
Padoan, S. A. and Bevilacqua, M. (2015), “Analysis of Random Fields
Using CompRandFld,” Journal of Statistical Software, 63, 1–27.
Page and Quintana (2016)
Page, G. L. and Quintana, F. A. (2016), “Spatial Product Partition
Models,” Bayesian Analysis, 11, 265–298.
Page and Quintana (2018)
— (2018), “Calibrating Covariate Informed Product Partition Models,”
Statistics and Computing, 28, 1009–1031.
Petrone et al. (2009)
Petrone, S., Guindani, M., and Gelfand, A. E. (2009), “Hybrid Dirichlet
Mixture Models for Functional Data,” Journal of the Royal Statistical
Society: Series B, 71, 755–782.
Quintana et al. (2018)
Quintana, F. A., Loschi, R. H., and Page, G. L. (2018), Bayesian
Product Partition Models, Wiley StatsRef: Statistics Reference Online, pp.
1–15.
R Core Team (2018)
R Core Team (2018), R: A Language and Environment for Statistical
Computing, R Foundation for Statistical Computing, Vienna, Austria.
Rand (1971)
Rand, W. M. (1971), “Objective Criteria for the Evaluation of
Clustering Methods,” Journal of the American Statistical
Association, 66, 846–850.
Savitsky (2016)
Savitsky, T. (2016), “Bayesian Nonparametric Multiresolution Estimation
for the American Community Survey,” Annals of Applied Statistics,
10, 2157–2181.
Sethuraman (1994)
Sethuraman, J. (1994), “A constructive definition of Dirichlet priors,”
Statistica Sinica, 4, 639–650.
Wade et al. (2014)
Wade, S., Walker, S. G., and Petrone, S. (2014), “A Predictive Study of
Dirichlet Process Mixture Models for Curve Fitting,” Scandinavian
Journal of Statistics, 41, 580–605.
Zanini et al. (2019)
Zanini, C. T. P., Müller, P., Ji, Y., and Quintana, F. A. (2019),
“A Bayesian Random Partition Model for Sequential Refinement and
Coagulation,” Biometrics, 75, 988–999.
Zhang et al. (2016)
Zhang, L., Guindani, M., Versace, F., Engelmann, J. M., and Vannucci, M.
(2016), “A Spatiotemporal Nonparametric Bayesian Model of
Multi-subject FMRI Data,” The Annals of Applied Statistics, 10,
638–666. |
C
A New Formulation Of
Lee-Wick Quantum Field Theory
Damiano Anselmi111damiano.anselmi@unipi.it and Marco Piva222marco.piva@df.unipi.it
Dipartimento di Fisica “Enrico
Fermi”, Università di Pisa,
Largo B. Pontecorvo 3, 56127 Pisa, Italy
and INFN, Sezione di Pisa,
Largo B. Pontecorvo 3, 56127 Pisa, Italy
Abstract
The Lee-Wick models are higher-derivative theories that are claimed to be
unitary thanks to a peculiar cancelation mechanism. In this paper, we
provide a new formulation of the models, to clarify several aspects that
have remained quite mysterious, so far. Specifically, we define them as
nonanalytically Wick rotated Euclidean theories. The complex energy plane is
divided into disconnected regions, which can be related to one another by
keeping the integration domain on the space momenta rigid when certain poles
cross it. Working in a generic Lorentz frame, the models are intrinsically
equipped with the right recipe to treat the pinchings of the Lee-Wick poles,
with no need of external ad hoc prescriptions. We describe these
features in detail by calculating the one-loop bubble diagram and explaining
how the key properties generalize to more complicated diagrams. The physical
results of our formulation are different from those of the previous ones.
The unusual behaviors of the physical amplitudes lead to interesting
phenomenological predictions.
1 Introduction
The Lee-Wick (LW) models are special higher-derivative theories, defined in
a peculiar way, which are claimed to lead to a perturbatively unitary $S$
matrix [1, 2, 3]. Precisely, the claim is that they are
equipped with well defined cutting equations, such that if we project the
initial and final states onto the subspace $V$ of physical degrees of
freedom, only states belonging to the same space $V$ propagate through the
cuts. Several properties of the models and aspects of their formulation have
not been clarified exhaustively, so far. In this paper we plan to overcome
those problems by reformulating the theories completely. We also show that
our formulation gives physical predictions that differ from those of the
previous ones.
It is well known that higher-derivative kinetic Lagrangian terms may improve
the ultraviolet behaviors of the Feynman diagrams and may turn
nonrenormalizable theories into renormalizable ones, as in the case of
higher-derivative gravity [4]. However, the higher-derivative
corrections, if not treated properly, lead to violations of unitarity or
even mathematical inconsistencies [5]. The Lee-Wick idea is
promising, because it claims to reconcile renormalizability and unitarity.
The propagators of the LW models contain extra poles, which we call LW poles, in addition to the poles corresponding to the physical degrees of
freedom and the poles corresponding to the gauge degrees of freedom (such as
the longitudinal and temporal components of the gauge fields and the poles
of the Faddeev-Popov ghosts). The LW poles come in complex conjugate pairs,
which we call LW pairs. Cutkosky et al. (CLOP) showed in
ref. [3] that the $S$ matrix is not analytic when pairs of LW poles
pinch. Analyticity is a property we are accustomed to, but not a fundamental
physical requirement. Nakanishi [6] showed that, if defined in
a certain way, the models violate Lorentz invariance. This problem is more
serious, but it can be avoided by defining the theories in a different way.
In ref. [3] it was proposed to treat the pinching of the LW poles
by means of a procedure of limit, which is known as CLOP prescription. In simple diagrams, the CLOP prescription gives an unambiguous and
unitary result, as confirmed by the calculations of Grinstein et al. [7] in the case of the bubble diagram. However, it is not
clear how to incorporate the CLOP prescription into a Lagrangian and
ambiguities are expected in high-order diagrams [3]. This leads us
to claim that some key issues concerning the formulation of the LW models
have remained open and are awaiting to be clarified.
It is more convenient to completely change approach and define the LW models
as nonanalytically Wick rotated Euclidean higher-derivative theories. First,
we know from ref. [5] that a Minkowski formulation of such a type of
higher-derivative theories is not viable, since in general it generates
nonlocal, non-Hermitian divergences that cannot be removed by any standard
approach. The Wick rotation from the Euclidean framework is thus expected to
play a crucial role, because it is the only viable path.
However, the Wick rotation of the higher-derivative theories we are
considering turns out to be nonanalytic, because of the LW pinching, to the
extent that the complex energy plane is divided into disjoint regions. The
Lorentz violation is avoided by working in a generic Lorentz frame and with
generic external momenta and then analytically continuing in each region
separately. We show that, if we do so, the models are intrinsically equipped
with all that is necessary to define them properly. In particular, there is
no need of the CLOP prescription, or any other prescription to handle the
pinching of the LW poles. Actually, the CLOP prescription should be dropped,
because it leads to physical results that differ from those of our
formulation, even in a simple case such as the bubble diagram with different
masses.
The behaviors of the amplitidues show some unexpected features, which lead
to interesting phenomenological predictions. In particular, the violation of
analyticity is quite apparent, when the amplitude is plotted. If ever
observed, this behavior could be the quickest way to determine the
experimental value of the energy scale $M$ associated with the
higher-derivative terms, which is the key physical constant of the LW models.
The Lee-Wick models have been also explored for their possible physical
applications, which include QED [2], the standard model [8], grand unified theories [9] and quantum
gravity [10].
The paper is organized as follows. In section 2, we outline the
formulation of the LW models as nonanalytically Wick rotated Euclidean
theories. In section 3 we study the LW pinching in detail, in the
case of the bubble diagram. In section 4, we describe the
calculations of the physical amplitudes in a neighborhood of the LW pinching and show that the CLOP and similar prescriptions are not
consistent with our approach. In section 5 we evaluate
the bubble diagram in the new formulation and show that the physical results
are in general different from those that follow from the CLOP and other
prescriptions. We also comment on the phenomenological relevance of the
results. In section 6 we explain how to deal with more
complicated diagrams.
2 Lee-Wick models as Wick rotated Euclidean theories
In this section we outline the new formulation of the LW models. We begin
by describing the class of higher-derivative theories that we are
considering. The higher-derivative Lagrangian terms are multiplied by
inverse powers of certain mass scales, which we call LW scales. For
simplicity, we can assume that there is just one LW scale, which we denote
by $M$, since the generalization to many LW scales is straightforward.
When $M$ tends to infinity, the propagators must tend to the ones of
ordinary unitary theories. Moreover, the extra poles that are present when $M<\infty$ must have nontrivial real and imaginary parts and come in complex
conjugate pairs.
A typical propagator of momentum $p$ is equal to the standard propagator
times a real function of $p^{2}$ that has no poles on the real axis. For
concreteness, we take
$$iD(p^{2},m^{2},\epsilon)=\frac{iM^{4}}{(p^{2}-m^{2}+i\epsilon)((p^{2})^{2}+M^{%
4})}.$$
(2.1)
More general propagators can be considered. In particular renormalization
may lead to structures such as
$$\frac{iM^{4}}{(p^{2}-m^{2}+i\epsilon)((p^{2}-\mu^{2})^{2}+M^{4})}.$$
However, the key features are already encoded in (2.1) and the
extension does not change the sense of our investigation.
The poles of (2.1) are
$$p^{0}=\pm\omega_{m}(\mathbf{p})\mp i\epsilon,\qquad p^{0}=\pm\omega_{M}(%
\mathbf{p}),\qquad p^{0}=\pm\omega_{M}^{\ast}(\mathbf{p}),$$
(2.2)
where $\omega_{m}(\mathbf{p})=\sqrt{\mathbf{p}^{2}+m^{2}}$ and $\omega_{M}(\mathbf{p})=\sqrt{\mathbf{p}^{2}+iM^{2}}$. Their locations are shown in the
picture
(2.3)
where the LW poles are denoted by means of an $\times$, while the standard
poles are denoted by a circled $\times$.
We can integrate $p^{0}$ along the real axis or along the imaginary axis.
The first choice defines the Minkowski theory, the second choice defines the
Euclidean theory. The two give different results, because, even if the
integration path at infinity does not contribute, some poles are located in
the first and third quadrants of the complex plane. In ref. [5] it
was shown that in general the Minkowski theories of this type are
inconsistent, because they are plagued with nonlocal, non-Hermitian
divergences that cannot be subtracted away. The bubble diagram in four
dimensions is one of the few convergent exceptions, but it becomes
nonlocally divergent as soon as nontrivial numerators are brought by the
vertices, which happens for example in higher-derivative gravity. This fact
forces us to proceed with the Euclidean theory.
Usually, the Wick rotation is an analytic operation, but in the Lee-Wick models the situation is different. The analytically Wick rotated Euclidean
theory gives the Lee-Wick theory only in a region of the complex energy
plane. It is the region that contains the imaginary axis, which we call
main region. The complex plane turns out to be divided into several
disconnected regions $\mathcal{A}_{i}$, which can be reached from the main
region in a nonanalytic way. The regions $\mathcal{A}_{i}$ are called
analytic regions.
In the light of this fact, the calculation of the correlation functions
proceeds as follows. The loop integrals are evaluated at generic (possibly
complex) external momenta, in each analytic region $\mathcal{A}_{i}$ of the
complex plane. For a reason that we will explain, we anticipate that it is
also necessary to work in a sufficiently generic Lorentz frame, because
special Lorentz frames may squeeze an entire region $\mathcal{A}_{i}$ to a
branch cut. The $\mathcal{A}_{i}$ subdomain where the calculation is done is
denoted by $\mathcal{O}_{i}$ and has to satisfy suitable properties. For
example, it must contain an accumulation point.
After the evaluation, the amplitude is analytically continued from $\mathcal{O}_{i}$ to the rest of the region $\mathcal{A}_{i}$. This procedure gives
the amplitude of the LW model, region by region. Since it is not possible
to relate the regions analytically, the Wick rotation is nonanalytic. Yet,
the regions are related by a well-defined computational procedure, which we
describe in the next sections.
We may condense their articulated definition by saying that the LW
models are nonanalytically Wick rotated Euclidean higher-derivative theories
of a special class.
Consider the propagator (2.1) and its poles (2.2). When the
imaginary axis is rotated to the real one, we get the integration path
(2.4)
The Wick rotation is less trivial when performed in Feynman diagrams. To be
explicit, consider the bubble diagram
(2.5)
The bubble diagram has two propagators, so the number of poles doubles. If
one propagator has momentum $k$ and the other propagator has momentum $k-p$,
we have a loop integral proportional to
$$\mathcal{J}(p)=\int\frac{\mathrm{d}^{D}k}{(2\pi)^{D}}D(k^{2},m_{1}^{2},%
\epsilon_{1})D((k-p)^{2},m_{2}^{2},\epsilon_{2}),$$
(2.6)
where $D$ is the spacetime dimension. When we vary the external momentum $p$, the poles of the first propagator are fixed [given by formula (2.2)
with $p\rightarrow k$, $m\rightarrow m_{1}$], while those of the second
propagator, which are
$$k^{0}=p^{0}\pm\omega_{m_{2}}(\mathbf{k-p})\mp i\epsilon,\qquad k^{0}=p^{0}\pm%
\omega_{M}(\mathbf{k-p}),\qquad k^{0}=p^{0}\pm\omega_{M}^{\ast}(\mathbf{k-p}),$$
(2.7)
move on the complex plane. With respect to the fixed poles, this sextet of
poles is translated by $p^{0}$ and deformed by $\mathbf{p}$. At some point,
the translation makes some poles cross the imaginary axis, which is the
integration path. To preserve analyticity, the integration path must be
deformed so that the crossing does not actually take place. Equivalently, we
can keep the main integration path on the imaginary axis and add integration
contours around the poles that have crossed the imaginary axis. In the end,
we obtain a figure like
(2.8)
where the thick poles are the moving ones. When we make the Wick rotation to
the real axis, we obtain an integration path like
(2.9)
or, depending on $p$,
(2.10)
In these pictures we have assumed for simplicity that the external space
momentum $\mathbf{p}$ vanishes.
A puzzling situation occurs when the right (respectively, left) LW pair of
the propagator $D(k^{2},m_{1}^{2},\epsilon_{1})$ hits the left (right) LW pair of $D((k-p)^{2},m_{2}^{2},\epsilon_{2})$, because in that case the
integration path gets pinched. We call this occurrence LW pinching. The integration paths before and after the LW pinching are illustrated in
the two pictures (2.9) and (2.10).
When we perform the Wick rotation, the analytic continuation is
straightforward in the situation (2.9), but we find an unexpected
behavior in the situation (2.10). The two situations correspond to
two independent regions $\mathcal{A}_{1}$ and $\mathcal{A}_{2}$ of the $p^{0}$ complex plane, separated by a branch cut. Each region $\mathcal{A}_{i}$ must be studied separately and gives a complex function $\mathcal{J}_{i}(p)$. The complex functions $\mathcal{J}_{1}(p)$ and $\mathcal{J}_{2}(p)$
are not related to each other by an analytic continuation. However they are
still related in a well defined, nonanalytic way.
We show that, with these caveats, the procedure to handle the LW pinching is
intrinsic to our definition of the theory, pretty much like the $i\epsilon$
prescription is intrinsic to the definition of a theory as a Wick rotated
Euclidean one. Moreover, it is consistent with perturbative unitarity.
The LW pinching motivated some authors to propose ad hoc
prescriptions to handle it. The CLOP prescription [3], for example,
amounts to deform the scale $M$ in one of the propagators of the integral (2.6) to a different value $M^{\prime}$. Under certain conditions,
the pinching is absent for $M^{\prime}\neq M$ and the regions we mentioned
above are analytically connected. In that situation, the Wick rotation is
analytic everywhere. After the calculation of the amplitude, the deformed
scale $M^{\prime}$ is sent to $M$. This operation cuts the complex plane
into disconnected regions.
The CLOP prescription is not sufficient to deal with the LW pinching in all
the diagrams, because higher-order diagrams are ambiguous, as shown in ref.
[3]. Moreover, it appears to be artificial. For example, there is no
obvious way to incorporate it into the Lagrangian or the Feynman rules. In
this paper, we also show that the CLOP prescription leads to physical
predictions that differ from the ones we obtain. We also show that, if we
strictly apply the rules that follow from the formulation of this paper, it
is possible to retrieve the correct result even starting from $M^{\prime}\neq M$ and letting $M^{\prime}$ tend to $M$ at the end. Then, however,
the CLOP prescription becomes redundant.
To summarize, we show that the theory is intrinsically equipped with the
procedure that allows us to handle the LW pinching. The amplitudes are well
defined and unambigous. The prescriptions that can be found in the existing
literature are either redundant or give predictions that are in
contradiction with ours.
In section 6 we explain how the results of this section extend
from the bubble diagram to more complicated diagrams.
3 LW pinching
In this section we describe the LW pinching in the case of the bubble
diagram (2.5), that is to say the loop integral (2.6).
First, we integrate on the loop energy $k^{0}$ by means of the residue
theorem along the imaginary axis. This operation leaves us with the integral
on the loop space momentum $\mathbf{k}$. Orienting the external space
momentum $\mathbf{p}$ along the vertical line, the integral on the azimuth
is trivial, so we remain with the integral on $k_{s}\equiv|\mathbf{k}|$
from $0$ to $\infty$ and the integral on $u\equiv\cos\theta$ from $-1$
to $1$, where $\theta$ is the zenith angle. To illustrate the problematics
involved in the LW pinching exhaustively, we consider two cases. In the
first case we work at $\mathbf{p}=0$, in the second case we work at $\mathbf{p}\neq 0$. Lorentz invariance suggests that there should be no big
difference between the two situations. It turns out that it is not so,
because the method of calculation we are using is not manifestly Lorentz
invariant. The calculation at $\mathbf{p}=0$ is equivalent to the one of
ref. [7], which is well suited to apply the CLOP prescription
and gives a Lorentz invariant result. However, this approach misses a
crucial point, which is visible only at $\mathbf{p}\neq 0$.
3.1 LW pinching at zero external space momentum
The LW pinching may involve pairs of LW poles (in which case it is called
pure LW pinching) or one LW pole and a standard pole (in which case
it is called mixed LW pinching). For the moment, we focus on the
pure LW pinching, because at one loop the mixed one cannot occur for real
external momenta.
There are two basic cases of pure LW pinching, shown in the figures
(3.1)
The first case involves the right LW pair of the first propagator and the
left LW pair of the second propagator. The second case involves the
upper-right LW pole of the first propagator and the bottom-left LW pole of
the second propagator. The other LW pinchings are the complex conjugates of
the ones just described and their reflections with respect to the imaginary
axis.
At $\mathbf{p}=0$, there is no $u$ dependence, so the $u$ integral is
trivial, the only nontrivial integration variable being $k_{s}$. The poles
relevant to the top pinching occurring in the left figure of (3.1) are
$$\frac{1}{k^{0}-p^{0}+\omega_{M}^{\ast}(\mathbf{k})}\frac{1}{k^{0}-\omega_{M}(%
\mathbf{k})},$$
(3.2)
while those relevant to the bottom pinching give the complex conjugate of
this expression. The pinching occurs when $k^{0}$ is such that the locations
of the two poles coincide, which gives the pinching equation
$$p^{0}=\sqrt{k_{s}^{2}+iM^{2}}+\sqrt{k_{s}^{2}-iM^{2}},$$
(3.3)
solved by
$$k_{s}^{2}=\frac{(p^{0})^{4}-4M^{4}}{4(p^{0})^{2}}.$$
(3.4)
The poles relevant to the pinching occurring in the right picture of (3.1) are
$$\qquad\frac{1}{k^{0}-p^{0}+\omega_{M}(\mathbf{k})}\frac{1}{k^{0}-\omega_{M}(%
\mathbf{k})}.$$
They give the pinching equations
$$\qquad p^{0}=2\sqrt{k_{s}^{2}+iM^{2}},$$
(3.5)
which are solved by
$$k_{s}^{2}=\frac{(p^{0})^{2}}{4}-iM^{2}.$$
(3.6)
We denote the $k_{s}$ integration path by $\gamma_{k}$. By default, we
expect it to be the positive real axis. For the moment, we stick to this,
but in a moment we will discover that we must deform $\gamma_{k}$ to
include complex values.
When $k_{s}$ is real and positive, the solution of (3.3) exists for $p^{2}$ real and larger than $2M^{2}$, while the solution of (3.5)
exists when $p^{2}-4iM^{2}$ is real and larger than zero. Performing the
integral in $\mathcal{J}(p)$, we find the LW branch cuts shown in the
picture
(3.7)
The middle branch point corresponds to the threshold $p^{2}=2M^{2}$, while
the other two branch points correspond to the thresholds $p^{2}=4iM^{2}$ and
$p^{2}=-4iM^{2}$. When we vary $p^{0}$ across a branch cut of figure (3.7), a pole $\nu$ of the $k_{s}$ integrand crosses the $k_{s}$
integration path $\gamma_{k}$ (which means that the imaginary part of the
pole becomes zero, while its real part stays positive), so the function $\mathcal{J}(p)$ is not analytic in that point. We have not shown the branch
cuts associated with the standard pinching and the mixed LW pinching.
For example, the right-hand side of (3.4) has vanishing imaginary
part and positive real part for $x\geqslant\sqrt{2}M$, $y=0$, where $x\equiv\mathrm{Re}[p^{0}]$, $y\equiv\mathrm{Im}[p^{0}]$. This gives the
middle branch cut of figure (3.7), which starts from the branch
point $p^{0}=\sqrt{2}M$. A mirror branch cut is obtained by reflecting with
respect to the imaginary axis.
On the other hand, the right-hand side of (3.6) has vanishing
imaginary part and positive real part when
$$xy=2M^{2},\qquad x^{2}\geqslant y^{2}.$$
(3.8)
This gives the branch cut shown in the first quadrant of figure (3.7), which starts from the branch point $p^{0}=\sqrt{2}M(1+i)$, and a
symmetric branch cut in the third quadrant. The complex conjugate LW
pinching gives the branch cut shown in the fourth quadrant of figure (3.7), with branch point $p^{0}=\sqrt{2}M(1-i)$, and a symmetric branch
cut in the second quadrant.
So far, we have described what happens when $\gamma_{k}$ is kept rigid. We
have seen that in that case certain poles $\nu$ of the integrand cross $\gamma_{k}$ when $p^{0}$ crosses the cuts of figure (3.7). There,
the function $\mathcal{J}(p)$ is not analytic. This is what naturally
happens if we make the integration numerically, since a generic integration
program of numerical integration does not know how to analytically deform
the integration paths.
If we want to turn $\mathcal{J}(p)$ into a function that is analytic in a
subdomain $\mathcal{O}$ that intersects the branch cuts of figure (3.7), we have to move those branch cuts away from $\mathcal{O}$. This is
done by deforming $\gamma_{k}$ when the poles $\nu$ approach it, so as to
prevent $\nu$ from crossing $\gamma_{k}$ in $\mathcal{O}$, and make the
crossing occur at different values of $p^{0}$. Equivalently, we can keep the
integration path $\gamma_{k}$ rigid, but add or subtract (depending on the
direction of motion of $\nu$) the residues of the moving poles $\nu$ after
the crossing. For example, in the equal mass case $m_{1}=m_{2}=m$, we can
recover analyticity on the real axis above $\sqrt{2}M$ by making the
replacement
$$\mathcal{J}(p)\rightarrow\mathcal{J}(p)-\frac{1}{16\pi}\frac{M^{4}}{m^{4}+M^{4%
}}\sqrt{1-\frac{4M^{4}}{(p^{2})^{2}}}\theta_{-}(p^{2}-2M^{2}),$$
when $p^{0}$ crosses the real axis above $\sqrt{2}M$ from the upper half
plane in the first quadrant (from the lower half plane in the third
quadrant), where $\theta_{-}(x)=1$ for $\mathrm{Re}[x]>0$, $\mathrm{Im}[x]<0$ and $\theta_{-}(x)=0$ in all other cases. In both sides of this
replacement the integration path $\gamma_{k}$ is the positive real axis.
Deforming the cuts with this procedure, we may obtain, for example, the
picture
(3.9)
Now the amplitude is mathematically well defined on the real axis, but it
has a nontrivial imaginary part for $p^{0}$ real and such that $(p^{0})^{2}>2M^{2}$, which violates unitarity. To preserve unitarity, we
must keep the branch cuts symmetric with respect to the real axis. At $\mathbf{p}=0$ this implies that a branch cut is necessarily on the real
axis, which makes the amplitude ill defined there.
3.2 LW pinching at nonzero external space momentum
At $\mathbf{p}\neq 0$ several interesting phenomena occur, which eventually
lead to the solution of the problem of properly handling the LW pinching.
The pinching equations (3.3) and (3.5) become
$$p^{0}=\sqrt{\mathbf{k}^{2}+iM^{2}}+\sqrt{(\mathbf{k-p})^{2}-iM^{2}},\qquad p^{%
0}=\sqrt{\mathbf{k}^{2}+iM^{2}}+\sqrt{(\mathbf{k-p})^{2}+iM^{2}},$$
(3.10)
respectively, plus their complex conjugates. Keeping $\mathbf{p}$ fixed, we
find the patterns
(3.11)
The first picture is obtained for smaller values of $|\mathbf{p|}$, the
second picture for larger values. The interiors of the regions $\mathcal{\tilde{A}}_{i}$ shown in these pictures are the values of $p^{0}$ that
satisfy the equations (3.10) for real $\mathbf{k}$. The cuts of (3.7) have enlarged into the regions $\mathcal{\tilde{A}}_{i}$, because
the right-hand sides of (3.10) now depend on two parameters, $k_{s}$
and $u$.
The point $P$, located at $p^{0}=\sqrt{2M^{2}+\mathbf{p}^{2}}\equiv E_{P}$,
corresponds to the threshold $p^{2}=2M^{2}$. The curve $\gamma$ is the
boundary of the region $\mathcal{\tilde{A}}_{P}$ that intersects the real
axis. Note that $\gamma$ does not cross the real axis in $P$, but in the
point $P^{\prime}$, which has energy
$$p^{0}=\sqrt{\frac{\mathbf{p}^{2}}{2}+\sqrt{\frac{(\mathbf{p}^{2})^{2}}{4}+4M^{%
4}}}\equiv E_{P^{\prime}}$$
(3.12)
and satisfies $\sqrt{2}M<E_{P^{\prime}}<E_{P}$. Thus, $\gamma$ can no
longer be associated with a true branch cut. If we define the amplitude
coherently with picture (3.11), Lorentz invariance is violated,
because the intersection between $\gamma$ and the real axis has no Lorentz
invariant meaning. This fact has already been noticed by Nakanishi in ref.
[6].
The intuitive reason behind the Lorentz violation is that, as shown in
figure (2.4), the loop energy is not everywhere real. By Lorentz
invariance, the loop momentum cannot be everywhere real. Thus, if we want to
restore Lorentz invariance, we cannot keep the $\mathbf{k}$ integration
domain rigid. If we insist on keeping it rigid, Lorentz invariance is
necessarily violated.
If we deform the $\mathbf{k}$ domain of integration to complex values, we
have several possibilities, but most of them violate unitarity, such as the
deformation (3.9). To preserve unitarity, we must keep the picture
symmetric with respect to the real axis. We deform the boundaries of the
surfaces that appear in figure (3.11) into the branch cuts
(3.13)
plus other cuts symmetric to these ones with respect to the imaginary axis.
In particular, the curve $\gamma$ is deformed so as to maximize its
distance from the origin. This means that it must cross the real axis in $p^{2}=2M^{2}$. The second thing to pay attention to is that the branch cuts
must not cross the real axis anywhere else. This way, both Lorentz
invariance and unitarity are preserved.
From the physical point of view, it does not really matter where the branch
cuts of figure (3.13) are located, as long as they are symmetric
with respect to the real axis and cross the real axis only in the threshold $p^{2}=2M^{2}$.
To summarize, at $\mathbf{p}\neq 0$ the cuts of figure (3.7) and
their reflections with respect to the imaginary axis are enlarged into
regions $\mathcal{\tilde{A}}_{i}$. Lorentz invariance is violated, but can
be restored by analytically deforming each region $\mathcal{\tilde{A}}_{i}$
into a new region $\mathcal{A}_{i}$. The extension must be maximal
“from below”, which means from smaller to
larger values of $p^{2}$. In the end, the complex plane is divided into
three disjoint analytic regions $\mathcal{A}_{i}$. The imaginary axis is
fully contained in one of them, called main analytic region. The
other two regions are symmetric with respect to the imaginary axis. The
right region can be reached from the main region in a nonanalytic way, by
crossing one of the branch cuts shown in (3.13), while keeping the $\mathbf{k}$ domain of integration rigid.
By means of this construction, the amplitude is unambiguous, Lorentz
invariant and satisfies unitarity. Now it is clear why we say that the
Lee-Wick models are formulated as nonanalytically Wick rotated Euclidean
higher-derivative theories.
4 Calculation around the LW pinching
In this section we show that the amplitudes are well defined without the
need of ad hoc prescriptions. Moreover, the nonanalyticity just
occurs on the boundaries of the surfaces shown in figure (3.11), but
not inside.
We focus on the pinching depicted in the left figure of (3.1).
The $\mathbf{k}$ integral has potential singularities of the form $1/D_{0}$
and $1/D_{0}^{\ast}$, where
$$D_{0}=p^{0}-\omega_{M}(\mathbf{k})-\omega_{M}^{\ast}(\mathbf{k-p}).$$
(4.1)
The top pinching occurs for $D_{0}=0$, i.e.
$$p^{0}=\sqrt{\mathbf{k}^{2}+iM^{2}}+\sqrt{(\mathbf{k}-\mathbf{p})^{2}-iM^{2}},$$
(4.2)
while the bottom pinching occurs for $D_{0}^{\ast}=0$. The crucial point of
the argument that follows is that the condition is complex for $\mathbf{p}\neq 0$ (while it becomes real at $\mathbf{p}=0$), so it splits into two
real conditions.
Our goal is to calculate the physical amplitude above the threshold of the
LW pinching, so we take a real $p^{0}>\sqrt{\mathbf{p}^{2}+2M^{2}}$.
Moreover, we take the loop space momentum $\mathbf{k}$ real, since in this
analytic region we can keep the $\mathbf{k}$ integration path rigid. Then,
the solution of (4.2) is a circle, equal to the intersection between a
sphere and a plane, given by
$$\mathbf{k}^{2}=\frac{(p^{0})^{4}-4M^{4}}{4(p^{0})^{2}},\qquad\mathbf{p}^{2}=2%
\mathbf{p}\cdot\mathbf{k}.$$
(4.3)
We can generalize the analysis to complex external energies $p^{0}$. The
main arguments are the same, but the solutions become more involved. It is
sufficient to extend the solution of (4.2) to the values of $p^{0}$
that are closed to the real axis, by making the substitution $p^{0}\rightarrow p^{0}\mathrm{e}^{i\varphi}$, with $\varphi$ small. Then
the denominator becomes
$$D_{\varphi}=p^{0}\mathrm{e}^{i\varphi}-\omega_{M}(\mathbf{k})-\omega_{M}^{\ast%
}(\mathbf{k-p}).$$
To simplify the formulas, we expand $D_{\varphi}$ around the solution (4.3) by means of the change of variables
$$k_{s}=\frac{\sigma_{-}}{2p^{0}}+\tau\frac{\sigma_{+}^{2}}{2\sigma_{-}(p^{0})^{%
2}}+\eta\frac{p_{s}\sigma_{+}^{2}}{4\sigma_{-}M^{2}},\qquad u=\frac{p_{s}}{2k_%
{s}}+\eta\frac{\sigma_{+}^{2}}{2\sigma_{-}M^{2}},$$
(4.4)
where $\sigma_{\pm}\equiv\sqrt{(p^{0})^{4}\pm 4M^{4}}$, $p_{s}\equiv|\mathbf{p}|$ and $u=\cos\theta$, $\theta$ being the angle between the
vectors $\mathbf{p}$ and $\mathbf{k}$. The fluctuations around the singular
loci (4.3) are parametrized by $\tau$ and $\eta$. The
parametrization (4.4) is chosen to simplify the behavior of the $\mathbf{k}$ integral around the solution. The integrand is proportional to
$$\frac{\mathrm{d}^{D-1}\mathbf{k}}{D_{\varphi}}\rightarrow-\frac{2\pi^{(D-2)/2}%
}{\Gamma\left(\frac{D}{2}-1\right)}\frac{k_{s}^{D-2}\mathrm{d}k_{s}(1-u^{2})^{%
(D-4)/2}\mathrm{d}u}{\tau-i(p^{0}\varphi+p_{s}\eta)},$$
(4.5)
where the arrow means that we integrate on all the angles besides $\theta$.
We have expanded the denominator to the first order in $\varphi$.
We see that the integral is regular as long as either $\varphi$ or $p_{s}$
are different from zero, which means that in those cases the potential
singularity at $D_{0}=0$ is integrable. If we keep $p_{s}\neq 0$ and reach $\varphi=0$, we obtain
$$\frac{\mathrm{d}^{D-1}\mathbf{k}}{D_{0}}=-\frac{2\pi^{(D-2)/2}}{\Gamma\left(%
\frac{D}{2}-1\right)}\frac{\sigma_{+}^{4}\left[\sigma_{-}^{2}-(p^{0})^{2}p_{s}%
^{2}\right]^{(D-4)/2}}{(2p^{0})^{D}M^{2}}\frac{\mathrm{d}\tau\mathrm{d}\eta}{%
\tau-ip_{s}\eta}.$$
(4.6)
It is interesting to study the limit $p_{s}\rightarrow 0$, which gives
$$-\frac{2\pi^{(D-2)/2}}{\Gamma\left(\frac{D}{2}-1\right)}\frac{\sigma_{+}^{4}%
\sigma_{-}^{D-4}}{(2p^{0})^{D}M^{2}}\mathrm{d}\tau\mathrm{d}\eta\left[\mathcal%
{P}\left(\frac{1}{\tau}\right)+i\pi\mathrm{sgn}(\eta)\delta(\tau)\right],$$
(4.7)
where $\mathcal{P}$ denotes the principal value and sgn is the sign
function. We learn that, basically, $p_{s}$ provides the prescription for
handling the integral. Note that at $p_{s}=0$ no $\eta$ dependence survives
in the integrand, besides the sign function of formula (4.7). If we
perform the trivial $\eta$ integration, we finally get
$$-\frac{8\pi^{(D-2)/2}}{\Gamma\left(\frac{D}{2}-1\right)}\frac{\sigma_{+}^{2}%
\sigma_{-}^{D-3}}{(2p^{0})^{D}}\mathrm{d}\tau\mathcal{P}\left(\frac{1}{\tau}%
\right).$$
(4.8)
In three and higher dimensions there is no singularity for $\sigma{}_{-}\rightarrow 0^{+}$.
To illustrate the procedure better, we check what happens when we pursue a
different strategy (which turns out to be unacceptable), by first setting $p_{s}=0$ at nonzero $\varphi$ and then send $\varphi$ to zero. To this
purpose, it is sufficient to replace the denominator $\tau-ip_{s}\eta$ of
formula (4.6) with $\tau-ip^{0}\varphi$. After integrating on $\eta$, we find
$$-\frac{8\pi^{(D-2)/2}}{\Gamma\left(\frac{D}{2}-1\right)}\frac{\sigma_{+}^{2}%
\sigma_{-}^{D-3}}{(2p^{0})^{D}}\frac{\mathrm{d}\tau}{\tau-ip^{0}\varphi}%
\underset{\varphi\rightarrow 0^{\pm}}{\longrightarrow}-\frac{8\pi^{(D-2)/2}}{%
\Gamma\left(\frac{D}{2}-1\right)}\frac{\sigma_{+}^{2}\sigma_{-}^{D-3}}{(2p^{0}%
)^{D}}\mathrm{d}\tau\left[\mathcal{P}\left(\frac{1}{\tau}\right)\pm i\pi\delta%
(\tau)\right].$$
(4.9)
This expression is also regular, but does not coincide with (4.8).
Formula (4.8) is certainly correct, because it is obtained by
approaching the real axis from $p_{s}\neq 0$, where the region $\mathcal{\tilde{A}}_{P}$ contains a neighborhood of the real axis above the threshold
$p^{2}=2M^{2}$. Instead, (4.9) is obtained by first squeezing the
region $\mathcal{\tilde{A}}_{P}$ to the real axis (which is a consequence of
letting $p_{s}\rightarrow 0$ first) and then approaching the real axis from
the complex plane. Thus, (4.9) cannot be correct.
We learn that the theory knows how to deal with the LW pinching
unambiguously. Moreover, the nonanalytic Wick rotation does work, is free of
singularities and consistent with unitarity. In section 5
we complete the calculation of the bubble diagram.
4.1 Comparison with the CLOP and other prescriptions
We have seen that the theory is intrinsically equipped with the right recipe
to handle the LW pinching, which is (4.8). This means that any
artificial prescription can potentially lead to wrong results. Now we
classify the whole set of unitary prescriptions, which includes the CLOP
one, and compare them with the results predicted by the formulation of this
paper. For definiteness, we work in four dimensions.
Consider the integrand of the loop integral (2.6) at $\mathbf{p}=0$.
We begin with the top pinching that appears in the left figure of (3.1), which is due to the poles (3.2). By means of the
expansion
$$k_{s}=\frac{\sigma_{-}}{2p^{0}}+\tau\frac{Mp^{0}}{\sigma_{-}},$$
(4.10)
we see that the integrand of $\mathcal{J}(p)$ behaves as
$$\frac{i}{(8\pi)^{2}}\frac{\sigma_{-}}{(p^{0})^{2}}\frac{M^{4}}{(M^{2}+im_{1}^{%
2})(M^{2}-im_{2}^{2})}\frac{\mathrm{d}\tau}{\tau}$$
(4.11)
around the singularity $\tau=0$.
We know that the formulation of this paper removes the singularity because,
working at nonvanishing $\mathbf{p}$ and letting $\mathbf{p}$ tend to zero
afterwards, (4.11) is replaced by
$$\frac{i}{(8\pi)^{2}}\frac{\sigma_{-}}{(p^{0})^{2}}\frac{M^{4}}{(M^{2}+im_{1}^{%
2})(M^{2}-im_{2}^{2})}\mathcal{P}\left(\frac{1}{\tau}\right)\mathrm{d}\tau.$$
More generally, we may have
$$\frac{i}{(8\pi)^{2}}\frac{\sigma_{-}}{(p^{0})^{2}}\frac{M^{4}}{(M^{2}+im_{1}^{%
2})(M^{2}-im_{2}^{2})}\left[\mathcal{P}\left(\frac{1}{\tau}\right)+ia\delta(%
\tau)\right]\mathrm{d}\tau,$$
(4.12)
where $a$ is an arbitrary real constant.
The LW poles come in conjugate pairs, so the pinching just considered is
accompanied by the complex conjugate one, which occurs when the residue
calculated in $k^{0}=p^{0}-\omega_{M}(\mathbf{k})$ hits the LW pole located
in $k^{0}=\omega_{M}^{\ast}(\mathbf{k})$. The contribution is minus the
complex conjugate of (4.12), because the $i$ factor that accompanies
the residue does not get conjugated. The total gives
$$\frac{2i}{(8\pi)^{2}}\frac{\sigma_{-}}{(p^{0})^{2}}\frac{M^{4}}{(M^{4}+m_{1}^{%
4})(M^{4}+m_{2}^{4})}\left[(M^{4}+m_{1}^{2}m_{2}^{2})\mathcal{P}\left(\frac{1}%
{\tau}\right)+aM^{2}(m_{1}^{2}-m_{2}^{2})\delta(\tau)\right]\mathrm{d}\tau.$$
We see that the contribution to the amplitude is regular and purely
imaginary. In particular, it does not affect the cutting equations, which
concern the sum of $\mathcal{J}(p)$ plus its conjugate. This result proves
that the prescription (4.12) is consistent with perturbative unitarity
for arbitrary $a$. However, the loop integral $\mathcal{J}(p)$ does depend
on $a$, at least when the two masses are different. This proves that no
prescription with nonvanishing $a$ is consistent with our formulation, which
predicts $a=0$.
In particular, the CLOP prescription gives $a=\pm\pi$. This result can be
proved by replacing the LW scale $M$ with $M^{\prime}=M+\delta$ in the
second propagator of (2.6). Modifying the expansion (4.10) into
$$k_{s}=\frac{\sigma_{-}}{2p^{0}}+\tau\frac{Mp^{0}}{\sigma_{-}}-2\delta\frac{M^{%
3}}{p^{0}\sigma_{-}},$$
the integrand $\mathcal{J}(p)$ behaves as
$$\frac{i}{(8\pi)^{2}}\frac{\sigma_{-}}{(p^{0})^{2}}\frac{iM^{4}}{(M^{2}+im_{1}^%
{2})(M^{2}-im_{2}^{2})}\frac{\mathrm{d}\tau}{\tau-i\delta},$$
around the top pinching of the left figure of (3.1). This
formula is equivalent to (4.12) with $a=\pi$sgn$(M^{\prime}-M)$.
Thus, the CLOP prescription should be dropped, because it is not consistent
with our formulation.
5 Complete bubble diagram
In this section, we complete the calculation of the bubble diagram. The main
goal is to describe what happens around the LW threshold. The threshold
associated with the physical poles is not the main focus of the calculation,
so, to avoid the superposition between the physical threshold and the LW
one, we assume that the mass $m$ is sufficiently large. For concreteness, we
take $m=3M$.
Another simplifying choice is to make the calculation at $\mathbf{p}=0$ and
resolve the singularity with the help of formula (4.8). We know that
this procedure is justified by starting from nonvanishing $\mathbf{p}$,
where the LW pinching is properly handled, and taking the limit $\mathbf{p}\rightarrow 0$ afterwards.
Setting $M=1$, the imaginary part of $\mathcal{J}(p)$ as a function of a
real $p^{0}$ has the behavior
(5.1)
while the real part vanishes, which confirms unitarity. We see that the
imaginary part is well defined and continuous, but not analytic. The
nonanalyticity that is visible at $p^{2}=2M^{2}$ is the remnant of the LW pinching. If in nature some physical processes are described by a LW theory,
the LW scale $M$ is the key physical quantity signaling the new physics. A
shape like (5.1) may be helpful to determine the magnitude of $M$
experimentally.
We stress again that the nonanalytically Wick rotated Euclidean theory does
not need a particular prescription. Because of this, we expect that it also
eliminates the ambiguities that the CLOP prescription is claimed to generate
in more complicated diagrams [3].
The CLOP prescription gives the same result, in the case just considered. As
explained in the previous section, we can appreciate the difference between
the predictions of our formulation and those of the CLOP prescription in
the bubble diagram with unequal masses. For example, we compare the cases $m_{1}=3$, $m_{2}=5$ and $m_{1}=m_{2}=4$. Using the CLOP prescription, we
take $M=1$ in the first propagator of formula (2.6) and $M=1+\delta$
in the second propagator, working at $\mathbf{p}=0$. Then we integrate $\mathcal{J}(p)$ numerically for smaller and smaller values of $\delta$,
till, say, $\delta=10^{-3}$. On the other hand, following the formulation
proposed here, we set $M=1$ in both propagators, but keep $p_{s}=|\mathbf{p}|$ different from zero. Then, we integrate numerically for smaller and
smaller values of $p_{s}$ till $p_{s}=10^{-3}$. The imaginary part of $\mathcal{J}(p)$ gives the pictures
$$\includegraphics[width=213.395669pt]{noCLOP.eps}\quad\includegraphics[width=21%
3.395669pt]{noCLOP2.eps}$$
(5.2)
while the real part still vanishes. The first figure refers to the case $m_{1}=3$, $m_{2}=5$, while the second figure refers to the case $m_{1}=m_{2}=4$. The upper graph is the one predicted by our formulation,
while the lower graph is the one coming from the CLOP prescription. The
match is very precise in the equal mass case, but there is a remarkable
discrepancy above the LW threshold in the unequal mass case. The result
confirms that the CLOP prescription is not compatible with the formulation
proposed here.
If we really want to retrieve our result from a procedure where the
propagators of formula (2.6) have two different LW scales $M$ and $M^{\prime}$, as in the CLOP prescription, we actually can, but in that case
the CLOP prescription becomes redundant. Instead of setting $p_{s}=0$ and
then letting $M^{\prime}$ tend to $M$, we must start from $p_{s}\neq 0$,
let $M^{\prime}$ approach $M$ while $p_{s}\neq 0$, work in a suitable
region $\mathcal{\tilde{A}}_{>}$, analytically continue “from above” and only at the end, if we want, let $p_{s}$
tend to zero.
The region $\mathcal{\tilde{A}}_{P}$ contained in the curve $\gamma$ of the
first figure shown in (3.11) splits into two regions $\mathcal{\tilde{A}}_{P}^{+}$ and $\mathcal{\tilde{A}}_{P}^{-}$, when $M^{\prime}=1+\delta$ is sufficiently different from $M$ (or $p_{s}$ sufficiently
large):
$$\includegraphics[width=199.169291pt]{CLOPa.eps}\hskip 28.452756pt%
\includegraphics[width=199.169291pt]{CLOPb.eps}$$
(5.3)
In both figures, we have $p_{s}=10^{-3}$ and $M=1$. In the first figure $\delta=5\cdot 10^{-3}$, while in the second figure $\delta=10^{-4}$.
We see that when $\delta$ is sufficiently large the real axis has no
intersection with $\mathcal{\tilde{A}}_{P}^{+}$ and $\mathcal{\tilde{A}}_{P}^{-}$. Instead, when $\delta$ becomes smaller, the region $\mathcal{\tilde{A}}_{>}\equiv\mathcal{\tilde{A}}_{P}^{+}\cap\mathcal{\tilde{A}%
}_{P}^{-}$ is nonempty. What the CLOP prescription requires is to cover the
entire real axis by analytic continuation from below (i.e. from the region
that contains the imaginary axis) and let $\delta$ tend to zero at the end.
What our formulation requires, instead, is to cover the portion of the real
axis that is located above the LW threshold by analytic continuation from
above, i.e. from $\mathcal{\tilde{A}}_{>}$. This is the crucial difference
between the two formulations, which explains the discrepancy shown in figure
(5.2).
6 More complicated diagrams
The arguments of the previous sections extend to more complicated diagrams,
but some remarks on how to deal with them are in order. In the one-loop
diagrams, the loop momentum remains unique, while the independent external
momenta become more than one. The mixed LW pinching cannot occur for real
external momenta. On the other hand, figure (2.10) shows that the
pure LW pinchings are entirely similar to the ones of the bubble and occur
between the right LW poles of any propagator and the left LW poles of any
other propagator.
At higher loops the pinching is not so different from the one we are
accustomed to in standard theories. There, if the propagators of the
internal legs of the diagram have masses $m_{i}$, the pinchings lead to
thresholds of the form $p^{2}=(m_{i_{1}}+m_{i_{2}}+m_{i_{3}}+\cdots)^{2}$,
where $p$ is a linear combination of incoming momenta. In the case of the
LW pinching, the formulas that give the thresholds are basically the same,
with the difference that some masses $m_{i}$ may be replaced by the complex
masses $M_{\pm}=(1\pm i)M/\sqrt{2}$ associated with the LW scales. We get
the thresholds
$$p^{2}=\left[(n_{+}+n_{-})\frac{M}{\sqrt{2}}+i(n_{+}-n_{-})\frac{M}{\sqrt{2}}+m%
_{i_{1}}+m_{i_{2}}+m_{i_{3}}+\cdots\right]^{2},$$
where the integers $n_{+}$ and $n_{-}$ are the numbers of masses $M_{+}$ and
$M_{-}$, respectively. The number of thresholds grows with the number of
loops and so does the number of disjoint analytic regions $\mathcal{A}_{i}$.
The thresholds that are relevant to the calculations of the physical
amplitudes are those that are located on the real axis, which are
$$p^{2}=(\sqrt{2}nM+m_{i_{1}}+m_{i_{2}}+m_{i_{3}}+\cdots)^{2},$$
where $n=n_{+}=n_{-}$.
The analytic regions $\mathcal{A}_{i}$ are determined as follows. Working in
a generic Lorentz frame, we find the regions $\mathcal{\tilde{A}}_{i}$ by
keeping the integrations on the loop space momenta rigid everywhere [see (3.11)]. Then, we maximally extend each $\mathcal{\tilde{A}}_{i}$
“from below”, which means from smaller to
larger values of the squared external momenta $p^{2}$, till we reach a
threshold [see (3.13)], where we must stop. Once the regions $\mathcal{A}_{i}$ are known we restrict the calculation of the loop integral
to subsets $\mathcal{O}_{i}\subset\mathcal{\tilde{A}}_{i}$ that contain at
least one accumulation point, where we can keep the integral on the space
momenta rigid. Then we analytically continue the result to $\mathcal{A}_{i}$.
7 Conclusions
The Lee-Wick models are higher-derivative theories that are claimed to
reconcile renormalizability and unitarity in a very nontrivial way. However,
several aspects of their formulation have remained unclear, so far. In this
paper, we have provided a new formulation of the models that overcomes the
major difficulties, by defining them as nonanalytically Wick rotated
Euclidean theories. Working in a generic Lorentz frame, the models are
intrinsically equipped with the right recipe to treat the pinchings of the
Lee-Wick poles, with no need of external ad hoc prescriptions. The
complex energy plane is divided into disconnected analytic regions, which
are related to one another by keeping the integration path on the space
momentum rigid when certain poles cross it.
The physical results that follow from our approach are quantitatively
different from those of the previous approaches. The nonanalytic behaviors
of the amplitudes may have interesting phenomenological consequences, which
may facilitate the measurements of some key physical constants of the
theories, such as the scales associated with the higher-derivative terms.
Acknowledgments
We are grateful to U.G. Aglietti and L. Bracci for useful discussions.
References
[1]
T.D. Lee and G.C. Wick, Negative metric and the unitarity
of the S-matrix, Nucl. Phys. B 9 (1969) 209.
[2]
T.D. Lee and G.C. Wick, Finite theory of quantum
electrodynamics, Phys. Rev. D 2 (1970) 1033.
[3]
R.E. Cutkosky, P.V. Landshoff, D.I. Olive, and J.C.
Polkinghorne, A non-analytic S-matrix, Nucl. Phys. B 12 (1969) 281.
[4]
K.S. Stelle, Renormalization of higher derivative quantum
gravity, Phys. Rev. D 16 (1977) 953;
E. S. Fradkin and A. A. Tseytlin, Renormalizable asymptotically free quantum
theory of gravity, Nucl. Phys. B 201 (1982) 469.
[5]
U.G. Aglietti and D. Anselmi, Inconsistency of Minkowski
higher-derivative theories, Eur. Phys. J. C 77 (2017) 84, 16A2 Renormalization.com and arXiv:1612.06510 [hep-th].
[6]
N. Nakanishi, Lorentz noninvariance of the complex-ghost
relativistic field theory, Phys. Rev. D 3, 811 (1971).
[7]
B. Grinstein, D. O’Connell and M.B. Wise, Causality as
an emergent macroscopic phenomenon: The Lee-Wick O(N) model, Phys. Rev. D 79
(2009) 105019 and arXiv:0805.2156
[hep-th].
[8]
B. Grinstein, D. O’Connell, and M.B. Wise, The
Lee-Wick standard model, Phys. Rev. D77 (2008) 025012 and arXiv:0704.1845 [hep-ph];
C.D. Carone and R.F. Lebed, Minimal Lee-Wick extension of the standard
model, Phys. Lett. B668 (2008) 221 and arXiv:0806.4555 [hep-ph];
J.R. Espinosa and B. Grinstein, Ultraviolet properties of the Higgs sector
in the Lee-Wick standard model, Phys. Rev. D83 (2011) 075019 and arXiv:1101.5538 [hep-ph];
C.D. Carone and R.F. Lebed, A higher-derivative Lee-Wick standard model,
JHEP 0901 (2009) 043 and arXiv:0811.4150 [hep-ph].
[9]
B. Grinstein and D. O’Connell, One-Loop
Renormalization of Lee-Wick Gauge Theory, Phys. Rev. D78 (2008) 105005 and
arXiv:0801.4034 [hep-ph];
C. D. Carone, Higher-derivative Lee-Wick unification, Phys. Lett. B677
(2009) 306,
and arXiv:0904.2359 [hep-ph].
[10]
E. Tomboulis, 1/N expansion and renormalization in quantum
gravity, Phys. Lett. B 70 (1977) 361;
E. Tomboulis, Renormalizability and asymptotic freedom in quantum gravity,
Phys. Lett. B 97 (1980) 77;
Shapiro and L. Modesto, Superrenormalizable quantum gravity with complex
ghosts, Phys. Lett. B755 (2016) 279-284 and arXiv:1512.07600 [hep-th];
L. Modesto, Super-renormalizable or finite Lee–Wick quantum gravity, Nucl.
Phys. B909 (2016) 584 and arXiv:1602.02421 [hep-th]. |
Conformal bridge
in a cosmic string background
Luis Inzunza and Mikhail S. Plyushchay
Departamento de Física,
Universidad de Santiago de Chile, Casilla 307, Santiago,
Chile
E-mails:
luis.inzunza@usach.cl,
mikhail.plyushchay@usach.cl
Abstract
Hidden symmetries of non-relativistic $\mathfrak{so}(2,1)\cong\mathfrak{sl}(2,\mathbb{R})$ invariant
systems in a cosmic string background are studied using the conformal bridge transformation.
Geometric properties of this background are analogous
to those of a conical surface with
a deficiency/excess
angle
encoded in the “geometrical parameter” $\alpha$,
determined by the linear positive/negative
mass density of the string.
The free particle and the harmonic oscillator
on this background
are shown to be related
by the conformal bridge transformation.
To identify the
integrals of the free system, we employ a local canonical transformation
that relates the model with its planar version.
The conformal bridge transformation is then
used to map the obtained integrals
to those of the harmonic oscillator on the cone.
Well-defined classical integrals in both models
exist only at $\alpha=q/k$ with $q,k=1,2,\ldots,$
which for $q>1$ are higher-order
generators of finite nonlinear algebras.
The systems are quantized for arbitrary values of $\alpha$; however,
the well-defined hidden symmetry operators associated with
spectral degeneracies only exist
when $\alpha$ is an integer, that reveals a quantum anomaly.
1 Introduction
Special characteristics of classical systems are reflected
by the
conserved quantities that canonically generate
the
symmetry transformations.
At the quantum level, these quantities
are promoted to the operators
that
carry the spectrum information.
In this context, hidden symmetries are associated
with the
non-obvious integrals of motion that
mix canonical
coordinates and
momenta
in a non-trivial way at the
classical level, while at the quantum level they are often responsible for the so-called “accidental degenerations” [1].
Integrals of this type are higher-order functions of the canonical momenta,
and they may
generate
nonlinear symmetry algebras [2, 3]. Some examples of integrals related to hidden symmetries
are the Laplace-Runge-Lenz vector for the Kepler-Coulomb problem [4],
the Fradkin tensor for the isotropic harmonic oscillator
[5],
the
analogs of these integrals
in a monopole background [6, 7], the higher-order symmetry generators of the
anisotropic harmonic oscillator with commensurable frequencies [2],
and the $N$ integrals in involution in the Calogero models of $N$ particles
[8, 9].
On the other hand, it is a quite general assertion that
the systems
in a curved space-time show special properties
related to the geometric background itself.
Some examples in this direction are the Hawking radiation
[10], the Unruh effect [11],
the conformal invariance of a
charged particle propagating near the horizon of
the extreme Reissner-Nordström black hole,
which
attracted attention in the context
of AdS/CFT
correspondence [12, 13, 14, 15]
(that, in turn, gave rise to a resurgence of interest to the
conformal model of de Alfaro, Fubini and Furlan [16]).
In a different but related line of research
we also mention here the
supersymmetric mechanics models
on curved spaces whose construction
is based on
the WDVV equation
formalism
[17, 18].
In the context of the present research,
the study of classical and quantum dynamics in curved spaces is interesting from the point of view of hidden symmetries.
An important result in this direction
is the proof
of the existence of a non-trivial conserved quantity that characterizes
the dynamics of a particle moving on a Kerr black hole background.
This quantity, known as the Carter integral [19],
is responsible for the complete integrability of the system.
Another interesting result
was reported in [20],
where it was shown that spinning particles
in the Kerr-Newman black hole background
are characterized by enhanced supersymmetry with
additional supercharges
of a nature different from a square root of the Hamiltonian of the system.
Geometrically, such supercharges are associated
with the so-called Yano and conformal Yano tensors [21, 22, 23, 24].
Some other examples related to the particle dynamics in curved spaces
can be found in references
[25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35].
In this article, we address the problem of studying the
classical and quantum
symmetry
aspects
of some conformal-invariant non-relativistic particle systems on a cosmic string background.
These strings are topological
defects whose creation in the early universe is predicted by quantum field theory arguments [36, 37, 38, 39],
and which, on the other hand, also appear in condensed matter physics
and wormholes [40, 41, 42, 43, 44].
Their effect is to introduce a conical singularity in the spatial part of the space-time metric [45, 43].
The effects of the presence of the cosmic string was examined in different physical contexts, see, e.g., refs.
[46, 47, 48, 49, 50, 27, 31, 33, 34, 35].
Our goal
here is to study
the influence of the geometrical properties of this background
(encoded in a “geometrical parameter” $\alpha$
given in terms of the linear mass density of the string)
on the dynamics of the systems from
the perspective of well-defined integrals of motion in
the phase space when considering classical cases,
and well-defined symmetry operators for
the corresponding quantum versions of the systems.
We are interested in the case of the free particle in the cone,
which has the $\mathfrak{so}(2,1)$ conformal symmetry,
and in the harmonic oscillators system in the same geometry,
which is characterized by
the $\mathfrak{sl}(2,\mathbb{R})\cong\mathfrak{so}(2,1)$ conformal Newton-Hooke
symmetry [51, 52, 53, 54, 55].
The key construction in
our investigation
is the so-called conformal bridge transformation [55], which in general is a mapping that
allows us to transform the complete set of symmetry generators
from an $\mathfrak{so}(2,1)$ invariant asymptotically free system, to those of a harmonically confined model
with the $\mathfrak{sl}(2,\mathbb{R})$ conformal symmetry.
The plan is to first characterize
the free case by identifying
the corresponding classical and quantum integrals of motion for different values of $\alpha$,
and then to obtain the complete information on the harmonic oscillator system
through the conformal bridge transformation.
The free dynamics and the
harmonic oscillator system in conical geometry were already studied in the literature, but under a different
perspective. In [25, 26]
the classical and quantum scattering on the cone were considered.
There, in particular, it was noticed that for the geodesic motion, there are special
values of the deficiency angle for which the particle experiences
a backward scattering, while for other values
the particle continues to move in the original direction, making several
revolutions around
the cone vertex. For the harmonic oscillator system, the structure of the wave-functions and the spectrum
for arbitrary values of $\alpha$ were investigated in
[28, 29, 32]. None of these works,
however, studied the systems in the light of hidden
symmetries which appear for special values of $\alpha$.
A notable result we present here
is that for the free particle on the cone characterized by the parameter
$\alpha$,
there are well-defined integrals of motion in the phase space
only when $\alpha$ is a rational number.
In the general case of rational $\alpha$ they are of higher-order in
momenta and produce a
finite nonlinear algebra.
At the quantum level we
reveal a quantum anomaly, because although the system can be quantized
for any real value of $\alpha$,
the
well-defined hidden symmetry operators in Hilbert physical space can only be constructed
when this
geometric parameter is an integer.
The same peculiarities
appear
for the harmonic oscillator system, whose
classical trajectories
are closed for rational values of $\alpha$, and at the quantum level,
only the cases with its integer values are anomaly-free,
and the hidden symmetry operators
reflect then the corresponding degeneracies of the spectrum.
The article is organized as follows.
We first review the geometry related to a cosmic string background in
Sec. 2 by explicitly showing that
its spatial part takes the form
of a two-dimensional conical metric,
whose
parameter
is defined
by the mass density of the string.
We also show that there is a set of local coordinates which allows us to formally
relate the conical metric to that of the Euclidean plane.
In Sec. 3, we consider some important aspects of the
non-relativistic free particle and the harmonic oscillator dynamics in $\mathbb{R}^{2}$.
This section serves as the basis from which we can build the symmetry algebra of the
corresponding versions of these systems in conical geometry by
using a locally defined canonical transformation. In Sec. 4, we explain
how the conformal bridge transformation works.
As an example, it is shown how to relate the Euclidean
free particle to the planar isotropic harmonic oscillator
by means of this mapping. We also establish there the
nontrivial relation between the hidden symmetry generators
of these two systems
and some sub-algebras generated by them.
In Sec. 5
we study the free particle on a cosmic string background.
Using the above-mentioned canonical transformation, we obtain the
explicit form of the solutions of the equations of motion,
as well as some conserved quantities which are generally of a formal nature,
but which serve to construct well-defined integrals in the phase space when the
geometric parameter is a rational number. At the quantum level,
we construct the symmetry operators and observe the
quantum anomaly by analyzing the action of these
operators on the physical eigenstates.
In Sec. 6, we employ
the conformal bridge transformation to get the symmetry generators of the harmonic oscillator
from the free particle system in the classical and quantum
cases. The properties of symmetry algebra for the
harmonically confined system, as well as the manifestation of the same quantum anomaly
are immediately obtained
by using the same transformation.
Finally, in Sec. 7 the
discussion of the obtained results and outlook are presented.
2 Cosmic string and conical geometry
Cosmic strings are hypothetical
one-dimensional topological defects which may have
formed in
the spontaneous symmetry-breaking
phase transition in the expanding Universe
[36, 37, 38, 39].
Such defects also are familiar
in condensed matter physics
[42, 43].
In this article we are interested in analyzing the
symmetries of a non-relativistic conformal dynamics, at the classical and
quantum levels, of a particle
in a two-dimensional space with properties of
a cosmic string background.
In this section, we
show that this problem is analogous to
studying the dynamics of a particle in a conical geometry
[25, 26],
and discuss some of its general properties
which will be important for the subsequent analysis.
Following [45, 38],
the
solution of the Einstein field equations
in (2+1) dimensions associated with cosmic string
is described by
the metric
$$\displaystyle dS^{2}=-c^{2}dt^{2}+ds^{2}\,,$$
(2.1)
$$\displaystyle ds^{2}=\left(1-\frac{8\mu G}{c^{2}}\ln(\frac{r}{r_{0}})\right)(%
dr^{2}+r^{2}d\varphi^{2})\,,$$
(2.2)
where $G$ is the Newton constant,
$c$ is the speed of light, $\mu$ is the linear mass density of the cosmic string,
and $r_{0}$ corresponds to its radius.
By introducing the new coordinate
$$\displaystyle r^{\prime 2}=\left(1-\frac{8\mu G}{c^{2}}\ln(\frac{r}{r_{0}})%
\right)r^{2}\,,\qquad\alpha^{2}dr^{\prime 2}=\left(1-\frac{8\mu G}{c^{2}}\ln(%
\frac{r}{r_{0}})\right)dr^{2}\,,$$
(2.3)
$$\displaystyle\alpha=\frac{1}{1-\frac{4\mu G}{c^{2}}}>0\,,$$
(2.4)
where higher-order terms in $\mu G/c^{2}$ are neglected in the computation of $dr^{\prime 2}$,
the spatial part
(2.2)
of the cosmic string metric (2.1)
takes the form
$$\displaystyle ds^{2}=\alpha^{2}dr^{2}+r^{2}d\varphi^{2}\,,$$
(2.5)
after changing
the notation $r^{\prime}\rightarrow r$.
Eq. (2.5)
corresponds to the two-dimensional metric of a conical geometry.
For $\alpha>1$,
it can be obtained by reducing
the three-dimensional Euclidean metric
$$ds_{E}^{2}=dr^{2}+r^{2}d\varphi^{2}+dz^{2}\,,$$
(2.6)
to the surface of a cone
$z=\lambda r$,
and identification
$\alpha^{2}=1+\lambda^{2}$.
Here $\lambda=\cot\beta$, $\beta$ is the aperture angle of the cone,
and
in
accordance with (2.4),
this metric is associated to a cosmic string whose mass density is positive.
On the other hand, the
metric (2.5) with
$0<\alpha<1$
can be
obtained from
the $(2+1)$-dimensional Minkowski space
$$ds_{M}^{2}=-c^{2}d\tau^{2}+dr^{2}+r^{2}d\varphi^{2}\,$$
(2.7)
by reduction to the cone surface
$c\tau=\lambda r$, $0<\lambda<1$,
$\alpha^{2}=1-\lambda^{2}$.
Formally, this is the reduction
to
the non-causal part of Minkowski space, where
particle’s
velocity
is
greater than $c$.
In
correspondence with (2.4),
these spaces are associated with
cosmic strings that have negative mass density
[40].
The case of the conic metric with
$0<\alpha<1$
also
describes topological defects in the physics of condensed matter and wormholes
[40, 41, 42, 43, 44].
From here one sees
that studying a
free non-relativistic particle
system in a
cosmic string background geometry with a given value of the parameter $\alpha$
is analogous to solving the problem of the dynamics of a particle on the surface of a cone
described by the action
$S=-mc^{2}\int\sqrt{1-\frac{1}{c^{2}}\left(\frac{ds}{dt}\right)^{2}}\,dt$
in the non-relativistic limit $c\rightarrow\infty$.
In
the remainder of
the
section, we explore some
geometric properties of the conical space
(2.5).
For this,
we express the metric $ds^{2}$ in different coordinate systems,
that
will allow us to
clearly see the singularity at the origin,
and clarify the corresponding
conformal properties of the metric (2.1).
First, we note that
in Cartesian coordinates,
$$ds^{2}=\frac{1}{(x^{2}+y^{2})}\left(\alpha^{2}(xdx+ydy)^{2}+(xdy-ydx)^{2}%
\right)\,,$$
(2.8)
the singularity of the metric
at the origin $x=y=0$ for $\alpha\neq 1$
becomes apparent.
Introducing a
new radial coordinate $r=r_{0}e^{\frac{\rho}{\alpha}}$,
the metric becomes
$$ds^{2}=r_{0}^{2}e^{\frac{2\rho}{\alpha}}(d\rho^{2}+d\varphi^{2})\,.$$
(2.9)
The variables $\rho$ and $\varphi$ correspond to the isothermal coordinates,
and we note that when $\alpha\rightarrow\infty$, the metric (2.9)
transforms into a cylinder’s metric.
From (2.9), as well as from (2.5),
the invariance of the metric
under rotations,
$\varphi\rightarrow\varphi+\varphi_{0}$, is obvious.
The corresponding Killing vector
is $\frac{\partial}{\partial\varphi}$.
Metric (2.9) also is conformally invariant
under transformations
$\rho\rightarrow\rho+\rho_{0}$. In polar coordinates, this corresponds to dilatation
in the radial coordinate
generated by
the
conformal Killing vector
$\frac{\partial}{\partial\rho}=\frac{r}{\alpha}\frac{\partial}{\partial r}\,.$
By introducing a pair of “regularized” Cartesian coordinates
$$X_{1}=\alpha r\cos\Phi\,,\qquad X_{2}=\alpha r\sin\Phi\,,\qquad\Phi=\varphi/%
\alpha\,,$$
(2.10)
the metric (2.5) is transformed into
$$ds^{2}=dX_{1}^{2}+dX_{2}^{2}\,.$$
(2.11)
This
formally
looks like
the metric of the Euclidean plane in Cartesian coordinates, but
$0\leq\Phi<2\pi/\alpha$ in (2.10), and the edges $\Phi=0$ and $\Phi=2\pi/\alpha$
have to be identified, that results in a conical singularity.
This singularity reveals itself, particularly, in
Riemann curvature tensor concentrated at $r=0$:
$\mathcal{R}^{r\varphi}_{r\varphi}=2\pi(1-\alpha^{-1})\delta(X_{1})\delta(X_{2})$
[45, 43].
To clarify further
the nature of coordinates (2.10), consider the complex combination
$$w=X_{1}+iX_{2}=\alpha re^{i\varphi/\alpha}\,,\qquad ds^{2}=dwdw^{*}\,.$$
(2.12)
From here it is seen that there may be problems for arbitrary values of $\alpha$
due to the exponential factor and the
associated
branch point.
When the rational case
$\alpha={q}/{k}$ with $q,k=1,2,\ldots$, is considered,
one can use instead the new coordinates
$\zeta=\zeta_{1}+i\zeta_{2}$,
$$\zeta=w^{q}=\left(\frac{qr}{k}\right)^{q}e^{ik\varphi}\quad\Rightarrow\quad%
\zeta_{1}=\left(\frac{qr}{k}\right)^{q}\cos(k\varphi)\,,\qquad\zeta_{2}=\left(%
\frac{qr}{k}\right)^{q}\sin(k\varphi)\,.$$
(2.13)
In their terms the metric
reads
$$\displaystyle ds^{2}=\frac{d\zeta_{1}^{2}+d\zeta_{2}^{2}}{q^{2}(\zeta_{1}^{2}+%
\zeta_{2}^{2})^{1-\frac{1}{q}}}\,.$$
(2.14)
For
$q=1$ $\Rightarrow\alpha=1/k$, it seems that
there is no singularity in the metric, and
the
coordinates $\zeta_{1}$, $\zeta_{2}$
themselves reveal no singularity. However, as it is seen from (2.13),
in this case
$\zeta_{1}$ and $\zeta_{2}$ cover conical space $k$ times
when $\alpha=1/k$ . The picture is similar to a
Riemann surface for the function $w=z^{1/k}$
where we pass from its one sheet to another when angle
increases in $2\pi$, while here the transition from
one sheet to another happens each time when $\varphi$ increases in $2\pi/k$.
Note also here that formally
metric (2.11) is invariant under
translations
$X_{i}\rightarrow X_{i}+a_{i}$,
$i=1,2$,
produced
by the vector fields
$$\displaystyle\frac{\partial}{\partial X_{1}}=\frac{1}{\alpha}\cos(\frac{%
\varphi}{\alpha})\frac{\partial}{\partial r}-\frac{1}{r}\sin(\frac{\varphi}{%
\alpha})\frac{\partial}{\partial\varphi}\,,\qquad\frac{\partial}{\partial X_{2%
}}=\frac{1}{\alpha}\sin(\frac{\varphi}{\alpha})\frac{\partial}{\partial r}+%
\frac{1}{r}\cos(\frac{\varphi}{\alpha})\frac{\partial}{\partial\varphi}\,.$$
(2.15)
The infinitesimal ($\delta_{1},\delta_{2}\sim 0$) form of transformations
generated by (2.15) is
$$\displaystyle\frac{\partial}{\partial X_{1}}:\quad\Rightarrow\quad r%
\rightarrow r_{1}^{\prime}=r+\frac{\delta_{1}}{\alpha}\cos(\frac{\varphi}{%
\alpha})\,,\qquad\varphi\rightarrow\varphi^{\prime}_{1}=\varphi-\frac{\delta_{%
1}}{r}\sin(\frac{\varphi}{\alpha})\,,$$
(2.16)
$$\displaystyle\frac{\partial}{\partial X_{2}}:\quad\Rightarrow\quad r%
\rightarrow r_{2}^{\prime}=r+\frac{\delta_{2}}{\alpha}\sin(\frac{\varphi}{%
\alpha})\,,\qquad\varphi\rightarrow\varphi_{2}^{\prime}=\varphi+\frac{\delta_{%
2}}{r}\cos(\frac{\varphi}{\alpha})\,.$$
(2.17)
In spite of that
these formal transformations are local isometries,
one sees their singular nature
at $r=0$
when $\alpha\not=1/k$.
Furthermore,
the corresponding global transformations,
$$\displaystyle r^{2}\rightarrow(r_{1}^{\prime})^{2}=r^{2}+\frac{2\delta_{1}r}{%
\alpha}\cos(\frac{\varphi}{\alpha})+\frac{\delta_{1}^{2}}{\alpha^{2}}\,,\qquad%
\varphi\rightarrow\varphi_{1}^{\prime}=\alpha\arctan(\frac{\alpha r\sin(\frac{%
\varphi}{\alpha})}{\alpha r\cos(\frac{\varphi}{\alpha})+\delta_{1}})\,,$$
(2.18)
$$\displaystyle r^{2}\rightarrow(r_{2}^{\prime})^{2}=r^{2}+\frac{2\delta_{2}r}{%
\alpha}\sin(\frac{\varphi}{\alpha})+\frac{\delta_{2}^{2}}{\alpha^{2}}\,,\qquad%
\varphi\rightarrow\varphi_{2}^{\prime}=\alpha\arctan(\frac{\alpha r\sin(\frac{%
\varphi}{\alpha})+\delta_{2}}{\alpha r\cos(\frac{\varphi}{\alpha})})\,,$$
(2.19)
reveal their singularity for $\alpha\not=1$.
The space-time metric (2.1) with spatial part presented in the form
(2.11) looks like the metric of (2+1)-dimensional
Minkowski space $dS^{2}=\eta_{\mu\nu}dX^{\mu}dX^{\nu}$,
$X^{0}=ct$, $\eta_{\mu\nu}=\text{diag}\,(-1,1,1)$. Locally, it is conformally invariant
under transformations of the conformal
$SO(3,2)$ group,
whose classical generators are $P^{\mu}$,
$J^{\mu\nu}=X^{\mu}P^{\nu}-X^{\nu}P^{\mu}$, $K^{\mu}=2X^{\mu}(XP)-X^{2}P^{\mu}$
and $D=XP$,
where $P_{\mu}=\eta_{\mu\nu}P^{\nu}$ are the momenta canonically conjugate to $X^{\mu}$.
Taking into account that $P_{1}=\frac{1}{\alpha}p_{r}\cos(\varphi/\alpha)-\alpha p_{\varphi}\sin(\varphi%
/\alpha)$,
$P_{2}=\frac{1}{\alpha}p_{r}\sin(\varphi/\alpha)+\alpha p_{\varphi}\cos(\varphi%
/\alpha)$,
where $p_{r}$ and $p_{\varphi}$ are the momenta canonically conjugate to $r$ and $\varphi$,
one finds that only the generators of the time translation, $P^{0}$,
the spatial rotation, $J^{12}=p_{\varphi}$, the dilatations, $D=-X^{0}P^{0}+rp_{r}$,
and special conformal transformations, $K^{0}=2X^{0}D-(\alpha^{2}r^{2}-(X^{0})^{2})P^{0}$, are globally
well-defined for
arbitrary values of the parameter $\alpha$,
while generators of the spatial translations, $P^{i}$, Lorentz boosts, $J^{0i}$,
and generators of special conformal transformations, $K^{i}$,
are globally well-defined only for $\alpha=1/k$.
After
the
appropriately taken non-relativistic limit
[56, 57, 58, 59, 60, 61, 62], as we shall see, the
rotation generator $p_{\varphi}$ and
the corresponding analogs of the generators $P^{0}$, $D$ and $K^{0}$
will play the key role in our subsequent analysis.
At the same time, in spite of the globally not well-defined nature
(in the general case of the parameter
$\alpha$ values) of
generators of the spatial translations and Lorentz boosts,
their corresponding non-relativistic analogs will be
employed by us for the construction of the globally
well-defined
generators of the hidden symmetries.
The
change of coordinates (2.10)
and the geodesic analysis presented in Sec. 5 will
show
that the non-relativistic dynamics in the cosmic string background
can be related to the free motion in the Euclidean plane.
Bearing this in mind,
instead of jumping directly to the
analysis of the dynamics in the
conical geometry, it is
appropriate
to review some important
characteristics related to the motion in $\mathbb{R}^{2}$.
3 Dynamics in the Euclidean plane
To understand
the complete symmetry algebra of a given mechanical system
in a cosmic string (conical) background,
both at the classical and quantum levels, it is instructive to remind the corresponding properties of such
a system in the flat Euclidean plane.
Later on, we will show that there is a formal
canonical transformation related to the change of coordinates (2.10)
which allows us to connect the dynamics in conical geometry
with the corresponding dynamics in $\mathbb{R}^{2}$.
We are interested in the free particle dynamics as well
as the dynamics of the particle in the harmonic trap, so this section contains all we
need to know
of these two systems in the Euclidean plane.
3.1 The free particle
Here we present the complete set of integrals of motion of order not higher than two
in momenta and display their explicit Lie algebra for a particle in Euclidean plane.
Next we use the conserved quantities to reconstruct the trajectory of the particle.
Finally we briefly describe the quantum theory of the system using the
polar coordinates.
The quadratic in momenta and coordinates integrals of motion are
$$\displaystyle H=\frac{1}{2m}p_{+}p_{-}\,,\qquad D=\frac{1}{4}(\chi_{+}p_{-}+p_%
{+}\chi_{+})\,,\qquad K=\frac{m}{2}\chi_{+}\chi_{-}\,,$$
(3.1)
$$\displaystyle J_{0}=\frac{i}{4}(\chi_{+}p_{-}-p_{+}\chi_{-})\,,\qquad J_{\pm}=%
\frac{1}{2}\chi_{\pm}p_{\pm}\,,$$
(3.2)
$$\displaystyle T_{\pm}=\frac{1}{2m}(p_{\pm})^{2}\,,\qquad S_{\pm}=\frac{m}{2}(%
\chi_{\pm})^{2}\,,$$
(3.3)
where
$$p_{\pm}=p_{1}\pm ip_{2}\,,\qquad\chi_{\pm}=\chi_{1}\pm i\chi_{2}\,$$
(3.4)
are the complex combinations of the canonical momenta $p_{i}$ and the Galilean boost generators
$\chi_{i}=x_{i}-\frac{1}{m}p_{i}t$.
The Hamiltonian $H$, the angular momentum
$p_{\varphi}=2J_{0}$, and the
integrals $p_{\pm}$, $T_{\pm}$
are the conserved quantities not depending explicitly on time $t$.
The integrals $J_{0}$, $p_{\pm}$ and $\chi_{\pm}$,
unlike the rest of the listed integrals,
do not mix coordinates $x_{i}$ and momenta $p_{i}$
when they act in the phase space
via Poisson brackets.
The integrals $H$, $D$ and $K$
correspond to the planar case $\alpha=1$
of the non-relativistic limit of the generators
$P^{0}$, $D$ and $K^{0}$ mentioned in the previous section,
while the Galilean boost generators $\chi_{i}$
appear as the non-relativistic limit of the Lorentz boosts
$J^{0i}$,
see refs. [57, 58, 61].
Note that the
integrals $J_{\pm}$, $T_{\pm}$ and $S_{\pm}$,
being quadratic in $p_{i}$,
correspond here
to generators of the hidden symmetries
[1].
The ten generators (3.1)-(3.3) satisfy the following
Poisson bracket relations
of the $\mathfrak{sp}(4,\mathbb{R})$ algebra,
$$\displaystyle\{D,H\}=H\,,\qquad\{D,K\}=-K\,,\qquad\{K,H\}=2D\,,$$
(3.5)
$$\displaystyle\{J_{0},J_{\pm}\}=\mp iJ_{\pm}\,,\qquad\{J_{-},J_{+}\}=-2iJ_{0}\,,$$
(3.6)
$$\displaystyle\{J_{0},T_{\pm}\}=\mp iT_{\pm}\,,\qquad\{J_{0},S_{\pm}\}=\mp iS_{%
\pm}\,,$$
(3.7)
$$\displaystyle\{H,S_{\pm}\}=-2J_{\pm}\,,\qquad\{H,J_{\pm}\}=-T_{\pm}\,,$$
(3.8)
$$\displaystyle\{K,T_{\pm}\}=2J_{\pm}\,,\qquad\{K,J_{\pm}\}=S_{\pm}\,,$$
(3.9)
$$\displaystyle\{D,T_{\pm}\}=T_{\pm}\,,\qquad\{D,S_{\pm}\}=-S_{\pm}\,,$$
(3.10)
$$\displaystyle\{S_{\pm},T_{\mp}\}=\mp 4i(J_{0}\pm iD)\,,$$
(3.11)
$$\displaystyle\{J_{\pm},T_{\mp}\}=2H\,,\qquad\{J_{\pm},S_{\mp}\}=2K\,.$$
(3.12)
By including the first order generators (3.4) with redefinition
$\xi_{\pm}=m\chi_{\pm}$,
we supplement the algebra (3.5)-(3.12) with the
Poisson bracket relations
$$\displaystyle\{\xi_{\pm},p_{\mp}\}=2m\,,$$
(3.13)
$$\displaystyle\{H,\xi_{\pm}\}=-p_{\pm}\,,\qquad\{D,\xi_{\pm}\}=-\frac{1}{2}\xi_%
{\pm}\,,\qquad\{J_{0},\xi_{\pm}\}=\mp\frac{i}{2}\xi_{\pm}\,,$$
(3.14)
$$\displaystyle\{K,p_{\pm}\}=\xi_{\pm}\,,\qquad\{D,p_{\pm}\}=\frac{1}{2}p_{\pm}%
\,,\qquad\{J_{0},p_{\pm}\}=\mp\frac{i}{2}p_{\pm}\,,$$
(3.15)
$$\displaystyle\{T_{\pm},\xi_{\mp}\}=-2p_{\pm}\,,\qquad\{S_{\pm},p_{\mp}\}=2\xi_%
{\pm}\,,$$
(3.16)
$$\displaystyle\{J_{\pm},\xi_{\mp}\}=-\xi_{\pm}\,,\qquad\{J_{\pm},p_{\mp}\}=p_{%
\pm}\,.$$
(3.17)
The
not displayed in (3.5)–(3.17)
Poisson brackets are equal to zero.
Relations
(3.13)–(3.17)
correspond to the
ideal sub-algebra generated by $\xi_{\pm}$ and $p_{\pm}$,
with mass $m$ playing a role of the central charge.
This also is an ideal sub-algebra
of the Schrödinger algebra $\mathfrak{sch}(2)$ [56], generated
by $H$, $D$, $K$, $p_{i}$, $\xi_{i}$ and $m$,
that, in turn, is a sub-algebra of the complete Lie algebra
(3.5)–(3.17).
Some remarkable properties of the presented
symmetry algebra are the following.
•
The algebraic relations (3.5)
correspond to the dynamical $\mathfrak{so}(2,1)\cong\mathfrak{sl}(2,\mathbb{R})$ conformal algebra.
Its Casimir element
is $D^{2}-HK=-J_{0}^{2}=-\frac{1}{4}p_{\varphi}^{2}$.
•
Relations
(3.6) correspond to another $\mathfrak{sl}(2,\mathbb{R})$ sub-algebra with
Casimir element $-J_{0}^{2}+J_{+}J_{-}=D^{2}$.
•
Each triplet of integrals $(J_{0},T_{\pm})$, $(J_{0},S_{\pm})$,
$(p_{\varphi},p_{\pm})$ and $(p_{\varphi},\xi_{\pm})$
generate Euclidean sub-algebra $\mathfrak{e}(2)$. The corresponding Casimirs
$T_{+}T_{-}$, $S_{+}S_{-}$, $p_{+}p_{-}$ and
$\xi_{+}\xi_{-}$ are
$H^{2}$, $K^{2}$, $2mH$ and $2mK$.
•
The two sets
$(\ell_{+}^{(+)}=\frac{1}{2\gamma}S_{+}$, $\ell_{-}^{(+)}=\frac{\gamma}{2}T_{-}$, $\ell_{0}^{(+)}=\frac{1}{2}(J_{0}+iD))$
and $(\ell_{+}^{(-)}=\frac{\gamma}{2}T_{+}$, $\ell_{-}^{(-)}=\frac{1}{2\gamma}S_{-}$, $\ell_{0}^{(-)}=\frac{1}{2}(J_{0}-iD))$
generate
the $\mathfrak{su}(2)\oplus\mathfrak{su}(2)$ sub-algebra,
where
$\gamma$ is a constant of dimension of squared length
that is introduced
to compensate the corresponding dimensions.
The Casimir elements of these two $\mathfrak{su}(2)$ sub-algebras are
$C^{(\pm)}=(\ell_{0}^{(\pm)})^{2}+\ell_{+}^{(\pm)}\ell_{-}^{(\pm)}=0$. Note that
$\big{(}\ell_{0}^{(+)}\big{)}^{*}=\ell_{0}^{(-)}$,
$\big{(}\ell_{+}^{(\pm)}\big{)}^{*}=\ell_{-}^{(\mp)}$.
•
Each integral is an eigenstate of $D$
in the sense of the Poisson bracket relation $\{D,A\}=\lambda A$ :
$(\lambda=1:H,T_{\pm})$,
$(\lambda=0:D,J_{0},J_{\pm})$,
$(\lambda=-1:K,S_{\pm})$,
$(\lambda=1/2:p_{\pm})$,
$(\lambda=-1/2:\xi_{\pm})$.
•
Analogously, each integral is an eigenstate of $J_{0}$:
$(\lambda=1:J_{+},T_{+},S_{+})$,
$(\lambda=0:J_{0},H,D,K)$,
$(\lambda=-1:J_{-},T_{-},S_{-})$,
$(\lambda=1/2:p_{+},\xi_{+})$,
$(\lambda=-1/2:p_{-},\xi_{-})$.
Introducing linear combinations
$\mathcal{J}_{0}=\frac{1}{2}(H+K)$,
$\mathcal{J}_{1}=\frac{1}{2}(H-K)$,
and denoting $\mathcal{J}_{2}=D$,
$\mathcal{J}_{\pm}=\mathcal{J}_{1}\pm i\mathcal{J}_{2}$,
conformal algebra (3.5)
can be presented in the form similar to
(3.6),
and its Casimir takes the form
$-\mathcal{J}_{0}^{2}+\mathcal{J}_{+}\mathcal{J}_{-}=-J_{0}^{2}$.
The generators
$\mathcal{J}_{\mu}$, $\mu=0,1,2$,
of the conformal algebra
describe the upper sheet of the two-sheeted hyperboloid,
that at $p_{\varphi}=0$ degenerates into the
cone with $\mathcal{J}_{0}\geq 0$.
The generators $J_{\mu}$ of the
$\mathfrak{sl}(2,\mathbb{R})$ algebra (3.6)
describe a one sheet hyperboloid
that at $\mathcal{J}_{2}=D=0$ degenerates into a double cone
with $J_{0}=2p_{\varphi}\in\mathbb{R}$.
Using the Casimir of conformal algebra
(3.5) and the explicit form of generators (3.1), one gets
$$\displaystyle r^{2}(t)=\frac{2}{m}(Ht^{2}+2Dt+K)=\frac{2H}{m}\left(\left(t+%
\frac{D}{H}\right)^{2}+\frac{p_{\varphi}^{2}}{4H^{2}}\right)\,,$$
(3.18)
from where we see that at the moment of time $t_{*}=-D/H$, the particle is in the “perihelion” of the
trajectory, $r(t_{*})\equiv r_{*}=p_{\varphi}/\sqrt{2mH}$.
On the other hand, from the angular and linear momenta integrals the
straight line trajectory is reconstructed in polar coordinates,
$$\displaystyle r(\varphi)=\frac{r_{*}}{\cos(\varphi-\varphi_{*})}\,,\qquad-\pi/%
2\leq\varphi-\varphi_{*}\leq\pi/2\,.$$
(3.19)
By means of (3.18) and (3.19) we also find
$$\displaystyle\varphi(t)=\arctan\left(\frac{1}{2\rho}\left(t+\frac{D}{H}\right)%
\right)+\varphi_{*}\,,\qquad\rho=\frac{p_{\varphi}}{2H}\,.$$
(3.20)
From $\chi_{i}$ and $p_{i}$
one can
construct a sort of Laplace-Runge-Lentz
vector,
$$\displaystyle\chi_{i}^{\bot}=x_{i}-p_{i}\frac{x_{j}p_{j}}{p_{k}p_{k}}$$
(3.21)
such that
$\chi_{i}^{\bot}p_{i}=0.$
This vector (with respect to $p_{\varphi}$)
integral specifies the coordinates of the perihelion,
$\chi_{1}^{\bot}=r_{*}\cos(\varphi_{*})=x_{1}(\varphi_{*})$,
$\chi_{2}^{\bot}=r_{*}\sin(\varphi_{*})=x_{2}(\varphi_{*}).$
The components $\chi_{i}^{\bot}$, however, are not independent integrals since they satisfy
$\chi_{1}^{\bot}=\frac{p_{\varphi}}{2mH}\,p_{2}$ and
$\chi_{2}^{\bot}=-\frac{p_{\varphi}}{2mH}\,p_{1}$.
In the quantum case, it is convenient here
to use the
polar coordinates, in which
the Hamiltonian operator is given by
$$\displaystyle\hat{H}=-\frac{\hbar^{2}}{2m}\left(\frac{1}{r}\frac{\partial}{%
\partial r}\left(r\frac{\partial}{\partial r}\right)+\frac{1}{r^{2}}\frac{%
\partial^{2}}{\partial\varphi}\right)\,.$$
(3.22)
Its eigenstates and eigenvalues are
$$\displaystyle\psi_{\kappa,l}^{\pm}(r,\varphi)=\sqrt{\frac{\kappa}{2\pi}}J_{l}(%
\kappa r)e^{\pm il\varphi}\,,\qquad E=\frac{\hbar^{2}\kappa^{2}}{2m}\,,\qquad l%
=0,1,\ldots\,,$$
(3.23)
where $J_{\beta}(\zeta)$ are the Bessel functions of the first kind.
With respect to the inner product
$$\displaystyle\bra{\Psi_{1}}\ket{\Psi_{2}}=\int_{0}^{\infty}rdr\int_{0}^{2\pi}d%
\varphi\Psi^{*}_{1}\Psi_{2}\,,$$
(3.24)
eigenstates (3.23)
satisfy the orthogonality relation
$\innerproduct{\psi_{\kappa,l}^{\pm}}{\psi_{\kappa^{\prime},l^{\prime}}^{\mp}}=%
\delta_{ll^{\prime}}\delta(\kappa-\kappa^{\prime})$,
and due to the property of the Bessel
functions $J_{-l}(\eta)=(-1)^{l}J_{l}(\eta)$, one has
$\psi_{\kappa,-l}^{\pm}=(-1)^{l}\psi_{\kappa,l}^{\mp}$.
The basic first order differential operators of the system in polar coordinates are
$$\displaystyle\hat{p}_{r}=-i\hbar\left(\frac{\partial}{\partial r}+\frac{1}{2r}%
\right)\,,\qquad\hat{p}_{\varphi}=-i\hbar\frac{\partial}{\partial\varphi}\,,$$
(3.25)
where $\hat{p}_{r}$ is symmetric with respect to (3.24), but not self-adjoint. We use them to construct
the well-defined operators
$$\displaystyle\hat{p}_{\pm}=-i\hbar e^{\pm i\varphi}\left[\frac{\partial}{%
\partial r}\pm i\frac{1}{r}\frac{\partial}{\partial\varphi}\right]\,,\qquad p_%
{+}^{\dagger}=\hat{p}_{-}\,,$$
(3.26)
which are the quantum version of the classical integrals $p_{\pm}$.
Their action on eigenstates can be found by using the recurrence relations
$\frac{2\beta}{\zeta}J_{\beta}(\zeta)=J_{\beta-1}(\zeta)+J_{\beta+1}(\zeta)\,,$
$2\frac{d}{d\zeta}J_{\zeta}(\zeta)=J_{\beta-1}(\zeta)-J_{\beta+1}(\zeta)\,,$
and we get
$$\displaystyle\hat{p}_{\pm}\psi_{\kappa,l}^{\pm}(r,\varphi)=i\hbar\kappa\psi_{%
\kappa,l+1}^{\pm}(r,\varphi)\,,\qquad\hat{p}_{\pm}\psi_{\kappa,l}^{\mp}(r,%
\varphi)=-i\hbar\kappa\psi_{\kappa,l-1}^{\mp}(r,\varphi)\,.$$
(3.27)
These relations
show that the operators $\hat{p}_{\pm}$
change the angular momentum quantum number of the wave-function
without changing the energy, that reflects
the infinite degeneracy of
the energy levels.
Quantum version of other integrals is obtained
by the substitution $p_{\pm}\rightarrow\hat{p}_{\pm}$
and using the Weyl (symmetric) ordering
in (3.1)-(3.3).
In the general case we
have dynamical integrals $\hat{A}(t)$
which include the explicit dependence on time,
$\frac{d}{dt}\hat{A}(t)=\frac{\partial}{\partial t}\hat{A}(t)+\frac{1}{i\hbar}[%
\hat{A}(t),\hat{H}]=0$.
For them we have
$$\hat{A}(t)\Psi(r,\varphi,t)=e^{-\frac{it\hat{H}}{\hbar}}\hat{A}|_{t=0}\Psi(r,%
\varphi,t=0)\,,$$
(3.28)
where $\Psi(r,\varphi,t)$ is a solution of the time dependent Schrödinger equation.
In particular, the quantum generators of dilatations and special conformal transformations,
$$\displaystyle\hat{D}=\frac{1}{4}(\hat{\chi}_{+}\hat{p}_{-}+\hat{p}_{+}\hat{%
\chi}_{-})=-i\frac{\hbar}{2}\left(r\frac{\partial}{\partial r}+1\right)-\hat{H%
}t\,,$$
(3.29)
$$\displaystyle\hat{K}=\frac{1}{2}m\hat{\chi}_{-}\hat{\chi}_{+}=\frac{1}{2}mr^{2%
}-2\hat{D}t-\hat{H}t^{2}\,,$$
(3.30)
are examples of dynamical symmetry operators. Together with the Hamiltonian, they generate the
quantum
$\mathfrak{sl}(2,\mathbb{R})$ algebra
$$\displaystyle[\hat{D},\hat{H}]=i\hbar\hat{H}\,,\qquad[\hat{D},\hat{K}]=-i\hbar%
\hat{K}\,,\qquad[\hat{K},\hat{H}]=2i\hbar\hat{D}\,.$$
(3.31)
On a Hilbert subspace with fixed value of the quantum number $l=0,1,\ldots$, the eigenstates (3.23)
correspond to an irreducible
infinite dimensional representation of
conformal $\mathfrak{sl}(2,\mathbb{R})$ algebra (3.31)
of the discrete type series $D^{+}_{j}$
characterized by
the Casimir operator value
$-\hat{\mathcal{J}}_{0}^{2}+\hat{\mathcal{J}}_{1}^{2}+\hat{\mathcal{J}}_{2}^{2}%
=\hat{D}^{2}-\frac{1}{2}(\hat{K}\hat{H}+\hat{H}\hat{K})=-\hbar^{2}j(j-1)$ with $j=\frac{1}{2}(l+1)$,
and eigenvalues of
the compact generator $\hat{\mathcal{J}}_{0}$ to be
$j+n$, $n=0,1,\ldots$.
The present free particle’s Hilbert space, in which
the non-compact $\mathfrak{sl}(2,\mathbb{R})$ generator
$\hat{\mathcal{J}}_{0}+\hat{\mathcal{J}}_{1}=\hat{H}$
is diagonal, corresponds to the so-called parabolic
realization of $D^{+}_{j}$ representation [63].
For quantum analog of the $\mathfrak{sl}(2,\mathbb{R})$ algebra
(3.6), the Casimir operator is
$\hat{J}_{0}(-\hat{J}_{0}+\hbar)+\hat{J}_{+}\hat{J}_{-}=\hat{D}^{2}+\frac{1}{4}%
\hbar^{2}$.
Operator $\hat{D}$ is self-adjoint with respect to the scalar product
(3.24), and at $t=0$ its eigenfunctions are
$\Psi_{\lambda}(r)=r^{2i\lambda-1}/\sqrt{2}\pi$,
$\hat{D}\Psi_{\lambda}=\lambda\Psi_{\lambda}$, $\lambda\in\mathbb{R}$,
$\bra{\Psi_{\lambda}}\ket{\Psi_{\lambda}^{\prime}}=\delta(\lambda-\lambda^{%
\prime})$.
One sees then that the quantum analog of the $\mathfrak{sl}(2,\mathbb{R})$
algebra (3.6) at fixed value of $\lambda$ corresponds here
to the principal continuous series representation
characterized by the Casimir invariant value $\hat{C}=\hbar^{2}(\lambda^{2}+1/4)\geq\hbar^{2}/4$,
in which the eigenvalues of the compact generator $\hat{J}_{0}$ are
$j_{0}=\hbar j$ with $j=0,\pm 1,\pm 2,\ldots$ on the subspace
with even values of $l$, and
$j=\pm 1/2,\pm 3/2,\ldots$ on the subspace with odd values of $l$
[64].
The difference between representations generated by
$\hat{\mathcal{J}}_{\mu}$ and $\hat{J}_{\mu}$ operators
is coherent with the difference of the above-mentioned corresponding
classical hyperboloids [63].
3.2 The isotropic harmonic oscillator
Here we consider some general properties of the harmonic oscillator in the Euclidean plane.
First, we construct the symmetry generators and the
symmetry algebra.
In the next step, we use these integrals of motion to
algebraically reproduce the orbit
of the particle from the integrals of motion,
and finally, we review the quantum picture of the model.
The classical Hamiltonian of the planar isotropic harmonic oscillator system
$$\displaystyle H_{\text{os}}=H_{1}+H_{2}=\frac{p_{r}^{2}}{2m}+\frac{p_{\varphi}%
^{2}}{2mr^{2}}+\frac{m\omega^{2}}{2}r^{2}$$
(3.32)
can be understood in Cartesian coordinates $x_{j}$, $j=1,2$, as the sum
of the two independent one-dimensional
harmonic oscillator Hamiltonians
$H_{j}=\frac{1}{2m}(p_{j}^{2}+m^{2}\omega^{2}x_{j}^{2})$
with the same
frequencies and masses, while
in the polar coordinates
it can be considered as a two-dimensional generalization
of the de Alfaro Fubini and Furlan conformal mechanics model
[16].
The quantities
$$\displaystyle a_{j}^{\pm}=\frac{1}{\sqrt{2}}e^{\mp i\omega t}\left(\sqrt{m%
\omega}x_{j}\mp i\frac{p_{j}}{\sqrt{m\omega}}\right)\,,\qquad j=1,2\,,$$
(3.33)
being classical analogs of the quantum ladder operators
multiplied by $e^{\mp i\omega t}$, are the basic dynamical integrals
expressed in Cartesian coordinates.
Their linear combinations
$$\displaystyle b_{1}^{-}=\frac{1}{\sqrt{2}}(a_{1}^{-}-ia_{2}^{-})=\frac{1}{2}e^%
{i(\omega t-\varphi)}\left(\sqrt{m\omega}r+\frac{p_{\varphi}}{\sqrt{m\omega}r}%
+i\frac{p_{r}}{\sqrt{m\omega}}\right)\,,\qquad b_{1}^{+}=(b_{1}^{-})^{*}\,,$$
(3.34)
$$\displaystyle b_{2}^{-}=\frac{1}{\sqrt{2}}(a_{1}^{-}+ia_{2}^{-})=\frac{1}{2}e^%
{i(\omega t+\varphi)}\left(\sqrt{m\omega}r-\frac{p_{\varphi}}{\sqrt{m\omega}r}%
+i\frac{p_{r}}{\sqrt{m\omega}}\right)\,,\qquad b_{2}^{+}=(b_{2}^{-})^{*}\,,$$
(3.35)
are more convenient, however, when we work in the polar coordinates.
They can be produced by a particular
classical canonical transformation, or
the corresponding unitary transformation at the quantum level,
that we consider below.
The ten second-order in these basic integrals
symmetry generators are
$$\displaystyle\mathcal{J}_{0}=\frac{1}{2}b_{j}^{+}b_{j}^{-}=\frac{1}{2\omega}H_%
{\text{os}}\,,\qquad\mathcal{L}_{2}=b_{1}^{+}b_{1}^{-}-b_{2}^{+}b_{2}^{-}=%
\frac{1}{2}p_{\varphi}\,,\qquad\mathcal{L}_{\pm}=b_{1}^{\pm}b_{2}^{\mp}\,,$$
(3.36)
$$\displaystyle\mathcal{J}_{\pm}=b_{1}^{\pm}b_{2}^{\pm}=\frac{1}{2}\left((a_{1}^%
{\pm})^{2}+(a_{2}^{\pm})^{2}\right)\,,\qquad\mathcal{B}_{j}^{\pm}=(b_{j}^{\pm}%
)^{2}\,.$$
(3.37)
Unlike the free particle case, only four integrals (3.36) here do not depend
explicitly on time.
The integrals (3.36) and (3.37) satisfy the following non-zero Poisson bracket relations
$$\displaystyle\{\mathcal{J}_{0},\mathcal{J}_{\pm}\}=\mp i\mathcal{J}_{\pm}\,,%
\qquad\{\mathcal{J}_{-},\mathcal{J}_{+}\}=-2i\mathcal{J}_{0}\,,$$
(3.38)
$$\displaystyle\{\mathcal{L}_{2},\mathcal{L}_{\pm}\}=\mp i\mathcal{L}_{\pm}\,,%
\qquad\{\mathcal{L}_{+},\mathcal{L}_{-}\}=-i2\mathcal{L}_{2}\,,$$
(3.39)
$$\displaystyle\{\mathcal{J}_{\pm},\mathcal{L}_{\mp}\}=\pm i\mathcal{B}_{2}^{\pm%
}\,,\qquad\{\mathcal{J}_{\pm},\mathcal{L}_{\pm}\}=\pm i\mathcal{B}_{1}^{\pm}$$
(3.40)
$$\displaystyle\{\mathcal{J}_{0},\mathcal{B}_{a}^{\pm}\}=\mp i\mathcal{B}_{a}^{%
\pm}\,,\qquad\{\mathcal{J}_{\mp},\mathcal{B}_{2}^{\pm}\}=2i\mathcal{L}_{\mp}\,%
,\qquad\{\mathcal{J}_{\mp},\mathcal{B}_{1}^{\pm}\}=2i\mathcal{L}_{\pm}\,,$$
(3.41)
$$\displaystyle\{\mathcal{L}_{2},\mathcal{B}_{j}^{\pm}\}=\mp i\mathcal{B}_{j}^{%
\pm}\,,\qquad\{\mathcal{L}_{\pm},\mathcal{B}_{2}^{\mp}\}=\pm 2i\mathcal{J}_{%
\pm}\qquad\{\mathcal{L}_{\mp},\mathcal{B}_{1}^{\pm}\}=\mp 2i\mathcal{J}_{\pm}\,,$$
(3.42)
$$\displaystyle\{\mathcal{B}_{2}^{-},\mathcal{B}_{2}^{+}\}=-4i\left(\mathcal{J}_%
{0}+\mathcal{L}_{2}\right)\,,\qquad\{\mathcal{B}_{1}^{-},\mathcal{B}_{1}^{+}\}%
=-4i\left(\mathcal{J}_{0}-\mathcal{L}_{2}\right)\,.$$
(3.43)
The brackets involving the basic integrals are
$$\displaystyle\{b_{i}^{-},b_{j}^{+}\}=-i\delta_{ij}\,,$$
(3.44)
$$\displaystyle\{\mathcal{J}_{0},b_{j}^{\pm}\}=\pm\frac{i}{2}b_{j}^{\pm}\,,%
\qquad\{\mathcal{J}_{\mp},b_{1}^{\pm}\}=\mp b_{2}^{\mp}\,,\qquad\{\mathcal{J}_%
{\mp},b_{2}^{\pm}\}=\mp b_{1}^{\mp}\,,$$
(3.45)
$$\displaystyle\{\mathcal{B}_{i}^{\pm},b_{j}^{\mp}\}=\pm 2i\delta_{ij}b_{j}^{\mp%
}\,,\qquad\{\mathcal{L}_{2},b_{1}^{\pm}\}=\mp\frac{i}{2}b_{1}^{\pm}\,,\qquad\{%
\mathcal{L}_{2},b_{2}^{\pm}\}=\pm\frac{i}{2}b_{2}^{\pm}\,,$$
(3.46)
$$\displaystyle\{\mathcal{L}_{\pm},b_{1}^{\mp}\}=\pm ib_{2}^{\mp}\,,\qquad\{%
\mathcal{L}_{\pm},b_{2}^{\pm}\}=\mp ib_{1}^{\pm}\,,$$
(3.47)
$$\displaystyle\{\mathcal{J}_{\pm},b_{j}^{\pm}\}=\{\mathcal{L}_{\pm},b_{1}^{\pm}%
\}=\{\mathcal{L}_{\pm},b_{2}^{\mp}\}=0\,.$$
(3.48)
Some properties
of the Lie algebra (3.38)–(3.48) are the following.
•
Dynamical integrals $b_{j}^{\pm}$ generate an ideal sub-algebra.
•
Relations (3.38) correspond to the conformal $\mathfrak{sl}(2,\mathbb{R})$ symmetry.
The Casimir is $-\mathcal{J}_{0}^{2}+\mathcal{J}_{+}\mathcal{J}_{-}=-\mathcal{L}_{2}^{2}$.
•
The not-depending explicitly on time
integrals of motion $\mathcal{L}_{2}$ and $\mathcal{L}_{\pm}=\mathcal{L}_{3}\pm i\mathcal{L}_{1}$
generate the $\mathfrak{su}(2)$ algebra (3.39) with the Casimir invariant
$\mathcal{L}_{1}^{2}+\mathcal{L}_{2}^{2}+\mathcal{L}_{3}^{2}=\mathcal{J}_{0}^{2%
}=\frac{1}{4\omega^{2}}H_{\text{os}}^{2}$.
•
Each triplet of integrals $(\mathcal{L}_{2},\mathcal{B}_{j}^{-})$, $(\mathcal{L}_{2},\mathcal{B}_{j}^{+})$,
$(p_{\varphi},b_{j}^{-})$ and $(p_{\varphi},b_{j}^{+})$
generate Euclidean sub-algebra $\mathfrak{e}(2)$. The corresponding Casimirs
$\mathcal{B}_{1}^{-}\mathcal{B}_{2}^{-}$, $\mathcal{B}_{1}^{+}\mathcal{B}_{2}^{+}$,
$b_{1}^{-}b_{2}^{-}$, and $b_{1}^{+}b_{2}^{+}$
are $\mathcal{J}_{-}^{2}$, $\mathcal{J}_{+}^{2}$, $\mathcal{J}_{-}$ and
$\mathcal{J}_{+}$, respectively.
•
The sets of integrals
($\mathscr{J}_{\pm}^{(+)}=\mathcal{B}_{1}^{\pm}/\sqrt{2}$, $\mathscr{J}_{0}^{(+)}=\mathcal{J}_{0}+\mathcal{L}_{2}$),
and
($\mathscr{J}_{\pm}^{(-)}=\mathcal{B}_{2}^{\pm}/\sqrt{2}$, $\mathscr{J}_{0}^{(-)}=\mathcal{J}_{0}-\mathcal{L}_{2}$),
generate the
$\mathfrak{sl}(2,\mathbb{R})\oplus\mathfrak{sl}(2,\mathbb{R})$ algebra.
Note that generators $\mathscr{J}_{0}^{\pm}$ can be
reinterpreted as
the Landau problem’s
Hamiltonians in the symmetric gauge, with
positive/negative magnetic field of the magnitude
$B=\frac{2cm\omega}{q}$, where $q$ and $c$ correspond to
the electric charge of the particle and the speed of light
[55].
•
Each integral is an eigenstate of $i\mathcal{J}_{0}$
in the sense of $\{i\mathcal{J}_{0},A\}=\lambda A$ :
$(\lambda=\pm 1:\mathcal{J}_{\pm},\mathcal{B}_{j}^{\pm})$,
$(\lambda=0:\mathcal{J}_{0},\mathcal{L}_{2},\mathcal{L}_{\pm})$,
$(\lambda=\pm 1/2:b_{j}^{\pm})$.
•
Analogously, each integral is an eigenstate of $i\mathcal{L}_{2}$:
$(\lambda=0:\mathcal{L}_{2},\mathcal{J}_{0},\mathcal{J}_{\pm})$,
$(\lambda=\pm 1:\mathcal{L}_{\pm},\mathcal{B}_{j}^{\pm})$,
$(\lambda=\pm 1/2:b_{1}^{\mp},b_{2}^{\pm})$.
Using the dynamical integrals $\mathcal{J}_{\pm}$ and integral
$\mathcal{J}_{0}$,
we define the generators of the Newton-Hooke conformal symmetry
[51, 52, 53, 54, 55]
$$\displaystyle\mathcal{K}=\frac{1}{2\omega}(\mathcal{J}_{+}+\mathcal{J}_{-}+2%
\mathcal{J}_{0})=\frac{1}{2}mr^{2}\cos(2\omega t)-\frac{1}{2\omega}rp_{r}\sin(%
2\omega t)+\frac{1}{\omega^{2}}H_{\text{os}}\sin^{2}(\omega t)\,,$$
(3.49)
$$\displaystyle\mathcal{D}=\frac{1}{2i}(\mathcal{J}_{-}-\mathcal{J}_{+})=\frac{1%
}{2}rp_{r}\cos(2\omega t)-\frac{1}{2\omega}(H_{\text{os}}-m\omega^{2}r^{2})%
\sin(2\omega t)\,.$$
(3.50)
Together with $H_{\text{os}}=2\omega\mathcal{J}_{0}$
they satisfy the Poisson bracket relations
$$\displaystyle\{\mathcal{D},H_{os}\}=H_{\text{os}}-2\omega^{2}\mathcal{K}\,,%
\qquad\{\mathcal{D},\mathcal{K}\}=-\mathcal{K}\,,\qquad\{H_{\text{os}},%
\mathcal{K}\}=2\mathcal{D}\,,$$
(3.51)
and in the limit $\omega\rightarrow 0$ take the form
of the free particle integrals (3.1).
From here, we obtain
$$\displaystyle r^{2}(t)=\frac{1}{m\omega^{2}}\Big{(}H_{\text{os}}+2\omega%
\mathcal{D}\sin(2\omega t)+(2\omega^{2}\mathcal{K}-H_{\text{os}})\cos(2\omega t%
)\Big{)}\,.$$
(3.52)
Taking into account the equivalent form
$\mathcal{D}^{2}+\omega^{2}\mathcal{K}^{2}-\mathcal{K}H_{\text{os}}=-\frac{1}{4%
}p_{\varphi}^{2}$
for the Casimir of the conformal
$\mathfrak{sl}(2,\mathbb{R})$ symmetry, equation (3.52)
allows us to find the radial turning points
$$\displaystyle r_{\pm}^{2}=\frac{H_{\text{os}}}{m\omega^{2}}\left(1\pm\delta%
\right)\,,\qquad\delta=\sqrt{1-\frac{\omega^{2}p_{\varphi}^{2}}{H_{\text{os}}^%
{2}}}\,,$$
(3.53)
which also can be found directly from the Hamiltonian
by applying the condition $p_{r}=0$.
On the other hand, by using the explicit form of the integrals $\mathcal{L}_{1}$ and $\mathcal{L}_{3}$ in polar coordinates,
$$\displaystyle\mathcal{L}_{1}=\sin(2\varphi)\left(\frac{1}{2\omega}H_{\text{os}%
}-\frac{1}{2m\omega r^{2}}p_{\varphi}^{2}\right)-\cos(2\varphi)\frac{1}{2m%
\omega r}p_{r}p_{\varphi}\,,$$
(3.54)
$$\displaystyle\mathcal{L}_{3}=\cos(2\varphi)\left(\frac{1}{2\omega}H_{\text{os}%
}-\frac{1}{2m\omega r^{2}}p_{\varphi}^{2}\right)+\sin(2\varphi)\frac{1}{2m%
\omega r}p_{r}p_{\varphi}\,,$$
(3.55)
one deduces the elliptic
trajectory for the isotropic planar harmonic oscillator,
$$\displaystyle r^{2}(\varphi)=\frac{r_{0}^{2}}{1+\delta\cos(2(\varphi-\varphi_{%
*}))}\,,\qquad r_{0}^{2}=\frac{p_{\varphi}^{2}}{mH_{\text{os}}}\,,$$
(3.56)
where $\varphi=\varphi_{*}$ corresponds to the angular position of one of the two
“perihelia” of the trajectory. Eq. (3.56) has a form of elliptic trajectory in Kepler
problem but with $r(\varphi)$ and $\varphi$ there changed for $r^{2}(\varphi)$ and
$2\varphi$ here.
By means of (3.52) and (3.56) we also get
$$\displaystyle\varphi(t)=\alpha\arctan(\frac{r_{+}}{r_{-}}\tan(\omega(t-t_{*}))%
)+\varphi_{*}\,,$$
(3.57)
where $t_{*}$ indicates the moment of time when the particle is in the corresponding perihelion.
Now let’s take a look at the quantum case. We first consider the Cartesian coordinates representation,
then we present the picture in the polar coordinates representation, and finally we
show how these
two representations are related to each other by a unitary transformation.
Cartesian coordinates representation
The quantum versions of (3.33) (at $t=0$) are given by
$$\displaystyle\hat{a}_{i}^{\pm}=\sqrt{\frac{m\omega}{2\hbar}}\left(x_{i}\mp%
\frac{\hbar}{m\omega}\frac{\partial}{\partial x_{i}}\right)\,,\qquad[\hat{a}_{%
i}^{-},\hat{a}_{j}^{+}]=1\,.$$
(3.58)
In terms of them, we construct the set of operators
(no summation in the repeated index),
$$\displaystyle\hat{\mathcal{J}}_{0}^{i}=\frac{1}{2\omega\hbar}\hat{H}_{i}=\frac%
{1}{4}(\hat{a}_{i}^{+}\hat{a}_{i}^{-}+\hat{a}_{i}^{-}\hat{a}_{i}^{+})\,,\qquad%
\hat{\mathcal{J}}^{i}_{\pm}=\frac{1}{2}(\hat{a}_{i}^{\pm})^{2}\,,\qquad i=1,2\,,$$
(3.59)
$$\displaystyle\hat{\mathcal{L}}_{1}=\frac{1}{2}(\hat{a}_{1}^{+}\hat{a}_{2}^{-}+%
\hat{a}_{2}^{+}\hat{a}_{1}^{-})\,,\qquad\hat{\mathcal{L}}_{2}=\frac{i}{2}(\hat%
{a}_{1}^{+}\hat{a}_{2}^{-}-\hat{a}_{2}^{+}\hat{a}_{1}^{-})=\frac{1}{2\hbar}%
\hat{p}_{\varphi}\,,$$
(3.60)
$$\displaystyle\hat{\mathcal{L}}_{3}=\frac{1}{2}(\hat{a}_{1}^{+}\hat{a}_{1}^{-}-%
\hat{a}_{2}^{+}\hat{a}_{2}^{-})=\hat{\mathcal{J}}_{0}^{1}-\hat{\mathcal{J}}_{0%
}^{2}\,,\qquad\hat{\mathcal{A}}_{\pm}=\hat{a}_{1}^{\pm}\hat{a}_{2}^{\pm}\,.$$
(3.61)
These ten operators satisfy the Lie algebra which (up to a unitary transformation) corresponds
to the quantum version of the classical
algebra (3.38)-(3.48). In this
representation, the set of physical eigenstates and the spectrum of the system are
$$\displaystyle\psi_{n_{1},n_{2}}(x_{1},x_{2})=\psi_{n_{1}}(x_{1})\psi_{n_{2}}(x%
_{2})\,,\qquad E_{n_{1},n_{2}}=\hbar(n_{1}+n_{2}+1)\,,\quad n_{1,2}=0,1,\ldots,$$
(3.62)
$$\displaystyle\psi_{n_{i}}(x_{i})=\frac{1}{\sqrt{2^{n}n!}}\left(\frac{m\omega}{%
\pi\hbar}\right)^{\frac{1}{4}}H_{n_{i}}\left(\sqrt{\frac{m\omega}{\hbar}}x_{i}%
\right)e^{-\frac{m\omega}{2\hbar}x_{i}^{2}}\,,$$
(3.63)
where $H_{n_{i}}$ is the Hermite polynomial of order $n_{i}$.
These wave-functions diagonalize simultaneously the operators
$\hat{\mathcal{J}}_{0}^{i}$,
$i=1,2$,
and, as a consequence, $\hat{H}_{\text{os}}$ and $\hat{\mathcal{L}}_{3}$.
Polar coordinates representation
The quantum versions of the classical dynamical
integrals (3.34),(3.35), (3.36) at $t=0$, and of the
integrals (3.37) are
$$\displaystyle\hat{b}_{1}^{-}=\frac{1}{\sqrt{2}}(\hat{a}_{1}^{-}-i\hat{a}_{2}^{%
-})=\frac{1}{2}e^{-i\varphi}\sqrt{\frac{m\omega}{\hbar}}\left(r+\frac{\hbar}{m%
\omega}\left(\frac{\partial}{\partial r}-\frac{i}{r}\frac{\partial}{\partial%
\varphi}\right)\right)\,,\qquad\hat{b}_{1}^{+}=(\hat{b}_{1}^{-})^{\dagger}\,,$$
(3.64)
$$\displaystyle\hat{b}_{2}^{-}=\frac{1}{\sqrt{2}}(a_{1}^{-}+ia_{2}^{-})=\frac{1}%
{2}e^{-i\varphi}\sqrt{\frac{m\omega}{\hbar}}\left(r+\frac{\hbar}{m\omega}\left%
(\frac{\partial}{\partial r}+\frac{i}{r}\frac{\partial}{\partial\varphi}\right%
)\right)\,,\qquad\hat{b}_{2}^{+}=(\hat{b}_{2}^{-})^{\dagger}\,,$$
(3.65)
$$\displaystyle\hat{\mathcal{J}}_{0}=\frac{1}{2\hbar\omega}\hat{H}_{\text{os}}=%
\frac{1}{4}([\hat{b}_{1}^{+},\hat{b}_{1}^{-}]_{{}_{+}}+[\hat{b}_{2}^{+},\hat{b%
}_{2}^{-}]_{{}_{+}})\,,\qquad\hat{\mathcal{L}}_{2}=(\hat{b}_{1}^{+}\hat{b}_{1}%
^{-}-\hat{b}_{2}^{+}\hat{b}_{2}^{-})=\frac{1}{2\hbar}\hat{p}_{\varphi}\,,$$
(3.66)
$$\displaystyle\hat{\mathcal{L}}_{\pm}=\hat{b}_{1}^{\pm}\hat{b}_{2}^{\mp}=\hat{%
\mathcal{L}}_{3}\pm i\hat{\mathcal{L}}_{1}=\frac{m\omega}{4\hbar}e^{\pm 2i%
\varphi}\left(r^{2}+\frac{\hbar^{2}}{m^{2}\omega^{2}}\left(\frac{1}{r^{2}}%
\frac{\partial^{2}}{\partial\varphi^{2}}-\frac{\partial^{2}}{\partial r^{2}}%
\pm i\frac{2}{r}\frac{\partial^{2}}{\partial r\partial\varphi}\right)\right)\,,$$
(3.67)
$$\displaystyle\hat{\mathcal{J}}_{\pm}=\hat{b}_{1}^{\pm}\hat{b}_{2}^{\pm}=-\frac%
{m\omega}{4\hbar}\left(\hat{H}_{\text{os}}-m\omega^{2}r^{2}\pm\hbar\omega(%
\frac{\partial}{\partial r}+1)\right)\,,\qquad\hat{\mathcal{B}}_{j}^{\pm}=(%
\hat{b}_{j}^{\pm})^{2}\,,$$
(3.68)
where $[,]_{{}_{+}}$
is the anti-commutator of the operators.
In this representation we diagonalize simultaneously
$\hat{H}_{\text{os}}=2\hbar\omega\hat{\mathcal{J}}_{0}$ and $\hat{p}_{\varphi}=2\hbar\hat{\mathcal{L}}_{2}$.
The eigenstates and spectrum are
$$\displaystyle\psi_{n_{r},l}^{\pm}(r,\varphi)=\left(\frac{m\omega}{\hbar}\right%
)^{\frac{1}{2}}\sqrt{\frac{n_{r}!}{2\pi\Gamma(n_{r}+l+1)}}\,\zeta^{l}L_{n_{r}}%
^{(l)}(\zeta^{2})e^{-\frac{\zeta^{2}}{2}\pm il\varphi}\,,\qquad\zeta=\sqrt{%
\frac{m\omega}{\hbar}}r\,,$$
(3.69)
$$\displaystyle E_{n,l}=\hbar\omega(2n_{r}+l+1)\,,\qquad n_{r}\,,l=0,1,\ldots\,,$$
(3.70)
where $L_{n_{r}}^{(l)}$ are the Laguerre polynomials.
Unitary transformation
The two representations corresponding to Cartesian and polar coordinates
can be related by the unitary operator
$$\displaystyle\hat{U}=\exp(-i\frac{2\pi}{3}\frac{1}{\sqrt{3}}(\hat{\mathcal{L}}%
_{1}+\hat{\mathcal{L}}_{2}+\hat{\mathcal{L}}_{3}))\,,$$
(3.71)
which produces the rotation of the $\mathfrak{su}(2)$ generators in the
“ambient
three-dimensional space”
for the angle $-2\pi/3$,
$$\hat{U}\hat{\mathcal{L}}_{1}\hat{U}^{\dagger}=\hat{\mathcal{L}}_{3}\,,\qquad%
\hat{U}\hat{\mathcal{L}}_{2}\hat{U}^{\dagger}=\hat{\mathcal{L}}_{1}\,,\qquad%
\hat{U}\hat{\mathcal{L}}_{3}\hat{U}^{\dagger}=\hat{\mathcal{L}}_{2}\,.$$
(3.72)
Then, from
$\hat{\mathcal{L}}_{3}\psi_{n_{1},n_{2}}=\frac{n_{1}-n_{2}}{2}\psi_{n_{1},n_{2}}$
we obtain
$\hat{\mathcal{L}}_{2}\hat{U}\psi_{n_{1},n_{2}}=\frac{n_{1}-n_{2}}{2}\hat{U}%
\psi_{n_{1},n_{2}}\,,$
and $\hat{U}\psi_{n_{1},n_{2}}$ are the states that diagonalize
the Hamiltonian $\hat{H}_{\text{os}}=2\hbar\omega\hat{\mathcal{J}}_{0}$ and
the operator
$\hat{p}_{\varphi}=2\hbar\hat{\mathcal{L}}_{2}$ in the
polar coordinates representation.
Acting on operators $\hat{a}_{i}^{\pm}$, the unitary transformation produces
$$\displaystyle\hat{U}\hat{a}_{j}^{\pm}\hat{U}^{\dagger}=e^{\pm i\frac{\pi}{4}}%
\hat{b}_{j}^{\pm}\,,$$
(3.73)
that corresponds to the spinor nature of the basic
integrals with respect to the action of the $\mathfrak{su}(2)$ generators
in the
ambient
three-dimensional space.
In the next section we show how these two different physical systems,
the free particle and harmonic oscillator in $\mathbb{R}^{2}$,
are related to each other by the
conformal bridge transformation [55]. This will serve as
a precedent
for
the procedure that we will use to extract all the information for a
system with the harmonic potential in conical geometry from the free
dynamics on
the same geometric background.
4 The conformal bridge transformation
In the general case, a mechanical system is governed by a symmetry algebra which encodes
its peculiarities. At the classical level, these
symmetries can be related with the geometric
properties of the trajectory,
and at the quantum level they encode the information related with the energy
spectrum.
According to Dirac [65],
a given symmetry algebra (up to isomorphisms) can represent different mechanical systems.
In this section we show how the two forms of dynamics
associated with the conformal algebra $\mathfrak{so}(2,1)\cong\mathfrak{sl}(2,\mathbb{R})$
are related to each other by means of a particular mapping,
the so-called conformal bridge transformation [55].
This also will allow us
to establish the interesting relation between the corresponding
symmetry sub-algebras of the free particle
and harmonic oscillator systems and
understand the change of nature of some of them
due to a non-unitarity of the conformal bridge
transformation.
Consider the classical $\mathfrak{so}(2,1)$ algebra
$$\{D,H\}=H\,,\qquad\{D,K\}=-K\,,\qquad\{K,H\}=2D\,,$$
(4.1)
without specifying the form of the generators themselves.
Now, let us introduce
the following complex linear combinations of them,
$$\mathcal{J}_{0}=\frac{1}{2}(\omega^{-1}H+\omega K)\,,\qquad\mathcal{J}_{\pm}=-%
\frac{1}{2\omega}(H-\omega^{2}K\pm i2\omega D)\,,$$
(4.2)
where $\omega$ is a constant that is introduced to compensate the dimensions of the generators.
These new complex quantities satisfy the classical $\mathfrak{sl}(2,\mathbb{R})$ algebra
$$\{\mathcal{J}_{0},\mathcal{J}_{\pm}\}=\mp i\mathcal{J}_{\pm}\,,\qquad\{%
\mathcal{J}_{-},\mathcal{J}_{+}\}=-2i\mathcal{J}_{0}\,.$$
(4.3)
Independently, both algebraic structures can represent different physical systems.
In the case where $H$ is the Hamiltonian of a certain
asymptotically free model with conformal symmetry (the free particle, for example),
we have that $K$ and $D$ are dynamical integrals that explicitly depend on $t$.
From here one has
$$\displaystyle K=T_{H}(t)(K_{0})\,,\qquad K_{0}=T_{H}(-t)(K)=K|_{t=0}\,,$$
(4.4)
$$\displaystyle D=T_{H}(t)(D_{0})\,,\qquad D_{0}=T_{H}(-t)(D)=D|_{t=0}\,,$$
(4.5)
where $T_{H}(\pm t)$ denotes a Hamiltonian flow, which is a canonical transformation
generated by the Hamiltonian itself.
The flow generated by a phase space function $F$ is
$$\displaystyle\exp(\gamma F)\star f(q,p):=f(q,p)+\sum_{n=1}^{\infty}\frac{%
\gamma^{n}}{n!}\{F,\{\ldots,\{F,f\underbrace{\}\ldots\}\}}_{n}=:T_{F}(\gamma)(%
f)\,.$$
(4.6)
In the same way, the compact generator $2\omega\mathcal{J}_{0}$
(ignoring the dependence on $t$ of $F$ in the definition (4.2)) can be interpreted as the
Hamiltonian of a harmonically trapped system (the harmonic oscillator, for example)
with frequency $\omega$, and the quantities $\mathcal{J}_{\pm}$ are its dynamical integrals that satisfy
$$\mathcal{J}_{\pm}=T_{2\omega\mathcal{J}_{0}}(\tau)(\mathcal{J}_{\pm}|_{t=0})\,%
,\qquad\mathcal{J}_{\pm}|_{t=0}=T_{2\omega\mathcal{J}_{0}}(-\tau)(\mathcal{J}_%
{\pm})$$
(4.7)
Both forms of dynamics are related to each other by the complex canonical transformation
$$\mathscr{T}(\tau,\beta,\delta,\gamma,t)=T_{2\omega\mathcal{J}_{0}}(\tau)\circ T%
_{\beta\delta\gamma}\circ T_{H}(-t)\,,$$
(4.8)
where
$$\displaystyle T_{\beta\delta\gamma}:=T_{K_{0}}(\beta)\circ T_{H}(\delta)\circ T%
_{D_{0}}(\gamma)=T_{K_{0}}(\delta)\circ T_{D_{0}}(\gamma)\circ T_{H}(2\delta)\,,$$
(4.9)
$$\displaystyle\delta=\frac{i}{2\omega}\,,\qquad\beta=-i\omega\,,\qquad\gamma=-%
\ln 2\,.$$
(4.10)
In this composition, the first transformation $T_{H}(-t)$ removes
the $t$ dependence in the dynamical integrals $D$ and $K$.
The second transformation
relates these $t=0$ generators with the generators of the $\mathfrak{sl}(2,\mathbb{R})$ algebra $\mathcal{J}_{0}$
and $\mathcal{J}_{\pm}$ taken at $\tau=0$.
The last transformation $T_{2\omega\mathcal{J}_{0}}(\tau)$ restores the $\tau$ dependence.
Explicitly one has
$$\displaystyle\mathscr{T}(\tau,\beta,\delta,\gamma,t)(H)=-\omega\mathcal{J}_{-}%
\,,\qquad\mathscr{T}(\tau,\beta,\delta,\gamma,t)(D)=-i\mathcal{J}_{0}\,,$$
(4.11)
$$\displaystyle\mathscr{T}(\tau,\beta,\delta,\gamma,t)(K)=\frac{1}{\omega}%
\mathcal{J}_{+}\,.$$
(4.12)
The corresponding inverse transformation is given by
$$\displaystyle(\mathscr{T}(\tau,\beta,\delta,\gamma,t))^{-1}=T_{H}(t)\circ(T_{%
\beta\delta\gamma})^{-1}\circ T_{2\omega\mathcal{J}_{0}}(-\tau)\,,$$
(4.13)
$$\displaystyle(T_{\beta\delta\gamma})^{-1}=T_{D_{0}}(-\gamma)\circ T_{H}(-%
\delta)\circ T_{K_{0}}(-\beta)\,.$$
(4.14)
This transformation is
a generalization of
the classical version of the quantum conformal bridge transformation introduced in [55],
and corresponds to an automorphism of algebra since it relates
the Wick rotated non-compact generator $iD$ and
compact generator $\mathcal{J}_{0}$ to each other,
the real-valued non-compact generators $H$ and $K$ with the non-compact
complex-valued
generators $-\omega\mathcal{J}_{-}$ and $\frac{1}{\omega}\mathcal{J}_{+}$, respectively.
In the particular case of the free particle
in the Euclidean space the generators of the $\mathfrak{so}(2,1)$
algebra are specified in (3.1).
Then the generators of the $\mathfrak{sl}(2,\mathbb{R})$ algebra
resulting from applying the transformation
(4.8)
are given by the planar harmonic oscillator generators $\mathcal{J}_{0}$ and $\mathcal{J}_{\pm}$
defined in (3.36)-(3.37).
The transformation also allows us to map the remaining symmetry generators
of the free particle system into the
corresponding generators of the harmonic oscillator.
In particular, by direct application of the transformation
to the generators $p_{\pm}$ and $\xi_{\pm}$ we get
$$\displaystyle\mathscr{T}(\tau,\beta,\delta,\gamma,t)(p_{-})=-i\sqrt{2m\omega}%
\,b_{1}^{-}\,,\qquad\mathscr{T}(\tau,\beta,\delta,\gamma,t)(p_{+})=-i\sqrt{2m%
\omega}\,b_{2}^{-}\,,$$
(4.15)
$$\displaystyle\mathscr{T}(\tau,\beta,\delta,\gamma,t)(\xi_{+})=\sqrt{\frac{2m}{%
\omega}}\,b_{1}^{+}\,,\qquad\mathscr{T}(\tau,\beta,\delta,\gamma,t)(\xi_{-})=%
\sqrt{\frac{2m}{\omega}}\,b_{2}^{+}\,,$$
(4.16)
where the relations (3.44)-(3.48) were used.
In the same vein,
we list the effects of the transformation on the remaining second order generators,
$$\displaystyle\mathscr{T}(\tau,\beta,\delta,\gamma,t)(J_{0})=\mathcal{L}_{2}\,,%
\qquad\mathscr{T}(\tau,\beta,\delta,\gamma,t)(J_{\pm})=-i\mathcal{L}_{\pm}\,,$$
(4.17)
$$\displaystyle\mathscr{T}(\tau,\beta,\delta,\gamma,t)(S_{+})=\frac{1}{\omega}%
\mathcal{B}_{1}^{+}\,,\qquad\mathscr{T}(\tau,\beta,\delta,\gamma,t)(T_{-})=-%
\omega\mathcal{B}_{1}^{-}\,,$$
(4.18)
$$\displaystyle\mathscr{T}(\tau,\beta,\delta,\gamma,t)(S_{-})=\frac{1}{\omega}%
\mathcal{B}_{2}^{+}\,,\qquad\mathscr{T}(\tau,\beta,\delta,\gamma,t)(T_{+})=-%
\omega\mathcal{B}_{2}^{-}\,.$$
(4.19)
In the following table we summarize the correspondence between generators and some subalgebras in both systems,
Note that the quantity $J_{0}=\mathcal{L}_{2}$ is the only object which is invariant under the transformation. This happens because
the angular momentum is the only conserved quantity which Poisson commutes with all the
$\mathfrak{sl}(2,\mathbb{R})$ conformal symmetry generators of the free particle.
On the other hand, the second $\mathfrak{sl}(2,\mathbb{R})$
subalgebra
is changed for the $\mathfrak{su}(2)$ symmetry algebra
after the transformation, and this is due to
the imaginary unit appearing in the second equation in
(4.17).
In the same vein, the two copies of the $\mathfrak{su}(2)$
algebra of the free particle with generators related
by the complex conjugation are mapped into the two copies
of the $\mathfrak{sl}(2,\mathbb{R})$ algebra associated with the Landau problem
due to the appearance of the minus sign
on the right hand side in the transformations of $T_{\pm}$
in (4.18) and (4.19).
The quantum analogue of the conformal bridge transformation corresponds to the non-unitary transformation produced by the operators
$$\displaystyle\hat{\mathfrak{S}}(t,\tau)=e^{-\frac{i}{\hbar}2\omega\hat{%
\mathcal{J}}_{0}\tau}e^{-\frac{\omega}{\hbar}\hat{K}_{0}}e^{\frac{\hat{H}}{2%
\hbar\omega}}e^{\frac{i}{\hbar}\ln(2)\hat{D}_{0}}e^{\frac{i}{\hbar}\hat{H}t}\,,$$
(4.20)
$$\displaystyle\hat{\mathfrak{S}}^{-1}(t,\tau)=e^{-\frac{i}{\hbar}\hat{H}t}e^{-%
\frac{i}{\hbar}\ln(2)\hat{D}_{0}}e^{-\frac{\hat{H}}{2\hbar\omega}}e^{\frac{%
\omega}{\hbar}\hat{K}_{0}}e^{\frac{i}{\hbar}2\omega\hat{\mathcal{J}_{0}}\tau}\,.$$
(4.21)
In this context, the Hamiltonian
flux is changed by employing the
Baker-Campbell-Hausdorff formula,
and in correspondence with
the established relation
between the two-dimensional free particle and the planar isotropic
harmonic oscillator in the Euclidean space, the classical relations
(4.11)-(4.12) and (4.15)-(4.19)
are preserved at the quantum level111We have considered dimensionless operators
for the harmonic oscillator at the quantum level in the previous section. To recover
these generators by the conformal bridge transformation
we must compensate
the multiplicative constants that appear in the quantum versions of the above-mentioned
relationships. The relations involving
the momenta operators linearly (quadratically) are multiplied by $\hbar^{1/2}$ ($\hbar$).
This is
taken into account in equations (4.20), (4.21).
.
In what concerns to eigenstates, in the general case
the transformation implies that
$$\displaystyle\hat{D}\ket{\lambda}=i\hbar\lambda\ket{\lambda}\quad\Rightarrow%
\quad\hat{\mathcal{J}}_{0}(\hat{\mathfrak{S}}\ket{\lambda})=\lambda\hat{%
\mathfrak{S}}\ket{\lambda}\,,$$
(4.22)
$$\displaystyle\hat{H}\ket{E}=E\ket{E}\quad\Rightarrow\quad\hat{\mathcal{J}}_{-}%
(\hat{\mathfrak{S}}\ket{E})=-\frac{E}{\hbar\omega}\hat{\mathfrak{S}}\ket{E}\,.$$
(4.23)
This means that the formal eigenstates
of the dilatation operator with imaginary eigenvalue are mapped to the
energy eigenstates of the harmonically confined system,
while
the asymptotic plane wave eigenstates of
the
asymptotically free Hamiltonian
$\hat{H}$ correspond to coherent states of
the system with the
additional harmonic potential term, which, in turn, are eigenstates of
the quadratic lowering operator.
Note that to have the physically acceptable solutions, the states
$\hat{\mathfrak{S}}\ket{\lambda}$ must be normalizable, and to ensure
this,
the three requirements must be met. First,
the series
$$\displaystyle e^{\frac{\hat{H}}{2\hbar\omega}}\ket{\lambda}=\sum_{n=0}^{\infty%
}\frac{1}{n!(2\hbar\omega)^{n}}(\hat{H})^{n}\ket{\lambda}\,,$$
(4.24)
has to reduce to a finite number of terms, i.e., $\ket{\lambda}$ should
be the Jordan states222The Jordan states are given by wave functions
that
satisfy $P(\hat{H})\Omega_{\lambda}=\psi_{\lambda}$, where $\hat{H}\psi_{\lambda}=\lambda\psi_{\lambda}$ and $P(\eta)$
represents a polynomial
[66, 67].
Here
we consider Jordan states satisfying
the relations $(\hat{H})^{\ell}\Omega_{\lambda}=\lambda\psi_{\lambda}$ with $\lambda=0$ for a certain
natural $\ell$. of $\hat{H}$ corresponding to zero energy [55].
Second, the function $\innerproduct{\boldmathe{r}}{\lambda}$ must not have singularities in the corresponding
operators domain.
And finally, these functions must be single-valued with respect to the angular coordinate.
In conclusion of this section, let us apply
the inverse conformal bridge
(similarity) transformation
to
the unitary
operator
(3.71). This yields us
the non-unitary operator
$$\displaystyle\hat{W}=\exp(-i\frac{2\pi}{3}\frac{1}{\sqrt{3}\hbar}(\hat{J}_{0}+%
i(\hat{J}_{1}-i\hat{J}_{2})))\,,$$
(4.25)
and from equations (3.73) one obtains the transformation relations
$$\displaystyle\hat{W}(\hat{\xi}_{1})\hat{W}^{-1}=\frac{1}{\sqrt{2}}e^{i\frac{%
\pi}{4}}\hat{\xi}_{+}\,,\qquad\hat{W}(\hat{\xi}_{2})\hat{W}^{-1}=\frac{1}{%
\sqrt{2}}e^{i\frac{\pi}{4}}\hat{\xi}_{-}\,,$$
(4.26)
$$\displaystyle\hat{W}(\hat{p}_{1})\hat{W}^{-1}=\frac{1}{\sqrt{2}}e^{-i\frac{\pi%
}{4}}\hat{p}_{+}\,,\qquad\hat{W}(\hat{p}_{2})\hat{W}^{-1}=\frac{1}{\sqrt{2}}e^%
{-i\frac{\pi}{4}}\hat{p}_{-}\,.$$
(4.27)
Equations (4.26), (4.27)
provide us with the
similarity transform of the Cartesian operators
$\hat{\xi}_{j}$ and $\hat{p}_{j}$
into the complex operators
$\hat{\xi}_{\pm}$ and $\hat{p}_{\pm}$ used
in the polar coordinates representation for the quantum free particle.
The conformal symmetry generators
of the free particle, $\hat{H}$, $\hat{D}$ and $\hat{K}$,
commute with the operator (4.25).
In the next section we finally pass over to the
study of the free
particle
dynamics in conical geometry.
Since the system possesses the conformal $\mathfrak{sl}(2,\mathbb{R})$ and rotational symmetries
for any
value of $\alpha$, then it will be possible to apply
the conformal bridge transformation
to analyze the dynamics of
the harmonic oscillator in the same geometry.
5 Free motion in a cosmic string background
As we have shown in Sec. 2,
the study of the dynamics in a cosmic string background
is analogous to
analyzing
the motion of a particle in conical geometry. A classical system in this space is governed
by the action
$$\displaystyle I=\int Ldt\,,\qquad L=\frac{m}{2}g_{ij}\frac{dx_{i}}{dt}\frac{dx%
_{j}}{dt}-V(\boldmathe{r})=\frac{m}{2}\left(\alpha^{2}\dot{r}^{2}+r^{2}\dot{%
\varphi}^{2}\right)-V(\boldmathe{r})\,.$$
(5.1)
In this section, we study the case of the
free motion ($V(\boldmathe{r})=0$) from the perspective of its symmetries.
We will show that
when $\alpha$ is a rational number, it is possible to construct higher-order
globally well-defined classical integrals of motion
that can be identified as generators of hidden symmetries [1].
Then
we study the system at the quantum level,
and show that the only cases in which these conserved
quantities can be promoted to the well-defined symmetry
operators
correspond to
integer values of $\alpha$.
Thus, we reveal here a kind of a quantum anomaly
in the case of
rational, non-integer values of $\alpha$.
5.1 Classical case
The classical dynamics of a free particle in a
conical geometry is governed by the Hamiltonian
$$\displaystyle H^{(\alpha)}=\frac{1}{2m}\left(\frac{p_{r}^{2}}{\alpha^{2}}+%
\frac{p_{\varphi}^{2}}{r^{2}}\right)\,.$$
(5.2)
Remarkably, the canonical transformation
$$\displaystyle r\rightarrow\alpha r\,,\qquad p_{r}\rightarrow\frac{p_{r}}{%
\alpha}\,,\qquad\varphi\rightarrow\frac{\varphi}{\alpha}\,,\qquad p_{\varphi}%
\rightarrow\alpha p_{\varphi}\,,$$
(5.3)
applied
to the Hamiltonian of
the Euclidean free particle
gives us the
Hamiltonian (5.2).
Also note that when we apply this transformation to the usual Cartesian coordinates
we get the “regularized” Cartesian coordinates (2.10). From the analysis
of Sec. 2
related to
those coordinates
it is clear that the canonical transformation
(5.3)
is well-defined only locally.
In spite of the indicated deficiency,
we can use the canonical transformation (5.3) to reconstruct
the solutions of the equations of motion
of the system
(5.2). The trajectory equation,
as well as the time dependence of the radial and angular variables
are immediately obtained from the corresponding
relations (3.18), (3.19) and (3.20)
for the free motion in the plane, and
are given by
$$\displaystyle r(\varphi)=\frac{r_{*}}{\cos\left((\varphi-\varphi_{*})/\alpha%
\right)}\,,\qquad r_{*}=\frac{p_{\varphi}}{\sqrt{2mH^{(\alpha)}}}\,,\qquad-%
\frac{\pi}{2}\alpha\leq\varphi-\varphi_{*}\leq\frac{\pi}{2}\alpha\,.$$
(5.4)
$$\displaystyle r^{2}(t)=\frac{2}{\alpha^{2}m}\left(H^{(\alpha)}t^{2}+2Dt+K%
\right)\,,\quad\varphi(t)=\alpha\arctan\left(\frac{1}{2\rho}(t+\frac{D}{H^{(%
\alpha)}})\right)+\varphi_{*}\,,$$
(5.5)
where $\rho=\alpha\frac{p_{\varphi}}{2H^{(\alpha)}}\,.$
Here $H^{(\alpha)}$,
$D$, $K$
and $p_{\varphi}$
are the integrals being generators of the $\mathfrak{sl}(2,\mathbb{R})\oplus\mathfrak{u}(1)$ symmetry of the
system (5.2),
see Eqs. (5.8), (5.9) below,
and we see that the scattering angle is $\varphi_{\text{scat}}=\varphi(+\infty)-\varphi(-\infty)=\alpha\pi$ under assumption $p_{\varphi}>0$.
Some pictures of the trajectory for different values of $\alpha$ are displayed in Fig 1.
From these figures one sees that when $\alpha$ is an even number,
$\alpha=2\ell$, the particle experiences a backward scattering,
that corresponds to the scattering angle $2\pi\ell$, while when
$\alpha$ is odd, $\alpha=2\ell+1$, the scattering angle is $2\pi\ell+\pi$, and the
particle continues asymptotic motion
in the initial direction
after realizing $\ell$ revolutions around the origin being the vertex of the cone.
This was already observed in ref. [26]333Note
that our parameter $\alpha$ corresponds to $\alpha^{-1}$ in
notations of [26], where only the case of positive mass
density of the cosmic string (that corresponds to $\alpha>1$ values
of our parameter) was considered..
To reveal the locally rectilinear
character of geodesics in the case $\alpha>1$,
it is convenient to consider the shape of trajectories in a cut and flattened cone
represented by the angular sector $-\frac{\pi}{2}\alpha\leq\varphi\leq\frac{\pi}{2}\alpha$
(wedge)
in which the symmetric points on the edges $\varphi=-\frac{\pi}{2}\alpha$ and
$\varphi=\frac{\pi}{2}\alpha$
correspond to the cut line and must be identified.
This is illustrated by Fig. 2.
From the diagrams shown in Fig. 2
one reveals the following special features in the case of integer $\alpha$:
1.
If we look for the case in which the asymptotes of the trajectory are parallel to symmetry
axis (going through the cone’s vertex and the perihelion),
we must impose the condition $\phi_{1}+\varepsilon=\pi/2$ (where $\varepsilon=\phi_{n}$),
which can be fulfilled if $\alpha=4n$, with $n=1,2,\ldots$.
2.
When $\alpha=2(2n+1)$, one has that the first incidence
angle is given by $\pi/(2n+1)$, which coincides with $\varphi_{s}$.
This implies that the asymptotes of the path are parallel to the
edges.
This also includes the case $\alpha=2$ for which $n=0$ and the trajectory
suffers no reflections at the edges.
3.
When $\alpha=4n-1$, the first incidence
angle corresponds to $\frac{1}{4n-1}\frac{\pi}{2}=\frac{\varphi_{s}}{4}$.
When
$\alpha=4n+1$, the first incidence
angle is given by $\frac{3}{4n+1}\frac{\pi}{2}=\frac{3}{4}\varphi_{s}$.
Now, let’s pass over to constructing the symmetry generators of the model
by
applying the transformation (5.3) to those of the Euclidean free particle
analyzed
in Sec. 3.
We
start with the complex combinations of the
locally defined translations
and the Galilean boost generators,
$$\displaystyle\Pi_{\pm}=\Pi_{1}\pm i\Pi_{2}=\left(\frac{p_{r}}{\alpha}\pm i%
\frac{p_{\varphi}}{r}\right)e^{\pm i\frac{\varphi}{\alpha}}\,,$$
(5.6)
$$\displaystyle\Xi_{\pm}=\Xi_{1}\pm i\Xi_{2}=\left[\alpha mr-t\left(\frac{p_{r}}%
{\alpha}\pm i\frac{p_{\varphi}}{r}\right)\right]e^{\pm i\frac{\varphi}{\alpha}%
}\,.$$
(5.7)
Note that the presence of the factor $\alpha^{-1}$ in the integrals (5.6) and (5.7) implies that they
are globally
well-defined functions on the phase space only in the case of natural values of $\alpha^{-1}=k=1,2,\ldots$, and otherwise they are formal
objects.
In spite of this deficiency,
these quantities will serve as the basis for constructing the
globally well-defined in
the phase space symmetry generators of the system.
For
arbitrary values of $\alpha$ we have
$$\displaystyle H^{(\alpha)}=\frac{1}{2m}\Pi_{+}\Pi_{-}\,,\qquad D=\frac{1}{4m}(%
\Xi_{+}\Pi_{-}+\Pi_{+}\Xi_{+})=\frac{rp_{r}}{2}-H^{(\alpha)}t\,,$$
(5.8)
$$\displaystyle K=\frac{1}{2m}\Xi_{+}\Xi_{-}=\frac{m}{2}\alpha^{2}r^{2}-2Dt-H^{(%
\alpha)}t^{2}\,,\qquad J_{0}=\frac{i}{4m}(\Xi_{+}\Pi_{-}-\Pi_{+}\Xi_{+})=\frac%
{\alpha}{2}p_{\varphi}\,,$$
(5.9)
which are the well-defined angular-independent generators of the $\mathfrak{sl}(2,\mathbb{R})\oplus\mathfrak{so}(2)$
symmetry,
$$\displaystyle\{D,H^{(\alpha)}\}=H^{(\alpha)}\,,\qquad\{D,K\}=-K\,,\qquad\{K,H^%
{(\alpha)}\}=2D\,,$$
(5.10)
$$\displaystyle\{J_{0},H^{(\alpha)}\}=\{J_{0},D\}=\{J_{0},K\}=0\,,$$
(5.11)
with the $\mathfrak{sl}(2,\mathbb{R})$ subalgebra Casimir element $D^{2}-H^{(\alpha)}K=-J_{0}^{2}$.
For rational values $\alpha=q/k$,
$q,k=1,2,\ldots$, local integrals
(5.6)-(5.7)
can be used to construct the higher-order
globally well-defined
integrals
$$\displaystyle\mathcal{O}_{\mu,\nu}^{\pm}=(\Xi_{\pm})^{\mu}(\Pi_{\pm})^{\nu}\,,%
\qquad\mu=0,1,\ldots,q,\qquad\nu=q-\mu\,,$$
(5.12)
$$\displaystyle\mathcal{S}_{\mu^{\prime},\nu^{\prime}}^{\pm}=(\Xi_{\pm})^{\mu^{%
\prime}}(\Pi_{\pm})^{\nu^{\prime}}\,,\qquad\mu^{\prime}=0,1,\ldots,2q,\qquad%
\nu^{\prime}=2q-\mu^{\prime}\,.$$
(5.13)
There are $2(q+1)$ conserved quantities of the type $\mathcal{O}^{\pm}_{\mu,\nu}$
and $2(2q+1)$ conserved quantities of the type $\mathcal{S}^{\pm}_{\mu^{\prime},\nu^{\prime}}$,
which have the angular dependence of the form $e^{\pm ik\varphi}$ and
$e^{\pm i2k\varphi}$, respectively.
So, these
are globally well-defined functions in the phase space.
In the Euclidean case
$q=k=1$,
the integrals (5.12) correspond to
the generators of the two-dimensional
Heisenberg algebra (3.13),
while
the integrals (5.13)
constitute the set of the second-order
integrals
$J_{0}$, $J_{\pm}$, $T_{\pm}$, $S_{\pm}$
defined by Eqs. (3.2), (3.3).
In the case of $q=1$ and $k=2,3,\ldots$,
i.e. when $\alpha$ is a unit fraction $1/k$ with $k>1$,
the integrals
$\Pi_{\pm}$ and $\Xi_{\pm}$ are
well-defined functions in the corresponding
phase space, and the set
of integrals (5.12), (5.13) is similar to that
of the free particle in Euclidean plane.
Together with the conformal symmetry generators $H^{(\alpha)}$, $D$ and $K$,
they generate
the same
Lie
algebra
as for the free particle in the plane
since the canonical transformation (5.3)
does not change the Poisson bracket relations.
To characterize the symmetry algebra for $q>1$, which will be nonlinear in the general case,
it is necessary to calculate some Poisson bracket relations, and for this we use the identity
$\{A,B^{n}\}=n\{A,B\}B^{n-1}$
together with the Poisson bracket
relations
$$\displaystyle\{\Xi_{\pm},\Pi_{\mp}\}=2m\,,$$
(5.14)
$$\displaystyle\{H^{(\alpha)},\Xi_{\pm}\}=-\Pi_{\pm}\,,\qquad\{D,\Xi_{\pm}\}=-%
\frac{1}{2}\Xi_{\pm}\,,\qquad\{J_{0},\Xi_{\pm}\}=\mp\frac{i}{2}\Xi_{\pm}\,,$$
(5.15)
$$\displaystyle\{K,\Pi_{\pm}\}=\Xi_{\pm}\,,\qquad\{D,\Pi_{\pm}\}=\frac{1}{2}\Pi_%
{\pm}\,,\qquad\{J_{0},\Pi_{\pm}\}=\mp\frac{i}{2}\Pi_{\pm}\,.$$
(5.16)
For integrals $\mathcal{O}_{\mu,\nu}^{\pm}$ we have then
$$\displaystyle\{J_{0},\mathcal{O}_{\mu,\nu}^{\pm}\}=\mp i\frac{\mu+\nu}{2}%
\mathcal{O}_{\mu,\nu}^{\pm}\,,\qquad\{D,\mathcal{O}_{\mu,\nu}^{\pm}\}=\frac{%
\nu-\mu}{2}\mathcal{O}_{\mu,\nu}^{\pm}\,,$$
(5.17)
$$\displaystyle\{H^{(\alpha)},\mathcal{O}_{\mu,\nu}^{\pm}\}=-\mu\mathcal{O}_{\mu%
-1,\nu+1}^{\pm}\,,\quad\{K,\mathcal{O}_{\mu,\nu}^{\pm}\}=\nu\mathcal{O}_{\mu+1%
,\nu-1}^{\pm}\,,\quad\{\mathcal{O}_{\mu,\nu}^{\pm},\mathcal{O}_{\lambda,\sigma%
}^{\pm}\}=0\,,$$
(5.18)
and by using the Jacobi identity we get
$$\displaystyle\{J_{0},\{\mathcal{O}_{\mu,\nu}^{+},\mathcal{O}_{\lambda,\sigma}^%
{-}\}\}=0\,,\qquad\{D,\{\mathcal{O}_{\mu,\nu}^{+},\mathcal{O}_{\lambda,\sigma}%
^{-}\}\}=\frac{\nu-\mu+\sigma-\lambda}{2}\{\mathcal{O}_{\mu,\nu}^{+},\mathcal{%
O}_{\lambda,\rho}^{-}\}\,,$$
(5.19)
$$\displaystyle\{H^{(\alpha)},\{\mathcal{O}_{\mu,\nu}^{+},\mathcal{O}_{\lambda,%
\sigma}^{-}\}\}=-\lambda\{\mathcal{O}_{\mu,\nu}^{+},\mathcal{O}_{\lambda-1,%
\sigma+1}^{-}\}-\mu\{\mathcal{O}_{\mu-1,\nu+1}^{+},\mathcal{O}_{\lambda,\sigma%
}^{-}\})\,,$$
(5.20)
$$\displaystyle\{K,\{\mathcal{O}_{\mu,\nu}^{+},\mathcal{O}_{\lambda,\sigma}^{-}%
\}\}\}=\sigma\{\mathcal{O}_{\mu,\nu}^{+},\mathcal{O}_{\lambda+1,\sigma-1}^{-}%
\}+\nu\{\mathcal{O}_{\mu+1,\nu-1}^{+},\mathcal{O}_{\lambda,\sigma}^{-}\}\,.$$
(5.21)
From
relations
(5.19)
it follows that the
Poisson brackets
$\{\mathcal{O}_{\mu,\nu}^{+},\mathcal{O}_{\lambda,\sigma}^{-}\}$ must be functions of the
globally well-defined generators
$m$, $D$, $J_{0}$, $H^{(\alpha)}$
and $K$ since they are the only generators
Poisson-commuting
with $J_{0}$.
We do not treat these
functions
as new, independent
generators,
but consider them as coefficients of a nonlinear algebra. On the other hand,
all the generators $\mathcal{O}_{\mu,\nu}^{\pm}$ are eigenstates of the generators $D$ and $J_{0}$,
in the sense of the Poisson bracket relations
$\{I,A\}=\lambda A$, $I=D,J_{0}$.
By taking their
Poisson brackets with $H^{(\alpha)}$ and $K$, we generate the complete list
of symmetry generators of this kind.
Thus,
the set $\mathcal{U}_{1}=\{H^{(\alpha)},D,K,J_{0},\mathcal{O}_{\mu,\nu}^{\pm}\}$
generates a finite nonlinear algebra.
In the same way, one can
see
that the integrals $\mathcal{S}_{\mu^{\prime},\nu^{\prime}}^{\pm}$ satisfy
the
relations similar to those shown above
for $\mathcal{O}_{\mu,\nu}^{\pm}$ but
with
the Greek indices
changed
for their primed versions.
This implies
that the set of generators
$\mathcal{U}_{2}=\{H^{(\alpha)},D,K,J_{0},\mathcal{S}_{\mu^{\prime},\nu^{\prime%
}}^{\pm}\}$ also produces a finite nonlinear algebra.
To study what happens when we mix both
sets of generators, we consider the relations
$$\displaystyle\{D,\{\mathcal{S}_{\mu^{\prime},\nu^{\prime}}^{\pm},\mathcal{O}_{%
\mu,\nu}^{\mp}\}\}=\frac{\mu-\nu-\mu^{\prime}-\nu^{\prime}}{2}\,\{\mathcal{S}_%
{\mu^{\prime},\nu^{\prime}}^{\pm},\mathcal{O}_{\mu,\nu}^{\mp}\}\,,$$
(5.22)
$$\displaystyle\{J_{0},\{\mathcal{S}_{\mu^{\prime},\nu^{\prime}}^{\pm},\mathcal{%
O}_{\mu,\nu}^{\mp}\}\}=\mp i\frac{\mu+\nu}{2}\{\mathcal{S}_{\mu^{\prime},\nu^{%
\prime}}^{\pm},\mathcal{O}_{\nu,\nu}^{\mp}\}\,,$$
(5.23)
where we have used the Jacobi identity one more time.
Comparing the last relation with the first equation in (5.17), we deduce that in the general case,
the integrals $\{\mathcal{S}_{\mu^{\prime},\nu^{\prime}}^{\pm},\mathcal{O}_{\mu,\nu}^{\mp}\}$
must be functions of the integrals
from the set $\mathcal{U}_{1}$.
This means
that the set of integrals $\mathcal{O}_{\mu,\nu}^{\mp}$
corresponds to generators of
an ideal nonlinear subalgebra.
5.2 Quantum case
The quantum Hamiltonian of the system is constructed by using the Laplace-Beltrami operator,
$$\displaystyle\hat{H}^{(\alpha)}=-\frac{\hbar^{2}}{2m}\frac{1}{\sqrt{g}}\frac{%
\partial}{\partial x^{i}}\sqrt{g}g^{ij}\frac{\partial}{\partial x^{j}}=-\frac{%
\hbar^{2}}{2m}\left(\frac{1}{\alpha^{2}r}\frac{\partial}{\partial r}\left(r%
\frac{\partial}{\partial r}\right)+\frac{1}{r^{2}}\frac{\partial^{2}}{\partial%
\varphi}\right)\,.$$
(5.24)
The corresponding eigenstates and spectrum are given by
$$\displaystyle\psi_{\kappa,l}^{\pm}(r,\varphi)=\sqrt{\frac{\kappa}{2\pi\alpha}}%
J_{\alpha l}(\kappa r)e^{\pm il\varphi}\,,\qquad E=\frac{\hbar^{2}\kappa^{2}}{%
2m\alpha^{2}}\,,\qquad l=0,1,\ldots\,.$$
(5.25)
These eigenstates satisfy the orthogonality relation $\innerproduct{\psi_{\kappa,l}^{\pm}}{\psi_{\kappa^{\prime},l^{\prime}}^{\mp}}=%
\delta_{ll^{\prime}}\delta(\kappa-\kappa^{\prime})$,
with respect to
the scalar product
$$\displaystyle\innerproduct{\Psi_{1}}{\Psi_{2}}=\int_{V}\Psi_{1}^{*}\Psi_{2}%
\sqrt{g}dV=\int_{0}^{\infty}\alpha rdr\int_{0}^{2\pi}d\varphi\Psi_{1}^{*}\Psi_%
{2}\,.$$
(5.26)
To address the problem of analyzing the quantum symmetry of the system, we consider the quantum versions
of the formal integrals $\Pi_{\pm}$ and $\Xi_{\pm}$, which have been
the basis of the algebraic construction in the classical case.
We start with
$$\displaystyle\hat{\Pi}_{\pm}=\hat{\Pi}_{1}\pm i\hat{\Pi}_{2}=e^{\pm\frac{i%
\varphi}{2\alpha}}\left(\frac{1}{\alpha}\hat{p}_{r}\pm\frac{i}{r}\hat{p}_{%
\varphi}\right)e^{\pm\frac{i\varphi}{2\alpha}}=-i\hbar\frac{1}{\alpha}e^{\pm i%
\frac{\varphi}{\alpha}}\left(\frac{\partial}{\partial r}\pm i\frac{\alpha}{r}%
\frac{\partial}{\partial\varphi}\right)\,,$$
(5.27)
where $\hat{p}_{r}$ and $\hat{p}_{\varphi}$
are the radial and angular momentum operators introduced in (3.25). Operators
$\hat{\Pi}_{\pm}$
are formal since their action on the eigenstates produces
$$\displaystyle\hat{\Pi}_{\pm}\psi_{\kappa,l}^{\pm}(r,\varphi)=i\frac{\hbar%
\kappa}{\alpha}\sqrt{\frac{\kappa}{2\pi\alpha}}J_{\alpha l+1}(\kappa r)e^{\pm i%
(l+\frac{1}{\alpha})\varphi}\,,$$
(5.28)
$$\displaystyle\hat{\Pi}_{\pm}\psi_{\kappa,l}^{\mp}(r,\varphi)=-i\frac{\hbar%
\kappa}{\alpha}\sqrt{\frac{\kappa}{2\pi\alpha}}J_{\alpha l-1}(\kappa r)e^{\pm i%
(l-\frac{1}{\alpha})\varphi}\,,$$
(5.29)
from where we explicitly see that they cannot be physical operators for arbitrary values of $\alpha$
since
in the general
case they can produce the functions outside the Hilbert space generated by
the states $\psi_{\kappa,l}^{\pm}$.
We also consider the quantum versions of the Galilean boosts (5.7), corresponding to the operators
$$\displaystyle\hat{\Xi}_{\pm}=\alpha mre^{\pm i\frac{\varphi}{\alpha}}\pm t\hat%
{\Pi}_{\pm}=e^{\pm i\frac{\varphi}{\alpha}}\left(m\alpha r-i\hbar\frac{\kappa}%
{\alpha}t\left[\frac{\partial}{\partial(\kappa r)}\pm i\frac{\alpha}{\kappa r}%
\frac{\partial}{\partial\varphi}\right]\right)\,.$$
(5.30)
Due to
appearance of $\hat{\Pi}_{\pm}$ in (5.30),
the operators $\hat{\Xi}_{\pm}$ inherit all the problems of the former operators.
Let us carefully study the case $\alpha=q/k$. As in the classical analysis of the previous subsection,
we can consider the powers of the $\hat{\Pi}_{\pm}$ and $\hat{\Xi}_{\pm}$
operators. The action of
$(\hat{\Pi}_{\pm})^{q}$
on the
Hamiltonian eigenstates produces
$$\displaystyle(\hat{\Pi}_{\pm})^{q}\psi_{\kappa,l}^{\pm}(r,\varphi)=\left(i%
\frac{k\hbar\kappa}{q}\right)^{q}\sqrt{\frac{k\kappa}{2\pi q}}J_{\frac{q}{k}(l%
+k)}(\kappa r)e^{\pm i(l+k)\varphi}=\left(i\frac{k\hbar\kappa}{q}\right)^{q}%
\psi_{\kappa,l+k}^{\pm}(r,\varphi)\,,$$
(5.31)
$$\displaystyle(\hat{\Pi}_{\pm})^{q}\psi_{\kappa,l}^{\mp}(r,\varphi)=\left(-i%
\frac{k\hbar\kappa}{q}\right)^{q}\sqrt{\frac{k\kappa}{2\pi q}}J_{\frac{q}{k}(l%
-k)}(\kappa r)e^{\pm i(l-k)\varphi}\,.$$
(5.32)
Last equation means that the functions
$$(\hat{\Pi}_{\pm})^{q}\psi_{\kappa,j}^{\mp}(r,\varphi)\propto J_{-\frac{q|k-j|}%
{k}}(\kappa r)e^{\mp i|k-j|\varphi}\,,\qquad j=1,\ldots,k-1\,,$$
are outside the Hilbert space
since the index of the Bessel function is negative and
non-integer.
On the other hand,
in the special case $k=1$ we have
$$\displaystyle(\hat{\Pi}_{\pm})^{q}\psi_{\kappa,l}^{\pm}(r,\varphi)=\left(i%
\frac{\hbar\kappa}{q}\right)^{q}\sqrt{\frac{\kappa}{2\pi q}}J_{q(l+1)}(\kappa r%
)e^{\pm i(l+1)\varphi}=\left(i\frac{\hbar\kappa}{q}\right)^{q}\psi_{\kappa,l+1%
}^{\pm}(r,\varphi)\,,$$
(5.33)
$$\displaystyle(\hat{\Pi}_{\pm})^{q}\psi_{\kappa,l}^{\mp}(r,\varphi)=\left(-i%
\frac{\hbar\kappa}{q}\right)^{q}\sqrt{\frac{\kappa}{2\pi q}}J_{q(l-1)}(\kappa r%
)e^{\pm i(l-1)\varphi}=\left(-i\frac{\hbar\kappa}{q}\right)^{q}\psi_{\kappa,l-%
1}^{\mp}(r,\varphi)\,.$$
(5.34)
Now, using the fact that $ql=\mathfrak{n}$ is a positive integer number and
$J_{\mathfrak{-n}}(\zeta)=(-1)^{\mathfrak{n}}J_{\mathfrak{n}}(\zeta)\,,$
one can show that
$\psi_{\kappa,\mp l}^{\pm}=(-1)^{ls}\psi_{\kappa,\pm l}^{\mp}$.
This
implies that the operators $(\hat{\Pi}_{\pm})^{q}$
always produce physical eigenstates
in the case of integer values $\alpha=q$.
They are a kind of the spectrum generating operators
which allow us to change the quantum
number $l$ of the states in $\pm 1$ without
changing their energies.
As in the classical case,
formal operators $\hat{\Pi}_{\pm}$ and $\hat{\Xi}_{\pm}$ are the basis in the construction
of the symmetry algebra, and we can built the quantum version
of the
generators of the $\mathfrak{so}(2,1)\oplus\mathfrak{u}(1)$
symmetry taking place
for arbitrary values of $\alpha$. We have
$$\displaystyle\hat{H}^{(\alpha)}=\frac{1}{2m}\hat{\Pi}_{+}\hat{\Pi}_{-}\,,%
\qquad\hat{D}=\frac{1}{4m}(\hat{\Xi}_{+}\hat{\Pi}_{-}+\hat{\Pi}_{+}\hat{\Xi}_{%
-})=\frac{\hbar}{2i}\left(r\frac{\partial}{\partial r}+1\right)-\hat{H}^{(%
\alpha)}t\,,$$
(5.35)
$$\displaystyle\hat{K}=\frac{1}{2m}\hat{\Xi}_{-}\hat{\Xi}_{+}=\frac{m}{2}\alpha^%
{2}r^{2}-2\hat{D}t-\hat{H}^{(\alpha)}t^{2}\,,\qquad\hat{J}_{0}=\frac{1}{2}(%
\hat{\Xi}_{1}\hat{\Pi}_{2}-\hat{\Xi}_{2}\hat{\Pi}_{1})=\frac{\alpha}{2}\hat{p}%
_{\varphi}\,.$$
(5.36)
These operators satisfy the commutation relations
$$\displaystyle[\hat{D},\hat{H}^{(\alpha)}]=i\hbar\hat{H}^{(\alpha)}\,,\qquad[%
\hat{D},\hat{K}]=-i\hbar\hat{K}\,,\qquad[\hat{K},\hat{H}^{(\alpha)}]=2i\hbar%
\hat{D}\,,$$
(5.37)
$$\displaystyle[\hat{J}_{0},\hat{D}]=[\hat{J}_{0},\hat{H}^{(\alpha)}]=[\hat{J}_{%
0},\hat{K}]=0\,.$$
(5.38)
However, the analysis related to the operators $\hat{\Pi}_{\pm}$ shows us that in contrast with the classical case,
only for
$\alpha=q$ we can have a well-defined additional symmetry generators,
that reveals a kind of the quantum anomaly in the case of rational non-integer values of $\alpha$.
For $\alpha=q$,
the corresponding symmetry operators are
$$\displaystyle\hat{\mathcal{O}}^{\pm}_{\mu,\nu}=(\hat{\Xi}_{\pm})^{\mu}(\hat{%
\Pi}_{\pm})^{\nu}\,,\qquad\mu+\nu=q\,.$$
(5.39)
$$\displaystyle\hat{\mathcal{S}}_{\mu^{\prime},\nu^{\prime}}^{\pm}=(\hat{\Xi}_{%
\pm})^{\mu^{\prime}}(\hat{\Pi}_{\pm})^{\nu^{\prime}}\,,\qquad\mu^{\prime}+\nu^%
{\prime}=2q\,,$$
(5.40)
being the quantum counterpart of the classical integrals (5.12) and (5.13) with $k=1$.
By means of the
commutation relations
$$\displaystyle[\hat{\Xi}_{\pm},\hat{\Pi}_{\mp}]=2im\hbar\,,\qquad[\hat{\Xi}_{%
\pm},\hat{\Pi}_{\pm}]=0\,,$$
(5.41)
$$\displaystyle[\hat{H}^{(\alpha)},\hat{\Xi}_{\pm}]=-i\hbar\hat{\Pi}_{\pm}\,,%
\qquad[\hat{D},\hat{\Xi}_{\pm}]=-\frac{i\hbar}{2}\hat{\Xi}_{\pm}\,,\qquad[\hat%
{J}_{0},\hat{\Xi}_{\pm}]=\pm\frac{\hbar}{2}\hat{\Xi}_{\pm}\,,$$
(5.42)
$$\displaystyle[\hat{K},\hat{\Pi}_{\pm}]=i\hbar\hat{\Xi}_{\pm}\,,\qquad[\hat{D},%
\hat{\Pi}_{\pm}]=\frac{i\hbar}{2}\hat{\Pi}_{\pm}\,,\qquad[\hat{J}_{0},\hat{\Pi%
}_{\pm}]=\pm\frac{\hbar}{2}\hat{\Pi}_{\pm}\,,$$
(5.43)
and the commutator identity
$[\hat{A},\hat{B}^{n}]=\sum_{j=1}^{n}\hat{B}^{j-1}[\hat{A},\hat{B}]\hat{B}^{n-j%
}\,,$
it can be shown that the properties of the classical nonlinear algebra generated by the integrals
$\mathcal{O}^{\pm}_{\mu,\nu}$ and $\mathcal{S}^{\pm}_{\mu^{\prime},\nu^{\prime}}$
with $k=1$ are preserved at the quantum level.
Some important algebraic relations are
$$\displaystyle[\hat{p}_{\varphi},\hat{\mathcal{O}}_{\mu,\nu}^{\pm}]=\pm\hbar%
\hat{\mathcal{S}}_{\mu,\nu}^{\pm}\,,\qquad[\hat{p}_{\varphi},\hat{\mathcal{S}}%
_{\mu^{\prime},\nu^{\prime}}^{\pm}]=\pm\hbar\hat{\mathcal{S}}_{\mu^{\prime},%
\nu^{\prime}}^{\pm}\,,$$
(5.44)
$$\displaystyle[\hat{D},\hat{\mathcal{O}}^{\pm}_{\mu,\nu}]=i\hbar\frac{\nu-\mu}{%
2}\hat{\mathcal{O}}^{\pm}_{\mu,\nu}\,,\qquad[\hat{D},\hat{\mathcal{S}}^{\pm}_{%
\mu,\nu}]=i\hbar\frac{\nu^{\prime}-\mu^{\prime}}{2}\hat{\mathcal{S}}^{\pm}_{%
\mu^{\prime},\nu^{\prime}}\,.$$
(5.45)
From (5.44) we learn that the operators $\hat{\mathcal{O}}^{\pm}_{\mu,\nu}$ ($\hat{\mathcal{S}}^{\pm}_{\mu^{\prime},\nu^{\prime}}$)
change the angular momentum quantum number by $\pm 1$ ($\pm 2$), and it its clear that
operators $\hat{\mathcal{O}}^{\pm}_{0,q}=(\hat{\Pi}_{\pm})^{q}$
are
responsible for the infinite degeneracy of the spectrum of the system.
6 Harmonic oscillator in a cosmic string background
To study the dynamics of the harmonic oscillator
in
a cosmic string background, we set the potential term in action (5.1)
to be $V(\boldmathe{r})=\frac{m\omega^{2}}{2}g_{ij}x_{i}x_{j}=\frac{1}{2}m\alpha^{2}%
\omega^{2}r^{2}$.
We will see that the particle dynamics has special characteristics for different values of $\alpha$,
and these peculiarities are coherently reflected in the classical orbits,
in the spectra of the system at the quantum level,
and in the construction of the symmetry algebra,
which again will reveal the quantum anomaly phenomenon.
We also show that this system is related to the free motion in the conical geometry
by means of the conformal bridge transformation, and we
use this to reconstruct the properties
of the complete classical symmetry algebra for the case $\alpha=q/k$ and the quantum
version in
the case $\alpha=q$.
6.1 Classical case
The Hamiltonian of the system corresponds to
$$\displaystyle H_{\text{os}}^{(\alpha)}=\frac{1}{2m}\left(\frac{p_{r}^{2}}{%
\alpha^{2}}+\frac{p_{\varphi}^{2}}{r^{2}}\right)+\frac{m\alpha^{2}\omega^{2}}{%
2}r^{2}\,,$$
(6.1)
and as it happened with the free particle, it
is directly related to the Hamiltonian
of the harmonic oscillator in the Euclidean plane,
which was reviewed in the subsection 3.2,
by means of the local canonical transformation (5.3).
From here, the solutions of the corresponding
equations of motion are immediately obtained,
$$\displaystyle r^{2}(\varphi)=\frac{p_{\varphi}^{2}}{mH_{\text{os}}^{(\alpha)}}%
\left(1+\delta\cos(\frac{2}{\alpha}(\varphi-\varphi_{*}))\right)^{-1}\,,$$
(6.2)
$$\displaystyle\varphi(\tau)=\alpha\arctan(\frac{r_{+}}{r_{-}}\tan(\omega(\tau-%
\tau_{*})))+\varphi_{*}\,,$$
(6.3)
$$\displaystyle r^{2}(\tau)=\frac{2}{m\omega^{2}}\big{(}\omega\mathcal{D}\sin(2%
\omega\tau)+\omega^{2}\mathcal{K}\cos(2\omega\tau)+H_{\text{os}}^{(\alpha)}%
\sin^{2}(\omega\tau)\big{)}\,.$$
(6.4)
Here $\mathcal{D}$ and $\mathcal{K}$ are given by
$$\displaystyle\mathcal{K}=\frac{m\alpha^{2}}{2}r^{2}\cos(2\omega\tau)+\frac{1}{%
\omega^{2}}H_{\text{os}}^{(\alpha)}\sin^{2}(\omega\tau)-\frac{1}{2\omega}rp_{r%
}\sin(2\omega\tau)\,,$$
(6.5)
$$\displaystyle\mathcal{D}=\frac{1}{2}rp_{r}\cos(2\omega\tau)-\frac{1}{2\omega}(%
H_{\text{os}}^{(\alpha)}-m\omega^{2}\alpha^{2}r^{2})\sin(2\omega\tau)\,.$$
(6.6)
The quantities
$$\displaystyle r_{\pm}^{2}=\frac{H_{\text{os}}^{(\alpha)}}{m\omega^{2}\alpha^{2%
}}\left(1\pm\delta\right)\,,\qquad\delta=\sqrt{1-\left(\frac{\omega\alpha p_{%
\varphi}}{H_{\text{os}}^{(\alpha)}}\right)^{2}}\,,$$
(6.7)
are identified with the
radial turning points of the trajectory, and
$\varphi_{*}$ and $\tau_{*}$
are
the angular position of one of the radial minima of the trajectory and
the corresponding moment of time
when the particle is in that place.
From the corresponding solutions of the equation of motion
one finds that the trajectory is closed
if and only if $\alpha$ is a rational number.
The images of the orbit
for some
rational and irrational values of $\alpha$
are shown in Fig. 3.
By using the canonical transformation (5.3),
one can also obtain the dynamical integrals representing the classical analogues of the ladder operators
in polar coordinates, which are:
$$\displaystyle\mathfrak{b}_{1}^{-}=\frac{1}{\sqrt{2}}(a_{1}^{-}-ia_{2}^{-})=%
\frac{1}{2}e^{i(\omega t-\frac{\varphi}{\alpha})}\left(\alpha\sqrt{m\omega}r+%
\frac{p_{\varphi}}{\sqrt{m\omega}r}+\frac{ip_{r}}{\alpha\sqrt{m\omega}}\right)%
\,,\qquad\mathfrak{b}_{1}^{+}=(\mathfrak{b}_{1}^{-})^{*}\,,$$
(6.8)
$$\displaystyle\mathfrak{b}_{2}^{-}=\frac{1}{\sqrt{2}}(a_{1}^{-}+ia_{2}^{-})=%
\frac{1}{2}e^{i(\omega t+\frac{\varphi}{\alpha})}\left(\alpha\sqrt{m\omega}r-%
\frac{p_{\varphi}}{\sqrt{m\omega}r}+\frac{ip_{r}}{\alpha\sqrt{m\omega}}\right)%
\,,\qquad\mathfrak{b}_{2}^{+}=(\mathfrak{b}_{2}^{-})^{*}\,.$$
(6.9)
In resemblance with the free particle system in a conical geometry, we also note that these dynamical integrals
are formal (locally defined only)
functions in the respective phase space when $\alpha$ takes
values different
from $1/k$.
However, we can use these formal integrals
to
construct the well-defined generators of the
$\mathfrak{sl}(2,\mathbb{R})\oplus\mathfrak{u}(1)$ symmetry algebra
of the system,
which
are given by
$$\mathcal{J}_{0}=\frac{1}{2}\mathfrak{b}_{a}^{+}\mathfrak{b}_{a}^{-}=\frac{1}{2%
\omega}H_{\text{os}}^{(\alpha)}\,,\qquad\mathcal{J}_{\pm}=\mathfrak{b}_{1}^{%
\pm}\mathfrak{b}_{2}^{\pm}\,,\qquad\mathcal{L}_{2}=(\mathfrak{b}_{1}^{+}%
\mathfrak{b}_{1}^{-}-\mathfrak{b}_{2}^{+}\mathfrak{b}_{2}^{-})=\frac{1}{2}%
\alpha p_{\varphi}\,.$$
(6.10)
In the context of the conformal bridge transformation, if we set the generators $H^{(\alpha)}$, $D$ and $K$ as
those in (5.8)-(5.9),
the transformation considered in Sec. 4 produces
the generators (6.10), in accordance with relations (4.11),
(4.12) and the first equation in (4.15).
To obtain the complete symmetry algebra for the case $\alpha=q/k$ we must apply the transformation to the higher-order
generators (5.12) and (5.13).
However, it is convenient first to consider the formal relations
$$\displaystyle\mathscr{T}(\tau,\beta,\delta,\gamma,t)(\Pi_{-})=-i\sqrt{2m\omega%
}\mathfrak{b}_{1}^{-}\,,\qquad\mathscr{T}(\tau,\beta,\delta,\gamma,t)(\Pi_{+})%
=-i\sqrt{2m\omega}\mathfrak{b}_{2}^{-}\,,$$
(6.11)
$$\displaystyle\mathscr{T}(\tau,\beta,\delta,\gamma,t)(\Xi_{+})=\sqrt{\frac{2m}{%
\omega}}\mathfrak{b}_{1}^{+}\,,\qquad\mathscr{T}(\tau,\beta,\delta,\gamma,t)(%
\Xi_{-})=\sqrt{\frac{2m}{\omega}}\mathfrak{b}_{2}^{+}\,,$$
(6.12)
where $\beta$, $\delta$ and $\gamma$ are still given by (4.10),
and the formal equations (5.14)-(5.16) are
employed.
These equations facilitate the application of the conformal bridge transformation
to
the mentioned
well-defined on the phase space integrals of motion.
This
yields us
$$\displaystyle\mathscr{T}(\tau,\beta,\delta,\gamma,t)(\mathcal{O}_{\mu,\nu}^{%
\pm})=(-i)^{\nu}(m)^{\frac{q}{2}}(\omega)^{\frac{\mu-\nu}{2}}\mathcal{G}_{\mu,%
\nu}^{\pm}\,,$$
(6.13)
$$\displaystyle\mathscr{T}(\tau,\beta,\delta,\gamma,t)(\mathcal{S}_{\mu^{\prime}%
,\nu^{\prime}}^{\pm})=(-i)^{\nu}(m)^{\frac{q}{2}}(\omega)^{\frac{\mu-\nu}{2}}%
\mathcal{F}_{\mu^{\prime},\nu^{\prime}}^{\pm}\,,$$
(6.14)
where
$$\displaystyle\mathcal{G}_{\mu,\nu}^{+}=(b_{1}^{+})^{\mu}(b_{2}^{-})^{\nu}\,,%
\quad(\mathcal{G}_{\mu,\nu}^{+})^{*}=\mathcal{G}_{\nu,\mu}^{-}\qquad\mathcal{F%
}_{\mu^{\prime},\nu^{\prime}}^{+}=(b_{1}^{+})^{\mu^{\prime}}(b_{2}^{-})^{\nu^{%
\prime}}\,,\quad(\mathcal{F}_{\mu^{\prime},\nu^{\prime}}^{+})^{*}=\mathcal{F}_%
{\nu^{\prime},\mu^{\prime}}^{-}\,,$$
(6.15)
and $\mu,\nu=0,1,\ldots q$, ($\mu^{\prime},\nu^{\prime}=0,1,\ldots 2q$) satisfy the restriction $\mu+\nu=q$ ($\mu^{\prime}+\nu^{\prime}=2q$).
The properties of the symmetry algebra generated by these integrals of motion
are retrieved
directly from the symmetry algebra of the free particle moving in the same conical geometry.
In summary we have:
•
The set of generators $\mathcal{U}_{1}=\{\mathcal{J}_{0},\mathcal{J}_{\pm},\mathcal{L}_{2},\mathcal{G%
}_{\mu,\nu}^{\pm}\}$
produces an ideal nonlinear subalgebra.
The (in general) dynamical integrals $\mathcal{G}_{\mu,\nu}^{\pm}$ are eigenstates of
$i\mathcal{L}_{2}=i\alpha p_{\varphi}/2$
with eigenvalue $\lambda=\pm q/2$,
in the sense
of the Poisson bracket relation
$\{\mathcal{L}_{2},A\}=\lambda A$, and, therefore,
they are
eigenstates of $ip_{\varphi}$, with eigenvalue $\pm k$.
In the same way, they are
eigenstates of $H_{\text{os}}^{(\alpha)}$ with eigenvalue $i\omega(\nu-\mu)$. Note that when
$\nu=\mu=q/2$
and
$q$ is an even number, we have two true
(not depending explicitly on time)
integrals of motion for the system.
Finally, the Poisson bracket action
of generators $\mathcal{J}_{\pm}$ corresponds to
$$\{\mathcal{J}_{-},\mathcal{G}_{\mu,\nu}^{\pm}\}=i\mu\mathcal{G}_{\mu-1,\nu+1}^%
{\pm}\,,\qquad\{\mathcal{J}_{+},\mathcal{G}_{\mu,\nu}^{\pm}\}=i\nu\mathcal{G}_%
{\mu+1,\nu-1}^{\pm}\,.$$
(6.16)
•
The integrals of the set $\mathcal{U}_{2}=\{\mathcal{J}_{0},\mathcal{J}_{\pm},\mathcal{L}_{2},\mathcal{F%
}_{\mu^{\prime},\nu^{\prime}}^{\pm}\}$
also produce a nonlinear subalgebra, but
which
is not an ideal. In this case the (in general) dynamical integrals $\mathcal{F}_{\mu^{\prime},\nu^{\prime}}^{\pm}$ are eigenstates of $i\mathcal{L}_{2}$
with eigenvalue $\lambda=\pm q$, implying that with respect to $ip_{\varphi}$, the eigenvalues are $\pm 2k$.
These objects
are also eigenstates of $H_{\text{os}}^{(\alpha)}$, with eigenvalue $i\omega(\nu^{\prime}-\mu^{\prime})$, and again,
when $\nu^{\prime}=\mu^{\prime}=q$ we have two true
integrals. In contrast with the previous case, these
true integrals can be constructed for any natural value of $q$,
but when $q$ is even, $q=2\tilde{q}$, $\tilde{q}=1,\ldots$,
we have the equalities $\mathcal{F}_{q,q}^{\pm}=(\mathcal{G}_{\tilde{q},\tilde{q}}^{\pm})^{2}$.
The Poisson bracket action of the integrals
$\mathcal{J}_{\pm}$ is
$$\{\mathcal{J}_{-},\mathcal{F}_{\mu^{\prime},\nu^{\prime}}^{\pm}\}=i\mu^{\prime%
}\mathcal{F}_{\mu^{\prime}-1,\nu^{\prime}+1}^{\pm}\,,\qquad\{\mathcal{J}_{+},%
\mathcal{F}_{\mu^{\prime},\nu^{\prime}}^{\pm}\}=i\nu^{\prime}\mathcal{F}_{\mu^%
{\prime}+1,\nu^{\prime}-1}^{\pm}\,.$$
(6.17)
When $\alpha=1$, we obtain the harmonic oscillator in the Euclidean plane,
and the dynamical integrals $\mathcal{G}_{\mu,\nu}^{\pm}$ correspond to the classical analogues
of the ladder operators themselves.
In this case, the true integrals $\mathcal{F}_{1,1}^{\pm}$
correspond
to the $\mathcal{L}_{\pm}$
generators of the $\mathfrak{su}(2)$ symmetry of the system, while $\mathcal{F}_{0,2}^{-}$ and
$\mathcal{F}_{2,0}^{+}$ are the mutually
complex conjugated dynamical integrals $\mathcal{B}^{\pm}$, see Section 3.2.
In the case $\alpha=1/k$, with $k=2,3,\ldots$
we still have the same symmetry algebra as for the Euclidean case,
since the generators $\mathfrak{b}_{a}^{\pm}$ are the globally
well-defined functions in the phase space.
6.2 Quantum case
Here we study the quantum theory corresponding to the classical system discussed in the previous section.
As we have seen, at the classical level the system has a large number of dynamical symmetries that
are eigenstates of the Hamiltonian in the sense of the Poisson
bracket relation
$\{H_{\text{os}}^{(\alpha)},\mathcal{C}\}=\lambda\mathcal{C}$
when $\alpha$
is rational.
Each of these integrals were obtained by applying the classical conformal bridge transformation to
the classical free particle system symmetry generators in conical geometry,
and in this section we follow the quantum version of that approach. In this context,
and remembering that at the quantum level the system of the free particle in conical space reveals
a
quantum anomaly for
rational, non-integer values of $\alpha$, it should not be surprising
that this anomaly is also present in the harmonically trapped system, and in this section we show how this happens.
As all the integrals that
admit a well-defined quantum extensions must be eigenstates of the
corresponding quantum Hamiltonian $\hat{H}_{\text{os}}^{(\alpha)}$
in the sense of $[\hat{H}_{\text{os}}^{(\alpha)},\hat{\mathcal{C}}]=i\hbar\lambda\hat{\mathcal{%
C}}$,
one concludes that the action
of these operators at $\tau=0$ on
a particular eigenstate of the system (and that actually is what we need to
calculate according to Eq. (3.28))
will produce another eigenstate. So, for the sake of simplicity, we assume that
all the evolution parameters in dynamical integrals are zero from now on.
At the quantum level, the system is governed by the Hamiltonian operator
$$\displaystyle\hat{H}_{\text{os}}^{(\alpha)}=-\frac{\hbar^{2}}{2m}\left(\frac{1%
}{\alpha^{2}r}\frac{\partial}{\partial r}\left(r\frac{\partial}{\partial r}%
\right)+\frac{1}{r^{2}}\frac{\partial^{2}}{\partial\varphi}\right)+\frac{%
\alpha^{2}m\omega^{2}}{2}r^{2}\,,$$
(6.18)
whose eigenstates and spectrum are given by
$$\displaystyle\psi_{n_{r},l}^{\pm}(r,\varphi)=\left(\frac{m\omega\alpha^{2}}{%
\hbar}\right)^{\frac{1}{2}}\sqrt{\frac{n_{r}!}{2\pi\alpha\Gamma(n_{r}+\alpha l%
+1)}}\,\zeta^{\alpha l}L_{n_{r}}^{(\alpha l)}(\zeta^{2})e^{-\frac{\zeta^{2}}{2%
}\pm il\varphi}\,,\qquad\zeta=\sqrt{\frac{m\alpha^{2}\omega}{\hbar}}r\,,$$
(6.19)
$$\displaystyle E_{n,l}=\hbar\omega(2n_{r}+\alpha l+1)\,,\qquad n_{r}\,,l=0,1,%
\ldots\,.$$
(6.20)
These eigenstates are orthonormal,
$\innerproduct{\psi_{n_{r},l}^{\pm}}{\psi_{n_{r}^{\prime},l^{\prime}}^{\mp}}=%
\delta_{n_{r},n_{r}^{\prime}}\delta_{l,l^{\prime}}$, with
respect to the scalar product (5.26).
Note that, contrary to the case of
the free particle, the degeneracy of the energy levels depends on the value of $\alpha$.
In the particular case of rational values $\alpha=q/k$ we distinguish two cases:
•
When $q$ is even, $q=2\tilde{q}$, the energy levels satisfy the
relation
$$\displaystyle E_{n_{r}+\tilde{q}j,l-kj}=E_{n_{r},l}\,,\qquad j=-[\frac{n_{r}}{%
\tilde{q}}],\ldots,[\frac{l}{k}]\,,$$
(6.21)
where $[.]$ indicates the integer part of the quotient. This
implies that all
the eigenstates $\psi_{n_{r}+\tilde{q}j,l-kj}^{\pm}$ have the same energy eigenvalue.
Counting the number of these eigenstates we get the following
value
for the degeneracy:
$$\displaystyle g(N)=2[\frac{N}{\tilde{q}}]+1\,,\qquad N=n_{r}+\frac{\tilde{q}}{%
k}l\,.$$
(6.22)
•
When $q$ is odd, we have
$$\displaystyle E_{n_{r}+qj,l-2kj}=E_{n_{r},l}\,,\qquad j=-[\frac{n_{r}}{q}],%
\ldots,[\frac{l}{2k}]\,,$$
(6.23)
and the degeneracy is given by
$$\displaystyle g(N)=[\frac{N}{q}]+1\,,\qquad N=2n_{r}+\frac{q}{k}l\,.$$
(6.24)
Like
the classical case,
we can use the quantum version of the conformal bridge transformation
to connect this system with the quantum version of the free particle
in the conical geometry with the same value of $\alpha$.
As we are interested in the operators at $\tau=0$, we consider
the stationary conformal bridge transformation
generated by
$\hat{\mathfrak{S}}(0,0)=\hat{\mathfrak{S}}_{0}$, which produces
$$\displaystyle\hat{\mathfrak{S}}_{0}(\hat{H}^{(\alpha)})\hat{\mathfrak{S}}_{0}^%
{-1}=-\omega\hbar\hat{\mathcal{J}}_{-}\,,\qquad\hat{\mathfrak{S}}_{0}(\hat{iD}%
_{0})\hat{\mathfrak{S}}_{0}^{-1}=\hbar\hat{\mathcal{J}}_{0}\,,\qquad\hat{%
\mathfrak{S}}_{0}(\hat{K}_{0})\hat{\mathfrak{S}}_{0}^{-1}=\frac{\hbar}{\omega}%
\hat{\mathcal{J}}_{-}\,,$$
(6.25)
where
$$\displaystyle\hat{\mathcal{J}}_{0}=\frac{1}{2\hbar\omega}\hat{H}_{\text{os}}^{%
(\alpha)}\,,\qquad\hat{\mathcal{J}}_{\pm}=-\frac{m\omega}{4\hbar}\left(\hat{H}%
_{\text{os}}^{(\alpha)}-m\omega^{2}\alpha^{2}r^{2}\pm\hbar\omega\left(\frac{%
\partial}{\partial r}+1\right)\right)\,.$$
(6.26)
Also, when the transformation acts on the formal operators $\hat{\Pi}_{\pm}$ and $\hat{\Xi}_{\pm}$,
one gets
$$\displaystyle\hat{\mathfrak{S}}_{0}(\hat{\Pi}_{-})\hat{\mathfrak{S}}_{0}^{-1}=%
-i\sqrt{2m\hbar\omega}\hat{\mathfrak{b}}_{1}^{-}\,,\qquad\hat{\mathfrak{S}}_{0%
}(\hat{\Pi}_{+})\hat{\mathfrak{S}}_{0}^{-1}=-i\sqrt{2m\hbar\omega}\hat{%
\mathfrak{b}}_{2}^{-}\,,$$
(6.27)
$$\displaystyle\hat{\mathfrak{S}}_{0}(\hat{\Xi}_{+})\hat{\mathfrak{S}}_{0}^{-1}=%
\sqrt{\frac{2m\hbar}{\omega}}\hat{\mathfrak{b}}_{1}^{+}\,,\qquad\hat{\mathfrak%
{S}}_{0}(\hat{\Xi}_{-})\hat{\mathfrak{S}}_{0}^{-1}=\sqrt{\frac{2m\hbar}{\omega%
}}\hat{\mathfrak{b}}_{2}^{+}\,,$$
(6.28)
where
$$\displaystyle\hat{\mathfrak{b}}_{1}^{-}=\frac{1}{2}e^{-i\frac{\varphi}{\alpha}%
}\sqrt{\frac{m\omega}{\hbar}}\left(\alpha r+\frac{\hbar}{m\omega\alpha}\left(%
\frac{\partial}{\partial r}-\frac{i\alpha}{r}\frac{\partial}{\partial\varphi}%
\right)\right)\,,\qquad\hat{\mathfrak{b}}_{1}^{+}=(\hat{\mathfrak{b}}_{1}^{-})%
^{\dagger}\,,$$
(6.29)
$$\displaystyle\hat{\mathfrak{b}}_{2}^{-}=\frac{1}{2}e^{i\frac{\varphi}{\alpha}}%
\sqrt{\frac{m\omega}{\hbar}}\left(\alpha r+\frac{\hbar}{m\omega\alpha}\left(%
\frac{\partial}{\partial r}+\frac{i\alpha}{r}\frac{\partial}{\partial\varphi}%
\right)\right)\,,\qquad\hat{\mathfrak{b}}_{2}^{+}=(\hat{\mathfrak{b}}_{2}^{-})%
^{\dagger}\,,$$
(6.30)
are the formal dimensionless ladder operators in polar coordinates representation. In terms of them, the generators
of the conformal algebra (6.26) and the quantum version of $\mathcal{L}_{2}$
take the form
$$\displaystyle\hat{\mathcal{J}}_{0}=\frac{1}{4}(\{\hat{\mathfrak{b}}_{1}^{+},%
\hat{\mathfrak{b}}_{1}^{-}\}+\{\hat{\mathfrak{b}}_{2}^{-},\hat{\mathfrak{b}}_{%
2}^{+}\})\,,\qquad\hat{\mathcal{J}}_{\pm}=\hat{\mathfrak{b}}_{1}^{\pm}\hat{%
\mathfrak{b}}_{2}^{\pm}\,,$$
(6.31)
$$\displaystyle\hat{\mathcal{L}}_{2}=(\hat{\mathfrak{b}}_{1}^{+}\hat{\mathfrak{b%
}}_{1}^{-}-\hat{\mathfrak{b}}_{2}^{+}\hat{\mathfrak{b}}_{2}^{-})=\frac{1}{2%
\hbar}\hat{p}_{\varphi}\,.$$
(6.32)
According to
relation (4.22), the
corresponding
eigenstates of the form
$\innerproduct{\boldmathe{r}}{\lambda}$ we are looking for correspond to
$$\Omega_{n_{r},l}^{\pm}(r,\varphi)=r^{2n_{r}+\alpha l}e^{\pm l\varphi}\,,$$
(6.33)
which satisfy the following
set of equations,
$$\displaystyle 2i\hat{D}_{0}\Omega_{n_{r},l}^{\pm}(r,\varphi)=\hbar(2n_{r}+%
\alpha l+1)\Omega_{n_{r},l}^{\pm}(r,\varphi)\,,$$
(6.34)
$$\displaystyle\hat{K}_{0}\Omega_{n_{r},l}^{\pm}(r,\varphi)=\frac{m\alpha^{2}}{2%
}\Omega_{n_{r}+1,l}^{\pm}(r,\varphi)\,,$$
(6.35)
$$\displaystyle\hat{H}^{(\alpha)}\Omega_{n_{r},l}^{\pm}(r,\varphi)=-\frac{2\hbar%
^{2}}{m\alpha^{2}}n_{r}(n_{r}+\alpha l)\Omega_{n_{r}-1,l}^{\pm}(r,\varphi)\,,$$
(6.36)
$$\displaystyle\hat{\Xi}_{\pm}\Omega_{n_{r},l}^{\pm}(r,\varphi)=\alpha m\Omega_{%
n_{r},l+\frac{1}{\alpha}}^{\pm}(r,\varphi)\,,\qquad\hat{\Xi}_{\pm}\Omega_{n_{r%
},l}^{\mp}(r,\varphi)=\alpha m\Omega_{n_{r}+1,l-\frac{1}{\alpha}}^{\mp}(r,%
\varphi)\,,$$
(6.37)
$$\displaystyle\hat{\Pi}_{\pm}\Omega_{n_{r},l}^{\pm}(r,\varphi)=-i\frac{2\hbar n%
_{r}}{\alpha}\Omega_{n_{r}-1,l+\frac{1}{\alpha}}^{\pm}(r,\varphi)\,,$$
(6.38)
$$\displaystyle\hat{\Pi}_{\pm}\Omega_{n_{r},l}^{\mp}(r,\varphi)=-i2(n_{r}+\alpha
l%
)\frac{\hbar}{\alpha}\Omega_{n_{r},l-\frac{1}{\alpha}}^{\mp}(r,\varphi)\,.$$
(6.39)
Here, the functions
$\Omega_{0,l}^{\pm}$ are the zero energy eigenstates of $\hat{H}^{(\alpha)}$, but only the function $\Omega_{0,0}^{+}=\Omega_{0,0}^{-}=1$ is
a physical eigenstate for the free particle system. Functions $\Omega_{n_{r},l}^{\pm}$ are the
rank $n_{r}$ Jordan states of zero energy which satisfy
$$\displaystyle(\hat{H}^{(\alpha)})^{j}\Omega_{n_{r},l}^{\pm}(r,\varphi)=\left(-%
2\frac{\hbar}{m\alpha^{2}}\right)^{j}(n_{r})_{j}(n_{r}+\alpha l)_{j}\Omega_{n_%
{r}-j,l}^{\pm}(r,\varphi)\,,\qquad j=1,\ldots n_{r}\,,$$
(6.40)
$$\displaystyle(g)_{j}=\prod_{i=0}^{j-1}(g-i)\,.$$
(6.41)
Another remarkable
property of these functions is that the eigenstates of the free particle admit the representation
$$\displaystyle\psi_{\kappa,l}^{\pm}(r,\varphi)=\sqrt{\frac{\kappa}{2\pi\alpha}}%
\sum_{n_{r}=0}^{\infty}\frac{(-1)^{n_{r}}(\kappa/2)^{2n_{r}+\alpha l}}{n_{r}!%
\Gamma(n_{r}+\alpha l+1)}\Omega_{n_{r},l}^{\pm}(r,\varphi)\,.$$
(6.42)
By direct application of the stationary
conformal bridge operator $\hat{\mathfrak{S}}_{0}$
to functions (6.33)
one gets
$$\displaystyle\hat{\mathfrak{S}}_{0}\Omega_{n_{r},l}^{\pm}(r,\varphi)=\mathcal{%
N}_{n_{r},l}\psi_{n_{r},l}^{\pm}(r,\varphi)\,,$$
(6.43)
$$\displaystyle\mathcal{N}_{n_{r},l}=(-1)^{n_{r}}(2)^{n_{r}+\alpha l}\left(\frac%
{\hbar}{m\omega\alpha^{2}}\right)^{n_{r}+\frac{\alpha l}{2}}\sqrt{2\alpha\pi n%
_{r}!\Gamma(n_{r}+\alpha l+1)}\,.$$
(6.44)
On the other hand, the action of the transformation on
the free particle eigenstates produces
the (non-normalized) coherent states of the
system corresponding to
$$\displaystyle\begin{array}[]{lcl}\Psi_{\kappa,\ell}^{\pm}(r,\varphi)&=&%
\mathfrak{S}_{0}\psi_{\kappa,\ell}^{\pm}(r,\varphi)=2^{\frac{1}{4}}e^{\frac{%
\mathcal{E}}{2}}\psi_{\kappa,\ell}^{\pm}(\sqrt{2}r,\varphi)e^{-\frac{m\omega r%
^{2}}{2\hbar}}\\
&=&\sqrt{\kappa}\sum_{n_{r}=0}^{\infty}\frac{\mathcal{E}^{n_{r}+\frac{\alpha l%
}{2}}}{2^{n_{r}}\sqrt{n_{r}!\Gamma(n_{r}+\alpha l+1)}}\psi_{n_{r},l}^{\pm}(r,%
\varphi)\,,\end{array}$$
(6.45)
where $\mathcal{E}=\frac{\hbar\kappa^{2}}{m\omega\alpha^{2}}$.
These states satisfy the relation
$\hat{\mathcal{J}}_{-}\Psi_{\kappa,\ell}^{\pm}(r,\varphi)=-\frac{\mathcal{E}}{2%
}\Psi_{\kappa,\ell}^{\pm}(r,\varphi)$.
From the equations
(6.36) we obtain the action of generators $\hat{\mathcal{J}}_{\pm}$,
$$\displaystyle\hat{\mathcal{J}}_{+}\psi_{n_{r},l}^{\pm}(r,\varphi)=-\sqrt{(n+1)%
(n+\alpha l+1)}\psi_{n_{r}+1,l}^{\pm}(r,\varphi)\,,$$
(6.46)
$$\displaystyle\hat{\mathcal{J}}_{-}\psi_{n_{r},l}^{\pm}(r,\varphi)=-\sqrt{n(n+%
\alpha l)}\psi_{n_{r}-1,l}^{\pm}(r,\varphi)\,,$$
(6.47)
and from equations (6.37)-(6.39) we get the formal relations
$$\displaystyle\hat{\mathfrak{b}}_{a}^{\pm}\,\psi_{n_{r},l}^{(a)}(r,\varphi)=%
\sqrt{2(n_{r}+\alpha l+\beta_{\pm})}\,\psi_{n_{r},l\pm\frac{l}{\alpha}}^{(a)}(%
r,\varphi)\,,$$
(6.48)
$$\displaystyle\hat{\mathfrak{b}}_{a}^{\pm}\,\psi_{n_{r},l}^{(b)}(r,\varphi)=-%
\sqrt{\frac{(n_{r}+\beta_{\pm})}{2}}\,\psi_{n_{r}\pm 1,l\mp\frac{l}{\alpha}}^{%
(b)}(r,\varphi)\,,$$
(6.49)
where
$\psi_{n_{r},l}^{(1)}=\psi_{n_{r},l}^{+}$,
$\psi_{n_{r},l}^{(2)}=\psi_{n_{r},l}^{-}$,
and $\beta_{\pm}=\frac{1}{2}(1\pm 1)$.
From
these equations it becomes obvious that these operators cannot be physical for arbitrary values of
$\alpha$, since they produce wave-functions outside
the Hilbert space generated by the physical eigenstates $\psi_{n_{r},l}^{\pm}$.
In fact, this is already observable
from the equations (6.37)-(6.39),
where it is seen that the produced functions on the
right hand side do not satisfy some of the criteria imposed at the end of Sec. 4:
these functions are not single-valued in the angular coordinate, and some of them are singular
at $r=0$.
Similarly to
the free particle system in the conical space,
we must study carefully the rational case $\alpha=q/k$, and to do that,
it is enough to
analyze
the relations
that imply a decrease in the angular momentum quantum number $l$
(the relations in which
this number is increasing have no problems).
First, consider the relation
$$\displaystyle(\hat{\mathfrak{b}}_{a}^{-})^{q}\,\psi_{n_{r},l}^{(a)}(r,\varphi)%
=2^{\frac{q}{2}}\sqrt{\frac{\Gamma(n_{r}+(q/k)l+1)}{\Gamma(n_{r}+(q/k)l-q+1)}}%
\,\psi_{n_{r},l-k}^{(a)}(r,\varphi)\,.$$
(6.50)
When $l<k$, that is $l-k=-j<0$,
the explicit form of the function
on the right hand side
is
$$\displaystyle\psi_{n_{r},-j}^{\pm}(r,\varphi)=\left(\frac{m\omega q^{2}}{\hbar
k%
^{2}}\right)^{\frac{1}{2}}\sqrt{\frac{kn_{r}!}{2\pi q\Gamma(n_{r}-(q/k)j+1)}}%
\,\zeta^{-\frac{q}{k}j}L_{n_{r}}^{(-\frac{q}{k}j)}(\zeta^{2})e^{-\frac{\zeta^{%
2}}{2}\mp ij\varphi}\,,$$
(6.51)
and because the upper index of the generalized Laguerre polynomial
is negative and rational, we conclude that
this function is outside
the Hilbert space generated by the physical eigenstates,
implying that the operators can not be observable.
Now, in the particular case when $k=1$, the upper index in the generalized
Laguerre polynomial is a negative integer number,
and due to the identity
$$\frac{(-\eta)^{i}}{i!}L_{n}^{(i-n)}(\eta)=\frac{(-\eta)^{n}}{n!}L_{i}^{(n-i)}(%
\eta)\,,\qquad i,n=0,1,\ldots,$$
(6.52)
one gets that in the case $n_{r}\geq ql$,
$$\psi_{n_{r},-l}^{\pm}(r,\varphi)=(-1)^{ql}\psi_{n_{r}-ql,l}^{\mp}(r,\varphi)\,,$$
(6.53)
while for $n_{r}<ql$ the right hand side in Eq. (6.51) vanishes due to
the poles in the Gamma function.
Therefore, Eq. (6.50) has no problems when $k=1$.
Note that
in these cases the normalization factor (6.51) also equals zero
by
the same reason.
Now, we consider the relation
$$\displaystyle(\hat{\mathfrak{b}}_{a}^{+})^{q}\,\psi_{n_{r},l}^{(b)}(r,\varphi)%
=(-1)^{q}2^{-\frac{q}{2}}\sqrt{\frac{\Gamma(n_{r}+1+q)}{\Gamma(n_{r}+1)}}\,%
\psi_{n_{r}+q,l-k}^{(b)}(r,\varphi)\,,$$
(6.54)
from where we see that the same problems for the cases $l<k$ appear again, and only in
the case $k=1$
the relation (6.53) ensures that the operator always
produces physical eigenstates.
The realized analysis of Eqs. (6.50) and (6.54)
reveals
that, again, we face the problem of the quantum anomaly in the general case of rational
values of $\alpha$, and
we can construct a well-defined symmetry operators only in the case
of integer values of $\alpha$,
that
we assume from now on.
Like
the classical case,
we construct the quantum symmetry generators by means of the conformal bridge transformation, obtaining
$$\displaystyle\hat{\mathfrak{S}}_{0}(\hat{\mathcal{O}}_{\mu,\nu}^{\pm})\hat{%
\mathfrak{S}}_{0}^{-1}=(-i)^{\nu}(m\hbar)^{\frac{q}{2}}(\omega)^{\frac{\mu-\nu%
}{2}}\hat{\mathcal{G}}_{\mu,\nu}^{\pm}\,,$$
(6.55)
$$\displaystyle\hat{\mathfrak{S}}_{0}(\hat{\mathcal{S}}_{\mu^{\prime},\nu^{%
\prime}}^{\pm})\hat{\mathfrak{S}}_{0}^{-1}=(-i)^{\nu}(m\hbar)^{q}(\omega)^{%
\frac{\mu^{\prime}-\nu^{\prime}}{2}}\hat{\mathcal{F}}_{\mu^{\prime},\nu^{%
\prime}}^{\pm}\,,$$
(6.56)
and the properties of the classical algebra (with $k=1$) are again preserved at the quantum level.
In conclusion of this discussion, let us
make some comments with respect to the differences between the cases
of even and odd values of $\alpha=q$.
In the even case $q=2\tilde{q}$, we have the true integrals of motion $\hat{\mathcal{G}}_{\tilde{q},\tilde{q}}^{\pm}$,
whose
explicit action on the eigenstates is
$$\displaystyle\hat{\mathcal{G}}_{\tilde{q},\tilde{q}}^{\pm}\psi_{n_{r},l}^{\pm}%
(r,\varphi)=(-1)^{\tilde{q}}\sqrt{\frac{\Gamma(n_{r}+1)\Gamma(n_{r}+2\tilde{q}%
l+1+\tilde{q})}{\Gamma(n_{r}+1-\tilde{q})\Gamma(n_{r}+2\tilde{q}l+1)}}\psi_{n_%
{r}-\tilde{q},l+1}^{\pm}(r,\varphi)\,,$$
(6.57)
$$\displaystyle\hat{\mathcal{G}}_{\tilde{q},\tilde{q}}^{\mp}\psi_{n_{r},l}^{\pm}%
(r,\varphi)=(-1)^{\tilde{q}}\sqrt{\frac{\Gamma(n_{r}+1+\tilde{q})\Gamma(n_{r}+%
2\tilde{q}l+1)}{\Gamma(n_{r}+1)\Gamma(n_{r}+2\tilde{q}l+1-\tilde{q})}}\psi_{n_%
{r}+\tilde{q},l-1}^{\pm}(r,\varphi)\,.$$
(6.58)
From Eq. (6.57) we learn that
the operators $\hat{\mathcal{G}}_{\tilde{q},\tilde{q}}^{\pm}$
annihilate the eigenstates $\psi_{n_{r},l}^{\pm}$
with $n_{r}<\tilde{q}-1$. Also, due to the relation (6.53), and the fact that $\psi_{n_{r},0}^{+}=\psi_{n_{r},0}^{-}:=\psi_{n_{r},0}$,
both equations (6.57) and (6.58)
are equivalent in
the case $l=0$ and $n_{r}>\tilde{q}-1$.
Otherwise, the pole in the Gamma function in (6.58)
produces zero. One also
notes that the index in the wave-functions
that appears on the right hand
side of these equations corresponds to
the same index in the equations (6.21) for the cases $j=\pm 1$.
On the other hand, for the odd case $q=2\tilde{q}+1$ we need to consider instead the integrals $\mathcal{F}_{q,q}^{\pm}$ which produce
$$\displaystyle\hat{\mathcal{F}}_{q,q}^{\pm}\psi_{n_{r},l}^{\pm}(r,\varphi)=(-1)%
^{q}\sqrt{\frac{\Gamma(n_{r}+1)\Gamma(n_{r}+ql+1+q)}{\Gamma(n_{r}+1-q)\Gamma(n%
_{r}+ql+1)}}\psi_{n_{r}-q,l+2}^{\pm}(r,\varphi)\,,$$
(6.59)
$$\displaystyle\hat{\mathcal{F}}_{q,q}^{\mp}\psi_{n_{r},l}^{\pm}(r,\varphi)=(-1)%
^{q}\sqrt{\frac{\Gamma(n_{r}+1+q)\Gamma(n_{r}+ql+1)}{\Gamma(n_{r}+1)\Gamma(n_{%
r}+ql+1-q)}}\psi_{n_{r}+q,l-2}^{\pm}(r,\varphi)\,.$$
(6.60)
These equations
reveal the properties similar to those described for the even case.
The index in the resulting wave-function is
in correspondence with the
index in the equation (6.23).
Then, we conclude that the existence of the true (not depending explicitly on time) integrals
$\hat{\mathcal{G}}_{\tilde{q},\tilde{q}}^{\pm}$ and $\hat{\mathcal{F}}_{q,q}^{\pm}$
reflects
the degeneracy of the system in the unique anomaly-free cases
of the even and odd values of $\alpha$, respectively.
7 Discussion and outlook
The premise of our research here was that on the one hand, geometric properties of space-time must be
reflected in the intrinsic characteristics of the physical systems that inhabit it, and on the other hand, these
peculiarities
must be encoded in a set of well-defined integrals of motion.
Bearing this in mind, we have studied different
non-relativistic forms of dynamics (in the sense of Dirac [65]) associated with
the $\mathfrak{so}(2,1)\cong\mathfrak{sl}(2,\mathbb{R})$ conformal symmetry, namely, the free particle
and the harmonic oscillator, on a cosmic string background [36, 37, 38, 39].
This is the analogous problem of
considering the non-relativistic
conformal invariant dynamical models
on
a two-dimensional cone surface, the deficiency/excess angle of which
is given in terms of the “geometrical parameter”
$\alpha={1}/(1-\frac{4\mu G}{c^{2}})\,,$
where $\mu$ is the linear mass
density of the cosmic string [45, 38] which can be positive,
or negative when topological defects in condensed matter physics and
wormholes are considered [40, 41, 42, 43, 44].
Based on the previous observations related to the shape of the trajectories of the systems
in this space [26, 28],
one might assume
that for some special values of the parameter $\alpha$,
the systems may
have well-defined
hidden
symmetry generators that adequately describe them.
Our results confirm this hypothesis.
The main tools used in this investigation were a local canonical
transformation and the conformal bridge transformation [55].
The first transformation relates the systems under investigation with
their version in the flat Euclidean plane, while the second one
allows us to map integrals from one form of conformal dynamics to another.
The strategy was to start with a free particle in $\mathbb{R}^{2}$
($\alpha=1$),
from where we obtain, by a local canonical transformation,
the solutions of the equations of
motion and the formal integrals of the free system in the cone.
These formal conserved quantities are the images of the Euclidean canonical momenta and the
Galilean boosts,
which are not globally
well-defined functions in phase space for arbitrary $\alpha$ due to their
peculiar angular
dependence.
However, no problems
appear for $\alpha=1/k$, $k=2,3,\ldots,$ while
in general case of rational values $\alpha=q/k$ with $q,k=1,2,\ldots$,
these formal conserved quantities can be used to construct well-defined functions on phase space.
After characterizing the complete symmetry algebra of the free particle in the cone,
we have proceeded to apply the conformal bridge transformation in order to obtain
the integrals and their
symmetry algebra
for the harmonic oscillator in the
same geometric background.
In order to know what to expect for systems on the cone, it is instructive to make
some comments about the models in $\mathbb{R}^{2}$.
In the case of the free particle we have ten second-order integrals that
generate the $\mathfrak{sp}(4,\mathbb{R})$ Lie algebra.
Also,
there are four first-order integrals, namely the canonical momenta and the Galilean boosts, that produce the
centrally extended two-dimensional Heisenberg algebra. The
first order integrals generate an ideal sub-algebra of the complete
Lie type symmetry of the system.
In addition to the principal,
conformal $\mathfrak{so}(2,1)$ algebra,
produced by the Hamiltonian $H$ and
generators of dilatations, $D$,
and special conformal transformations, $K$,
one also can identify some other sub-algebraic structures:
•
A secondary
$\mathfrak{sl}(2,\mathbb{R})$ algebra, generated by the angular momentum and two
dynamical integrals
that commute with the dilatation generator.
•
An
$\mathfrak{su}(2)\oplus\mathfrak{su}(2)$ dynamical symmetry, produced by complex dynamical generators.
When considering the planar isotropic harmonic oscillator, one also finds 10 second-order symmetry generators.
They produce an algebraic structure that is isomorphic to the $\mathfrak{sp}(4,\mathbb{R})$ algebra
(up to complex linear combinations of generators). Furthermore, the system possesses an ideal two-dimensional
Heisenberg sub-algebra, produced by the
four first-order ladder operators.
As for the free particle system, here we also
have
the principal
$\mathfrak{sl}(2,\mathbb{R})$ algebra
of the conformal Newton-Hooke symmetry
generated by the Hamiltonian and the second order radial ladder operators.
Other
sub-algebraic structures
which we would like to highlight are:
•
The very well known $\mathfrak{su}(2)$ symmetry, generated by the angular momentum and two other
true
integrals of motion that commute with the corresponding Hamiltonian.
•
The $\mathfrak{sl}(2,\mathbb{R})\oplus\mathfrak{sl}(2,\mathbb{R})$ symmetry,
whose generators can be identified with the integrals of
the associated Landau problem in symmetric gauge [55].
After applying the conformal bridge transformation to the Euclidean free particle
system, we find that the principal
$\mathfrak{so}(2,1)$ conformal symmetry is mapped into
the
$\mathfrak{sl}(2,\mathbb{R})$ algebra of the conformal Newton-Hooke
symmetry of the harmonic oscillator.
In a similar way,
the mentioned secondary
$\mathfrak{sl}(2,\mathbb{R})$ symmetry of the free dynamics is mapped
to the above-mentioned
$\mathfrak{su}(2)$ symmetry of the
harmonically trapped particle; the complex algebra
$\mathfrak{su}(2)\oplus\mathfrak{su}(2)$ of the free case
is transformed into the
$\mathfrak{sl}(2,\mathbb{R})\oplus\mathfrak{sl}(2,\mathbb{R})$ sub-symmetry of the harmonically
confined system,
and there
is a non-unitary correspondence between the Heisenberg algebras of both models.
In the general case
the conformal bridge transformation maps sub-algebras of one system into
sub-algebras of the another. This
happens due to its nature of the complex canonical transformation
at the classical level, and of the non-unitary similarity transformation at the quantum
level, which is the non-unitary
automorphism of the
conformal $\mathfrak{so}(2,1)\cong\mathfrak{sl}(2,\mathbb{R})$ algebra
[55].
It is important to reinforce the fact that in both cases the Lie algebraic structure
described
here is greater than the Schrödinger symmetry presented in Niederer’s early works
[56, 51].
There, only the generators of the conformal symmetry, the Heisenberg symmetry and rotations were considered.
Let us note here
that at the quantum level, the ten second-order generators in both systems
can be built by taking anti-commutators of
the linear integrals.
Then,
if we consider
a nonlocal operator
of the rotation in $\pi$, $\mathscr{R}=\exp(i\pi\hat{p}_{\varphi})$,
due to relations
$\mathscr{R}x_{i}=-x_{i}\mathscr{R}$ and
$\mathscr{R}^{2}=1$,
it can be identified as a grading operator.
As a result, the Lie symmetry algebras we considered
can be reinterpreted as Lie superalgebras $\mathfrak{osp}(1|4,\mathbb{R})$ of the corresponding
two-dimensional systems with no fermionic degrees of freedom
[68].
This corresponds to the
so-called systems with
the bosonized (hidden) supersymmetry, see
[69, 70, 71, 72, 73, 74]
and references therein.
In one-dimensional quantum case,
the origin of such type of hidden supersymmetries
can be understood in terms of reduction of
supersymmetric systems with fermionic degrees of freedom [72, 74].
It would be interesting to
investigate in a similar way
the origin of the indicated
two-dimensional hidden (bosonized) supersymmetry.
The conformal and rotational symmetries
of the free particle on the cone
are described by the integrals
which are well-defined phase space functions
for arbitrary value of the geometric parameter $\alpha>0$,
and generate
the $\mathfrak{so}(2,1)\oplus\mathfrak{u}(1)$ algebra.
They are quadratic in the above-mentioned
formal in general case basic integrals.
For the case of rational $\alpha=q/k$ ($q,k=1,2,\ldots$ with no common divisors),
the formal basic generators can be used to build the new well-defined integrals of motion:
•
When $q=1$ and $k=2,3,\ldots$ the symmetry algebra
is the same as for the free particle in the flat Euclidean plane ($\alpha=1$).
•
When $q=2,3\ldots$ and $k=1,2,\ldots$, it is possible to construct two different sets of higher-order
integrals of motion: a set $\mathcal{U}_{1}$, consisting of $2q+2$ integrals of order $q$, and the other set
$\mathcal{U}_{2}$, which contains $4q+2$ integrals of order $2q$. We have verified that both sets
generate independent finite nonlinear algebras, and together they produce a larger finite nonlinear algebraic structure.
We have also shown that the nonlinear algebra generated by $\mathcal{U}_{1}$ is an ideal sub-algebra of the complete
nonlinear symmetry.
After analyzing the classical system, we have considered the quantum case.
The system can be quantized for arbitrary values of $\alpha$, and one could expect that
for the rational case, the corresponding quantum versions of the (in general) higher-order hidden symmetry
integrals
will be the spectrum generating operators.
However,
we have revealed a quantum anomaly, since only in the case
of integer values $\alpha=q=2,3,\ldots$ such
spectrum generating integrals indeed can be constructed,
while in the case of rational non-integer values of $\alpha$
the quantum analogs of the classically well-defined hidden symmetry generators
take out the states from the physical Hilbert space.
Here, there is a couple of open interesting questions.
First,
knowing that via the corresponding non-relativistic limit [57, 58, 61],
one can relate conformal symmetry of the free particle
on the cone with the corresponding Killing and conformal Killing
vector fields of the cosmic string space-time background,
a natural question is what geometrical objects correspond
to the considered hidden symmetry generators.
We speculate that they can be related
to some Killing and conformal Killing tensors [1, 75]
of the cosmic string space-time.
Second, it would be interesting to look what happens with the quantum anomaly
under perspective of unconventional boundary conditions, which were
considered for quantum systems in the cone in [27].
Finally, we reconstruct the complete information on the symmetry algebra for the harmonic
oscillator at the classical and quantum levels through the conformal bridge transformation.
From here we have learned that the generators of the $\mathfrak{sl}(2,\mathbb{R})\oplus\mathfrak{u}(1)$
algebra are well-defined for any value of $\alpha$, but only
in the rational case $\alpha=q/k$ there
is an extended set of the well-defined integrals of motion.
This reflects the peculiarity of the geometry of the trajectories of the harmonically trapped particle
on the cone: the trajectories are closed only when $\alpha=q/k$.
The number of radial minima/maxima on the trajectory is given by
$\mathscr{N}_{\text{min}/\text{max}}=k(q\,\text{mod}\,2+1)$.
In particular,
for $k=1$ one has $\mathscr{N}_{\text{min}/\text{max}}=1$ when $\alpha$ is even and
$\mathscr{N}_{\text{min}/\text{max}}=2$
when $\alpha$ is odd. At the quantum level it is seen that the
value of $\alpha$ explicitly determines the
degeneracy of the energy spectrum of the system, while
the symmetry operators
and their action on the corresponding eigenstates of the system are obtained directly
from those of the free particle
by employing the quantum version of the conformal bridge transformation.
From there, the quantum anomaly is revealed automatically: only when $\alpha=q=2,3,\ldots$,
it is possible
to have the well-defined higher-order differential
operators corresponding to hidden symmetries,
which reflect the spectral degeneracy.
In conclusion, let us indicate some other problems for which
the results and ideas employed in this article can be used.
First, we note that the local
canonical transformation that relates the cone with the flat Euclidean plane
can be applied for the analysis of other
central potentials on the conical background.
One can expect that the well-defined hidden symmetry generators
in such systems can appear only at special values of $\alpha$, and
that quantum anomaly can also emerge there.
In particular, all the analysis presented here can immediately be transferred and
generalized for the case of conformal mechanics on the cone
as it was done in [55] in the flat Euclidean plane.
In the same vein, the results of [55] can be employed and generalized
immediately for the Landau problem on the cone.
Second, it would be interesting to employ the conformal bridge transformation
for the systems in different geometries, such as the Lobachevsky plane and
the non-commutative plane.
These both geometries are used in the description of anyons
[76, 77, 78],
and on the other hand, one has to bare in mind that anyons can directly be related
with the cone geometry [79, 80, 81].
Finally, as it was shown in [55],
the non-unitary generator $\hat{\mathfrak{S}}$ of the conformal bridge transformation is
the fourth order root of the space reflection operator $\mathcal{P}$,
which in the present two-dimensional case has to be substituted for
the above-mentioned non-local operator $\mathscr{R}=\exp(i\pi\hat{p}_{\varphi})$.
At the same time, it is easy to see from the explicit form (4.20)
of $\hat{\mathfrak{S}}$
in terms of the $\mathfrak{sl}(2,\mathbb{R})$ generators of conformal symmetry
that it is $\mathcal{PT}$ symmetric,
where $\mathcal{T}$ corresponds to
the time reversal (anti-linear) transformation
[82, 83, 84, 86, 85].
Taking also into account that its classical analog
generates complex canonical transformations
in the phase space, and that conformal symmetry
plays important role in some $\mathcal{PT}$-symmetric
systems with peculiar properties [86, 87, 88],
it would be very interesting to look at the conformal bridge transformations in the
light of the $\mathcal{PT}$ symmetry.
Acknowledgements
The work was partially supported by
the FONDECYT Project 1190842.
References
[1]
M. Cariglia,
“Hidden symmetries of dynamics in classical and quantum physics,”
Rev. Mod. Phys. 86 (2014) 1283
[arXiv: 1411.1262 [math-th]].
[2]
J. de Boer, F. Harmsze and T. Tjin,
“Nonlinear finite W symmetries and applications
in elementary systems,”
Phys. Rept. 272 (1996) 139
[arXiv: hep.th/9503161].
[3]
J. Beckers, Y. Brihaye and N. Debergh, “On realizations of ‘nonlinear’ Lie algebras
by differential operators,”
J. Phys. A 32 (1999) 2791
[arXiv: hep.th/9803253].
[4]
W. Pauli,
“Über das wasserstoffspektrum vom standpunkt der neuen quantenmechanik,”
Z. Physik 36 (1926) 336.
[5]
D. M. Fradkin,
“Three-dimensional isotropic harmonic oscillator and SU${}_{3}$,”
Am. J. Phys. 33 (1965) 207.
[6]
M. S. Plyushchay and A. Wipf,
“Particle in a self-dual dyon background: hidden free nature, and exotic superconformal symmetry,”
Phys. Rev. D 89 (2014) 045017
[arXiv: 1311.2195 [hep-th]].
[7]
L. Inzunza, M. S. Plyushchay and A. Wipf,
“Hidden symmetry and (super)conformal mechanics in a monopole background,”
JHEP 04 (2020) 028
[arXiv: 2002.04341 [hep-th]].
[8]
S. Wojciechowski,
“Superintegrability of the Calogero-Moser system,”
Phys. Lett. A 95 (1983) 279.
[9]
V. B. Kuznetsov,
“Hidden symmetry of the quantum Calogero-Moser system,”
Phys.
Lett. A 218 (1996) 212
[arXiv: solv-int/9509001].
[10]
S. W. Hawking,
“Black hole explosions,”
Nature 248 (1974) 30.
[11]
W. G. Unruh,
“Notes on black hole evaporation,”
Phys. Rev. D 14 (1976) 870.
[12]
R. Britto-Pacumio, J. Michelson, A. Strominger and A. Volovich,
“Lectures on superconformal quantum mechaics and multi-black hole moduli spaces,”
NATO Sci. Ser. C 556 (2000) 255
[arXiv: hep-th/9911066].
[13]
P. Claus, M. Derix, R. Kallosh, J. Kumar, P. K. Townsend and A. Van Proeyen,
“Black holes and superconformal mechanics,”
Phys. Rev. Lett. 81 (1998) 4553
[arXiv: hep-th/9804177].
[14]
J. A. de Azcárraga, J. M. Izquierdo, J. C. Pérez Bueno and P. K. Townsend,
“Superconformal mechanics, black holes, and nonlinear realizations,”
Phys. Rev. D 59 (1999) 084015
[arXiv: hep-th/9810230].
[15]
G. W. Gibbons and P. K. Townsend,
“Black holes and Calogero models,”
Phys. Lett. B 454 (1999) 187
[arXiv: hep-th/9812034].
[16]
V. de Alfaro, S. Fubini and G. Furlan,
“Conformal invariance in quantum mechanics,”
Nuovo Cim. A 34 (1976) 569.
[17]
A. Galajinsky, K. Polovnikov and O. Lechtenfeld,
“${\cal N}{=}\,4$ superconformal Calogero models,”
JHEP 11 (2007) 008
[arXiv: 0708.1075 [hep-th]].
[18]
N. Kozyrev, S. Krivonos, O. Lechtenfeld and A. Sutulin,
“SU(2$|$1) supersymmetric mechanics on curved spaces,”
JHEP 05 (2018) 175
[arXiv: 1712.09898 [hep-th]].
[19]
B. Carter,
“Axisymmetric black hole has only two degrees of freedom,”
Phys. Rev. Lett. 26 (1971) 331.
[20]
G. W. Gibbons, R. H. Rietdijk and J. W. van Holten,
“SUSY in the sky,”
Nucl. Phys. B 404 (1993) 42
[arXiv: hep-th/9303112].
[21]
M. Cariglia,
“Quantum mechanics of Yano tensors: Dirac equation in curved spacetime,”
Class. Quant. Grav. 21 (2004) 1051
[arXiv: hep-th/0305153].
[22]
V. P. Frolov and D. Kubiznak,
“Hidden symmetries of higher dimensional rotating black holes,”
Phys. Rev. Lett. 98 (2007) 011101
[arXiv: gr-qc/0605058].
[23]
V. P. Frolov and D. Kubiznak,
“Higher-dimensional black holes: Hidden symmetries and separation of variables,”
Class. Quant. Grav. 25 (2008) 154005
[arXiv: 0802.0322 [hep-th]].
[24]
V. Frolov, P. Krtous and D. Kubiznak,
“Black holes, hidden symmetries, and complete integrability,”
Living Rev. Rel. 20 (2017) no.1, 6
[arXiv: 1705.05482 [gr-qc]].
[25]
G.’t Hooft, “Non-perturbative 2 particle scattering amplitudes in 2+1 dimensional quantum gravity,”
Commun. Math. Phys. 117 (1988) 685.
[26]
S. Deser and R. Jackiw,
“Classical and quantum scattering on a cone,”
Comm.
Math. Phys 118 (1988) 495.
[27]
B.S. Kay, and U.M. Studer, “Boundary conditions for quantum mechanics on cones and fields around cosmic strings,”
Commun. Math. Phys. 139 (1991) 103.
[28]
C. Furtado and F. Moraes,
“Harmonic oscillator interacting with conical singularities,”
J. Phys. A 33 (2000) 5513.
[29]
J. L. A. Coelho and R. L. P. G. Amaral,
“Coulomb and quantum oscillator problems in conical spaces with arbitrary dimensions,”
J. Phys. A 35 (2002) 5255
[arXiv: gr-qc/0111114].
[30]
C. C. Barros Jr,
“Quantum mechanics in curved space-time,”
Eur. Phys. J. C 42 (2005) 119
[arXiv: 0409064 [physics.gen-ph]].
[31]
G. De A.Marques, V. B. Bezerra and S. G. Fernandes,
“Exact solution of the Dirac equation for a Coulomb and scalar potentials in the gravitational field of a cosmic string,”
Phys. Lett. A 341 (2005) 39.
[32]
K. Kowalski and J. Rembieliński,
“On the dynamics of a particle on a cone,”
Annals of Physics 329 (2013) 146
[arXiv: 1304.4412 [quant-ph]].
[33]
F. M. Andrade and E. O. Silva,
“Effects of spin on the dynamics of the 2D Dirac oscillator in the magnetic cosmic string background,”
Eur. Phys. J. C 74 (2014) 3187
[arXiv: 1403.4113 [hep-th]].
[34]
M. Hosseinpour, F. M. Andrade, E. O. Silva and H. Hassanabadi,
“Scattering and bound states for the Hulthén potential in a cosmic string background,”
Eur. Phys. J. C 77 (2017) 373
[arXiv: 1608.03558 [hep-th]].
[35]
F. Ahmed,
“Relativistic quantum dynamics of spin-0 massive charged particle in the presence of external fields in 4D curved space-time with a cosmic string,”
Eur. Phys. J. Plus 135 (2020) 108.
[arXiv:1910.12700 [hep-th]].
[36]
T. Kibble,
“Topology of cosmic domains and strings,”
J. Phys. A 9 (1976) 1387.
[37]
A. Vilenkin, “Gravitational field of vacuum domain walls and strings,”
Phys. Rev. D 23 (1981) 852.
[38]
A. Vilenkin,
“Cosmic strings and domain walls,”
Phys. Rept. 121 (1985) 263.
[39]
J. S. Dowker,
“Quantum field theory on a cone,”
J. Phys. A: Math. Gen. 10 (1977) 115.
[40]
M. Visser,
“Traversable wormholes: Some simple examples,”
Phys. Rev. D 39 (1989) 3182.
[41]
J. G. Cramer, R. L. Forward, M. S. Morris, M. Visser, G. Benford, and G. A. Landis,
“Natural wormholes as gravitational lenses,”
Phys. Rev. D 51 (1995) 3117 .
[42]
M. O. Katanaev and I. V. Volovich,
“Theory of defects in solids and three-dimensional gravity,”
Annals Phys.
216 (1992) 1.
[43]
G. E. Volovik, The universe in a helium droplet
(Oxford Science Publications, 2003).
[44]
N. S. Manton,
“Five vortex equations,”
J. Phys. A 50 (2017) 125403
[arXiv: 1612.06710 [hep-th]].
[45]
D. D. Sokolov and A. A. Starobinsky,
“On the structure of curvature tensor on conical singularities,”
Dokl. Akad. Nauk
234
(1977) 1043 [Sov. Phys. - Dokl. 22 (1977) 312].
[46]
M. Aryal, L. H. Ford and A. Vilenkin,
“Cosmic strings and black holes,”
Phys. Rev. D 34 (1986) 2263.
[47]
C. Furtado, B. G. C. da Cunha, F. Moraes, E. R. B. de Mello, V. B. Bezzerra,
“Landau levels in the presence of disclinations,”
Phys. Lett. A 195 (1994) 90.
[48]
S. N. Solodukhin,
“Conical singularity and quantum corrections to the entropy of a black hole,”
Phys. Rev. D 51 (1995) 609
[arXiv: hep-th/9407001].
[49]
M. G. Germano, V. B. Bezerra, and E. R. B. de Mello,
“Gravitational effects due to a cosmic string in Schwarzschild spacetime,”
Class. Quant. Grav. 13 (1996) 2663.
[50]
E. R. B. de Mello and A. A. Saharian,
“Vacuum polarization induced by a cosmic string in anti-de Sitter spacetime,”
J. Phys. A 45 (2012) 115002
[arXiv: 1110.2129 [hep-th]].
[51]
U. Niederer,
“The maximal kinematical invariance group of the harmonic oscillator,”
Helv. Phys. Acta 46 (1973) 191.
[52]
P. D. Alvarez, J. Gomis, K. Kamimura, and M. S. Plyushchay,
“(2+1)D Exotic Newton-Hooke symmetry, duality and projective phase,”
Annals Phys. 322 (2007) 1556
[arXiv: hep-th/0702014].
[53]
A. Galajinsky,
“Conformal mechanics in Newton-Hooke spacetime,”
Nucl. Phys. B 832 (2010) 586
[arXiv: 1002.2290 [hep-th]].
[54]
K. Andrzejewski,
“Conformal Newton-Hooke algebras, Niederer’s transformation and Pais-Uhlenbeck oscillator,”
Phys. Lett. B 738 (2014) 405
[arXiv: 1409.3926 [hep-th]].
[55]
L. Inzunza, M. S. Plyushchay, and A. Wipf,
“Conformal bridge between asymptotic freedom and confinement,”
Phys. Rev. D 101 (2020) 105019
[arXiv: 1912.11752 [hep-th]].
[56]
U. Niederer,
“Maximal kinematical invariance group of the free Schrödinger equation,”
Helv. Phys. Acta 45 (1972) 802.
[57]
C. R. Hagen, (1972).
“Scale and conformal transformations in Galilean-covariant field Theory,”
Phys. Rev. D 5 (1972) 377.
[58]
A. O. Barut,
“Conformal group $\rightarrow$ Schrödinger group
$\rightarrow$
dynamical group - the maximal kinematical group of the massive
Schrödinger particle,”
Helv. Phys. Acta 46 (1973) 496.
[59]
C. Leiva and M. S. Plyushchay,
“Conformal symmetry of relativistic and nonrelativistic systems and AdS/CFT correspondence,”
Annals Phys. 307 (2003) 372
[arXiv: hep-th/0301244].
[60]
M. Henkel and J. Unterberger,
“Schrödinger invariance and space-time symmetries,”
Nucl. Phys. B 660 (2003) 407
[arXiv: hep-th/0302187].
[61]
D. T. Son,
“Toward an AdS/cold atoms correspondence:
A geometric realization of the Schrödinger symmetry,”
Phys. Rev. D 78 (2008) 046003
[arXiv: 0804.3972 [hep-th]].
[62]
A. Bagchi and R. Gopakumar,
“Galilean conformal algebras and AdS/CFT,”
JHEP 0907 (2009) 037
[arXiv: 0902.1385 [hep-th]].
[63]
M. S. Plyushchay,
“Quantization of the classical $SL(2,R)$ system and representations of $\overline{SL(2,R)}$ group,”
J. Math. Phys. 34 (1993) 3954 .
[64]
V. Bargmann,
“Irreducible unitary representations of the Lorentz group,”
Annals of Mathematics, 48 (1947) 568.
[65]
P. A. M. Dirac,
“Forms of relativistic dynamics,”
Rev. Mod. Phys. 21 (1949) 392.
[66]
F. Correa, V. Jakubsky and M. S. Plyushchay,
“PT-symmetric invisible defects and
confluent Darboux-Crum transformations,”
Phys. Rev. A 92 (2015) 023839
[arXiv: 1506.00991 [hep-th]].
[67]
L. Inzunza and M. S. Plyushchay,
“Klein four-group and Darboux duality in conformal mechanics,”
Phys. Rev. D 99 (2019) 125016
[arXiv: 1902.00538 [hep-th]].
[68]
M. de Crombrugghe and V. Rittenberg,
“Supersymmetric quantum mechanics,”
Annals Phys.
151 (1983) 99.
[69]
M. S. Plyushchay,
“Deformed Heisenberg algebra, fractional spin fields and supersymmetry without fermions,”
Annals Phys. 245 (1996) 339
[arXiv: hep-th/9601116].
[70]
M. S. Plyushchay,
“Hidden nonlinear supersymmetries in pure parabosonic systems,”
Int. J. Mod. Phys. A 15
(2000) 3679
[arXiv: hep-th/9903130].
[71]
F. Correa and M. S. Plyushchay,
“Hidden supersymmetry in quantum bosonic systems,”
Annals Phys. 322 (2007) 2493
[arXiv: hep-th/0605104].
[72]
V. Jakubsky, L. M. Nieto and M. S. Plyushchay,
“The origin of the hidden supersymmetry,”
Phys. Lett. B 692 (2010) 51
[arXiv: 1004.5489 [hep-th]].
[73]
R. Bonezzi, O. Corradini, E. Latini, and A. Waldron,
“Quantum mechanics and hidden superconformal symmetry,”
Phys. Rev. D 96 (2017) 126005
[arXiv: 1709.10135 [hep-th]].
[74]
L. Inzunza and M. S. Plyushchay,
“Hidden superconformal symmetry: Where does it come from?,”
Phys. Rev. D 97 (2018) 045002
[arXiv: 1711.00616 [hep-th]].
[75]
M. Crampin, “Hidden symmetries and Killing tensors,”
Reports on Math. Phys 20 (1984) 31.
[76]
M. S. Plyushchay,
“The model of relativistic particle with torsion,”
Nucl. Phys. B 362 (1991) 54 .
[77]
P. A. Horvathy and M. S. Plyushchay,
“Non-relativistic anyons, exotic Galilean symmetry and noncommutative plane,”
JHEP 06 (2002) 033
[arXiv: hep-th/0201228].
[78]
P. A. Horvathy and M. S. Plyushchay,
“Anyon wave equations and the noncommutative plane,”
Phys. Lett. B 595 (2004) 547
[arXiv: hep-th/0404137].
[79]
J. M. Leinaas and J. Myrheim,
“On the theory of identical particles,”
Nuovo Cim. B 37 (1977) 1.
[80]
J. M. Leinaas and J. Myrheim,
“Intermediate statistics for vortices in superfluid films,”
Phys. Rev. B 37 (1988) 9286.
[81]
R. MacKenzie and F. Wilczek,
“Peculiar spin and statistics in two space dimensions,”
Int. J. Mod. Phys. A 3 (1988) 2827.
[82]
C. M. Bender and S. Boettcher,
“Real spectra in non-Hermitian Hamiltonians having PT symmetry,”
Phys. Rev. Lett. 80 (1998) 5243
[arXiv: physics/9712001].
[83]
A. Mostafazadeh,
“Pseudo-Hermiticity versus PT symmetry. The necessary condition for the reality of the spectrum,”
J. Math. Phys. 43 (2002) 205
[arXiv: math-ph/0107001].
[84]
C. M. Bender,
“Making sense of non-Hermitian Hamiltonians,”
Rept. Prog. Phys. 70 (2007) 947
[arXiv: hep-th/0703096 [hep-th]].
[85]
A. Fring,
“PT-symmetric deformations of integrable models,”
Phil. Trans. Roy. Soc. Lond. A 371 (2013) 20120046
[arXiv: 1204.2291 [hep-th]].
[86]
P. Dorey, C. Dunning and R. Tateo,
“From PT-symmetric quantum mechanics to conformal field theory,”
Pramana 73 (2009) 217
[arXiv: 0906.1130 [hep-th]].
[87]
J. Mateos Guilarte and M. S. Plyushchay,
“Perfectly invisible $\mathcal{PT}$-symmetric zero-gap systems, conformal
field theoretical kinks, and exotic nonlinear supersymmetry,”
JHEP 12 (2017) 061
[arXiv: 1710.00356 [hep-th]].
[88]
J. Mateos Guilarte and M. S. Plyushchay,
“ Nonlinear symmetries of perfectly invisible PT-regularized conformal and superconformal
mechanics systems,”
JHEP 01 (2019) 194
[arXiv: 1806.08740 [hep-th]]. |
Knotted nodal lines in superpositions of Bessel-Gaussian light beams
Tomasz Radożycki
t.radozycki@uksw.edu.pl
Faculty of Mathematics and Natural Sciences, College of Sciences, Institute of Physical Sciences, Cardinal Stefan Wyszyński University, Wóycickiego 1/3, 01-938 Warsaw, Poland
Abstract
A simple analytical way of creating superpositions of Bessel-Gaussian light beams with knotted nodal lines is proposed. It is based on the equivalence between the paraxial wave equation and the two-dimensional Schrödinger equation for a free particle. The $2D$ Schrödinger propagator is expressed in terms of Bessel functions, which allows to obtain directly superpositions of beams with a desired topology of nodal lines. Four types of knots are constructed in the explicit way: the unknot, the Hopf link, the Borromean rings and the trefoil. It is also shown, using the example of the figure-eight knot, that more complex structures require larger number of constituent beams as well as high precision both from the numerical and the experimental side. A tiny change of beam’s intensity can lead to the knot “switching”.
I Introduction
In recent decades, it has become possible to analytically “design” and experimentally generate beams of light with given topological properties. This marriage of optical phenomena and topology has led to the birth of the so-called topological optics. Topological effects have entered into light propagation in several instances.
One intensively studied family constitute phase singularities of helical character. Such waves, apart from spin, are also endowed with orbital angular momentum. Particular interest of researchers was attracted here by Laguerre-Gaussian lg ; lg2 ; arlt2 or Bessel arlt2 ; durnin1 ; durnin2 ; vg ; ibb1 ; tr3 beams possessing a property of vorticity and an associated “charge” or topological index.
An example that can also be classified as topological are polarization singularities connected with the inability to fully specify the polarization at certain places nye ; nyh ; freu ; ddd ; car . Yet another idea is that of the “knotted” light. This primarily involves electric or magnetic field lines which can get entangled ran ; ir ; besi ; db ; kedia ; arr ; ho ; arr2 ; kpi , but also the nodal lines of wave intensity or, in other words, optical vortex lines which can develop topologically non-trivial structures both in exact and paraxial regimes bd ; den ; bd2 ; bkj ; kle ; deklerk ; su . This latter case covering some kind of doubly-topological structures (i.e., knotted vortices, which per se are topological entities mentioned above) is our main concern in this work. From an experimental point of view, these types of beams can be produced, which opens up many research and application possibilities leach ; leach1 ; sha ; wil .
Quite paradoxically, while dealing with light beams it is not bright regions but those of darkness, where the exact destructive interference occurs, that are of main interest. They are connected with the presence of phase singularities, as stated above, since the phase is undetermined for vanishing field. Such areas of suppressed intensity can serve as traps for both polarizable neutral particles with negative polarization constant (such as blue-detuned atoms) dav ; odtna ; sheng ; frie ; trb , for charged particles like electrons through the ponderomotive potential ibb1 , and even for micrometer-sized objects. In case of neutral atoms the gradient forces arising from the inhomogeneities of the electric field occur due to the Stark effect dk and for larger objects the trapping appears via Mie scattering mie ; neu ; dho1 ; die .
Generating nodal lines with highly non-trivial geometrical properties and studying knotted structures as such, can prove useful for instance for manipulating particles using this mechanism of trapping and guiding or through implementations into other physical systems. Many potential practical applications range from physics through chemistry to biology or medicine liu ; ste ; fazal ; pad ; woe ; bowpa ; grier1 ; tka ; mar ; hall ; brad ; dan .
The question of constructing a kind of a knotted trap for silica spheres was undertaken in sha and for neutral atoms in trc . In this latter work we proposed a simple method to design “arbitrary” traps of this sort from the superposition of simple Gaussian beams. It was also shown in the numerical way that the trajectories of neutral polarizable particles are really confined on these knotted nodal lines.
However, in principle this kind of traps can be made of more complex beams, as for instance Laguerre-Gaussian (LG) beams or Bessel-Gaussian (BG) beams for which some knotted nodal lines are successfully obtained bd1 ; bd3 ; lea ; king ; pad2 . Therefore, in the present work we would like to extend to BG beams the method advocated in our previous paper trc . From the experimentalist point of view it seems essential that knots could be formed of a variety of beam types. Contrary to some earlier attempts the present approach should allow, in principle, to construct in an easy and straighforward way any knots (composed of nodal lines) that can be obtained from the so-called Milnor polynomial. Moreover, the BG beams are already routinely achievable in experiments. The procedure leading to this result will be discussed in detail in Section II.
When dealing with laser beams it has proved convenient to introduce dimensionless coordinates through the relations
$$\xi_{x}=kx,\;\;\;\xi_{y}=ky,\;\;\;\xi=\sqrt{\xi_{x}^{2}+\xi_{y}^{2}},\;\;\;%
\zeta=kz.$$
(1)
This corresponds to measuring distances in the units $k^{-1}=\lambda/2\pi$ and ensures that our formulas become as simple as possible. It should be noted that for the third component the symbol $\zeta$ instead of $\xi_{z}$ is used. This is due to the fact that this coordinate plays some special role in our considerations, i.e. that of the “time” in the corresponding Schrödinger equation. Therefore, in what follows the bold mathematical symbols stand for only two-dimensional vectors (i.e., $\bm{\xi}=[\xi_{x},\xi_{y}]$ and $\bm{r}=[x,y]$).
A real wave close to the propagation axis satisfies the so called paraxial equation which in the dimensionless coordinates has the form:
$$\mathcal{4}\Psi(\bm{\xi},\zeta)+2i\partial_{\zeta}\Psi({\bm{\xi}},\zeta)=0,$$
(2)
with $\mathcal{4}$ representing the two-dimensional Laplace operator. In order to construct superpositions of BG beams on one hand satisfying Eq. (2) and on the other displaying a given topological structure the clear similarity of (2) to the Schrödinger equation for a free particle in $2D$ will be exploited as in trc . The latter equation has the form:
$$-\frac{\hbar^{2}}{2m}\,\mathcal{4}\Psi(\bm{r},t)=i\hbar\partial_{t}\Psi(\bm{r}%
,t),$$
(3)
and becomes identical to (2) upon the the identification:
$$mc^{2}=\hbar\omega,\;\;\;\;\zeta=ct.$$
(4)
This allows the well-known Schrödinger propagator to be made use of, as described in general terms in the next section. In Section III this general procedure is implemented in five subsequent examples: the unknot, the Hopf link, the Borromean rings, the trefoil and the figure-eight knot. In a systematic way, the concrete superpositions of coaxial BG beams are derived, which yield these nodal structures.
II General procedure
The two-dimensional free Schrödinger propagator may be written in terms of Bessel functions $J_{n}(x)$ with the use of our dimensionless variables (1) as follows:
$$\displaystyle K(\xi,\phi,\zeta;\xi^{\prime},\phi^{\prime},\zeta^{\prime})=%
\frac{1}{2\pi}\,\sum\limits_{n=-\infty}^{\infty}e^{in(\phi-\phi^{\prime})}$$
$$\displaystyle\;\;\;\;\times\int\limits_{0}^{\infty}\mathrm{d}t\,tJ_{n}(t\xi)J_%
{n}(t\xi^{\prime})e^{-i(\zeta-\zeta^{\prime})t^{2}/2}.$$
(5)
One can verify by a direct calculations that the paraxial equation (2) (or Schrödinger equation) is satisfied by the above expression. It is inessential whether $K(\xi,\phi,\zeta;\xi^{\prime},\phi^{\prime},\zeta^{\prime})$ itself meets the condition of the paraxial approximation, which for certain function $f(\bm{\xi},\zeta)$ in our units reads
$$\partial^{2}_{\zeta}f(\bm{\xi},\zeta)\ll\partial_{\zeta}f(\bm{\xi},\zeta),$$
(6)
since $K$ is only a tool for obtaining a beam envelope $\Psi(\bm{\xi},\zeta)$ of the physical significance for our considerations. As we will see in the following sections, this envelope is virtually a superposition of BG paraxial modes.
When $\zeta\rightarrow\zeta^{\prime}$, one obtains
$$\lim_{\zeta\rightarrow\zeta^{\prime}}K(\xi,\phi,\zeta;\xi^{\prime},\phi^{%
\prime},\zeta^{\prime})=\delta^{(2)}(\bm{\xi}-\bm{\xi}^{\prime}),$$
(7)
as it is expected. All one needs is to apply the identity
$$\delta\left(\phi-\phi^{\prime}\right)=\frac{1}{2\pi}\sum_{n=-\infty}^{\infty}e%
^{in(\phi-\phi^{\prime})}$$
(8)
together with arfken
$$\delta\left(x-x^{\prime}\right)=x\int_{0}^{\infty}\mathrm{d}t\,tJ_{n}\left(xt%
\right)J_{n}\left(x^{\prime}t\right),$$
(9)
and then exploiting the property of the Dirac delta function in two dimensions:
$$\frac{1}{\xi}\,\delta(\phi-\phi^{\prime})\delta(\xi-\xi^{\prime})=\delta^{(2)}%
(\bm{\xi}-\bm{\xi}^{\prime}).$$
(10)
leads to (7).
Now we move on to the construction of a knot of a certain required topology. In this work we focus on four chosen knotted structures: the unknot (or simple ring), the Hopf link, the Borromean rings and the trefoil and at the end it is shown using the example of the figure-eight knot what kind of complications occur for more complex windings. The details of the construction are the subject of the knot theory to be found elsewhere bra ; bkj ; king ; bode and stay beyond the scope of this work. The main steps are listed below.
A knot is in general a closed curve in $\mathbb{R}^{3}$ which constitutes a homeomorphic image of $S^{1}$. Contrary to our everyday’s meaning of a knot, it forms a closed loop. Typically, this curve is a nodal line of a certain complex-valued function of the spatial variables $x,y,z$ (or $\xi_{x},\xi_{y},\zeta$ in our case). If it is composed of several disjoint loops, that are tangled up, it is called a link, as the Hopf link for instance. The other example of the same family are the Borromean rings. Both of these cases, among others, will be dealt with in the next section.
The construction under consideration proceeds as follows. As a first step, one creates a polynomial $q(u,v)$ of two complex variables $u$ and $v$ which satisfy the condition for the three-dimensional sphere: $|u|^{2}+|v|^{2}=1$. The examples of such polynomials are considered in the following section. All the nodal points (i.e, those where where $q(u,v)=0$) represent an algebraic knot. For the first three cases dealt with in the following section, these polynomials have the form:
$$q(u,v)=\prod\limits_{k=0}^{n-1}(u-\varepsilon_{n}^{(k)}v),$$
(11)
with $\varepsilon_{n}^{(k)}$, $k=0,1,2,\ldots,n-1$, denoting the subsequent $n$th roots of unity.
Such a knots, however, would be located on $S^{3}$, which is important for their classification, but inappropriate for our purposes. We are concerned about knots in ${\mathbb{R}}^{3}$, so the next step is to exploit the stereographic projection by means of the relations:
$$\displaystyle u(\bm{\xi},\zeta)$$
$$\displaystyle=\frac{\bm{\xi}^{2}+\zeta^{2}-1+2i\zeta}{\bm{\xi}^{2}+\zeta^{2}+1},$$
(12a)
$$\displaystyle v(\bm{\xi},\zeta)$$
$$\displaystyle=\frac{2(\xi_{x}+i\xi_{y})}{\bm{\xi}^{2}+\zeta^{2}+1}.$$
(12b)
The obtained equation $q(u(\bm{\xi},\zeta),v(\bm{\xi},\zeta))=0$ defines a knot curve as an intersection of two surfaces in three-dimensional space: $\operatorname{Re}q(u(\bm{\xi},\zeta),v(\bm{\xi},\zeta))=0$ and $\operatorname{Im}q(u(\bm{\xi},\zeta),v(\bm{\xi},\zeta))=0$. Recalling that $q(u,v)$ was a polynomial, $q(u(\bm{\xi},\zeta),v(\bm{\xi},\zeta))$ can again be treated as a polynomial (the so called Milnor polynomial milnor ), upon removing the common denominator appearing in (12). The appropriate Milnor polynomials will be below denoted with $q_{M}(\bm{\xi},\zeta)$. These polynomials will constitute the basis for obtaining superpositions of BG beams with identical topology of nodal lines. The “initial” envelope $\Psi(\bm{\xi},0)$ will be created as a sum of such BG modes, that for small $\xi$ exhibit the same bahaviour as $q_{M}(\bm{\xi},0)$, i.e. the following replacement will be made:
$$q_{M}(\bm{\xi},0)\mapsto\Psi(\bm{\xi},0)=e^{-\kappa\xi^{2}}\sum_{l,m}\alpha_{%
lm}e^{2im\phi}J_{m}(\chi_{lm}\xi),$$
(13)
together with the appropriate choice of the coefficients $\alpha_{lm}$. The summations with respect to $l$ and $m$ run over a range dictated by the form of a given Milnor polynomial for $\zeta=0$, and $\kappa>0$. It is outlined in detail in the next section.
The values of coefficients $\alpha_{lm}$ are chosen so as to exactly (but apart from the Gaussian factor $e^{-\kappa\xi^{2}}$) reproduce the polynomial $q_{M}(\bm{\xi},0)$ as $\xi\ll 1$. On the other hand, the values of $\chi_{lm}$ can be set arbitrarily, dependent on the specific beams used in an experiment, and are connected with half-aperture of the appropriate beam cone (see formula (25b)). Consequently $\alpha_{lm}$’s which can be related to the beams’ intensities, are functions of $\chi_{lm}$’s. Later, when considering specific examples, we prefer to denote the coefficients $\alpha_{lm}$ with $\alpha_{l}$, $\beta_{l}$, $\gamma_{l}$ and so on, in order to avoid too many indices. Usually, at least for the simplest knots, $m$ runs from 1 to say 2, 3 or maximally 4, so the symbols $\alpha_{l}$, $\beta_{l}$, $\gamma_{l}$, $\ldots$ are sufficient. For the same reason the symbols $\chi_{lm}$’s are modified in an obvious way.
Finally the full beam is found according to the formula
$$\Psi(\bm{\xi},\zeta)=\int d^{2}\xi^{\prime}K({\bm{\xi}},\zeta;{\bm{\xi}}^{%
\prime},0)\Psi(\bm{\xi}^{\prime},0)$$
(14)
and automatically satisfies the paraxial equation (2).
III Specific knotted beams
In this section five concrete examples of knotted beams are dealt with. The suggested technique allows in principle to create BG beams with any knotted topology derived from a Milnor polynomial $q_{M}$. Yet, more complex knots require superpositions of larger number of mods, the structure of the nodal lines becomes shallower, and the computer time necessary to visualize them grows significantly. For this reason, our analysis will be restricted to relatively simple knots, although – apart from the first – far non-trivial ones.
III.1 The unknot
The simplest knot, called the unknot is a simple ring. The polynomial $q(u,v)$, that yields such a ring, can be obtained from (11) when setting $n=1$:
$$q(u,v)=u-v.$$
(15)
Upon substitution of $u$ and $v$ as given in (12), the Milnor polynomial at $\zeta=0$ is found in the form
$$q_{M}(\bm{\xi},0)=-1+\xi_{x}^{2}+\xi_{y}^{2}-2(\xi_{x}+i\xi_{y}),$$
(16)
In order to preserve the capability of resizing the knot and adjust its dimension to the conditions of the paraxial approximation, a parameter $\gamma$ will be introduced, which can be called a “scaling parameter”, thereby modifying the form of the Milnor polynomial at $\zeta=0$ to
$$q_{M}(\bm{\xi},0)=-1+\gamma^{2}(\xi_{x}^{2}+\xi_{y}^{2})-2\gamma(\xi_{x}+i\xi_%
{y}),$$
(17)
without, however, altering the emerging-knot topology. Larger values of $\gamma$ lead to smaller knots. In the figures presented underneath, the value of this parameter is set to $10$, which results in typical knot sizes to be small fractions of a wavelength. For optical frequencies, they can be then called “nano-knots”.
Let us now replace (17) with
$$q_{M}(\bm{\xi},0)\mapsto\alpha_{1}J_{0}(\chi_{1}\xi)+\alpha_{2}J_{0}(\chi_{2}%
\xi)+\beta e^{i\phi}J_{1}(\chi\xi)$$
(18)
according to the formula (13), where instead of Cartesian $\xi_{x},\xi_{y}$, the polar coordinates $\xi,\phi$ have been introduced and with the aforementioned and obvious renaming of constants $\alpha_{lm}$ and $\chi_{lm}$. These latter values will, in general, be small since they define the aperture half-angles of the beam cones.
Using now the well-known power expansion of the Bessel functions:
$$J_{n}(z)=\left(\frac{z}{2}\right)^{n}\sum_{k=0}^{\infty}\frac{(-1)^{k}(z/2)^{2%
k}}{k!(n+k)!}$$
(19)
it is easy to demonstrate that with the appropriate choice of the coefficients, the first few terms of the Maclaurin expansion of expression (18) in $\xi$ accurately reproduce $q_{M}(\bm{\xi},0)$ as given by the formula (17). This proper choice is as follows:
$$\alpha_{1}=\frac{\chi_{2}^{2}-4\gamma^{2}}{\chi_{1}^{2}-\chi_{2}^{2}},\;\;\;\;%
\alpha_{2}=\frac{\chi_{1}^{2}-4\gamma^{2}}{\chi_{2}^{2}-\chi_{1}^{2}},\;\;\;\;%
\beta=-\frac{4\gamma}{\chi}.$$
(20)
Applying (14) together with (5) one obtains the paraxial envelope in the form
$$\displaystyle\Psi(\bm{\xi},\zeta)=$$
$$\displaystyle\frac{1}{2\pi}\sum_{n=-\infty}^{\infty}\int\limits_{0}^{\infty}dt%
\,t\int\limits_{0}^{\infty}d\xi^{\prime}\xi^{\prime}\int\limits_{0}^{2\pi}d%
\phi^{\prime}e^{in(\phi-\phi^{\prime})}e^{-\frac{i\zeta t^{2}}{2}}$$
(21)
$$\displaystyle\times J_{n}(t\xi)J_{n}(t\xi^{\prime})\big{[}\alpha_{1}J_{0}(\chi%
_{1}\xi^{\prime})+\alpha_{2}J_{0}(\chi_{2}\xi^{\prime})$$
$$\displaystyle+\beta e^{i\phi^{\prime}}J_{1}(\chi\xi^{\prime})\big{]}e^{-\kappa%
\xi^{\prime 2}},$$
Integral with respect to $\phi^{\prime}$ can be easily taken and leads to the two Kronecker deltas: $\delta_{n0}$ (the first two terms) and $\delta_{n1}$ (the last one), which reduces the infinite sum over $n$ to two terms only:
$$\displaystyle\Psi(\bm{\xi},\zeta)=$$
$$\displaystyle\int\limits_{0}^{\infty}dt\,te^{-\frac{i\zeta t^{2}}{2}}\int%
\limits_{0}^{\infty}d\xi^{\prime}\xi^{\prime}e^{-\kappa\xi^{\prime 2}}\big{[}J%
_{0}(t\xi)J_{0}(t\xi^{\prime})$$
$$\displaystyle\times\sum_{l=1}^{2}\alpha_{l}J_{0}(\chi_{l}\xi^{\prime})+\beta e%
^{i\phi}J_{1}(t\xi)J_{1}(t\xi^{\prime})J_{1}(\chi\xi^{\prime})\big{]},$$
The remaining integrals with respect to $\xi^{\prime}$ and $t$ are of similar character and can be executed subsequently with the use of the formulas gr :
$$\displaystyle\int\limits_{0}^{\infty}dx\,x\,e^{-px^{2}}J_{n}(ax)J_{n}(bx)=$$
$$\displaystyle\frac{1}{2p}\,e^{-\frac{a^{2}+b^{2}}{4p}}I_{n}\Big{(}\frac{ab}{2p%
}\Big{)},$$
$$\displaystyle\mathrm{for}\;\;p\in\mathbb{R}_{+}$$
(23a)
$$\displaystyle\int\limits_{0}^{\infty}dx\,x\,e^{-qx^{2}}J_{n}(ax)I_{n}(bx)=$$
$$\displaystyle\frac{1}{2q}\,e^{-\frac{a^{2}-b^{2}}{4q}}J_{n}\Big{(}\frac{ab}{2q%
}\Big{)},$$
$$\displaystyle\mathrm{for}\;\;\mathrm{Re}\,q\in\mathbb{R}_{+}$$
(23b)
As a result of these two integrations, the complete paraxial envelope, applicable for $\zeta\neq 0$, is found as a sum of three coaxial BG modes:
$$\displaystyle\Psi(\bm{\xi},\zeta)=$$
$$\displaystyle\frac{1}{c(\zeta)}e^{-\frac{\kappa\xi^{2}}{c(\zeta)}}\bigg{[}\sum%
_{l=1}^{2}\alpha_{l}e^{-i\frac{\chi_{l}^{2}\zeta}{2c(\zeta)}}J_{0}\Big{(}\frac%
{\chi_{l}\xi}{c(\zeta)}\Big{)}$$
(24)
$$\displaystyle+\beta e^{i\phi}e^{-i\frac{\chi^{2}\zeta}{2c(\zeta)}}J_{1}\Big{(}%
\frac{\chi\xi}{c(\zeta)}\Big{)}\bigg{]},$$
where $c(\zeta)=1+2i\kappa\zeta$.
The nodal line of $\Psi(\bm{\xi},\zeta)$, can now be easily drawn. It is done in Fig. 1. As expected, it represents a ring. What should be especially emphasized, it is entirely constructed of physical beams of BG type.
The form of $\Psi(\bm{\xi},\zeta)$ as given in (24) may be somewhat illegible to an optical physicist due to the notation used. This notation is highly convenient for conducting calculations, as many of the expressions greatly simplify, but below the result is rewritten in a traditional form in which the standard coaxial BG modes can be easily recognized.
The connection will be easily established if one observes that
$$\displaystyle c(\zeta)$$
$$\displaystyle=1+i\frac{z}{z_{R}},\;\;\;\;\frac{1}{c(\zeta)}=\frac{w_{0}}{w(z)}%
e^{-i\psi(z)},$$
(25a)
$$\displaystyle\chi_{l}$$
$$\displaystyle=\sin\theta_{l},\;\;\;\;\chi=\sin\theta$$
(25b)
$$\displaystyle\kappa$$
$$\displaystyle=\frac{1}{k^{2}w_{0}^{2}},$$
(25c)
where $z_{R}$ denotes the Rayleigh length, $w_{0}$ is the beam waist, $w(z)=w_{0}\sqrt{1+(z/z_{R})^{2}}$ stands for the beam radius, $R(z)=z(1+(z_{R}/z)^{2})$ denotes the wave-front curvature, $\psi(z)=\arctan(z/z_{R})$ is the Gouy phase and $\theta_{l}$ (and similarly $\theta$) denotes the angular half-aperture of the appropriate beam cone.
For small values of these latter quantities expressed in radians, just as it is in this work, they are practically equal to the parameters $\chi$.
As can be seen from (24) and with the application of the definitions (25), the desired nodal line of Fig. 1 can be constructed as a result of the superposition of three standard BG beams, readily obtainable in experiments, of the form:
$$\displaystyle\Psi_{n}(\bm{r},z)=\frac{w_{0}}{w(z)}e^{in\phi}J_{n}\Big{(}\frac{%
kr\sin\theta}{1+iz/z_{R}}\Big{)}$$
(26)
$$\displaystyle\;\;\;\;\;\;\exp\left[-\frac{r^{2}}{w(z)^{2}}-i\frac{kr^{2}}{2R(z%
)}-i\frac{z/z_{R}}{1+iz/z_{R}}\sin^{2}\theta-i\psi(z)\right],$$
with $n=0$ or $n=1$ and different values of the aperture angles $\theta_{1},\,\theta_{2}$ and $\theta$, and with intensities determined by the coefficients $\alpha_{l}$ and $\beta$. Since only relative values matter, $\alpha_{1}$ may be by definition set to $1$ and then, for the data of Fig. 1 one gets
$\alpha_{2}=-1,\,\beta\approx-0.00019$. It is interesting to note that the $\phi$-dependent contribution is much weaker than the principal ones containing Bessel functions of the zeroth order. They are, however, necessary to ensure the correct shape of the nodal line, close to which the contributions from the “large” terms strongly decrease. This means that creating a knotted line through the mechanism of the destructive interference is a delicate matter requiring some precision and the nodal line itself is a rather shallow structure. This observation is confirmed by our further examples. The values of the wavelength, beam waist and Rayleigh length are identical for all three beams and can be adapted to the experimental requirements.
In Fig. 2 the phases of the outgoing light beam are depicted in four selected planes $\zeta=\mathrm{const}$. In particular, one can see the phase change from the value of $-\pi$ (black color) to the value of $\pi$ (white color) when walking around the nodal line (which is – roughly speaking –perpendicular to the plane at that point) as shown in the figure. This is typical for vortex lines from which a knot is formed. The same observation can be made in the following figures.
III.2 The Hopf link
The Hopf link is the first nontrivial knot and represents two rings linked to each other. It can again be obtained from (11) upon setting $n=2$:
$$q(u,v)=(u-v)(u+v).$$
(27)
The corresponding Milnor polynomial, obtained by substituting $u$ and $v$ as defined by (12), and next reduced to the plane $\zeta=0$, takes the form
$$q_{M}(\bm{\xi},0)=(1-\xi_{x}^{2}-\xi_{y}^{2})^{2}-4(\xi_{x}+i\xi_{y})^{2}.$$
(28)
and that with the scaling factor
$$q_{M}(\bm{\xi},0)=[1-\gamma^{2}(\xi_{x}^{2}+\xi_{y}^{2})]^{2}-4\gamma^{2}(\xi_%
{x}+i\xi_{y})^{2}.$$
(29)
Analyzing the presence of the factors $e^{im\phi}$ in the expression above (where $\xi_{x}+i\xi_{y}\mapsto\xi e^{i\phi}$) one easily notices, that now Bessel functions of two orders ($m=0$ and $m=2$) are required. On the other hand even for $J_{0}$ one needs to fix terms up to $\xi^{4}$. Since $J_{0}$ in an even function, this means that the sum with respect to $l$ in (13) now extends to $l=3$. Therefore, one can replace $q_{M}(\bm{\xi},0)$ with
$$\displaystyle q_{M}(\bm{\xi},0)\mapsto\sum_{l=1}^{3}\alpha_{l}J_{0}(\chi_{l}%
\xi)+\beta e^{2i\phi}J_{2}(\chi\xi)$$
(30)
The values of coefficients are established similarly as it was done in the former subsection. They turn out to be
$$\alpha_{l}=\prod_{j=1\atop j\neq l}^{3}\frac{\chi_{j}^{2}-8\gamma^{2}}{\chi_{l%
}^{2}-\chi_{j}^{2}},\;\;\;\;\beta=-\frac{32\gamma^{2}}{\chi^{2}}.$$
(31)
Following the procedure outlined in the case of the unknot, one now finds
$$\displaystyle\Psi(\bm{\xi},\zeta)=\frac{1}{2\pi}\sum_{n=-\infty}^{\infty}\int%
\limits_{0}^{\infty}dt\,t\int\limits_{0}^{\infty}d\xi^{\prime}\xi^{\prime}\int%
\limits_{0}^{2\pi}d\phi^{\prime}e^{in(\phi-\phi^{\prime})}e^{-\frac{i\zeta t^{%
2}}{2}}$$
$$\displaystyle\;\;\;\;\times J_{n}(t\xi)J_{n}(t\xi^{\prime})\bigg{[}\sum_{l=1}^%
{3}\alpha_{l}J_{0}(\chi_{l}\xi^{\prime})+\beta e^{2i\phi^{\prime}}J_{2}(\chi%
\xi^{\prime})\bigg{]}e^{-\kappa\xi^{\prime 2}}.$$
Performing the trivial integration over $\phi^{\prime}$ together with the $n$ summation and next those with respect to $\xi^{\prime}$ and $t$ according to (23), one finally finds
$$\displaystyle\Psi(\bm{\xi},\zeta)=$$
$$\displaystyle\frac{1}{c(\zeta)}e^{-\frac{\kappa\xi^{2}}{c(\zeta)}}\bigg{[}\sum%
_{l=1}^{3}\alpha_{l}e^{-i\frac{\chi_{l}^{2}\zeta}{2c(\zeta)}}J_{0}\Big{(}\frac%
{\chi_{l}\xi}{c(\zeta)}\Big{)}$$
(33)
$$\displaystyle+\beta e^{2i\phi}e^{-i\frac{\chi^{2}\zeta}{2c(\zeta)}}J_{2}\Big{(%
}\frac{\chi\xi}{c(\zeta)}\Big{)}\bigg{]}.$$
The obtained envelope turns out to be a superposition of four BG modes defined with (26) with $n=0$ and $n=2$. The nodal lines are depicted in Fig. 3 and clearly constitute the Hopf link.
The normalized (i.e. after having set $\alpha_{1}=1$) values of coefficients are: $\alpha_{2}\approx-1.89,\,\alpha_{3}\approx 0.89,\,\beta\approx-2.82\times 10^{%
-8}$, and allow to establish the relative intensity of the combined beams. Again, a very tiny value of the coefficient of the angle-dependent term is worth noting. Obviously, it can be modified within certain limits by a suitable choice of $\chi$’s, but this coefficient is invariably significantly smaller. In turn, the negative values account for the relative phases of the modes.
It is noteworthy that the knot lines produced by this method turn out to be pretty smooth (cf. e.g. bkj ), which may have practical significance.
The phases of $\Psi(\bm{\xi},\zeta)$ are drawn in Fig. 4. Again the change of the phase by $2\pi$ can be observed, when encircling the vortex line.
III.3 The Borromean rings
Proceeding further towards the higher complexity of considered knots, we will now examine that composed of three loops, each of which is linked to both others. Such a knotted construction bears the name of the Borromean rings, and is again obtainable from (11) when setting $n=3$:
$$q(u,v)=(u-v)(u-e^{2\pi i/3}v)(u-e^{4\pi i/3}v).$$
(34)
In consequence, upon applying the outlined procedure the corresponding Milnor polynomial if found to be
$$q_{M}(\bm{\xi},0)=(-1+\gamma^{2}(\xi_{x}^{2}+\xi_{y}^{2}))^{3}-8\gamma^{3}(\xi%
_{x}+i\xi_{y})^{3},$$
(35)
where the scaling factor has already been introduced. Upon the expansion, this expression contains the following powers of $\xi$: $\xi^{0},\xi^{2},\xi^{4}$ and $\xi^{6}$, as well as the term with the factor $e^{3i\phi}$. This indicates that four zero-order Bessel functions together with $J_{3}$ will be needed:
$$q_{M}(\bm{\xi},0)\mapsto\sum_{l=1}^{4}\alpha_{l}J_{0}(\chi_{l}\xi)+\beta e^{3i%
\phi}J_{3}(\chi\xi),$$
(36)
where coefficients can be calculated according to the following formulas:
$$\displaystyle\alpha_{l}=$$
$$\displaystyle\frac{\prod_{j=1\atop j\neq l}^{4}(\chi_{j}^{2}-12\gamma^{2})+48%
\gamma^{4}(\sum_{j=1\atop j\neq l}^{4}\chi_{j}^{2}-12\gamma^{2})}{\prod_{j=1%
\atop j\neq l}^{4}(\chi_{l}^{2}-\chi_{j}^{2})},$$
$$\displaystyle\beta=$$
$$\displaystyle-\frac{384\gamma^{3}}{\chi^{3}}.$$
(37)
This allows to write the integral form of the enevelope:
$$\displaystyle\Psi(\bm{\xi},\zeta)=\frac{1}{2\pi}\sum_{n=-\infty}^{\infty}\int%
\limits_{0}^{\infty}dt\,t\int\limits_{0}^{\infty}d\xi^{\prime}\xi^{\prime}\int%
\limits_{0}^{2\pi}d\phi^{\prime}e^{in(\phi-\phi^{\prime})}e^{-\frac{i\zeta t^{%
2}}{2}}$$
$$\displaystyle\;\;\;\;\times J_{n}(t\xi)J_{n}(t\xi^{\prime})\bigg{[}\sum_{l=1}^%
{4}\alpha_{l}J_{0}(\chi_{l}\xi^{\prime})+\beta e^{3i\phi^{\prime}}J_{3}(\chi%
\xi^{\prime})\bigg{]}e^{-\kappa\xi^{\prime 2}}.$$
Executing the integrals and the sum as in the previous cases, one arrives at
$$\displaystyle\Psi(\bm{\xi},\zeta)=$$
$$\displaystyle\frac{1}{c(\zeta)}e^{-\frac{\kappa\xi^{2}}{c(\zeta)}}\bigg{[}\sum%
_{l=1}^{4}\alpha_{l}e^{-i\frac{\chi_{l}^{2}\zeta}{2c(\zeta)}}J_{0}\Big{(}\frac%
{\chi_{l}\xi}{c(\zeta)}\Big{)}$$
(39)
$$\displaystyle+\beta e^{3i\phi}e^{-i\frac{\chi^{2}\zeta}{2c(\zeta)}}J_{3}\Big{(%
}\frac{\chi\xi}{c(\zeta)}\Big{)}\bigg{]}$$
The nodal lines of $\Psi(\bm{\xi},\zeta)$ are demonstrated in Fig. 5. Obviously, they represent a system of three linked loops, i.e. the Boromean rings. The superposition yielding such a rather sophisticated knot consists of five BG mods (39), each defined by the formula (26).
It is quite a challenge to produce these lines because in places of their close passing, the calculations require great precision. It is related to the earlier observation that the factor $\beta$ is much smaller than $\alpha_{l}$’s, which applies in the present case. This means again that generating such a structure needs a precise adjustment of the intensity of the beam bearing orbital angular momentum. For better visualization the values of $\chi_{1,2,3,4}$ in Fig. 5 have been increased and that of $\chi$ decreased. This makes all coefficients comparable in size and the graphic representation is easier to obtain in a numerical manner. Large differences between the coefficient values, reaching many orders of magnitude, require very high calculation accuracy.
The normalized values of the coefficients are now as follows:
$$\displaystyle\alpha_{2}\approx-2.00001,\;\;\;\;\alpha_{3}\approx 1.28574,$$
(40a)
$$\displaystyle\alpha_{4}\approx-0.285723,\;\;\;\;\beta\approx-0.937557.$$
(40b)
They are recorded with the accuracy of a few significant digits to illustrate how crucial is the precision for obtaining a particular knot. This is related to our discussion concerning the Boromean rings, so we will reconsider these values later.
When decreasing values of $\chi_{i}$’s in the beams, $\beta$ (to put it precisely, the normalized value of $\beta$, i.e. $\beta/\alpha_{1}$) becomes extremely small, as in previous cases which reflects the complicated nodal structure close to the passing points.
In Fig. 6 the phases of (39) are drawn in four planes. The same effects as before are observed.
III.4 The trefoil
In order to obtain the knot called the trefoil one has to start with the polynomial
$$q(u,v)=u^{2}-v^{3}.$$
(41)
which does not belong to the class described with the formula (11) but falls into a wide family of knots that can be derived from the expressions of the type
$$q(u,v)=u^{2}-v^{n},$$
(42)
with $n\in\mathbb{N}$ king . The same family includes for example the cinquefoil knot (for $n=5$) or the septafoil knot (for $n=7$) etc. Setting $n=3$ the following Milnor polynomial (with the scaling factor $\gamma$ included) is found:
$$q_{M}(\bm{\xi},0)=[1+\gamma^{2}(\xi_{x}^{2}+\xi_{y}^{2})][1-\gamma^{2}(\xi_{x}%
^{2}+\xi_{y}^{2})]^{2}-8\gamma^{3}(\xi_{x}+i\xi_{y})^{3},$$
(43)
The wave envelope can now be constructed from $q_{M}(\bm{\xi},0)$ following the steps of the previous subsections. Again the expression (39) is obtained, since $q_{M}(\bm{\xi},0)$ contains the same terms (apart from the multiplicative constants) as in the case of the Borromean rings. Only the values of the coefficients alter (i.e. the relative intensities of the beams involved) and they are currently expressed as follows
$$\displaystyle\alpha_{l}=$$
$$\displaystyle-\frac{\prod_{j=1\atop j\neq l}^{4}(\chi_{j}^{2}-4\gamma^{2})+48%
\gamma^{4}(\sum_{j=1\atop j\neq l}^{4}\chi_{j}^{2}+48\gamma^{2})+64\gamma^{6}}%
{\prod_{j=1\atop j\neq l}^{4}(\chi_{l}^{2}-\chi_{j}^{2})},$$
$$\displaystyle\beta=$$
$$\displaystyle-\frac{384\gamma^{3}}{\chi^{3}},$$
(44)
where minuses are obviously inessential.
The nodal line obtained now from (39) has really the form of a trefoil and is depicted in Fig. 7. The values of parameters (i.e. the beams used) are identical as in the case of the Borromean rings. Only their relative intensities differ. This means that by setting up experimentally an identical set of BG beams, various knots of entirely different topology can be obtained by simply modifying their relative intensities. Formulas (44) yield now the following values of the ratios:
$$\displaystyle\alpha_{2}\approx-1.99999583,\;\;\;\;\alpha_{3}\approx 1.28571,$$
(45a)
$$\displaystyle\alpha_{4}\approx-0.285711,\;\;\;\;\beta\approx-0.937481.$$
(45b)
They should be compared to (40). The differences between these values apparently seem negligible, but the topology of the knots is completely distinct. Tiny changes in the relative intensities of the beams may lead to a different reconnection of nodal lines passing very close to one another and to a completely different knot. This property seems to be general: for any knot with a more complex structure, there may happen a kind of “switching” of nodal lines.
Fig. 8 presents the phases in the planes of constant $\zeta$ for the trefoil knot. The similarities to the Borromean rings case should be noted.
III.5 More complicated knots
We now move on to more complex knots which are simultaneously more challenging to generate numerically and experimentally using BG light beams. As an example let us consider the figure-eight knot as its shape – if open, as usually is in climbing or sailing – with the appropriate line arrangement in space, resembles number $8$. Contrary to that, the mathematical knot is closed, according to the definition provided in Section II. The polynomial $q(u,v)$ is in this case considerably more complicated, and in addition to the complex variables $u$ and $v$ depends as well on the conjugated quantity $\bar{v}$ bkj ; bode :
$$q(u,v)=64u^{3}-12u[2(v^{2}-\bar{v}^{2})+3]-14(v^{2}+\bar{v}^{2})-(v^{4}-\bar{v%
}^{4})$$
(46)
The Minor polynomial contains now many terms, which means that in order to create the beam of this topology, relatively many BG modes have to be superimposed. When rewritten in polar variables it contains expressions up to $\xi^{8}$ and also angular factors $e^{\pm}2i\phi$ and $e^{\pm}4i\phi$:
$$\displaystyle q_{M}(\bm{\xi},0)$$
$$\displaystyle=$$
$$\displaystyle 28[\gamma^{8}(\xi_{x}^{2}+\xi_{y}^{2})^{4}-1]$$
$$\displaystyle-200\gamma^{2}(\xi_{x}^{2}+\xi_{y}^{2})[\gamma^{4}(\xi_{x}^{2}+%
\xi_{y}^{2})^{2}-1]$$
$$\displaystyle+\gamma^{2}(\xi_{x}^{2}+\xi_{y}^{2})[-152\gamma^{4}(\xi_{x}^{2}+%
\xi_{y}^{2})^{2}$$
$$\displaystyle-112\gamma^{2}(\xi_{x}^{2}+\xi_{y}^{2})+40]e^{2i\phi}$$
$$\displaystyle+\gamma^{2}(\xi_{x}^{2}+\xi_{y}^{2})[40\gamma^{4}(\xi_{x}^{2}+\xi%
_{y}^{2})^{2}$$
$$\displaystyle-112\gamma^{2}(\xi_{x}^{2}+\xi_{y}^{2})-152]e^{-2i\phi}$$
$$\displaystyle-16\gamma^{4}(\xi_{x}^{2}+\xi_{y}^{2})^{2}e^{4i\phi}+16\gamma^{4}%
(\xi_{x}^{2}+\xi_{y}^{2})^{2}e^{-4i\phi}.$$
It may be checked by the Taylor expansion of Bessel functions that the field envelope has to be written the form:
$$\displaystyle\Psi(\bm{\xi},\zeta)=\frac{1}{2\pi}\sum_{n=-\infty}^{\infty}\int%
\limits_{0}^{\infty}dt\,t\int\limits_{0}^{\infty}d\xi^{\prime}\xi^{\prime}\int%
\limits_{0}^{2\pi}d\phi^{\prime}e^{in(\phi-\phi^{\prime})}e^{-\frac{i\zeta t^{%
2}}{2}}$$
$$\displaystyle\;\;\;\;\times J_{n}(t\xi)J_{n}(t\xi^{\prime})\bigg{[}\sum_{l=1}^%
{5}\alpha_{l}J_{0}(\chi_{l}\xi^{\prime})+\sum_{l=1}^{3}\beta_{l}e^{2i\phi^{%
\prime}}J_{2}(\chi_{l}^{\prime}\xi^{\prime})$$
$$\displaystyle\;\;\;\;+\sum_{l=1}^{3}\gamma_{l}e^{-2i\phi^{\prime}}J_{-2}(\chi_%
{l}^{\prime\prime}\xi^{\prime})+\delta_{1}e^{4i\phi^{\prime}}J_{4}(\chi\xi^{%
\prime})$$
$$\displaystyle\;\;\;\;+\delta_{2}e^{-4i\phi^{\prime}}J_{-4}(\chi^{\prime}\xi^{%
\prime})\bigg{]}e^{-\kappa\xi^{\prime 2}},$$
(48)
where the coefficients have to satisfy the sets of equations listed below. The first one for $\alpha_{l}$’s comprises the following five equations:
$$\displaystyle\alpha_{1}+\alpha_{2}+\alpha_{3}+\alpha_{4}+\alpha_{5}$$
$$\displaystyle=28,$$
(49a)
$$\displaystyle\alpha_{1}\chi_{1}^{2}+\alpha_{2}\chi_{2}^{2}+\alpha_{3}\chi_{3}^%
{2}+\alpha_{4}\chi_{4}^{2}+\alpha_{5}\chi_{5}^{2}$$
$$\displaystyle=800\gamma^{2},$$
(49b)
$$\displaystyle\alpha_{1}\chi_{1}^{4}+\alpha_{2}\chi_{2}^{4}+\alpha_{3}\chi_{3}^%
{4}+\alpha_{4}\chi_{4}^{4}+\alpha_{5}\chi_{5}^{4}$$
$$\displaystyle=0,$$
(49c)
$$\displaystyle\alpha_{1}\chi_{1}^{6}+\alpha_{2}\chi_{2}^{6}+\alpha_{3}\chi_{3}^%
{6}+\alpha_{4}\chi_{4}^{6}+\alpha_{5}\chi_{5}^{6}$$
$$\displaystyle=-460800\gamma^{6},$$
(49d)
$$\displaystyle\alpha_{1}\chi_{1}^{8}+\alpha_{2}\chi_{2}^{8}+\alpha_{3}\chi_{3}^%
{8}+\alpha_{4}\chi_{4}^{8}+\alpha_{5}\chi_{5}^{8}$$
$$\displaystyle=-4128768\gamma^{8}.$$
(49e)
That for $\beta_{l}$’s is composed of three equations:
$$\displaystyle\beta_{1}{\chi^{\prime}}_{1}^{2}+\beta_{2}{\chi^{\prime}}_{2}^{2}%
+\beta_{3}{\chi^{\prime}}_{3}^{2}$$
$$\displaystyle=320\gamma^{2},$$
(50a)
$$\displaystyle\beta_{1}{\chi^{\prime}}_{1}^{4}+\beta_{2}{\chi^{\prime}}_{2}^{4}%
+\beta_{3}{\chi^{\prime}}_{3}^{4}$$
$$\displaystyle=10752\gamma^{4},$$
(50b)
$$\displaystyle\beta_{1}{\chi^{\prime}}_{1}^{6}+\beta_{2}{\chi^{\prime}}_{2}^{6}%
+\beta_{3}{\chi^{\prime}}_{3}^{6}$$
$$\displaystyle=-466944\gamma^{6},$$
(50c)
and is similar to the set for $\gamma_{l}$’s:
$$\displaystyle\gamma_{1}{\chi^{\prime\prime}}_{1}^{2}+\gamma_{2}{\chi^{\prime%
\prime}}_{2}^{2}+\gamma_{3}{\chi^{\prime\prime}}_{3}^{2}$$
$$\displaystyle=-1216\gamma^{2},$$
(51a)
$$\displaystyle\gamma_{1}{\chi^{\prime\prime}}_{1}^{4}+\gamma_{2}{\chi^{\prime%
\prime}}_{2}^{4}+\gamma_{3}{\chi^{\prime\prime}}_{3}^{4}$$
$$\displaystyle=10752\gamma^{4},$$
(51b)
$$\displaystyle\gamma_{1}{\chi^{\prime\prime}}_{1}^{6}+\gamma_{2}{\chi^{\prime%
\prime}}_{2}^{6}+\gamma_{3}{\chi^{\prime\prime}}_{3}^{6}$$
$$\displaystyle=122880\gamma^{6},$$
(51c)
Solutions to all of these equations can be found in an obvious way. We do not write them out explicitly in order to avoid listing lenghty expressions. Unlike, the last two parameters can directly be found:
$$\delta_{1}=-\frac{6144\gamma^{4}}{\chi^{4}},\;\;\;\;\delta_{2}=\frac{6144%
\gamma^{4}}{{\chi^{\prime}}^{4}}$$
(52)
Now, we are in the position to present the result for the envelope:
$$\displaystyle\Psi(\bm{\xi},\zeta)=$$
$$\displaystyle\frac{1}{c(\zeta)}e^{-\frac{\kappa\xi^{2}}{c(\zeta)}}\bigg{[}\sum%
_{l=1}^{5}\alpha_{l}e^{-i\frac{\chi_{l}^{2}\zeta}{2c(\zeta)}}J_{0}\Big{(}\frac%
{\chi_{l}\xi}{c(\zeta)}\Big{)}$$
(53)
$$\displaystyle+\sum_{l=1}^{3}\beta_{l}e^{-i\frac{{\chi^{\prime}}_{l}^{2}\zeta}{%
2c(\zeta)}}e^{2i\phi}J_{2}\Big{(}\frac{\chi^{\prime}_{l}\xi}{c(\zeta)}\Big{)}$$
$$\displaystyle+\sum_{l=1}^{3}\gamma_{l}e^{-i\frac{{\chi^{\prime\prime}}_{l}^{2}%
\zeta}{2c(\zeta)}}e^{-2i\phi}J_{-2}\Big{(}\frac{\chi^{\prime\prime}_{l}\xi}{c(%
\zeta)}\Big{)}$$
$$\displaystyle+\delta_{1}e^{-i\frac{\chi^{2}\zeta}{2c(\zeta)}}e^{4i\phi}J_{4}%
\Big{(}\frac{\chi\xi}{c(\zeta)}\Big{)}$$
$$\displaystyle+\delta_{2}e^{-i\frac{\chi^{\prime 2}\zeta}{2c(\zeta)}}e^{-4i\phi%
}J_{-4}\Big{(}\frac{\chi^{\prime}\xi}{c(\zeta)}\Big{)}\bigg{]}.$$
As one can see, the procedure can formally be continued in a systematic way. It should be noted, however, that the complexity of this expression has significantly increased. In order to generate such a knot, 13 BG beams are now needed! Unlike this, for the trefoil or the Boromean rings, 5 beams were sufficient. Thus, producing more and more complicated knots proves to be quite a challenge. It is a problem from the numerical point of view as well, since with many nodal lines passing close to one another, extremely high precision of calculations is required, which, combined with the large number of terms in the formula, results in long computer runtime.
IV Summary
The creation of knotted vortex lines in light beams is one of the most interesting challenges in topological optics. The current paper presents a systematic method of obtaining superpositions of Bessel-Gaussian mods such that the nodal lines (where field intensity drops to zero) form given knot structures. We have selected five geometries as examples: the unknot, the Hopf link, the Borromean rings, the trefoil and the figure-eight knot. It is quite a wide spectrum of choice which indicates a certain universality of the approach. However, with the increasing complexity of a knot, a growing number of BG wave components were needed. In the case of the unknot, $3$ were sufficient; for the most complex knot (figure-eight), $13$ were needed. However, because of the fact that BG beams are relatively easy to obtain in experiments, one can imagine entangled traps for particles obtained by means of these waves.
The proposed method is systematic: in principle, it can be relatively easily applied to increasingly complex structures. Yet, it must be understood that these superpositions will comprise a very large number of component waves. This is reflected in the numerical calculations such as those presented in this paper, where computer time dramatically increases for complex superpositions. There is also another reason for this increase: for complex knots several nodal lines are passing close to one another, which requires high accuracy. This was demonstrated by comparing the Borromean rings and the trefoil.
Using a different representation of the Schrödinger propagator than that given in formula (5), the proposed method can readily be applied to other types of beams. Moreover, it seems that the nodal lines obtained from this method are much smoother and contain fewer “fringes” (naturally with identical topology) than those formerly produced with other methods. This may be of practical importance, as keeping the particles in such traps or guiding them along knotted lines presents some kind of a real challenge, and the smoother the trapping-potential valley is, the easier is to achieve it.
References
(1)
L. Allen, M.W. Beijersbergen, R.J.C. Spreeuw and J.P. Woerdman, Phys. Rev. A 45, 8185(1992).
(2)
M. Padgett, J. Arlt, N. Simpson and L. Allen, Am. J. Phys. 64, 114(1995).
(3)
J. Arlt, T. Hitomi and K. Dholakia, Appl. Phys. B 71, 549(2000).
(4)
J. Durnin, J. Opt. Soc. Am. A 4, 651(1987).
(5)
J. Durnin, J.J. Miceli, and J. Eberly, Phys. Rev. Lett. 58, 1449(1987).
(6)
K. Volke-Sepúlveda, V. Garcés-Chávez, S. Chávez-Cerda, J. Arlt and K. Dholakia, J. Opt. B 4, S82(2002).
(7)
I. Białynicki-Birula, Z. Białynicka-Birula and N. Drozd, Trapping of charged particles by Bessel beams in: The Angular Momentum of Light, ed. D.L. Andrews and M. Babiker, Cambridge University Press (Cambridge 2012).
(8)
T. Radożycki, Phys. Rev. A 100, 063412 (2019).
(9)
J.F. Nye, Proc. Roy. Soc. Lond. A 389, 279(1983).
(10)
J.F. Nye and J.V. Hajnal, Proc. Roy. Soc. Lond. A 409, 21(1987).
(11)
I. Freund, Opt. Commun. 201, 251(2002).
(12)
M.R. Dennis, K. O’Holleran and M.J.Padgett, Progress in Optics 53, 293(2009).
(13)
F. Cardano, E. Karimi, L. Marrucci, C. de Lisio and E. Santamato, Opt. Express 21, 8815(2013).
(14)
A.F. Rañada, Lett. Math. Phys. 18, 97 (1989).
(15)
W.T.M. Irvine and D. Bouwmeester, Nat. Phys. 4, 716(2008).
(16)
I.M. Besieris and A.M. Shaarawi, Opt. Lett. 34, 3887(2009).
(17)
J. W. Dalhuisen and D. Bouwmeester, J. Phys. A 45, 135201(2010).
(18)
H. Kedia, I. Białynicki-Birula, D. Peralta-Salas and W.T.M. Irvine, Phys. Rev. Lett. 111, 150404 (2013).
(19)
M. Arrayás and J.L. Trueba, J. Phys. A 48, 025203(2014).
(20)
C. Hoyos, N. Sircar, and J. Sonnenschein, J. Phys. A 48, 255204(2015).
(21)
M. Arrayás, D. Bouwmeester and J.L. Trueba, Phys. Rep. 667, 1(2017).
(22)
H. Kedia, D. Peralta-Salas and W.T.M. Irvine, J. Phys. A 51, 025204(2018).
(23)
M.V. Berry and M.R. Dennis, Proc. Roy. Soc. Lond. A 457, 2251(2001).
(24)
M.R. Dennis, New J. Phys. 5, 134(2003).
(25)
M.V. Berry and M.R. Dennis, J. Phys. A 40, 65(2007).
(26)
M.R. Dennis, R.P. King, B. Jack, K. OâHolleran and M.J. Padgett, Nat. Phys. 6, 118(2010).
(27)
D. Kleckner and W.T.M. Irvine, Nat. Phys. 9, 253(2013).
(28)
A.J.J.M. de Klerk, R.I. van der Veen, J.W. Dalhuisen and D. Bouwmeester, Phys. Rev A95, 053820 (2017).
(29)
D. Sugic and M.R. Dennis, J. Opt. Soc. Am. A, 35, 1987(2018).
(30)
J. Leach, M.R. Dennis, J. Courtial, M.J. Padgett, Nature 432, 165(2004).
(31)
J. Leach, M.R. Dennis, J. Courtial and M.J. Padgett, New Journal of Physics 7, 55(2005).
(32)
E.R. Shanblatt and D.G. Grier, Optics Express 19, 5833(2011).
(33)
S.J. Tempone-Wiltshire, S.P. Johnstone and K. Helmerson, Sci. Rep. 6, 24463(2016).
(34)
N. Davidson, H.J. Lee, C.S. Adams, M. Kasevich and S. Chu, Phys. Rev. Lett. 74, 1311(1995).
(35)
R. Grimm, M. Weidemüller, Y.B. Ovchinnikov, Adv. Atom. Mol. Opt. Phys. 42, 95(2000).
(36)
D. Sheng, J. Zhang and L. A. Orozco, Phys. Rev A 87, 063412(2013).
(37)
N. Friedman, A. Kaplan and N. Davidson, Adv. At. Mol. Opt. Phys. 48, 99(2002).
(38)
T. Radożycki, Phys. Rev. A 100, 063412(2019)
(39)
N.B. Delone and V.P. Krainov, Phys.-Usp. 42, 669(1999).
(40)
G. Mie, Annalen der Physik, 330,377(1908).
(41)
K.C. Neuman and S.M. Block, Rev. Sci. Instrum. 75, 2787(2004).
(42)
K. Dholakia, P. Reece and M. Gu, Chem. Soc. Rev. 37, 42(2008).
(43)
M. Dienerowitz, M. Mazilu and K. Dholakia, J. Nanophoton. 2, 021875(2008).
(44)
L.F. Liu, J.L. Davis and R. Calendar, Nucleic Acids Res. 9, 3979(1981).
(45)
D.J. Stevenson, F.J. Gunn-Moore, K. Dholakia, J. Biomed. Opt. 15, 041503(2010).
(46)
F.M. Fazal and S.M. Block, Nat. Photon. 5, 318(2011).
(47)
M. Padgett, J. Molloy and D. McGloin (eds.), Optical Tweezers: Methods and Applications, (Series in Optics and Optoelectronics), CRC Press, Taylor and Francis, (Boca Raton, London, New York, 2010).
(48)
M. Woerdemann, Structured Light Fields: Applications in Optical Trapping, Manipulation, and Organisation, Springer, (Berlin, Heidelberg, 2012).
(49)
R.W. Bowman and M.J. Padgett, Rep. Prog. Phys. 76, 026401(2013).
(50)
D.G. Grier, Nature 424, 810(2003).
(51)
U. Tkalec, M. Ravnik, S. Copar, S. Z̆umer, and I. Mus̆evic, Science 333, 62(2011).
(52)
A. Martinez, M. Ravnik, B. Lucero, R. Visvanathan, S. Z̆umer and I. I. Smalyukh, Nat. Mater. 13, 258(2014).
(53)
D.S. Hall, M.W. Ray, K. Tiurev, E. Ruokokoski, A.H. Gheorghe and M. Möttönen, Nat. Phys. 12, 478(2016).
(54)
D.S. Bradshaw and D.L. Andrews, Eur. J. Phys. 38 034008(2017).
(55)
J.J. Danon, A.Krüger, D.A. Leigh, J.-F. Lemonnier, A.J. Stephens, I.J. Vitorica-Yrezabal and S.L. Woltering, Science 355, 159(2017).
(56)
T. Radożycki, arXiv:2008.13045 (2020).
(57)
M.V. Berry and M.R. Dennis, Proc. Roy. Soc. Lond. A 457, 2251(2001).
(58)
M.V. Berry and M.R. Dennis, J. Phys. A 34, 8877(2001).
(59)
J. Leach, M.R. Dennis, J. Courtial and M.J. Padgett, New J. Phys. 7, 55(2005).
(60)
R.P. King, Knotting of Optical Vortices, PhD Thesis, Univ. of Southampton, 2010.
(61)
M.J. Padgett, K. O’Holleran, R.P. King and M.R. Dennis, Contemporary Physics 52, 265(2011).
(62)
G.B. Arfken and H.J. Weber, Mathematical Methods for Physicists, Academic Press (oxford 2013).
(63)
K. Brauner, Abh. Math. Semin. Univ. Hambg 6, 1(1928).
(64)
B. Bode, M.R. Dennis, D. Foster and R.P. King, Proc. Roy. Soc. A 473, 20160829(2017).
(65)
J.W. Milnor, Singular points of complex hypersurfaces, Annals of Mathematics Studies, Princeton University Press, Princeton, NJ; University of Tokyo Press, (Tokyo, 1968).
(66)
I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals Series and Products, Academic Press, (New York, 2015). |
Negative Poisson’s Ratio in Single-Layer Graphene Ribbons
Jin-Wu Jiang
Shanghai Institute of Applied Mathematics and Mechanics, Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai University, Shanghai 200072, People’s Republic of China
jiangjinwu@shu.edu.cn
Harold S. Park
Department of Mechanical Engineering, Boston University, Boston, Massachusetts 02215, USA
Abstract
The Poisson’s ratio characterizes the resultant strain in the lateral direction for a material under longitudinal deformation. Though negative Poisson’s ratios (NPR) are theoretically possible within continuum elasticity, they are most frequently observed in engineered materials and structures, as they are not intrinsic to many materials. In this work, we report NPR in single-layer graphene ribbons, which results from the compressive edge stress induced warping of the edges. The effect is robust, as the NPR is observed for graphene ribbons with widths smaller than about 10 nm, and for tensile strains smaller than about 0.5%, with NPR values reaching as large as -1.51. The NPR is explained analytically using an inclined plate model, which is able to predict the Poisson’s ratio for graphene sheets of arbitrary size. The inclined plate model demonstrates that the NPR is governed by the interplay between the width (a bulk property), and the warping amplitude of the edge (an edge property), which eventually yields a phase diagram determining the sign of the Poisson’s ratio as a function of the graphene geometry.
keywords: Graphene, Negative Poisson’s Ratio, Edge Effect, Warping Effect
The Poisson’s ratio ($\nu$) characterizes the resultant strain in the lateral direction for a material under longitudinal deformation. Most materials contract (expand) laterally when they are stretched (compressed), so that the Poisson’s ratio is positive for these materials. A negative Poisson’s ratio (NPR) is allowed by classical elasticity theory, which sets a range of $-1<\nu<0.5$ for the Poisson’s ratio for isotropic materials.1
The NPR has been found to exist intrinsically in some materials, and various models have been proposed for the explanation.2, 3, 4, 5 Milstein and Huang reported NPR for some face centered cubic structures, in which the Poisson’s ratio is calculated using the elastic modulus.6 Baughman et al. explored the correlation between the NPR and the work function in many face centered cubic metals, and the NPR was interpreted from a structural point of view.7 In contrast to individual materials, many more examples of NPR phenomena have been reported in composites and other engineered structures since the seminal work of Lakes in 1987.8 In this experiment, the NPR was induced by a permanent compression of a conventional low-density open cell polymer foam, which was explained by the re-entrant configuration of the cell. Materials with NPR have become known as auxetic, as coined by Evans.9
The study of NPR phenomena has focused on bulk, engineered auxetic structures.3, 4, 10, 11 However, in the past 3 years reports of NPR phenomena have emerged for low-dimensional nanomaterials. For example, the NPR for metal nanoplates was found due to a surface-induced phase transformation.12 NPR was found to be intrinsic to single-layer black phosphorus due to its puckered configuration, which leads to NPR in the out-of-plane direction.13 NPR was also predicted for few-layer orthorhombic arsenic using first-principles calculations.14 For graphene, we are aware of one report of NPR by Grima et al., which occurred due to the introduction of many vacancy defects, and the resulting rippling curvature, in bulk graphene sheets.15
One of the key defining physical characteristics of nanomaterials is their large ratio of surface area to volume (for 1D nanomaterials like nanowires), or edge length to area (for 2D nanomaterials like graphene). Because of this surface, or edge effects can play a fundamental role in impacting the mechanical properties of these nanomaterials. For example, as mentioned above surface stress-induced phase transformations were the mechanism enabling NPR in metal nanoplates.12 As graphene is a 2D nanomaterial with the thinnest possible (one atom thick) thickness, its physical properties are very sensitive to free edge effects, for example edge warping due to compressive edge stresses.16 The warped free edges are also the origin for localized edge phonon modes that are responsible for the edge reconstruction of graphene17, 18 or edge induced energy dissipation in graphene nanoresonators.19, 20
In this letter, we report, using molecular statics simulations, an intrinsic NPR induced by the warped free edges in single-layer graphene ribbons. The effect is robust, as NPR as large as -1.51 is observed for graphene ribbons with widths smaller than about 10 nm, and for tensile strains smaller than about 0.5%. The NPR is explained analytically using an inclined plate model, which is able to predict the Poisson’s ratio for graphene sheets of arbitrary size. The inclined plate model demonstrates that the NPR is governed by the interplay between the width (a bulk property), and the warping amplitude of the edge (an edge property), which eventually yields a phase diagram determining the sign of the Poisson’s ratio as a function of the graphene geometry.
Results. We start by first briefly characterizing the Poisson’s ratio in bulk graphene. To do so, we stretch graphene with periodic boundary conditions (PBC) in both the x and y-directions. Graphene is stretched in the x-direction as shown in Fig. 1a and the resultant strain in the y-direction is recorded. Fig. 1b shows the $\epsilon_{y}$-$\epsilon_{x}$ relation for graphene of dimensions $46.86\times 49.19$ Å and $97.98\times 98.38$ Å, while Fig. 1c shows the corresponding Poisson’s ratio, where a negligible difference in results from these two differently sized sheets is observed. Fig. 1c shows that Poisson’s ratio in graphene decreases with applied tensile strain, which agrees with the findings of previous continuum mechanics21 and first-principles calculations.22 In the small strain region, the Poisson’s ratio value is 0.34, which matches values previously found using the Brenner potential.21 The salient point is that the Poisson’s ratio of graphene is always positive if PBCs, which eliminate edge warping effects, are applied in the y-direction, or the direction normal to the stretching direction.
A characteristic feature for free edges in graphene is the warped configuration that is induced by the compressive edge stress as shown in Fig. 2a. The warped structure can be described by the surface function16 $z(x,y)=Ae^{-y/l_{c}}\sin(\pi x/\lambda)$, where $\lambda=L/n$, with $L$ being the length of the graphene ribbon and $n$ being the warping number. The graphene ribbon shown in Fig. 2a has dimensions of $195.96\times 199.22$ Å, resulting in the fitting parameters for the warped free edges as the warping amplitude $A=2.26$ Å, penetration depth $l_{c}=8.55$ Å, and half wave length $\lambda=32.01$ Å. We note that $\lambda$ is about one sixth of the length $L$, i.e., $\lambda=L/6$.
We study five sets of graphene structures with free boundary conditions in the y-direction. Set I: graphene is 195.96 Å in length, and the warped edge has a warping number $n=6$. Set II: graphene is 195.96 Å in length, and the warped edge has a warping number $n=8$. Set III: graphene is 195.96 Å in length, and the warped edge has a warping number $n=10$. Set IV: graphene is 195.96 Å in length, and the warped edge has a warping number $n=12$. Set V: graphene is 97.98 Å in length, and the warped edge has a warping number $n=2$. For each simulation set, we consider eight different widths of 29.51, 39.35, 49.19, 59.03, 78.70, 98.38, 147.57, and 199.22 Å. We will demonstrate that the NPR phenomena we report is robust, and is observed for different warping periodicities.
Fig. 3a shows the strain dependence for the Poisson’s ratio of Set I, where the width of the graphene ribbon increases for data from the bottom to the top in the figure. The occurrence of a NPR is clearly observed for small strains, and for the narrower width ribbons. Furthermore, the Poisson’s ratio changes from negative to positive at some critical strain $\epsilon_{c}$. This critical strain is more clearly illustrated in the inset, which shows a critical strain of $\epsilon_{c}=0.005$ in the $\epsilon_{y}$-$\epsilon_{x}$ relation for graphene with width 29.51 Å. For $\epsilon_{x}<\epsilon_{c}$, graphene expands in the y-direction when it is stretched in the x-direction; i.e., the NPR phenomenon occurs.
The critical strain $\epsilon_{c}$ represents a structural transition for the warped edge. To reveal this structural transition, we show in Fig. 3b the z position of two atoms from different warped edge regions. One atom is at the peak of the warped edge (shown by the red arrow in the inset), while the other atom is at the valley of the warped edge (shown by blue arrow). Fig. 3b clearly shows that both atoms fall into the xy plane at the critical strain $\epsilon_{x}=\epsilon_{c}$. In other words, the warped edge transitions at the critical point from a three-dimensional, out-of-plane warping configuration into a two-dimensional planar configuration due to the externally applied tensile strain. The connection of the critical strain in the disappearance of the NPR in Fig. 3a and the transition to the two-dimensional planar configuration in Fig. 3b implies that the NPR is connected to the flattening of the warped edges, as the Poisson’s ratio becomes positive after the structural transition of the warped edge. We note that the z-coordinates of the atoms in the warped edges in Fig. 3b can be well fitted to the functions $z=\pm b_{0}\sin(\theta_{0}(1-\epsilon/\epsilon_{c}))$ for $\epsilon<\epsilon_{c}$.
The width dependence for the Poisson’s ratio is displayed in Fig. 3c. The Poisson’s ratio is strain dependent as shown in Fig. 3a, so we compute an averaged Poisson’s ratio using data in the strain range $[0,\epsilon_{c}]$, which is equivalent to extracting the Poisson’s ratio value by a linear fitting for the $\epsilon_{y}$-$\epsilon_{x}$ relation in $[0,\epsilon_{c}]$. Fig. 3c shows this averaged Poisson’s ratio value for graphene with different widths. We note that the critical strain varies for graphene with different width as indicated by the inset in Fig. 3c, where the critical strain is fitted to the function $\epsilon_{c}=0.0082-0.092/W$. The critical strain is smaller in narrower graphene, because some interactions occur between the free ($\pm$y) edges for narrower widths. These edge interactions enable the tension-induced structural transition of the warped edges for narrower ribbons to occur at lower values of tensile strains, because the warping directions for the free ($\pm$y) edges are different, forming a see-saw like configuration. The saturation value $\epsilon_{c}=0.0082$ at $W\rightarrow\infty$ can be regarded as the actual value of the critical strain for an isolated warped free edge.
The Poisson’s ratio in Fig. 3c increases with increasing width ($W$), and can be fitted to the function $\nu=0.34-31.71/W$. According to this result, graphene can be regarded as the integration of one central region (with bulk Poisson’s ratio $\nu_{0}$) and two edge regions (with Poisson’s ratio $\nu_{e}$). The size of each edge region is $l_{c}$, which is the penetration depth of the warped configuration in Fig. 2. The size of the remaining central region is $W-2l_{c}$. Simple algebra gives the effective Poisson’s ratio for the graphene ribbon as
$$\displaystyle\nu=\nu_{0}-\frac{2l_{c}}{W}\left(\nu_{0}-\nu_{e}\right).$$
(1)
Comparing equation (1) with the fitting function in Fig. 3c, we get $\nu_{0}=0.34$ and $\nu_{e}=-1.51$.
This result shows that for the Set I ribbon geometries, in the limit of an ultra-narrow, edge-dominated graphene ribbon, the value of the NPR can be as large as -1.51. Perhaps more importantly, according to equation (1), the NPR phenomenon can be observed in graphene sheets with widths up to about 10 nm. Such width nanoribbons are not small, and are regularly studied experimentally.23 Furthermore, Fig. 3a shows that the NPR phenomenon is most significant for tensile strains smaller than about 0.5%. These strain values are important as they fall within the strain range of [0, 0.8%] that has already been achievable in many experimental strain engineering investigations.24, 25 We thus expect that this NPR phenomenon can readily be observed experimentally in the near future.
Discussion. We have demonstrated in the above discussion the connection between the NPR phenomenon and the warped free edges in graphene. We now present an analytic model to describe the relationship between the NPR phenomenon and the warped free edges. In Fig. 4a, the warped edge is represented by an inclined plate (IP) (gray area). During the tensile deformation, the IP falls into the xy plane. The side view in the dashed ellipse illustrates the mechanism enabling the NPR clearly. Specifically, it shows that the projection ($b_{y}$) of the IP on the y-axis increases during the falling down of the IP, resulting in the NPR phenomenon.
This IP model is inspired by the strain-dependent z (out-of-plane) coordinates of atoms in the warped edges shown in Fig. 3b, where the z-coordinates are fitted to the functions $z=\pm b_{0}\sin(\theta_{0}(1-\epsilon/\epsilon_{c}))$, in which the parameters are restricted by $b_{0}=z_{0}/\sin\theta_{0}$. This function describes exactly the trajectory of the tip of the IP (displayed by blue arrow in Fig. 4a) during its falling down process in Fig. 3b. This function also indicates that the IP’s tilting angle $\theta$ is a linear function of the applied tensile strain $\epsilon$,
$$\displaystyle\theta=\theta_{0}\left(1-\frac{\epsilon}{\epsilon_{c}}\right),$$
(2)
where $\theta_{0}$ is the initial tilting angle. This expression gives $\theta=0$ at the critical strain $\epsilon=\epsilon_{c}$, as required by the definition of the critical strain in Fig. 3b. For the applied tensile strain $\epsilon$ in the x-direction, the resulting strain in the y-direction is
$$\displaystyle\epsilon_{y}=\frac{\cos\theta-\cos\theta_{0}}{\cos\theta_{0}}\approx\frac{\theta_{0}^{2}}{\epsilon_{c}}\epsilon,$$
yielding the Poisson’s ratio of the edge
$$\displaystyle\nu_{e}$$
$$\displaystyle=$$
$$\displaystyle-\frac{\epsilon_{y}}{\epsilon_{x}}=-\frac{\theta_{0}^{2}}{\epsilon_{c}}.$$
(3)
We now determine the initial tilting angle $\theta_{0}$ for the IP. The tilting angle with respect to the y-axis for the tangent plane at point $\left(x,y,z\right)$ on the warped surface is
$$\displaystyle\phi\left(x,y\right)$$
$$\displaystyle=$$
$$\displaystyle\tan^{-1}\left(\frac{\partial w}{\partial y}\right)\approx\frac{A}{l_{c}}e^{-y/l_{c}}\sin\frac{\pi x}{\lambda},$$
in which the tilting angle is assumed to be small. This assumption is reasonable as will be shown below. The average tilting angle for the warping area $x\in[0,\lambda]$ and $y\in[0,l_{c}]$ is
$$\displaystyle\bar{\phi}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{\lambda l_{c}}\int_{0}^{\lambda}dx\int_{0}^{l_{c}}dy\phi\left(x,y\right)=\frac{A}{l_{c}}\frac{2}{\pi}\left(1-\frac{1}{e}\right).$$
(4)
Inserting the value of $A$ and $l_{c}$ from Fig. 2, we get an average tilting angle $\bar{\phi}=0.106$. We use this average tilting angle as the initial tilting angle for the IP, i.e., $\theta_{0}=\bar{\phi}=0.106$. As a result, we obtain the Poisson’s ratio for the warped edge
$$\displaystyle\nu_{e}=-\frac{\theta_{0}^{2}}{\epsilon_{c}}=-\left(\frac{A}{l_{c}}\frac{2}{\pi}\left(1-\frac{1}{e}\right)\right)^{2}/\epsilon_{c}=-1.37$$
in which the critical strain $\epsilon_{c}=0.0082$ is the saturation value from the inset of Fig. 3c, which should be used here in the discussion of the Poisson’s ratio for an isolated warped edge. The Poisson’s ratio of -1.37 for the Set I geometries that is obtained in the limit of an ultra-narrow graphene ribbon using the IP model is very close to the value of -1.51 obtained via the molecular statics calculations in Fig. 3c.
We find in Fig. 4b that the critical strains for Sets I-V obey the same relation $\epsilon_{c}=0.0082-0.092/W$, where again we use the saturation value of $\epsilon_{c}=0.0082$ for the calculation of the effective Poisson’s ratio of graphene ribbons with warped edges. The critical strain is directly related to the compressive edge stress, which generates compressive strain ($\epsilon_{e}$) in the edge region, and thus various warped configurations (with different local minimum potential energies). From equation (2), the magnitude of the compressive edge strains equals the critical strain $\epsilon_{c}$ when the warped edges have been completely flattened into a planar structure, i.e. $\epsilon_{e}=\epsilon_{c}=0.0082$. Using the calculated values for the Young’s modulus of the edge region as $E_{e}=11.15$8 eVÅ${}^{-2}$, while the compressive edge stress density is16 $\sigma_{e}=1.05$ eVÅ${}^{-1}$, we can estimate the width ($W_{e}$) of the region that will be compressed by the edge strain according to $W_{e}=\sigma_{e}/\left(E_{e}\epsilon_{e}\right)=11.48$ Å. Taking a representative value of the penetration depth $l_{c}=8.55$ Å from simulation Set I, we find that the penetration depth of the resultant warped configuration is close to $W_{e}$. These analysis illustrate that the saturation value of the critical strain is an intrinsic property for the free edge, which should not depend on the detailed warping configuration.
From equations (1), (3) and (4), we can thus obtain the following general formula for the Poisson’s ratio in graphene ribbons of arbitrary width,
$$\displaystyle\nu=\nu_{0}-\frac{2}{\tilde{W}}\left(\nu_{0}+\frac{1}{\epsilon_{c}}\tilde{A}^{2}C_{0}^{2}\right),$$
(5)
where $C_{0}=\frac{2}{\pi}\left(1-\frac{1}{e}\right)$ is a universal constant and $\nu_{0}=0.34$ is the Poisson’s ratio for bulk graphene. The dimensionless quantity $\tilde{W}=W/l_{c}$ is the width with reference to the penetration depth $l_{c}$. This is a bulk related quantity, and a larger $\tilde{W}$ correlates with moving the Poisson’s ratio in the positive direction. The other dimensionless quantity $\tilde{A}=A/l_{c}$ is the warping amplitude with reference to the penetration depth. This quantity is an edge related property, which tunes the Poisson’s ratio in the negative direction. The sign and also the value of the Poisson’s ratio are determined by the competition between these two effects, while the length of graphene has no effect. Fig. 4c shows a three-dimensional plot for the Poisson’s ratio predicted by the IP model based on equation (5). The numerical results for all of the five simulation Sets are also shown in the figure, which agree quite well with the IP model.
We note that the effective Poisson’s ratio defined in equation (5) is intrinsically width-dependent, due to the width-dependence of the warping amplitude $A$ and the penetration depth $l_{c}$. Thus, by using the dimensionless quantities $\tilde{W}$ and $\tilde{A}$, the effect of the warping number is intrinsically included in the expression for the effective Poisson’s ratio. Furthermore, the resultant equation (5) is a general expression for the Poisson’s ratio, which is an explicit function of the ribbon geometry. Hence, equation (5) could be readily extended to describe the Poisson’s ratio in other similar atomic-thick materials.
By equating equation (5) to be zero, we get
$$\displaystyle\tilde{W}=2+\frac{2}{\nu_{0}}\frac{C_{0}^{2}}{\epsilon_{c}}\tilde{A}^{2}.$$
(6)
This function is plotted in Fig. 4d, and serves to delineate the positive and negative Poisson’s ratio regions. This figure serves as a phase diagram for NPR phenomenon in the parameter space of $\tilde{A}$ and $\tilde{W}$. The NPR phenomenon occurs for graphene with parameters in the region below the curve. In particular, if the width $\tilde{W}<2$, then the Poisson’s ratio is negative irrespective of the value for the other parameter $\tilde{A}$. The numerical results for all of the five simulation Sets can be correctly categorized into positive or negative regions in this phase diagram, further validating the analytical IP model we have presented.
Conclusion. We have used molecular statics simulations to demonstrate the occurrence of negative Poisson’s ratios in graphene ribbons. The negative Poisson’s ratios occur due to the warping of the free edges due to the compressive edge stresses, and can be observed for graphene ribbons with widths smaller than about 10 nm, and for tensile strains smaller than about 0.5%. An inclined plate model was developed, which revealed the link between the warped free edge, and the negative Poisson’s ratios. The model also led to an analytic formula for the structural dependence for the Poisson’s ratio of graphene. Specifically, we find that the value of the Poisson’s ratio is determined by the interplay between the width (bulk property) and the warping amplitude (edge property), while the length does not play a role. The analytic phase diagram for the sign and value of the Poisson’s ratio accurately explained the numerical results. We expect these results to further augment the already interesting suite of physical properties and potential applications for graphene ribbons.17, 18
Methods. The NPR studies of single-layer graphene ribbons were performed using classical molecular statics simulations, where the interactions between carbon atoms in graphene were described by the second generation Brenner potential 26, which has been widely used to study the mechanical response of graphene.27 The Cartesian coordinate system is displayed in Fig. 1a. Periodic boundary conditions (PBCs) were applied in the x-direction, while either PBCs or free boundary condition (FBCs) were applied in the y-direction.
To obtain stable freestanding graphene ribbons with edge warping, following Shenoy et al.16, we introduce a small out-of-plane perturbation $0.5e^{-y/10.0}\sin(\pi x/\lambda)$ to the z-coordinate of edge atoms in the ideal planar graphene structure. This perturbed structure is then relaxed by conjugate gradient (CG) energy minimization. The relaxation is performed until the relative energy change is less than $10^{-12}$. We note that the half wave length $\lambda=L/n$, with the warping number $n$ being an even number corresponding to the PBCs applied in the x-direction and $L$ being the length of the graphene ribbon. In other words, if a perturbation with $\lambda=L/6$ is used, then the relaxed structure shown in Fig. 2a is obtained. If the perturbation with $\lambda=L/8$ is used, then the relaxed warped edge has $\lambda=L/8$. The effect of the warping number $n$ is that if $n$ is larger, there are more warping segments at the edge, though each segment has a smaller warping amplitude.
We find that, for the graphene ribbons of length 195.96 Å, the warped configurations with warping number $n\geq 14$ are not energetically favorable, as they transition into warped configurations with smaller warping number during tensile deformation. On the other hand, the energy difference among these warped configurations with warping number $n\leq 4$ is very small, so the graphene ribbons will assume multiple warping configurations during tensile deformation. We thus have focused our investigations into configurations with warping number $6\leq n\leq 12$.
Once the equilibrium warped configuration is obtained, the tensile loading is applied. This is done by stretching the relaxed graphene ribbon in the x-direction as illustrated in Fig. 1a by deforming the simulation box in the x-direction. The deformed structure is then relaxed by the CG energy minimization procedure, in which the structure also allowed to be fully relaxed in the lateral directions. All simulations were performed using the publicly available simulation code LAMMPS 28, 29. The OVITO package was used for visualization 30.
Corresponding to the applied external tension $\epsilon_{x}=\epsilon$ in the x-direction, the resultant strain in the y-direction is computed as
$$\displaystyle\epsilon_{y}$$
$$\displaystyle=$$
$$\displaystyle\frac{W-W_{0}}{W_{0}},$$
(7)
where the width is $W=y_{\rm top}-y_{\rm bot}$ with $y_{\rm top}$ ($y_{\rm bot}$) being the averaged y-coordinate for atoms from the top (bottom) group in Fig. 1a. The Poisson’s ratio is then calculated by its definition
$$\displaystyle\nu=-\frac{\epsilon_{y}}{\epsilon_{x}}.$$
(8)
In our numerical calculation, equation (8) is realized using the finite difference method. More specifically, the Poisson’s ratio at strain $\epsilon_{x}=\epsilon^{j}$ is obtained by
$$\displaystyle\nu_{j}=-\frac{\epsilon_{y}^{j+1}-\epsilon_{y}^{j-1}}{\epsilon^{j+1}-\epsilon^{j-1}},$$
(9)
where the integer j=1, 2, …, represents the strain increment number.
Acknowledgements The work is supported by the Recruitment Program of Global Youth Experts of China, the National Natural Science Foundation of China (NSFC) under Grant No. 11504225 and the start-up funding from Shanghai University. HSP acknowledges the support of the Mechanical Engineering department at Boston University.
Author contributions J.W.J performed the calculations and discussed the results with H.S.P. J.W.J and H.S.P co-wrote the paper.
Competing financial interests The authors declare no competing financial interests.
References
Landau and Lifshitz 1995
Landau, L. D.; Lifshitz, E. M. Theory of Elasticity; Pergamon,Oxford,
1995
Rothenburg et al. 1991
Rothenburg, L.; Berlint, A. A.; Bathurst, R. J. Nature 1991,
354, 470
Lakes 1993
Lakes, R. Advanced Materials 1993, 5, 293–296
Evans and Alderson 2000
Evans, K. E.; Alderson, A. Advanced Materials 2000, 12,
617
Lethbridge et al. 2010
Lethbridge, Z. A.; Walton, R. I.; Marmier, A. S.; Smith, C. W.; Evans, K. E.
Acta Materialia 2010, 58, 6444–6451
Milstein and Huang 1979
Milstein, F.; Huang, K. Physical Review B 1979, 19,
2030
Baughman et al. 1998
Baughman, R. H.; Shacklette, J. M.; Zakhidov, A. A.; Stafstrom, S.
Nature 1998, 392, 362–365
Lakes 1987
Lakes, R. Science 1987, 235, 1038
Evans 1991
Evans, K. E. Endeavour 1991, 15, 170–174
Yang et al. 2004
Yang, W.; Li, Z.-M.; Shi, W.; Xie, B.-H.; Yang, M.-B. Journal of
Materials Science 2004, 39, 3269–3279
Greaves et al. 2011
Greaves, G. N.; Greer, A. L.; Lakes, R. S.; Rouxel, T. Nature Materials
2011, 10, 823
Ho et al. 2014
Ho, D. T.; Park, S.-D.; Kwon, S.-Y.; Park, K.; Kim, S. Y. Nature
Communications 2014, 5, 3255
Jiang and Park 2014
Jiang, J.-W.; Park, H. S. Nature Communications 2014, 5,
4727
Han et al. 2015
Han, J.; Xie, J.; Zhang, Z.; Yang, D.; Si, M.; Xue, D. Applied Physics
Express 2015, 8, 041801
Grima et al. 2015
Grima, J. N.; Winczewski, S.; Mizzi, L.; Grech, M. C.; Cauchi, R.; Gatt, R.;
Attard, D.; Wojciechowski, K. W.; Rybicki, J. Advanced Materials
2015, 27, 1455–1459
Shenoy et al. 2008
Shenoy, V. B.; Reddy, C. D.; Ramasubramaniam, A.; Zhang, Y. W. Physical
Review Letters 2008, 101, 245501
Jia et al. 2009
Jia, X.; Hofmann, M.; Meunier, V.; Sumpter, B. G.; Campos-Delgado, J.;
Romo-Herrera, J. M.; Son, H.; Hsieh, Y.-P.; Reina, A.; Kong, J.
et al. Science 2009, 323, 1701
Engelund et al. 2010
Engelund, M.; Furst, J. A.; Jauho, A. P.; Brandbyge, M. Physical Review
Letters 2010, 104, 036807
Kim and Park 2009
Kim, S. Y.; Park, H. S. Nano Letters 2009, 9,
969–974
Jiang and Wang 2012
Jiang, J.-W.; Wang, J.-S. Journal of Applied Physics 2012,
111, 054314
Reddy et al. 2006
Reddy, C. D.; Rajendran, S.; Liew, K. M. Nanotechnology 2006,
17, 864
Liu et al. 2007
Liu, F.; Ming, P.; Li, J. Physical Review B 2007, 76,
064120
Jiao et al. 2009
Jiao, L.; Zhang, L.; Wang, X.; Diankov, G.; Dai, H. Nature
2009, 458, 877
Ni et al. 2008
Ni, Z. H.; Yu, T.; Lu, Y. H.; Wang, Y. Y.; Feng, Y. P.; Shen, Z. X. ACS
Nano 2008, 2, 2301–2305
Mohiuddin et al. 2009
Mohiuddin, T. M. G.; Lombardo, A.; Nair, R. R.; Bonetti, A.; Savini, G.;
Jalil, R.; Bonini, N.; Basko, D. M.; Galiotis, C.; Marzari, N. et al.
Physical Review B 2009, 79, 205433
Brenner et al. 2002
Brenner, D. W.; Shenderova, O. A.; Harrison, J. A.; Stuart, S. J.; Ni, B.;
Sinnott, S. B. Journal of Physics: Condensed Matter 2002,
14, 783–802
Mo et al. 2009
Mo, Y.; Turner, K. T.; Szlufarska, I. Nature 2009, 457,
1116
Plimpton 1995
Plimpton, S. J. Journal of Computational Physics 1995,
117, 1–19
Lammps 2012
Lammps, http://www.cs.sandia.gov/$\sim$sjplimp/lammps.html
2012,
Stukowski 2010
Stukowski, A. Modelling and Simulation in Materials Science and
Engineering 2010, 18, 015012 |
Quantum Cosmology and the Inflationary Spectra from a Non-Minimally Coupled Inflaton
Alexander Y. Kamenshchik
Alexander.Kamenshchik@bo.infn.it
Alessandro Tronconi
Alessandro.Tronconi@bo.infn.it
Dipartimento di Fisica e Astronomia and INFN, Via Irnerio 46,40126 Bologna,
Italy
Giovanni Venturi
Giovanni.Venturi@bo.infn.it
Dipartimento di Fisica e Astronomia and INFN, Via Irnerio 46,40126 Bologna,
Italy
Abstract
We calculate the quantum gravitational corrections to the Mukhanov-Sasaki equation obtained by the canonical quantization of the inflaton-gravity system. Our approach, which is based on the Born-Oppenheimer decomposition of the resulting Wheeler-DeWitt equation, was previously applied to a minimally coupled inflaton. In this article we examine the case of a non minimally coupled inflaton and, in particular, the induced gravity case is also discussed. Finally, the equation governing the quantum evolution of the inflationary perturbations is derived on a de Sitter background. Moreover the problem of the introduction of time is addressed and a generalized method, with respect to that used for the minimal coupling case, is illustrated. Such a generalized method can be applied to the universe wave function when, through the Born-Oppenheimer factorization, we decompose it into a part which contains the minisuperspace degrees of freedom and another which describes the perturbations.
1 Introduction
Inflation [1] was originally introduced to overcome the fine tuning problems affecting the old hot big bang cosmology. Today, 40 years after its introduction, by inflation we mean a very articulated framework potentially capable of connecting many aspects of the very early Universe to the present day, low energy, observations [2]. Since the microphysics behind inflation is still unknown, people generically speak of the inflationary paradigm and its theoretical description has been declined in many different ways.
Any inflationary model describing the cosmological evolution during the very early stages of our Universe must supply at least 60 e-folds of accelerated expansion and, during such a phase, the quantum fluctuations of the vacuum, which are believed to generate the seed of the large scale structure we observe today, are stretched beyond the causal horizon giving rise to a nearly scale independent spectrum of perturbations [3]. Any successful model of inflation must provide a dynamical mechanism which, independently of the initial conditions, satisfies to the above requirements.
Moreover, depending on the formulation chosen for the inflationary paradigm, other observable outcomes may be generated by the accelerated phase such as primordial gravitational waves and black holes (which today may constitute part of the Dark Matter content of our Universe [4]). Due to its high energy origin inflation can provide an answer to many fundamental problems of modern physics such as the origin of the Dark components of the Universe or the description of Quantum Gravity and can fill the huge gap between the physics at Planck scales down to the Standard Model and classical General Relativity (GR).
It is an accepted belief that GR is an effective description of gravity at large distances (low energy). At Planck energies the classical description provided by GR must include new effects arising from quantum mechanics and a new, description of the microscopic world at the Planck length could even be possible [5]. Quantum effects could generate new operators, irrelevant at low energies, such as higher powers of curvature and any theory sector containing a scalar field (such as the Higgs field [6]) may couple to gravity non-minimally, drive inflation and/or dynamically affect the “effective” Newton’s constant. The latter case has been investigated many years before inflation was introduced and was originally called Induced Gravity (IG) since the gravitational field equations were a consequence of (were “induced by”) the quantum behaviour of some scalar field on the curved background [7]. Later on [8], it was realised that such a model could in principle drive inflation and become GR at low energies (in the presence of a suitable potential).
Induced gravity is thus a natural candidate for the description of the very early Universe, including inflation and quantum effects. The addition of non-perturbative quantum gravitational corrections to such a class of models would lead to an even more complete description of inflationary physics close to Planck energies. In this article we shall consider such a possibility as we canonically quantize the minisuperspace (homogeneous) degrees of freedom (DOF) and then study the evolution of the vacuum fluctuations on the homogeneous background. The method employed was already applied to the GR case in a series of articles [9] and is based on the quantization of the Hamiltonian constraint leading to the Wheeler-DeWitt (WdW) equation for the wave function of the universe [10]. After a decomposition à la Born-Oppenheimer (BO) [11] for the total universe wave function one is led to a quantum equation for the homogeneous degrees of freedom, which includes the back-reaction of the quantum fluctuations, and an equation for the wave function of each mode of the quantum fluctuations which also depends on the minisuperspace variable. In this context we illustrate a general method for the introduction of the classical time in this latter equation, based on the solution of the Hamilton-Jacobi equation for the minisuperspace variables. This method is a generalisation of that used in the GR case and can be applied to a class of solutions of the homogeneous WdW equation which cannot be non trivially decomposed into a gravitational part and a homogeneous inflaton part [12].
Finally we apply our method to the de Sitter case which, for IG, is given by a quartic potential. The equation governing the evolution of each mode of the vacuum fluctuation is found to be the same as is obtained for GR and de Sitter (with a constant potential for the scalar field). This result is derived by evaluating the non adiabatic effects emerging from the BO decomposition perturbatively and showing that, at least in the de Sitter case, the inflationary spectra are invariant w.r.t. the Jordan to Einstein frame transition even when the quantum gravitational corrections are included. This result is non trivial and is complementary to that obtained in a previous article, [12], where such an equivalence was shown only for the homogeneous DOF.
The article is organized as follows. In section 2 we review the basic equations and introduce the formalism. In section 3 we first apply the BO decomposition to the inflaton gravity system and we then illustrate how time can be introduced in this context so as to finally derive the MS equation with quantum gravitational corrections included. In section 3 we apply the formalism to the de Sitter case and finally in section 4 we illustrate our conclusions.
2 Beyond the Minisuperspace approximation
Let us consider a non minimally coupled scalar field on a curved, spatially flat, spacetime described by the following action
$$S=\int d^{4}x\sqrt{-g}\left[-\frac{U}{2}R+\frac{1}{2}\partial_{\mu}\phi%
\partial^{\mu}\phi-V(\phi)\right]$$
(1)
where $U=\left(M^{2}+\xi\phi^{2}\right)$.
The above action can be decomposed into a homogeneous part plus fluctuations around it. In what follows we shall only consider the scalar fluctuations of the metric. They are associated with the scalar field and can be collectively described in terms of a single, Mukhanov-Sasaki (MS) field $v(x,t)$.
The full lagrangian density governing the evolution of the homogeneous variables and perturbations is given by
$$\mathcal{L}=-L^{3}\left(3U\frac{a\dot{a}^{2}}{N}+6\xi\phi\dot{\phi}\frac{a^{2}%
\dot{a}}{N}-\frac{a^{3}\dot{\phi}^{2}}{2N}+a^{3}NV\right)+\sum_{k}\mathcal{L}_%
{k}$$
(2)
where the dot denotes the derivative w.r.t. a generic time variable associated with the lapse function $N$ and $\mathcal{L}_{k}$ is the lagrangian of the k-mode of the Mukhanov-Sasaki variable $v_{k}$ which here describes scalar perturbations111A formally identical contribution can be added to describe the tensor perturbations.. Let us note that, on working in a flat 3-space, and considering both homogeneous and inhomogeneous quantities, one must introduce an unspecified length $L$ (see [9] for more details). In what follows we shall set $L=1$. The lagrangian $\mathcal{L}_{k}$ takes the form
$$\mathcal{L}_{k}=\frac{1}{2}\left(v^{\prime}_{k}-\omega_{k}^{2}v_{k}^{2}\right)$$
(3)
with
$$\omega_{k}^{2}=k^{2}-\frac{z^{\prime\prime}}{z}$$
(4)
when the conformal time ($N=a$) is chosen. The time dependent mass term for the scalar perturbations $z^{\prime\prime}/z$ in this context is defined in terms of the homogeneous classical DOF as
$$z\equiv\frac{a^{2}\phi^{\prime}}{a^{\prime}}\left(1+\frac{6\xi^{2}\phi^{2}}{U}%
\right)^{1/2}\left(1+\frac{\xi a\phi\phi^{\prime}}{a^{\prime}U}\right)^{-1}$$
(5)
and is a function of time. Moreover the MS variable $v_{k}$, in the uniform curvature gauge, is
$$v_{k}=\frac{z\,a^{\prime}}{a\,\phi^{\prime}}\delta\phi_{k},$$
(6)
where $\delta\phi_{k}$ is the Fourier transform of the inflaton field fluctuations.
The definition of the momenta mixes the velocities of the homogeneous DOF in the minisuperspace approximation (see [12]) and one has
$$\pi_{a}=-6Ua^{\prime}-6\xi\phi\phi^{\prime}a,\;\pi_{\phi}=-6\xi\phi aa^{\prime%
}+a^{2}\phi^{\prime},\;\pi_{k}=v^{\prime}_{k}.$$
(7)
Correspondingly the velocities are
$$\phi^{\prime}=\frac{U\pi_{\phi}-\xi a\phi\pi_{a}}{a^{2}\left(U+6\xi^{2}\phi^{2%
}\right)},\;a^{\prime}=-\frac{a\pi_{a}+6\xi\phi\pi_{\phi}}{6a\left(U+6\xi^{2}%
\phi^{2}\right)},\;v^{\prime}_{k}=\pi_{k}.$$
(8)
The system Hamiltonian is finally
$$\mathcal{H}=\frac{\pi_{\phi}^{2}}{2a^{2}}\frac{U}{U+6\xi^{2}\phi^{2}}-\frac{%
\xi\phi\,\pi_{a}\pi_{\phi}}{a\left(U+6\xi^{2}\phi^{2}\right)}-\frac{\pi_{a}^{2%
}}{12\left(U+6\xi^{2}\phi^{2}\right)}+a^{4}V+\sum_{k}\mathcal{H}_{k}$$
(9)
with $\mathcal{H}_{k}=\frac{1}{2}\left(\pi_{k}^{2}+\omega_{k}^{2}v_{k}^{2}\right)$. Given the invariance of the system w.r.t. time reparametrisation, the Hamiltonian $\mathcal{H}$ is zero.
The canonical quantisation of the matter-gravity system, then leads to the following Wheeler-DeWitt equation, in the coordinate representation, where, for simplicity, we consider a suitable ordering for the kinetic terms:
$$\displaystyle\left\{\frac{1}{12U}\partial_{A}^{2}+\frac{\xi}{U}\partial_{A}%
\partial_{F}-\frac{1}{2\phi^{2}}\partial_{F}^{2}+a^{6}\left(1+\frac{6\,\xi^{2}%
\phi^{2}}{U}\right)V\right.$$
$$\displaystyle\left.+a^{2}\left(1+\frac{6\,\xi^{2}\phi^{2}}{U}\right)\sum_{k}%
\hat{\mathcal{H}}_{k}\right\}\Psi(a,\phi,\left[v_{k}\right])=0$$
(10)
with $A\equiv\ln a$, $F\equiv\ln\phi$. In the limit $\xi\rightarrow 0$ ($U\rightarrow{\rm M_{\rm P}}^{2}$) the above equation becomes
$$\displaystyle\left\{\frac{1}{12a^{2}{\rm M_{\rm P}}^{2}}\partial_{A}^{2}-\frac%
{1}{2a^{2}\phi^{2}}\partial_{F}^{2}+a^{4}V+\sum_{k}\hat{\mathcal{H}}_{k}\right%
\}\Psi(a,\phi,\left[v_{k}\right])=0$$
(11)
which is its correct GR limit. On the other hand in the limit ${\rm M_{\rm P}}\rightarrow 0$ ($U\rightarrow\xi\phi^{2}$) the WdW equation (2) becomes
$$\displaystyle\left\{\frac{1}{12\xi}\partial_{A}^{2}+\partial_{A}\partial_{F}-%
\frac{1}{2}\partial_{F}^{2}+a^{6}\left(1+6\,\xi\right)\phi^{2}V\right.$$
$$\displaystyle\left.+a^{2}\phi^{2}\left(1+6\,\xi\right)\sum_{k}\hat{\mathcal{H}%
}_{k}\right\}\Psi(a,\phi,\left[v_{k}\right])=0$$
(12)
and its correct IG limit is recovered.
Let us now perform the following Born-Oppenheimer (BO) decomposition where the homogeneous DOF are factorised with respect to the wave function of the perturbations:
$$\Psi(a,\phi,\left[v_{k}\right])=\Psi_{0}(A,F)\prod_{k}\chi_{k}(A,F,v_{k}).$$
(13)
Let us note that each mode of the perturbations is described by the corresponding wave function which also depends on the homogeneous DOF.
2.1 BO decomposition
The BO decomposition was originally applied in atomic physics and consists of factorising the total wave function of atoms and molecules in a part for the “slow” DOF (nuclei) and a part for the “fast” DOF (electrons), the latter depending on the “slow” variables as well. To the leading order in the adiabatic approximation the BO decomposition then leads to a system of coupled Schrödinger equations which can be solved analytically. Non adiabatic terms, at the next to leading order, determine non adiabatic transitions between quantum levels, otherwise neglected in the adiabatic approximation. The same BO approach has been successfully applied to the inflaton-gravity system by usually associating to the scale factor the role played by the nucleus in atomic physics and to matter (homogeneous inflaton and perturbations) that of the electrons. The non adiabatic contributions which arise in the decompositions are, in this context, associated with the quantum gravitational effects. Such effects in the common semiclassical treatment of the evolution of inflationary perturbations are neglected.
In contrast with [9], where only the scale factor dependence was factorised, here we followed a more general approach which, in principle, can be applied to systems where the wave function of the “slow” (gravitational) DOF cannot be, non trivially, factorised. Moreover in scalar-tensor theories the role of the scalar field (besides being the inflaton) is tightly intertwined with gravity since it dynamically determines Newton’s constant and “induces” its dynamics through quantum effects.
In order to proceed with the BO decomposition let us first rewrite the WdW equation in a compact form as
$$\left\{\sum_{\alpha,\beta=1,2}G^{\alpha\beta}\partial_{\alpha}\partial_{\beta}%
+a_{0}^{6}{\rm e}^{6A}{\mathcal{V}}+a_{0}^{2}{\rm e}^{2A}h\sum_{k}\hat{%
\mathcal{H}}_{k}\right\}\Psi(a,\phi,\left[v_{k}\right])=0$$
(14)
where $X=(A,F)$ ($X^{1}=A$, $X^{2}=F$), $\partial_{\alpha}\equiv\partial_{X^{\alpha}}$ and
$$G\equiv\frac{1}{2}\left(\begin{array}[]{cc}\left(6\xi\right)^{-1}&1\\
1&-g\end{array}\right)$$
(15)
is the metric of the homogeneous minisuperspace.
Moreover let us set
$$g=\frac{U}{\xi\phi^{2}},\;h=\frac{U+6\,\xi^{2}\phi^{2}}{\xi},\;{\mathcal{V}}=hV$$
(16)
and from here on we shall use the Einstein summation convention in order to keep the notation as compact as possible.
The BO decomposition is performed by splitting the total wave function using the ansatz (13). Then an equation for the homogeneous wave function $\Psi_{0}$ can be obtained by projecting out the inhomogeneous DOF i.e. by contracting the WdW equation with $\prod_{k}\chi_{k}^{*}(A,F,v_{k})$ and integrating over $\prod_{k}{\rm d}v_{k}$. The resulting equation is
$$\displaystyle G^{\alpha\beta}\left\{\partial_{\alpha}\partial_{\beta}+\sum_{k}%
\left[2\langle\chi_{k}|\partial_{\alpha}\chi_{k}\rangle\left(\partial_{\beta}+%
\sum_{j\neq k}\langle\chi_{j}|\partial_{\beta}\chi_{j}\rangle\right)+\langle%
\chi_{k}|\partial_{\alpha}\partial_{\beta}\chi_{k}\rangle\right]\right\}\Psi_{0}$$
$$\displaystyle+\left(a_{0}^{6}{\rm e}^{6A}{\mathcal{V}}+a_{0}^{2}{\rm e}^{2A}h%
\sum_{k}\langle\chi_{k}|{\hat{\mathcal{H}}}_{k}|\chi_{k}\rangle\right)\Psi_{0}=0$$
(17)
where
$$\langle\chi_{k}|\hat{O}|\chi_{k}\rangle\equiv\int_{-\infty}^{+\infty}{\rm d}v_%
{k}\,\chi_{k}^{*}(a,\phi,v_{k})\,\mathcal{R}(\hat{O})\,\chi_{k}(a,\phi,v_{k})$$
(18)
and $\mathcal{R}(\hat{O})$ is the coordinate representation of the operator $\hat{O}$. Henceforth, in order to keep the notation compact, we shall use the same notation for quantum operators independetly of the representation used.
This latter equation correctly reproduces that for minisuperspace [12] when one neglects the back-reaction of the inhomogeneities on the homogeneous part in the above equation (2.1). In the present context, the back-reaction is given by the semiclassical contribution of the energy density of the inhomogeneities and consists in the sum of the averaged hamiltonians $H_{k}$ plus the non adiabatic contributions which describe the quantum gravitational effects. These contributions are expected to be small during inflation when the homogeneous inflaton energy density is usually assumed to be, by far, the leading contribution.
One then finds the equations for the modes $\chi_{k}$. These equations can be obtained by multiplying the gravitational equation by $\chi_{k}$ and then subtracting the WdW equation multiplicated by $\prod_{j\neq k}\chi_{j}^{*}$ and integrated over $\prod_{j\neq k}{\rm d}v_{j}$.
The resulting equation is
$$\displaystyle\!\!\!\!\!\!\!\!\!\!G^{\alpha\beta}\left\{\phantom{\frac{A}{B}}\!%
\!\!\!\!\!2\left(\partial_{\alpha}\Psi_{0}\right)\left(\partial_{\beta}-%
\langle\chi_{k}|\partial_{\beta}\chi_{k}\rangle\right)\chi_{k}+\Psi_{0}\left(%
\partial_{\alpha}\partial_{\beta}-\langle\chi_{k}|\partial_{\alpha}\partial_{%
\beta}\chi_{k}\rangle\right)\chi_{k}\right.$$
$$\displaystyle\!\!\!\!\!\!\!\!\!\!\left.+2\Psi_{0}\left(\sum_{i\neq k}\langle%
\chi_{i}|\partial_{\alpha}\chi_{i}\rangle\right)\left(\partial_{\alpha}-%
\langle\chi_{k}|\partial_{\alpha}\chi_{k}\rangle\right)\partial_{\beta}\chi_{k%
}\right\}$$
$$\displaystyle\!\!\!\!\!\!\!\!\!\!+a_{0}^{2}{\rm e}^{2A}h\Psi_{0}\left(\hat{%
\mathcal{H}}_{k}-\langle\chi_{k}|\hat{\mathcal{H}}_{k}|\chi_{k}\rangle\right)%
\chi_{k}=0.$$
(19)
Let us now define the recurrent expression $\langle\chi_{k}|\hat{O}|\chi_{k}\rangle\equiv\langle\hat{O}\rangle_{k}$. The expression (2.1) is the equation for the wave function of the $k$-mode of the MS field and in the present form also contains the dependence on the modes different from $k$. The equations (2.1) and (2.1) are equivalent to the WdW equation (14). They can be simplified by re-phasing $\Psi_{0}$ and $\chi_{k}$ as follows:
$$\Psi_{0}\equiv\tilde{\Psi}_{0}{\rm e}^{i\theta(A,F)}\,,\chi_{k}\equiv{\rm e}^{%
-i\theta_{k}(A,F)}\tilde{\chi}_{k}$$
(20)
with
$$\theta(A,F)\equiv i\sum_{j}\int^{A,F}\langle\partial_{\alpha}\rangle_{j}d\bar{%
X}^{\alpha}\,,\theta_{k}(A,F)\equiv i\int^{A,F}\langle\partial_{\alpha}\rangle%
_{k}d\bar{X}^{\alpha}.$$
(21)
Let us note that the above line integrals are independent of the contour of integration chosen provided no singularities are present in the domain of integration (and this is generally the case) [11]. Moreover one has
$$\partial_{\alpha}\theta(X)=i\sum_{j}\langle\partial_{\alpha}\rangle_{j}\,,\;%
\partial_{\alpha}\theta_{k}(X)=i\langle\partial_{\alpha}\rangle_{k}.$$
(22)
In terms of the re-defined wave functions the homogeneous equation (2.1) takes the form
$$G^{\alpha\beta}\partial_{\alpha}\partial_{\beta}\tilde{\Psi}_{0}+\left({\rm e}%
^{6A}{\mathcal{V}}+{\rm e}^{2A}h\sum_{k}\langle\hat{\widetilde{\mathcal{H}}}_{%
k}\rangle_{k}\right)\tilde{\Psi}_{0}=G^{\alpha\beta}\sum_{k}\langle\partial_{%
\alpha}\tilde{\chi}_{k}|\partial_{\beta}\tilde{\chi}_{k}\rangle\tilde{\Psi}_{0}$$
(23)
where $\langle\hat{\widetilde{O}}\rangle_{k}\equiv\langle\tilde{\chi}_{k}|\hat{O}|%
\tilde{\chi}_{k}\rangle$ and the r.h.s. contains the quantum gravitational effects on the total back-reaction of inhomogeneities for the homogeneous background. On neglecting such inhomogeneities one recovers the WdW equation for the minisuperspace variables.
The equation for the perturbations finally becomes
$$\left\{G^{\alpha\beta}\left[2\frac{\partial_{\alpha}\tilde{\Psi}_{0}}{\tilde{%
\Psi}_{0}}\partial_{\beta}+\left(\partial_{\alpha}\partial_{\beta}-\langle%
\widetilde{\partial_{\alpha}\partial_{\beta}}\rangle_{k}\right)\right]+a_{0}^{%
2}{\rm e}^{2A}h(F)\left(\hat{\mathcal{H}}_{k}-\langle\hat{\widetilde{\mathcal{%
H}}}_{k}\rangle_{k}\right)\right\}\tilde{\chi}_{k}=0$$
(24)
and, in contrast with (2.1), it only contains a single $k$-mode. Therefore, from here on, we shall omit the external subscript $k$ to keep the notation compact ($\langle\hat{\widetilde{O}}\rangle_{k}\rightarrow\langle\hat{\widetilde{O}}\rangle$).
Let us note that the expression $G^{\alpha\beta}(2\partial_{\alpha}\tilde{\Psi}_{0}/\tilde{\Psi}_{0})\partial_{\beta}$ is related to the introduction of time [9]. It is given by 4 contributions:
$$2G^{\alpha\beta}\frac{\partial_{\alpha}\tilde{\Psi}_{0}}{\tilde{\Psi}_{0}}%
\partial_{\beta}=\frac{1}{6\xi}\frac{\partial_{A}\tilde{\Psi}_{0}}{\tilde{\Psi%
}_{0}}\partial_{A}+\frac{\partial_{F}\tilde{\Psi}_{0}}{\tilde{\Psi}_{0}}%
\partial_{A}+\frac{\partial_{A}\tilde{\Psi}_{0}}{\tilde{\Psi}_{0}}\partial_{F}%
-g(F)\frac{\partial_{F}\tilde{\Psi}_{0}}{\tilde{\Psi}_{0}}\partial_{F}.$$
(25)
We observe that
$$\left(\frac{1}{6\xi}\frac{\partial_{A}\tilde{\Psi}_{0}}{\tilde{\Psi}_{0}}+%
\frac{\partial_{F}\tilde{\Psi}_{0}}{\tilde{\Psi}_{0}}\right)\partial_{A}=\frac%
{i}{\tilde{\Psi}_{0}}\left(\frac{a\hat{\pi}_{a}+6\xi\phi\hat{\pi}_{\phi}}{6\xi%
}\tilde{\Psi}_{0}\right)\;a\,\partial_{a}$$
(26)
and
$$\left(\frac{\partial_{A}\tilde{\Psi}_{0}}{\tilde{\Psi}_{0}}-g(F)\frac{\partial%
_{F}\tilde{\Psi}_{0}}{\tilde{\Psi}_{0}}\right)\partial_{F}=\frac{i}{\tilde{%
\Psi}_{0}}\left(\frac{\xi\phi a\hat{\pi}_{a}-U\hat{\pi}_{\phi}}{\xi\phi}\tilde%
{\Psi}_{0}\right)\phi\,\partial_{\phi}.$$
(27)
In the semiclassical limit, to the leading order in $\hbar$, quantum operators can be replaced by their classical counterparts leading to
$$\left(a\hat{\pi}_{a}+6\xi\phi\hat{\pi}_{\phi}\right)\tilde{\Psi}_{0}\simeq-6%
\xi a\,a^{\prime}h\,\tilde{\Psi}_{0}\,,\;\left(\xi\phi a\hat{\pi}_{a}-U\hat{%
\pi}_{\phi}\right)\tilde{\Psi}_{0}\simeq-\xi a^{2}\phi^{\prime}h\,\tilde{\Psi}%
_{0},$$
(28)
where the quantities on the r.h.s. in (28) are the classical (time dependent) variables.
Therefore, in such a limit,
$$\left(\frac{1}{6\xi}\frac{\partial_{A}\tilde{\Psi}_{0}}{\tilde{\Psi}_{0}}+%
\frac{\partial_{F}\tilde{\Psi}_{0}}{\tilde{\Psi}_{0}}\right)\partial_{A}=G^{%
\alpha 1}\frac{\partial_{\alpha}\tilde{\Psi}_{0}}{\tilde{\Psi}_{0}}\partial_{1%
}\simeq-i\left(a^{2}a^{\prime}h\right)\,\partial_{a}$$
(29)
$$\left(\frac{\partial_{A}\tilde{\Psi}_{0}}{\tilde{\Psi}_{0}}-g(F)\frac{\partial%
_{F}\tilde{\Psi}_{0}}{\tilde{\Psi}_{0}}\right)\partial_{F}=G^{\alpha 2}\frac{%
\partial_{\alpha}\tilde{\Psi}_{0}}{\tilde{\Psi}_{0}}\partial_{2}\simeq-i\left(%
a^{2}\phi^{\prime}h\right)\partial_{\phi}.$$
(30)
and
$$2G^{\alpha\beta}\frac{\partial_{\alpha}\tilde{\Psi}_{0}}{\tilde{\Psi}_{0}}%
\partial_{\beta}\tilde{\chi}_{k}(a,\phi,v_{k})\simeq-ia^{2}h\frac{\partial}{%
\partial\eta}\tilde{\chi}_{k}(a(\eta),\phi(\eta),v_{k}),$$
(31)
where the homogeneous variables must be evaluated on the classical trajectory $\tilde{\chi}_{k}(a(\eta),\phi(\eta),v_{k})\equiv\tilde{\chi}_{k}(\eta,v_{k})$.
2.2 Introduction of time
As we already pointed out the emergence of time in equation (24) is related to the derivative of the homogeneous wave function and is therefore a consequence of the BO decomposition. We have also observed (see (31)) that the emerging “flow” of the time is defined by the trajectories in the $(A,F)$ manifold (i.e. the configuration space of the homogeneous variables) described by the tangent vector
$$\partial_{\eta}=\eta^{A}\partial_{A}+\eta^{F}\partial_{F}\equiv\eta^{\alpha}%
\partial_{\alpha},$$
(32)
where $\eta^{A}$ and $\eta^{F}$ are functions defined on the configuration space and corresponding to the classical velocities $\eta^{A}=\frac{\partial A_{\rm cl}}{\partial\eta}$ and $\eta^{F}=\frac{\partial F_{\rm cl}}{\partial\eta}$. The integral curves $(A(\eta),F(\eta))$, solutions of the system
$$\left\{\begin{array}[]{l}\frac{{\rm d}A}{{\rm d}\eta}=\eta^{A}(A,F)\\
\frac{{\rm d}F}{{\rm d}\eta}=\eta^{F}(A,F)\end{array}\right.,$$
(33)
represent the classical solutions and the corresponding tangent defines the (classical) time flow. The solutions of (33) depend on two integration constants. The resulting curves form a congruence on the configuration space (minisuperspace).
Let us note that the “emergence” of time is associated with some classical limit of the state described by the homogeneous wave function $\Psi_{0}$. If the matter-gravity system maintains a purely quantum behaviour a classical time cannot be introduced and no real advantage can be obtained from the BO approach. For example, a well defined classical behaviour in minisuperspace is recovered in the leading order of the WKB approximation or in the large $a$ limit for some quantum solutions to the homogeneous WdW equation [12]. The introduction of time depends on the quantum fluctuations around the classical trajectory due to the intrinsic quantum nature of the matter-gravity system. When such fluctuations are small they can be treated perturbatively and the classical limit is well defined. The presence of large quantum fluctuation destroys the classical evolution and signals that the system is in a highly quantum (non-classical) state.
2.3 Hamilton-Jacobi equation
In order to obtain the classical flow of the time, one needs the functions $\eta^{\alpha}$, defined over the configuration space (and corresponding to the minisuperspace). These functions can be calculated from the general solution of the classical Hamilton-Jacobi (HJ) equation. From the classical Hamiltonian $\mathcal{H}=\mathcal{H}\left(\pi_{a},\pi_{\phi},a,\phi\right)$, given by expression (9) (without including the inhomogeneities), we derive the following HJ equation for the HJ function $W(a,\phi)$
$$\mathcal{H}\left(\partial_{a}W,\partial_{\phi}W,a,\phi\right)=0.$$
(34)
An exact general solution for (34) can be obtained in the IG case for potentials of the form $V=\lambda M^{4-n}\phi^{n}$ starting from the ansatz
$$W=\nu\ln\frac{a}{a_{0}}+\ln\omega(x),$$
(35)
where $x\equiv a^{3}\phi^{\frac{n+2}{2}}$. Thus,
$$\partial_{A}W=\nu+3\frac{{\rm d}\ln\omega}{{\rm d}\ln x},\;\partial_{F}W=\frac%
{n+2}{2}\frac{{\rm d}\ln\omega}{{\rm d}\ln x}$$
(36)
and the HJ equation becomes
$$\displaystyle\left(\frac{{\rm d}\ln\omega}{{\rm d}\ln x}\right)^{2}\left[\frac%
{3}{4\xi}-\frac{\left(n-4\right)^{2}}{8\left(1+6\xi\right)}\right]+\left(\frac%
{{\rm d}\ln\omega}{{\rm d}\ln x}\right)\left[\frac{1}{\xi}+\frac{n-4}{1+6\xi}%
\right]\frac{\nu}{2}+$$
$$\displaystyle+\left[\frac{\nu^{2}}{12\xi\left(1+6\xi\right)}-\lambda M^{4-n}x^%
{2}\right]=0.$$
(37)
This first order differential equation can be solved algebraically for ${\rm d}\ln\omega/{\rm d}\ln x$ and then integrated to obtain
$$\omega(x)=\widetilde{D}\,x^{\widetilde{A}}\exp\left[\pm\left(\sqrt{\widetilde{%
B}+\widetilde{C}x^{2}}-\sqrt{\widetilde{B}}\tanh^{-1}\sqrt{1+\frac{\widetilde{%
C}}{\widetilde{B}}x^{2}}\right)\right]$$
(38)
with
$$\widetilde{A}=-\frac{\left[1+\frac{\left(n-4\right)\xi}{1+6\xi}\right]}{\left[%
1-\frac{\xi\left(n-4\right)^{2}}{6\left(1+6\xi\right)}\right]}\frac{\nu}{3},$$
(39)
$$\widetilde{B}=\widetilde{A}^{2}-\frac{2\nu^{2}}{3\left[6\left(1+6\xi\right)-%
\xi\left(n-4\right)^{2}\right]},$$
(40)
$$\widetilde{C}=\frac{4\xi\lambda M^{4-n}}{3\left[1-\frac{\xi\left(n-4\right)^{2%
}}{6\left(1+6\xi\right)}\right]}$$
(41)
and $\widetilde{D}$ is an integration constant. In the $n=4$ limit the expressions above are further simplified and, in particular, one obtains
$$\frac{{\rm d}\ln\omega}{{\rm d}\ln x}=-\frac{\nu}{3}\pm\sqrt{\frac{2\xi\nu^{2}%
}{3\left(1+6\xi\right)}+\frac{4\xi\lambda x^{2}}{3}}.$$
(42)
The classical velocities can be obtained from (8) with $\pi_{a}=\partial_{a}W$ and $\pi_{\phi}=\partial_{\phi}W$. For the $n=4$ case one has
$$\frac{\phi^{\prime}}{\phi}=-\frac{\nu}{x^{2/3}\left(1+6\xi\right)},\;\frac{a^{%
\prime}}{a}=-\frac{1}{6\xi x^{2/3}}\left(\frac{\nu}{1+6\xi}+3\frac{{\rm d}\ln%
\omega}{{\rm d}\ln x}\right).$$
(43)
The constant $\nu$ parametrises different sets of trajectories on the configuration space, the de Sitter (inflationary) attractor trajectory corresponding to $\nu=0$. The expressions (43) then take the form
$$\frac{\phi^{\prime}}{\phi}=0,\;\frac{a^{\prime}}{a}=\sqrt{\frac{\lambda}{3\xi}%
}\,a\,\phi.$$
(44)
2.4 Auxiliary vector
We already observed that
$$\partial_{\eta}\sim\frac{2iG^{\alpha\beta}}{a^{2}h}\frac{\partial_{\alpha}%
\tilde{\Psi}_{0}}{\tilde{\Psi}_{0}}\partial_{\beta}$$
(45)
where the approximate equality becomes exact in the semiclassical limit and to the leading order in $\hbar$. Higher order contributions in $i\partial_{\alpha}\tilde{\Psi}_{0}/\tilde{\Psi}_{0}$ must be interpreted as quantum gravitational effects related to the definition of time and, if small, can be treated perturbatively. To the leading order one has
$$\partial_{\eta}\equiv\frac{2G^{\alpha\beta}}{a^{2}h}\left(\partial_{\alpha}W%
\right)\partial_{\beta}=\eta^{\beta}\partial_{\beta},$$
(46)
where $W$ is a solution to the HJ equation, and the so-called WKB time is recovered.
On the other hand the above definition of the time flow can be applied to more general solutions of the homogeneous equation which substantially differ from the semiclassical ones and then need a more careful treatment.
Once time is formally introduced by (45) one recovers, to the leading order, the Schrödinger equation governing the evolution for the wave function of the inflationary perturbations. Such an equation is equivalent to the Mukhanov-Sasaki (MS) equation for the operator $\hat{v}_{k}$ calculated on a classical background (indeed it is the same equation but in the Schrödinger representation).
If one is interested in calculating the quantum gravitational corrections to the semiclassical MS equation things are more involved. From the definition (32) of the time flow one can introduce an “auxiliary” vector satisfying
$$\partial_{\tau}\equiv\tau^{A}\partial_{A}+\tau^{F}\partial_{F}\equiv\tau^{%
\alpha}\partial_{\alpha}\;,{\rm with}\quad\left[\partial_{\eta},\partial_{\tau%
}\right]=0$$
(47)
where the normalization of $\partial_{\tau}$ is unspecified and can be fixed arbitrarily. Let us note that $\partial_{\tau}$ is not defined in a unique way and, for example, $\partial_{\tau}+c\,\partial_{\eta}$ still satisfies the condition (47).
The components of the auxiliary vector, by definition, must satisfy the following equations
$$\left\{\begin{array}[]{l}\eta^{A}\left(\partial_{A}\tau^{F}\right)+\eta^{F}%
\left(\partial_{F}\tau^{F}\right)-\tau^{A}\left(\partial_{A}\eta^{F}\right)-%
\tau^{F}\left(\partial_{F}\eta^{F}\right)=0\\
\eta^{A}\left(\partial_{A}\tau^{A}\right)+\eta^{F}\left(\partial_{F}\tau^{A}%
\right)-\tau^{A}\left(\partial_{A}\eta^{A}\right)-\tau^{F}\left(\partial_{F}%
\eta^{A}\right)=0.\end{array}\right.$$
(48)
In the $n=4$ case $\eta^{\alpha}=\eta^{\alpha}(x)$ with $x=(a\phi)^{3}$ and therefore
$$\partial_{A}\eta^{\alpha}=\frac{\partial\ln x}{\partial\ln a}\frac{{\rm d}\eta%
^{\alpha}}{{\rm d}\ln x}=3\frac{{\rm d}\eta^{\alpha}}{{\rm d}\ln x}=\frac{%
\partial\ln x}{\partial\ln\phi}\frac{{\rm d}\eta^{\alpha}}{{\rm d}\ln x}=%
\partial_{F}\eta^{\alpha}.$$
(49)
The conditions (48) can be then satisfied by setting $\tau^{A}=-\tau^{F}=\tau_{0}^{-1}={\rm const}$. One then has the following auxiliary vector field
$$\partial_{\tau}=\tau_{0}^{-1}\left(\partial_{A}-\partial_{F}\right)$$
(50)
which is associated with a new coordinate. The coordinates $(\eta,\tau)$ can now be adopted to parametrise the configuration space and can be related to $(A,F)$ by the following change of variable
$$A=\frac{\tau}{\tau_{0}}+A_{\rm cl}(\eta),\;F=-\frac{\tau}{\tau_{0}}+F_{\rm cl}%
(\eta)$$
(51)
where
$$A+F=A_{\rm cl}(\eta)+F_{\rm cl}(\eta)$$
(52)
is a function of $\eta$ only.
By inverting the relations (46,50) one has
$$\partial_{A}=\frac{1}{\eta^{A}+\eta^{F}}\left(\partial_{\eta}+\eta^{F}\,\tau_{%
0}\partial_{\tau}\right),\;\partial_{F}=\frac{1}{\eta^{A}+\eta^{F}}\left(%
\partial_{\eta}-\eta^{A}\,\tau_{0}\partial_{\tau}\right).$$
(53)
Let us note that while $\partial_{\eta}$ is a vector tangent to the classical trajectories in minisuperspace, $\partial_{\tau}$ is not associated to any particular direction.
The two vectors are necessary in order to estimate the quantum gravitational corrections to the MS equation originally parametrised by $(a,\phi)$. Locally, given $\partial_{\eta}$, one can always find a vector orthogonal to it (here the orthogonality means the orthogonality w.r.t. the minisuperspace supermetric) and the quantum gravitational effects may be “projected” on these two directions. Physically $\partial_{\eta}$ generates the time flow on the classical trajectory and the associated quantum corrections are the fluctuations along such a trajectory. On the other hand the quantum corrections on the orthogonal direction describe the fluctuations away from a given classical trajectory. When one performs the BO decomposition factorising only one homogeneous degree of freedom and then using it as the “classical clock” for the rest of the system, by construction only the quantum fluctuations along the classical trajectory are present.
If one now considers the de Sitter attractor (44) the above expressions are simplified and one obtains $\eta_{F}=0$, $\eta_{A}=\sqrt{\frac{\lambda}{3\xi}}\,a\,\phi$ and
$$\partial_{A}=\frac{1}{\eta^{A}}\partial_{\eta},\;\partial_{F}=\frac{1}{\eta^{A%
}}\partial_{\eta}-\tau_{0}\partial_{\tau}.$$
(54)
For such a case $\partial_{\tau}$ given by (50) is orthogonal to $\partial_{\eta}$ globally. We shall adopt this definition of $\tau$ in the following sections.
2.5 The modified MS Equation
The equation for the wave function of the perturbations is (24). In such an equation the terms related to the introduction of the time are
$$-\frac{2G^{\alpha\beta}}{a_{0}^{2}{\rm e}^{2A}h(F)}\frac{\partial_{\alpha}%
\widetilde{\Psi}_{0}}{\widetilde{\Psi}_{0}}\partial_{\beta}\tilde{\chi}_{k}=i%
\left(\eta^{\alpha}\partial_{\alpha}+q^{\alpha}\partial_{\alpha}\right)\tilde{%
\chi}_{k}\equiv i\left(\partial_{\eta}+q^{\alpha}\partial_{\alpha}\right)%
\tilde{\chi}_{k}$$
(55)
and contributions proportional to $i\left(q^{\alpha}\partial_{\alpha}\right)\tilde{\chi}_{k}$ should be considered as quantum corrections emerging from the definition of time. Eq. (24) also contains “pure” quantum gravitational contributions (originating from non-adiabatic effects) given by
$$\left(\hat{Q}-\langle\hat{\widetilde{Q}}\rangle\right)\tilde{\chi}_{k}=\frac{G%
^{\alpha\beta}}{a_{0}^{2}{\rm e}^{2A}h(F)}\left[\left(\partial_{\alpha}%
\partial_{\beta}-\langle\widetilde{\partial_{\alpha}\partial_{\beta}}\rangle%
\right)\right]\tilde{\chi}_{k}.$$
(56)
Solving the full quantum equation (24) is a hopeless task. One still may search for a solution perturbatively. To the leading order one has
$$\left\{-i\frac{\partial}{\partial\eta}+\left(\hat{\mathcal{H}}_{k}-\langle\hat%
{\mathcal{H}}_{k}\rangle\right)\right\}\tilde{\chi}_{k}(\eta,v_{k})=0$$
(57)
where, in $\hat{\mathcal{H}}_{k}$, $a=a(\eta)$ and $\phi=\phi(\eta)$ are the classical trajectories on minisuperspace. One may now redefine
$$\chi_{k,s}\equiv\exp\left[-i\int{\rm d}\eta^{\prime}\langle\hat{\mathcal{H}}_{%
k}\rangle\right]\tilde{\chi}_{k}$$
(58)
and obtain the standard MS equation
$$\left(i\frac{\partial}{\partial\eta}-\hat{\mathcal{H}}_{k}\right)\chi_{k,s}=0.$$
(59)
This equation can be solved exactly in some cases (for example on a de Sitter background) or in the Slow Roll approximation. On then following a perturbative approach, the quantum gravitational corrections can be evaluated using the leading order solution. Let us note that the solutions of (59) are functions of $\eta$ and not of the auxiliary parameter $\tau$. Therefore the quantum gravitational effects are only generated by the derivatives with respect to the classical time flow $\partial_{\eta}$ and not $\partial_{\tau}$. In the first order equation, all the contributions arising from $\partial_{\tau}$ can be ignored. If $\partial_{\tau}$ is chosen as the direction orthogonal to the classical trajectory we conclude that, within the perturbative approach, the fluctuations away from the classical trajectory must be zero.
3 Application to de Sitter evolution
One may apply the method described above to de Sitter evolution and IG. Such a case is relevant as it describes inflation to the leading order in the slow roll approximation and many expressions simplify. One may then easily check how quantum gravitational corrections to the primordial power spectra can be calculated without having too complicated expressions. In IG the stable de Sitter attractor exists only for a quartic potential. Correspondingly the scalar field (at least classically) is constant and takes a value which is arbitrary and only depends on the initial conditions. We shall only consider the solutions corresponding to the above mentioned attractor and ignore those which describe the transient phase with the scalar field slowing down and approaching the attractor asymptotically.
Classically the formulae relevant for this case have been presented in the sections 2 and 3. The auxiliary vector has been already calculated and its relation with the coordinate basis vectors on the configuration space are given by (54).
The general full perturbation equation is
$$\left[-i\left(\eta^{\alpha}+q^{\alpha}\right)\partial_{\alpha}+\left(\hat{%
\mathcal{H}}_{k}-\langle\hat{\mathcal{H}}_{k}\rangle\right)+\left(\hat{Q}-%
\langle\hat{\widetilde{Q}}\rangle\right)\right]\tilde{\chi}_{k}=0$$
(60)
where $\hat{Q}$ is defined by (56) and $q^{\alpha}$ is defined implicitly by (55).
For the de Sitter attractor, the time derivative is $\partial_{\eta}=\eta^{\alpha}\partial_{\alpha}$ with
$$\eta^{F}=0,\;\eta^{A}=\sqrt{\frac{\lambda}{3\xi}}\,a\,\phi\equiv a\,H,$$
(61)
$H=\sqrt{\frac{\lambda}{3\xi}}\phi$, and the corresponding auxiliary vector is $\partial_{\tau}$ and they are related to $\partial_{A}$ and $\partial_{F}$ by the following relations
$$\partial_{A}=\frac{1}{aH}\partial_{\eta},\;\partial_{F}=\frac{1}{aH}\left(%
\partial_{\eta}-aH\partial_{\tau}\right).$$
(62)
Let us note that $aH=\sqrt{\frac{\lambda}{3\xi}}\exp\left(A+F\right)$ and, see (52), is a function of $\eta$ only.
Moreover when $\partial_{F}$ acts on a function of $\eta$ (and not $\tau$) one has $\partial_{F}f(\eta)=\frac{1}{aH}\partial_{\eta}f(\eta)$.
Therefore
$$G^{\alpha\beta}\partial_{\alpha}\partial_{\beta}=\frac{1+6\xi}{12\xi}\left(%
\frac{1}{a^{2}H^{2}}\partial_{\eta}^{2}-\frac{1}{aH}\partial_{\eta}\right)$$
(63)
and the perturbations equation then becomes
$$\displaystyle\left\{-i\left(1+\frac{q^{A}+q^{F}}{aH}\right)\partial_{\eta}+%
\left(\hat{\mathcal{H}}_{k}-\langle\hat{\mathcal{H}}_{k}\rangle\right)\right.$$
$$\displaystyle\left.+\frac{1}{12\xi a^{4}H^{2}\phi^{2}}\left[\partial_{\eta}^{2%
}-\langle\widetilde{\partial_{\eta}^{2}}\rangle-aH\left(\partial_{\eta}-%
\langle\widetilde{\partial_{\eta}}\rangle\right)\right]\tilde{\chi}_{k}\right\},$$
(64)
where the quantities $a\,H$ and $a\,\phi$ are functions of $\eta$ evaluated on the inflationary attractor. Let us note that, to the leading order, the above equation becomes (57) and $\langle\widetilde{\partial_{\eta}}\rangle=0$.
Let us now evaluate the quantum corrections associated with the introduction of time. They are given by
$$\left(\hat{T}-\langle\hat{\widetilde{T}}\rangle\right)\tilde{\chi}_{k}\equiv-i%
\left(\frac{q^{A}+q^{F}}{aH}\right)\left(\partial_{\eta}-\langle\widetilde{%
\partial}_{\eta}\rangle\right)\tilde{\chi}_{k},$$
(65)
where the quantities $q^{\alpha}$ are implicitly defined by (55) and are thus given by
$$q^{\beta}=\frac{2i}{a^{2}h}G^{\alpha\beta}\frac{\partial_{\alpha}\tilde{\Psi}_%
{0}}{\tilde{\Psi}_{0}}-\eta^{\beta}.$$
(66)
On the inflationary attractor $\tilde{\Psi}_{0}$ is a function of $x\equiv a^{3}\phi^{3}$ (see [12] for the details). Let us rephase it as
$$\tilde{\Psi}_{0}\equiv\psi_{q}\exp\left(iW\right),$$
(67)
where $W$ is the Hamilton Jacobi function which satisfies (34) with $\nu=0$ and $\tilde{\Psi}_{0}$ satisfies the homogeneous WdW (23) where, for simplicity, we neglect the backreaction of the perturbations (recovering the WdW equation in minisuperspace). The rephased wave function $\psi_{q}$ satisfies the following equation
$$2i\frac{{\rm d}W}{{\rm d}\ln x}\frac{{\rm d}\ln\psi_{q}}{{\rm d}\ln x}+i\frac{%
{\rm d}^{2}W}{{\rm d}\ln x^{2}}+\frac{{\rm d}^{2}\ln\psi_{q}}{{\rm d}\ln x^{2}%
}+\left(\frac{{\rm d}\ln\psi_{q}}{{\rm d}\ln x}\right)^{2}=0$$
(68)
where the first two terms usually give the leading contribution in the semiclassical ($\hbar\rightarrow 0$) expansion and lead to the van Vleck determinant. One finally obtains
$$q^{A}=\frac{i}{2\xi a^{2}\phi^{2}}\frac{{\rm d}\ln\psi_{q}}{{\rm d}\ln x},\;q^%
{F}=0$$
(69)
for the inflationary attractor we are considering.
In contrast, on neglecting the last two terms in (68) (and then following the prescription for the standard WKB approximation), the expression for $\psi_{q}$ can be easily calculated in terms of $W$ and one has ${\rm d}\ln\psi_{q}/{\rm d}\ln x=-1/2$. In this latter case, the quantum effects are inversely proportional to $\xi\phi^{2}$. Let us note that $q^{A}$ only depends on $x$ and is therefore a function of $\eta$ only.
The existence of exact solutions for IG and a power law potential also allows an explicit calculation of the above quantum gravitational corrections. In [12] we found the following exact solution for IG with a quartic potential:
$$\tilde{\Psi}_{0}=\left(\frac{a}{a_{0}}\right)^{\nu}\chi(x),$$
(70)
where
$$\chi(x)=x^{q}\left[c_{1}J_{r}(Ax)+c_{2}Y_{r}(Ax)\right]$$
(71)
with $x=a^{3}\phi^{3}$, $A=\left(\frac{4}{3}\xi\lambda\right)^{1/2}$, $r=q=0$, and $J_{r}$, $Y_{r}$ are Bessel functions. Let us note that the superpositions of the Bessel functions generally mix contracting and expanding universes. The solution corresponding to the classical evolution on the de Sitter attractor corresponds to $\nu=0$. In the limit for large ${\zeta}\equiv Ax=2\xi\phi^{2}a^{3}H$ one has
$$J_{r}({\zeta})\sim\frac{1}{\sqrt{2\pi{\zeta}}}\left[{\rm e}^{i\left({\zeta}-%
\frac{\pi}{4}\right)}\left(1-\frac{i}{8{\zeta}}-\frac{9}{128{\zeta}^{2}}+%
\mathcal{O}\left(\frac{1}{{\zeta}^{3}}\right)\right)+\rm{c.c.}\right]$$
(72)
and
$$Y_{r}({\zeta})\sim\frac{1}{\sqrt{2\pi{\zeta}}}\left[-i{\rm e}^{i\left({\zeta}-%
\frac{\pi}{4}\right)}\left(1-\frac{i}{8{\zeta}}-\frac{9}{128{\zeta}^{2}}+%
\mathcal{O}\left(\frac{1}{{\zeta}^{3}}\right)\right)+\rm{c.c.}\right].$$
(73)
Let us now consider the linear combination with $c_{2}=-ic_{1}$ which corresponds to the expanding phase.
Then, following the procedure for the introduction of time, we find
$$-\frac{2G^{\alpha\beta}}{a^{2}h}\frac{\partial_{\alpha}\tilde{\Psi}_{0}}{%
\tilde{\Psi}_{0}}\partial_{\beta}=-\frac{{\zeta}}{2\xi a\phi^{2}}\frac{%
\partial_{\zeta}\chi}{\chi}\partial_{a}.$$
(74)
If we keep the leading and next to leading contribution in
$$\frac{\partial_{\zeta}\chi}{\chi}\sim-i\left(1-\frac{i}{2{\zeta}}+\dots\right)$$
(75)
for large ${\zeta}$ then
$$-\frac{{\zeta}}{2\xi a\phi^{2}}\frac{\partial_{\zeta}\chi}{\chi}\partial_{a}%
\simeq ia^{2}H\left(1-\frac{i}{2{\zeta}}\right)\partial_{a}\simeq i\partial_{%
\eta}+\frac{1}{2{\zeta}}\partial_{\eta}$$
(76)
and thus
$$iq^{\alpha}\partial_{\alpha}=\frac{1}{2{\zeta}}\partial_{\eta}.$$
(77)
This last contribution is proportional to $\left(\xi\phi^{2}\right)$ and is identical to that obtained with the WKB approximation.
Let us note that the quantum gravitational corrections evaluated in terms of $\eta$ and the auxiliary variable $\tau$ only depend on $a\phi$, which is a function of $\eta$ only, and on the first and second order derivatives in $\partial_{\eta}$ and $\partial_{\tau}$. Therefore the resulting “modified” MS equation admits solutions of the form $\chi_{k,s}=\chi_{k,s}(\eta)$ (without any functional dependence on $\tau$). Therefore, to the leading order, one recovers the usual MS equation having solutions which depend on the classical time $\eta$. To the next to leading order the quantum gravitational corrections are evaluated perturbatively with the leading order solution and therefore the perturbed solution is a function of $\eta$ only. Thus the quantum gravitational corrections associated to the direction orthogonal to the time flow are necessarily zero and one is left with those parallel to the classical trajectory.
Finally one can rephase $\tilde{\chi}_{k}$ according to the prescription (58), express (3) in terms of $\chi_{k,s}$ (which satisfies the conventional Schrödinger equation (59) to leading order) and obtain
$$\displaystyle\left\{\left(-i\frac{\partial}{\partial\eta}+\hat{\mathcal{H}_{k}%
}\right)-\frac{i}{2\xi a^{3}H\phi^{2}}\frac{{\rm d}\ln\psi_{q}}{{\rm d}\ln x}%
\left(\hat{\mathcal{H}_{k}}-\langle\hat{\mathcal{H}_{k}}\rangle_{s}\right)+\right.$$
$$\displaystyle\left.\frac{1}{12\xi a^{4}H^{2}\phi^{2}}\left(\langle i\partial_{%
\eta}\hat{\mathcal{H}_{k}}\rangle_{s}-i\partial_{\eta}\hat{\mathcal{H}_{k}}%
\right)-\left(\hat{\mathcal{H}_{k}}-\langle\hat{\mathcal{H}_{k}}\rangle_{s}%
\right)^{2}+\right.$$
$$\displaystyle\left.\langle\hat{\mathcal{H}^{2}}\rangle_{s}-\langle\hat{%
\mathcal{H}_{k}}\rangle_{s}^{2}+iaH\left(\hat{\mathcal{H}_{k}}-\langle\hat{%
\mathcal{H}_{k}}\rangle_{s}\right)\right\}\chi_{k,s}=0,$$
(78)
where now $\langle\hat{O}\rangle_{s}\equiv\langle\chi_{k,s}|\hat{O}|\chi_{k,s}\rangle$.
On also considering the van Vleck contribution to the introduction of time and defining $\xi\phi^{2}=\tilde{m}_{\rm P}^{2}/6$ (the effective value of the Planck mass in the IG framework) one obtains
$$\left(i\frac{\partial}{\partial\eta}-\hat{\mathcal{H}_{k}}\right)\chi_{s}=%
\frac{1}{2\tilde{m}_{\rm P}^{2}}\left(\hat{\Omega}_{k}-\langle\hat{\Omega}_{k}%
\rangle_{s}\right)\chi_{s},$$
(79)
where
$$\hat{\Omega}_{k}=\frac{1}{a^{4}H^{2}}\left[2\langle\hat{\mathcal{H}_{k}}%
\rangle_{s}\hat{\mathcal{H}_{k}}-\hat{\mathcal{H}_{k}}^{2}-i\frac{{\rm d}\hat{%
\mathcal{H}_{k}}}{{\rm d}\eta}+4\left(aH\right)\hat{\mathcal{H}_{k}}\right].$$
(80)
Let us note that formally this result is the same as the one for the de Sitter solution in GR with the identification of $\widetilde{m}_{\rm P}$ and $m_{\rm P}\equiv\sqrt{6}{\rm M_{\rm P}}$ where the former is proportional to the effective Planck mass, which depends on the expectation value of the scalar (inflaton) field and the latter is proportional to the Planck mass.
Furthermore on expressing the modified MS equation (79) in terms of the Einstein Frame DOF $\tilde{a}$, $\tilde{\phi}$, $\tilde{\eta}$, $\tilde{v}_{k}$ with
$$\tilde{a}=\frac{\sqrt{6\xi}}{m_{\rm P}a\,\phi},\;\tilde{\phi}=\sqrt{\frac{1+6%
\xi}{6\xi}}m_{\rm P}\ln\frac{\phi}{{\rm M_{\rm P}}},\;\tilde{H}=\frac{m_{\rm P%
}}{\sqrt{6\xi}}\frac{H}{\phi}$$
(81)
we observe that
$$a\,H=\tilde{a}\tilde{H}\Rightarrow\eta=\tilde{\eta}\quad{\rm and}\quad v_{k}=%
\tilde{v}_{k}$$
(82)
and therefore one recovers exactly the equation already found for GR [9].
We can therefore conclude that, on even including the quantum gravitational corrections and in the “pure” de Sitter case, the primordial spectra are invariant w.r.t. the Jordan to Einstein frame transformation. Indeed the de Sitter evolution is invariant with respect to frame transformations and the primordial spectra calculated without the quantum gravitational corrections are the same (this latter property of the primordial spectra is valid independently of the background evolution chosen). Such an invariance holds also when quantum gravitational corrections are included. Let us note that the fact that such an equivalence holds at the quantum level (at least for the de Sitter case) also confirms the consistency of the approach adopted here for the introduction of time in a matter-gravity system with two minisuperspace variables playing the role of the “classical clock”.
4 Conclusions
Non-minimally coupled scalar fields are ubiquitous in cosmology, in particular when energies become very high since a non-minimal coupling generally emerges from quantum effects. It seems therefore natural to study their quantum behaviour (in particular during inflation with the scalar field playing the role of the inflaton) in the presence of the quantum gravitational effects which are usually ignored (or included in an effective description) in the inflationary era and calculate the evolution of the inflationary spectra. Theories with non-minimally coupled scalar fields are usually included in the class of modified gravity theories since such scalar fields affect Newton’s constant and can modify gravitational attraction even at long distances. Furthermore there exists a mapping between the DOF of such theories and those of General Relativity with a minimally coupled scalar field, which is called Jordan to Einstein frame mapping. The mapping is often used since performing calculations in the Einstein Frame is usually easier and the results can be finally translated into the Jordan frame through the inverse mapping. This “equivalence” is known to hold at classical and semiclassical levels but at the full quantum level the complete equivalence of the two frames is not clear [13].
In this article the technique already employed in a series of articles [11] for a minimally coupled inflaton and standard General Relativity is applied to inflation with a non-minimally coupled inflaton. Such a technique leads to a MS equation with quantum gravitational corrections. The resulting quantum corrections can then be calculated explicitly for different inflationary models and the resulting inflationary spectra obtained. Moreover the full quantum equivalence between the Einstein and the Jordan frame can be investigated case by case (at least in the canonical quantisation context and within the approximation scheme followed). As an application we calculated the corrections on a de Sitter background. The MS equation obtained reproduces correctly that of [11] in the minimally coupled limit and the resulting spectra are invariant in both frames. This latter result is a consequence of the fact that the de Sitter evolution is frame invariant and is non-trivial.
Furthermore we discussed the problem of the introduction of time in the context of quantum cosmology by generalising the approach adopted in [11]. The full scheme presented can be applied to cases more general than pure de Sitter, in particular more general inflationary potentials should be considered and more realistic inflationary evolutions (including first order correction in the slow roll approximation) studied as was already done for the minimally coupled case [9].
5 Acknowledgements
Alexander Y. Kamenshchik is supported in part by the RFBR grant 18-52-45016.
References
[1]
A.A. Starobinsky. Springer. in H.J. De Vega and N. Sanchez (eds.) Current trends in field theory quantum gravity and strings, Lecture Notes in Physics 246 Verlag, Heidelberg, 1986), pp. 107-126.
A.D. Linde. Academic. Particle Physics and Inflationary Cosmology (Harwood New York, 1990).
[2]
P. A. R. Ade et al. [Planck Collaboration],
arXiv:1502.02114 [astro-ph.CO].
P. A. R. Ade et al. [Planck Collaboration],
arXiv:1502.01589 [astro-ph.CO].
Y. Akrami et al. [Planck Collaboration],
arXiv:1807.06211 [astro-ph.CO].
N. Aghanim et al. [Planck Collaboration],
arXiv:1807.06209 [astro-ph.CO].
[3]
V.F. Mukhanov, Sov. Phys. JETP 68, 1297 (1988);
J. M. Maldacena,
JHEP 0305 (2003) 013;
V. F. Mukhanov, H. A. Feldman and R. H. Brandenberger,
Phys. Rept. 215 (1992) 203.
V.F. Mukhanov, Phys. Lett. B 218, 17 (1989);
J. M. Bardeen,
Phys. Rev. D 22, 1882 (1980).
doi:10.1103/PhysRevD.22.1882
M. Sasaki,
Prog. Theor. Phys. 70 (1983) 394.
doi:10.1143/PTP.70.394
[4]
B. J. Carr,
Astrophys. J. 201 (1975) 1.
doi:10.1086/153853
M. Sasaki, T. Suyama, T. Tanaka and S. Yokoyama,
Class. Quant. Grav. 35 (2018) no.6, 063001
doi:10.1088/1361-6382/aaa7b4
[arXiv:1801.05235 [astro-ph.CO]].
A. Y. Kamenshchik, A. Tronconi, T. Vardanyan and G. Venturi,
Phys. Lett. B 791, 201 (2019)
doi:10.1016/j.physletb.2019.02.036
[arXiv:1812.02547 [gr-qc]].
[5]
K. G. Wilson,
Phys. Rev. B 4 (1971) 3174.
doi:10.1103/PhysRevB.4.3174
K. G. Wilson,
Phys. Rev. B 4 (1971) 3184.
doi:10.1103/PhysRevB.4.3184
S. Weinberg,
PoS CD 09, 001 (2009)
[arXiv:0908.1964 [hep-th]].
M. Reuter,
Phys. Rev. D 57, 971 (1998)
[hep-th/9605030].
A. Tronconi,
JCAP 1707 (2017) no.07, 015
doi:10.1088/1475-7516/2017/07/015
[arXiv:1704.05312 [gr-qc]].
A. Bonanno, A. Platania and F. Saueressig,
Phys. Lett. B 784 (2018) 229
doi:10.1016/j.physletb.2018.06.047
[arXiv:1803.02355 [gr-qc]].
A. Eichhorn,
Found. Phys. 48 (2018) no.10, 1407
doi:10.1007/s10701-018-0196-6
[arXiv:1709.03696 [gr-qc]].
[6]
F. L. Bezrukov and M. Shaposhnikov,
Phys. Lett. B 659 (2008) 703
doi:10.1016/j.physletb.2007.11.072
[arXiv:0710.3755 [hep-th]].
F. L. Bezrukov, A. Magnin and M. Shaposhnikov,
Phys. Lett. B 675 (2009) 88
doi:10.1016/j.physletb.2009.03.035
[arXiv:0812.4950 [hep-ph]].
A. O. Barvinsky, A. Y. Kamenshchik and A. A. Starobinsky,
A. O. Barvinsky, A. Y. Kamenshchik, C. Kiefer, A. A. Starobinsky and C. Steinwachs,
JCAP 0912 (2009) 003
doi:10.1088/1475-7516/2009/12/003
[arXiv:0904.1698 [hep-ph]].
A. De Simone, M. P. Hertzberg and F. Wilczek,
Phys. Lett. B 678 (2009) 1
doi:10.1016/j.physletb.2009.05.054
[arXiv:0812.4946 [hep-ph]].
[7]
A. D. Sakharov, Dokl. Akad. Nauk. SSSR 117, 70 (1967);
[Sov. Phys. Dokl. 12, 1040 (1967)];
A. Zee,
Phys. Rev. D 23, 858 (1981);
[8]
F. Cooper and G. Venturi,
Phys. Rev. D 24 (1981) 3338.
doi:10.1103/PhysRevD.24.3338
F. Finelli, A. Tronconi and G. Venturi,
Phys. Lett. B 659 (2008) 466
doi:10.1016/j.physletb.2007.11.053
[arXiv:0710.2741 [astro-ph]].
A. Tronconi and G. Venturi,
Phys. Rev. D 84 (2011) 063517
doi:10.1103/PhysRevD.84.063517
[arXiv:1011.3958 [gr-qc]].
A. Y. Kamenshchik, A. Tronconi and G. Venturi,
Phys. Lett. B 713 (2012) 358
doi:10.1016/j.physletb.2012.06.035
[arXiv:1204.2625 [gr-qc]].
A. Cerioni, F. Finelli, A. Tronconi and G. Venturi,
Phys. Lett. B 681 (2009) 383
doi:10.1016/j.physletb.2009.10.066
[arXiv:0906.1902 [astro-ph.CO]];
A. Cerioni, F. Finelli, A. Tronconi and G. Venturi,
Phys. Rev. D 81 (2010) 123505
doi:10.1103/PhysRevD.81.123505
[arXiv:1005.0935 [gr-qc]].
[9]
A. Y. Kamenshchik, A. Tronconi and G. Venturi,
Phys. Lett. B 726 (2013) 518
doi:10.1016/j.physletb.2013.08.067
[arXiv:1305.6138 [gr-qc]].
A. Y. Kamenshchik, A. Tronconi and G. Venturi,
Phys. Lett. B 734 (2014) 72
doi:10.1016/j.physletb.2014.05.028
[arXiv:1403.2961 [gr-qc]].
A. Y. Kamenshchik, A. Tronconi and G. Venturi,
JCAP 1504 (2015) no.04, 046
doi:10.1088/1475-7516/2015/04/046
[arXiv:1501.06404 [gr-qc]].
A. Y. Kamenshchik, A. Tronconi and G. Venturi,
Phys. Rev. D 94 (2016) no.12, 123524
doi:10.1103/PhysRevD.94.123524
[arXiv:1609.02830 [gr-qc]].
A. Y. Kamenshchik, A. Tronconi and G. Venturi,
Class. Quant. Grav. 35 (2018) no.1, 015012
doi:10.1088/1361-6382/aa8fb3
[arXiv:1709.10361 [gr-qc]].
[10]
B.S. DeWitt, Phys. Rev. 160, 113 (1967)
[11]
M. Born and J.R. Oppenheimer, Ann. Physik 84, 457 (1927);
C. A. Mead and D. G. Truhlar, J. Chem. Phys. 70, 2284 (1979);
C. A. Mead, Chem. Phys 49, 23 (1980)
C. A. Mead, Chem. Phys 49, 33 (1980)
R. Brout and G. Venturi, Phys. Rev. D 39, 2436 (1989);
G. Venturi, Class. Quantum Grav. 7, 1075 (1990);
G. L. Alberghi, R. Casadio, A. Tronconi,
Phys. Rev. D 74 (2006) 103501;
G. L. Alberghi, C. Appignani, R. Casadio, F. Sbisa, A. Tronconi,
Phys. Rev. D 77 (2008) 044002;
A. Tronconi, G. P. Vacca and G. Venturi,
Phys. Rev. D 67, 063517 (2003)
A. Tronconi, G. P. Vacca and G. Venturi,
Phys. Rev. D 67 (2003) 063517
doi:10.1103/PhysRevD.67.063517
[gr-qc/0302030].
A. Y. Kamenshchik, A. Tronconi, T. Vardanyan and G. Venturi,
Phys. Rev. D 97 (2018) no.12, 123517
doi:10.1103/PhysRevD.97.123517
[arXiv:1804.10075 [gr-qc]].
A. Y. Kamenshchik, A. Tronconi, T. Vardanyan and G. Venturi,
Int. J. Mod. Phys. D 28 (2019) no.06, 1950073
doi:10.1142/S0218271819500731
[arXiv:1809.08083 [gr-qc]].
[12]
A. Y. Kamenshchik, A. Tronconi and G. Venturi,
Phys. Rev. D 100 (2019) no.2, 023521
doi:10.1103/PhysRevD.100.023521
[arXiv:1905.02454 [gr-qc]].
[13]
A. Y. Kamenshchik, E. O. Pozdeeva, S. Y. Vernov, A. Tronconi and G. Venturi,
Phys. Rev. D 94 (2016) no.6, 063510
doi:10.1103/PhysRevD.94.063510
[arXiv:1602.07192 [gr-qc]].
A. Y. Kamenshchik, E. O. Pozdeeva, A. Tronconi, G. Venturi and S. Y. Vernov,
Class. Quant. Grav. 31 (2014) 105003
doi:10.1088/0264-9381/31/10/105003
[arXiv:1312.3540 [hep-th]].
A. Y. Kamenshchik, E. O. Pozdeeva, A. Tronconi, G. Venturi and S. Y. Vernov,
Class. Quant. Grav. 33 (2016) no.1, 015004
doi:10.1088/0264-9381/33/1/015004
[arXiv:1509.00590 [gr-qc]].
A. Y. Kamenshchik and C. F. Steinwachs,
Phys. Rev. D 91 (2015) no.8, 084033
doi:10.1103/PhysRevD.91.084033
[arXiv:1408.5769 [gr-qc]]. |
Federated Random Reshuffling with Compression and Variance Reduction
Grigory Malinovsky
Peter Richtárik
Abstract
Random Reshuffling (RR), which is a variant of Stochastic Gradient Descent (SGD) employing sampling without replacement, is an immensely popular method for training supervised machine learning models via empirical risk minimization. Due to its superior practical performance, it is embedded and often set as default in standard machine learning software. Under the name FedRR, this method was recently shown to be applicable to federated learning (Mishchenko et al., 2021), with superior performance when compared to common baselines such as Local SGD. Inspired by this development, we design three new algorithms to improve FedRR further: compressed FedRR and two variance reduced extensions: one for taming the variance coming from shuffling and the other for taming the variance due to compression. The variance reduction mechanism for compression allows us to eliminate dependence on the compression parameter, and applying additional controlled linear perturbations for Random Reshuffling, introduced by Malinovsky et al. (2021) helps to eliminate variance at the optimum. We provide the first analysis of compressed local methods under standard assumptions without bounded gradient assumptions and for heterogeneous data, overcoming the limitations of the compression operator. We corroborate our theoretical results with experiments on synthetic and real data sets.
Machine Learning, ICML
1 Introduction
The primary approach for training supervised machine learning models in the modern machine learning world is Empirical Risk Minimization. While the ultimate goal of supervised learning is to train models that generalize well to unseen data, in practice, only a finite data set is available during training. ERM formulation leads to the following finite-sum optimization problem:
$$\displaystyle\min_{x\in\mathbb{R}^{d}}\left[f(x)=\frac{1}{M}\sum_{m=1}^{M}g_{m}(x)\right],$$
(1)
where each function we also have finite-sum structure
$$\displaystyle g_{m}=\frac{1}{n}\sum_{i=1}^{n}f_{m,i}(x).$$
Big machine learning models are typically trained in a distributed setting. The training data is distributed across several workers, which compute local updates and then communicate them to the server. We are particularly interested in the Federated Learning setting. Federated Learning (Konečnỳ et al., 2016) is a subarea of distributed machine learning, where the number of devices $n$ is enormous. Usually, millions of local devices are heterogeneous to local data and computational and memory resources. Also, users want to keep their privacy, so the algorithm should do training locally. Moreover, communication between workers should be conducted via a trusted aggregation server, which is very expensive.
Communication as the bottleneck.
In literature, we have two strategies to overcome communication issues in federated learning. The first one is communication compression, where our goal is to reduce the number of communicated bits using gradient compression scheme (Mishchenko et al., 2019; Gorbunov et al., 2020) and compressed iterates (Khaled & Richtárik, 2019; Chraibi et al., 2019). There are many compression techniques such as quantization (Alistarh et al., 2017; Bernstein et al., 2018; Ramezani-Kebrya et al., 2019), sparsification (Aji & Heafield, 2017; Lin et al., 2017; Wangni et al., 2017; Alistarh et al., 2018) and other approaches (Shamir et al., 2014; Vogels et al., 2019; Wu et al., 2018). The second strategy to tackle this issue is increasing the number of local steps between the communication rounds. The most popular algorithm — FedAvg (McMahan et al., 2017)— is based on this idea. Many papers provide theoretical justifications for special cases of FedAvg such as local GD (Khaled et al., 2019) and local SGD (Khaled et al., 2020; Gorbunov et al., 2020; Stich, 2018; Lin et al., 2018). The natural union of communication compression and local computations is presented in Basu et al. (2020); Haddadpour et al. (2021). However, the theory provided in these papers is limited due to unrealistic assumptions.
Sampling without replacement.
Stochastic first-order algorithms, in particular, have attracted much attention in the machine learning world. Of these, stochastic gradient descent (SGD) is perhaps the best known and the most basic. SGD has a long history (Robbins & Monro, 1951) and is therefore well-studied and well-understood (Gower et al., 2019). However, methods based on data permutations (Bottou, 2009), when data points are shuffled randomly and processed in order, without replacement, show better performance than SGD (Recht & Ré, 2013). Also, this method has software implementation advantages since these methods are friendly to cache locality (Bengio, 2012). The most popular model in this class is Random Reshuffling (Recht & Ré, 2012). Shuffle-Once (Safran & Shamir, 2020) uses a similar approach, but shuffling occurs only once, at the very beginning, before the training begins. Random Reshuffling and Shuffle-Once have a long history, and many theoretical works try to show the advantages of Random Reshuffling (Gürbüzbalaban et al., 2019; Haochen & Sra, 2019; Nagaraj et al., 2019) and Shuffle Once (Rajput et al., 2020).
Federated Random Reshuffling. Recent advances of Mishchenko et al. (2020) and providing extension for Federated Learning (Mishchenko et al., 2021) allow us to consider this technique as a particular variant of FedAvg with a fixed number of local computations and sampling without replacement.
Variance Reduction. Compression operators help to reduce the number of transmitted bits. However, at the same time, it starts to be a source of variance, which increases the neighborhood of the optimal solution. This variance can sufficiently slow down the algorithm. In order to overcome this challenge, we need to use a variance reduction mechanism. The idea of this approach is based on shifted compression operator, and firstly it was proposed for compressed gradients (Mishchenko et al., 2019). For compressed iterates, the variance reduction mechanism was proposed in Chraibi et al. (2019). Moreover, stochastic first-order methods become a source of variance due to their random nature. Hopefully, another variance reduction mechanism can help with this type of variance. There are many variance-reduced methods which use sampling with replacement such as SVRG (Johnson & Zhang, 2013), L-SVRG (Kovalev et al., 2020),SAGA (Defazio et al., 2014a),SAG (Roux et al., 2012),Finito(Defazio et al., 2014b) etc. For permutation-based algorithms, we have only a few variance-reduced methods (Ying et al., 2019; Park & Ryu, 2020; Mokhtari et al., 2018). A recent paper of Malinovsky et al. (2021) introduced linear perturbation reformulation that allows getting better rates for variance reduced Random Reshuffling.
2 Contributions
This section outlines our work’s key contributions and offers explanations and clarifications regarding some of the development.
Compressed FedRR. We propose the first method, which combines three ideas: compression, local steps, and sampling without replacement. This method is compressed federated random reshuffling. The basic approach is to apply the compression operator to the iterates after each epoch and then aggregate compressed updates. Applying compression to the iterates can significantly worsen convergence properties. We prove the following rate:
$$\displaystyle\mathbb{E}\|x_{T}-x_{*}\|^{2}\leq(1-\gamma\mu)^{\frac{nT}{2}}\mathbb{E}\|x_{0}-x_{*}\|^{2}$$
$$\displaystyle+\frac{2\omega}{M}\frac{1}{\gamma\mu}\frac{1}{M}\sum_{m=1}^{M}\|x^{n}_{*,m}\|^{2}$$
$$\displaystyle+\frac{2}{\mu}\left(1+\frac{2\omega}{M}\right)\gamma^{2}L\frac{1}{M}\sum_{m=1}^{M}\left(\|\nabla F_{m}(x_{*})\|^{2}+\frac{n}{4}\sigma_{*,m}^{2}\right)$$
As we can see, we have a part with linear rate: $(1-\gamma\mu)^{\frac{nT}{2}}\mathbb{E}\|x_{0}-x_{*}\|^{2}$. Also there are three sources of variance in the optimum. The first one is $\frac{2\omega}{M}\frac{1}{\gamma\mu}\frac{1}{M}\sum_{m=1}^{M}\|x^{n}_{*,m}\|^{2}$ and it caused by compression. It is equal to zero only if $\omega=0$. This term cannot be elimated by decreasing step-sizes strategies. The second term is $\frac{2}{\mu}\left(1+\frac{2\omega}{M}\right)\gamma^{2}L\frac{1}{M}\sum_{m=1}^{M}\frac{n}{4}\sigma_{*,m}^{2}$. This source of variance is caused by stochasticity of Random Reshuffling method. This variance can be decreased by decreasing step-sizes. The third term is $\frac{2}{\mu}\left(1+\frac{2\omega}{M}\right)\gamma^{2}L\frac{1}{M}\sum_{m=1}^{M}\|\nabla F_{m}(x_{*})\|^{2}$. This source of variance is caused by heterogeneity of data. In other words, if we have the same optimum for all functions $F_{m}(x)$, we can get rid of this term. In heterogenious regime we need to use additional mechanism for controlling client drift such as SCAFFOLD (Karimireddy et al., 2019).
Variance-reduced Compressed FedRR. In the previous section, it was shown that using compressed iterates causes additional variance, which cannot be vanished by decreasing step-sizes strategies. Moreover, it forces us to have an additional assumption on the compression operator. We need to have very small compression parameter $\omega\leq\frac{M\gamma\mu\varepsilon}{\frac{2}{M}\|x^{n}_{*,m}\|^{2}}$. In order to fix it, we propose Variance-reduced compressed FedRR (FedCRR-VR) that utilizes shifted compressed updates with learning shifts. The similar mechanism is used in Mishchenko et al. (2019) and Gorbunov et al. (2020).
Double Variance-reduced Compressed FedRR.
We propose a modification of Variance-reduced Compressed Federated Random Reshuffling, which allows eliminating variance caused by stochasticity. We armed Variance-reduced Compressed FedRR with the linear permutation approach proposed by Malinovsky et al. (2021). Now we get both variance reduction mechanisms in one algorithm.
3 Preliminaries
3.1 $L$-smooth and $\mu$-strongly convex functions
Before introducing our convergence results, let us first formulate all concepts that we use throughout the paper. Firstly, we consider a class of $mu$-strongly convex and $L$-smooth functions.
Definition 1.
A differentiable function $f$ is $\mu$-strongly convex if
$$f(y)\geq f(x)+\nabla f(x)^{T}(y-x)+\frac{\mu}{2}\|y-x\|^{2}$$
for $\mu>0$ and all $x,y$.
Definition 2.
A differentiable function $f$ is $L$-smooth if
$$f(y)\leq f(x)+\nabla f(x)^{T}(y-x)+\frac{L}{2}\|y-x\|^{2}$$
for some $L>0$ and all $x,y$.
There is the first assumption that we use in all theorems.
Assumption 1.
Each $f_{m,i}$ is $\mu$-strongly convex and $L$-smooth.
We also need to define Bregman divergence, which is often used in the analysis.
Definition 3.
The Bregman divergence with respect to $f$ is the mapping $D_{f}:\mathbb{R}^{d}\times\mathbb{R}^{d}\rightarrow\mathbb{R}$ defined as follows:
$$D_{f}(x,y)\stackrel{{\scriptstyle\text{ def }}}{{=}}f(x)-f(y)-\langle\nabla f(y),x-y\rangle.$$
3.2 Compression operator
In order to overcome communication issues, we apply a compression operator to the iterates. Now we are going to extend Federated Random Reshuffling using compression. Let us define the concept of compressors.
Definition 4.
We say that a randomized map $\mathcal{C}:\mathbb{R}^{d}\to\mathbb{R}^{d}$ is in class $\mathbb{B}^{d}(\omega)$ if there exists a constant $\omega\geq 0$ such that the following relations hold for all $x\in\mathbb{R}^{d}$:
$$\mathbb{E}\left[\mathcal{C}(x)\right]=x,\quad\mathbb{E}\left[\|\mathcal{C}(x)\|^{2}\right]\leq(\omega+1)\|x\|^{2}.$$
Assumption 2.
All compression operators are in class $\mathbb{B}^{d}(\omega)$.
This class of compressor operators is classical in literature (Mishchenko et al., 2019; Horvath et al., 2019; Basu et al., 2020).
3.3 Random Reshuffling, Shuffle Once
In order to conduct analysis for sampling without replacement we need to establish specific notions. We sample a random permutation $\left\{\pi_{0},\pi_{1},\ldots,\pi_{n-1}\right\}$ of the set $\{1,2,\ldots,n\}$, and proceed with $n$ iterates of the form
$x_{t,m}^{i+1}=x_{t,m}^{i}-\gamma\nabla f_{m,\pi_{i}}\left(x_{t,m}^{i}\right)$ at each machine locally. We also consider option when we have only one random permutation, at the very beginning, and then algorithm uses this permutation during the whole process.
For a constant stepsize and a fixed permutation, we define intermediate limit point:
$$\displaystyle x_{*}^{i}\stackrel{{\scriptstyle\text{ def }}}{{=}}x_{*}-\gamma\sum_{j=0}^{i-1}\nabla f_{\pi_{j}}\left(x_{*}\right),\quad i=1,\ldots,n-1.$$
To measure the closeness between $x_{*}$ and $x_{*}^{n}$ we use definition from Mishchenko et al. (2021) of Shuffling radius.
Definition 5.
For given a stepsize $\gamma>0$ and a random permutation $\pi$ of $\{1,2,\ldots,n\}$ shuffling radius is defined by
$$\sigma_{\mathrm{rad}}^{2}\stackrel{{\scriptstyle\text{ def }}}{{=}}\max_{i=1,\ldots,n-1}\left[\frac{1}{\gamma^{2}}\mathbb{E}_{\pi}\left[D_{f_{\pi_{i}}}\left(x_{*}^{i},x_{*}\right)\right]\right].$$
We also need to define the most popular parameter for method’s stochastisity.
Definition 6.
Variance at the optimum:
$$\displaystyle\sigma_{*}^{2}\stackrel{{\scriptstyle\text{ def }}}{{=}}\frac{1}{n}\sum_{i=1}^{n}\left\|\nabla f_{i}\left(x_{*}\right)-\nabla f\left(x_{*}\right)\right\|^{2}.$$
The shuffling radius for permutation-based algorithms is natural, and it is more convenient to work with this concept. However, we need to have an upper bound in terms of $\sigma^{2}_{*}$ to compare different methods. To get an upper bound for shuffling radius, we need to use a lemma in Mishchenko et al. (2020) that bounds variance of sampling without replacement.
Theorem 1.
For any stepsize $\gamma>0$ and any random permutation $\pi$ of $\{1,2\ldots,n\}$ we have
$$\sigma_{\mathrm{rad}}^{2}\leq\frac{L_{\max}}{2}n\left(n\left\|\nabla f\left(x_{*}\right)\right\|^{2}+\frac{1}{2}\sigma_{*}^{2}\right).$$
In case when we have only one node we obtain that $\left\|\nabla f\left(x_{*}\right)\right\|^{2}=0$. However, in multuple node case we will need this term.
3.4 Lifted problem reformulation
Let us consider a bigger product space by introducing dummy variables and the constraint $x_{1}=x_{2}=\ldots=x_{M}$. We need to define regularizer for this reformulation:
$$\displaystyle\psi\left(x_{1},\ldots,x_{M}\right)=\left\{\begin{array}[]{ll}0,&x_{1}=\cdots=x_{M}\\
+\infty,&\text{ otherwise .}\end{array}\right.$$
Using this regularizer, we can establish the reformulated problem:
$$\displaystyle\min_{x_{1},\ldots,x_{M}\in\mathbb{R}^{d}}\frac{1}{nM}$$
$$\displaystyle\sum_{m=1}^{M}F_{m}\left(x_{m}\right)+\psi\left(x_{1},\ldots,x_{M}\right)$$
$$\displaystyle F_{m}(x)=$$
$$\displaystyle\sum_{j=1}^{n}f_{mj}(x).$$
We need to have an upper bound for the shuffling radius for this reformulated problem. First, we need to define the variance of the method’s stochasticity in distributed case.
Definition 7.
The variance of local gradients:
$$\displaystyle\sigma_{m,*}^{2}\stackrel{{\scriptstyle\text{ def }}}{{=}}\frac{1}{n}\sum_{j=1}^{n}\left\|\nabla f_{mj}\left(x_{*}\right)-\frac{1}{n}\nabla F_{m}\left(x_{*}\right)\right\|^{2}.$$
Now we need to use a lemma from (Mishchenko et al., 2021) to bound shuffled radius for the reformulated problem:
Lemma 1.
The shuffling radius $\sigma^{2}_{rad}$ of lifted problem is upper bounded by
$$\displaystyle\sigma_{\mathrm{rad}}^{2}\leq L\sum_{m=1}^{M}\left(\left\|\nabla F_{m}\left(x_{*}\right)\right\|^{2}+\frac{n}{4}\sigma_{m,*}^{2}\right).$$
We can see that there are two parts of variance. First one depends on the sum of local variances $\sum_{m=1}^{M}\sigma_{m,*}^{2}$. The second part depends on sum of local gradient norms $\sum_{m=1}^{M}\left\|\nabla F_{m}\left(x_{*}\right)\right\|^{2}$. Both of these terms appear in analysis of local SGD (Khaled et al., 2020).
4 Compressed Federated Random Reshuffling
In this section, we propose a direct application of compressed iterates to federated Random Reshuflling. In this procedure server distributes the current point to workers, then each worker computes the full epoch according to its sampled permutation locally. After that, the final iterate $x^{n}_{t,m}$ is compressed and transmitted to the server, where all updates are aggregated by taking the average.
Theorem 2.
Suppose that Assumption 2 and Assumption 1 hold. Additionally assume that compression parameter is sufficiently small: $\omega\leq\frac{M}{2}\frac{1-\left(1-\gamma\mu\right)^{\frac{n}{2}}}{\left(1-\gamma\mu\right)^{\frac{n}{2}}}$. If the stepsize satisfies $\gamma\leq\frac{1}{L}$, the iterates generated by FedCRR or FedCSO (Algorithm 1) satisfy
$$\displaystyle\mathbb{E}\left[\|x_{t+1}-x_{*}\|^{2}\right]\leq(1-\gamma\mu)^{\frac{nT}{2}}\|x_{0}-x_{*}\|^{2}$$
$$\displaystyle+\frac{2}{\mu}\gamma^{2}L_{\max}\frac{1}{M}\sum_{m=1}^{M}\left(\|F_{m}(x_{*})\|^{2}+\frac{n}{4}\sigma_{*,m}^{2}\right)$$
$$\displaystyle+\frac{2\omega}{M}\frac{1}{\gamma\mu}\frac{1}{M}\sum_{m=1}^{M}\|x^{n}_{*,m}\|^{2}.$$
We can see that the last term makes the largest contribution to the size of the neighborhood, and decreasing stepsizes cannot help. Now we establish communication complexity.
Corollary 1.
Let the assumptions in the Theorem 2 hold. Also assume that
$\omega\leq\frac{M\gamma\mu\varepsilon}{\frac{2}{M}\|x^{n}_{*,m}\|^{2}}$.
Then the communication complexity of Algorithm 1 is
$$\textstyle T=\tilde{\mathcal{O}}\left(\left(\kappa+\frac{\sqrt{\kappa}}{\mu\sqrt{\varepsilon}}\Delta\right)\log\left(\frac{1}{\varepsilon}\right)\right),$$
where $\Delta=\frac{1}{M}\sum_{m=1}^{M}\left(\|\nabla F_{m}(x_{*})\|+\sqrt{n}\sigma_{*,m}\right)$
5 Variance Reduced Compressed Federated Random Reshuffling
In this section, we introduce a variance reduction mechanism for compression in order to upgrade Algorithm 1. The main part of the algorithm remains the same. However, after each epoch, we apply the compression operator to the difference between local iterates and learning shifts. After that, at each node, we compute updates of learning shifts. To control the learning process of shifts, we use additional parameter $\alpha$. To get convergence we need to satisfy $\alpha\leq\frac{1}{\omega+1}$. After that server aggregates updates by using a convex combination of previous iterate and average of updates. To control this convex combination, we use additional parameter $\eta$. The next theorem shows that this mechanism helps to get rid of compression variance and the additional assumptions. To get the convergence rate, we introduce the Lyapunov function.
Theorem 3.
Suppose that Assumption 1 and Assumption 2 hold. Then provided the stepsize satisfies $\gamma\leq\frac{1}{L}$, $\alpha\leq\frac{1}{\omega+1}$ and $\eta\leq\min\left(1,\frac{\left(1-\left(1-\gamma\mu\right)^{n}\right)M}{12\omega\left(1-\gamma\mu\right)^{n}}\right)$ the iterates generated by FedCRR-VR or FedCSO-VR (Algorithm 2) satisfy
$$\displaystyle\textstyle\mathbb{E}\Psi_{T}$$
$$\displaystyle\leq\left(1-\frac{\min\left(\alpha,\eta(1-(1-\gamma\mu)^{n})\right)}{2}\right)^{T}\Psi_{0}$$
$$\displaystyle+\frac{2\left(\alpha+\eta+\frac{2\eta^{2}\omega}{M}\right)\gamma^{3}L_{\max}}{M\left(\alpha,\eta(1-(1-\gamma\mu)^{n})\right)}\sum_{m=1}^{M}\delta_{m},$$
where Lyapunov function is defined as $\Psi_{t}=\|x_{t}-x_{*}\|^{2}+\frac{4\eta^{2}\omega}{\alpha M}\frac{1}{M}\sum_{m=1}^{M}\left\|h_{t,m}-x^{n}_{*,m}\right\|^{2}$ and $\delta_{m}=\left(\left\|\nabla F_{m}\left(x_{*}\right)\right\|^{2}+\frac{n}{4}\sigma_{m,*}^{2}\right).$
Now there is no compression variance term anymore. Next corollary demonstrates communication complexity.
Corollary 2.
Let the assumptions in the Theorem 3 hold. Then the communication complexity of Algorithm 2 is
$$\textstyle T=\mathcal{O}\left(\left(\frac{(\omega+1)\left(1-\frac{1}{\kappa}\right)^{n}}{\left(1-\left(1-\frac{1}{\kappa}\right)^{n}\right)^{2}}+\frac{\sqrt{\kappa}}{\mu\sqrt{\varepsilon}}\Delta\right)\log\left(\frac{1}{\varepsilon}\right)\right),$$
where $\Delta=\frac{1}{M}\sum_{m=1}^{M}\left(\|\nabla F_{m}(x_{*})\|+\sqrt{n}\sigma_{*,m}\right).$
We have the same second term, which depends on the sum of local gradients and local variances. Also, the linear rate is slightly worse since we can use any compression operator, and we also need to learn shifts. However, the main advantage of this method is the possibility of using any compression parameter $\omega$.
6 Double Variance Reduced Compressed Federated Random Reshuffling
This section proposes another variance reduction mechanism to eliminate local variances caused by the method’s stochasticity. To achieve this goal, we need to use inner product reformulation introduced by Malinovsky et al. (2021). We can get an equivalent form of the local function. Let $a_{i},\ldots,a_{n}\in\mathbb{R}^{d}$ are vectors that sum to zero $\sum_{i=1}^{n}a_{i}=0$:
$$\displaystyle F_{m}(x)=\sum_{i=1}^{n}\left(f_{i,m}+\left\langle a_{i,m},x\right\rangle\right)=\sum_{i=1}^{n}\tilde{f}_{i,m}.$$
(2)
Let us consider the following gradient estimate:
$$g\left(x_{t}^{i},y_{t}\right)=\nabla f_{\pi_{i},m}\left(x_{t}^{i}\right)-\nabla f_{\pi_{i},m}\left(y_{t}\right)+\frac{1}{n}\nabla F_{m}\left(y_{t}\right).$$
Obviously, the sum of these vectors is equal to zero:
$$\sum_{i=1}^{n}a_{i,m}=-\sum_{i=1}^{n}\nabla f_{\pi_{i},m}\left(y_{t}\right)+\frac{1}{n}\sum_{i=1}^{n}\nabla F_{m}\left(y_{t}\right)=0.$$
Now we are ready to formulate the theorem of convergence guarantees.
Theorem 4.
Suppose that Assumption 2 and Assumption 1 hold. Then provided the stepsize satisfies $\gamma\leq\frac{1}{8L}\sqrt{\frac{\mu}{nL}}$, $\alpha\leq\frac{1}{\omega+1}$, $\eta\leq\min\left(1,\frac{\left(1-\left(1-\gamma\mu\right)^{\frac{n}{2}}\right)M}{12\omega\left(1-\gamma\mu\right)^{\frac{n}{2}}}\right)$ and $\frac{1}{8}\leq(1-\gamma\mu)^{\frac{n}{2}}\left(1-(1-\gamma\mu)^{\frac{n}{2}}\right),$ the iterates generated by FedCRR-VR-2 or FedCSO-VR-2 (Algorithm 3) satisfy
$$\displaystyle\textstyle\mathbb{E}\Psi_{T}$$
$$\displaystyle\leq\left(1-\frac{1}{2}\min\left(\alpha,\eta(1-(1-\gamma\mu)^{\frac{n}{2}})\right)\right)^{T}\Psi_{0}$$
$$\displaystyle+\frac{2\left(\alpha+\eta+\frac{2\eta^{2}\omega}{M}\right)\gamma^{3}L\sum_{m=1}^{M}\left(\|\nabla F_{m}(x_{*})\|^{2}\right)}{M\min\left(\alpha,\eta(1-(1-\gamma\mu)^{\frac{n}{2}})\right)},$$
where Lyapunov function is defined as $\Psi_{t}=\|x_{t}-x_{*}\|^{2}+\frac{4\eta^{2}\omega}{\alpha M}\frac{1}{M}\sum_{m=1}^{M}\left\|h_{t,m}-x^{n}_{*,m}\right\|^{2}.$
We need to use smaller stepsize since we applied variance reduction mechanism. However, we managed to vanish sum of local variances. The next theorem shows the communication complexity of Algorithm 3.
Corollary 3.
Let the assumptions in the Theorem 3 hold. Then the communication complexity of Algorithm 3 is
$$\textstyle T=\mathcal{O}\left(\left(\frac{(\omega+1)\left(1-\frac{1}{\kappa\sqrt{\kappa n}}\right)^{\frac{n}{2}}}{\left(1-\left(1-\frac{1}{\kappa\sqrt{\kappa n}}\right)^{\frac{n}{2}}\right)^{2}}+\frac{\sqrt{\kappa}}{\mu\sqrt{\varepsilon}}\Delta^{\prime}\right)\log\left(\frac{1}{\varepsilon}\right)\right),$$
where $\Delta^{\prime}=\frac{1}{M}\sum_{m=1}^{M}\left(\|\nabla F_{m}(x_{*})\|\right).$
7 Experiments
Model.
In our experiments we solve the regularized ridge regression problem, which has the form 1 with
$\textstyle f_{im}(x)=\frac{1}{2}\left\|A^{m}_{i,:}x-y^{m}_{i}\right\|^{2}+\frac{\lambda}{2}\|x\|^{2},$
where $A^{m}\in\mathbb{R}^{n\times d},y^{m}\in\mathbb{R}^{n}$ and $\lambda>0$ is regularization parameter. Consider concatenated matrix $A\in\mathbb{R}^{mn\times d}$. This problem satisfies Assumption 1 for $L=\max_{i}\left\|A_{i,:}\right\|^{2}+\lambda$ and $\mu=\frac{\rho_{\min}\left(A^{\top}A\right)}{n}+\lambda$, where $\rho_{\min}$ is the smallest eigenvalue. In our experiments we set $\lambda=\frac{1}{n}$. In all plots $x$-axis is the number of communicated bits, and $y$-axis is the squared norm of difference between current iterate and solution.
Compression operator. In all experiments we used random sparsification as compression operator:
$$C(x)=\frac{d}{k}\sum_{i\in S}x_{i}e_{i},$$
where $S$ is a random subset of $\left\{1,2,\ldots,d\right\rangle$ of cardinality $k$ chosen uniformly at random, and $e_{i}$ is the $i$-th standard unit basis vector in $\mathbb{R}^{d}$.
Hardware and software. We use real datasets from
open LIBSVM corpus (Chang & Lin, 2011) (Modified BSD License www.csie.ntu.edu.tw/ cjlin/libsvm/) and synthetic datasets from scikit-learn.datasets (Pedregosa et al., 2011) (BSD License https://scikit-learn.org). We implemented all algorithms in Python. All methods were evaluated on a computer with an Intel(R) Xeon(R) Gold 6146 CPU at 3.20GHz, having 24 cores. You can find more details and additional experiments in supplementary materials.
Results. We have a very tight match between our theory and the numerical results. As we can see, Compressed Federated Random Reshuffling cannot get appropriate accuracy since we have a huge compression variance term. It means that this method can be used only if the required accuracy is not high. However, we can see that variance-reduced methods show better convergence. While FedRR-VR has the same linear rate, the solution’s neighborhood is smaller in comparison to other methods. FedRR-VR-2 has a slower linear rate because of stepsize requirements, but the neighborhood of the solution is the smallest and allows to get a much better solution to the problem.
8 Conclusion
In this work, we propose three new algorithms: Compressed Federated Random Reshuffling and two variance-reduced variants. These methods are first-of-its-kind algorithms that include three popular approaches: periodic aggregation, compressed updates, and sampling without replacement. Moreover, we sequentially applied variance reduction mechanisms for compression and then for Random Reshuffling. We provide the first analysis under general assumptions. Experimental results confirm our theoretical findings. Thus, we gain a deeper theoretical understanding of how these algorithms work and hope that this will inspire researchers to develop further and analyze methods for Federated learning. In future work, we desire to get rid of the client drift term in the neighborhood of the solution and get an algorithm that will converge linearly to the exact solution. We also want to allow a method to have partial participation of clients since it is essential for a Federated Learning setting. We also believe that our theoretical and practical results can be applied to other aspects of machine learning and federated learning, leading to improvements in current and future applications.
References
Aji & Heafield (2017)
Aji, A. F. and Heafield, K.
Sparse communication for distributed gradient descent.
arXiv preprint arXiv:1704.05021, 2017.
Alistarh et al. (2017)
Alistarh, D., Grubic, D., Li, J., Tomioka, R., and Vojnovic, M.
Qsgd: Communication-efficient sgd via gradient quantization and
encoding.
Advances in Neural Information Processing Systems,
30:1709–1720, 2017.
Alistarh et al. (2018)
Alistarh, D., Hoefler, T., Johansson, M., Khirirat, S., Konstantinov, N., and
Renggli, C.
The convergence of sparsified gradient methods.
arXiv preprint arXiv:1809.10505, 2018.
Basu et al. (2020)
Basu, D., Data, D., Karakus, C., and Diggavi, S.
Qsparse-local-sgd: Distributed sgd with quantization, sparsification,
and local computations.
IEEE Journal on Selected Areas in Information Theory,
1:217–226, 2020.
Bengio (2012)
Bengio, Y.
Practical recommendations for gradient-based training of deep
architectures.
Neural Networks: Tricks of the Trade, pp. 437–478, 2012.
ISSN 1611-3349.
doi: 10.1007/978-3-642-35289-8˙26.
Bernstein et al. (2018)
Bernstein, J., Wang, Y.-X., Azizzadenesheli, K., and Anandkumar, A.
signsgd: Compressed optimisation for non-convex problems.
In International Conference on Machine Learning, pp. 560–569. PMLR, 2018.
Bottou (2009)
Bottou, L.
Curiously fast convergence of some stochastic gradient descent
algorithms.
2009.
Chang & Lin (2011)
Chang, C.-C. and Lin, C.-J.
Libsvm: a library for support vector machines.
ACM Transactions on Intelligent Systems and Technology (TIST),
2(3):1–27, 2011.
Chraibi et al. (2019)
Chraibi, S., Khaled, A., Kovalev, D., Richtárik, P., Salim, A., and
Takáč, M.
Distributed fixed point methods with compressed iterates.
arXiv preprint arXiv:1912.09925, 2019.
Defazio et al. (2014a)
Defazio, A., Bach, F., and Lacoste-Julien, S.
SAGA : A fast incremental gradient method with support for
non-strongly convex composite objectives.
arXiv preprint arXiv:1407.0202, 2014a.
Defazio et al. (2014b)
Defazio, A., Domke, J., and Caetano.
Finito: A faster, permutable incremental gradient method for big data
problems.
In Xing, E. P. and Jebara, T. (eds.), Proceedings of the 31st
International Conference on Machine Learning, volume 32 of Proceedings
of Machine Learning Research, pp. 1125–1133, Bejing, China, 22–24 Jun
2014b. PMLR.
Gorbunov et al. (2020)
Gorbunov, E., Hanzely, F., and Richtárik, P.
A unified theory of sgd: Variance reduction, sampling, quantization
and coordinate descent.
In International Conference on Artificial Intelligence and
Statistics, pp. 680–690. PMLR, 2020.
Gower et al. (2019)
Gower, R. M., Loizou, N., Qian, X., Sailanbayev, A., Shulgin, E., and
Richtárik, P.
SGD: General analysis and improved rates.
In International Conference on Machine Learning, pp. 5200–5209. PMLR, 2019.
Gürbüzbalaban et al. (2019)
Gürbüzbalaban, M., Ozdaglar, A., and Parrilo, P. A.
Why random reshuffling beats stochastic gradient descent.
Mathematical Programming, pp. 1–36, 2019.
Haddadpour et al. (2021)
Haddadpour, F., Kamani, M. M., Mokhtari, A., and Mahdavi, M.
Federated learning with compression: Unified analysis and sharp
guarantees.
In International Conference on Artificial Intelligence and
Statistics, pp. 2350–2358. PMLR, 2021.
Haochen & Sra (2019)
Haochen, J. and Sra, S.
Random shuffling beats SGD after finite epochs.
In Chaudhuri, K. and Salakhutdinov, R. (eds.), Proceedings of
the 36th International Conference on Machine Learning, volume 97 of
Proceedings of Machine Learning Research, pp. 2624–2633, Long
Beach, California, USA, 09–15 Jun 2019. PMLR.
Horvath et al. (2019)
Horvath, S., Ho, C.-Y., Horvath, L., Sahu, A. N., Canini, M., and Richtarik, P.
Natural compression for distributed deep learning.
arXiv preprint arXiv:1905.10988, 2019.
Johnson & Zhang (2013)
Johnson, R. and Zhang, T.
Accelerating stochastic gradient descent using predictive variance
reduction.
Advances in Neural Information Processing Systems,
26:315–323, 2013.
Karimireddy et al. (2019)
Karimireddy, S. P., Kale, S., Mohri, M., Reddi, S. J., Stich, S. U., and
Suresh, A. T.
SCAFFOLD: Stochastic Controlled Averaging for On-Device Federated
Learning.
arXiv preprint arXiv:1910.06378, 2019.
Khaled & Richtárik (2019)
Khaled, A. and Richtárik, P.
Gradient descent with compressed iterates.
arXiv preprint arXiv:1909.04716, 2019.
Khaled et al. (2019)
Khaled, A., Mishchenko, K., and Richtárik, P.
First analysis of local gd on heterogeneous data.
arXiv preprint arXiv:1909.04715, 2019.
Khaled et al. (2020)
Khaled, A., Mishchenko, K., and Richtárik, P.
Tighter theory for local sgd on identical and heterogeneous data.
In International Conference on Artificial Intelligence and
Statistics, pp. 4519–4529. PMLR, 2020.
Konečnỳ et al. (2016)
Konečnỳ, J., McMahan, H. B., Yu, F. X., Richtárik, P., Suresh,
A. T., and Bacon, D.
Federated learning: Strategies for improving communication
efficiency.
arXiv preprint arXiv:1610.05492, 2016.
Kovalev et al. (2020)
Kovalev, D., Horváth, S., and Richtárik, P.
Don’t jump through hoops and remove those loops: SVRG and
katyusha are better without the outer loop.
In Algorithmic Learning Theory, pp. 451–467. PMLR, 2020.
Lin et al. (2018)
Lin, T., Stich, S. U., Patel, K. K., and Jaggi, M.
Don’t use large mini-batches, use local sgd.
arXiv preprint arXiv:1808.07217, 2018.
Lin et al. (2017)
Lin, Y., Han, S., Mao, H., Wang, Y., and Dally, W. J.
Deep gradient compression: Reducing the communication bandwidth for
distributed training.
arXiv preprint arXiv:1712.01887, 2017.
Malinovsky et al. (2021)
Malinovsky, G., Sailanbayev, A., and Richtárik, P.
Random reshuffling with variance reduction: New analysis and better
rates.
arXiv preprint arXiv:2104.09342, 2021.
McMahan et al. (2017)
McMahan, B., Moore, E., Ramage, D., Hampson, S., and y Arcas, B. A.
Communication-efficient learning of deep networks from decentralized
data.
In Artificial Intelligence and Statistics, pp. 1273–1282.
PMLR, 2017.
Mishchenko et al. (2019)
Mishchenko, K., Gorbunov, E., Takáč, M., and Richtárik, P.
Distributed learning with compressed gradient differences.
arXiv preprint arXiv:1901.09269, 2019.
Mishchenko et al. (2020)
Mishchenko, K., Khaled, A., and Richtárik, P.
Random reshuffling: Simple analysis with vast improvements.
Advances in Neural Information Processing Systems, 33, 2020.
Mishchenko et al. (2021)
Mishchenko, K., Khaled, A., and Richtárik, P.
Proximal and federated random reshuffling.
arXiv preprint arXiv:2102.06704, 2021.
Mokhtari et al. (2018)
Mokhtari, A., Gürbüzbalaban, M., and Ribeiro, A.
Surpassing gradient descent provably: A cyclic incremental method
with linear convergence rate.
SIAM Journal on Optimization, 28(2):1420–1447, 2018.
Nagaraj et al. (2019)
Nagaraj, D., Jain, P., and Netrapalli, P.
SGD without replacement: sharper rates for general smooth convex
functions.
In Chaudhuri, K. and Salakhutdinov, R. (eds.), Proceedings of
the 36th International Conference on Machine Learning, volume 97 of
Proceedings of Machine Learning Research, pp. 4703–4711, Long
Beach, California, USA, 09–15 Jun 2019. PMLR.
Park & Ryu (2020)
Park, Y. and Ryu, E. K.
Linear convergence of cyclic SAGA.
Optimization Letters, 14(6):1583–1598,
2020.
Pedregosa et al. (2011)
Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel,
O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., Vanderplas, J.,
Passos, A., Cournapeau, D., Brucher, M., Perrot, M., and Duchesnay, E.
Scikit-learn: Machine learning in Python.
Journal of Machine Learning Research, 12:2825–2830,
2011.
Rajput et al. (2020)
Rajput, S., Gupta, A., and Papailiopoulos, D.
Closing the convergence gap of SGD without replacement.
arXiv preprint arXiv:2002.10400, 2020.
Ramezani-Kebrya et al. (2019)
Ramezani-Kebrya, A., Faghri, F., and Roy, D. M.
Nuqsgd: Improved communication efficiency for data-parallel sgd via
nonuniform quantization.
arXiv preprint arXiv:1908.06077, 2019.
Recht & Ré (2012)
Recht, B. and Ré, C.
Toward a noncommutative arithmetic-geometric mean inequality:
conjectures, case-studies, and consequences.
In Conference on Learning Theory, pp. 11–1. JMLR Workshop
and Conference Proceedings, 2012.
Recht & Ré (2013)
Recht, B. and Ré, C.
Parallel stochastic gradient algorithms for large-scale matrix
completion.
Mathematical Programming Computation, 5(2):201–226, 2013.
Robbins & Monro (1951)
Robbins, H. and Monro, S.
A stochastic approximation method.
The annals of mathematical statistics, pp. 400–407, 1951.
Roux et al. (2012)
Roux, N. L., Schmidt, M., and Bach, F.
A stochastic gradient method with an exponential convergence rate for
finite training sets.
arXiv preprint arXiv:1202.6258, 2012.
Safran & Shamir (2020)
Safran, I. and Shamir, O.
How good is SGD with random shuffling?
In Conference on Learning Theory, pp. 3250–3284. PMLR,
2020.
Shamir et al. (2014)
Shamir, O., Srebro, N., and Zhang, T.
Communication-efficient distributed optimization using an approximate
newton-type method.
In International conference on machine learning, pp. 1000–1008. PMLR, 2014.
Stich (2018)
Stich, S. U.
Local sgd converges fast and communicates little.
arXiv preprint arXiv:1805.09767, 2018.
Vogels et al. (2019)
Vogels, T., Karimireddy, S. P., and Jaggi, M.
Powersgd: Practical low-rank gradient compression for distributed
optimization.
arXiv preprint arXiv:1905.13727, 2019.
Wangni et al. (2017)
Wangni, J., Wang, J., Liu, J., and Zhang, T.
Gradient sparsification for communication-efficient distributed
optimization.
arXiv preprint arXiv:1710.09854, 2017.
Wu et al. (2018)
Wu, J., Huang, W., Huang, J., and Zhang, T.
Error compensated quantized sgd and its applications to large-scale
distributed optimization.
In International Conference on Machine Learning, pp. 5325–5333. PMLR, 2018.
Ying et al. (2019)
Ying, B., Yuan, K., Vlaski, S., and Sayed, A. H.
Stochastic learning under random reshuffling with constant
step-sizes.
In IEEE Transactions on Signal Processing, volume 67, pp. 474–489, 2019.
Supplementary materials
Appendix A Basic Facts
Proposition 1.
Let $f:\mathbb{R}^{d}\to\mathbb{R}$ be continuously differentiable and let $L\geq 0$. Then the following statements are equivalent:
•
$f$ is $L$-smooth
•
$2D_{f}(x,y)\leq L\|x-y\|^{2}\text{ for all }x,y\in\mathbb{R}^{d}$
•
$\langle\nabla f(x)-\nabla f(y),x-y\rangle\leq L\|x-y\|^{2}\text{ for all }x,y\in\mathbb{R}^{d}$
Proposition 2.
Let $f:\mathbb{R}^{d}\to\mathbb{R}$ be continuously differentiable and let $\mu\geq 0$. Then the following statements are equivalent:
•
$f$ is $\mu$-strongly convex
•
$2D_{f}(x,y)\geq\mu\|x-y\|^{2}\text{ for all }x,y\in\mathbb{R}^{d}$
•
$\langle\nabla f(x)-\nabla f(y),x-y\rangle\geq\mu\|x-y\|^{2}\text{ for all }x,y\in\mathbb{R}^{d}$
Note that the $\mu=0$ case reduces to convexity.
Proposition 3.
Let $f:\mathbb{R}^{d}\to\mathbb{R}$ be continuously differentiable and $L>0$. Then the following statements are equivalent:
•
$f$ is convex and $L$-smooth
•
$0\leq 2D_{f}(x,y)\leq L\|x-y\|^{2}\text{ for all }x,y\in\mathbb{R}^{d}$
•
$\frac{1}{L}\|\nabla f(x)-\nabla f(y)\|^{2}\leq 2D_{f}(x,y)\text{ for all }x,y\in\mathbb{R}^{d}$
•
$\frac{1}{L}\|\nabla f(x)-\nabla f(y)\|^{2}\leq\langle\nabla f(x)-\nabla f(y),x-y\rangle\text{ for all }x,y\in\mathbb{R}^{d}$
Proposition 4 (Jensen’s inequality).
Let $f:\mathbb{R}^{d}\to\mathbb{R}$ be a convex function, $x_{1},\ldots,x_{m}\in\mathbb{R}^{d}$ and $\lambda_{1},\ldots,\lambda_{m}$ be nonnegative real numbers adding up to 1. Then
$$f\left(\sum_{i=1}^{m}\lambda_{i}x_{i}\right)\leq\sum_{i=1}^{m}\lambda_{i}f\left(x_{i}\right)$$
Proposition 5.
For all $a,b\in\mathbb{R}^{d}$ and $t>0$ the following inequalities holds:
$$\displaystyle\langle a,b\rangle$$
$$\displaystyle\leq\frac{\|a\|^{2}}{2t}+\frac{t\|b\|^{2}}{2}$$
$$\displaystyle\|a+b\|^{2}$$
$$\displaystyle\leq 2\|a\|^{2}+2\|b\|^{2}$$
$$\displaystyle\frac{1}{2}\|a\|^{2}-\|b\|^{2}$$
$$\displaystyle\leq\|a+b\|^{2}$$
Appendix B General lemmas
B.1 Proposition 1
We need to prove a basic fact which will be used later.
Proposition 6.
Let us consider
$$x^{n}_{*,m}=x_{*}-\gamma\sum_{i=0}^{n-1}\nabla f_{\pi_{i},m}(x_{*}),$$
then
$$\frac{1}{M}\sum_{m=1}^{M}x^{n}_{*,m}=x_{*}.$$
Proof.
We start from the definition:
$$\displaystyle\frac{1}{M}\sum_{m=1}^{M}x^{n}_{*,m}$$
$$\displaystyle=\frac{1}{M}\sum_{m=1}^{M}\left(x_{*}-\gamma\sum_{i=0}^{n-1}\nabla f_{\pi_{i},m}(x_{*})\right)$$
$$\displaystyle=\frac{1}{M}\sum_{m=1}^{M}x_{*}-\frac{1}{M}\sum_{m=1}^{M}\sum_{i=0}^{n-1}\nabla f_{\pi_{i},m}(x_{*})$$
$$\displaystyle=x_{*}-\nabla f(x_{*})$$
$$\displaystyle=x_{*}.$$
∎
B.2 Proof of Theorem 1
For completeness we include the proof of important theorem introduced in Mishchenko et al. (2020).
Proof.
$$\displaystyle\mathbb{E}\left[D_{f_{\pi_{i}}}\left(x_{*}^{i},x_{*}\right)\right]\leq\mathbb{E}\left[\frac{L}{2}\left\|x_{*}^{i}-x_{*}\right\|^{2}\right]$$
$$\displaystyle\leq\frac{L_{\max}}{2}\mathbb{E}\left[\left\|x_{*}^{i}-x_{*}\right\|^{2}\right]$$
$$\displaystyle=\frac{\gamma^{2}L_{\max}}{2}\mathbb{E}\left[\left\|\sum_{j=0}^{i-1}\nabla f_{\pi_{j}}\left(x_{*}\right)\right\|^{2}\right]$$
$$\displaystyle=\frac{\gamma^{2}L_{\max}i^{2}}{2}\mathbb{E}\left[\left\|\frac{1}{i}\sum_{j=0}^{i-1}\nabla f_{\pi_{j}}\left(x_{*}\right)\right\|^{2}\right]$$
$$\displaystyle=\frac{\gamma^{2}L_{\max}i^{2}}{2}\mathbb{E}\left[\left\|\bar{X}_{\pi}\right\|^{2}\right],$$
where $\bar{X}_{\pi}=\frac{1}{j}\sum_{j=0}^{i-1}X_{\pi_{j}}\text{ with }X_{j}\stackrel{{\scriptstyle\text{ def }}}{{=}}\nabla f_{j}\left(x_{*}\right)\text{ for }j=1,2,\ldots,n\text{ . Since }\bar{X}=\nabla f\left(x_{*}\right),$ by applying Lemma 1 in (Mishchenko et al., 2020).
$$\mathbb{E}\left[\left\|\bar{X}_{\pi}\right\|^{2}\right]=\|\bar{X}\|^{2}+\mathbb{E}\left[\left\|\bar{X}_{\pi}-\bar{X}\right\|^{2}\right]=\left\|\nabla f\left(x_{*}\right)\right\|^{2}+\frac{n-i}{i(n-1)}\sigma_{*}^{2}.$$
It remains to combine both terms and use the bounds $i^{2}\leq n^{2}$ and $i(n-i)\leq\frac{n(n-1)}{2}$, which holds for all $i\in\left\{1,2,\ldots,n-1\right\}$, and divide both sides of the resulting inequality by $\gamma^{2}$.
∎
B.3 Proof of Lemma 1
Proof.
We start from Theorem 1. Then for reformulated problem we have
$$n\sigma_{*}^{2}\stackrel{{\scriptstyle\text{ def }}}{{=}}\sum_{i=1}^{n}\left\|\nabla f_{i}\left(\boldsymbol{x}_{*}\right)-\nabla f\left(\boldsymbol{x}_{*}\right)\right\|^{2}=\sum_{i=1}^{n}\sum_{m=1}^{M}\left\|\nabla f_{mi}\left(x_{*}\right)-\frac{1}{n}\nabla F_{m}\left(x_{*}\right)\right\|^{2}.$$
For inner sum we have a bound from Mishchenko et al. (2021):
$$\sum_{i=1}^{n}\left\|\nabla f_{mi}\left(x_{*}\right)-\frac{1}{n}\nabla F_{m}\left(x_{*}\right)\right\|^{2}\leq n\sigma_{m,*}^{2}+\left\|\nabla F_{m}\left(x_{*}\right)\right\|^{2}.$$
Also, we have
$$n^{2}\left\|\nabla f\left(\boldsymbol{x}_{*}\right)\right\|^{2}=n^{2}\left\|\frac{1}{n}\sum_{i=1}^{n}\nabla f_{i}\left(\boldsymbol{x}_{*}\right)\right\|^{2}=\sum_{m=1}^{M}\left\|\sum_{i=1}^{n}\nabla f_{mi}\left(x_{*}\right)\right\|^{2}=\sum_{m=1}^{M}\left\|\nabla F_{m}\left(x_{*}\right)\right\|^{2}.$$
Plugging the last two inequalities back inside the first bound on $\sigma_{\mathrm{rad}}^{2}$, we get the lemma’s statement.
∎
Appendix C Analysis of Algorithm 1
C.1 Proof of Theorem 1
Proof.
We start from conditional expectation
$$\displaystyle\mathbb{E}\left[\|x_{t+1}-x_{*}\|^{2}\mid x^{n}_{t,m}\right]$$
$$\displaystyle=\mathbb{E}\left[\left\|\frac{1}{M}\sum_{m=1}^{M}C(x^{n}_{t,m})-x_{*}\right\|^{2}\bigg{|}x^{n}_{t,m}\right]$$
$$\displaystyle\leq\mathbb{E}\left\|\frac{1}{M}\sum_{m=1}^{M}C(x^{n}_{t,m})-\frac{1}{M}\sum_{m=1}^{M}x^{n}_{t,m}\bigg{|}x^{n}_{t,m}\right\|^{2}+\left\|\frac{1}{M}\sum_{m=1}^{M}x^{n}_{t,m}-x_{*}\right\|^{2}$$
$$\displaystyle\leq\frac{\omega}{M^{2}}\sum_{m=1}^{M}\|x^{n}_{t,m}\|^{2}+\frac{1}{M}\sum_{m=1}^{M}\|x^{n}_{t,m}-x^{n}_{*,m}\|^{2}$$
$$\displaystyle\leq\frac{2\omega}{M^{2}}\sum_{m=1}^{M}\|x^{n}_{t,m}-x^{n}_{*,m}\|^{2}+\frac{1}{M}\sum_{m=1}^{M}\|x^{n}_{t,m}-x^{n}_{*,m}\|^{2}+\frac{2\omega}{M^{2}}\sum_{m=1}^{M}\|x^{n}_{*,m}\|^{2}$$
$$\displaystyle\leq(1-\gamma\mu)^{n}\left(1+\frac{2\omega}{M}\right)\|x_{t}-x_{*}\|^{2}+\frac{2\omega}{M}\frac{1}{M}\sum_{m=1}^{M}\|x^{n}_{*,m}\|^{2}$$
$$\displaystyle+2\left(1+\frac{2\omega}{M}\right)\gamma^{3}\sigma^{2}_{rad}\left(\sum_{j=0}^{n-1}(1-\gamma\mu)^{j}\right).$$
Using tower property we get
$$\mathbb{E}\left[\|x_{t+1}-x_{*}\|^{2}\right]=\mathbb{E}\left[\mathbb{E}\left[\|x_{t+1}-x_{*}\|^{2}\mid x^{n}_{t,m}\right]\right].$$
Utilizing this property we have
$$\displaystyle\mathbb{E}\left[\|x_{t+1}-x_{*}\|^{2}\right]$$
$$\displaystyle\leq(1-\gamma\mu)^{n}\left(1+\frac{2\omega}{M}\right)\mathbb{E}\left[\|x_{t}-x_{*}\|^{2}\right]+\frac{2\omega}{M}\frac{1}{M}\sum_{m=1}^{M}\mathbb{E}\|x^{n}_{*,m}\|^{2}$$
$$\displaystyle+2\left(1+\frac{2\omega}{M}\right)\gamma^{3}\sigma^{2}_{rad}\left(\sum_{j=0}^{n-1}(1-\gamma\mu)^{j}\right).$$
Unrolling this recursion we get
$$\displaystyle\mathbb{E}\left[\|x_{t+1}-x_{*}\|^{2}\right]$$
$$\displaystyle\leq\left((1-\gamma\mu)^{n}\left(1+\frac{2\omega}{M}\right)\right)^{T}\|x_{0}-x_{*}\|^{2}$$
$$\displaystyle+\sum_{i=0}^{T-1}\left((1-\gamma\mu)^{n}\left(1+\frac{2\omega}{M}\right)\right)^{i}\frac{2\omega}{M}\frac{1}{M}\sum_{m=1}^{M}\mathbb{E}\|x^{n}_{*,m}\|^{2}$$
$$\displaystyle+2\sum_{i=0}^{T-1}\left((1-\gamma\mu)^{n}\left(1+\frac{2\omega}{M}\right)\right)^{i}\left(1+\frac{2\omega}{M}\right)\gamma^{3}\sigma^{2}_{rad}\left(\sum_{j=0}^{n-1}(1-\gamma\mu)^{j}\right).$$
Using assumption of compression operator we have
$$(1-\gamma\mu)^{n}\left(1+\frac{2\omega}{M}\right)\leq(1-\gamma\mu)^{\frac{n}{2}}.$$
Also let us look at last term:
$$\displaystyle\left(1+\frac{2\omega}{M}\right)\left(\sum_{i=0}^{T-1}(1-\gamma\mu)^{i}\right)$$
$$\displaystyle\leq\sum_{i=0}^{T-1}(1-\gamma\mu)^{i}\left(1+\frac{2\omega}{M}\right)^{i}$$
$$\displaystyle\leq\sum_{j=0}^{T-1}\left((1-\gamma\mu)\left(1+\frac{2\omega}{M}\right)\right)^{i}$$
$$\displaystyle\leq\sum_{i=0}^{T-1}(1-\gamma\mu)^{\frac{nj}{2}}.$$
Moreover, we have this bound for geometric sequence:
$$\displaystyle\left(\sum_{i=0}^{T-1}(1-\gamma\mu)^{\frac{nj}{2}}\right)\left(\sum_{j=0}^{n-1}(1-\gamma\mu)^{j}\right)\leq\sum_{i=0}^{T-1}\sum_{j=0}^{n-1}(1-\gamma\mu)^{\frac{ni}{2}+j}\leq\frac{1}{\gamma\mu}.$$
The same bound we have for the second sum:
$$\left(\sum_{i=0}^{T-1}(1-\gamma\mu)^{\frac{nj}{2}}\right)\leq\frac{1}{\gamma\mu}.$$
Finally, we have the following:
$$\displaystyle\mathbb{E}\left[\|x_{t+1}-x_{*}\|^{2}\right]\leq(1-\gamma\mu)^{\frac{nT}{2}}\|x_{0}-x_{*}\|^{2}+\frac{2}{\mu}\gamma^{2}\sigma^{2}_{rad}+\frac{2\omega}{M}\frac{1}{\gamma\mu}\frac{1}{M}\sum_{m=1}^{M}\mathbb{E}\|x^{n}_{*,m}\|^{2}.$$
Using Lemma we have
$$\displaystyle\mathbb{E}\left[\|x_{t+1}-x_{*}\|^{2}\right]$$
$$\displaystyle\leq(1-\gamma\mu)^{\frac{nT}{2}}\|x_{0}-x_{*}\|^{2}+\frac{2}{\mu}\gamma^{2}L_{\max}\frac{1}{M}\sum_{m=1}^{M}\left(\|F_{m}(x_{*})\|^{2}+\frac{n}{4}\sigma_{*,m}^{2}\right)$$
$$\displaystyle+\frac{2\omega}{M}\frac{1}{\gamma\mu}\frac{1}{M}\sum_{m=1}^{M}\mathbb{E}\|x^{n}_{*,m}\|^{2}.$$
∎
Appendix D Analysis of Algorithm 2 and Algorithm 3
D.1 Lemma 2
For Algorithm 2 and Algorithm 3 the following inequality holds:
$$\mathbb{E}\left[\|x_{t+1}-x_{*}\|^{2}\mid x_{t},h_{t,m}\right]\leq\frac{\eta^{2}}{M^{2}}\omega\sum_{m=1}^{M}\|x^{n}_{t,m}-h_{t,m}\|^{2}+(1-\eta)\|x_{t}-x_{*}\|^{2}+\eta\frac{1}{M}\sum_{m=1}^{M}\|x^{n}_{t,m}-x^{n}_{*,m}\|^{2}$$
Proof.
Let us use property of compression operator:
$$\displaystyle\mathbb{E}\left[\|x_{t+1}-x_{*}\|^{2}\mid x_{t},h_{t,m}\right]$$
$$\displaystyle=\mathbb{E}\left[\left\|(1-\eta)x_{t}+\eta\frac{1}{M}\sum_{m=1}^{M}\left(C(x^{n}_{t,m}-h_{t,m})+h_{t,m}\right)\right\|^{2}\Bigg{|}x_{t},h_{t,m}\right]$$
$$\displaystyle=\mathbb{E}\left[\left\|\eta\frac{1}{M}\sum_{m=1}^{M}\mathcal{C}(x^{n}_{t,m}-h_{t,m})-\eta\frac{1}{M}\sum_{m=1}^{M}(x^{n}_{t,m}-h_{t,m})\right\|^{2}\Bigg{|}x_{t},h_{t,m}\right]$$
$$\displaystyle+\left\|(1-\eta)x_{t}+\eta\frac{1}{M}\sum_{m=1}^{M}x^{n}_{t,m}-x_{*}\right\|^{2}$$
$$\displaystyle\leq\frac{\eta^{2}}{M^{2}}\omega\sum_{m=1}^{M}\|x^{n}_{t,m}-h_{t,m}\|^{2}+(1-\eta)\|x_{t}-x_{*}\|^{2}+\eta\frac{1}{M}\sum_{m=1}^{M}\|x^{n}_{t,m}-x^{n}_{*,m}\|^{2}.$$
∎
D.2 Lemma 3
For Algorithm 2 and Algorithm 3 the following inequality holds:
$$\displaystyle\mathbb{E}\left[\|h_{t+1,m}-x^{n}_{*,m}\|^{2}\right]\leq(1-\alpha)\mathbb{E}\|h_{t,m}-x^{n}_{*,m}\|^{2}+\alpha\mathbb{E}\|x^{n}_{t,m}-x^{n}_{*,m}\|^{2}.$$
Proof.
Let us start from conditional expectation:
$$\displaystyle\mathbb{E}\left[\|h_{t+1,m}-x^{n}_{*,m}\|^{2}\mid x^{n}_{t,m},h_{t,m}\right]$$
$$\displaystyle=\mathbb{E}\left[\left\|h_{t,m}+\alpha q_{t,m}-x^{n}_{*,m}\right\|^{2}\mid x^{n}_{t,m},h_{t,m}\right]$$
$$\displaystyle\leq\|h_{t,m}-x_{*,m}^{n}\|^{2}+2\alpha\left\langle h_{t,m}-x^{n}_{*,m},\mathbb{E}[q_{t,m}]\right\rangle+\alpha^{2}\mathbb{E}\left[\|q_{t,m}\|^{2}\right]$$
$$\displaystyle\leq\|h_{t,m}-x_{*,m}^{n}\|^{2}+2\alpha\left\langle h_{t,m}-x^{n}_{*,m},x^{n}_{t,m}-h_{t,m}\right\rangle$$
$$\displaystyle+\alpha^{2}(\omega+1)\|x^{n}_{t,m}-h_{t,m}\|^{2}$$
$$\displaystyle\leq\|h_{t,m}-x_{*,m}^{n}\|^{2}+2\alpha\left\langle h_{t,m}-x^{n}_{*,m},x^{n}_{t,m}-h_{t,m}\right\rangle$$
$$\displaystyle+\alpha\|x^{n}_{t,m}-h_{t,m}\|^{2}$$
$$\displaystyle=\|h_{t,m}-x_{*,m}^{n}\|^{2}+\alpha\left\langle 2h_{t,m}-2x^{n}_{*,m}+x^{n}_{t,m}-h_{t,m},x^{n}_{t,m}-h_{t,m}\right\rangle.$$
Let us consider last term:
$$\displaystyle\left\langle 2h_{t,m}-2x^{n}_{*,m}+x^{n}_{t,m}-h_{t,m},x^{n}_{t,m}-h_{t,m}\right\rangle$$
$$\displaystyle=\left\langle h_{t,m}-x^{n}_{*,m}+x^{n}_{t,m}-x^{n}_{*,m},x^{n}_{t,m}-x^{n}_{*,m}-\left(h_{t,m}-x^{n}_{*,m}\right)\right\rangle$$
$$\displaystyle=-\|h_{t,m}-x^{n}_{*,m}\|^{2}+\|x^{n}_{t,m}-x^{n}_{*,m}\|^{2}.$$
∎
Using this result and previous inequlaity we get th following:
$$\mathbb{E}\left[\|h_{t+1,m}-x^{n}_{*,m}\|^{2}\mid x^{n}_{t,m},h_{t,m}\right]\leq(1-\alpha)\|h_{t,m}-x^{n}_{*,m}\|^{2}+\alpha\|x^{n}_{t,m}-x^{n}_{*,m}\|^{2}.$$
Taking full expectation we finish the proof.
D.3 Lemma 6
For completeness we include the proof of important theorem introduced in Mishchenko et al. (2021). Suppose that each $f_{i}$ is $L$-smooth and $\mu$-strongly convex. Then the inner iterates satisfy
$$\mathbb{E}\left[\left\|x_{t}^{i+1}-x_{*}^{i+1}\right\|^{2}\right]\leq(1-\gamma\mu)\mathbb{E}\left[\left\|x_{t}^{i}-x_{*}^{i}\right\|^{2}\right]-2\gamma\left(1-\gamma L\right)\mathbb{E}\left[D_{f_{\pi_{i}}}\left(x_{t}^{i},x_{*}\right)\right]+2\gamma^{3}\sigma_{\mathrm{rad}}^{2}.$$
Proof.
By definition of $x^{i+1}_{t}$ and $x^{i+1}_{*}$, we have
$$\displaystyle\mathbb{E}\left[\left\|x_{t}^{i+1}-x_{*}^{i+1}\right\|^{2}\right]=\mathbb{E}$$
$$\displaystyle\left[\left\|x_{t}^{i}-x_{*}^{i}\right\|^{2}\right]-2\gamma\mathbb{E}\left[\left\langle\nabla f_{\pi_{i}}\left(x_{t}^{i}\right)-\nabla f_{\pi_{i}}\left(x_{*}\right),x_{t}^{i}-x_{*}^{i}\right\rangle\right]$$
$$\displaystyle+\gamma^{2}\mathbb{E}\left[\left\|\nabla f_{\pi_{i}}\left(x_{t}^{i}\right)-\nabla f_{\pi_{i}}\left(x_{*}\right)\right\|^{2}\right].$$
Note that the third term can be bounded as
$$\left\|\nabla f_{\pi_{i}}\left(x_{t}^{i}\right)-\nabla f_{\pi_{i}}\left(x_{t}^{i}\right)\right\|^{2}\leq 2L\cdot D_{f_{\pi_{i}}}\left(x_{t}^{i},x_{*}\right).$$
Using the three-point identity we get
$$\left\langle\nabla f_{\pi_{i}}\left(x_{t}^{i}\right)-\nabla f_{\pi_{i}}\left(x_{*}\right),x_{t}^{i}-x_{*}^{i}\right\rangle=D_{f_{\pi_{i}}}\left(x_{*}^{i},x_{t}^{i}\right)+D_{f_{\pi_{i}}}\left(x_{t}^{i},x_{*}\right)-D_{f_{\pi_{i}}}\left(x_{*}^{i},x_{*}\right).$$
Combining these bounds we have
$$\displaystyle\mathbb{E}\left[\left\|x_{t}^{i+1}-x_{*}^{i+1}\right\|^{2}\right]\leq\mathbb{E}\left[\left\|x_{t}^{i}-x_{*}^{i}\right\|^{2}\right]$$
$$\displaystyle-2\gamma\cdot\mathbb{E}\left[D_{f_{\pi_{i}}}\left(x_{*}^{i},x_{t}^{i}\right)\right]+2\gamma\cdot\mathbb{E}\left[D_{f_{\pi_{i}}}\left(x_{*}^{i},x_{*}\right)\right]$$
$$\displaystyle-2\gamma\left(1-\gamma L\right)\mathbb{E}\left[D_{f_{\pi i}}\left(x_{t}^{i},x_{*}\right)\right].$$
Using $\mu$-strong convexity of $f_{\pi_{i}}$ , we derive
$$\frac{\mu}{2}\left\|x_{t}^{i}-x_{*}^{i}\right\|^{2}\leq D_{f_{\pi_{i}}}\left(x_{*}^{i},x_{t}^{i}\right).$$
Using definition of shuffling radius we have
$$\mathbb{E}\left[D_{f_{\pi_{i}}}\left(x_{*}^{i},x_{*}\right)\right]\leq\max_{i=1,\ldots,n-1}\mathbb{E}\left[D_{f_{\pi_{i}}}\left(x_{*}^{i},x_{*}\right)\right]=\gamma^{2}\sigma_{\mathrm{rad}}^{2}.$$
Putting all bounds together we get result.
∎
D.4 Proof of Theorem 3
Proof.
Let us define the Lyapunov function:
$$\Psi_{t}=\|x_{t}-x_{*}\|^{2}+\frac{4\eta^{2}\omega}{\alpha M}\frac{1}{M}\sum_{m=1}^{M}\left\|h_{t,m}-x^{n}_{*,m}\right\|^{2}.$$
Now we use Lemma 2 and Lemma 3:
$$\displaystyle\mathbb{E}\left[\Psi_{t+1}\right]$$
$$\displaystyle=\mathbb{E}\|x_{t+1}-x_{*}\|^{2}+\frac{4\eta^{2}\omega}{\alpha M}\frac{1}{M}\sum_{m=1}^{M}\mathbb{E}\left\|h_{t+1,m}-x^{n}_{*,m}\right\|^{2}$$
$$\displaystyle\leq\frac{2\eta^{2}}{M^{2}}\omega\sum_{m=1}^{M}\mathbb{E}\|x^{n}_{t,m}-x^{n}_{*,m}\|^{2}+\frac{2\eta^{2}}{M^{2}}\omega\sum_{m=1}^{M}\mathbb{E}\|h_{t,m}-x^{n}_{*,m}\|^{2}+(1-\eta)\mathbb{E}\|x_{t}-x_{*}\|^{2}$$
$$\displaystyle+\eta\frac{1}{M}\sum_{m=1}^{M}\mathbb{E}\|x^{n}_{t,m}-x^{n}_{*,m}\|^{2}+\frac{4\eta^{2}\omega}{\alpha M}(1-\alpha)\sum_{m=1}^{M}\mathbb{E}\left\|h_{t,m}-x^{n}_{*,m}\right\|^{2}+\frac{4\eta^{2}\omega}{\alpha M}\alpha\sum_{m=1}^{M}\mathbb{E}\|x^{n}_{t,m}-x^{n}_{*,m}\|^{2}.$$
Using Lemma 6 and Theorem 2 from Mishchenko et al. (2021) we have
$$\displaystyle\mathbb{E}\left[\Psi_{t+1}\right]$$
$$\displaystyle\leq\frac{4\eta^{2}\omega}{\alpha M}\left(1-\frac{\alpha}{2}\right)\frac{1}{M}\sum_{m=1}^{M}\mathbb{E}\left\|h_{t,m}-x^{n}_{*,m}\right\|^{2}+\left(1-\eta+\eta(1-\gamma\mu)^{n}+\frac{6\eta^{2}\omega}{M}(1-\gamma\mu)^{n}\right)\mathbb{E}\|x_{t}-x_{*}\|^{2}$$
$$\displaystyle+\left(\alpha+\eta+\frac{2\eta^{2}\omega}{M}\right)2\gamma^{3}\sigma^{2}_{rad}\left(\sum_{j=0}^{n-1}(1-\gamma\mu)^{j}\right)$$
Using the condition $\eta\leq\min\left(1,\frac{\left(1-(1-\gamma\mu)^{n}\right)M}{12\omega(1-\gamma\mu)^{n}}\right)$ we have
$$\displaystyle\mathbb{E}\left[\Psi_{t+1}\right]\leq\max\left(1-\frac{\alpha}{2},1-\frac{\eta\left(1-(1-\gamma\mu)^{n}\right)}{2}\right)\mathbb{E}\left[\Psi_{t}\right]+\left(\alpha+\eta+\frac{2\eta^{2}\omega}{M}\right)2\gamma^{3}\sigma^{2}_{rad}\left(\sum_{j=0}^{n-1}(1-\gamma\mu)^{j}\right)$$
Note that we have the following inequality:
$$\displaystyle 1-\gamma\mu\leq 1-\frac{1}{2}\eta\left(1-(1-\gamma\mu)^{n}\right)$$
$$\displaystyle-\gamma\mu\leq-\frac{1}{2}\eta\left(1-(1-\gamma\mu)^{n}\right)$$
$$\displaystyle\gamma\mu\geq\frac{1}{2}\eta\left(1-(1-\gamma\mu)^{n}\right).$$
We have it since $0<\eta\leq 1$ and $n>1$, so we have
$$\displaystyle\gamma\mu$$
$$\displaystyle\geq\frac{1}{2}\left(1-(1-\gamma\mu)^{n}\right)$$
$$\displaystyle\gamma\mu$$
$$\displaystyle\geq\frac{1}{2}\left(1-(1-\gamma\mu)\right)$$
$$\displaystyle 1$$
$$\displaystyle\geq\frac{1}{2}.$$
Unrolling this recursion finishes the proof.
∎
Lemma 4
For completeness we include the proof of important theorem introduced in Malinovsky et al. (2021).
Suppose that the functions $f_{1},\ldots,f_{n}$ are $\mu$-strongly convex and $L$-smooth. Fix constant $0<\delta<1$. If the
stepsize satisfies $\gamma\leq\frac{\delta}{L}\sqrt{\frac{\mu}{2nL}}$ and if number of functions is sufficiently big:
$$n>\log\left(\frac{1}{1-\delta^{2}}\right)\cdot\left(\log\left(\frac{1}{1-\gamma\mu}\right)\right)^{-1}$$
and
$$\delta^{2}\leq(1-\gamma\mu)^{\frac{n}{2}}\left(1-(1-\gamma\mu)^{\frac{n}{2}}\right).$$
$$\mathbb{E}\left[\left\|x^{n}_{t}-x^{n}_{*}\right\|^{2}\right]\leq(1-\gamma\mu)^{\frac{n}{2}}\mathbb{E}\left[\left\|x_{t}-x_{*}\right\|^{2}\right],$$
then we have
$$\mathbb{E}\left[\left\|x^{n}_{t}-x^{n}_{*}\right\|^{2}\mid x_{t}\right]\leq(1-\gamma\mu)^{n}\left\|x_{t}-x_{*}\right\|^{2}+\frac{\gamma^{3}Ln}{2}\sigma_{*}^{2}\left(\sum_{i=0}^{n-1}(1-\gamma\mu)^{i}\right)$$
Proof.
We start from Theorem 1 in Mishchenko et al. (2020):
$$\mathbb{E}\left[\left\|x^{n}_{t}-x^{n}_{*}\right\|^{2}\mid x_{t}\right]\leq(1-\gamma\mu)^{n}\left\|x_{t}-x_{*}\right\|^{2}+\frac{\gamma^{3}Ln}{2}\sigma_{*}^{2}\left(\sum_{i=0}^{n-1}(1-\gamma\mu)^{i}\right)$$
Using property of geometric progression we can have an upper bound $\sum_{i=0}^{n-1}(1-\gamma\mu)^{i}\leq\frac{1}{\gamma\mu}:$
$$\displaystyle\mathbb{E}\left[\left\|x^{n}_{t}-x^{n}_{*}\right\|^{2}\mid x_{t}\right]$$
$$\displaystyle\leq(1-\gamma\mu)^{n}\left\|x_{t}-x_{*}\right\|^{2}+\frac{\gamma^{2}Ln}{2\mu}\sigma_{*}^{2}.$$
Using Lemma 1 in Malinovsky et al. (2021) we get
$$\displaystyle\mathbb{E}\left[\left\|x^{n}_{t}-x^{n}_{*}\right\|^{2}\mid x_{t}\right]$$
$$\displaystyle\leq\left((1-\gamma\mu)^{n}+\frac{2\gamma^{2}L^{3}n}{\mu}\right)\left\|x_{t}-x_{*}\right\|^{2}.$$
Let us use $\gamma\leq\frac{\delta}{L}\sqrt{\frac{\mu}{2nL}}$. To get convergence we need
$$(1-\gamma\mu)^{n}+\delta^{2}<1.$$
This leads to the following inequality:
$$n>\log\left(\frac{1}{1-\delta^{2}}\right)\cdot\left(\log\left(\frac{1}{1-\gamma\mu}\right)\right)^{-1}.$$
Also assume
$$\delta^{2}\leq(1-\gamma\mu)^{\frac{n}{2}}\left(1-(1-\gamma\mu)^{\frac{n}{2}}\right).$$
Putting this into bound finishes the proof.
∎
D.5 Lemma 5
For completeness we include the proof of important lemma introduced in Malinovsky et al. (2021).
Assume that each $f_{i}$ is $L$-smooth and convex. If we apply the linear perturbation reformulation 6, then the variance of reformulated problem satisfies the following inequality:
$$\tilde{\sigma}_{*}^{2}\leq 4L^{2}\|y_{t}-x_{*}\|^{2}.$$
Proof.
$$\tilde{\sigma}_{*}^{2}=\frac{1}{n}\sum_{i=1}^{n}\left\|\nabla f_{i}\left(x_{*}\right)-\nabla f_{i}\left(y_{t}\right)+\nabla f\left(y_{t}\right)-\nabla f\left(x_{*}\right)\right\|^{2}$$
Using Young’s inequality we have
$$\displaystyle\tilde{\sigma}_{*}^{2}$$
$$\displaystyle\leq\frac{1}{n}\sum_{i=1}^{n}\left(2\left\|\nabla f_{i}\left(y_{t}\right)-\nabla f_{i}\left(x_{*}\right)\right\|^{2}+2\left\|\nabla f\left(y_{t}\right)-\nabla f\left(x_{*}\right)\right\|^{2}\right)$$
$$\displaystyle\leq\frac{1}{n}\sum_{i=1}^{n}4L_{i}D_{f_{i}}\left(y_{t},x_{*}\right)+\frac{1}{n}\sum_{i=1}^{n}4LD_{f}\left(y_{t},x_{*}\right)$$
$$\displaystyle\leq 4LD_{f}\left(y_{t},x_{*}\right)+4LD_{f}\left(y_{t},x_{*}\right)$$
$$\displaystyle=8LD_{f}\left(y_{t},x_{*}\right)$$
$$\displaystyle\leq 4L^{2}\left\|y_{t}-x_{*}\right\|^{2}$$
∎
D.6 Proof of Theorem 4
The proof is similar to the proof of 3.
Proof.
Let us define the Lyapunov function:
$$\Psi_{t}=\|x_{t}-x_{*}\|^{2}+\frac{4\eta^{2}\omega}{\alpha M}\frac{1}{M}\sum_{m=1}^{M}\left\|h_{t,m}-x^{n}_{*,m}\right\|^{2}.$$
Now we use Lemma 2 and Lemma 3:
$$\displaystyle\mathbb{E}\left[\Psi_{t+1}\right]$$
$$\displaystyle=\mathbb{E}\|x_{t+1}-x_{*}\|^{2}+\frac{4\eta^{2}\omega}{\alpha M}\frac{1}{M}\sum_{m=1}^{M}\mathbb{E}\left\|h_{t+1,m}-x^{n}_{*,m}\right\|^{2}$$
$$\displaystyle\leq\frac{2\eta^{2}}{M^{2}}\omega\sum_{m=1}^{M}\mathbb{E}\|x^{n}_{t,m}-x^{n}_{*,m}\|^{2}+\frac{2\eta^{2}}{M^{2}}\omega\sum_{m=1}^{M}\mathbb{E}\|h_{t,m}-x^{n}_{*,m}\|^{2}+(1-\eta)\mathbb{E}\|x_{t}-x_{*}\|^{2}$$
$$\displaystyle+\eta\frac{1}{M}\sum_{m=1}^{M}\mathbb{E}\|x^{n}_{t,m}-x^{n}_{*,m}\|^{2}+\frac{4\eta^{2}\omega}{\alpha M}(1-\alpha)\sum_{m=1}^{M}\mathbb{E}\left\|h_{t,m}-x^{n}_{*,m}\right\|^{2}+\frac{4\eta^{2}\omega}{\alpha M}\alpha\sum_{m=1}^{M}\mathbb{E}\|x^{n}_{t,m}-x^{n}_{*,m}\|^{2}.$$
Using Lemma 5 and Theorem 3 from Malinovsky et al. (2021) we have
$$\displaystyle\mathbb{E}\left[\Psi_{t+1}\right]$$
$$\displaystyle\leq\frac{4\eta^{2}\omega}{\alpha M}\left(1-\frac{\alpha}{2}\right)\frac{1}{M}\sum_{m=1}^{M}\mathbb{E}\left\|h_{t,m}-x^{n}_{*,m}\right\|^{2}+\left(1-\eta+\eta(1-\gamma\mu)^{\frac{n}{2}}+\frac{6\eta^{2}\omega}{M}(1-\gamma\mu)^{\frac{n}{2}}\right)\mathbb{E}\|x_{t}-x_{*}\|^{2}$$
$$\displaystyle+\left(\alpha+\eta+\frac{2\eta^{2}\omega}{M}\right)2\gamma^{3}L\sum_{m=1}^{M}\left\|\nabla F_{m}\left(x_{*}\right)\right\|^{2}\left(\sum_{j=0}^{n-1}(1-\gamma\mu)^{j}\right)$$
Using the condition $\eta\leq\min\left(1,\frac{\left(1-(1-\gamma\mu)^{\frac{n}{2}}\right)M}{12\omega(1-\gamma\mu)^{\frac{n}{2}}}\right)$ we have
$$\displaystyle\mathbb{E}\left[\Psi_{t+1}\right]$$
$$\displaystyle\leq\max\left(1-\frac{\alpha}{2},1-\frac{\eta\left(1-(1-\gamma\mu)^{\frac{n}{2}}\right)}{2}\right)\mathbb{E}\left[\Psi_{t}\right]$$
$$\displaystyle+\left(\alpha+\eta+\frac{2\eta^{2}\omega}{M}\right)2\gamma^{3}L\sum_{m=1}^{M}\left\|\nabla F_{m}\left(x_{*}\right)\right\|^{2}\left(\sum_{j=0}^{n-1}(1-\gamma\mu)^{j}\right)$$
Unrolling this recursion as we did previously finishes the proof.
∎ |
Interpolating compact binary waveforms using the singular value decomposition
Kipp Cannon
kipp.cannon@ligo.org
Canadian Institute for Theoretical Astrophysics, 60 St.
George Street, University of Toronto, Toronto, ON M5S 3H8, Canada
Chad Hanna
chad.hanna@ligo.org
Perimeter Institute for Theoretical Physics, Waterloo,
Ontario N2L 2Y5, Canada
Drew Keppel
drew.keppel@ligo.org
Albert-Einstein-Institut, Max-Planck-Institut für
Gravitationsphysik, D-30167 Hannover, Germany
Leibniz Universität Hannover, D-30167 Hannover, Germany
Abstract
Compact binary systems with total masses between tens and hundreds of solar
masses will produce gravitational waves during their merger phase that are
detectable by second-generation ground-based gravitational-wave detectors. In
order to model the gravitational waveform of the merger epoch of compact binary
coalescence, the full Einstein equations must be solved numerically for the
entire mass and spin parameter space. However, this is computationally
expensive. Several models have been proposed to interpolate the results of
numerical relativity simulations. In this paper we propose a numerical
interpolation scheme that stems from the singular value decomposition. This
algorithm shows promise in allowing one to construct arbitrary waveforms within
a certain parameter space given a sufficient density of numerical simulations
covering the same parameter space. We also investigate how similar approaches
could be used to interpolate waveforms in the context of parameter estimation.
I Introduction
Searches for gravitational waves from binary black holes with total masses
between tens and hundreds of solar masses benefit from the complete model of
the gravitational waveform obtained by numerical
relativity Flanagan and Hughes (1998); Hanna (2010). Numerically solving
Einstein’s equations is now quite routine Pretorius (2005); Campanelli et al. (2006); Baker et al. (2006); Pretorius (2009); Husa (2007); Hannam (2009); Hinder (2010), yet still computationally burdensome. Reference
Baumgarte et al. (2008) suggests that there is a finite density of numerical
simulations that would adequately cover the parameter space for certain
ground-based detectors. In this work we explore this concept and extend the
numerical techniques presented in Cannon et al. (2010) and
Cannon et al. (2011), to interpolation of template waveforms using the
singular value decomposition. This should allow for the construction of
gravitational waveforms with parameters between the numerically generated
waveforms.
The idea of interpolating gravitational waveforms has existed for over a
decade. Interpolation of waveforms generated by post-Newtonian techniques was
described in Croce et al. (2000) and Mitra et al. (2005). In these references
analytic formulae for waveform interpolation were derived for particular PN
models. Since 2005 the numerical relativity community has been generating a
substantial number of gravitational waveforms for the coalescence of binary
black holes Pretorius (2005); Campanelli et al. (2006); Baker et al. (2006); Pretorius (2009); Husa (2007); Hannam (2009); Hinder (2010). Interpolation of
these waveforms has been accomplished primarily by (i) phenomenologically
fitting the simulations to closed-form expressions Ajith et al. (2007, 2008, 2009) or (ii) by numerically solving simpler
differential equations that capture the orbital dynamics combined with
numerical stitching of the ringdown phase Buonanno et al. (2007a); Damour et al. (2008); Buonanno et al. (2007b); Damour et al. (2008, 2008); Damour and Nagar (2009); Buonanno et al. (2009). In this work
we propose a different approach to interpolate a set of template waveforms.
This approach does not require careful tuning of fitting formulae or stitching
of waveforms and can be applied to any waveform set of sufficient density.
This paper is organized as follows. First, we describe the technique for
interpolating waveforms via the singular value decomposition. Second we apply
the technique to a set of waveforms containing all phases of the compact binary
coalescence, inspiral, merger and ringdown. Third we discuss how these results
might be applied to the contruction of waveform families, ongoing gravitational
wave searches, and parameter estimation.
II Interpolation technique
It was shown in Cannon et al. (2010) that the singular value decomposition (SVD) reduces the number of
template waveforms needed to search a given parameter space. Additionally,
Cannon et al. (2011) showed that arbitrary waveforms within the parameter
space could be reconstructed from the SVD of a sufficiently dense template
bank. Here we demonstrate a method to directly obtain approximate
reconstruction coefficients for arbitrary waveforms in the parameter space via
interpolation. Consider a waveform family ${\bf h}(x,y)$ described by the
physical parameters $(x,y)$, and consider a set of these waveforms enumerated
by the index $\alpha$ drawn from a region of the parameter space,
${\bf h}(x_{\alpha},y_{\alpha})$. Recall that a SVD of these waveforms
allows each to be written as a linear combination of basis waveforms
${\bf u}_{\mu}$ with coefficients $M_{\mu}(x_{\alpha},y_{\alpha})$
$${\bf h}(x_{\alpha},y_{\alpha})=\sum_{\mu}M_{\mu}(x_{\alpha},y_{\alpha})\,{\bf u%
}_{\mu},$$
(1)
where, in the formalism of Cannon et al. (2010) and Cannon et al. (2011),
$M_{\mu}(x_{\alpha},y_{\alpha}):=\sigma_{\mu}(v_{(2\alpha-1)\mu}+\mathrm{i}v_{(%
2\alpha)\mu})$ is the $\alpha$th combination of singular values
$\sigma_{\mu}$ and reconstruction coefficients $v_{(2\alpha-1)\mu}$ and
$v_{(2\alpha)\mu}$. Recall also that waveforms with arbitrary physical
parameters from the same region of parameter space can also be reconstructed
using the basis vectors ${\bf u}_{\mu}$ by projecting the waveforms onto the
basis vectors to obtain the reconstruction coefficients—a computationally
expensive procedure,
$${\bf h}(x,y)\approx\sum_{\mu}({\bf h}(x,y)\cdot{\bf u}_{\mu})\,{\bf u}_{\mu}.$$
(2)
This can be used to define the arbitrary reconstruction coefficients as
$$M_{\mu}(x,y)={\bf h}(x,y)\cdot{\bf u}_{\mu}.$$
(3)
We seek the set of interpolated reconstruction coefficients $M_{\mu}^{\prime}(x,y)$ that
can approximately reconstruct an arbitrary waveform from that region of
parameter space.
Compact binary gravitational waveforms with negligible effects from spin and
eccentricity are characterized by their component masses. We will assume for
concreteness a two parameter family of waveforms ${\bf h}(x,y)$ where $x$ and
$y$ are $M$ and $q$, respectively, where $M=m_{1}+m_{2}$ is the total mass of the
system and $q=m_{1}/m_{2}$ is the mass ratio of the system.
Chebyshev polynomials of the first kind are known to be good for interpolation,
however other interpolation schemes are also possible. We start with a set of
basis vectors ${\bf u}_{j}$ covering the desired region of parameter space. We
choose a net of points, scaled such that each dimension covers the interval
$[-1,1]$, located at the ${J_{\mathrm{max}}}$th order Chebyshev nodes. For a single
dimension, these nodes occur at the locations
$$x_{j}=\cos\left(\pi\frac{j+\frac{1}{2}}{{J_{\mathrm{max}}}+1}\right),$$
(4)
where $j$ ranges from 0 to ${J_{\mathrm{max}}}$. This choice of net reduces Runge’s
phenomenon when used with the Chebyshev polynomials, which, for a single
dimension, are given as
$$T_{J}(x)=\frac{(x-\sqrt{x^{2}-1})^{J}+(x+\sqrt{x^{2}-1})^{J}}{2w},$$
(5)
where $w=\sqrt{(1+\delta_{J0})({J_{\mathrm{max}}}+1)/2}$ is a normalization factor for the
polynomials and $\delta_{J0}$ is the Kroenecker delta. Both $x_{j}$ and $w$
depend on the choice of ${J_{\mathrm{max}}}$, however for ease of notation we will leave
this implied. The polynomials $T_{J}(x)$ satisfy the discrete orthogonality
condition
$$\sum_{j=0}^{{J_{\mathrm{max}}}}T_{I}(x_{j})T_{J}(x_{j})=\delta_{IJ}.$$
(6)
It is straightforward to extend this to higher dimensions.
In order to obtain the reconstruction coefficients for these locations, we
project waveforms from these locations onto the basis vectors.
From the values on this net, we interpolate to other positions in parameter
space using 2D-Chebyshev interpolation for each set of reconstruction
coefficients $M_{\mu}(x,y)$. Specifically, these values are projected onto the
Chebyshev polynomials
$$C_{KL\mu}=\sum_{k=0}^{{K_{\mathrm{max}}}}\sum_{l=0}^{{L_{\mathrm{max}}}}T_{K}(%
x_{k})T_{L}(y_{l})M_{\mu}(x_{k},y_{l}).$$
(7)
This results in coefficients for the 2D-Chebyshev polynomials which can
be used to evaluate the interpolated reconstruction coefficients $M_{\mu}^{\prime}(x,y)$
at other points in parameter space
$$M_{\mu}^{\prime}(x,y)=\sum_{K=0}^{{K_{\mathrm{max}}}}\sum_{L=0}^{{L_{\mathrm{%
max}}}}C_{KL\mu}T_{K}(x)T_{L}(y).$$
(8)
In the next section we explore this approximation technique using gravitational
waveforms containing all three phases of binary coalescence, inspiral, merger
and ringdown.
II.1 Reconstruction Errors
Errors in reconstructing these waveforms come in two types: errors due to
SVD truncation, and errors due to reconstruction coefficient
interpolation. The truncation errors have previously been shown to take the
form
$$\left(\frac{\delta\rho(x,y)}{\rho(x,y)}\right)_{\rm trunc}=\frac{1}{4}\sum_{%
\mu=N^{\prime}+1}^{N}|M_{\mu}(x,y)|^{2},$$
(9)
where the sum is over the basis vectors that are discarded. The interpolation
errors have a similar form
$$\left(\frac{\delta\rho(x,y)}{\rho(x,y)}\right)_{\rm interp}=\frac{1}{4}\sum_{%
\mu=1}^{N^{\prime}}|M_{\mu}(x,y)-M_{\mu}^{\prime}(x,y)|^{2}.$$
(10)
It should be noted that here the sum is over the basis vectors that are kept
from the SVD. By setting the reconstruction coefficients with $\mu>N^{\prime}$ to
zero, these can be combined into a single expression
$$\frac{\delta\rho(x,y)}{\rho(x,y)}=\frac{1}{4}\sum_{\mu=1}^{N}|M_{\mu}(x,y)-M_{%
\mu}^{\prime}(x,y)|^{2}.$$
(11)
III Results
We apply this procedure in two ways. In section III.1 we
investigate using this approach in the context of interpolating whitened
waveforms. This would be useful in the context of parameter estimation.
Specifically, one could obtain reconstruction coefficients that would be used
for constructing filter outputs associated with arbitrary points in parameter
space using the filter outputs from the SVD basis vectors.
In section III.2, we apply similar techniques to interpolate
raw waveforms. This is done in the context of waveforms one would receive from
numerical relativity simulations (i.e., time series of $\Psi_{2}(t)=\partial_{t}^{2}h_{+}(t)+\mathrm{i}\partial_{t}^{2}h_{\times}(t)$ that are restricted to lie
along lines of constant $M$). This approach could be taken to extend numerical
relativity waveform catalogs at greatly reduced computational cost.
III.1 Whitened waveforms
We apply this procedure to non-spinning phenomenological inspiral-merger-ringdown (IMR)
waveforms Ajith et al. (2009) with $M\in[60M_{\odot},80M_{\odot}]$, $q\in[1,10]$, whitened with an initial LIGO amplitude spectral density (ASD), and transformed to the time
domain. We generate a stochastic template bank Harry et al. (2009) with $99\%$
minimal match for this range of parameters. Since we are working with IMR
waveforms, there is no well defined end of the waveform. We choose to align
the whitened waveforms according to their peak amplitudes and compute
the SVD basis vectors from these waveforms using the procedure described
in Cannon et al. (2010). At this intermediate stage, if we were to look at how
the resulting reconstruction coefficients vary in parameter space, we would see
high frequency features that would be difficult to resolve with interpolation
without a high density interpolation net.
Fortunately, these features can be ameliorated by a complex rotation of the
input waveforms, which is equivalent to a complex rotation of the
reconstruction coefficients,
$$M_{\mu}(x,y)\rightarrow e^{-\mathrm{i}\arg M_{1}(x,y)}M_{\mu}(x,y).$$
(12)
This rotation is chosen such that $\Im\left[M_{1}(x,y)\right]=0$.
Fig. 1 shows the reconstruction coefficients associated with the
$3^{\rm rd}$ and $21^{\rm st}$ basis vectors after this complex rotation. The
smoothness of these reconstruction coefficients indicates that interpolation
should be possible.
In order to perform the interpolation, waveforms from the (12,12) order
2D-Chebyshev net are then projected onto these basis vectors to obtain the
interpolation coefficients, as described by (7), and rotated
as described above. $40\times 40$ test waveforms from within the parameter
space, laid out in a grid, are used for computing mismatches between the
interpolated waveforms, given by (8), and the original
waveforms. Fig. 2 compares the fitting factor residual, which
we define to be one minus the fitting factor, obtained from using the net
waveforms as templates with the interpolation mismatches associated with the
test waveforms. We see that the largest interpolation mismatch is more than an
order of magnitude smaller than the fitting factor residual from the net
waveforms.
III.2 Raw waveforms
We apply similar techniques to waveforms of a type that would be provided by
numerical relativity simulations. Specifically, we use non-spinning
phenomenological IMR waveforms Ajith et al. (2009) with $M\in[60M_{\odot},80M_{\odot}]$, $q\in[1,6]$, multiplied by $f^{2}$, which is
equivalent to taking two time-derivatives, and transformed to the time domain.
We use the same alignment and rotation techniques described in
section III.1 to prepare the waveforms for interpolation.
To generate the basis vectors that enclose this parameter space, we construct a
stochastic template bank with an additional constraint. The mass ratios of the
templates are restricted to take on values $q\in\{q_{j}=5x_{j}+1|j\in[1,6]\}$, where $x_{j}$ are the nodes associated with the 10th order Chebyshev
polynomial.
With the basis vectors in hand, we project the waveforms from an interpolation
net consisting of the (20,10) 2D-Chebyshev nodes onto the basis vectors to
obtain the reconstruction coefficients. These complex coefficients are rotated
as described above, and then used to obtain the interpolation coefficients.
Again, $40\times 40$ test waveforms from within the parameter space, laid out
in a grid, are used for computing mismatches between the interpolated waveforms
and the original waveforms. We find comparable interpolation mismatches for
these non-whitened waveforms, shown in figure 3, as for the
whitened waveforms.
IV Conclusion
Using the procedure described above, we have shown it is possible to produce
gravitational waveforms for arbitrary points in parameter space by
interpolating reconstruction coefficients from the SVD of a set of
waveforms uniformly covering the space.
Results have been presented for both whitened waveforms, and raw waveforms.
The former could be useful in the context of parameter estimation associated
with compact binary coalescence (CBC) gravitational-wave (GW) signals, which frequently uses Monte Carlo Markov Chain
methods to measure the likelihood ratio from many points in parameter space.
This requires the generation of the waveforms for each point in parameter space
and the overlap computation between the waveform and the data. Using the
interpolated reconstruction coefficients, the same computation could be
approximately performed with generating a subset of the total waveforms,
reconstructing the overlap by appropriately recombining the filter outputs from
the basis vectors. The latter could be used to accurately interpolate waveforms
that are computationally costly to produce, as is the case for numerical
relativity waveforms.
For future work, these techniques should be expanded to include additional
dimensions of parameter space (e.g., binary object spin parameters). In
addition, other interpolation schemes that use equispaced or random points in
parameter space might be found to be favorable for different applications. We
also note that these techniques could be applied to other gravitational
waveforms such as supernova waveforms where singular value decomposition has
also been applied Heng (2009), or where other methods have been used to
reduce the rank of the parameter space Field et al. (2011).
Acknowledgements.We would like to thank Christian Röver and Ilya Mandel for discussions and
comments related to this work. Research at Perimeter Institute is supported
through Industry Canada and by the Province of Ontario through the Ministry of
Research & Innovation. KC was supported by the National Science and
Engineering Research Council, Canada. DK was supported from the Max Planck
Gesellschaft. This work has LIGO document number LIGO-P1100101-v2.
References
Flanagan and Hughes (1998)
E. E. Flanagan and S. A. Hughes, Phys. Rev. D 57, 4535 (1998).
Hanna (2010)
C. Hanna (LIGO Scientific Collaboration and Virgo
Scientific Collaboration), Class. Quantum Grav 27, 114003 (2010).
Pretorius (2005)
F. Pretorius, Phys. Rev. Lett. 95, 121101 (2005), arXiv:gr-qc/0507014
.
Campanelli et al. (2006)
M. Campanelli, C. O. Lousto, P. Marronetti,
and Y. Zlochower, Phys. Rev. Lett. 96, 111101 (2006), arXiv:gr-qc/0511048 .
Baker et al. (2006)
J. G. Baker, J. Centrella,
D.-I. Choi, M. Koppitz, and J. van Meter, Phys. Rev. Lett. 96, 111102 (2006), arXiv:gr-qc/0511103 .
Pretorius (2009)
F. Pretorius, in Physics of
Relativistic Objects in Compact Binaries: from Birth to Coalescence, edited by M. Colpi, P. Casella, V. Gorini, U. Moschella, and A. Possenti (Springer, Heidelberg, Germany, 2009) arXiv:0710.1338 .
Husa (2007)
S. Husa, Eur. Phys. J. ST 152, 183 (2007), arXiv:0812.4395 [gr-qc] .
Hannam (2009)
M. Hannam, Class.
Quantum Grav. 26, 114001
(2009), arXiv:0901.2931 .
Hinder (2010)
I. Hinder, (2010), arXiv:1001.5161 [gr-qc] .
Baumgarte et al. (2008)
T. Baumgarte, P. R. Brady, J. D. E. Creighton, L. Lehner,
F. Pretorius, and R. DeVoe, Phys. Rev. D 77, 084009 (2008).
Cannon et al. (2010)
K. Cannon, A. Chapman,
C. Hanna, D. Keppel, A. C. Searle, and A. J. Weinstein, Phys.
Rev. D 82, 044025
(2010).
Cannon et al. (2011)
K. Cannon, C. Hanna, and D. Keppel, ArXiv e-prints
(2011), arXiv:1101.4939 [gr-qc] .
Croce et al. (2000)
R. P. Croce, T. Demma,
V. Pierro, I. M. Pinto, and F. Postiglione, Phys.
Rev. D 62, 124020
(2000).
Mitra et al. (2005)
A. S. Mitra, S. V. Dhurandhar, and L. S. Finn, Phys.
Rev. D 72, 102001
(2005).
Ajith et al. (2007)
P. Ajith et al., Class. Quantum Grav. 24, S689 (2007), arXiv:0704.3764 .
Ajith et al. (2008)
P. Ajith et al., Phys. Rev. D 77, 104017 (2008), arXiv:arXiv:0710.2335
[gr-qc] .
Ajith et al. (2009)
P. Ajith et al., (2009), arXiv:0909.2867 [gr-qc]
.
Buonanno et al. (2007a)
A. Buonanno, G. B. Cook,
and F. Pretorius, Phys. Rev. D 75, 124018 (2007a).
Damour et al. (2008)
T. Damour, A. Nagar,
E. N. Dorband, D. Pollney, and L. Rezzolla, Phys.
Rev. D 77, 084017
(2008).
Buonanno et al. (2007b)
A. Buonanno et al., Phys. Rev. D 76, 104049 (2007b), arXiv:0706.3732 [gr-qc] .
Damour et al. (2008)
T. Damour, A. Nagar,
M. Hannam, S. Husa, and B. Brügmann, Phys. Rev. D 78, 044039 (2008).
Damour and Nagar (2009)
T. Damour and A. Nagar, Phys. Rev. D 79, 081503 (2009).
Buonanno et al. (2009)
A. Buonanno, Y. Pan,
H. P. Pfeiffer, M. A. Scheel, L. T. Buchman, and L. E. Kidder, Phys. Rev. D 79, 124028 (2009).
Harry et al. (2009)
I. W. Harry, B. Allen, and B. S. Sathyaprakash, Phys. Rev. D 80, 104014 (2009).
Heng (2009)
I. S. Heng, Classical and Quantum Gravity 26, 105005 (2009).
Field et al. (2011)
S. E. Field, C. R. Galley, F. Herrmann, J. S. Hesthaven, E. Ochsner, and M. Tiglio, Physical Review Letters 106, 221102 (2011), arXiv:1101.3765 [gr-qc]
. |
Unruh effect of nonlocal field theories with a minimal length
Yongwan Gim
yongwan89@sogang.ac.kr
Department of Physics, Sogang University, Seoul, 04107,
Republic of Korea
Research Institute for Basic Science, Sogang University,
Seoul, 04107, Republic of Korea
Hwajin Um
um16@sogang.ac.kr
Department of Physics, Sogang University, Seoul, 04107,
Republic of Korea
Wontae Kim
wtkim@sogang.ac.kr
Department of Physics, Sogang University, Seoul, 04107,
Republic of Korea
(December 4, 2020)
Abstract
The nonlocal field theory commonly requires a minimal length, and so it appears to
formulate the nonlocal theory in terms of
the doubly special relativity which makes the speed of light and the minimal length
invariant simultaneously.
We set up a generic nonlocal model having the same set of solutions as the local theory but allowing Lorentz violations due to the minimal length.
It is exactly corresponding to the model with the modified dispersion relation in
the doubly special relativity.
For this model, we calculate the modified Wightman function and the rate of response function
by using the Unruh-DeWitt detector method.
It turns out that the Unruh effect should be corrected by the minimal length
related to the nonlocality in the regime of the doubly special relativity.
However, for the Lorentz-invariant limit, it is shown that
the Wightman function and the Unruh effect remain the same as those of the local theory.
I Introduction
There has been much attention to nonlocal theories
in light of a low energy description of fundamental nonlocal interactions.
Initially, nonlocality was mostly considered
in the context of axiomatic quantum field theory Efimov:1967pjn ; Efimov:1968flw ; Iofa:1969ex ; Fainberg:1978cc ; Fainberg:1992jt .
And then many efforts have been devoted to studying various aspects of nonlocality in connection with gravity Tomboulis:1997gg ; Moffat:2010bh ; Biswas:2011ar ; Isi:2013cxa ; Briscese:2013lna ; Calcagni:2014vxa ; Frolov:2015bta
and cosmology Deser:2007jk ; Barnaby:2007ve ; Barnaby:2008fk ; Barnaby:2010kx ; Deser:2013uya as well as the role of nonlocality in the framework of string theory
Witten:1985cc ; Brekke:1988dg ; Eliezer:1989cr ; Ghoshal:2000dd ; Moeller:2003gg ; Ghoshal:2006te ; Fuchs:2008cc ; Calcagni:2013eua .
In the nonlocal field theories,
the nonlocality is commonly accompanied by a length scale $\ell$
because of the presence of higher derivative terms in the equations of motion.
There has been an intriguing issue on the Unruh effect
in a specific nonlocal field theory.
In the particular nonlocal theory
obeying the field equation of $\square e^{-\ell^{2}\square/2}\phi=0$
with the minimal length $\ell$, the rate of response function
was calculated by using the Unruh-DeWitt detector method,
and then it was claimed
that there are significant modifications in the Unruh effect due to the
the modification of the Feynman propagator originated from the nonlocality Nicolini:2009dr .
However,
it was proposed that, for a wide class of nonlocal theory obeying
the field equation given by $\square f(\ell^{2}\square)\phi=0$ with the everywhere nonzero and analytic function $f$,
the Bogoliubov coefficients should be exactly the same as
the case of a local theory
and thus
the Unruh effect should remain unchanged
Kajuri:2017jmy .
Recently, for the specific nonlocal model of $\square e^{-\ell^{2}\square/2}\phi=0$,
it was shown that the modified Feynman propagator due to the nonlocality consists of
the Wightman function along with the complementary error function
instead of the conventional step function Modesto:2017ycz .
It means that the nonlocality is responsible for the error function
rather than the modification of the Wightman function,
so that the Unruh effect relying on the form of the Wightman function
naturally remains the same as
that of the local theory.
On the other hand,
if the nonlocal theory should respect the special theory of relativity,
the length $\ell$ in the nonlocal theory
will be no longer minimal length due to the length contraction depending on the inertial frame.
So, the nonlocal field theory having the minimal length should be implemented by
the doubly special relativity of the extended version of Einstein’s special relativity AmelinoCamelia:2000ge ; AmelinoCamelia:2000mn ,
where the minimal length as well as the speed of light is an observer-independent scale.
In the framework of the doubly special relativity,
the existence of the minimal length
would necessarily lead to
the modification of the dispersion relation such as
$f(\ell^{2}k_{0}^{2},\ell^{2}k_{i}^{2})k^{\mu}k_{\mu}=m^{2}$,
where $k^{\mu}k_{\mu}$ is related to the invariant speed of light and
$f(\ell^{2}k_{0}^{2},\ell^{2}k_{i}^{2})$ makes the minimal length $\ell$ an invariant scale
under any inertial frames Magueijo:2001cr ; Magueijo:2002am ; Kimberly:2003hp .
If one were to consider the nonlocal model implemented by
the minimal length allowing the Lorentz violation, then
the field equation would be $\square f((i\ell\partial_{0})^{2},(i\ell\partial_{i})^{2})\phi=m^{2}$.
So, it would be interesting to study the Unruh effect for the nonlocal field theory
in the regime of the double special
relativity which makes the minimal length invariant.
In Sec. II,
we will recapitulate the Unruh-DeWitt detector method for the Unruh effect in the local theory.
Then, in Sec. III we will consider a massless nonlocal model of
$\square f((i\ell\partial_{0})^{2},(i\ell\partial_{i})^{2})\phi=0$ which
respects the doubly special relativity, where
the function $f$ is an analytic function and everywhere nonzero.
Using the Unruh-DeWitt detector method presented in Sec. II,
we will obtain the modified Wightman function compatible with the doubly special relativity
and show that the Unruh effect receives some corrections due to the presence of
the minimal length in the nonlocal field theory.
If the nonlocal theory respects the Lorentz symmetry,
in other words, for the Lorentz-invariant limit of $\square f(\ell^{2}\square)\phi=0$,
it turns out that the Unruh effect is the same as that of the local case as was shown
in Ref. Kajuri:2017jmy ; Modesto:2017ycz .
Finally, conclusion and discussion will be given in Sec. IV.
II Unruh-DeWitt detector method
We would like to encapsulate the Unruh-DeWitt detector method for the Unruh effect in the local field theory
presented in Ref. Unruh:1983ms .
Let us consider a detector moving in Minkowski spacetime along a trajectory $x^{\mu}(\tau)$
with the proper time $\tau$,
and assume that it moves through a region permeated by a quantum scalar field $\phi(x)$.
The minimal interaction between the detector and the scalar field
is given by the Lagrangian of $L_{\rm int}=g\mu(\tau)\phi\left(x(\tau)\right)$,
where $g$ is the small coupling constant and the detector operator $\mu(\tau)$ is approximated by
$\mu(\tau)=e^{iH_{0}\tau}\mu(0)e^{-iH_{0}\tau}$.
The detector will measure the energy transition
from the energy $E_{0}$ of the ground state to the energy $E$ of an excited state.
Since the coupling constant $g$ is small,
the transition probability is written as $P=\int dE|\mathcal{A}^{(1)}|^{2}$,
where the first order amplitude $\mathcal{A}^{(1)}$ is given by
$$\displaystyle\mathcal{A}^{(1)}$$
$$\displaystyle=i\langle E;\psi|\int^{\tau}_{\tau_{\rm i}}L_{\rm int}d\tau|E_{0}%
;0\rangle$$
(1)
$$\displaystyle=ig\langle E|\mu(0)|E_{0}\rangle\int^{\tau}_{\tau_{\rm i}}d\tau e%
^{-i\tau\Delta E}\langle\psi|\phi(x)|0\rangle.$$
(2)
Note that $|0\rangle$ is the Minkowski vacuum and $|\psi\rangle$ is the excited state,
and the energy difference between them is denoted by $\Delta E=E-E_{0}$.
Then, the transition probability $P$ is obtained as
$$P=g^{2}\int dE|\langle E|\mu(0)|E_{0}\rangle|^{2}R(\Delta E),$$
(3)
where the response function $R(\Delta E)$ is written as
$$\displaystyle R(\Delta E)$$
$$\displaystyle=\int^{\tau^{\prime}}_{\tau^{\prime}_{\rm i}}d\tau^{\prime}\int^{%
\tau}_{\tau_{\rm i}}d\tau e^{-i(\tau-\tau^{\prime})\Delta E}\langle 0|\phi%
\left(x(\tau)\right)\phi\left(x(\tau^{\prime})\right)|0\rangle$$
(4)
$$\displaystyle=\int^{\Delta\tau^{+}}_{\Delta\tau^{+}_{\rm i}}d\Delta\tau^{+}%
\int^{\Delta\tau^{-}}_{\Delta\tau^{-}_{\rm i}}d\Delta\tau^{-}e^{-i\Delta\tau^{%
-}\Delta E}G^{+}(\Delta\tau^{+},\Delta\tau^{-})$$
(5)
with $\Delta\tau^{\pm}=\tau\pm\tau^{\prime}$, and the positive frequency
Wightman function $G^{+}$ is defined as
$$G^{+}(\Delta\tau^{+},\Delta\tau^{-})=\langle 0|\phi\left(x(\tau)\right)\phi%
\left(x(\tau^{\prime})\right)|0\rangle.$$
(6)
So, one can obtain the transition probability per unit time as
$$\dot{P}=g^{2}\int dE|\langle E|\mu(0)|E_{0}\rangle|^{2}\dot{R}(\Delta E)$$
(7)
with the rate of the response function
$$\displaystyle\dot{R}(\Delta E)=\int^{\infty}_{-\infty}d\Delta\tau^{-}e^{-i%
\Delta\tau^{-}\Delta E}G^{+}(\Delta\tau^{+},\Delta\tau^{-}),$$
(8)
where the integration range of $\Delta\tau^{-}$ is extended up to $\pm\infty$.
In the local field theory, the field equation of the free scalar field $\phi$ is given by
$$\square\phi=0,$$
(9)
and the solution is written as
$$\phi(x)=\int\frac{d^{3}k}{(2\pi)^{3}2\omega}\left[a_{k}e^{ik_{\mu}x^{\mu}}+a^{%
\dagger}_{k}e^{-ik_{\mu}x^{\mu}}\right],$$
(10)
where the coefficients $a_{k}$ and $a^{\dagger}_{k}$ are operators in such a way that
the Minkowski vacuum is annihilated by the operator $a_{k}$,
i.e., $a_{k}|0\rangle=0$.
By imposing the equal-time commutation relations,
$[\phi(\vec{x},t),\pi(\vec{y},t)]=i\delta^{(3)}(\vec{x}-\vec{y})$ and $[\phi(\vec{x},t),\phi(\vec{y},t)]=[\pi(\vec{x},t),\pi(\vec{y},t)]=0$
with the conjugate momentum $\pi=\dot{\phi}$,
we obtain the following quantization rules,
$$\displaystyle[a_{k},a^{\dagger}_{k^{\prime}}]=(2\pi)^{3}2\omega\delta^{(3)}(%
\vec{k}-\vec{k}^{\prime}),\qquad[a_{k},a_{k^{\prime}}]=[a^{\dagger}_{k},a^{%
\dagger}_{k^{\prime}}]=0.$$
(11)
Plugging Eqs. (10) and (11)
into Eq. (6),
one can also obtain the positive frequency Wightman function in the local field theory as
$$\displaystyle G^{+}(x,x^{\prime})$$
$$\displaystyle=\int\frac{d^{3}kd^{3}k^{\prime}}{(2\pi)^{6}2\omega 2\omega^{%
\prime}}\langle 0|[a_{k},a^{\dagger}_{k^{\prime}}]|0\rangle e^{i(k_{\mu}x^{\mu%
}-k^{\prime}_{\mu}x^{\prime\mu})}$$
(12)
$$\displaystyle=\int\frac{d^{3}k}{(2\pi)^{3}2\omega}e^{ik_{\mu}(x^{\mu}-x^{%
\prime\mu})}.$$
(13)
In the Minkowski spacetime with the coordinates of $x^{\mu}=(t,x,y,z)$,
the hyperbolic trajectory describes a uniformly accelerated detector along the $x$-axis
with the proper acceleration $a$ and the proper time $\tau$,
which is given by
$$t(\tau)=\frac{1}{a}\sinh(a\tau),\qquad x(\tau)=\frac{1}{a}\cosh(a\tau)$$
(14)
with the fixed $y$ and $z$.
By using Eqs. (8) and (13) in the hyperbolic trajectory (14),
the rate of response function of the uniformly accelerated detector can be calculated as
$$\dot{R}(\Delta E)=\frac{\Delta E}{2\pi}\frac{1}{e^{\frac{2\pi\Delta E}{a}}-1}.$$
(15)
Eventually, the temperature can be read off
from Eq. (15)
as
$$\displaystyle T_{\rm loc}=\frac{a}{2\pi},$$
(16)
which is the well-known Unruh effect in the local field theory Unruh:1983ms .
In the next section, we shall discuss the Unruh effect in the nonlocal field theory
in order to figure out
how the conventional Unruh effect is modified by the nonlocality.
III Modified Wightman function and Unruh effect
There appears a minimal length in nonlocal formulation of theories with
higher derivatives, so that
the special relativity related to the Lorentz symmetry
could be promoted to the doubly special relativity where
the minimal length as well as the speed of light is invariant in any inertial frames.
Let us introduce a nonlocal model where the minimal length $\ell$ is invariant
under any inertial frames allowing the
conventional Lorentz violation Kimberly:2003hp ,
which is generically described by the field equation of
$$\square f\left((i\ell\partial_{0})^{2},(i\ell\partial_{i})^{2}\right)\phi_{\rm
NL%
}=0,$$
(17)
where $f$ is an analytic function and everywhere nonzero.
It is worth noting that the dispersion relation corresponding to Eq. (17) is
written as $f(\ell^{2}k_{0}^{2},\ell^{2}k_{i}^{2})k^{\mu}k_{\mu}=0$,
where $k^{\mu}k_{\mu}$ gives the invariant speed of light and
$f(\ell^{2}k_{0}^{2},\ell^{2}k_{i}^{2})$ makes the minimal length $\ell$ an invariant scale
under any inertial frames Magueijo:2001cr ; Magueijo:2002am ; Kimberly:2003hp .
Also, we assumed that the function $f$ is nonzero,
so that the frequency $\omega$ satisfies $\omega=|\vec{k}|$, which
means that the speed of light is still invariant in spite of
the modified dispersion relation in the light of the doubly special relativity.
According to Ref. Barnaby:2007ve ,
the number of independent solutions to an infinite order differential equation is equal to the number of poles in its propagator.
Since the number of poles for the propagator of the nonlocal model (17) is the same as that of the local theory,
the complete set of solutions to the nonlocal field equation (17) is just that to the field equation (9) of the local theory.
So, combining the plane wave solution of $e^{\pm ik_{\mu}x^{\mu}}$ as the set of the solutions to Eq. (17)
gives
the field expansion as
$$\phi_{\rm NL}(x)=\int\frac{d^{3}k}{(2\pi)^{3}2\omega}\left[b_{k}e^{ik_{\mu}x^{%
\mu}}+b^{\dagger}_{k}e^{-ik_{\mu}x^{\mu}}\right].$$
(18)
Now, one might wonder where the nonlocal effects including the Lorentz violation are reflected in the field expansion (18).
In the case of the nonlocal model (17), it has the same set of the solutions as that of
the local theory,
so that nonlocal effects are not in the plane wave solutions
but in the coefficients $b_{k}$ and $b^{\dagger}_{k}$,
and the commutation relations between them would be modified.
Let us see how the nonlocal effects are taken into account
in the coefficients $b_{k}$ and $b^{\dagger}_{k}$ and what the commutation relations are.
Actually, the equal-time commutator between the field and its momentum
is unclear in the nonlocal field theory, since it is difficult to define the conjugate momentum for the field $\phi_{\rm NL}(x)$ Barnaby:2007ve ; Modesto:2017ycz .
So, we redefine the field as
$\phi_{\rm NL}=f^{-1}\left((i\ell\partial_{0})^{2},(i\ell\partial_{i})^{2}%
\right)\phi$
to obtain the commutation relation between the operators $b_{k}$ and $b^{\dagger}_{k}$,
and thus the field equation (17) can be rewritten as $\square\phi=0$.
Now the nonlocal field $\phi_{\rm NL}$ is written in terms of the local field $\phi$
by using Eq. (10),
$$\displaystyle\phi_{\rm NL}(x)$$
$$\displaystyle=\frac{1}{f((i\ell\partial_{0})^{2},(i\ell\partial_{i})^{2})}\phi%
(x)$$
$$\displaystyle=\int\frac{d^{3}k}{(2\pi)^{3}2\omega}\left[\frac{a_{k}}{f(\ell^{2%
}\omega^{2},\ell^{2}\vec{k}^{2})}e^{ik_{\mu}x^{\mu}}+\frac{a^{\dagger}_{k}}{f(%
\ell^{2}\omega^{2},\ell^{2}\vec{k}^{2})}e^{-ik_{\mu}x^{\mu}}\right].$$
(19)
By comparing Eq. (19) with the field expansion (18),
the relations between the local operators of $a_{k}$ and nonlocal operators of $b_{k}$ are obtained as
$$b_{k}=\frac{a_{k}}{f(\ell^{2}\omega^{2},\ell^{2}\vec{k}^{2})},\qquad b^{%
\dagger}_{k}=\frac{a^{\dagger}_{k}}{f(\ell^{2}\omega^{2},\ell^{2}\vec{k}^{2})}.$$
(20)
From the commutation relation (11) between $a_{k}$ and $a^{\dagger}_{k^{\prime}}$,
we can obtain a relation between $b_{k}$ and $b^{\dagger}_{k^{\prime}}$ as
$$[b_{k},b^{\dagger}_{k^{\prime}}]=\frac{(2\pi)^{3}2\omega\delta^{3}(\vec{k}-%
\vec{k}^{\prime})}{f(\ell^{2}\omega^{2},\ell^{2}\vec{k}^{2})f(\ell^{2}\omega^{%
2},\ell^{2}\vec{k}^{\prime 2})},$$
(21)
where the vacuum of the nonlocal field theory is the same as that of the local theory as $b_{k}|0\rangle=f^{-1}(\ell^{2}\omega^{2},\ell^{2}\vec{k}^{2})a_{k}|0\rangle=0$.
Next, from the definition of the positive frequency Wightman function (6),
we can find the modified Wightman function corresponding to Eq. (17) as
$$\displaystyle G^{+}(x,x^{\prime})$$
$$\displaystyle=\int\frac{d^{3}kd^{3}k^{\prime}}{(2\pi)^{6}2\omega 2\omega^{%
\prime}}\langle 0|[b_{k},b^{\dagger}_{k^{\prime}}]|0\rangle e^{i(k_{\mu}x^{\mu%
}-k^{\prime}_{\mu}x^{\prime\mu})}$$
$$\displaystyle=\int\frac{d^{3}k}{(2\pi)^{3}2\omega}\frac{1}{f^{2}(\ell^{2}%
\omega^{2},\ell^{2}\vec{k}^{2})}e^{ik_{\mu}(x^{\mu}-x^{\prime\mu})}.$$
(22)
Since $f$ is everywhere nonzero and analytic function,
it can be represented by a power series
with the help of $\omega=|\vec{k}|$,
$$\frac{1}{f^{2}(\ell^{2}\omega^{2},\ell^{2}\vec{k}^{2})}=\frac{1}{f^{2}(\ell^{2%
}\omega^{2},\ell^{2}\omega^{2})}=\sum^{\infty}_{n=0}\alpha_{n}(\ell^{2}\omega^%
{2})^{n},$$
(23)
where the coefficient $\alpha_{0}$ is fixed by $\alpha_{0}=1$
to recover the local case when the nonlocal effects disappear.
Then, the positive frequency Wightman function (22) is
explicitly calculated as
$$\displaystyle G^{+}(x,x^{\prime})=\sum^{\infty}_{n=0}\alpha_{n}\ell^{2n}\frac{%
(-1)^{n}(2n)!}{8\pi^{2}\Delta x}\frac{(\Delta t+\Delta x)^{2n+1}-(\Delta t-%
\Delta x)^{2n+1}}{(-\Delta t^{2}+\Delta x^{2})^{2n+1}},$$
(24)
where $\Delta t=t-t^{\prime}$ and $\Delta x=|\vec{x}-\vec{x}^{\prime}|$.
When the detector follows the hyperbolic trajectory (14),
the modified Wightman function is obtained as
$$G^{+}(\tau,\tau^{\prime})=\sum^{\infty}_{n=0}\sum^{2n}_{m=0}\alpha_{n}\ell^{2n%
}\frac{(-1)^{n+1}(2n)!}{4\pi^{2}}\frac{e^{a(n-m)\Delta\tau^{+}}}{\left(\frac{2%
}{a}\sinh\left(\frac{a}{2}\Delta\tau^{-}\right)\right)^{2(n+1)}}$$
(25)
with $\Delta\tau^{\pm}=\tau\pm\tau^{\prime}$.
So,
the rate of response function in the nonlocal model (17) is calculated as
$$\dot{R}(\Delta E)=\frac{\Delta E}{2\pi}\frac{1}{e^{\frac{2\pi\Delta E}{a}}-1}%
\left[1+\sum^{\infty}_{n=1}\sum^{2n}_{m=0}\frac{\alpha_{n}\ell^{2n}}{2n+1}e^{a%
(n-m)\Delta\tau_{+}}\prod^{n}_{s=1}(s^{2}a^{2}+\Delta E^{2})\right],$$
(26)
where it reduces to the standard local limit for $\ell\to 0$.
The similar rate of response function can be found
in the Lorentz-violating cases Rinaldi:2008qt ; Gutti:2010nv ; MartinMartinez:2012th ; Harikumar:2012yu ; Harikumar:2012ff ; Majhi:2013koa .
Consequently,
in the generic nonlocal model (17),
the Wightman function and the rate of response function were corrected by the minimal length related to the nonlocality,
so that
the Unruh effect could be modified.
For example,
let us see the Unruh effect for a nonlocal model
defined by the doubly special relativity in Ref. Kimberly:2003hp ,
which is given by
$$\frac{\square}{1-\ell^{2}\partial_{0}^{2}}\phi_{\rm NL}=0,$$
(27)
where the nonlocal term is $f=(1+\ell^{2}\omega^{2})^{-1}$ in the momentum space
and thus $f^{-2}=1+2\ell^{2}\omega^{2}+\ell^{4}\omega^{4}$.
So, one can easily
identify the coefficients $\alpha_{n}$ related to the nonlocality in Eq. (26) as
$\alpha_{0}=1$, $\alpha_{1}=2$, $\alpha_{2}=1$ and $\alpha_{n>2}=0$,
and thus the rate of response function is obtained as
$$\displaystyle\dot{R}(\Delta E)=\frac{\Delta E}{2\pi}\frac{1}{e^{2\pi\frac{%
\Delta E}{a}}-1}$$
$$\displaystyle\left[1+\frac{2\ell^{2}a^{2}}{3}A_{1}(a,\Delta\tau_{+})\left(%
\frac{\Delta E^{2}}{a^{2}}+1\right)\right.$$
$$\displaystyle+\left.\frac{\ell^{4}a^{4}}{5}A_{2}(a,\Delta\tau_{+})\left(\frac{%
\Delta E^{2}}{a^{2}}+1\right)\left(\frac{\Delta E^{2}}{a^{2}}+4\right)\right],$$
(28)
with $A_{1}(a,\Delta\tau_{+})=1+2\cosh(a\Delta\tau_{+})$,
$A_{2}(a,\Delta\tau_{+})=1+2\cosh(a\Delta\tau_{+})+2\cosh(2a\Delta\tau_{+})$.
Here, we note that Eq. (28) depends on the proper time
and
such time-dependent rate of response functions also appear in
the case of superluminal dispersion relations Rinaldi:2008qt
and $\kappa$-deformations Harikumar:2012yu ; Harikumar:2012ff .
The dependency of the proper time
means that the system is not in a global thermodynamic equilibrium.
So,
we take the approximation as $a\ll\Delta E$ to make Eq. (28)
as a slight deviation out of the global thermodynamic equilibrium,
and thus we assume a local thermodynamic equilibrium.
Finally, we can read off the Unruh temperature
by using the relation between the Planck distribution and the rate of response function of $\dot{R}(\Delta E)=(\Delta E/2\pi)(e^{\Delta E/T}-1)^{-1}$ Unruh:1983ms
as
$$\displaystyle T=\frac{\Delta E}{\ln\left(1+\frac{\Delta E}{2\pi\dot{R}(\Delta E%
)}\right)}.$$
(29)
This procedure is very similar to derivation of the Hawking temperature from the scattering amplitude Ghoroku:1994np ; Natsuume:1996wf ; Kim:2002nc ; Clement:2007tw .
Then, the temperature can be read off from Eq. (29) as
$$T_{\rm NL}=\frac{a}{2\pi}+\frac{a^{2}}{2\pi\Delta E}\ln\left(1+\frac{2\ell^{2}%
\Delta E^{2}}{3}A_{1}(a,\Delta\tau_{+})+\frac{\ell^{4}\Delta E^{4}}{5}A_{2}(a,%
\Delta\tau_{+})\right),$$
(30)
which shows that the Unruh effect could be corrected by the nonlocality.
Next, let us see the Unruh effect for
a wide class of the nonlocal model
respecting the Lorentz symmetry such as
$$\square f(\ell^{2}\square)\phi_{\rm NL}=0,$$
(31)
which was already considered by using the Bogoliubov transformation in Ref. Kajuri:2017jmy .
By replacing $f(\ell^{2}\omega^{2},\ell^{2}\vec{k}^{2})$ with $f(\ell^{2}k^{\mu}k_{\mu})$
in Eqs. (17) and (22),
the positive frequency Wightman function is modified
as
$$\displaystyle G^{+}(x,x^{\prime})=\int\frac{d^{3}k}{(2\pi)^{3}2\omega}\frac{1}%
{f^{2}(\ell^{2}k^{\mu}k_{\mu})}e^{ik_{\mu}(x^{\mu}-x^{\prime\mu})}.$$
(32)
Note that the nonlocal effect actually disappears because
$f(\ell^{2}k^{\mu}k_{\mu})=f(0)=1$ due to the fact that $k^{\mu}k_{\mu}=-\omega^{2}+\vec{k}^{2}=0$,
which in essence amounts to the limit of $\ell\rightarrow 0$.
Then, the positive frequency Wightman function of the nonlocal model (31)
is found as
$$G^{+}(x,x^{\prime})=\int\frac{d^{3}k}{(2\pi)^{3}2\omega}e^{ik_{\mu}(x^{\mu}-x^%
{\prime\mu})},$$
(33)
which is nothing but the standard Wightman function (13) for the local theory.
Therefore,
the rate of response function is also coincident with Eq. (15),
so that
the Unruh effect remains the same as the local case.
This result is consistent with the result
obtained from the Bogoliubov transformation Kajuri:2017jmy .
Finally, it is worth noting that
there is an advantage of our formalism in the sense that
one could discuss the Wightman function (22) directly
in connection with the Unruh effect.
For the specific Lorentz-invariant nonlocal field theory of
$$\square e^{-\ell^{2}\square/2}\phi_{\rm NL}=0,$$
(34)
where the Feynman propagator $G_{\rm F}$ is given by
$$G_{\rm F}(x)=\int\frac{d^{4}k}{(2\pi)^{4}}\frac{e^{-\frac{\ell^{2}}{2}k^{2}}}{%
k^{2}}e^{ik^{\mu}x_{\mu}}.$$
(35)
In Ref.
Modesto:2017ycz , the Feynman propagator (35) could be nicely expressed as
$$G_{\rm F}(x)=\int\frac{d^{3}k}{(2\pi)^{3}}\left[\frac{1}{2}{\rm erfc}\left(%
\frac{\ell}{\sqrt{2}}\omega-\frac{t_{\rm E}}{\sqrt{2}\ell}\right)G^{+}_{k}+%
\frac{1}{2}{\rm erfc}\left(\frac{\ell}{\sqrt{2}}\omega+\frac{t_{\rm E}}{\sqrt{%
2}\ell}\right)G^{-}_{k}\right],$$
(36)
where $t_{\rm E}$ is the Euclidean time and $G^{\pm}_{k}=(2\omega)^{-1}e^{\mp\omega t_{\rm E}\pm i\vec{k}\cdot\vec{x}}$, and the complementary error function is defined as
$${\rm erfc}(x)=1-\frac{2}{\sqrt{\pi}}\int^{x}_{0}d\xi e^{-\xi^{2}}.$$
From the fact that the complementary error functions in the Feynman propagator (36)
become the $\theta$-function
for the limit of $\ell\rightarrow 0$,
it was proposed that the positive frequency Wightman function
should be written as Modesto:2017ycz
$$G^{+}(x)=\int\frac{d^{3}k}{(2\pi)^{3}}G^{+}_{k}.$$
(37)
However, as seen from Eq. (36), the complementary error function could not be
factorized if $\ell$ is finite.
Thus, it would be more reasonable to treat the Wightman function directly such as
Eq. (22) or Eq. (32).
Eventually, our formalism tells us that the Wightman function is the same as that of the local theory
as long as the nonlocal model is Lorentz-invariant, so that the Unruh temperature is accordingly unchanged.
IV Conclusion and discussion
We calculated the Wightman function and the rate of the response function
with respect to the generic nonlocal model (17) allowing the Lorentz violation
to make the minimal length an invariant scale under any inertial frames.
In this model, by using the Unruh-DeWitt detector method, we showed that the Unruh effect could
be corrected by the minimal length generically.
Additionally, taking the limit of the Lorentz-invariant model,
we showed that the Unruh effect always remains unchanged,
which is compatible with the result using the Bogoliubov transformation in Ref. Kajuri:2017jmy .
Our calculations also explain why the Wightman function read off from the Feynman propagator (36)
is invariant
for the particular Lorentz-invariant nonlocal model (34)
in Ref. Modesto:2017ycz .
Therefore, it turned out that the Wightman function and the Unruh effect
remain unchanged if the nonlocal model respects the Lorentz symmetry,
while they should be corrected when the
doubly special relativity is preferred
to make the minimal length an invariant scale.
Finally,
one might wonder
how the temperature (30) could be defined
even though the rate of the response function depends on the proper time.
In the local theory and the Lorentz-invariant nonlocal theory,
the rate of response functions were invariant under the time translation,
which means that
the number of quanta absorbed by the detector per unit proper time $\tau$ is constant,
so that the detector is in a global thermodynamic equilibrium with the scalar field Birrell:1982ix .
So, the temperature could be read off
from the rate of response function
by using the Planck distribution.
However,
in the Lorentz-violating nonlocal theory,
the rate of response function depends on the proper time,
which implies that the system is not in global thermodynamic equilibrium.
So, one should assume that the system is in a local thermodynamic equilibrium,
where the temperature varies with the spacetime
but the system is in equilibrium with the neighborhood for each point.
In the local equilibrium,
the scalar field is locally distributed according to the Planck distribution for a certain temperature near a given time when it is observed by the detector.
We could read off the temperature for the Lorentz-violating nonlocal model
approximately in order to figure out how much the thermal temperature deviates
from the standard one.
This issue deserves further attention in this direction.
Acknowledgements.We would like to thank Myungseok Eune and Sang-Heon Yi for exciting discussions.
W. Kim, Y. Gim, and H. Um were supported by
the National Research Foundation of Korea(NRF) grant funded by the
Korea government(MSIP) (2017R1A2B2006159).
W. Kim was in part supported by the Sogang University Research Grant (201819009.01).
References
(1)
G. V. Efimov, Non-local quantum theory of the scalar field,
Commun. Math. Phys. 5 (1967) 42–56.
(2)
G. V. Efimov, On a class of relativistic invariant distributions,
Commun. Math. Phys. 7 (1968) 138–151.
(3)
M. Z. Iofa and V. Ya. Fainberg, The wightman formulation for
nonlocalizable theories. 2.,
Teor. Mat. Fiz. 1
(1969) 187–199.
(4)
V. Ya. Fainberg and M. A. Solovev, How Can Local Properties Be
Described in Field Theories Without Strict Locality?,
Annals Phys.
113 (1978) 421–447.
(5)
V. Ya. Fainberg and M. A. Solovev, Nonlocalizability and
asymptotical commutativity,
Theor. Math. Phys. 93 (1992) 1438–1449, [hep-th/9211099].
(6)
E. T. Tomboulis, Superrenormalizable gauge and gravitational theories,
hep-th/9702146.
(7)
J. W. Moffat, Ultraviolet Complete Quantum Gravity,
Eur. Phys. J. Plus
126 (2011) 43, [1008.2482].
(8)
T. Biswas, E. Gerwick, T. Koivisto and A. Mazumdar, Towards singularity
and ghost free theories of gravity,
Phys. Rev.
Lett. 108 (2012) 031101, [1110.5249].
(9)
M. Isi, J. Mureika and P. Nicolini, Self-Completeness and the
Generalized Uncertainty Principle,
JHEP 11 (2013)
139, [1310.8153].
(10)
F. Briscese, L. Modesto and S. Tsujikawa, Super-renormalizable or finite
completion of the Starobinsky theory,
Phys. Rev. D89 (2014) 024029, [1308.1413].
(11)
G. Calcagni and L. Modesto, Nonlocal quantum gravity and M-theory,
Phys. Rev. D91 (2015) 124059, [1404.2137].
(12)
V. P. Frolov, Mass-gap for black hole formation in higher derivative and
ghost free gravity,
Phys. Rev.
Lett. 115 (2015) 051102, [1505.00492].
(13)
S. Deser and R. P. Woodard, Nonlocal Cosmology,
Phys. Rev.
Lett. 99 (2007) 111301, [0706.2151].
(14)
N. Barnaby and N. Kamran, Dynamics with infinitely many derivatives: The
Initial value problem,
JHEP 02
(2008) 008, [0709.3968].
(15)
N. Barnaby and J. M. Cline, Predictions for Nongaussianity from Nonlocal
Inflation,
JCAP 0806
(2008) 030, [0802.3218].
(16)
N. Barnaby, A New Formulation of the Initial Value Problem for Nonlocal
Theories,
Nucl. Phys.
B845 (2011) 1–29, [1005.2945].
(17)
S. Deser and R. P. Woodard, Observational Viability and Stability of
Nonlocal Cosmology,
JCAP 1311
(2013) 036, [1307.6639].
(18)
E. Witten, Noncommutative Geometry and String Field Theory,
Nucl. Phys. B268 (1986) 253–294.
(19)
L. Brekke, P. G. O. Freund, M. Olson and E. Witten, Nonarchimedean
String Dynamics,
Nucl. Phys. B302 (1988) 365–402.
(20)
D. A. Eliezer and R. P. Woodard, The Problem of Nonlocality in String
Theory, Nucl.
Phys. B325 (1989) 389.
(21)
D. Ghoshal and A. Sen, Tachyon condensation and brane descent relations
in p-adic string theory,
Nucl. Phys.
B584 (2000) 300–312, [hep-th/0003278].
(22)
N. Moeller and M. Schnabl, Tachyon condensation in open closed p adic
string theory,
JHEP 01
(2004) 011, [hep-th/0304213].
(23)
D. Ghoshal, p-adic string theories provide lattice discretization to the
ordinary string worldsheet,
Phys. Rev.
Lett. 97 (2006) 151601,
[hep-th/0606082].
(24)
E. Fuchs and M. Kroyter, Analytical Solutions of Open String Field
Theory, Phys.
Rept. 502 (2011) 89–149, [0807.4722].
(25)
G. Calcagni and L. Modesto, Nonlocality in string theory,
J. Phys. A47 (2014) 355402, [1310.4957].
(26)
P. Nicolini and M. Rinaldi, A Minimal length versus the Unruh effect,
Phys. Lett.
B695 (2011) 303–306, [0910.2860].
(27)
N. Kajuri, Unruh Effect in nonlocal field theories,
Phys. Rev. D95 (2017) 101701, [1704.03793].
(28)
L. Modesto, Y. S. Myung and S.-H. Yi, Universality of the Unruh
effect, Phys.
Rev. D97 (2018) 044016, [1710.04367].
(29)
G. Amelino-Camelia, Testable scenario for relativity with minimum
length, Phys.
Lett. B510 (2001) 255–263,
[hep-th/0012238].
(30)
G. Amelino-Camelia, Relativity in space-times with short distance
structure governed by an observer independent (Planckian) length scale,
Int. J. Mod. Phys.
D11 (2002) 35–60, [gr-qc/0012051].
(31)
J. Magueijo and L. Smolin, Lorentz invariance with an invariant energy
scale, Phys.
Rev. Lett. 88 (2002) 190403,
[hep-th/0112090].
(32)
J. Magueijo and L. Smolin, Generalized Lorentz invariance with an
invariant energy scale,
Phys. Rev. D67 (2003) 044017, [gr-qc/0207085].
(33)
D. Kimberly, J. Magueijo and J. Medeiros, Nonlinear relativity in
position space,
Phys. Rev. D70 (2004) 084007, [gr-qc/0303067].
(34)
W. G. Unruh and R. M. Wald, What happens when an accelerating observer
detects a Rindler particle,
Phys. Rev. D29
(1984) 1047–1056.
(35)
M. Rinaldi, Superluminal dispersion relations and the Unruh effect,
Phys. Rev. D77 (2008) 124029, [0802.0618].
(36)
S. Gutti, S. Kulkarni and L. Sriramkumar, Modified dispersion relations
and the response of the rotating Unruh-DeWitt detector,
Phys. Rev. D83 (2011) 064011, [1005.1807].
(37)
E. Martin-Martinez, M. Montero and M. del Rey, Wavepacket detection with
the Unruh-DeWitt model,
Phys. Rev. D87 (2013) 064038, [1207.3248].
(38)
E. Harikumar, A. K. Kapoor and R. Verma, Uniformly accelerating observer
in $\kappa$-deformed space-time,
Phys. Rev. D86 (2012) 045022, [1206.6179].
(39)
E. Harikumar and R. Verma, Uniformly accelerated detector in the
$\kappa$-deformed Dirac vacuum,
Mod. Phys. Lett.
A28 (2013) 1350063, [1211.4304].
(40)
B. R. Majhi and E. C. Vagenas, Modified Dispersion Relation, Photon’s
Velocity, and Unruh Effect,
Phys. Lett.
B725 (2013) 477–480, [1307.4195].
(41)
K. Ghoroku and A. L. Larsen, Hawking temperature and string scattering
off the (2+1) black hole,
Phys. Lett. B328 (1994) 28–35, [hep-th/9403008].
(42)
M. Natsuume, N. Sakai and M. Sato, String scattering off the
(2+1)-dimensional rotating black hole,
Mod. Phys. Lett.
A11 (1996) 1467–1474,
[hep-th/9603173].
(43)
W. T. Kim, J. J. Oh and K. H. Yee, Scattering amplitudes and thermal
temperatures of the Schwarzschild-de Sitter black holes,
Phys. Rev. D66 (2002) 044017, [hep-th/0201117].
(44)
G. Clement, J. C. Fabris and G. T. Marques, Hawking radiation of linear
dilaton black holes,
Phys. Lett.
B651 (2007) 54–57, [0704.0399].
(45)
N. D. Birrell and P. C. W. Davies, Quantum Fields in Curved Space.
Cambridge Monographs on Mathematical Physics. Cambridge Univ. Press,
Cambridge, UK, 1984. |
Water at an electrochemical interface - a simulation study
Adam P. Willard${}^{2}$
Stewart K. Reed${}^{1}$
Paul A. Madden${}^{1}$
David Chandler${}^{2}$
${}^{1}$School of Chemistry, University of Edinburgh, Edinburgh EH9 3JJ, UK
${}^{2}$Department of Chemistry, University of California, Berkeley,California 94720
(November 21, 2020)
Abstract
The results of molecular dynamics simulations of the properties of water in an
aqueous ionic solution close to an interface with a model metallic electrode are
described. In the simulations the electrode behaves as an ideally polarizable
hydrophilic metal, supporting image charge interactions with charged species, and
it is maintained at a constant electrical potential with respect to the solution so
that the model is a textbook representation of an electrochemical interface through
which no current is passing. We show how water is strongly attracted to and ordered
at the electrode surface. This ordering is different to the structure that might
be imagined from continuum models of electrode interfaces. Further, this ordering
significantly affects the probability of ions reaching the surface. We describe
the concomitant motion and configurations of the water and ions as functions of the
electrode potential, and we analyze the length scales over which ionic atmospheres
fluctuate. The statistics of these fluctuations depend upon surface structure and
ionic strength. The fluctuations are large, sufficiently so that the mean ionic
atmosphere is a poor descriptor of the aqueous environment near a metal surface.
The importance of this finding for a description of electrochemical reactions is
examined by calculating, directly from the simulation, Marcus free energy profiles
for transfer of charge between the electrode and a redox species in the solution
and comparing the results with the predictions of continuum theories. Significant
departures from the electrochemical textbook descriptions of the phenomenon are
found and their physical origins are characterized from the atomistic perspective
of the simulations.
I Introduction
The altered solvation, dielectric and dynamical properties of water molecules close
to electrode surfaces have an important influence on electrochemical reactions.
There have been numerous simulation studies of aqueous solutions close to charged
solid surfaces which have cast light on the ordering of water molecules by the
solid surface and begun to make the connection between the layers with altered
dielectric characteristics invoked in continuum models of the electrode capacitance
and the ordered molecular films which are known from surface science. References
neurock contain an excellent summary of work to date, and
neurock1 ; spohr a more longstanding review. To complete the link between
molecular behaviour and electrochemical observations we need a realistic
representation of the electrochemical interface and a direct way of calculating the
electrochemical observable, namely the dependence of the rate of the
electrochemical electron transfer on the potential applied to the electrode
bockris .
What is required to achieve the second of these objectives is suggested by the
Marcus theory of electron transfer marcus1 ; marcus2 . In a Marcus description
of the oxidation of some solution species R to an oxidised species O by transfer of
an electron to an electrode maintained at some potential $V$ with respect to the
solution we construct curves, as illustrated schematically in figure
1, which describe how the free energies of O and R depend upon
some reaction coordinate, which is envisaged as reflecting the influence of
fluctuating solvent degrees of freedom. The Marcus expression for the rate of
electron transfer can be calculated from the probability that the system will
access the configuration where the two curves cross. Note that the free energy
curve for O includes the potential energy of the electron on the electrode ($eV$),
so that the position and height of the crossing point depend on the electrode
potential. The two curves are coupled by a term ($\gamma$) which reflects the
tunneling of the electron between the redox centre and the electrode, and this is
expected to depend exponentially on its distance from the electrode surface. We
should, therefore, be thinking about the dependence of the Marcus curves on the
proximity to the electrode surface, to which two factors contribute. Firstly, the
difference between the direct interactions of the O and R species
with the charged surface itself will produce a differential shift on them, and
therefore affect the crossing point. Secondly, if the redox species is close to the
electrode, the competing interactions of the water molecules in its coordination
shell with the surface and the solute itself may result in a change in the
character of the fluctuations of the reaction coordinate. Both of these factors may
be affected by the potential applied to the electrode. The Marcus curves contain
the information which is required to calculate the electron transfer rate for a
redox ion at a given distance from the electrode surface, but to complete the
calculation of the rate we also need to know the probability of the reactant
reaching this position, and this too will be affected by the potential exerted by
the electrode and its influence on the solvating properties of the water molecules.
Blumberger, Sprik and co-workers sprik1 ; sprik2 have demonstrated how the
Marcus curves can be calculated for a homogeneous electron transfer reaction within
an ab initio molecular dynamics scheme. Following Warshel warshel they
emphasize the advantages for computation of choosing the “vertical energy-gap” as
the reaction coordinate. The vertical energy gap for oxidation is calculated for a
single configuration in a simulation by switching the identity (and all associated
interaction parameters) of a redox species initially in its reduced form R to its
oxidised form without allowing any changes in the nuclear coordinates (hence a
“vertical” transition in the Franck-Condon sense) and evaluating the energy
difference between the final and initial states. The Marcus curves for the oxidised
and reduced species may then be estimated from the probability distributions of the
vertical energy gaps obtained by repeatedly sampling through the course of an
molecular dynamics (MD) simulation and assuming this distribution is Gaussian. The
Gaussian assumption is required by this method because regions of the distribution
pertinent to the charge transfer reaction are not generally accessed in a
straightforward simulation. In the calculations reported here, the Gaussian
approximation is tested and shown to be accurate. More generally, the Gaussian
approximation has been tested and found to be accurate chandler1 ; chandler2
provided proper account is taken of molecular boundary conditions chandler3
Use of this scheme to study the electrochemical
electron transfer process in a simulation is illustrated in figure 2.
An aqueous solution is contained between two crystalline arrays of atoms which
comprise the (metallic) electrodes, these are maintained at a definite electrical
potential. The solution contains the redox species in its reduced and oxidised
forms and, periodically during the simulation, an ion (the “redox target”) is
selected and its redox state is switched and the energy difference between final
and initial states is evaluated. By selecting ions at different distances from the
electrodes and by examining how the vertical energy gaps depend on the applied
potential we can build up the necessary information to study how the nature of the
water at the electrode surface affects the electrochemical electron transfer rate
via the Marcus construction.
In order to make contact with experimental studies the calculation needs to be done
with as realistic representation of the constant-potential electrode and
interfacial water as can be managed. Ideally, it would be done within an ab initio MD
scheme, as this would enable the difficult-to-characterise interactions
between the solution and the electrode to be modelled without the introduction of
interaction potentials neurock ; halley ; voth . However, the time and length
scales involved in the relaxation of the solution in the vicinity of the electrodes
(which we will characterise below) are far too long to allow a full self-consistent
description of the screening of the electrode potential within an ab initio
scheme halley-acs . Recently we introduced a way of incorporating some of the
essential physical effects necessary for a realistic description of the interfacial
charge-transfer process into a simulation which uses interaction potentials and
therefore enables simulations of much larger time and length scales than is
possible ab initio reed1 . In particular, model metallic
electrodes maintained at a constant electrical potential may be introduced
into such a simulation, following a technique introduced by Siepmann and Sprik
siepmann . Because the electrodes behave as ideally polarizable metals they
support image-charge interactions between charged species and the electrode; these,
as we shall see, have an important influence on the electron transfer process.
Because the electrodes are maintained at a constant potential when the charge of
the redox species is changed to sample the vertical energy gap, that charge is
transferred in full to the electrodes, so that the source of the dependence of the
electron-transfer rate on the electrode potential is included in the calculation.
The electrode potential and the potential felt by the molecules and ions in the
solution region are calculated self-consistently. Calculations using these methods
have already been performed to examine the Marcus curves in simulations of redox
active molten salts reed2 .
We begin with a brief description of the methods and interaction potentials used to
simulate pure water and aqueous solutions of LiCl and the Ru${}^{2+}$/Ru${}^{3+}$
couple confined between model platinum electrodes. We then examine the structure
and dynamical properties of the electrode-adsorbed water and the way they are
affected by the application of a potential to the electrode. In sections
III and IV we consider the consequences of this adsorbed water for
the approach of ions to the electrode surface and the effect of the adsorbed water
and the ionic atmosphere for the electrical potential in the vicinity of the
electrode. In classical models of electrochemical charge transfer this potential is
invoked to represent the dependence of the energies of the oxidised and reduced
species on the proximity to the electrode. We then present preliminary results for
the Marcus curves for the Ru${}^{2+}$/Ru${}^{3+}$ system and discuss the physical
factors which determine their dependence on the applied potential and the proximity
to the electrode.
II Scope of the model.
The electrochemical interface is affected by many phenomena and a comprehensive
representation of all of them within a single simulation is beyond current
capabilities. Our focus here is on the solution side of the interface, on the
properties of the water molecules at the interface and their influence on the
electrical potential. As such we will present a significantly simplified model of
the electrode itself in which we ignore the motion of the electrode atoms and
therefore neglect effects like the restructuring of the electrode surface under
chemical or electrical influences kolb . Furthermore the representation of
the electrode as a metal is a simplified one, designed to capture the correct
macroscopic response to an electrical potential appropriate to a metal rather than
to deal with a correct microscopic description of surface electronic states etc.
As illustrated in figures 2 and 3, the electrodes each
consist of three layers of atoms arranged in an fcc lattice with the 100 face
exposed to the solution; the lattice parameter is appropriate to Pt. Following
Siepmann and Sprik siepmann each electrode atom $i$ carries a Gaussian
charge distribution of fixed width but variable amplitude ($q_{i}$). These charges
are coulombically coupled to all other charges in the system. They are treated as
additional dynamical degrees of freedom whose values are adjusted at each timestep
in the molecular dynamic procedure in order to variationally minimise an
appropriate energy functional. The energy functional is chosen siepmann so
that at its minimum the electrical potential on every electrode atom is the same
(as approriate to a metal) and equal to some pre-set value $V_{0}$. The use of a
variational principle allows forces and response behaviour to be calculated
straightforwardly via an application of the Hellmann-Feynman Theorem, as used to
good effect in ab initio MD simulations. The methodology for
the simulation of the electrodes is described in great detail in ref reed1 ,
where it is shown that the electrodes become polarised in the presence of a charge
in the solution region in a way which corresponds to the classical image-charge
response. We illustrate this response in figure 3 where the charge
induced on the electrode atoms by the instantaneous configuration of the charges in
solution is shown by colour-coding the electrode atoms. The bright blue region, for
example, is caused by the presence of an anion within the first molecular layer of
the solution close to this position. Furthermore, if the charge on one of the ions
in solution is changed, the charge difference is fully transferred to the
electrodes to maintain charge neutrality with a constant electrode potential
reed2 : it is this feature which enables us to calculate Marcus curves for
electrochemical charge transfer.
To describe the interactions between water molecules we use the SPC/E potential
SPCE , which is known to give a dielectric constant for water close to the
experimental value. The interactions of water molecules with a metallic electrode
are complex, and cannot be modelled accurately with a simple two-body potential;
since the behaviour of water at the interface is the central purpose of our study
we were concerned to represent this interaction as carefully as is possible through
the introduction of a potential. Experiment thiel and ab-initio studies
holloway have shown that water molecules interact with a crystalline
platinum surface by adsorbing on top sites and orienting their dipole along the
plane of the electrode. In their study of water at an STM tip, Siepmann and Sprik
siepmann parameterized a two- and three-body potential to describe the
adsorption of water molecules on a platinum surface; their potentials are
particularly appropriate for our study since they do not include the consequences
of image interactions, which are dealt with through the polarizable electrode model
as in our simulations. We have used these potentials exactly as described in their
paper.
We have not attempted the same level of realism with the interactions between the
ions and the electrode surface. Experimental studies show that anions interact
quite strongly with transition metal surfaces to the extent that complete surface
coverage of ordered layers of Cl${}^{-}$ is observed on positively charged
single-crystal electrodes above about 0.5 V from molar solutions magnussen .
Guymon et al have shown how suitable potentials to describe these strong
interactions could be obtained from ab initio calculations guymon .
However, adsorption of this strength would present a significant problem for our
simulations since it would mean that the solution region would be strongly depleted
in Cl${}^{-}$ ions. To represent the interface under these conditions we would need to
equilibrate our system in the presence of a reservoir of electrolyte; furthermore
equilibrating this system would be very slow, as we shall see. We have,
therefore, for the present study introduced only weakly attractive interaction
potentials between the ions and the electrode surface – we use exponential-6
potentials to represent the short-range interactions with the parameters chosen as
if the atoms of the metallic walls were themselves Cl${}^{-}$ ions. These potentials
are too weakly attractive, compared to the water-electrode interactions, to lead to
the kind of anion adsorption phenomena seen in the experimental studies of
Cl${}^{-}$-containing electrolytes. We note that fluoride ions are not thought to form
adsorbed layers magnussen and so the picture of the interface we present may
be more representative of a fluoride than a chloride-containing solution.
The interactions between the other species present in solution were modelled with
pair potentials; this too reflects a compromise in the realism of the calculations
as it is known that polarisation effects have a significant influence on the way
that ions interact and coordinate water polarizable-ion-water . The water-ion
interactions were modelled with a Lennard-Jones potential acting between the ion and
the oxygen center of the water molecule. The parameters used in these interactions
were adapted from Lynden-Bell bell1 . The ruthenium ion - water interactions
were parameterized with a purely repulsive potential,
$$U_{\mathrm{Ru},\mathrm{O}}=A/|\textbf{r}_{\mathrm{Ru}}-\textbf{r}_{\mathrm{O}}%
|^{9},$$
(1)
with the parameter ($A=49977.9k\mathrm{Jmol^{-1}\AA^{9}}$ for both Ru${}^{2+}$ and
Ru${}^{3+}$) chosen so that the first peaks of the ion-water radial distribution
agreed with those obtained in an ab initio MD study sprik_ru . The ions
interact with each other with exponential-6 potentials with the Ru-Cl potentials
taken from lanthanides of corresponding ionic size morgan ; hutchinson .
III Pure water results
We begin by showing, in figure 4, results obtained for the
profiles across the cell of the mean electrical (or “Poisson”) potential, $\Psi$,
in pure water. This is obtained by integrating Poisson’s equation,
$$\nabla^{2}\Psi=-\frac{\rho}{\varepsilon_{0}}\;,$$
(2)
with the mean charge density, $\rho$, calculated from the simulation for different
values of the applied electrode potential, $V_{0}$, as the source term
($\varepsilon_{0}$ is the permittivity of free space). The Poisson potential is the
potential used in describing the potential at the electrochemical interface in
classical theories and is therefore an important point of contact between our
calculations and textbook descriptions of the electrochemical interface
bockris ; kornyshev . The potential is constant on the interior of the
electrodes and equal to the applied potential. It then drops rapidly and oscillates
across an interfacial region about 12 Å wide, for reasons we will discuss in
detail below, before settling down to acquire the constant slope appropriate to the
behaviour of the potential in a bulk dielectric subject to an external potential.
Notice that, other than at $V_{0}=0$, the potential drops across the two interfaces
are not symmetrical because of the different microscopic arrangements of the water
molecules at positively and negatively charged surfaces.
We can calculate a value for the dielectric constant of water from the behaviour of
the potential across the bulk region. By integrating the mean charge in the region
of the cell to the left of 20 Å we can obtain a value for the charge, Q, on one
plate of a virtual parallel plate capacitor placed at this position; the region to
the right of 61.5 Å has an equal and opposite charge and can be regarded as the
other plate. The potential drop between the two plates is $\Delta\Psi$ and we can
obtain values for the capacitance $C$ from $Q=C\Delta\Psi$ for the different
applied potentials. This calculated capacitance $C$ can be compared with
theoretical expression for capacitance of parallel plates filed with a medium of
dielectric constant $\varepsilon$,
$$C=\varepsilon\varepsilon_{0}A/d,$$
(3)
where $A$ is the cross-sectional area of the cell and $d$ the distance between the
plates. The calculation shows that the dielectric constant for SPC/E water depends
on the applied electrode potential. Specifically, when the electrode potential has
the values $V_{0}=0.27$ V, $V_{0}=1.36$ V, and $V_{0}=2.72$ V, the dielectric
constant for the bulk water is $\epsilon=75.07$, $\epsilon=61.50$, and
$\epsilon=57.30$ respectively. The low voltage result is in reasonable agreement
with direct simulation studies (68$\pm$5.8 steinhauser ), which is good
confirmation that the potential and the response of the water molecules to it are
correct (see also reference reed1 ). The reduction in the apparent
dielectric constant at higher voltages is consistent with a saturation effect
berkowitz , note that the potential difference of $\Delta\Psi\simeq$ 1.1 V,
obtained with $V_{0}$=2.72 V, across our virtual capacitor of width 41.5 Å is
equivalent to an electric field of $\simeq 2.6\times 10^{8}$ Vm${}^{-1}$.
The rapid oscillation of the potential close to the interfaces is due to the strong
adsorption of a layer of water molecules at the electrode surfaces. This is
illustrated in figure 5. The oxygen atoms of the water molecules
form a commensurate layer on the top-sites of the underlying 100 fcc surface. With
zero applied potential the water molecules lie with at least one O-H bond in the
plane of the interface with the H-atom pointing towards a neighbouring oxygen and
form ordered domains which reorient on a long timescale. Although the predominant
orientation is in-plane, at zero applied potential there is a net orientation of
the negative ends of the molecular dipoles towards the surface and this induces a
small positive charge on the electrode atoms berkowitz .
When the potential is applied to the cell, the water molecules in the first layer
partially reorient in the interfacial field. This is illustrated in the right-most
panel of figure 5 where one of the OH bonds of the water molecules
at the negatively charged electrode may point towards the electrode surface (note
that this potential is very large for aqueous electrochemistry). The consequences
of this for the Poisson potential can be see by reference to the right-hand
interface in figure 4. Whereas at $V_{0}=0$ the potential initially
drops on moving into the electrolyte, consistent with the negatively charged oxide
ions being closest to the surface, at the applied potential of $V_{0}=-2.72$ V, the
potential now rises showing an excess of positive charge lying close to the
surface.
We can track these changes by showing the probability distributions of the
orientation of O-H bonds in the first and second layers of water molecules as the
applied potential is changed. We compute the probability distribution function
$P[\cos(\theta)]$, where $\theta$ is the angle between an OH bond vector (the
vector extending from the oxygen centre of a water molecule to the centre of one of
the associated hydrogen atoms) and the the outward normal vector to the electrode
surface. Fig 6 shows the distribution $P[\cos(\theta)]$ for
different values of applied potential for the two electrodes. The left-hand panel of
figure 6 shows the results at the positively charged electrode
and the right-hand panel at the negatively charged one. At all values of the potential, the
probability distribution is peaked around $\cos(\theta)=0$, corresponding to
configurations for which the OH vector of a water molecule is aligned with the
plane of the electrode surface. At zero applied potential the distribution
$P[\cos(\theta)]$ for the adsorbed molecules is the same for the two electrodes,
but even at $V_{0}=0$ there is an excess of outward (negative $cos(\theta)$) over
inward pointing OH bonds, which give rise to the potential drop between the
electrode and solution noted above. At different values of applied potential the
distribution of OH vectors is changed significantly. At the positive electrode
(left panel of Fig. 6) the main effect of the electrode potential
is to deplete the population of OH bonds pointing into the electrode ($\cos(\theta)<0$). At the negative electrode (right panel of Fig. 6) however,
at increased electrode potential there emerges a large population of OH vectors
which point into the electrode. This change in orientational structure at the
negative electrode is yet another demonstration of the asymmetry between the
solvent structure at the positive and negative electrode.
The orientation of the adsorbed water molecules influences the charge induced on
the electrode atoms on which they sit. Figure 7 shows the
distributions of the charges induced on the atoms which make up the outermost layer
of atoms on the negatively charged electrode. At zero applied potential the average
charge is small and positive, for the reasons discussed above, but there is a
significant number of negatively charged atoms which are associated with an
adsorbed water molecule with an inward-pointing O-H bond. As the electrode
potential becomes increasingly negative, so does the mean charge, but the bimodal
character of the distribution becomes even more pronounced as more water molecules
flip an O-H bond towards the electrode.
One question which arises from these results is the extent to which they are
influenced by the inclusion of image charge interactions in the potential model. A
way of addressing this question is to carry out constant charge simulations in
which the values of the charges on the electrode atoms are fixed at the values of
the average charges on the first, second and third layers of the electrode atoms
obtained in constant potential runs with an electrode potential $V_{0}$. These static
charge distributions generate similar electrode potentials to the $V_{0}$ values used
in the constant potential runs used to generate them. It can be seen, from the
right-hand panel of figure 7, that the dynamical nature of the
charges in the constant potential simulations has only a small effect on the mean
orientational distributions of the water molecules in the adsorbed layer and none
on the second layer. However, as illustrated in figure 7
left-hand panel, there is a local response of the electrode in the constant
potential simulations and this is responsible for the emergence of a significant
population of electrode-pointing OH bonds in the directly adsorbed molecules; it is
not observed in the simulations run at constant charge. The relatively small effect
of the image charges on average interfacial structure parallels the findings
in the molten salt simulations reed1 .
The layering of solvent and the molecular orientations within the layers adjacent
to the electrode affect the capacitance of the electrode, which can be measured
experimentally. The capacitance of the first two layers of solvent can be
calculated through the differential capacitance, $C=(\partial q_{m}/\partial\Delta\Psi)$, where $q_{m}$ is the charge density on the electrode and $\Delta\Psi$ is the
potential drop across the first two layers of water. Figure 8
shows the dependence of $\Delta\Psi$ on $q_{m}$ for several values of the applied
potential. The plot reveals that the potential of zero charge (pzc) for our
simulated system is at -0.8 V. Comparing this quantity with experiment is not
straightforward, since in the experimental measurements the potential is quoted
with respect to a reference electrode whereas we can access directly the potential
difference between the interior of the electrode and the solution. The experimental
value with respect to a standard hydrogen electrode is +0.41 V bockris . The
capacitance for electrode potentials which are on the positive side of the
potential of zero charge ($C=8.39\mu\mathrm{F/cm^{2}}$) is larger than for negative
potentials, $C=5.20\mu\mathrm{F/cm^{2}}$. Both values are considerably lower than
predicted through experimental data which measures the the double layer capacitance
in the range of $C=20-50\mu\mathrm{F/cm^{2}}$ kolb ; parsons close to the
pzc. The substantial difference between our calculated value and experiment is not
surprising as our model does not include a realistic description of the electron
density at the metallic surface; the surface dipole potential arises from the
extension of the metal electrons into the interface beyond the nuclei and makes a
large contribution to the capacitance parsons . This effect can be included
in a jellium model for the metal, as included in the theory of Schmickler and
Henderson schmickler .
IV Results for electrolyte solutions
In figure 9 we show the Poisson potential for an approximately one
molar solution of LiCl. In contrast to the pure water case (figure
4) in the bulk region away from the interfaces, the Poisson
potential is now constant as a consequence of the screening by the ions present in
the solution. The oscillations in the potential across the interfacial region
closely resemble those in pure water at the same values of the applied potential.
That the Poisson potential is exhibiting perfect screening is quite surprising as
the profiles of the ion density obtained by averaging over the whole simulation
runs are manifestly not well equilibrated (see figure 10). Even
at $V_{0}=0$ we see an excess of ions at the left-hand side of the cell, whereas the
equilibrium ion density profile should be constant, except close to the electrodes.
It would appear that efficient screening can be caused by an appropriate
local arrangement of cations and anions, relaxation of the whole ion density
profile is not necessary.
The failure to reach a fully equilibrated ion density profile arises primarily
because of the slow rate of relaxation of the concentration by diffusion (and,
perhaps, an inappropriate initialisation of the ion positions in the simulations).
The rate should depend on the diffusion coefficient divided by the square of the
distance between the electrodes, and because we have used a large cell in the hope
of seeing the interfaces well separated by bulk, the relaxation times have become
extremely long. There is a second slow relaxation process, however. Examination of
the Li${}^{+}$ profile close to the left-hand (positively charged) electrode shows a
sharp peak in the region associated with the adsorbed layer of water. This peak
arises from the presence of a single cation ion in this layer throughout the
$V_{0}=0$ and $V_{0}=0.27$ V runs; it was placed there in the initial configuration and
remained until the electrode potential was increased to 1.36 V. That this process
is so slow is because exchange of an ion between the strongly adsorbed layer of
water and the bulk is very slow, effectively because the ion cannot carry its
coordinating water molecules between the two regions.
We can examine the barrier which arises to prevent exchange between the adsorbed
layer and the bulk by umbrella sampling techniques. We compute the mean force,
$F(z)$, in the $z$ direction perpendicular to the electrode on an atom $i$ in the
simulation by constraining the atom $i$ at some position $z_{0}$ in a harmonic
potential $U_{z_{0}}(z)={\frac{k}{2}}(z-z_{0})^{2}$, where $k$ is the force constant
of the harmonic well, taken to be $100~{}k_{\mathrm{B}}T\mathrm{\AA}^{-2}$. The mean force on
atom $i$ at position $z_{0}$ can be estimated as $F(z_{0})=-k(\bar{z}-z_{0})$ where
$\bar{z}$ is the average value of $z$ for species $i$ constrained to $U_{z_{0}}(z)$
during a simulation. We performed the mean force calculations on a $\mathrm{Li^{+}}$
ion and an oxygen centre of a water molecule in a LiCl(aq) solvent at zero applied
potential ($V_{0}=0.00$). In addition we computed $F(z)$ for the oxygen centre in a
pure water with electrode potentials $V_{0}=\pm 2.72$ V. This method for generating
the mean force (and subsequently the potential of mean force (PMF) by integration)
can be sensitive to the set of initial conditions blue-moon . One set of
initial conditions, which corresponds to electrode desorption, was initiated by
choosing an already adsorbed species setting $z_{0}$ at the adsorption distance and
equilibrating with $U_{z_{0}}(z)$ for 1 picosecond. The next member of this set of
initial conditions was created by setting $z_{0}\rightarrow z_{0}+0.26\mathrm{\AA}$
and again equilibrating with $U_{z_{0}}(z)$ for 1 picoseconds. This process is
continued, in increments of $0.26~{}\mathrm{\AA}$ for approximately $5~{}\mathrm{\AA}$. Another set of initial conditions, corresponding to electrode
adsorption were generated in an analogous fashion by selecting an atom in the bulk
and moving $z_{0}$ towards the electrode in $0.26~{}\mathrm{\AA}$ steps. The mean
force was computed by averaging $\bar{z}$ over a 20 picosecond trajectory.
The potentials of mean force obtained by integration over $F(z)$ show a large
degree of hysteresis, which often arises when there is a free energy barrier in
the chosen coordinate ($z$) frustrating equilibration on short timescales. In
other words, the reaction mechanism for electrode adsorption is not correctly
characterized simply by a species distance from the electrode. For the atom to
move into the adlayer it is necessary for an already adsorbed water molecule to
vacate an adsorption site, thus we might expect that a more suitable reaction
coordinate would describe the collective rearrangement which this entails.
Nonetheless, the calculated potential of mean force curves are informative.
If we focus firstly on the adsorption PMF for Li${}^{+}$, we see that there is a
substantial barrier at about 5 Å for the movement of the ion from the bulk into
a relatively stable position where the cation sits between the first and second
adlayers located about 4.4 Å from the electrode surface; this position is
illustrated at the left-hand electrode in figure 2. This barrier
arises from the reorganisation of the solvation shell of the Li${}^{+}$ ion which is
necessary for it to be accommodated in this layer. Note that a similar barrier
appears in the desorption pathway. There is then a second barrier before the ion
is adsorbed at the electrode surface. In this region the hysteresis in the two
curves is pronounced. On the adsorption pathway the ion must force an already
adsorbed water molecule out of the way, so the energy increases steeply. On
desorption from the adlayer there is also a large energy increase as the process
leaves an empty adsorption site on the electrode surface.
The PMF for water shows no barrier for exchange of water molecules between the bulk
and the second adlayer. A large degree of hysteresis then sets in, associated with
the replacement of an already adsorbed water molecule by a molecule from the bulk.
The barrier to desorption from the first adlayer suggested by these data is of the
order of 10 $k_{\mathrm{B}}T$, sufficient to lead to very slow exchange of the adsorbed water
and the bulk.
V Calculation of the Marcus curves for electron transfer.
In order to examine how the electrical potential and the water structure in the
interfacial region affect the rate constants for electrochemical charge transfer we
have followed the scheme illustrated in figure 2 for the aqueous
Ru${}^{2+}$/Ru${}^{3+}$ couple close to the model metallic electrode. Similar
calculations have been reported recently for a redox-active molten salt system
reed2 , where the problems caused by the very slow equilibration of the
concentration profiles we have noted above are not so marked and where the
statistical precision necessary to validate the calculations was relatively easily
obtained. We refer the reader to that paper for full details of the calculation and
merely recapitulate some essential details here.
We calculate the probability distribution functions for the vertical transition
energy between the two redox states, $\delta E_{{\rm Ru}^{2+}\to{\rm Ru}^{3+}}$
for oxidation and $\delta E_{{\rm Ru}^{3+}\to{\rm Ru}^{2+}}$ for reduction. The
vertical transition consists of changing the identity (i.e. charge and
interaction potentials) of a single ion at some configuration along an MD
trajectory with the electrode potentials set at some value $V_{0}$ and, without
changing the atomic positions (as befits the vertical or diabatic nature of the
Marcus curves), relaxing the electrode charges. The vertical transition energy is
the difference in the total interaction energy between the initial and final
states. There is a constant term in this energy gap that depends upon the metal of
which the electrode is made (through its work function) and reacting ion (through
the gas-phase ionisation energy (Ru${}^{2+}~{}\to$ Ru${}^{3+}$+e${}^{-}$), but independent of
the electrolyte solution. We have arbitrarily set the value of this constant to
make the mean energy gap of $\mathrm{Ru^{2+}}$ to be the negative of the that for
$\mathrm{Ru^{3+}}$ when $V_{0}=0$; the consequences of this will be illustrated
below.
In both oxidation and reduction, as discussed in detail in reference
reed2 a balancing charge is transferred to the electrodes and the energy of
this, which depends on the potential applied to the electrodes, is included in the
vertical transition energy footnote . We then return the identity of the ion
to its initial value and continue the MD trajectory. By repeatedly sampling these
transition processes for redox species found at a given distance from the
electrodes we may build up probability distributions for the energy gaps, $P_{{\rm Ru}^{2+}}(\delta E_{{\rm Ru}^{2+}\to{\rm Ru}^{3+}})$ and $P_{{\rm Ru}^{3+}}(\delta E_{{\rm Ru}^{3+}\to{\rm Ru}^{2+}})$, at different positions in
the cell. Examples of the probability distributions are shown in figure
12. They are calculated for a sample of ions located in the middle
of the simulation cell, and compared to those of ions adjacent to the electrode.
The mean positions of the distributions and their widths are found to depend quite
strongly on the position of the redox ion in the cell, as we will discuss below.
The distributions are found to be rather accurately gaussian, which is the
expectation from Marcus theory if the surrounding medium responds linearly to the
change in the identity of the redox species. Our potentials describing the
interaction of the Ru${}^{2+}$ and Ru${}^{3+}$ with water were chosen so that both
cations had similar coordination shells and, as previous studies of redox processes
in the bulk have shown benjamin ; sprik2 ; sprik_ru , under these conditions it
is likely that the linear response limit is applicable. Our data seems to be
consistent with linear response even when we consider the redox process for ions
close to the electrode surface, despite the strength of the interactions and the
restricted nature of the water molecules in the first adsorbed layer.
Following Sprik and co-workers sprik1 ; sprik2 , and making use of the special
properties of the mean vertical gap for oxidation ($\Delta E=\delta E_{{\rm Ru}^{2+}\to{\rm Ru}^{3+}}=-\delta E_{{\rm Ru}^{3+}\to{%
\rm Ru}^{2+}}$) as a
reaction coordinate marcus2 ; warshel , we may evaluate the free energies of
the Ru${}^{2+}$ and Ru${}^{3+}$ ions along this reaction coordinate from the
probability distributions
$$A_{{\rm Ru}^{2+}}(\Delta E)=-k_{B}T\ln P_{{\rm Ru}^{2+}}(\Delta E=\delta E_{{%
\rm Ru}^{2+}\to{\rm Ru}^{3+}})+{\bar{A}_{{\rm Ru}^{2+}}}$$
(4)
and
$$A_{{\rm Ru}^{3+}}(\Delta E)=-k_{B}T\ln P_{{\rm Ru}^{3+}}(\Delta E=-\delta E_{{%
\rm Ru}^{3+}\to{\rm Ru}^{2+}})+{\bar{A}_{{\rm Ru}^{3+}}},$$
(5)
where ${\bar{A}_{{\rm Ru}^{2+}}}$ corresponds to the free-energy at the minimum of
the Ru${}^{2+}$ curve. Furthermore, when the vertical energy gap is taken as the
reaction coordinate the two free energy curves are linearly dependent, i.e. tachiya ; sprik2
$$A_{{\rm Ru}^{3+}}(\Delta E)-A_{{\rm Ru}^{2+}}(\Delta E)=\Delta E.$$
(6)
This apparently simple relationship is remarkably powerful; it means that we can
establish a relationship between the origins of the two curves (${\bar{A}}_{{\rm Ru}^{3+}}$ and ${\bar{A}}_{{\rm Ru}^{2+}}$) and also sample the free energy
surfaces for values of the reaction coordinate which are well away from the most
stable configurations simply by calculating the energy gap in the free-running
simulation. The ability to sample the curves away from their minima means that we
can obtain information on the Marcus curves in the vicinity of their crossing
point, which is the region which determines the kinetics of the electron transfer
event.
The data points obtained from (4)- (6) are plotted in
figure 13 for $V_{0}=0$. Note that our choice of the arbitrary
energy added to the gap to represent the work function and ionisation energy has
resulted in only a small difference between the mean free energies of the oxidised
and reduced forms for the mid-cell position: experimentally, the reduction
potential for this couple is 0.249 V with respect to the standard hydrogen
electrode, so the relative positions of the minima in the curves should be similar
to reality and the electron transfer in the “normal” Marcus régime at $V_{0}=0$.
If the probability distributions really are Gaussian, equations 4 and
5 show that the Marcus free-energy curves will be harmonic about the
mean values of the reaction coordinate for the oxidation and reduction processes,
i.e. the peak positions of the respective probability distributions $\langle\Delta E_{1}\rangle$ and $\langle\Delta E_{2}\rangle$, respectively. It was shown by
Tachiya tachiya that under this Gaussian assumption all properties of the
Marcus curves can be predicted simply from a knowledge of $\langle\Delta E_{1}\rangle$ and $\langle\Delta E_{2}\rangle$; the necessary manipulations are
described in the previous paper reed2 ; sprik2 . These predicted curves are
shown by solid lines in figure 13 and are seen to provide an
accurate representation of the data. This applies both for the data obtained for
redox ions close to the centre of the cell and also close to the electrode
surfaces, despite the fact that the values of $\langle\Delta E_{1}\rangle$ and
$\langle\Delta E_{2}\rangle$ themselves depend quite strongly upon the distance
from the electrode. The position-dependence of the widths of the probability
distributions which we noted in discussing figure 12 is therefore
seen to be contained within the Gaussian description of the fluctuations in the
reaction coordinate and related to the position-dependence of $\langle\Delta E_{1}\rangle$ and $\langle\Delta E_{2}\rangle$.
The parameters which are normally used to describe the shapes of the Marcus curves
are $\Delta A$ and the reorganization energy $\lambda$, see figure
1. In the Gaussian/linear response régime, both may be
written in terms of the mean energy gaps tachiya ; sprik2 :
$$\Delta A={\frac{1}{2}}(\langle\Delta E_{1}\rangle+\langle\Delta E_{2}\rangle)$$
(7)
and
$$\lambda={\frac{1}{2}}(\langle\Delta E_{1}\rangle-\langle\Delta E_{2}\rangle),$$
(8)
with $\lambda=\lambda^{\prime}$ in figure 1. In this reǵime
the activation free energy for electron transfer is given by the famous expression
marcus1
$$\Delta A^{\ddagger}={\frac{(\Delta A+\lambda)^{2}}{4\lambda}}.$$
(9)
The dependence of $\Delta A$ and $\lambda$ on the position of the redox ion in the
cell and on the applied potential is illustrated in figure
14. The behaviour of these parameters parallels that seen in
the molten salt simulations reed2 and we refer to that paper to fully
vindicate the interpretations of the data which we offer below.
The reorganisation energy $\lambda$ is seen to be virtually independent of the
applied potential, but strongly dependent on the position of the ion in the cell.
The latter is associated with the way in which the polarization of the electrodes
(image charge effect) contributes to the vertical energy gap. In passing from the
initial state, say Ru${}^{2+}$, to the final state Ru${}^{3+}$ we create a unit
positive charge at the location of the redox ion. When we allow the relaxation of
the electrode charges to re-establish the constant potential condition we not only
allow the transfer of one unit of negative charge to the electrodes, we also allow
the electrode to be polarized by the newly-created positive charge. The interaction
between the newly created image charge and the change in the charge of the redox
ion is not screened because the positions of the electrolyte atoms do not
relax after the excitation event in a diabatic description of the charge transfer
process. Since the image effect contributes to $\langle\Delta E_{1}\rangle$ and $\langle\Delta E_{2}\rangle$ with equal magnitude but opposite sign, it does not
affect the value of $\Delta A$, which is seen to be $z$-independent. The
reorganization energy is, however, strongly affected by the image effect. Marcus
Marcus_image obtained an expression for the reorganization energy
appropriate to an ion in a dielectric fluid at a distance $d$ from a single
metallic surface,
$$\lambda_{Marcus}(d)=\delta q^{2}\left[{\frac{1}{\varepsilon_{\infty}}}-{\frac{%
1}{\varepsilon_{s}}}\right]\left({\frac{1}{2a}}-{\frac{1}{2d}}\right),$$
(10)
where $\delta q$ is the charge difference between the reduced and oxidised species.
When $d$ is large, this expression gives the reorganization energy for a redox
process in the bulk fluid: it contains the contribution the non-electronic part of
the dielectric response of the fluid (i.e. that caused by reorganization of
the nuclear positions) to the change in the charge of a redox species with radius
$a$; as such it involves (in the first bracket) the difference between the static
$\varepsilon_{s}^{-1}$ and infinite frequency $\varepsilon_{\infty}^{-1}$
longitudinal dielectric susceptibilities. We cannot compare directly with this
expression because our sample geometry has two metallic surfaces and is
periodic in the transverse direction. However, we can compare directly with the
position-dependent energy of a charge introduced into an empty simulation cell,
this is the effective image interaction energy in our periodic system reed2
when the newly created charge is in a vacuum. Away from the interfaces, any
difference between this quantity and the reorganisation energy should reflect the
effective dielectric screening function of our simulated electrolyte (i.e.
the factor analogous to the square-bracketed term in equation
10). In fact, we see that the two curves coincide well showing
that the factor is indistinguishable from one. In our simulated system the water
molecules and ions are not polarizable, so $\varepsilon_{\infty}$ is just unity and
since for SPC/E water $\varepsilon_{s}$ is about 70 we can only conclude that our
data is consistent with the Marcus expression. Close to the electrodes, the
reorganisation energy does appear to depart from the modified Marcus expression,
and this could be associated with the effect of the proximity of the electrode on
the solvation characteristics of the water molecules there. However, the statistics
in this domain are not good, as the ruthenium ions are even more reluctant to
reorganise their solvation shells and approach the electrode than were the Li${}^{+}$
ones. Better sampling methods for the vertical energy gaps are required before
firm conclusions may be drawn.
In the central part of the simulation cell, the reaction free-energy $\Delta A$
varies linearly with the potential applied to the electrode to which the electron
is transferred. This reflects the change in the energy of the electron which is
transferred to the electrode, which, as we have emphasised, contributes to the
free-energy of the oxidised state. As the redox species approaches the electrode
surface, the reaction free-energy seems to be remarkably constant. It might have
been expected to show the kind of fluctuating behaviour evident in the Poisson
potential, since conventional electrostatic considerations would suggest that this
potential should influence the relative energies of the doubly and triply charged
ions. However, as discussed in the molten salt context reed2 , this is
not the potential which should be used to discuss the changes in the energy
levels of an ion. Rather, we should be considering the potential at the ion’s
centre due only to the other charges present in the system: this might be
better called a Madelung potential. The difference between the two potentials is
surprisingly large, as illustrated in figure 15 where we show the
$z$-dependence of the mean Madelung potentials experienced by the Ru${}^{2+}$ and
Ru${}^{3+}$ ions compared with the Poisson potential.
Not only does the Madelung potential depend on the identity of the species on
which the potential is evaluated, it is seen to be constant across the simulation
cell except in the immediate vicinity of the electrode surfaces - its behaviour
illustrates perfect screening much closer to the electrode surface. The
$z$-independence of the Madelung potential therefore provides a much better
explanation of the insensitivity of $\Delta A$ to $z$ than does the Poisson
potential.
VI Summary and Conclusion
The methods described have allowed a full, self-consistent calculation of the
liquid structure and electrical potentials for an aqueous ionic solution close to a
model metallic wall maintained at a constant electrical potential. The simulation
is a direct realisation of a model electrochemical interface, as it appears in text
books. Using a realistic potential for water-platinum interactions, we find a
strongly absorbed layer of water molecules on the electrode with the molecules
oriented in the plane of the interface at zero potential, in common with earlier
studies berkowitz . Despite the strength of the absorption, the water
molecules do reorient as the electrode potential is changed and this affects the
behaviour of the electrical potential across the interface and the differential
capacitance of the electrode. The absorption of cations at the electrode is
strongly inhibited by the requirement for them to reorganise their hydration shells
to approach the electrode surface.
We have begun to characterise how the interfacial water affects the rate constant
for electrochemical charge transfer by directly calculating the Marcus free energy
curves for the oxidised and reduced species at different positions in the cell with
a particular choice of reaction coordinate. The fluctuations in the solvation
structure which influence these curves were shown to be accurately Gaussian for the
modelled Ru${}^{2+}$/Ru${}^{3+}$ couple, consistent with linear response of the solvent
to the charge state of the redox ion. The reorganisation energy was strongly
dependent on the distance of the redox species from the electrode surface and
independent of the electrode potential. The effect was traced to image charge
interactions with the metal surface. With the statistics available to us at
present, we could not detect an effect of the altered dynamical characteristics of
the absorbed water on the solvation fluctuations when the redox species was close
to the electrode surface. The reaction free energy $\Delta A$ measures the
difference in the free energies of the oxidised and reduced states with the redox
ion at a given distance from the electrode surface. Contrary to textbook
expectations, its position dependence does not resemble the mean electrical
potential. It only deviates from the bulk value in the immediate vicinity of the
interface where the competition between solvating the electrode and solvating the
redox species becomes a factor.
VII Acknowledgments
This work was supported by EPSRC, through Grant GR/T23268/01, as well as by the
Director, Office of Science, Office of Basic Energy Sciences, Chemical Sciences,
Geosciences, and Biosciences Division, U.S. Department of Energy under Contract No.
DE-AC02-05CH11231.
References
[1]
C.D. Taylor, S.A. Wasileski, J.S. Filhol, and M. Neurock
Phys. Rev. B, 73, 165402 (2006).
[2]
C.D. Taylor, S.A. Wasileski and M. Neurock
Curr. Opinion in Sol. Stat. and Mat. Sci., 9 , 49 (2005).
[3]
E. Spohr, Electrochim. Acta, 49, 23 (2003).
[4]
J. O’M. Bockris, A.K.N. Reddy, and M. Gamboa-Aldeco, Modern
Electrochemistry 2A Kluwer, New York, 2000.
[5]
R.A. Marcus, J. Chem. Phys., 24, 966 (1956), Rev. Mod. Phys., 65, 599 (1993).
[6]
R.A. Marcus, Disc. Faraday Soc., 29, 21 (1960).
[7]
J. Blumberger, I. Tavernelli, M.L. Klein and M.Sprik, J. Chem. Phys., 124, (2006);
J. Blumberger, Y. Tateyama Y, and M. Sprik M Computer Physics Communications,
169 256 (2005).
[8]
J. Blumberger and M. Sprik, ”Redox free-energies from
vertical energy gaps: ab initio MD implementation”, page 481 in Computer
Simulations in Condensed Matter: from Materials to Chemical Biology, vol 2 Eds. M.
Ferrario, G. Ciccotti and K. Binder, Lecture Notes in Physics vol. 704, Springer,
2006
[9]
A. Warshel, J. Phys. Chem., 86, 2218 (1982).
[10]
R.A. Kuharski, J.S. Bader, D. Chandler, M. Sprik, M.L. Klein, R.W. Impey, J. Chem. Phys., 89, 3248-3257, (1988).
[11]
J.S. Bader, D. Chandler, Chem. Phys. Lett., 157, 501-504, (1989).
[12]
P.L. Geissler, D. Chandler, J. Chem. Phys., 113, 815408160, (2000).
[13]
D.L. Price, and J.W. Halley, J. Chem. Phys., 102, 6603
(1995).
[14]
S. Izvekov, A. Mazzolo, K. VanOpdorp, and G.A. Voth J. Chem.
Phys., 114, 3248 (2001) and loc. cit..
[15]
S. Walbran and J.W. Halley in Solid-Liquid Interface
Theory - ACS Symposium 789, Ed. J.W. Halley, American Chemical Society, Washington
D.C., (2001).
[16]
S.K. Reed, O.J. Lanning, and P.A. Madden J. Chem. Phys, 126,
084704 (2007).
[17]
J.I. Siepmann, and M. Sprik, J. Chem. Phys., 102,
511 (1995).
[18]
S.K. Reed, P.A. Madden, and A. Papadopoulos J. Chem. Phys 128, 124701
(2008).
[19]
D.M. Kolb, Surf. Sci., 500, 722 (2002).
[20]
H.J.C. Berendsen, J.R. Grigera, T.p. Straatsma, J. Phys Chem.,91,(1987).
[21]
P.A. Thiel and T.E. Madey, Surf. Sci. Rep., 7, 385
(1987).
[22]
S. Holloway and K.H. Bennemann, Surf. Sci., 101, 327
(1980).
[23]
O.M. Magnussen, Chem. Rev., 102, 679 (2002).
[24]
C.G. Guymon, R.L. Rowley, J.N. Harb and D.R. Wheeler, Condensed Matter Physics 8, 335 (2005).
[25]
D. Spangberg and K. Hermansson, J. Chem.
Phys., 120, 4829 (2004)
[26]
S. Koneshan, J.C. Rasaiah, R.M. Lynden-Bell, S.H. Lee, J. Phys. Chem. B,102,(1998)
[27]
J. Blumberger and M. Sprik, J. Phys. Chem. B, 109,
6793 (2005)
[28]
B. Morgan and P.A. Madden J. Chem. Phys. 120, 1402 (2004)
[29]
F. Hutchinson, M. Wilson, and P. A. Madden, Molec. Phys. 99, 811 (2001)
[30]
A.A. Kornyshev, J. Phys. Chem. B, 111, 5545
(2007).
[31]
P. Höchtl, S. Boresch, W. Bitomsky & O. Steinhauser, J. Chem. Phys., 109, 4927 (1998).
[32]
I.C. Yeh and M.L. Berkowitz, J. Electroanalytical Chem.
450, 313 (1998).
[33]
R. Parsons, Solid-State Ionics, 94, 91 (1997).
[34]
W. Schmickler and D. Henderson, J. Chem. Phys.,85,
1650 (1986).
[35]
M. Sprik and G. Ciccotti, J. Chem. Phys., 109, 7737
(1998).
[36]
As discussed at length in reference [18],
because of the adiabatic nature of our calculations with respect to the electrode charges,
a fraction of the charge is transferred to each electrode, whereas in the normal
description of electrochemical charge transfer we envisage a whole electron being
transferred to a designated electrode. The value of the fraction depends
systematically on the position of the ion in the cell and we may therefore correct
our vertical transition energies to correspond to electron transfer to the anode,
as decsribed in detail [18]. We therefore quote our vertical gap energies
as if the electron transfer event is occurring at the anode.
[37]
D.A. Rose and I. Benjamin, J. Chem. Phys., 100, 3545 (1994).
[38]
M. Tachiya, J. Phys. Chem. 93, 7050 (1989), ibid 97, 5911 (2003).
[39]
R.A. Marcus, J. Chem. Phys., 43, 679 (1965);
ibid. 94, 1050 (1990). |
\MakeSorted
figure
\MakeSortedtable
The Magnetic Field Effect on Thermodynamics of Hot QCD Matter using Extensive and non-Extensive Statistics.
Essam Tarek
M.M.Ahmed
Asmaa G. Shalaby
Abstract
We study in detail the thermodynamics of the quantum chromodynamics (QCD) matter utilizing two different statistics, extensive and non-extensive. The thermodynamics such as (pressure, number density, energy density , entropy and magnetization) are determined from both statistics at zero and non-zero magnetic field, $eB=0,0.2,0.3\hskip 1.42271ptGeV^{2}$. The magnetic field effect appears by adding a vacuum contribution to the free energy along side the thermal contribution. The extensive thermodynamics can be emerged from the resonance hadron gas model which is incorporated in the present work. Accordingly, we repeat our calculations at zero and non-zero magnetic field for the non-extensive statistics. The theoretical results of thermodynamical quantities calculated from both statistics are confronted to the lattice results which show a reasonable agreement with the extensive thermodynamics, but overrated in non-extensive thermodynamics at high temperature only and higher entropic index. The response of the QCD matter due to the magnetic field is additionally examined, and it is concluded that QCD matter is considered as a paramagnetic matter.
1 Introduction
One of the most interesting questions that, the scientists are trying to answer is ”How the universe was created?”. The answer might start with the theory of the big bang which states that, the universe was born due to a massive explosion creating fireball. The firball was composed of a dense and hot quarks and gluons plasma (QGP), then, the fireball began to expand and cool off and the elementary particles began to recombine. Due to freezing of QGP, the quarks and gluons combined creating the ordinary matter beginning with the main building block of the Hydrogen atom, protons, neutrons which after that they are categorized as hadrons. After more decreasing of the temperature, the nuclei created which in turn combined with each other creating the ordinary matter [1].
Many efforts have been done to study the phase transition between the hadronic matter and QGP theoretically and experimentally. Starting with a collision of heavy ions or two hadrons accelerated to extremely high energies such as pp, and Pb-Pb collision at the Large Hadron Collider (LHC), and the Au-Au collision at Relativistic Heavy Ion Collider (RHIC) [2]. Unfortunately, the QGP matter can exist for a very short period of time (time span) begins to expand leading to a recombination of quarks and gluons forming the hadronic matter again at a critical temperature $\mathrm{T_{c}}$ [3, 4, 5].
The freezing-out process passes through different stages, the thermal freeze-out and chemical freeze-out.
The chemical freeze-out is the point at which the inelastic collisions stop and no new species create. Then the thermal freeze-out stage begins when the elastic collisions cease and the particles freely fly to the detectors [6, 7]. For more details about the freeze-out conditions and the thermal parameters characterize these stages [8, 9, 10].
The hadrons are interacting strongly and can be described by the quantum chromodynamic (QCD) [11, 12]. Additionally, the properties of the hot hadronic matter are best described as the properties of a statistical system, at which we can extract the thermodynamical quantities of such system depending on the Hadron Resonance Gas Model (HRG) [13] and HRG studied at finite chemical potential [14, 15, 16]. As a matter of fact, the QCD can be described non-perturbativly in which the coupling constant approaches the unity [17, 18], this non-perturbative method is called lattice QCD (LQCD) [19, 20, 21, 22, 23, 24]
HRG is a powerful tool for the description of the thermal properties of the QCD matter and to reproduce the LQCD results at low temperature [25]. However, it is in disagreement with LQCD at high temperature, this might be a consequent of the negligence of the interaction among the particles in which the interaction is significantly effective at high temperature [26, 27]. In addition to that, the HRG model is studied to study the effect of the magnetic field on the QCD matter in a series of papers e.g., [28, 29, 30, 31, 32], a good review in the effect of magnetic field in vacuum [33] and also in in the a lattice QCD such as [34, 35, 36].
An external magnetic field B is generated by the spectators particles (particles do not contribute in the interaction) in the non-central heavy-ion collision. According to the illustration of the non-central heavy ion collision the induced magnetic field can be estimated [37, 38].
Numerical estimation of the magnetic field Au-Au in (RHIC) at $\sqrt{S}=200\hskip 1.42271ptGeV$ Which produces a magnetic field $(\sim 10^{18}−10^{19})$ Gauss and for Pb-Pb in (LHC) at $\sqrt{S}=2.76\hskip 2.27626ptTeV\sim 10^{20}$ Gauss $\sim 10\hskip 1.42271ptm^{2}_{\pi}\hskip 1.42271ptGeV^{2}$ [39, 40, 41] 111 The pion mass $m^{2}_{\pi}$ in $GeV^{2}$ is taken as the unit of eB, where e is the electron charge and $m_{\pi}≈140\hskip 0.85355ptMeV$, where $1MeV^{2}=e\cdot 1.6904\times 10^{14}$ Gauss with ($\hbar=c=1$).
1.1 Motivation
The effect of magnetic catalysis (MC) and $/$ or inverse magnetic catalysis (IMC) mechanism still lacks from the complete understanding in particular on the thermodynamics of the quantum chromodynamics (QCD) matter. This is the main motivation of the present work to make more investigation in the non-extensive statistics along with the extensive one. Beside the recent lattice results have intriguing properties and characteristics.
The effect of magnetic field on QCD matter which is known as the magnetic catalysis (MC) in which it exhibits an increasing in the quark condensate at very low temperature much below the $T_{c}$, this phenomenon is attributed to the valence and the sea contributions to the quark condensate, therefore both contributions are increasing as a function of the magnetic field at ($T=0$). As a matter of fact, whatever the kind of charges (valence or sea), the magnetic field affects the motion of them (catalyzing effect) and makes a restriction on the motion of the charged particles to move in a direction perpendicular to the magnetic field, in other words it makes a ”dimensional reduction” [42] which is treated mathematically in various works and here in the present. However, in contrary to this, it is found that MC effect converted to inverse MC (inverse magnetic catalysis) near the critical temperature Tc and at ($eB<1\hskip 1.42271ptGeV^{2}$) and especially for increasing the pion mass [43]. In the latter means that the quark condensate decreases with the magnetic field [44]. Lattice calculations also have further investigation for both MC and IMC [45, 46, 35].
Strong magnetic field has an important role in physical systems, such as the early universe explanation, the phase transition, cosmology [47]. Additionally, the magnetic field that generated in the peripheral heavy ion collisions at the Relativistic Heavy Ion Collider (RHIC) or the Large Hadron Collider (LHC) [48, 38]. The analysis of the particle production at different energies are studied in [49, 50], extensive and non-extensive is applied on black hole in which the black hole appeared as an extensive system [51]. Recently, and up to three decades the non-extensive statistics has been applied to many physical systems and showed a great success in the description of QCD matter [52], neutron stars [53] and cosmology [54].
The work is organized as follows: section 2 represents the hadron resonance gas model and the thermodynamical quantities of the hadronic system at zero and non-zero magnetic field. SubSection 2.1, discussed the vacuum and thermal free energy contributions. Section 3, is devoted to study the non-extensive statistics in details. The results and discussion is represented in section 4. Finally, the concluding remarks is introduced in section 5, followed by two appendices A, B.
2 Hadron Resonance Gas model (HRG)
The focus of this work is to examine the extensive and non-extensive
thermodynamics in the hadronic system. Beginning with the extensive thermodynamics which can be emerged within the HRG model. The foremost known form of the entropy is the Boltzmann-Gibbs (BG) for various and discrete states $W$, as follows,
$$S_{BG}=-k_{B}\sum_{i=1}^{W}P_{i}\hskip 2.84544ptln\hskip 2.84544ptP_{i}\,,$$
(2.1)
with $k_{B}$ is the Boltzmann constant, and the sum $\sum_{i=1}^{W}P_{i}=1$ takes into account all the possible microscopic configurations $W$, and the probability of each state $i$ is $P_{i}$. For the particular case $P_{i}=1/W$
and for equal probabilities Eq. (2.1) can be rewritten [55, 56],
$$S_{BG}=k_{B}\hskip 2.84544ptln\hskip 2.84544ptW\,.$$
(2.2)
In the thermal equilibrium, the probability of the system is defined in terms of the temperature, T
$$P_{i}=\frac{e^{-\beta E_{i}}}{Z_{BG}}\,,$$
(2.3)
where, $E_{i}$ is the energy of the $i^{th}$ state of the system, $\beta=\frac{1}{k_{B}T}$, and $Z_{BG}$ is the partition function, the latter is defines as [57],
$$Z_{BG}=\sum_{i}^{W}e^{-\beta E_{i}}\,.$$
(2.4)
The canonical partition function has been studied by different methods, e.g. Refs.[58, 59, 60, 61, 62]. The most remarkable point in Boltzmann-Gibbs statistics is the additive property, where a system $A$ composed of two subsystems
$A_{1}$ and $A_{2}$, so that the total entropy can be written as,
$$S_{BG}(A)=S_{BG}(A_{1})+S_{BG}(A_{2})\,.$$
(2.5)
In the grand-canonical ensemble, the partition function of an ideal gas consisting of hadrons and resonances for the i particle is given as [63]
$$\mathrm{\ln Z_{i}=\pm\frac{V\hskip 0.56917ptg_{i}}{(2\pi)^{3}}\int d^{3}p\ln\left[1\pm\lambda_{i}\exp\left(\frac{-E_{i}(p)}{T}\right)\right]},$$
(2.6)
with the grand canonical partition function is just a sum over all resonances $(ln\hskip 0.85355ptZ=\sum_{i}ln\hskip 0.85355ptZ_{i})$
where $\pm$ refer to fermions and bosons, respectively, $\mathrm{g_{i}}$ is degeneracy, $\lambda_{i}$ is defined as [64]
$$\lambda_{i}(\mu,T)=exp\left(\frac{\mu_{s}S_{i}+\mu_{B}B_{i}+\mu_{q}Q_{i}}{T}\right),$$
(2.7)
where the chemical potential is defined $\mathrm{\mu_{s}}$, $\mathrm{\mu_{B}}$, $\mathrm{\mu_{q}}$ are the strange, baryon and quark chemical potential, respectively, multiplied by corresponding quantum numbers S, B, Q, $\mathrm{E_{i}=\sqrt{\mathrm{p^{2}+m_{i}^{2}}}}$ is the relativistic particle energy, in which ($\hbar=c=k_{B}=1$).
In the present work, we are going to study the thermodynamics at zero and non-zero magnetic field in both statistics. Now We represent the free energy formula followed by the free energy with the effect of the magnetic field in order to derive the thermodynamics of the hadronic system. First, the free energy reads [65]
$$\mathrm{F=F_{vac}+F_{therm}}$$
(2.8)
Where $\mathrm{F_{vac}}$ and $\mathrm{F_{therm}}$ are vacuum and thermal energies respectively. The free energy F is given in terms of the total internal energy U [1],
$$F(V,T)=U-T\hskip 0.85355ptS$$
(2.9)
Then replacing all the quantities by their corresponding densities, (e.g. $s=\frac{S}{V},\varepsilon=\frac{U}{V},f=\frac{F}{V}=-p$). The general form would be the Gibbs-Duham relation which is given by eq. (2.10) [1] in addition of the presence of the magnetic field,
$$\mathrm{\varepsilon=T\hskip 0.85355pts+B\hskip 0.85355ptm_{B}-p}$$
(2.10)
Where $(m_{B}=\frac{M_{B}}{V})$, and the magnetization $M_{B}$ is defined as [66]
$$q\hskip 0.56917ptM_{B}=\left(-\frac{\partial F}{\partial B}\right).$$
(2.11)
Once the partition partition function is known, all thermodynamical observables can be calculated e.g. the pressure, P, number density, n, energy density, $\varepsilon$, and entropy for particle i [1]. Moreover, the free energy is directly related to the partition function so that the thermodynamical quantities can be got from the free energy too as $F_{thermal}(V,T)=-\beta^{-1}ln\hskip 0.85355ptZ(V,\beta)$:
$$p=\left(\frac{-\partial F}{\partial V}\right),\quad N=\left(\frac{-\partial F}{\partial\mu}\right),\quad S=\left(-\frac{\partial F}{\partial T}\right)$$
(2.12)
It is known that, the partition function $ln\hskip 0.85355ptZ$ and $F$ are extensive quantities which leads to that, the derivative with respect the V converts to ($\frac{1}{V}$) this satisfied at the thermodynamic limit (i.e at $V\longrightarrow\infty$).
2.1 The Free energy density contributions
In this part we introduce the free energy different contributions according to eq.(2.8) at zero and non-zero magnetic field, in other words for the neutral and charged particles respectively.
•
Firstly, Free energy density at zero magnetic field $B=0$ [65, 67].
$$f(s)=\mp\sum_{i}\sum_{s_{z}}\int\frac{d^{3}\textbf{p}}{(2\pi)^{3}}\left(\frac{E_{i}(\textbf{p})}{2}+T\log\left[1\pm exp\left(\frac{\mu_{i}-E_{i}(\textbf{p})}{T}\right)\right]\right);\quad\hskip 0.85355ptq_{i}=0,q_{i}\neq 0$$
(2.13)
Again, the relativistic energy for particle i in three-dimension momentum p ,
$$E_{i}(\textbf{p})=\sqrt{\textbf{p}^{2}+m_{i}^{2}}$$
(2.14)
Where $E_{\textbf{p}}$ is the relativistic energy for neutral and charged particles, in units ($\hbar=c=k_{B}=1$).
•
Free energy density at non-zero magnetic field $B\neq 0$ [65, 68].
The free energy at the presence of the magnetic field ($B\neq 0$).
$$f_{charge}(s)=\mp\sum_{i}\sum_{s_{z}}\sum_{k=0}^{\infty}\frac{|q_{i}|B}{(2\pi)^{2}}\int dp_{z}\left(\frac{E_{i}(p_{z},k,s_{z})}{2}+T\log\left[1\pm\exp\left(\frac{\mu_{i}-E_{i}(p_{z},k,s_{z})}{T}\right)\right]\right);q_{i}\neq 0$$
(2.15)
Where the integral over $d^{3}p$ in eq. (2.6) is polarized in z-direction for $i^{th}$ particle,
$$\mathrm{\int d^{3}p\rightarrow 2\pi|Q_{i}|eB_{z}\sum_{k}\sum_{s_{z}}\int dp_{z}}$$
(2.16)
In comparison with eq. (2.8), the first part is the contribution due to vacuum and the second part due to the thermal contribution. Where $\mp$ corresponds to bosons (lower sign) and to fermions (upper sign) which in turn due to the spin sectors bosons with integer spin and for fermions is half-integer. The spin takes the values, $(\mathrm{s_{z}=-s,..,s})$ is the z-component of the particle spin (s). Where the modified energy due to Landau levels, k, is defined as [69],
$$E_{i_{z}}(p_{z},k,s_{z})=\sqrt{p_{z}^{2}+m_{i}^{2}+2|q_{i}|\hskip 0.85355ptB\hskip 0.85355pt\left(k-s_{z}+\frac{1}{2}\right)}$$
(2.17)
With $q_{i}=Q_{i}\hskip 0.56917pte$ is the charge of the particle i, mass $m_{i}$, and the electron charge e.
222In eqs. (2.15, 2.13) we add another summation over all the hadronic particles i, and also we can add the chemical potential in the exponential of the thermal part
. In the present work we performed the numerical calculations for Particle Data Group (PDG) with masses up to 10 GeV [70].
It is found that the vacuum parts in both eqs. (2.15, 2.13) are ultraviolet divergent and need to be normalized through dimensional regularization method [71], and we followed the steps the details in [65] to get the following:
As mentioned above, the free energy is composed of two parts, vacuum part and thermal part. We are going to define each part at zero and non-zero magnetic field. Firstly, at $B=0$ the vacuum part is defined as,
$$f_{vac}(s,B=0)=\pm(2s+1)\frac{(q\hskip 0.85355ptB)^{2}}{8\pi^{2}}\hskip 0.85355ptv^{2}\left[\frac{1}{\epsilon}+\frac{3}{4}-\frac{\gamma}{2}-\frac{1}{2}\log\left(\frac{2q\hskip 0.85355ptB}{4\pi\alpha^{2}}\right)-\frac{1}{2}\log(v)\right]$$
(2.18)
with parameter $\epsilon$ and scale $\alpha$ and Euler Lagrange $\gamma$, where $v=\frac{m^{2}}{2q\hskip 0.85355ptB}$ removes the dependence on B itself.
And the normalized free energy vacuum part at $B\neq 0$ is defined as,
$$f_{vac}(s,B\neq 0)=\pm\frac{(q\hskip 0.85355ptB)^{2}}{8\pi^{2}}\sum_{a}\left[\left(-\frac{2}{\epsilon}+\gamma+\log\left(\frac{2q\hskip 0.85355ptB}{4\pi\alpha^{2}}\right)-1\right)\left(-\frac{1}{12}-\frac{(v+a)^{2}}{2}+\frac{v+a}{2}\right)-\zeta^{\backprime}(-1,v+a)\right]$$
(2.19)
where $\zeta$ is the Hurwitz function (see A) emerged from the conversion of the sum over k, and $a=1/2+s_{z}$.
The change in the free energy density due to the magnetic field, one subtract the B = 0 eq.(2.18). All details of the normalization see refs.[65, 72], we write down the final normalized free energies as obtained in [65]. In the present work we have performed the calculation for different sectors of the spin of particles (s $=$ 0, 1/2 and 1). Consequently, the renormalized free energy for those sectors of spins are defined as:
$$\Delta f^{vac}(0)=\frac{(q\hskip 0.56917ptB)^{2}}{8\pi^{2}}\left[\zeta^{\backprime}(-1,v+\frac{1}{2})+\frac{v^{2}}{4}-\frac{v^{2}}{2}\hskip 0.56917ptlog(v)+\frac{log(v)+1}{24}\right]$$
(2.20)
$$\Delta f^{vac}(1/2)=-\frac{(q\hskip 0.56917ptB)^{2}}{4\pi^{2}}\left[\zeta^{\backprime}(-1,v)+\frac{v}{2}log(v)+\frac{v^{2}}{4}-\frac{v^{2}}{2}log(v)-\frac{log(v)+1}{12}\right]$$
(2.21)
$$\Delta f^{vac}(1)=\frac{3(q\hskip 0.56917ptB)^{2}}{8\pi^{2}}\Bigg{[}\zeta^{\backprime}(-1,v-\frac{1}{2})+\frac{1}{3}(v+\frac{1}{2})\hskip 0.56917ptlog(v+\frac{1}{2})+\frac{2}{3}(v-\frac{1}{2})\hskip 0.56917ptlog(v-\frac{1}{2})+\frac{v^{2}}{2}\left(\frac{1}{2}-log(v)\right)-7\frac{log(v)+1}{24}\Bigg{]}$$
(2.22)
3 Essential Features of Non-extensive Statistics
In this section we represent the non-extensive thermodynamics which is calculated at vanishing and non-zero magnetic field.
The great success of the well-established BG statistics for more than a century ago is the fact that it can be generalized for complex systems [73]. For complex systems, it is not so easy to extract their properties easily, since there are time-dependent interactions among their numerous constituents. Complex systems are not used only in the domain of physics but also in other domains like biology, economics, and so on. [74]. For more details about the non-extensive statistics application in various fields in physics , cosmology and astronomy see [75, 76, 77, 78, 79, 54].
Specifically, many systems deviate from the BG statistics, and it was necessary to generalize the original ones. These systems are, for example, systems with long-range interactions, ferromagnetism related systems pure-electron plasma two-dimensional turbulence systems. Complex systems effects can be found in the peculiar velocities of galaxies, black holes, high energy collisions of elementary particles, quantum entanglement, see, for example, Ref. [80] and references therein.
In 1988, Tsallis suggested the non-extensive [81] in other words the non-additive property of entropy eq. (2.5)is no longer satisfied. Therefore the non-extensive entropy is represented as [75],
$$S_{q}(A_{1}+A_{2})=S_{q}(A_{1})+S_{q}(A_{2})+(1-q)\hskip 1.13791ptS_{q}(A_{1})S_{q}(A_{2})$$
(3.1)
Where q is the non-extensive parameter or the entropic index, in the limit ($q\rightarrow 1$), one recovers the BG-statistics. Additionally, q should be greater than one, the limit of q is discussed in [82, 83, 84, 85].
The probability and the partition function can be re-defined,
$$P_{i}^{q}=\frac{[1-\beta(q-1)E_{i}]^{1/(q-1)}}{Z_{q}}\,,\qquad{\textrm{w}here}\qquad Z_{q}=\sum_{i=1}^{W}[1-\beta(q-1)E_{i}]^{1/(q-1)}\,.$$
(3.2)
Now we turn to define the grand canonical partition function for the hadronic system based on the non-extensive statistics. Starting with the following definitions;
•
The q-exponential
$$\begin{split}e_{q}^{(+)}(x)=[1+(q-1)x]^{\frac{1}{q-1}},\qquad x\geqslant 0\\
e_{q}^{(-)}(x)=[1+(1-q)x]^{\frac{1}{1-q}},\qquad x<0\end{split}$$
(3.3)
•
The q-logarithmic
$$\begin{split}\log_{q}^{(+)}(x)=\frac{x^{q-1}-1}{q-1},\\
\log_{q}^{(-)}(x)=\frac{x^{1-q}-1}{1-q},\end{split}$$
(3.4)
Then the grand-canonical partition function is defined [79, 86].
$$\log\Xi_{q}(V,T,\mu_{i})=-\xi V\int\frac{d^{3}p}{(2\pi)^{3}}\sum_{r=\pm}\Theta(rx)\log_{q}^{(-r)}\left(\dfrac{e_{q}^{(r)}(x)-\xi}{e_{q}^{(r)}(x)}\right)$$
(3.5)
Where $x=\beta(E-\mu)$, with the particle energy of mass m is given by $E=\sqrt{p^{2}+m^{2}}$, and $\mu$ is the chemical potential. The step function $\Theta(rx)$ within $r=>-$ applies for fermions only which is satisfied when $\mu\leq m$, and $\xi=\pm 1$ refer to bosons and fermions respectively.
The thermodynamical non-extensive quantities can all be derived as in eq.(2.12) by replacing $lnZ$ by the non-extensive partition function for a hadronic system eq.(3.5).
Then, the pressure, number density, energy density and entropy are , eqs.(3.6, 3.7, 3, 3). :
$$P_{i}=-\xi T\int\frac{d^{3}p}{(2\pi)^{3}}\sum_{r=\pm}\Theta(rx)\log_{q}^{(-r)}\left(\frac{e_{q}^{(r)}(x)-\xi}{e_{q}^{(r)}(x)}\right)$$
(3.6)
$$n_{i}=\left[C_{N,q}(\mu,\beta,m)+\int\frac{d^{3}p}{(2\pi)^{3}}\sum_{r=\pm}\Theta(rx)\left(\frac{1}{e_{q}^{(r)}(x)-\xi}\right)^{\tilde{q}}\right]$$
(3.7)
Where
$$C_{N,q}(\mu,\beta,m)=\frac{1}{2\pi^{2}}\frac{\mu\sqrt{\mu^{2}-m^{2}}}{\beta}\frac{2^{q-1}+2^{1-q}-2}{q-1}\Theta(\mu-m)$$
(3.8)
Where the part in eq.(3.8) emerged from the fact that the integral in eq.(3.5) is discontinuous at $x=0$, this leads to that the momentum $p=\sqrt{\mu^{2}-m^{2}}$ see details in Appendix B
$$\displaystyle\varepsilon$$
$$\displaystyle=$$
$$\displaystyle-\frac{\partial}{\partial\beta}\log\Xi_{q}\Bigg{|}_{\mu}+\mu\hskip 0.85355ptT\hskip 0.85355pt\frac{\partial}{\partial\mu}\log\Xi_{q}\Bigg{|}_{\beta}$$
$$\displaystyle=$$
$$\displaystyle\left[C_{E,q}(\mu,\beta,m)+\int\frac{d^{3}p}{(2\pi)^{3}}\sum_{r=\pm}\Theta(rx)E\left(\frac{1}{e_{q}^{(r)}(x)-\xi}\right)^{\tilde{q}}\right],\,\,.$$
where $C_{E,q}(\mu,\beta,m)=\mu C_{N,q}(\mu,\beta,m)$
$$\displaystyle s$$
$$\displaystyle=$$
$$\displaystyle-\beta^{2}\hskip 1.13791pt\frac{\partial}{\partial\beta}\left(\frac{\log\Xi_{q}}{\beta}\right)$$
$$\displaystyle=$$
$$\displaystyle\int\frac{d^{3}p}{(2\pi)^{3}}\sum_{r=\pm}\Theta(rx)\left[-[\overline{n}_{q}^{(r)}(x)]^{\tilde{q}}\log_{q}^{(-r)}\left(\overline{n}_{q}^{(r)}(x)\right)+\xi[1+\xi\overline{n}_{q}^{(r)}(x)]^{\tilde{q}}\log_{q}^{(-r)}\left(1+\xi\overline{n}_{q}^{(r)}(x)\right)\right],\,\,.$$
Where $\mathrm{\overline{n}_{q}^{(r)}(x)\equiv\left[n_{q}^{(r)}(x)\right]^{1/\tilde{q}}}$
$$\mathrm{n_{q}^{(r)}(x)=\left(\frac{1}{e_{q}^{(r)}(x)-\xi}\right)^{\tilde{q}},\qquad with\qquad\tilde{q}}=\left\{\begin{array}[]{ll}\mathrm{q,\qquad x\geqslant 0}\\
\mathrm{2-q,\qquad x<0}\end{array}\right.$$
(3.11)
The next step is to modify the non-extensive partition function due to the magnetic field and all steps of the thermodynamics at non-zero magnetic field is repeated.
$$\log\Xi^{B_{z}}_{q}(V,T,\mu)=\frac{-\xi V}{2\pi^{2}}|Q|eB_{z}\sum_{k}\sum_{s_{z}}\int_{0}^{\infty}dp_{z}\sum_{r=\pm}\Theta(rx)\log_{q}^{(-r)}\left(\frac{e_{q}^{(r)}(x)-\xi}{e_{q}^{(r)}(x)}\right)$$
(3.12)
following the conversion in eq.(2.16) and the energy in z-direction in eq.(2.17).
4 Results and discussion
We represent the results of two different thermodynamic statistics, extensive and non-extensive thermodynamics. The impact of the magnetic field is studied for both statistics, in which the thermodynamical observables such as,( pressure, energy density, entropy and magnetization) are investigated. Both statistics are studied at zero and non-zero magnetic field. The response of the QCD matter to the magnetic field is examined through the magnetization which exhibits as paramagnetic material. All our results are studied at vanishing baryon chemical potential.
4.1 Extensive Individual Particle Contribution and Thermodynamics
Figure (1) shows the pressure for the particles $p,\rho^{0,\pm},\pi^{0,\pm}$ the left panel (1(a)) for vanishing magnetic field and the right panel (1(b)) for magnetic field $(eB=0.2\hskip 1.13791ptGeV^{2})$ versus the temperature T. It is noticed that, the charged particles pressure differs from zero magnetic field to the one at finite magnetic field. This difference appears at temperature ($100<T<140\hskip 1.13791ptMeV$), in which $\rho^{\pm}$ shifted by $\sim 0.2\times 10^{-4}\hskip 1.13791ptGeV^{4}$, and $\pi^{\pm}$ shifted by $\sim 0.5\times 10^{-4}\hskip 1.13791ptGeV^{4}$. The domination of the charged pions is clearly appeared in both cases. It is noticeable that, the pressure for the charged particles are shifted because of the vacuum term included and confirms the effect of the magnetic field only on the charged hadrons.
Figure (2) represents the EoS of QCD matter at ($eB=0,0.2,0.3\hskip 1.9919ptGeV^{2}$). The pressure fig.(2(a)) versus the temperature shows a reasonable agreement with the lattice results [35], the pressure also exhibits an increasing by increasing the magnetic field. Figure (2 represents the energy density, one can see the increasing in the energy density with increasing the magnetic field. The entropy and magnetization of the QCD matter fig. (3) are calculated at ($eB=0.2,0.3\hskip 1.42271ptGeV^{2}$), it is remarkable that, the magnetization increases with increasing the magnetic field as appeared in (3), while the increasing in the entropy is slight as shown in fig.3(a)).
4.2 Non-Extensive Individual Particle Contribution and thermodynamics
In a similar case the thermodynamical quantities are studied by the non-extensive statistics with zero and non-zero magnetic field.
The individual pressure of charged and neutral particles are represented in fig. (4). This figure is calculated by the non-extensive statistics, one can see a similar results for the one in fig.(1), and the effect of vacuum term that appeared at $T\geq 0$ or at very low temperature. This is obvious in the shifting of charged pressure at $B\neq 0$.
Accordingly, Fig.(5) pressure (5), energy density 5, and fig. (6) in which entropy 6, and magnetization 6, all the results are confronted to the lattice results. It is surprisingly noticed at $(eB=0\hskip 1.42271ptGeV^{2})$ that, the thermodynamics overestimated the one at $(eB=0.2,0.3\hskip 1.42271ptGeV^{2})$ in particular at higher temperature ($100<T<200\hskip 1.42271ptMeV$).
In order to do more investigation, we have studied the number density Fig.7 with changing the entropic index ($q=1.001,1.01,1.1,1.2$) with increasing the magnetic field for each case, in which the deviation between zero and non-zero magnetic field occurs at high temperature. his ensures the last result for the presser, energy density and entropy at higher temperature. However we can conclude that increasing the degree of non-extensivity exhibits a reverse magnetic catalysis only at higher temperature
5 Conclusion
In the present work, we have explored the effect of the magnetic field on the extensive and non-extensive thermodynamics.
The free energy is separated into two parts, the vacuum and thermal contributions. We have applied the magnetic field on both terms in which the vacuum free energy needs to be regularized according to the reduction in the dimension from 3 to 1. Accordingly, the relativistic energy of the charged particles has been modified with the Landau levels and polarized spin in z-direction.
The response of the QCD matter to the magnetic field is examined through the magnetization via the temperature and the magnetic field values. In conclusion the magnetization showed a negligibly change which slightly increases at low temperature, however it increases with positive values dramatically and rapidly with increasing the magnetic field and the temperature which indicates that, QCD is a paramagnetic matter. All results from both statistics are confronted to the available lattice results at zero and non-zero magnetic field which show a reasonable agreement with the extensive thermodynamics, but overrated in non-extensive thermodynamics at high temperature only, and higher entropic index.
Appendix A Appendix A
These are the sample of the tables that we have used in our calculations, and includes up to mass $\sim 10.876\hskip 1.9919ptGeV$. Table (1) contains, the mass code, mass in GeV, width, degeneracy, baryon, strangeness, charmness, bottomness quantum numbers, third component of isospin, charge and the number of decay channels respectively. Table (1) also includes information for the daughter in case of the decay of the same particle with ”Br” the branching ratio and ”Nrs” the number of resonances. In the present work we did not include the decay channels. In addition, the degeneracy appears in table (1), then the spin can be extracted from the relation between the degeneracy and spin eq.(A.1),
$$g_{i}=2\hskip 0.56917pts_{i}+1$$
(A.1)
•
Some useful definitions
The vacuum part can be solved by using the standard integration in $d$ dimensions (Ref. [87])
$$\displaystyle\int_{-\infty}^{\infty}\frac{d^{d}p}{(2\pi)^{d}}\sqrt{p^{2}+M^{2}}=\frac{1}{(4\pi)^{d/2}}\frac{\Gamma(-1/2-d/2)}{\Gamma(-1/2)}(M^{2})^{1/2+d/2},$$
(A.2)
with $\Gamma(-1/2)=-2\sqrt{\pi}$.
•
The Hurwitz $\zeta$ function is defined as
$$\displaystyle\sum_{k=0}^{\infty}\frac{1}{(v+k)^{z}}=\zeta(z,v),$$
(A.3)
with the expansion and asymptotic behavior [88]
$$\displaystyle\zeta(-1+\epsilon/2,v)\approx-\frac{1}{12}-\frac{v^{2}}{2}+\frac{v}{2}+\frac{\epsilon}{2}\zeta^{\prime}(-1,v)+\mathcal{O}(\epsilon^{2}),$$
(A.4)
$$\displaystyle\zeta^{\prime}(-1,v)=\frac{1}{12}-\frac{v^{2}}{4}+\left(\frac{1}{12}-\frac{v}{2}+\frac{v^{2}}{2}\right)\log(v)+\mathcal{O}(v^{-2}).$$
(A.5)
Appendix B Appendix B
Beginning with the definition of the momentum again at $x=0$, $p=\sqrt{\mu^{2}+m^{2}}$
so that the integrand in eq.(3.5) will be discontinuous at ($x=0$). In order to avoid and solve this:-
Firstly, Using the general definition of the average number of particles i as,
$$\langle N\rangle=T\hskip 1.42271pt\frac{\partial}{\partial\mu}log\hskip 1.42271pt\Xi_{q}\Bigg{|}_{\beta}$$
(B.1)
$$\langle N\rangle=V\left[C_{N,q}(\mu,\beta,m)+\int\frac{d^{3}p}{(2\pi)^{3}}\sum_{r=\pm}\Theta(rx)\bigg{(}\frac{1}{e_{q}^{(r)}(x)-\xi}\bigg{)}^{\tilde{q}}\right]\,,$$
(B.2)
where
$$\tilde{q}=\begin{cases}&q\qquad\quad\;\;\,\,,\,\,\,x\geq 0\,,\\
&2-q\qquad\,,\,\,\,x<0\,.\end{cases}$$
(B.3)
Let the integrand in eq. (B.2) to be composed of two parts $F^{(-)},F^{(+)}$, and by converting the three integrals to one integral as;
$$\displaystyle\int d^{3}p\rightarrow 4\hskip 0.85355pt\pi\hskip 0.85355pt\int_{0}^{\infty}p^{2}\hskip 0.85355ptdp$$
$$\displaystyle\langle N\rangle$$
$$\displaystyle=$$
$$\displaystyle T\frac{\partial}{\partial\mu}\left[\int_{0}^{\sqrt{\mu^{2}-m^{2}}}dp\,F^{(-)}(p,\mu)+\int_{\sqrt{\mu^{2}-m^{2}}}^{\infty}dp\,F^{(+)}(p,\mu)\right]$$
$$\displaystyle=$$
$$\displaystyle-\frac{\mu}{\beta\sqrt{\mu^{2}-m^{2}}}\left[F^{(+)}(x=0^{+})-F^{(-)}(x=0^{-})\right]+\beta^{-1}\int_{0}^{\infty}dp\,\sum_{r=\pm 1}\frac{\partial}{\partial\mu}F^{(r)}(p,\mu)\,\,.$$
To get the last part in eq. (B), one can use Leibniz’ Rule:
$$\frac{d}{dx}\int_{u(x)}^{v(x)}F(x,t)dt=F(x,v(x))\frac{dv}{dx}-F(x,u(x))\frac{du}{dx}+\int_{u(x)}^{v(x)}\frac{\partial F}{\partial x}dt$$
(B.5)
Applying the rule for both parts in the first line of eq. (B). For the first part, it is straightforward to get,
$$\frac{\partial}{\partial\mu}\int_{0}^{\sqrt{\mu^{2}-m^{2}}}F^{-}(p,\mu)\hskip 0.85355ptdp=\frac{\mu}{\sqrt{\mu^{2}-m^{2}}}\hskip 1.42271ptF^{-}(\mu,\sqrt{\mu^{2}-m^{2}})+\int_{0}^{\sqrt{\mu^{2}-m^{2}}}\frac{\partial}{\partial\mu}F^{-}(\mu,p)\hskip 0.85355ptdp$$
(B.6)
And the second part in the first line of eq. (B),
$$\frac{\partial}{\partial\mu}\int_{\sqrt{\mu^{2}-m^{2}}}^{\infty}F^{+}(p,\mu)\hskip 0.85355ptdp=-\frac{\mu}{\sqrt{\mu^{2}-m^{2}}}\hskip 1.42271ptF^{+}(\mu,\sqrt{\mu^{2}-m^{2}})+\int_{\sqrt{\mu^{2}-m^{2}}}^{\infty}\frac{\partial}{\partial\mu}F^{+}(\mu,p)\hskip 0.85355ptdp$$
(B.7)
Substituting by both parts in eq. eq. (B)
$$\displaystyle\langle N\rangle$$
$$\displaystyle=$$
$$\displaystyle\frac{\mu}{\sqrt{\mu^{2}-m^{2}}}\hskip 1.42271ptF^{-}(\mu,\sqrt{\mu^{2}-m^{2}})+\int_{0}^{\sqrt{\mu^{2}-m^{2}}}\frac{\partial}{\partial\mu}F^{-}(\mu,p)\hskip 0.85355ptdp$$
$$\displaystyle-$$
$$\displaystyle\frac{\mu}{\sqrt{\mu^{2}-m^{2}}}\hskip 1.42271ptF^{+}(\mu,\sqrt{\mu^{2}-m^{2}})+\int_{\sqrt{\mu^{2}-m^{2}}}^{\infty}\frac{\partial}{\partial\mu}F^{+}(\mu,p)\hskip 0.85355ptdp\,\,.$$
The discontinuous point at $x=0$, $p=\sqrt{\mu^{2}+m^{2}}$ , the above eq. (B) can be rewritten in a compact form,
$$\langle N\rangle=C_{N,q}(\mu,T,m)+T\int_{0}^{\infty}\frac{\partial}{\partial\mu}\sum_{r=\pm}F^{(r)}(\mu,p)dp$$
(B.9)
where
$$C_{N,q}(\mu,T,m)=-\frac{T\hskip 0.85355pt\mu}{\sqrt{\mu^{2}-m^{2}}}\hskip 1.42271pt\left[F^{+}(x=0^{+})-F^{-}(x=0^{-})\right]$$
(B.10)
Now exploiting eqs. (3.3, 3.4, and 3.5).
Then one can easily get,
$$\displaystyle F^{+}(x=0^{+})$$
$$\displaystyle=$$
$$\displaystyle\frac{(\mu^{2}-m^{2})}{2\hskip 0.56917pt\pi^{2}}\hskip 1.42271pt\frac{2^{1-q}-1}{1-q}\hskip 1.42271pt\Theta(\mu-m),\quad F^{-}(x=0^{-})=\frac{(\mu^{2}-m^{2})}{2\hskip 0.56917pt\pi^{2}}\hskip 1.42271pt\frac{2^{q-1}-1}{q-1}\hskip 1.42271pt\Theta(\mu-m)\,\,.$$
Collect altogether we obtain $\langle N\rangle$.
References
[1]
J. Letessier, and J. Rafelski. Hadrons and quark–gluon plasma. Vol. 18. Cambridge University Press, (2002).
[2]
R. S. Bhalerao, Relativistic heavy-ion collisions, CERN-2014-001 and KEK-Proceedings (2013) 219. ArXiv:1404.3294 [nucl-th].
[3]
L. P. Csernai, Introduction to relativistic heavy ion collisions,© John Wiley and Sons Ltd; Chichester (United Kingdom) (1994).
[4]
A. K. Chaudhuri, A short course on relativistic heavy ion collisions, © IOP Publishing Ltd (2014).
[5]
J. Rafelski, Melting Hadrons, Boiling Quarks: From Hagedorn Temperature to Ultra-Relativistic Heavy-Ion Collisions at CERN With a Tribute to Rolf Hagedorn, © Springer Cham Heidelberg New York Dordrecht London (2016).
[6]
V. Magas, H. Satz, Conditions for Confinement and Freeze-Out, Eur.Phys.J. C32 (2003) 115. ArXiv:hep-ph/0308155.
[7]
J. Xu, and C. Ming Ko., Chemical freeze-out in relativistic heavy-ion collisions, Phys. Lett. B772, (2017) 290.
[8]
J. Cleymans, H. Oeschler, K. Redlich and S. Wheaton, Phys. Rev. C73, 034905 (2006). ArXiv:hep-ph/0511094.
[9]
F. Becattini, J. Manninen and M. Gazdzicki, Phys. Rev. C73, (2006) 044905. ArXiv:hep-ph/0511092.
[10]
A. Andronic, P. Braun-Munzinger, K. Redlich and J. Stachel, J. Phys. G 38, 124081 (2011). ArXiv:nucl-th/1106.6321.
[11]
G. Altarelli, Collider Physics within the Standard Model: A Primer © Springer (2017).
[12]
N. Brambilla, et al., QCD and strongly coupled gauge theories: challenges and perspectives, Eur.Phys.J. C74, 10 (2014) 1.
[13]
R. Hagedorn, Suppl. Nuovo Cimento III, (1965) 147.
[14]
S. Subhasis, S. Chatterjee, and B. Mohanty, Exploring the hadron resonance gas phase on the QCD phase diagram, J.Phys. G: Nucl.Part.Phys. 46, 6 (2019) 065106.
[15]
E. Megias, E. R. Arriola, and L. L. Salcedo, The hadron resonance gas model: thermodynamics of QCD and Polyakov loop, Nucl.Phys.B Proceedings Supplements, 234 (2013) 313.
[16]
R. Hagedorn, and J. Rafelski, Hot hadronic matter and nuclear collisions, Phys. Lett. B97, 1 (1980) 136.
[17]
E. Shuryak, Lectures on nonperturbative QCD (Nonperturbative Topological Phenomena in QCD and Related Theories), arXiv:hep-ph/1812.0150.
[18]
G. Sterman, et al., Handbook of perturbative QCD, Rev. Mod. Phys. 67, 1 (1995) 157.
[19]
R. Gupta, Introduction to lattice QCD, Lectures given at the LXVIII Les Houches Summer School ”Probing the Standard Model of Particle Interactions”, July 28-Sept 5, 1997, arXiv:hep-lat/9807028.
[20]
S. Gupta, X. Luo, B. Mohanty, H.G. Ritter, Nu Xu, Scale for the phase diagram of quantum chromodynamics, Science 332, (2011) 1525. arXiv/hep:1105.3934
[21]
Y. Aoki, G. Endrodi, Z. Fodor, S.D. Katz, K.K. Szabo, The order of the quantum chromodynamics transition predicted by the standard model of particle physics, Nature 443, (2006) 675. ArXiv:hep-lat/0611014.
[22]
G. Endrodi, Z. Fodor, S. D. Katz, and K. K. Szabo, The QCD phase diagram at nonzero quark density
JHEP, 04 (2011) 001.
[23]
O. Kaczmarek et al.,Phase boundary for the chiral transition in (2+1) -flavor QCD at small values of the chemical potential, Phys. Rev. D83, 014504 (2011).ArXiv:hep-lat/1011.3130.
[24]
C. Bonati, M. D’Elia, M. Mariti, M. Mesiti, F. Negro, and F. Sanfilippo, Curvature of the chiral pseudocritical line in QCD, Phys. Rev.D90, (2014) 114025.
[25]
R. Bellwied, S. Borsányi, Z. Fodor, S. D. Katz, A. Pásztor, C. Ratti, and K. K. Szabó, Fluctuations and correlations in high temperature QCD, Phys.Rev.D92, 11 (2015) 114505.
[26]
S.Subhasis, and B. Mohanty, Criticality in a hadron resonance gas model with the van der Waals interaction, Phys. Rev. C97, 1 (2018) 015201.
[27]
V. Volodymyr, M.I. Gorenstein, and H. Stoecker, Van der Waals interactions in hadron resonance gas: from nuclear matter to lattice QCD, Phys. Rev.Lett.118, 18 (2017) 182301.
[28]
N.O. Agasian, Phase structure of the QCD vacuum in a magnetic field at low temperature, Phys. Lett. B488, (2000) 39.
[29]
G. Endrodi, QCD equation of state at nonzero magnetic fields in the Hadron Resonance Gas model, JHEP 07 (2015) 173.
[30]
A. Ayala, C. A.Dominguez, L.A.Hernández, M.Loewe,and R.Zamora, Inverse magnetic catalysis from the properties of the QCD coupling in a magnetic field, Phys.Lett.B759 (2016) 99.
[31]
A. Tawfik, A. Diab, N. Ezzelarab, and Asmaa. G. Shalaby, QCD Thermodynamics and Magnetization in
Nonzero Magnetic Field, Adv. High. Eng.Phys. 2016, Article ID 1381479, (2016).
[32]
V. V. Braguta, M. N. Chernodub, A. Yu. Kotov, A. V. Molochkov and A. A. Nikolaev, Finite-density QCD transition in a magnetic background field, Phys.Rev. D100, (2019) 114503.
[33]
E. Shuryak, Theory and phenomenology of the QCD vacuum, Phys. Rep.115, (1984) 151.
[34]
G. S. Bali, F. Bruckmann, G. Endrodi, Z. Fodor, S. D. Katz, S. Krieg, A. Schafer, and K. K. Szabo, The QCD phase diagram for external magnetic fields, JHEP, 2012 (2012) 44.
[35]
G. S. Bali, F. Bruckmann, G. Endrodi, S. D. Katz, and A. Schafer, The QCD equation of state in background magnetic fields, JHEP 2014, (2014) 1408.
[36]
P. V. Buividovich, M.N. Chernodub, E.V. Luschevskaya, M.I. Polikarpov, Lattice QCD in strong magnetic fields, talk given by M. N. Chernodub at the 10th Workshop on Non-Perturbative QCD, June 8-12, (2009), Paris, France. ArXiv:hep-ph/0909.1808.
[37]
Huang, Xu-Guang, Electromagnetic fields and anomalous transports in heavy-ion collisions: A pedagogical review, Rep.Prog.Phys. 79 (2016) 076302.
[38]
Skokov, V. V., A. Yu Illarionov, and V. D. Toneev, Estimate of the magnetic field strength in heavy-ion collisions, Int.J. Mod.Phys.A24, 31 (2009) 5925.
[39]
A. Bzdaka, V. Skokov, Event-by-event fluctuations of magnetic and electric fields in heavy ion collisions, Phys.Lett.B710, (2012) 171. ArXiv:hep-ph/1111.1949.
[40]
Li Ou and Bao-An Li, Magnetic effects in heavy-ion collisions at intermediate energies, Phys.Rev.C84, (2011) 064605. ArXiv:nucl-th/1107.3192.
[41]
D. E.Kharzeev, L.D. McLerran, and Harmen J.Warringa, The effects of topological charge change in heavy ion collisions: “Event by event $\mathcal{P}$ and $\mathcal{CP}$ violation, Nucl.Phys.A803, (2008) 227. ArXiv:hep-ph/0711.0950.
[42]
D.E. Kharzeev, K. Landsteiner, A. Schmitt, and Ho-Ung Yee, Strongly interacting matter in magnetic fields, © Springer-Verlag Berlin Heidelberg (2013).
[43]
G. Endrődi, M. Giordano, S. D. Katz, T. G. Kovács F. Pittler, Magnetic catalysis and inverse catalysis for heavy pions, JHEP, 2019 (2019) 7.
[44]
M. D’Elia, F. Manigrasso, F. Negro and F. Sanfilippo, QCD phase diagram in a magnetic
background for different values of the pion mass, Phys.Rev.D98 (2018) 054509. ArXiv:hep.lat/1808.07008.
[45]
E.M. Ilgenfritz, M. M$\ddot{u}$ller-Preussker, B. Petersson and A. Schreiber, Magnetic catalysis (and
inverse catalysis) at finite temperature in two-color lattice QCD, Phys.Rev.D89 (2014) 054512. ArXiv:hep-lat/1310.7876.
[46]
V.G. Bornyakov, P.V. Buividovich, N. Cundy, O.A. Kochetkov and A. Schafer, Deconfinement transition in two-flavor lattice QCD with dynamical overlap fermions in an external magnetic field, Phys.Rev.D90 (2014) 034501. ArXiv:hep-lat/1312.5628.
[47]
G. S. Bali, F. Bruckmann, G. Endrődi, Z. Fodor, S. D. Katz, and A. Schäfer, QCD quark condensate in external magnetic fields, Phys.Rev.D86, 071502(R).
[48]
V. Voronyuk, V. D. Toneev, W. Cassing, E. L. Bratkovskaya, V. P. Konchakovski, and S. A. Voloshin, Electromagnetic field evolution in relativistic heavy-ion collisions
, Phys. Rev. C83, (2011) 054911. ArXiv:nucl.th/1103.4239.
[49]
F. Becattini, M. Gaydzicki, A. Keranen, J. Manninen, R. Stock, Chemical equilibrium study in nucleus-nucleus collisions at relativistic energies, Phys. Rev. C69 (2004) 024905.
[50]
G. Kadam, S. Pal, A. Bhattacharyya, Interacting hadron resonance gas model in magnetic field and the fluctuations of conserved charges, J.Phys.G47 (2020) 125106. ArXiv:hep-ph/1908.10618v2.
[51]
Asmaa G. Shalaby, Extensive and non-extensive thermodyamics, Acta Phyica.Polon B47 (2016).
[52]
Jacek Rożynek and Grzegorz Wilk, Nonextensive Quasiparticle Description of QCD Matter, Symmetry, 11(3), (2019) 401.
[53]
N.K. Patra, T. Malik, D. Sen, T.K. Jha, and H. Mishra, An Equation of State for Magnetized Neutron Star Matter and Tidal Deformation in Neutron Star Mergers, Astrophys.J. 900 (2020) 1, 49.
[54]
Asmaa G. Shalaby, Vasilis K. Oikonomou, and Gamal G. L. Nashed, Non-Extensive Thermodynamics Effects in the Cosmology of f(T) Gravity., Symmetry, 13(1), 75 (2021).
[55]
L. Boltzmann, Kinetische Theorie II, (1970) 115.
[56]
J. W. Gibbs, Elementary principles in statistical mechanics: developed with especial reference to the rational foundations of thermodynamics, © C. Scribner’s sons (1902).
[57]
F. Reif, Fundamentals of statistical and thermal physics, © Waveland Press (2009).
[58]
A. Bohr and B. R. Mottelson, Nuclear Structure © W. A Benjamin, Inc. (1969).
[59]
B.K. Jennings, L. Satpathy, and S. Das Gupta, Approximate method of including finite particle number effects in relativistic heavy ion collisions, Phys.Rev.C24, (1981) 440.
[60]
K.C. Chase and A.Z. Mekjian, Exact methods for expectation values in canonical fragmentation models
, Phys.Rev.C52, (1995) 2339.
[61]
S. Das Gupta and A.Z. Mekjian, Phase transition in a statistical model for nuclear multifragmentation, Phys. Rev.C57, (1998) 1361.
[62]
A. Keranen, F. Becattini, Chemical factors in canonical statistical models for relativistic heavy ion collisions, Phys.Rev.C65, (2002) 044901, Erratum Phys.Rev.C68 (2003) 059901.
[63]
A. Bhattacharyya, S. K. Ghosh, R.Ray, and S. Samanta. Exploring effects of magnetic field on the Hadron Resonance Gas, EPL 115 (2016) 62003. Arxiv:hep-ph/1504.04533.
[64]
A. Tawfik, Loutfy I. Abou-Salem, Asmaa G. Shalaby, M. Hanafy, A. Sorin, O. Rogachevsky, and W. Scheinast, Particle production and chemical freezeout from the hybrid UrQMD approach at NICA energies, Eur. Phys. J. A52 (2016) 324.
[65]
G. Endrödi, QCD equation of state at nonzero magnetic fields in the Hadron Resonance Gas model
, JHEP 04, 023 (2013).
[66]
Jens O. Andersen, William R. Naylor, Anders Tranberg, Phase diagram of QCD in a magnetic field: A review, Rev. Mod. Phys. 88, 025001 (2016). ArXiv:hep-ph/1411.7176v3
[67]
J. Kapusta and C. Gale, Finite-Temperature Field Theory: Principles and Applications. Cambridge monographs on mechanics and applied mathematics, © Cambridge University Press, (2006).
[68]
E. S. Fraga and A. J. Mizher, Chiral transition in a strong magnetic background, Phys.Rev. D78 (2008) 025016. ArXiv:hep-ph/0804.1452.
[69]
L. Landau and E. Lifshits, Quantum Mechanics: Non-Relativistic Theory. Course of theoretical physics (Vol 3.) (Landau, L. D, 1908-1968) © Pergamon Press, (1977).
[70]
K. Olive et al., Summary tables of particle properties, Chin. Phys. C38, 090001 (2014).
[71]
G. Leibbrandt, Introduction to the technique of dimensional regularization
, Rev. Mod. Phys. 47 (1975) 849.
[72]
D.P.Menezes, M. Benghi Pinto, S.S. Avancini, A. Perez Martinez, and C. Providencia, Quark matter under strong magnetic fields in the Nambu–Jona-Lasinio Model, Phys.Rev.C79 (2009) 035807. ArXiv:nucl-th/0811.3361.
[73]
M. Gell-Mann, The Quark and the Jaguar: Adventures in the Simple and the Complex, Macmillan (1995).
[74]
Y. Holovatch, R. Kenna, and S. Thurner, Complex systems: physics beyond physics
, Eur.J.Phys.38 (2017) 023002. ArXiv:Physics/1610.01002.
[75]
C. Tsallis, Introduction to nonextensive statistical mechanics: approaching a complex world © Springer Science BusinessMedia (2009).
[76]
C. Tsallis, Nonextensive statistical mechanics: Applications to high energy physics, Eur. Phys.J. Web of Conferences, 13 (2011) 05001.
[77]
A. Plastino and A. Plastino, Stellar polytropes and Tsallis’ entropy, Phys.Lett.A174 (1993) 384.
[78]
A. Plastino and A. R. Plastino, Tsallis Entropy and Jaynes’ information theory formalism, Braz. J.Phys. 29 (1999) 50.
[79]
E. Megias, D. P. Menezes, and A. Deppman, Nonextensive thermodynamics with finite chemical potentials and protoneutron stars, Eur. Phys.J. Web of Conferences, 80 (2014) 00040. ArXiv:hep-ph/1407.8044.
[80]
S. Abe and Y. Okamoto, Nonextensive statistical mechanics and its applications, Vol. 560 © Springer Science Business Media, (2001).
[81]
Tsallis, C. Possible generalization of Boltzmann-Gibbs statistics. J. Stat.Phys. 52 (1988) 479.
[82]
S. Umarov, C. Tsallis, S. Steinberg, On a q-Central Limit Theorem Consistent with Nonextensive Statistical Mechanics, Milan J. Math. 76, 307 (2008).
[83]
R. Rath, S. Tripathy, B. Chatterjee, R. Sahoo, S. Kumar Tiwari, and A. Nath, Violation of Wiedemann-Franz Law for Hot Hadronic Matter created at NICA, FAIR and RHIC Energies using Non-extensive Statistics, Eur. Phys. J. A 55, 125 (2019). ArXiv:hep-ph/1902.07922.
[84]
C. Tsallis, Nonextensive statistical mechanics: A brief review of its present status, Annals of the Brazilian Academy of Sciences, 74(3) (2002) 393. ArXiv:cond-mat/0205571.
[85]
C. Tsallis, Computational applications of nonextensive statistical mechanics, J.Comp.Appl.Math. 227 (2009) 51.
[86]
E. Megias, D.P. Menezes, A. Deppman, Non extensive thermodynamics for hadronic matter with finite chemical potentials, Physica A421 (2015) 15.
[87]
M. Peskin and D. Schroeder, An Introduction To Quantum Field Theory © Advanced Book Program. Basic Books, (1995).
[88]
http://dlmf.nist.gov. |
ON THE PROPAGATION OF NON STATIONARY
PRESSURE WAVES IN STELLAR INTERIORS
Patryk Mach
e-mail: mach@th.if.uj.edu.pl
Institute of Physics, Jagiellonian University,
Reymonta 4, 30-059 Kraków, Poland
Abstract
An analysis of the propagation of non stationary waves in the adiabatic region of stellar interior is presented. An equation of motion with an effective potential is derived, similar to the Zerilli equation known in the propagation of gravitational waves. The Huyghens principle is violated in this case and the energy diffusion outward null cones is expected. Numerical calculations demonstrate that the diffusion is weak for the case of standard Solar model; thus no significant effect corresponding to quasinormal modes can be expected. The likely reason for the absence of stronger features is the restriction of our analysis to adiabatic regions only, where the breakdown of the Huyghens principle is insignificant.
1 Motivation
In the standard helioseismology (or asteroseismology, to be more general) one usually deals with stationary perturbations of the gas pressure (or density) in stellar interiors. This issue has been well investigated so far, both from the theoretical and observational side (see \eg[2]). Here, by stationary approach we understand posing a hydrodynamical boundary problem leading to some characteristic frequencies that can be subsequently compared with the frequencies of observed stellar oscillations. There is no reason, however, why not to consider the propagation of non stationary waves in stellar interiors. It is known that waves propagating in an inhomogeneous medium can produce some non stationary effects such as, for instance, appearance of the quasinormal modes (for an example taken from the theory of perturbations of the Schwarzschild space-time see [6]). These are especially important from the observational point of view as their frequencies and damping coefficients are independent of the wave profile but depend on the characteristics of the medium. It is also known that in some cases the quasinormal modes can dominate [4].
In this paper we perform a simplified analysis of the problem which appears in the astrophysics of stellar interiors. The order of this work is as follows. In Section 2 we recall some basic formalism commonly used in the theory of stellar oscillations. Section 3 gives the description of the propagation of non stationary waves in the adiabatic region of the stellar interior together with the derivation of the exact form of the equation of motion. We transform this equation to the form of the Zerilli equation [7], [8] which one encounters in the theory of gravitational waves propagating in the Schwarzschild background metric (a case known to give significant quasinormal modes). The Lagrangian formulation of the problem is given in Section 4 while in Section 5 we deal with the Noether’s energy density and its diffusion through the characteristics. In Section 6 we present the effective potential occurring in the equation of motion, obtained for the case of standard solar model. Finally Section 7 shows the results of some numerical calculation of the propagation of non stationary pressure waves in the Sun. Some final remarks and conclusions are given in Section 8.
2 The formalism
In this section we shall remind some basic equations known from the Newtonian theory of stellar oscillations (or asteroseismology). We will not give any precise derivation here as it can be easily found in other papers. In turn, we will try to focus our attention mainly on putting down all the assumptions leading to the mentioned equations and we will explain all the notation we use.
We will consider the motion of gas in the star ruled by the Euler equation
$$\varrho\partial_{t}\mathbf{v}+\varrho\mathbf{v}\nabla\mathbf{v}=-\nabla p+%
\varrho\mathbf{g}$$
(1)
together with the continuity equation
$$\partial_{t}\varrho+\nabla(\mathbf{v}\varrho)=0.$$
(2)
Here $\varrho$ denotes the density, $p$ the pressure and $\mathbf{v}$ the velocity field of the gas. The term $\mathbf{g}$ in the equation (1) stands for the gravitational acceleration. In further considerations it will be, however, much more convenient to use the gravitational potential $\Phi$ instead of $\mathbf{g}$. We shall assume that a non minus convention $\mathbf{g}=\nabla\Phi$ holds. Then, the potential $\Phi$ satisfies Poisson’s equation of the form
$$\nabla^{2}\Phi=-4\pi G\varrho.$$
(3)
The above set of equations should be completed with one more, namely energy conservation equation (or the first law of thermodynamics)
$$\frac{dq}{dt}=\frac{1}{\varrho(\Gamma_{3}-1)}\left(\frac{dp}{dt}-\frac{\Gamma_%
{1}p}{\varrho}\frac{d\varrho}{dt}\right).$$
(4)
Here $q$ denotes specific heat (\ieheat per unit mass) and we have used standard thermodynamic notation (see \eg[3])
$$\Gamma_{1}=\left(\frac{\partial\ln p}{\partial\ln\varrho}\right)_{S},\>\Gamma_%
{3}-1=\left(\frac{\partial\ln T}{\partial\ln\varrho}\right)_{S}.$$
The next step is to derive a set of linearized equations describing the evolution of small perturbations of the equilibrium structure of the star. Under assumptions of adiabaticity of the motion and the spherical symmetry of the undisturbed medium one may obtain:
$$\varrho_{0}\partial_{t}^{2}\xi=-\partial_{r}p^{\prime}+\varrho_{0}\partial_{r}%
\Phi^{\prime}-\varrho^{\prime}g_{0},$$
(5)
$$-\partial_{t}^{2}\left(\varrho^{\prime}+\frac{1}{r^{2}}\partial_{r}(r^{2}%
\varrho_{0}\xi)\right)=-\nabla_{h}^{2}p^{\prime}+\varrho_{0}\nabla_{h}^{2}\Phi%
^{\prime},$$
(6)
$$\varrho^{\prime}=\frac{\varrho_{0}}{\Gamma_{1,0}p_{0}}p^{\prime}+\varrho_{0}%
\xi\left(\frac{1}{\Gamma_{1,0}}\frac{d\ln p_{0}}{dr}-\frac{d\ln\varrho_{0}}{dr%
}\right),$$
(7)
$$\frac{1}{r^{2}}\partial_{r}(r^{2}\partial_{r}\Phi^{\prime})+\nabla_{h}^{2}\Phi%
^{\prime}=-4\pi G\varrho^{\prime}.$$
(8)
The notation used here requires some explanation. The primed quantities denote Eulerian perturbations and are functions of both position $\mathbf{r}$ and time $t$, whereas quantities with zero indices describe the undisturbed medium (\iean equilibrium structure of the star) and are only functions of the distance from the center of the star, due to the spherical symmetry we have assumed. Thus, for instance
$$\displaystyle\varrho(\mathbf{r},t)$$
$$\displaystyle=$$
$$\displaystyle\varrho_{0}(r)+\varrho^{\prime}(\mathbf{r},t),$$
$$\displaystyle p(\mathbf{r},t)$$
$$\displaystyle=$$
$$\displaystyle p_{0}(r)+p^{\prime}(\mathbf{r},t),\dots$$
(9)
By $\xi$ we have denoted the radial part of the gas displacement, \ie
$$\delta\mathbf{r}=\xi\mathbf{e}_{r}+\mathbf{\xi}_{h},$$
(10)
where $\mathbf{e}_{r}$ stands for the unit vector in the radial direction and $\mathbf{\xi}_{h}$ is a horizontal part of the displacement vector. Finally $\nabla_{h}$ denotes the horizontal part of the gradient operator. The reader interested in rigorous derivation of the above equations may consult [2].
3 Non stationary perturbations, effective potential
The equations introduced in the preceding section possess a class of stationary solutions, which is actually one of the main interests of the theory of stellar oscillations. We may, however, try to look for the non stationary solutions that could, in fact, have some physical meaning. The aim of this section is to derive some simplified equations in the form suitable for further numerical search for such solutions.
We proceed with the separation of variables. All perturbations of our interest, such as $\xi$, $p^{\prime}$, $\varrho^{\prime}$ may be expanded in the series of spherical harmonics. To simplify the notation we will drop the adequate spherical harmonics indices in the amplitudes. Due to the linearity of the obtained equations it is sufficient to write
$$\displaystyle\xi(r,\theta,\phi,t)$$
$$\displaystyle=$$
$$\displaystyle\tilde{\xi}(r,t)Y_{lm}(\theta,\phi),$$
$$\displaystyle p^{\prime}(r,\theta,\phi,t)$$
$$\displaystyle=$$
$$\displaystyle\tilde{p}(r,t)Y_{lm}(\theta,\phi),\dots$$
Taking into the account that
$$\nabla_{h}^{2}Y_{l}=-\frac{l(l+1)}{r^{2}}Y_{l},$$
and substituting the above expressions to the equations (5), (6), (7) and (8) we get
$$\varrho_{0}\partial_{t}^{2}\tilde{\xi}=-\partial_{r}\tilde{p}+\varrho_{0}%
\partial_{r}\tilde{\Phi}-\tilde{\varrho}g_{0},$$
(11)
$$-\partial_{t}^{2}\tilde{\varrho}-\frac{1}{r^{2}}\partial_{r}(r^{2}\varrho_{0}%
\partial_{t}^{2}\tilde{\xi})=\frac{l(l+1)}{r^{2}}(\tilde{p}-\varrho_{0}\tilde{%
\Phi}),$$
(12)
$$\frac{1}{r^{2}}\partial_{r}(r^{2}\partial_{r}\tilde{\Phi})-\frac{l(l+1)}{r^{2}%
}\tilde{\Phi}=-4\pi G\tilde{\varrho}$$
(13)
and
$$\tilde{\varrho}=\frac{\varrho_{0}}{\Gamma_{1,0}p_{0}}\tilde{p}+\varrho_{0}%
\tilde{\xi}\left(\frac{1}{\Gamma_{1,0}}\frac{d\ln p_{0}}{dr}-\frac{d\ln\varrho%
_{0}}{dr}\right).$$
(14)
The mentioned stationary solutions can now be obtained by setting
$$\tilde{f}(r,t)=\hat{f}(r)e^{-i\omega t}$$
for each amplitude $\tilde{f}(r,t)$ of a hydrodynamical variable $f$. By completing ordinary differential equations obtained this way with the suitable boundary conditions we can determine characteristic frequencies $\omega$ of the oscillation modes.
We will, however, try to proceed in a different way. Instead of posing an eigenvalue, boundary problem, we will try to formulate some Cauchy problem with just one, second order, partial differential equation, describing the time evolution of an initial perturbation. Additionally, we will not consider any boundary conditions and thus we will treat a star as a formally infinitely distributed medium. Of course, we will not consider the propagation of the perturbations beyond the assumed radius of the star.
We shall now formulate two important, simplifying assumptions:
1.
We will put $\tilde{\Phi}\equiv 0$ everywhere. This assumption was examined for the first time by Cowling. It means simply neglecting the changes in the gravitational field that arise due to small perturbations of the density in the star.
2.
We will assume that the medium is adiabatic, \ie
$$\frac{1}{\Gamma_{1,0}}\frac{d\ln p_{0}}{dr}-\frac{d\ln\varrho_{0}}{dr}=0.$$
(15)
With this assumption we eliminate the propagation of the internal gravity waves, focusing attention only on the pressure waves in the star. Now, of course, we will have to examine, whether the star region in which the perturbation propagates is indeed adiabatic. The convective zone in the stellar atmosphere can serve as an example of such region.
Under above assumptions the considered set of equations reduces to the following form
$$-\partial_{t}^{2}\tilde{\varrho}-\frac{1}{r^{2}}\partial_{r}(r^{2}\varrho_{0}%
\partial_{t}^{2}\tilde{\xi})=\frac{l(l+1)}{r^{2}}\tilde{p},$$
(16)
$$\varrho_{0}\partial_{t}^{2}\tilde{\xi}=-\partial_{r}\tilde{p}-\tilde{\varrho}g%
_{0},$$
(17)
$$\tilde{\varrho}=\frac{\varrho_{0}}{\Gamma_{1,0}p_{0}}\tilde{p}.$$
(18)
We may now put the equation (18) into the equation (16) to obtain
$$-\frac{\varrho_{0}}{\Gamma_{1,0}p_{0}}\partial_{t}^{2}\tilde{p}-\frac{1}{r^{2}%
}\partial_{r}(r^{2}\varrho_{0}\partial_{t}^{2}\tilde{\xi})=\frac{l(l+1)}{r^{2}%
}\tilde{p}.$$
The term $\partial_{t}^{2}\tilde{\xi}$ in the last equation can be subsequently eliminated using (17). Taking into account that $-g_{0}\varrho_{0}=dp_{0}/dr$ and remembering the condition (15) we get after some calculations
$$\displaystyle-\frac{\varrho_{0}}{\Gamma_{1,0}p_{0}}\partial_{t}^{2}\tilde{p}+%
\partial_{r}^{2}\tilde{p}+\left(\frac{2}{r}-\frac{d\ln\varrho_{0}}{dr}\right)%
\partial_{r}\tilde{p}+$$
$$\displaystyle-\left(\frac{2}{r}\frac{d\ln\varrho_{0}}{dr}+\frac{d^{2}\ln%
\varrho_{0}}{dr^{2}}+\frac{l(l+1)}{r^{2}}\right)\tilde{p}$$
$$\displaystyle=$$
$$\displaystyle 0.$$
We can now get rid of the term containing the derivative $\partial_{r}\tilde{p}$ by introducing a new dynamical variable $P$, defined with the equation
$$\tilde{p}=\frac{\sqrt{\varrho_{0}}}{r}P.$$
Indeed, after some calculations we obtain an equation of the form
$$-\frac{1}{c^{2}(r)}\partial_{t}^{2}P+\partial_{r}^{2}P-V(r)P=0,$$
(19)
where
$$c^{2}(r)=\frac{\Gamma_{1,0}p_{0}}{\varrho_{0}}.$$
(20)
By $V$ we have denoted here a function playing the role of an effective potential
$$V(r)=\frac{1}{r}\frac{d\ln\varrho_{0}}{dr}+\frac{1}{2}\frac{d^{2}\ln\varrho_{0%
}}{dr^{2}}+\frac{1}{4}\left(\frac{d\ln\varrho_{0}}{dr}\right)^{2}+\frac{l(l+1)%
}{r^{2}}.$$
(21)
Equation (19) can be still transformed into even more convenient form. We will eliminate the term $c^{-2}$ standing before the time derivative of $P$ by introducing a new coordinate
$$r^{\ast}=\int\limits_{0}^{r}\frac{dr^{\prime}}{c(r^{\prime})}.$$
(22)
Thus an obvious relation
$$\frac{dr^{\ast}}{dr}=\frac{1}{c(r)}$$
holds and the equation (19) may be written as
$$-\partial_{t}^{2}P+\partial_{r^{\ast}}^{2}P-\frac{d\ln c}{dr^{\ast}}\partial_{%
r^{\ast}}P-c^{2}VP=0.$$
We will get rid of the term proportional to $\partial_{r^{\ast}}P$ in a way similar to that we had used before. We define a function $\Pi$ with a relation
$$P=\sqrt{c}\Pi$$
to derive the final version of our equation of motion
$$-\partial_{t}^{2}\Pi+\partial_{r^{\ast}}^{2}\Pi-\tilde{V}\Pi=0$$
(23)
in which a new effective potential
$$\tilde{V}=c^{2}V+\frac{1}{4}\left(\frac{d\ln c}{dr^{\ast}}\right)^{2}-\frac{1}%
{2}\frac{d^{2}\ln c}{dr^{\ast 2}}$$
(24)
has been introduced.
4 Lagrangian description, energy
One may notice that the equation (23) can be obtained from the variational principle
$$\delta_{\Pi}S=0$$
by taking an action $S$ of the form
$$S=\int\mathcal{L}dtdr^{\ast}=-\frac{1}{2}\int\left((\partial_{t}\Pi)^{2}-(%
\partial_{r^{\ast}}\Pi)^{2}-\tilde{V}\Pi^{2}\right)dtdr^{\ast}.$$
(25)
The equation (23) appears then as the Euler–Lagrange equation for the Lagrangian density $\mathcal{L}$, \ie
$$\partial_{\Pi}\mathcal{L}-\partial_{t}\frac{\partial\mathcal{L}}{\partial%
\partial_{t}\Pi}-\partial_{r^{\ast}}\frac{\partial\mathcal{L}}{\partial%
\partial_{r^{\ast}}\Pi}=0.$$
(26)
We can now make use of the first Noether theorem applied to the action (25), what should allow us to define an energy for the $\Pi$ amplitudes. We will present this issue in a little detail due to some subtle matters that appear here.
We begin with considering an infinitesimal time translation of some domain $\Omega$
$$\psi\colon\mathbb{R}^{2}\supset\Omega\ni(t,r)\mapsto(t^{\prime},r)=(t-%
\varepsilon,r)\in\mathbb{R}^{2},$$
which will be assumed to be a symmetry which means that a variation of the action (25) caused by this transformation vanishes in the domain $\Omega$. If in addition we assume that the motion happens to be real (the Euler–Lagrange equation (26) is satisfied) then after some calculations we obtain
$$\delta S=\varepsilon\int_{\partial\Omega}\left(\left(\mathcal{L}-\frac{%
\partial\mathcal{L}}{\partial\partial_{t}\Pi}\partial_{t}\Pi\right)dr^{\ast}-%
\left(\frac{\partial\mathcal{L}}{\partial\partial_{r^{\ast}}\Pi}\partial_{t}%
\Pi\right)dt\right)=0.$$
(27)
The amplitude $\Pi$ is defined in a half plane $r^{\ast}\geqslant 0$. As a domain $\Omega$, over which we proceed with integration we may now take a part of that half plane enclosed between two constant time lines, given by the equations $t=t_{1}$ and $t=t_{2}$.
Let us notice, that the amplitudes $\Pi$ have appeared in the separation of variables in the $(3+1)$ dimensional problem and, therefore, have to satisfy some additional conditions. In particular $\tilde{p}$ needs to be finite in its domain and thus also at $r^{\ast}=r=0$. It follows simply that an equality $\Pi(r^{\ast}=0,t)=0$, and so $\partial_{t}\Pi(r^{\ast}=0,t)=0$ must hold. Therefore, for the $\Omega$ chosen above we have
$$\int_{\partial\Omega}\left(\frac{\partial\mathcal{L}}{\partial\partial_{r^{%
\ast}}\Pi}\right)dt=\int\limits_{t_{1}}^{t_{2}}\left(\frac{\partial\mathcal{L}%
}{\partial\partial_{r^{\ast}}\Pi}\partial_{t}\Pi\right)_{r^{\ast}=0}dt=0.$$
Finally it follows that the quantity
$$\int\limits_{0}^{\infty}\left(\mathcal{L}-\frac{\partial\mathcal{L}}{\partial%
\partial_{t}\Pi}\partial_{t}\Pi\right)dr^{\ast}$$
is conserved, \ieconstant in time. The expression
$$\mathcal{E}=\mathcal{L}-\frac{\partial\mathcal{L}}{\partial\partial_{t}\Pi}%
\partial_{t}\Pi=\frac{1}{2}\left((\partial_{t}\Pi)^{2}+(\partial_{r^{\ast}}\Pi%
)^{2}+\tilde{V}\Pi^{2}\right)$$
(28)
can thus be interpreted as an energy density.
In consistency with the comment made earlier, we have assumed here that a medium in which the waves propagate is infinite and all perturbations vanish at least at infinity.
5 Energy diffusion
Let us consider now $\gamma$, being a part of the incoming characteristic of the equation (23), with an origin in the point with the coordinates $r^{\ast}=r_{1}$, $t=t_{1}$. The characteristic $\gamma$ divides the domain $\Omega$ enclosed between two constant time lines $t=t_{1}$ and $t=t_{2}$ into two subdomains: the inner one, $\Omega_{1}$, and the outer one $\Omega_{2}$ (Fig. 1). Let us consider next a point with the coordinate $r^{\ast}=R$ lying on the $\gamma$. The expression
$$j(R,t)=(-\partial_{R}+\partial_{t})\int\limits_{R}^{\infty}\mathcal{E}dr^{\ast}$$
(29)
may be interpreted as a rate of the energy change along $\gamma$. A straightforward calculation making use of equation (28) and of the motion equation (23) shows that
$$j(R,t)=\frac{1}{2}\left((\partial_{t}\Pi-\partial_{r^{\ast}}\Pi)^{2}+\tilde{V}%
\Pi^{2}\right)_{r^{\ast}=R}.$$
Calculations leading to the above result may be simplified even more by noticing that
$$\partial_{t}\frac{1}{2}\int\limits_{R}^{\infty}\mathcal{E}dr^{\ast}=\left(-%
\frac{\partial\mathcal{L}}{\partial\partial_{r^{\ast}}\Pi}\partial_{t}\Pi%
\right)_{r^{\ast}=R},$$
what in fact we had obtained earlier by writing formula (27).
The whole energy $\Delta E$, that diffused from a domain $\Omega_{1}$ to the domain $\Omega_{2}$ during a time between $t_{1}$ and $t_{2}$ may be calculated as an integral of the expression (29) over $\gamma$
$$\displaystyle\Delta E$$
$$\displaystyle=$$
$$\displaystyle\int\limits_{t_{1}}^{t_{2}}j(R,t)_{R=-t+r_{1}+t_{1}}dt=$$
$$\displaystyle=$$
$$\displaystyle\int\limits_{t_{1}}^{t_{2}}\frac{1}{2}\left((\partial_{t}\Pi-%
\partial_{r^{\ast}}\Pi)^{2}+\tilde{V}\Pi^{2}\right)_{r^{\ast}=-t+r_{1}+t_{1}}dt.$$
Let us now imagine that an incoming perturbation of compact support enclosed initially in the domain $\Omega_{1}$ moves along the characteristic $\gamma$. The quantity $\Delta E$ given by the expression (5) would represent the whole energy scattered outwards (into the domain $\Omega_{2}$) during the period between $t_{1}$ and $t_{2}$. This scattering is mathematically due to the non-vanishing potential $\tilde{V}$ and, to be more precise, to a potential that differs from a single term proportional to the $l(l+1)/r^{2}$ which will always arise from the separation of variables in the wave problem of spherical symmetry.
Experience tells us that robust energy diffusion signals the presence of quasinormal modes. Since quasinormal modes are rather difficult to be found numerically, we prefer to deal with examining the energy diffusion instead [4].
In the next sections of this paper we will concern ourselves with examining such energy scattering for a realistic case, namely standard solar model.
6 Effective potential for a standard solar model
We will now apply the results of the preceding sections to a standard solar model. We have already stated, that a fundamental assumption of our simplified model is that of adiabaticity of the region in which the waves propagate, and that validity of this assumption need to be carefully examined. The standard solar model has been chosen because of the existence of the convective zone in which the condition (15) is satisfied up to a high degree.
All our numerical calculations were made on the basis of a standard solar model computed by Bahcall, Pinsonneault & Basin [1]. The obtained relation between the variable $r^{\ast}$ and the radius $r$ is shown on the Fig. 2. It should be noticed here, that the $r^{\ast}$ variable was normalized in such a way that it changes between the values 0 and 1 for the used data range. Next, the function
$$\frac{N^{2}}{g_{0}}=\frac{1}{\Gamma_{1,0}}\frac{d\ln p_{0}}{dr}-\frac{d\ln%
\varrho_{0}}{dr}$$
is plotted versus $r^{\ast}$ in Fig. 3. We have adopted the notation $N^{2}/g_{0}$ here, as $N$ corresponds to the well known Brunt–Väisälä frequency. This plot shows, that we may regard the area with approximately $r^{\ast}\gtrapprox 0.58$ as satisfying our assumption of adiabaticity. The effective potential, i.e. $\tilde{V}$ for $l=0$ is, in turn, plotted in Fig. 4. Unfortunately the only one interesting feature of this potential, that is a clear peak at $r^{\ast}\sim 0.5$, remains outside the adiabaticity area. Therefore we can not consider any effects caused by the existence of this peak as being physically meaningful.
One remark should be made here concerning the figures presented in this section. It is not possible to differentiate the data presented by Bahcall et al.. in any straightforward way to obtain any smooth enough functions and therefore some smoothing procedure appears to be necessary. A slight modification of the so called Savitzky–Golay filter (see \eg[5]) was used here to obtain presented result.
7 Example of some numerical calculations
Finally it was possible to examine, how an initially incoming wave package evolves. Fig. 5 presents an example result of the numerical experiment explained in Section 5.
For simplicity, only a spherically symmetric perturbation \iethe case with $l=0$ is considered here. Propagation of all other modes may be examined in exactly the same way by taking the potential $\tilde{V}$ for an arbitrary $l$ number.
Here, the characteristic $\gamma$ was chosen to originate at $r^{\ast}=0.845$ and the initial (\iefor $t=0$), purely incoming perturbation was taken to be a function of the shape defined by
$$\Pi(r^{\ast},t=0)=\left\{\begin{array}[]{cl}A\sin^{2}\left(\frac{\pi(r^{\ast}-%
a)}{b-a}\right),&\mbox{if }r^{\ast}\in[a,b],\\
0,&\mbox{otherwise,}\end{array}\right.$$
thus being just one, bell like part of the squared sine function, centered on a compact support $[a,b]$. This is, of course, a continuous and differentiable function.
We have also examined the case with an initial perturbation in the form of standard $C^{\infty}$ class function of a compact support. Such function may be constructed in a well known way with an use of the exponential function. It has also a bell like shape but it gives lower FWHM to support length ratio. It appears that the squared sine function is much more useful for our purpose as, basically, the amount of scattered energy increases with an increase of the FWHM of the initial perturbation.
In the presented case, $a$ and $b$ were given the values $a=0.73$ and $b=0.845$ which correspond to the initial data support lying entirely in the $\Omega_{1}$ domain (see Fig. 1). The plot presented on Fig. 5 shows the time derivative of the energy that has diffused from the domain $\Omega_{1}$ to the outside domain $\Omega_{2}$ divided by the whole initial perturbation energy. The variable $r^{\ast}$ and the units of $t$ (approximately 2500 s per unit) are defined in such a way that the sound speed expressed in these coordinates equals unity. Thus, looking at Fig. 5 it is easy to see how far could the initial perturbation arrive for a given time. Clearly, the large peak in the scattered energy corresponds to the mentioned bump in the effective potential which, as it has been already stated, cannot be considered in a convincing way as giving any results of physical meaning. It is, however, a good example of effects that may arise in the propagation of the non stationary waves due to the inhomogeneity of the medium.
In our considerations we are of course restricted to the area where the adiabaticity assumption is satisfied. In fact, even in this area some diffusion of energy does occur but on a negligible scale. Fig. 6 shows the energy that has diffused through the characteristic as a function of time. These are in fact the same data which we have already plotted on the Fig. 5 but this time restricted to the times lower than 0.2 what corresponds to the propagation in the adiabatic zone.
8 Final remarks
It is already well known that non stationary waves can carry interesting information about an inhomogeneous medium in which they propagate [6], [7], [8], [4]. In this paper we have examined the propagation of such waves in the Solar convective zone by looking at the energy diffusion process. It appears that the energy scattering occurs with rather negligible efficiency and, consequently, we expect the quasinormal modes to be absent. This may, however, follow from the fact that we have restricted ourselves to the, perhaps not interesting, simplified case of adiabatic media. It is possible that the investigation of the full model which takes into account all important physical aspects would lead to some positive results. It is also possible that positive results can be obtained by repeating the calculations presented in this paper for the models of some other stars.
Acknowledgements
I wish to thank Professor Edward Malec for showing me the issue of non stationary waves effects and his great help in doing this work.
References
[1]
J. N. Bahcall, M. H. Pinsonneault, S. Basin, ApJ. 555, 990 (2001).
[2]
J. Christensen–Dalsgaard, Lecture Notes on Stellar Oscillations, http://astro.phys.au.dk/~jcd/oscilnotes/, 2003.
[3]
J. P. Cox, R. T. Giuli, Principles of Stellar Structure, Gordon and Breach, New York 1968.
[4]
J. Karkowski, K. Roszkowski, Z. Świerszczyński, E. Malec, Phys. Rev. D 67, 064024 (2003).
[5]
W. H. Press, S. A. Teukolsky, W. T. Vetterling et al., Numerical Recipes in C, Cambridge University Press, 2002.
[6]
C. V. Vishveshwara, Phys. Rev. D 1, 2870 (1970).
[7]
F. J. Zerilli, Phys. Rev. Lett. 24, 737 (1970).
[8]
F. J. Zerilli, Phys. Rev. D 2, 2141 (1970). |
Iterative detection and decoding for SCMA systems with LDPC codes
Baicen Xiao, Kexin Xiao, Shutian Zhang, Zhiyong Chen, Bin Xia and Hui Liu
Department of Electronic Engineering, Shanghai Jiao Tong University, Shanghai, P. R. China
Email: {xinzhiniepan, kexin.xiao, zhangshutian, zhiyongchen, bxia, huiliu}@sjtu.edu.cn
Abstract
Sparse code multiple access (SCMA) is a promising multiplexing approach to achieve high system capacity. In this paper, we develop a novel iterative detection and decoding scheme for SCMA systems combined with Low-density Parity-check (LDPC) decoding. In particular, we decompose the output of the message passing algorithm (MPA) based SCMA multiuser detection into intrinsic part and prior part. Then we design a joint detection and decoding scheme which iteratively exchanges the intrinsic information between the detector and the decoder, yielding a satisfied performance gain. Moreover, the proposed scheme has almost the same complexity compared to the traditional receiver for LDPC-coded SCMA systems. As numerical results demonstrate, the proposed scheme has a substantial gain over the traditional SCMA receiver on AWGN channels and Rayleigh fading channels.
I Introduction
Sparse code multiple access (SCMA) has attracted much attention since it is capable of supporting massive connections simultaneously, yielding a competitive candidate for Fifth Generation (5G) communications[1]. Commonly, SCMA can be viewed as a generalization of sparse spread CDMA[2], with a few numbers of nonzero elements within a signature. For an uplink SCMA system, when each user is assigned a specific codebook, the multiplexing becomes a superposition scheme which will obtain the shaping gain. However, on the receiver side serious multiple address interference (MAI) is the main obstacle to implement multiuser detection. The optimum maximum a posterior (MAP) algorithm obviously shows the best performance with considerable complexity. In order to tackle the high complexity of MAP algorithm, some low complexity algorithm are proposed to handle this NP-complete problem[3] within tolerable performance loss. Especially, thanks to the sparse structure of SCMA, the complex MAP formula can be solved iteratively with sum-product algorithm or message passing algorithm (MPA)[4].
Lately, in order to improve the bit error rate (BER) performance of SCMA, [5] has introduced the Turbo-principle, which is widely used in detection and decoding problems such as joint source-channel coding [6, 7] and multiuser detection [8, 9], to exchange information between the SCMA detector and the channel decoder. However, the proposed Turbo-like scheme in [5] doesn’t take full advantage of the iterative structure of SCMA detection and suffers from high complexity proportional to the number of outer iterations.
For the sake of taking full advantage of the iterative characteristic of SCMA detection, we need a kind of channel coding which applies iterative decoding. Low-Density Parity-Check (LDPC) codes which are excellent error correcting codes providing a large coding gain [10] and adopt iterative decoding, are especially suitable for our requirments.
The goal of this paper is to apply a novel Turbo-like combination of SCMA multiuser detection and LDPC decoding. The difference from [5] should be noted that this paper do a novel Turbo-like combination of iterative detection and iterative decoding, i.e, during each outer loop only partial inner iterations in detector and decoder are implemented, to obtain a satisfied performance gain with almost the same complexity compared to the receiver without Turbo-like scheme. Firstly, we investigate in detail the MPA-based SCMA multiuser detector from the perspective of solving marginal function and then deduce the SCMA multiuser detection algorithm in logarithmic form. Furthermore, the intrinsic information is decomposed from the output of both SCMA multiuser detector and LDPC decoder and the way intrinsic information interacts between detector and decoder is presented. As numerical results show, this scheme achieves a 0.9 dB performance gain in terms of BER with almost the same complexity for both AWGN channels and Rayleigh fading channels compared to traditional receiver for LDPC-coded SCMA systems.
For the sake of clarity, throughout this paper, the sets of binary and complex numbers are denoted by $\mathbb{B}$ and $\mathbb{C}$, respectively. Upper-case calligraphic symbols $\mathcal{X}$ denote constellation sets and log$(\cdot)$ denotes natural logarithm. To represent a scalar, a vector and a matrix, we use $x$, $\mathbf{x}$ and $\mathbf{X}$, respectively.
II System model
We consider an uplink LDPC-coded SCMA system with $J$ users and $K$ resources, and signaling through fading channels with additive white Gaussian noise (AWGN), as shown in Fig. 1.
For each user $j$, $j$ = 1, $\cdots$ , $J$, data bits {$d_{n_{0}}^{j}\mid n_{0}=1,2,\cdots,n$} are first encoded into {$b_{m_{0}}^{j}\mid m_{0}=1,2,\cdots,m$} by an LDPC encoder with code rate $R_{j}=n/m$. In order to reduce error bursts and take advantage of diversity gain, the coded bits are permuted by an interleaver $\pi_{j}$. Every log${}_{2}$$(M)$ interleaved coded bits $\{b_{m^{\pi}}^{~{}j}|m^{\pi}=1,2,\cdots,\text{log}_{2}(M)\}$ is grouped together and then mapped by SCMA mapper $f_{j}$ into a $K$-dimensional complex symbol as $f_{j}$: $\mathbb{B}^{\text{log}_{2}(M)}\rightarrow{\mathbf{x}}_{j}\in\mathcal{X}^{j}%
\subset\mathbb{C}^{K}$ with cardinality $|\mathcal{X}^{j}|$ = $M$. Because of the sparsity of SCMA, a $K$-dimensional symbol ${\mathbf{x}}_{j}$ consists of $N_{j}$ $<$ $K$ non-zero elements, each corresponding to an OFDMA tone or other resources. For the receiver end, the received siganls are the superposition of $J$ users’ signals and ambient noise, which can be written in a discrete-form, if memoryless channel considered, as
$$\ \mathbf{y}=\sum_{j=1}^{J}{\text{diag}(\mathbf{h}_{j}){\mathbf{x}}_{j}}+%
\mathbf{n}\text{,}$$
(1)
where $\mathbf{y}=(y_{1},\cdots,y_{K})^{\text{T}}$ is the received signal vector, $\mathbf{h}_{j}=(h_{1j},\cdots,h_{Kj})^{\text{T}}$ is the channel vector for user $j$, $\mathbf{x}_{j}=(x_{1j},\cdots,x_{Kj})^{\text{T}}$ is the symbol transmitted by user $j$, and $\mathbf{n}$ is a white Gaussian noise vector subject to $\mathcal{CN}(0,N_{0}\mathbf{I})$. Apparently, from (1) the received signal at resource $k$ can be written as
$$\ y_{k}=\sum_{j=1}^{J}{h_{k,j}x_{k,j}}+n_{k},k=1,\cdots,K.$$
(2)
It is easy to recognize from (2) that each user sees interference from other $K-1$ users. However, since the signal vector from arbitrary user is sparse, i.e., not all users contribute to the k-th resource, the interference is hence reduced and (2) can be re-written as
$$\ y_{k}=\sum_{j\in{\partial k}}{h_{k,j}x_{k,j}}+n_{k},k=1,\cdots,K.$$
(3)
where $\partial k$ denotes the users contributing to the $k$-th resource, called the neighborhood of node $k$, and this relationship which is decided by the SCMA mapper can be presented by factor graph and indicator matrix $\mathbf{C}$. Let $d_{j}$ and $d_{k}$ be the number of resources occupied by user $j$ and the number of users resource $k$ is connected, respectively. For the sake of clarity, we give an example of SCMA factor graph. Assuming $d_{j}=2$ for all $j$ and $d_{k}=3$ for all $k$, the factor representation is shown as Fig. 2, and the corresponding indicator matrix is
$$\mathbf{C}=\left[\begin{array}[]{cccccc}1&1&1&0&0&0\\
1&0&0&1&1&0\\
0&1&0&1&0&1\\
0&0&1&0&1&1\end{array}\right]$$
therein, $c_{k,i}=1$ means resource $k$ is occupied by user $i$.
It’s observed that this factor graph is regular, i.e., all the user nodes have the same degree, so do the resource nodes. It should be emphasized that this regular structure may not be the best, in another word, it’s possible for a irregular structure to play a better performance if there are diversities between the quality of different resources. In this paper, however, our attention is only focused on regular SCMA structure, but some of the results can be easily extended to the irregular scenario and the irregular structure will be our future work.
III Iterative detection and decoding
In this section, we analyse SCMA detection based on MPA iterative algorithm, and propose an effective combination method of SCMA multiuser detection and LDPC decoding.
III-A SCMA detection
Firstly, we describe why the MPA algorithm can be applied to SCMA detection from the perspective of solving marginal function. Given received signal $\mathbf{y}$ and assuming ideal channel estimation, a SCMA detection based on MAP is to choose a matrix $\hat{\mathbf{X}}=(\mathbf{x}_{1},\cdots,\mathbf{x}_{J})$ to maximizing the joint $a~{}posterior$ pmf, which is expressed as
$$\ \hat{\mathbf{X}}=\arg\max_{\mathbf{X}\in{\mathcal{X}^{KJ}}}p(\mathbf{X}|%
\mathbf{y})\text{,}$$
(4)
where $\mathcal{X}^{KJ}$ denotes the set of all possible symbols, i.e., the $j$-th column of $\mathcal{X}^{KJ}$ is the set $\mathcal{X}^{j}$ described in Section II. In order to estimate the information of user $j$, we can choose a $\mathbf{\hat{x}}_{j}$ to maximize the marginal $a~{}posterior$ pmf with respect to $\mathbf{x}_{j}$ as
$$\ \hat{\mathbf{x}}_{j}=\arg\max_{z\in\mathcal{X}^{j}}\sum_{\begin{subarray}{c}%
\mathbf{X}\in{\mathcal{X}^{K\!J}}\\
{\mathbf{x}_{j}=z}\end{subarray}}{p(\mathbf{X}|\mathbf{y})}\text{.}$$
(5)
Applying Bayes’s rule, we can get
$$\ p(\mathbf{X}|\mathbf{y})=\frac{p(\mathbf{y}|\mathbf{X})p(\mathbf{X})}{p(%
\mathbf{y})}$$
(6)
$$\ \propto p(\mathbf{y}|\mathbf{X})p(\mathbf{X})=p(\mathbf{X})\prod_{k=1}^{K}{p%
(y_{k}|\mathbf{X})}\text{.}$$
(7)
The last equation follows the fact that the elements of noise vector is identically independent distributed (i.i.d.) and uncorrelated with transmitted symbols, hence once the transmitted symbols are given, different dimension of received signal $\mathbf{y}$ are independent.
Since $y_{k}$ is influenced by parts of users, i.e., users from $\partial k$, equation (5) can be reduced to
$$\ \hat{\mathbf{x}}_{j}=\arg\max_{z\in\mathcal{X}^{j}}\sum_{\begin{subarray}{c}%
\mathbf{X}\in{\mathcal{X}^{K\!J}}\\
{\mathbf{x}_{j}=z}\end{subarray}}{p(\mathbf{X})\prod_{k=1}^{K}{p(y_{k}|\mathbf%
{x}_{p},p\in{\partial k})}}~{}~{}\forall j\text{.}$$
(8)
Furthermore transmitted symbols from different users are independent, $p(\mathbf{X})$ can be written in a product form as $p(\mathbf{X})=\prod_{q=1}^{J}{p(\mathbf{x}_{q})}$, and it’s clear that the sum terms in equation (8) can be given by
$$\ f(\mathbf{x}_{1},\cdots,\mathbf{x}_{J})=\prod_{q=1}^{J}{p(\mathbf{x}_{q})}%
\prod_{k=1}^{K}{p(y_{k}|\mathbf{x}_{p},p\in{\partial k})}\text{.}$$
(9)
Traditionally, to solve equation (8) needs $J$ operations where redundant computing exits. Because of the product form (9), thanks to the method in [4], we can solve a marginal function problem of multivariable function like (8) iteratively based on factor graph, which is usually called sum-product algorithm or MPA algorithm. Each node in the factor graph sends “belief message” to its neighbors during each iteration and the “belief message” shouldn’t be sent back during the next iteration, hence the inference can be made sufficiently in the graph after some iterations. It should be emphasized that when the factor graph of equation (9) is cycle-free MPA algorithm is able to produce the accurate marginal function [2], but for the factor graph with cycles this algorithm is suboptimal.
The iterative SCMA multiuser detection algorithm based on MPA is presented in algorithm 1.
Therein£¬ “$\partial k\backslash j$” denotes the neighborhood of node $k$ excluding node $j$ and the notation “$\sum_{(\mathbf{x}_{p})\partial k\backslash j}$” denotes sum over all possible values of $\mathbf{x}_{p}\in\mathcal{X}^{p}$ for all $p\in\partial k\backslash j$. In this algorithm, we assume all possible symbols are equiprobable, that is to say there is no prior information. When prior information is provided by channel decoder as described later, $p(\mathbf{x}_{j})$ should be updated. For the convenience of implementation, we then derive the SCMA detection algorithm in logarithmic form, given prior information in the form of the log-likelihood ratio. Note that one of our goal is to decompose the output of SCMA detector into intrinsic information and prior information in order to do a novel Turbo-like combination with LDPC decoder in subsection $C$.
Once the prior information $\{L^{s,p}(\mathcal{X}^{j})\}_{j=1,\cdots,J}$ is given, here the superscript “s” denotes the LLR is with respect to symbol, the posterior probability of $\mathbf{X}$ can be written as
$$\begin{split}\displaystyle\ p(\mathbf{X}|\mathbf{y},&\displaystyle\{L^{s,p}(%
\mathcal{X}^{j})\}_{j=1,\cdots,J})=\\
&\displaystyle C\cdot p(\mathbf{y}|\mathbf{X})p(\mathbf{X}|\{L^{s,p}(\mathcal{%
X}^{j})\}_{j=1,\cdots,J})\text{,}\end{split}$$
(12)
where $C=\frac{p(\{L^{s,p}(\mathcal{X}^{j})\}_{j=1,\cdots,J})}{p(\mathbf{y},\{L^{s,p}%
(\mathcal{X}^{j})\}_{j=1,\cdots,J})}$ is a constant during each inner loop of SCMA detection. Comparing equation (6) with (12), it’s easy to show that only the initialization part of algorithm 1 should be modified when prior information is available. Then “$p(\mathbf{x}_{j})\leftarrow\frac{1}{M}$” should be replaced by “$p(\mathbf{x}_{j})\leftarrow p(\mathbf{x}_{j}|L^{s,p}(\mathbf{x}_{j}))$”.
Let us fix a reference point $\mathbf{\tilde{x}}_{j}$ , which denotes all “1” transmitted to facilitate the following deduction, for each user $j$. Let $LV_{j\rightarrow k}^{t}(\mathbf{x}_{j})$ and $LU_{k\rightarrow j}^{t}(\mathbf{x}_{j})$ be the information in logarithmic form from user node $j$ to resource node $k$ and from resource node $k$ to user node $j$, respectively, then,
$$\begin{split}\displaystyle LV_{j\rightarrow k}^{t}(\mathbf{x}_{j})&%
\displaystyle=\text{log}\frac{V_{j\rightarrow k}^{t}(\mathbf{x}_{j})}{V_{j%
\rightarrow k}^{t}(\mathbf{\tilde{x}}_{j})}\\
&\displaystyle=\text{log}\frac{p(\mathbf{x}_{j}|L^{s,p}(\mathbf{x}_{j}))\prod_%
{s\in{\partial j}\backslash k}{U_{s\rightarrow j}^{t-1}(\mathbf{x}_{j})}}{p(%
\mathbf{\tilde{x}}_{j}|L^{s,p}(\mathbf{\tilde{x}}_{j})\prod_{s\in{\partial j}%
\backslash k}{U_{s\rightarrow j}^{t-1}(\mathbf{\tilde{x}}_{j})}}\\
&\displaystyle=L^{s,p}(\mathbf{x}_{j})+\sum_{s\in{\partial j}\backslash k}LU_{%
k\rightarrow j}^{t-1}(\mathbf{x}_{j})\text{,}\end{split}$$
(13)
$$\begin{split}\displaystyle LU_{k\rightarrow j}^{t}&\displaystyle(\mathbf{x}_{j%
})=\text{log}\frac{\sum_{(\mathbf{x}_{p})\partial k\backslash j}\text{exp}[f_{%
k}(\mathbf{x}_{j})]\prod_{p\in{\partial k}\backslash j}V_{p\rightarrow k}^{t}(%
\mathbf{x}_{p})}{\sum_{(\mathbf{x}_{p})\partial k\backslash j}\text{exp}[f_{k}%
(\mathbf{\tilde{x}}_{j})]\prod_{p\in{\partial k}\backslash j}V_{p\rightarrow k%
}^{t}(\mathbf{x}_{p})}\\
&\displaystyle=\text{log}\frac{\sum_{(\mathbf{x}_{p})\partial k\backslash j}%
\text{exp}[f_{k}(\mathbf{x}_{j})+\sum_{p\in{\partial k}\backslash j}LV_{p%
\rightarrow k}^{t}(\mathbf{x}_{p})]}{\sum_{(\mathbf{x}_{p})\partial k%
\backslash j}\text{exp}[f_{k}(\mathbf{\tilde{x}}_{j})+\sum_{p\in{\partial k}%
\backslash j}LV_{p\rightarrow k}^{t}(\mathbf{x}_{p})]}\text{.}\end{split}$$
(14)
In equation (14), $f_{k}(\mathbf{x}_{j})=\frac{1}{N_{0}}\parallel y_{k}-h_{k,j}x_{k,j}-\sum_{p\in%
\partial k\backslash j}{h_{k,p}x_{k,p}}\parallel^{2}$. And in the final round, the output of SCMA detector is
$$LV_{j}(\mathbf{x}_{j})=L^{s,p}_{j}(\mathbf{x}_{j})+\sum_{s\in{\partial j}}LU_{%
k\rightarrow j}^{T}(\mathbf{x}_{j})\text{.}$$
(15)
Hence, the output can be decomposed into two parts: one is the prior information and another is the intrinsic information derived from the structure of SCMA factor graph. From another perspective, the LLR of $\mathbf{x}_{j}$ given prior information can be written as
$$\begin{split}\displaystyle L^{s}_{j}&\displaystyle(\mathbf{x}_{j})=\text{log}%
\frac{p(\mathbf{x}_{j}|\mathbf{y},\{L^{s,p}(\mathcal{X}^{j})\}_{j=1,\cdots,J})%
}{p(\mathbf{\tilde{x}}_{j}|\mathbf{y},\{L^{s,p}(\mathcal{X}^{j})\}_{j=1,\cdots%
,J})}\\
&\displaystyle=\text{log}\frac{p(\mathbf{y}|\mathbf{x}_{j},\{L^{s,p}(\mathcal{%
X}^{j})\}_{j=1,\cdots,J})}{p(\mathbf{y}|\mathbf{\tilde{x}}_{j},\{L^{s,p}(%
\mathcal{X}^{j})\}_{j=1,\cdots,J})}+\text{log}\frac{p(\mathbf{x}_{j}|L^{s,p}(%
\mathbf{x}_{j}))}{p(\mathbf{\tilde{x}}_{j}|L^{s,p}(\mathbf{\tilde{x}}_{j}))}\\
&\displaystyle=L^{s,i}_{j}(\mathbf{x}_{j})+L^{s,p}_{j}(\mathbf{x}_{j})\text{.}%
\end{split}$$
(16)
It is apparent that $L^{s}_{j}(\mathbf{x}_{j})$ also consists of two parts, the intrinsic parts and prior parts. when using MPA SCMA detection, we apply the term $\sum_{s\in{\partial j}}LU_{k\rightarrow j}^{T}(\mathbf{x}_{j})$ to approximate $L^{s,i}_{j}(\mathbf{x}_{j})$.
III-B LDPC decoding
In this section, we briefly introduce the decoding process of LDPC for the sake of clear description of the combination of SCMA detection with LDPC decoder in the next section.
For LDPC is a sparse liner block code, it can be effectively expressed as factor graph which is a bipartite graph with check nodes and variable nodes. Take into consideration the complexity of implementation, LDPC decoder usually adopts BP algorithm using LLRs to reduce multiplication and avoid normalization. During each iteration, belief information is exchanged between variable nodes and check nodes based on check matrix. Let $L_{1,j}^{b,p}(b_{i})$ be the LLR of the $i$-th bit of user $j$ input to the decoder, i.e., $L_{1,j}^{b,p}(b_{i})=\text{log}\frac{p(b_{i}=0|\mathbf{y})}{p(b_{i}=1|\mathbf{%
y})}$, then the output of LDPC decoder is
$$L_{2,j}^{b}(b_{i})=L_{1,j}^{b,p}(b_{i})+\sum_{m\in\partial i}L_{j,m\rightarrow
i%
}\text{,}$$
(17)
where the notation $\partial i$ denotes the set of parity check functions variable $i$ belongs to and $L_{j,m\rightarrow i}$ is the belief information passed from check node $m$ to variable node $i$ in the final iteration. And the summation $\sum_{m\in\partial i}L_{j,m\rightarrow i}$ can be seen LDPC intrinsic information $L_{2,j}^{b,i}(b_{i})$ based on LDPC structure. Hence, the output of LDPC decoder can also be decomposed to intrinsic information parts and prior information parts. Note that to distinguish the information generated by SCMA detector and LDPC decoder, we use subscript “1” and “2” respectively.
In the following section, we’ll cope with some obstacle to implement the novel combination of SCMA detection and LDPC decoding and present the iterative scheme.
III-C Combination of LDPC decoding and SCMA detection
From equation (15), the output of SCMA detector is at symbol level, but unfortunately the bit LLRs are required for input of LDPC decoder. In order to do a combination of LDPC decoding and SCMA detection, one problem we should deal with is the transformation between symbol LLRs and bit LLRs. The input to LDPC decoder should be intrinsic information parts of the multiuser detector output for a better performance like the decoding method for Turbo codes. Then another problem arises that the prior information part for a symbol comprise several bits’ prior information and for a specific bit of this symbol, the other bits’ prior information can be seen as intrinsic information, so we should decompose the output of SCMA detector into two parts, prior bit LLR and intrinsic bit LLR.
Here we assume that an SCMA symbol consists of $n=\text{log}_{2}(M)$ bits. Without loss of generality, the interleaved coded bits for a symbol can be considered independent. For a specific symbol $\mathbf{x}_{j}$, let $\mathcal{X}_{j}^{+}=\{i~{}|\text{if the i-th bit of $\mathbf{x}_{j}$ is}``1"\}$ and $\mathcal{X}_{j}^{-}=\{i~{}|\text{if the i-th bit of $\mathbf{x}_{j}$ is}``0"\}$. Then, the symbol LLR can be calculated based on bit LLRs as
$$L^{s,p}_{2,j}(\mathbf{x}_{j})=\sum_{i\in\mathcal{X}_{j}^{-}}L_{2,j}^{b,p}(b_{i%
})\text{.}$$
(18)
Let $\mathcal{X}_{i}^{j,0}$ and $\mathcal{X}_{i}^{j,1}$ denote the set of the symbols of which the $i$-th bit corresponding to “0” and “1” for user $j$, respectively. The bit LLR can be written as
$$\begin{split}&\displaystyle L_{1,j}^{b}(b_{i})=\text{log}\frac{\sum_{\mathbf{x%
}_{j}\in\mathcal{X}_{i}^{j,0}}\text{exp}[LV_{j}(\mathbf{x}_{j})]}{\sum_{%
\mathbf{x}_{j}\in\mathcal{X}_{i}^{j,1}}\text{exp}[LV_{j}(\mathbf{x}_{j})]}\\
&\displaystyle=\text{log}\frac{\sum_{\mathbf{x}_{j}\in\mathcal{X}_{i}^{j,0}}%
\text{exp}[L^{s,p}_{2,j}(\mathbf{x}_{j})+\sum_{s\in{\partial j}}LU_{k%
\rightarrow j}^{T}(\mathbf{x}_{j})]}{\sum_{\mathbf{x}_{j}\in\mathcal{X}_{i}^{j%
,1}}\text{exp}[L^{s,p}_{2,j}(\mathbf{x}_{j})+\sum_{s\in{\partial j}}LU_{k%
\rightarrow j}^{T}(\mathbf{x}_{j})]}\\
&\displaystyle=L_{1,j}^{b,i}(b_{i})+L_{2,j}^{b,p}(b_{i})\text{.}\end{split}$$
(19)
The last equation follows the fact that $L^{s,p}_{2,j}(\mathbf{x}_{j})$ is a combination of $\{{L_{2,j}^{b,p}(b_{i})}\}_{i=1,\cdots,n}$ and we can extract a same product term $\text{exp}[L_{2,j}^{b,p}(b_{i})]$ out of each sum term in the numerator.
Let $\text{I}_{T}$ and $\text{I}_{L}$ be the number of iterations of SCMA detector and LDPC decoder, respectively. The scheme of the novel iterative combination is described as follows. Besides outer iterations, this scheme consists of two inner stages: an SCMA detection stage with $\text{I}_{T}$ iterations, followed by LDPC decoding stage with $\text{I}_{L}$ iterations and the number of inner iterations of both stages can be much less than the number of iterations required by the SCMA receiver with or without traditional Turbo-like outer iterations. The two inner stages are connected by a connection module containing LLR converters, interleavers and deinterleavers. During each outer iteration, the symbol LLRs $\{L^{s,p}_{1,j}(\mathbf{x}_{j})\}$ output by SCMA detector are converted into bit LLRs $\{L_{1,j}^{b}(b_{i})\}$ by LLR converter.In order to obtain a better performance, the intrinsic bit LLRs $\{L_{1,j}^{b,i}(b_{i})\}$ from SCMA detector should be extracted and then feed to the deinterleaver to get $\{L_{1,j}^{b,p}(b_{i})\}$ which is fed into LDPC decoder as prior information. The process to feed information from LDPC decoder output to SCMA detector is similar. During each outer iteration, the detector and decoder only do a few number of iterations, less than the receiver without Turbo-like scheme need, although the total number of iterations may be the same. We call this characteristics partial inner iterations. For example, numerical results in the next section show that to obtain a satisfied BER performance 8 iterations are needed in traditional SCMA receiver, but 2 inner iterations are enough in our scheme with 4 outer iterations and the BER performance is even better. Hence this iterative combination can achieve a more satisfied BER performance gain with almost the same complexity as shown in the next section.
The connection module and the information flow of a special case where $\text{I}_{T}=\text{I}_{L}=1$ is shown as Fig.3. We call this special case mode 1. In particular, in mode 1 the detector does 1 demapping iteration and then the LLR goes to LDPC decoder through connection modules. The LDPC decoder only does 1 iteration and fed back the LLR to the detector. Then we goes to the next iteration between detector and LDPC decoder. The best number of inner iterations under the constraint of total iteration number is unsolved and will be our future work. Intuitively, the more outer iterations the more intrinsic information exchanged, and hence the better performance. However, it will be shown in the next section that when take the extreme case where $\text{I}_{T}=\text{I}_{L}=1$ the BER performance is poor. One reason may lie in the fact that if $\text{I}_{T}=1$, it can be seen from Fig.3 that the information interacts in SCMA detector is far from sufficient, i.e., there is no red arrow line connecting user node 1 with user node 6. Hence, in this scenario the information given out from SCMA detector is particularly poor so that the information cannot be corrected by LDPC decoder with only 1 iteration.
III-D Complexity analysis
In this section, considering regular SCMA systems, we evaluate the complexity of traditional scheme for LDPC-coded SCMA receiver and our proposed scheme, i.e., the average number of operations we need for each symbol. Since the main complexity is introduced by multiplication, division, logarithm and exponent operations, we mainly consider these four kinds of operations for real number. From equation (13) and (14), it is clear that each inner iteration of SCMA detection needs $(2d_{k}KM^{d_{k}}+4{d_{k}}^{2}KM)/J$ multiplication operations, $d_{k}KM^{d_{k}}/J$ division operations $d_{k}KM^{d_{k}}/J$ exponent operations and $KMd_{k}/J$ logarithm operations. For regular LDPC codes with 1/2 code rate, $(11P-9)\text{log}_{2}(M)$ multiplications and $(P+1)\text{log}_{2}(M)$ divisions are needed during each iteration of decoding, [11], where P denotes the degree of variable nodes. For other kind of code rates, the order of complexity is the same [12]. $M\text{log}_{2}(M)$ exponent operations and $2\text{log}_{2}(M)$ logarithm operations are needed in connection module during each outer iterations and these operations are caused by the converting from symbol LLRs to bit LLRs.
It is now obvious that the main complexity is introduced by SCMA detection. Let $\text{I}_{O}$ be the number of outer iterations. From the deduction above and ignore the lower oder terms, it is easy to get that the total number of multiplication operations is approximately equal to $2I_{O}I_{T}d_{k}KM^{d_{k}}/J$, division and exponent operations are both approximately equal to $I_{O}I_{T}d_{k}KM^{d_{k}}/J$.
Here, with a slight abuse of notation, let the superscript t and p denote traditional scheme and proposed scheme, respectively. For the traditional scheme $I^{t}_{O}=1$, and for the proposed scheme $I^{p}_{O}I^{p}_{T}$ can be approximately equal to $I^{t}_{T}$. Hence, the complexity of our proposed scheme is almost the same as traditional receiver for LDPC-coded SCMA systems.
IV Numerical results
In this section, numerical simulations are conducted to evaluate the performance of our proposed scheme. This simulation is based on the LDPC codes with finite block size. The regular systematic LDPC code in China Mobile Multimedia Broadcasting (CMMB) systems is used in the simulations [13]. The code length we adopted is 9216 bits and code rate is $R=1/2$. Bit error rates (BER) over AWGN channel and Rayleigh fading channel with $J=6,~{}K=4,~{}M=4$, and $N_{j}=2$ for all $j$ are presented. The user-specific codebooks are designed according to [14]. The channel coefficient $h$ is set to unit power for both AWGN channel and Rayleigh fading channel, i.e., $E(h^{2})=1$. To keep numerically stable, when computing the term like $\text{log}(e^{a}+e^{b})$, we adopt the equation as $\text{log}(e^{a}+e^{b})=\text{max}(a,b)+\text{log}(1+e^{-|a-b|})$ [15].
Here, we compare four different modes, of which the total number of iterations is almost the same excepting the mode 1 “$\text{I}_{T}=\text{I}_{L}=1,\text{I}_{O}=32$”. The purpose to compare this mode with great difference in total iteration number is to show that the BER performance of this extreme case deteriorates seriously as shown in Fig. 4 and Fig. 5. This poor BER performance of Mode 1 is consistent with the analysis in the last section. Fig. 4 shows that the required ${E_{s}/N_{0}}$ of the Mode 2 to achieve a BER of $10^{-4}$ is 4.5 dB on AWGN channels and 5.4 dB on Rayleigh fading channels, obtaining 0.3 dB gain over Mode 3 and 0.9 dB gain over Mode 4 on both kinds of channels. Note that Mode 4 is the traditional SCMA receiver scheme, i.e., there is no information fed back from decoder to detector. It can also be recognized that the BER waterfall region of Mode 3 is more narrow than Mode 4 and that of Mode 2 is even more narrow than Mode 3. The phenomenon that a better BER performance and more narrow waterfall region of Mode 2 and Mode 3 than Mode 4 owes to the ability of iterative combination that more diversity gain can be achieved. Note that the complexity of the last three modes is almost the same if ignore the complexity caused by interleaver. If let Mode 2 and Mode 4 have the same BER performance, Mode 2 should reduce the number of iterations. Hence the proposed iterative scheme can be viewed as a method to reduce complexity from another perspective.
V Conclusions
In this paper, we have proposed a novel Turbo-like combination scheme of SCMA detector and LDPC decoding with almost the same complexity compared to traditional SCMA receiver (non-iterative structure). Firstly, we have investigated the SCMA detection algorithm based on MPA from the aspect of solving marginal functions and shown this MPA-based detection is an approximation of MAP criterion since the factor graph of SCMA is usually not cycle-free. Then the MPA-based SCMA detection in Logarithmic form as well as the intrinsic information of the output have also been deduced in order to apply a novel soft Turbo-like combination with LDPC decoder. The combination scheme is described in detail and the information flows of a special case are presented. Numerical results show that the proposed scheme can achieve an excellent performance gain and more narrow waterfall region in terms of BER compared with traditional SCMA receiver for both AWGN channels and Rayleigh fading channels.
References
[1]
H. Nikopour and H. Baligh, “Sparse code multiple access,” in Proc. IEEE
24th Int’l. Symp. Personal Indoor & Mobile Radio Commun. (PIMRC), 2013.
[2]
D. Guo and C.-C. Wang, “Multiuser detection of sparsely spread CDMA,”
IEEE J. Sel. Areas Commun., vol. 26, no. 3, pp. 421–431, 2008.
[3]
S. Verdu, Multiuser detection. Cambridge University Press, 1998.
[4]
F. R. Kschischang, B. J. Frey, and H.-A. Loeliger, “Factor graphs and the
sum-product algorithm,” IEEE Trans. Inform. Theory, vol. 47, no. 2,
pp. 498–519, 2001.
[5]
S. Z. Yiqun Wu and Y. Chen, “ Iterative Multiuser Receiver in Sparse Code
Multiple Access Systems,” in Proc. IEEE ICC, Jun. 2015.
[6]
J. Hagenauer, “The turbo principle: Tutorial introduction and state of the
art,” in Proc. Int’l. Symp. Turbo Codes & Related Topics, 1997.
[7]
J. Hagenauer and N. Gortz, “The turbo principle in joint source-channel
coding,” in Proc. IEEE Inform. Theory Workshop, 2003.
[8]
X. Wang and H. V. Poor, “Iterative (turbo) soft interference cancellation and
decoding for coded CDMA,” IEEE Trans. Commun., vol. 47, no. 7,
pp. 1046–1061, 1999.
[9]
A. Sanderovich, M. Peleg, and S. Shamai, “LDPC coded MIMO
multiple access with iterative joint decoding,” IEEE Trans. Inform.
Theory, vol. 51, no. 4, pp. 1437–1450, 2005.
[10]
T. Richardson and R. Urbanke, “The renaissance of gallager’s low-density
parity-check codes,” IEEE Commun. Mag., vol. 41, no. 8, pp. 126–131,
2003.
[11]
M. P. Fossorier, M. Mihaljević, and H. Imai, “Reduced complexity iterative
decoding of low-density parity check codes based on belief propagation,”
IEEE Trans. Commun., vol. 47, no. 5, pp. 673–680, 1999.
[12]
D. J. MacKay, “Good error-correcting codes based on very sparse matrices,”
IEEE Trans. Inform. Theory, vol. 45, no. 2, pp. 399–431, 1999.
[13]
Z. Hu and H. Liu, “A low-complexity LDPC decoding algorithm for
hierarchical broadcasting: design and implementation,” IEEE Trans.
Veh. Technol., vol. 62, no. 4, pp. 1843–1849, 2013.
[14]
M. Taherzadeh, H. Nikopour, A. Bayesteh, and H. Baligh, “SCMA codebook
design,” in Proc. IEEE 80th Veh. Technol. Conf. (VTC Fall), 2014.
[15]
R. Hoshyar, F. P. Wathan, and R. Tafazolli, “Novel low-density signature for
synchronous CDMA systems over AWGN channel,” IEEE Trans.
Signal Process., vol. 56, no. 4, pp. 1616–1626, 2008. |
[
[
1. A. A. Kharkevich Institute for Information Transmission Problems, Russian Academy of Sciences,
Bolshoy Karetny per. 19, 127994, Moscow, Russia;
2. N. E. Bauman Moscow State Technical University,
ul. Baumanskay 2-ya, 5, 105005, Moscow, Russia.
dm.shirokov@gmail.com
(September 11, 2004)
Abstract
In this paper we consider some expressions (sums) in Clifford algebras which we call contractions or averaging.
We study full contractions, contractions by adjoint sets of multi-indices, simple contractions. We present the relation between simple contractions and projection operations onto fixed subspaces of Clifford algebras.
Using method of contractions we present solutions of system of commutator equations.
keywords: Clifford algebra, contraction, projection, adjoint sets of multi-indices, simple contraction, full contraction, commutator equation
Method of contractions in Clifford algebras]
Method of Contractions
in Clifford Algebras
D. S. Shirokov]D. S. Shirokov
††thanks: This work was supported by Russian Science Foundation (project RSF 14-11-00687, Steklov Mathematical Institute).
\subjclass
15A66
1 Introduction
In this paper we consider expressions in Clifford algebras
$$\sum_{A\in S}e_{A}Ue^{A},\qquad e_{A}=(e^{A})^{-1}$$
where $e^{A}$ are basis elements and $S\subseteq{\rm I}$ is the subset of the set of all ordered multi-indices $A$ of the length from $0$ to $n$. We call them contractions (or averaging) in Clifford algebras.
There is a relationship between contractions in Clifford algebras and representation theory of finite groups (see [1], [2]).
We study full contractions, contractions by adjoint sets of multi-indices, simple contractions. We present the relation between simple contractions and projection operations onto fixed subspaces of Clifford algebras.
Using method of contractions we present solutions of system of commutator equations
$$e^{A}X+\epsilon Xe^{A}=q^{A},\qquad A\in S\subseteq{\rm I},\qquad\epsilon\neq 0%
\in{\mathbb{C}}$$
for unknown element $X\in{C}\!\ell(p,q)$ and known elements $q^{A}\in{C}\!\ell(p,q)$.
2 Clifford algebras, ranks, projection operations
Consider complex Clifford algebra ${C}\!\ell(p,q)$ (or real ${C}\!\ell^{\mathbb{R}}(p,q)$) with $p+q=n$, $n\geq 1$.
The construction of Clifford algebra is discussed in details in [3] or [4].
Let $e$ be the identity element and let $e^{a}$, $a=1,\ldots,n$ be generators of the Clifford algebra ${C}\!\ell(p,q)$,
$$e^{a}e^{b}+e^{b}e^{a}=2\eta^{ab}e,$$
where $\eta=||\eta^{ab}||=||\eta_{ab}||$ is the diagonal matrix with $p$ pieces of $+1$ and $q$ pieces of $-1$ on the diagonal. Elements
$$e^{a_{1}\ldots a_{k}}=e^{a_{1}}\ldots e^{a_{k}},\quad a_{1}<\ldots<a_{k},\,k=1%
,\ldots,n,$$
together with the identity element $e$, form the basis of the Clifford
algebra. The number of basis elements is equal to $2^{n}$.
Let denote the set of ordered multi-indices of the length from $0$ to $n$ by
$$\displaystyle{\rm I}=\{-,\,1,\,\ldots,\,n,\,12,\,13,\,\ldots,\,1\ldots n\},$$
(2.1)
where ${}^{\prime\prime}-^{\prime\prime}$ is an empty multi-index. So, we have a basis of Clifford algebra $\{e^{A},\,A\in{\rm I}\}$, where $A$ is an arbitrary ordered multi-index. Let denote the length of multi-index $A$ by $|A|$.
Below we also consider different subsets of $S\subseteq{\rm I}$:
$${\rm I}_{{\rm Even}}=\{A\in{\rm I},\,|A|-\mbox{even}\},\qquad{\rm I}_{{\rm Odd%
}}=\{A\in{\rm I},\,|A|-\mbox{odd}\}.$$
We have $e_{a}=\eta_{ab}e^{b}$, $e^{a}=\eta^{ab}e_{b}$, where we use Einstein summation convection (there is a sum over index $b$).
Also we have
$$e_{a_{1}\ldots a_{k}}=\eta_{a_{1}b_{1}}\ldots\eta_{a_{k}b_{k}}e^{b_{k}}\ldots e%
^{b_{1}}=e_{a_{k}}\ldots e_{a_{1}}=(e^{a_{1}\ldots a_{k}})^{-1},\quad a_{1}<%
\ldots<a_{k}.$$
Any Clifford algebra element $U\in{C}\!\ell(p,q)$ can be written in the form
$$\displaystyle U=ue+u_{a}e^{a}+\sum_{a_{1}<a_{2}}u_{a_{1}a_{2}}e^{a_{1}a_{2}}+%
\ldots+u_{1\ldots n}e^{1\ldots n}=u_{A}e^{A},$$
(2.2)
where we have a sum over ordered multi-index $A$ and $\{u_{A}\}=\{u,u_{a},u_{a_{1}a_{2}},\ldots,u_{1\ldots n}\}$ are complex (real) numbers.
We denote by ${C}\!\ell_{k}(p,q)$ the vector spaces that span over the basis elements
$e^{a_{1}\ldots a_{k}}$. Elements of ${C}\!\ell_{k}(p,q)$ are said to be
elements of rank $k$. We have
$$\displaystyle{C}\!\ell(p,q)=\bigoplus_{k=0}^{n}{C}\!\ell_{k}(p,q).$$
(2.3)
We consider projection operators on the vector subspaces ${C}\!\ell_{k}(p,q)$
$$\displaystyle\pi_{k}:{C}\!\ell(p,q)\to{C}\!\ell_{k}(p,q),\qquad\pi_{k}(U)=\sum%
_{a_{1}<\ldots<a_{k}}u_{a_{1}\ldots a_{k}}e^{a_{1}\ldots a_{k}}.$$
(2.4)
Clifford algebra ${C}\!\ell(p,q)$ is a superalgebra. It is represented as the direct sum of even and odd subspaces
$$\displaystyle{C}\!\ell(p,q)={C}\!\ell_{{\rm Even}}(p,q)\oplus{C}\!\ell_{{\rm
Odd%
}}(p,q),$$
$${C}\!\ell_{{\rm Even}}(p,q)=\bigoplus_{k-even}{C}\!\ell_{k}(p,q),\qquad{C}\!%
\ell_{{\rm Odd}}(p,q)=\bigoplus_{k-odd}{C}\!\ell_{k}(p,q).$$
3 Full contractions
We have the following well-known statement about center $cen{C}\!\ell(p,q)=\{U\in{C}\!\ell(p,q)\,|\,UV=VU\quad\forall V\in{C}\!\ell(p,q)\}$ of Clifford algebra.
Theorem 3.1.
The center $cen{C}\!\ell(p,q)$ of Clifford algebra ${C}\!\ell(p,q)$ of dimension $n=p+q$ is subspace ${C}\!\ell_{0}(p,q)$ in the case of even $n$ and subspace ${C}\!\ell_{0}(p,q)\oplus{C}\!\ell_{n}(p,q)$ in the case of odd $n$:
$$\displaystyle{\rm cen}{C}\!\ell(p,q)=\left\{\begin{array}[]{ll}{C}\!\ell_{0}(p%
,q),&\mbox{$n$ is even};\\
{C}\!\ell_{0}(p,q)\oplus{C}\!\ell_{n}(p,q),&\mbox{$n$ is odd.}\end{array}\right.$$
(3.1)
Let consider the following contraction
$$F(U)=\frac{1}{2^{n}}e_{A}Ue^{A},$$
where we have a sum over all multi-indices $A\in{\rm I}$. We call this expression full contraction.
Theorem 3.2.
We have
$$\displaystyle F(U)=\frac{1}{2^{n}}e_{A}Ue^{A}=\left\{\begin{array}[]{ll}\pi_{0%
}(U),&\parbox{86.724pt}{if $n$ is even;}\\
\pi_{0}(U)+\pi_{n}(U),&\parbox{86.724pt}{if $n$ is odd,}\end{array}\right.$$
(3.2)
where $\pi_{0}$ and $\pi_{n}$ are projection operations (see (2.4)) onto the subspaces of fixed ranks. Operator $F$ is a projector $F^{2}=F$ (on the center of Clifford algebra).
Proof.
We have
$$(e^{a})^{-1}F(U)e^{a}=\sum_{A}(e^{A}e^{a})^{-1}F(U)(e^{A}e^{a})=\sum_{B}(e^{B}%
)^{-1}F(U)e^{B}=F(U).$$
So, $F(U)$ is in the center of Clifford algebra (see Theorem 3.1).
For elements $U$ of ranks $k=1,\ldots,n-1$ (and $k=n$ in the case of even $n$) we have $F(U)=0$.
In other particular cases we have $e_{A}e^{A}=2^{n}e$ and (in the case of odd $n$) $e_{A}e^{1\ldots n}e^{A}=2^{n}e^{1\ldots n}$.
It is also easy to verify that $F^{2}=F$.
∎
Let consider system of $2^{n}$ commutator equations in Clifford algebra.
Theorem 3.3.
Let unknown element $X\in{C}\!\ell(p,q)$ satisfy system of equations with known $q^{A}\in{C}\!\ell(p,q)$
$$\displaystyle e^{A}X+\epsilon Xe^{A}=q^{A}\quad\forall A,\qquad\epsilon\neq 0%
\in{\mathbb{C}}.$$
(3.3)
If $\epsilon=-1$ (commutator case), then this system of equations has not got solutions or has unique solution up to element of center:
$$\displaystyle X=-\frac{1}{2^{n}}q^{A}e_{A}+Z,\qquad Z\in{\rm cen}{C}\!\ell(p,q).$$
(3.4)
If $\epsilon\neq-1$, then this system of equations has not got solutions or has unique solution
$$\displaystyle X=\left\{\begin{array}[]{ll}\frac{1}{2^{n}\epsilon}(q^{A}e_{A}-%
\frac{1}{(\epsilon+1)}\pi_{0}(q^{A}e_{A})),&\mbox{if $n$ is even},\\
\\
\frac{1}{2^{n}\epsilon}(q^{A}e_{A}-\frac{1}{(\epsilon+1)}(\pi_{0}(q^{A}e_{A})+%
\pi_{n}(q^{A}e_{A})),&\mbox{if $n$ is odd}.\end{array}\right.$$
(3.5)
Proof.
Let multiply each equation by $e_{A}$ on the right and add them (see Theorem 3.2):
$$e^{A}Xe_{A}+\epsilon Xe^{A}e_{A}=q^{A}e_{A}\quad\Rightarrow\quad 2^{n}\pi_{%
centr}(X)+\epsilon X2^{n}=q^{A}e_{A},$$
where $\pi_{centr}$ is the projection on the center of Clifford algebra. Using $X=\sum_{k=0}^{n}\pi_{k}(X)$ and Theorem 3.1, we obtain statement of the theorem.
∎
Note that we have solution or have not solution of system of commutator equations. It depends on elements $q^{A}$ (it suffices to substitute solution in equation and check the equality).
4 Adjoint sets of multi-indices
We call ordered multi-indices $a_{1}\ldots a_{k}$ and $b_{1}\ldots b_{l}$ adjoint multi-indices if they have no common indices and they form multi-index $1\ldots n$ of the length $n$. We write $b_{1}\ldots b_{l}=\widetilde{a_{1}\ldots a_{k}}$ and $a_{1}\ldots a_{k}=\widetilde{b_{1}\ldots b_{l}}$. We call corresponding basis elements
$e^{a_{1}\ldots a_{m}}$, $e^{b_{1}\ldots b_{l}}$ adjoint and write $e^{b_{1}\ldots b_{l}}=e^{\widetilde{a_{1}\ldots a_{m}}}$, $e^{a_{1}\ldots a_{m}}=e^{\widetilde{b_{1}\ldots b_{l}}}$. We can write also that $e^{a_{1}\ldots a_{m}}e^{b_{1}\ldots b_{l}}=\pm e^{1\ldots n}$ and
$\star e^{a_{1}\ldots a_{m}}=\pm e^{b_{1}\ldots b_{l}}$, where $\star$ is Hodge operator111It is the analogue of Hodge operator in Clifford algebra $\star U=U^{\sim}e^{1\ldots n}$, where $\sim$ is the standard reverse operation in Clifford algebra..
We denote the sets of corresponding multi-indices by ${\rm I}_{{\rm Adj}}$ and $\widetilde{{\rm I}_{{\rm Adj}}}={\rm I}\setminus{\rm I}_{{\rm Adj}}$. We have
$$\displaystyle\{e^{A}\,|\,A\in{\rm I}\}=\{e^{A}\,|\,A\in{\rm I}_{{\rm Adj}}\}%
\cup\{e^{A}\,|\,A\in\widetilde{{\rm I}_{{\rm Adj}}}\}.$$
(4.1)
For Clifford algebra ${C}\!\ell(p,q)$ of dimension $n=p+q$ we have $2^{2^{n-1}-1}$ different partitions of the form (4.1). For example,
$${\rm I}_{{\rm Adj}}={\rm I}_{{\rm First}},\qquad\widetilde{{\rm I}_{{\rm Adj}}%
}={\rm I}\setminus{\rm I}_{{\rm First}}={\rm I}_{{\rm Last}},$$
where ${\rm I}_{{\rm First}}$ consist of the first (in the order) $2^{n-1}$ multi-indices of the set ${\rm I}$ (2.1).
In the case of odd $n$ we can write
$${\rm I}_{{\rm First}}=\{A\in{\rm I},\quad|A|\leq\frac{n-1}{2}\},\qquad{\rm I}_%
{{\rm Last}}=\{A\in{\rm I},\quad|A|\geq\frac{n+1}{2}\}.$$
In the case of odd $n$ we can consider the following adjoint sets
$${\rm I}_{{\rm Adj}}={\rm I}_{{\rm Even}},\qquad\widetilde{{\rm I}_{{\rm Adj}}}%
={\rm I}_{{\rm Odd}}.$$
5 Commutative properties of basis elements
Theorem 5.1.
Consider real or complex Clifford algebra ${C}\!\ell(p,q)$ and the set of basis elements $e^{A}=\{e^{b_{1}\ldots b_{m}}\}$. Let denote arbitrary element of this set by $e^{a_{1}\ldots a_{k}}$.
Then element $e^{a_{1}\ldots a_{k}}$ (if it is not $e$ or $e^{1\ldots n}$) commutes with $2^{n-2}$ even ($|A|$ - even) elements of the set, commutes with $2^{n-2}$ odd elements of the set, anticommutes with $2^{n-2}$ even elements of the set and anticommutes with $2^{n-2}$ elements of the set $e^{A}$.
Element $e$ commutes with all elements of the set $e^{A}$.
1.
if $n$ - even, then $e^{1\ldots n}$ commutes with all $2^{n-1}$ even elements of the set and anticommutes with all $2^{n-1}$ odd elements of the set $e^{A}$;
2.
if $n$ - odd, then $e^{1\ldots n}$ commutes with all $2^{n}$ elements of the set $e^{A}$.
Proof.
Let $i$ be the number of coincident indices in multi-indices $a_{1}\ldots a_{k}$ and $b_{1}\ldots b_{m}$. Then for any $i$ number of sets $b_{1}\ldots b_{m}$ for fixed set $a_{1}\ldots a_{k}$ equals $C^{i}_{k}C_{n-k}^{m-i}$, where $C_{n}^{k}=\left(\begin{array}[]{l}n\\
k\end{array}\right)=\frac{n!}{k!(n-k)!}$ is binomial coefficient (we have $C_{n}^{k}=0$ for $k>n$). When we swap element $e^{a_{1}\ldots a_{k}}$ with element $e^{b_{1}\ldots b_{m}}$ we obtain coefficient $(-1)^{km-i}$.
If $k$ is even and does not equal $0$ and $n$, then number of even and odd elements $e^{b_{1}\ldots b_{m}}$ that commutes with $e^{a_{1}\ldots a_{k}}$ ($km-i$ is even, and, so $i$ is even) respectively equals
$$\sum_{m-even}\sum_{i-even}C^{i}_{k}C_{n-k}^{m-i}=2^{n-2},\qquad\sum_{m-odd}%
\sum_{i-even}C^{i}_{k}C_{n-k}^{m-i}=2^{n-2}.$$
If $k$ is odd and does not equal $n$, then number of even and odd elements $e^{b_{1}\ldots b_{m}}$ that anticommutes with $e^{a_{1}\ldots a_{k}}$ ($m-i$ is even) respectively equals
$$\sum_{m-even}\sum_{i-even}C^{i}_{k}C_{n-k}^{m-i}=2^{n-2},\qquad\sum_{m-odd}%
\sum_{i-odd}C^{i}_{k}C_{n-k}^{m-i}=2^{n-2}.$$
The cases $k=0$ and $k=n$ are trivial (see Theorem 3.1).
∎
Also we have the following theorem about adjoint sets of multi-indices.
Theorem 5.2.
Consider real or complex Clifford algebra ${C}\!\ell(p,q)$ and the set of basis elements $e^{A}=\{e^{b_{1}\ldots b_{m}}\}$. Let the set of basis elements is a sum of 2 adjoint sets of multi-indices ${\rm I}={\rm I}_{{\rm Adj}}\cup\widetilde{{\rm I}_{{\rm Adj}}}$.
If $n$ is even then any even (not odd!) basis element (if it is not $e$) commutes with $2^{n-2}$ basis elements from $\{e^{A}\,|\,A\in{\rm I}_{{\rm Adj}}\}$, anticommutes with $2^{n-2}$ basis elements from $\{e^{A}\,|\,A\in{\rm I}_{{\rm Adj}}\}$, commutes with $2^{n-2}$ basis elements from $\{e^{A}\,|\,A\in\widetilde{{\rm I}_{{\rm Adj}}}\}$ and anticommutes with $2^{n-2}$ basis elements from $\{e^{A}\,|\,A\in\widetilde{{\rm I}_{{\rm Adj}}}\}$.
If $n$ is odd then any basis element (if it is not $e$ and $e^{1\ldots n}$) commutes with $2^{n-2}$ basis elements from $\{e^{A}\,|\,A\in{\rm I}_{{\rm Adj}}\}$, anticommutes with $2^{n-2}$ basis elements from $\{e^{A}\,|\,A\in{\rm I}_{{\rm Adj}}\}$, commutes with $2^{n-2}$ basis elements from $\{e^{A}\,|\,A\in\widetilde{{\rm I}_{{\rm Adj}}}\}$ and anticommutes with $2^{n-2}$ basis elements from $\{e^{A}\,|\,A\in\widetilde{{\rm I}_{{\rm Adj}}}\}$.
Note that in the case of odd $n$ we can take ${\rm I}_{{\rm Adj}}={\rm I}_{{\rm Even}}$, $\widetilde{{\rm I}_{{\rm Adj}}}={\rm I}_{{\rm Odd}}$ and obtain the statement from the Theorem 5.1.
Proof.
If $n$ is odd then $e^{1\ldots n}$ is in the center of Clifford algebra. So if basis element commutes with some basis element, then it commutes with adjoint basis element. But we know from Theorem 5.1 that basis elements (except $e$ and $e^{1\ldots n}$) commutes with $2^{n-1}$ basis elements and anticommutes with $2^{n-1}$ basis elements. So we obtain the statement of theorem for the case of odd $n$.
If $n$ is even then even (not odd) basis element commutes with $e^{1\ldots n}$. So if even basis element commutes with some basis element, then it commutes with adjoint basis element.
∎
Let represent the commutative property of basis elements in the following tables. At the intersection of two basis elements is a sign ’’+’’ if they commute and the sign ’’–’’ if they anticommute. For small dimensions we have the following tables:
$$n=1$$
$$e$$
$$e^{1}$$
$$e$$
+
+
$$e^{1}$$
+
+
$$n=2$$
$$e$$
$$e^{1}$$
$$e^{2}$$
$$e^{12}$$
$$e$$
+
+
+
+
$$e^{1}$$
+
+
–
–
$$e^{2}$$
+
–
+
–
$$e^{12}$$
+
–
–
+
$$n=3$$
$$e$$
$$e^{1}$$
$$e^{2}$$
$$e^{3}$$
$$e^{12}$$
$$e^{13}$$
$$e^{23}$$
$$e^{123}$$
$$e$$
+
+
+
+
+
+
+
+
$$e^{1}$$
+
+
–
–
–
–
+
+
$$e^{2}$$
+
–
+
–
–
+
–
+
$$e^{3}$$
+
–
–
+
+
–
–
+
$$e^{12}$$
+
–
–
+
+
–
–
+
$$e^{13}$$
+
–
+
–
–
+
–
+
$$e^{23}$$
+
+
–
–
–
–
+
+
$$e^{123}$$
+
+
+
+
+
+
+
+
6 Simple contractions
We denote the corresponding square symmetric matrices of size $2^{n}$ from the previous section (with elements $1$ and $-1$, see tables) by $M_{n}=||m_{AB}||$. For arbitrary element of these matrices we have $m_{AB}=(e^{A})^{-1}e^{B}e^{A}(e^{B})^{-1}$, $A,B\in{\rm I}$, $e\equiv 1$.
We have
$$\displaystyle m_{AB}=m_{BA}=\left\{\begin{array}[]{lll}1,&\mbox{if $[e^{A},e^{%
B}]=0$};\\
-1,&\mbox{if $\{e^{A},e^{B}\}=0$},\end{array}\right.$$
(6.1)
In the case of odd $n$ we also consider symmetric matrix $L$ of size $2^{n-1}$ $L_{n}=||l_{AB}||$, $l_{AB}=m_{AB}$, $A,B\in{\rm I}_{{\rm First}}=\{A\in{\rm I},\quad|A|\leq\frac{n-1}{2}\}.$
Theorem 6.1.
Matrix $M_{n}$ is invertible in the case of even $n$ and $M_{n}^{-1}=\frac{1}{2^{n}}M_{n}$.
Matrix $M_{n}$ is not invertible in the case of odd $n$.
Matrix $L_{n}$ is invertible in the case of odd $n$ and $L_{n}^{-1}=\frac{1}{2^{n-1}}L_{n}$.
Proof.
Matrices are symmetric $M_{n}^{T}=M_{n}$, $N_{n}^{T}=N_{n}$ by definition. Let multiply matrix $M_{n}$ by itself. For two arbitrary rows we have
$$\sum_{B}m_{AB}m_{CB}=\sum_{B}((e^{A})^{-1}(e^{B})^{-1}e^{A}e^{B})((e^{B})^{-1}%
e^{C}e^{B}(e^{C})^{-1})=$$
$$=(e^{A})^{-1}(\sum_{B}(e^{B})^{-1}e^{A}e^{C}(e^{B})^{-1})(e^{C})^{-1}.$$
In the last expression sum is equal zero if $A\neq C$ and (in the case of odd $n$) $A,C$ are not adjoint multi-indices (see below), because contraction $e_{B}Ue^{B}$ is projection onto the center of Clifford algebra (see Theorem 3.2). In the other cases the last expression equals $2^{n}$. In the case of odd $n$ we must use matrix $L_{n}$ because we does not have adjoint multi-indices in this matrix.
∎
Let consider simple contractions $F_{e^{A}}(U)=(e^{A})^{-1}Ue^{A}$.
Theorem 6.2.
For simple contraction $F_{e^{A}}(U)=(e^{A})^{-1}Ue^{A}$ we have
$$\displaystyle F_{e^{A}}(U)=\sum_{B}m_{AB}\pi_{e^{B}}(U),$$
(6.2)
where $\pi_{e^{B}}$ is a projection onto subspace spanned over element $e^{B}$.
We have $F_{e^{A}}(F_{e^{A}}(U))=U$.
Proof.
The statement follows from the definition of matrix $M_{n}=||m_{AB}||$ and definition of simple contraction. ∎
Fixed multi-index $A$ divides the set ${\rm I}$ into 2 sets ${\rm I}={\rm I}_{[A]}\cup{\rm I}_{\{A\}},$ where $e^{B}$, $B\in{\rm I}_{[A]}$ commutes with $e^{A}$, and $e^{B}$, $B\in{\rm I}_{\{A\}}$ anticommutes with $e^{A}$. Denote the corresponding subspaces of Clifford algebra by ${C}\!\ell_{[A]}(p,q)$ è ${C}\!\ell_{\{A\}}(p,q)$ and corresponding projection operations by $\pi_{[A]}$ and $\pi_{\{A\}}$. We have ${C}\!\ell(p,q)={C}\!\ell_{[A]}(p,q)\oplus{C}\!\ell_{\{A\}}(p,q)$ and
$$F_{e^{A}}(U)=(e^{A})^{-1}Ue^{A}=\pi_{[A]}(U)-\pi_{\{A\}}(U),\qquad\forall A.$$
Theorem 6.3.
For arbitrary Clifford algebra element $U$ we have
$$\pi_{[A]}(U)=\frac{1}{2}(U+(e^{A})^{-1}Ue^{A}),\qquad\pi_{\{A\}}(U)=\frac{1}{2%
}(U-(e^{A})^{-1}Ue^{A}).$$
Proof.
Using
$$(e^{A})^{-1}Ue^{A}=\pi_{[A]}(U)-\pi_{\{A\}}(U),\qquad U=\pi_{[A]}(U)+\pi_{\{A%
\}}(U)$$
we obtain the statement of theorem.
∎
For empty multi-index $A=-$ we have $m_{-,B}=1$ for all $B$, ${\rm I}={\rm I}_{[A]}$, ${\rm I}_{\{A\}}=\o$.
For multi-index $A=1\ldots n$ we have $m_{1\ldots n,B}=1$ for all $B$ in the case of odd $n$ and
$$\displaystyle m_{1\ldots n,B}=\left\{\begin{array}[]{lll}1,&\mbox{if $B$ is %
even};\\
-1,&\mbox{if $B$ is odd},\end{array}\right.$$
(6.3)
and $e_{1\ldots n}Ue^{1\ldots n}=\pi_{{\rm Even}}(U)-\pi_{{\rm Odd}}(U)$ in the case of even $n$, where $\pi_{{\rm Even}}$ and $\pi_{{\rm Odd}}$ are projection operations onto the even and odd subspaces of Clifford algebra. In the other cases (when $A$ is not empty and in not $1\ldots n$) we have $2^{n-1}$ elements in each of the sets
${\rm I}_{[A]}$, ${\rm I}_{\{A\}}$ (see Theorem 5.1) i.e. we have
$\dim{C}\!\ell_{[A]}(p,q)=\dim{C}\!\ell_{\{A\}}(p,q)=2^{n-1}$ in these cases.
In particular case we obtain the following statement (for $A=1\ldots n$):
in the case of even $n$ we have
$$\pi_{{\rm Even}}(U)=\frac{1}{2}(U+e_{1\ldots n}Ue^{1\ldots n}),\qquad\pi_{{\rm
Odd%
}}(U)=\frac{1}{2}(U-e_{1\ldots n}Ue^{1\ldots n}).$$
Let consider commutator equation. We have the following theorem.
Theorem 6.4.
Let unknown element $X\in{C}\!\ell(p,q)$ satisfy the following equation with known element $q^{A}\in{C}\!\ell(p,q)$
$$\displaystyle e^{A}X+\epsilon Xe^{A}=q^{A},\qquad\epsilon\neq 0\in{\mathbb{C}}.$$
(6.4)
If $\epsilon\neq\pm 1$, then
$$X=\sum_{B}\frac{1}{1+\epsilon m_{AB}}\pi_{B}((e^{A})^{-1}q^{A}).$$
If $\epsilon=-1$ (commutator case), then:
•
if $A$ is empty or $A=1\ldots n$ (in the case of odd $n$) and $q^{A}\neq 0$, then there is no solutions,
•
if $A$ is empty or $A=1\ldots n$ (in the case of odd $n$) and $q^{A}=0$, then solution is arbitrary Clifford algebra element,
•
in other cases we have solution
$$\frac{1}{2}\pi_{\{A\}}((e^{A})^{-1}q^{A})+\pi_{[A]}(U),$$
where $U$ is arbitrary Clifford algebra element.
If $\epsilon=1$ (anticommutator case), then
we have solution
$$\frac{1}{2}\pi_{[A]}((e^{A})^{-1}q^{A})+\pi_{\{A\}}(U),$$
where $U$ is arbitrary Clifford algebra element.
Proof.
Multiply equation on the left by $(e^{A})^{-1}=e_{A}$ and use Theorem 6.2:
$$X+\epsilon(e^{A})^{-1}Xe^{A}=(e^{A})^{-1}q^{A}\quad\Rightarrow\quad X+\epsilon%
\sum_{B}m_{AB}\pi_{e^{B}}(X)=(e^{A})^{-1}q^{A}.$$
Using $X=\sum_{B}\pi_{e^{B}}(X)$ we obtain
$$\sum_{B}(1+\epsilon m_{AB})\pi_{e^{B}}(X)=\sum_{B}\pi_{B}((e^{A})^{-1}q^{A}).$$
In the case $\epsilon\neq\pm 1$ we obtain the statement of the theorem for this case.
Let $\epsilon=-1$. If $A$ is empty or $A$ is $1\ldots n$ in the case of odd $n$, then $m_{AB}=1$ for all $B$. We has not got solution if $q^{A}\neq 0$, or solution is arbitrary element if $q^{A}=0$. In the other cases $(1+\epsilon m_{AB})=(1-m_{AB})$ equals $0$ or $2$ and we obtain formula from the statement.
In the case $\epsilon=1$ proof is similar.
∎
We have the following relation between projection operations onto subspaces of fixed basis element and simple contractions.
Theorem 6.5.
In the case of even $n$ we have
$$\displaystyle\pi_{e^{A}}(U)=\frac{1}{2^{n}}\sum_{B}m_{AB}(e^{B})^{-1}Ue^{B}.$$
(6.5)
In the case of odd $n$ we have
$$\displaystyle\pi_{e^{A},\widetilde{e^{A}}}(U)=\pi_{e^{A}}(U)+\pi_{\widetilde{e%
^{A}}}(U)=\frac{1}{2^{n-1}}\sum_{B\in{\rm I}_{{\rm First}}}l_{AB}(e^{B})^{-1}%
Ue^{B}.$$
(6.6)
Note that we can use instead ${\rm I}_{{\rm First}}$ any adjoint set ${\rm I}_{{\rm Adj}}$.
Proof.
From (6.2) we obtain
$$\left(\begin{array}[]{l}F_{e}(U)\\
F_{e^{a}}(U)\\
\ldots\\
F_{e^{1\ldots n}}(U)\end{array}\right)=M_{n}\left(\begin{array}[]{l}\pi_{e}(U)%
\\
\pi_{e^{1}}(U)\\
\ldots\\
\pi_{e^{1\ldots n}}(U)\end{array}\right).$$
Using Theorem 6.1 in the case of even $n$ we obtain
$$\left(\begin{array}[]{l}\pi_{e}(U)\\
\pi_{e^{1}}(U)\\
\ldots\\
\pi_{e^{1\ldots n}}(U)\end{array}\right)=\frac{1}{2^{n}}M_{n}\left(\begin{%
array}[]{l}F_{e}(U)\\
F_{e^{1}}(U)\\
\ldots\\
F_{e^{1\ldots n}}(U)\end{array}\right).$$
In the case of odd $n$ we have
$$\displaystyle F_{e^{A}}(U)=(e^{A})^{-1}Ue^{A}=\sum_{B}m_{AB}\pi_{e^{B}}(U)=%
\sum_{B\in{\rm I}_{{\rm First}}}l_{AB}\pi_{e^{B},\widetilde{e^{B}}}(U).$$
(6.7)
and use Theorem 6.1.
∎
Let’s give some examples.
In the case $n=1$ we have
$$\displaystyle M_{1}=\begin{pmatrix}1&1\cr 1&1\end{pmatrix},\qquad N_{1}=\begin%
{pmatrix}1\end{pmatrix},\qquad F_{e}(U)=F_{e^{1}}(U)=U,\qquad\pi_{e,e^{1}}(U)=U.$$
In the case $n=2$ we have
$M_{2}=\begin{pmatrix}1&1&1&1\cr 1&1&-1&-1\cr 1&-1&1&-1\cr 1&-1&-1&1\end{%
pmatrix},$
$$\displaystyle F_{e}(U)=U,\qquad F_{e^{1}}(U)=\pi_{e}(U)+\pi_{e^{1}}(U)-\pi_{e^%
{2}}(U)-\pi_{e^{12}}(U),$$
$$\displaystyle F_{e^{2}}(U)=\pi_{e}(U)-\pi_{e^{1}}(U)+\pi_{e^{2}}(U)-\pi_{e^{12%
}}(U),$$
$$\displaystyle F_{e^{12}}(U)=\pi_{e}(U)-\pi_{e^{1}}(U)-\pi_{e^{2}}(U)+\pi_{e^{1%
2}}(U).$$
$$\displaystyle\pi_{e}(U)$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{4}(e_{A}Ue^{A}),$$
$$\displaystyle\pi_{e^{1}}(U)$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{4}(eUe+(e^{1})^{-1}Ue^{1}-(e^{2})^{-1}Ue^{2}-(e^{12})^{-%
1}Ue^{12}),$$
$$\displaystyle\pi_{e^{2}}(U)$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{4}(eUe-(e^{1})^{-1}Ue^{1}+(e^{2})^{-1}Ue^{2}-(e^{12})^{-%
1}Ue^{12}),$$
$$\displaystyle\pi_{e^{12}}(U)$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{4}(eUe-(e^{1})^{-1}Ue^{1}-(e^{2})^{-1}Ue^{2}+(e^{12})^{-%
1}Ue^{12}).$$
In the case $n=3$ we have
$N_{2}=\begin{pmatrix}1&1&1&1\cr 1&1&-1&-1\cr 1&-1&1&-1\cr 1&-1&-1&1\cr\end{%
pmatrix},$
$$\displaystyle\pi_{e,e^{123}}(U)$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{4}(eUe+(e^{1})^{-1}Ue^{1}+(e^{2})^{-1}Ue^{2}+(e^{3})^{-1%
}Ue^{12}),$$
$$\displaystyle\pi_{e^{1},e^{23}}(U)$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{4}(eUe+(e^{1})^{-1}Ue^{1}-(e^{2})^{-1}Ue^{2}-(e^{3})^{-1%
}Ue^{3}),$$
$$\displaystyle\pi_{e^{2},e^{13}}(U)$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{4}(eUe-(e^{1})^{-1}Ue^{1}+(e^{2})^{-1}Ue^{2}-(e^{3})^{-1%
}Ue^{3}),$$
$$\displaystyle\pi_{e^{3},e^{12}}(U)$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{4}(eUe-(e^{1})^{-1}Ue^{1}-(e^{2})^{-1}Ue^{2}+(e^{3})^{-1%
}Ue^{3}).$$
7 Contractions by adjoint set
Consider contractions by adjoint set
$$F_{{\rm Adj}}(U)=\frac{1}{2^{n-1}}\sum_{A\in{\rm I}_{{\rm Adj}}}e_{A}Ue^{A}.$$
Theorem 7.1.
Consider an arbitrary Clifford algebra element $U$. Let we have 2 adjoint sets ${\rm I}={\rm I}_{{\rm Adj}}\cup\widetilde{{\rm I}_{{\rm Adj}}}$.
In the case of arbitrary $n$ we have
$$F_{{\rm Adj}}(U)=F(U).$$
Proof.
If $n$ is odd, then $e^{1\ldots n}$ is in the center of Clifford algebra,
$(e^{a_{1}\ldots a_{m}})^{-1}Ue^{a_{1}\ldots a_{m}}=e^{1\ldots n}(e^{1\ldots n}%
)^{-1}(e^{a_{1}\ldots a_{m}})^{-1}Ue^{a_{1}\ldots a_{m}}=(e^{\widetilde{a_{1}%
\ldots a_{m}}})^{-1}Ue^{\widetilde{a_{1}\ldots a_{m}}},$
and
$$\displaystyle e_{A}Ue^{A}=2\sum_{A\in{\rm I}_{{\rm Adj}}}e_{A}Ue^{A}.$$
(7.1)
If $n$ is even, then $e^{1\ldots n}$ anticommutes with all odd basis elements and commutes with all even basis elements (see Theorem 5.1). So
if $U=U_{0}+U_{1}$, $U_{0}\in{C}\!\ell_{{\rm Even}}(p,q)$, $U_{1}\in{C}\!\ell_{{\rm Odd}}(p,q)$, then for $k=0,1$ we have
$$(e^{a_{1}\ldots a_{m}})^{-1}U_{k}e^{a_{1}\ldots a_{m}}=e^{1\ldots n}(e^{1%
\ldots n})^{-1}(e^{a_{1}\ldots a_{m}})^{-1}U_{k}e^{a_{1}\ldots a_{m}}=$$
$$=(-1)^{2m+k}(e^{a_{1}\ldots a_{m}})^{-1}(e^{1\ldots n})^{-1}U_{k}e^{1\ldots n}%
e^{a_{1}\ldots a_{m}}=(-1)^{k}(e^{\widetilde{a_{1}\ldots a_{m}}})^{-1}U_{k}e^{%
\widetilde{a_{1}\ldots a_{m}}},$$
and we obtain (7.1) again.
∎
So we can use contraction $F_{{\rm Adj}}$ (with $2^{n-1}$ summands) instead of full contraction $F(U)$ (with $2^{n}$ summands) in all calculations.
8 Conclusion
In the present paper we consider full contractions, simple contractions and contractions by adjoint sets of multi-indices. We can also consider another contractions (for other subsets $S\subseteq{\rm I}$).
In [4] we consider generator contractions in Clifford algebra ${C}\!\ell(p,q)$
$$F_{1}(U)=e_{a}Ue^{a}$$
and prove that $e_{a}Ue^{a}=\sum_{k=0}^{n}(-1)^{k}(n-2k)\pi_{k}(U)$.
In [10] we present relation between generator contractions and projections onto subspaces of fixed ranks. We use this relation to present new class of gauge invariant solutions of Yang-Mills equations.
We can consider contractions $\sum_{A\in S}e_{A}Ue^{A}$ by another subsets $S\subseteq{\rm I}$:
$$\displaystyle{\rm I}_{{\rm Even}}=\{A\in{\rm I},\,|A|-\mbox{even}\},\qquad{\rm
I%
}_{{\rm Odd}}=\{A\in{\rm I},\,|A|-\mbox{odd}\},$$
$$\displaystyle{\rm I}_{k}=\{A\in{\rm I},\quad|A|=k\},\qquad k=0,1,\ldots,n,$$
$$\displaystyle{\rm I}_{\overline{k}}=\{A\in{\rm I},\quad|A|=m\mod 4\},\qquad m=%
0,1,2,3.$$
We call them even and odd contractions, contractions of ranks $k$ (in particular case $k=1$ we obtain generator contraction) and contractions of quaternion types $m$. There is a relation between these contractions and projective operations onto fixed subspaces of Clifford algebras. This is a subject for further research.
Note that in all theorems of this paper we can consider not basis elements of Clifford algebra elements but arbitrary set of Clifford algebra elements $\gamma^{a}\in{C}\!\ell(p,q)$ that satisfy conditions $\gamma^{a}\gamma^{b}+\gamma^{b}\gamma^{a}=2\eta^{ab}e$. This set may generate another basis of Clifford algebra ${C}\!\ell(p,q)$ (but in some cases of odd dimension $n$ this set does not generate basis of Clifford algebra element ${C}\!\ell(p,q)$, see [11]).
In [11] we consider generalized contractions $\sum_{A}\gamma^{A}U\beta_{A}$ when we have 2 such different sets $\gamma^{a}$, $\beta^{a}$ in Clifford algebra. We use these contractions to prove generalized Pauli’s theorem and some other problems about spin groups (see [5], [6], [7], [8], [9]).
The results of this article (especially about relation between projection operations and contractions; solving commutator equations) may be used in computer calculations.
Acknowledgment
The author is grateful to N.G.Marchuk for fruitful discussions.
References
[1]
Dixon J. D., Computing irreducible representations of groups, Math. of comp., (1970).
[2]
Babai L., Friedl K., Approximate representation theory of finite groups, Found. of Comp. Science, (1991).
[3]
Lounesto P., Clifford Algebras and Spinors. Vol. 239 / L.M.S. Lecture Notes. Cambridge: Cambridge Univ. Press, 306 pp. (1997).
[4]
N. G. Marchuk, D. S. Shirokov, Unitary spaces on Clifford algebras, Adv. Appl. Clifford Algebr., 18:2 (2008), 237-254.
[5]
Shirokov D.S., Extension of Pauli’s theorem to Clifford algebras, Dokl. Math., 84, 2, 699-701 (2011).
[6]
Shirokov D.S., Pauli theorem in the description of n-dimensional spinors in the Clifford algebra formalism, Theoret. and Math. Phys., 175:1 (2013), 454-474.
[7]
Shirokov D.S. The use of the generalized Pauli’s theorem for odd elements of Clifford algebra to analyze relations between spin and orthogonal groups of arbitrary dimensions, Vestn. Samar. Gos. Tekhn. Univ. Ser. Fiz.-Mat. Nauki, 1(30), 2013, p. 279-287.
[8]
Shirokov D.S., Calculations of elements of spin groups using generalized Pauli’s theorem, Advances in Applied Clifford Algebras, to appear, 2014, arXiv:1409.2449 [math-ph].
[9]
Marchuk N.G., Shirokov D.S., Local Generalized Pauli’s Theorem, (2012), arXiv:1201.4985 [math-ph].
[10]
N.G. Marchuk, D. S. Shirokov, New class of gauge invariant solutions of Yang-Mills equations, arXiv:1406.6665 [math-ph].
[11]
D.S. Shirokov, Method of generalized contractions and Pauli’s theorem in Clifford algebras, 2014, 14 pp., arXiv:1409.8163. |
An Ising model for galaxy bias
Andrew Repp & István Szapudi
Institute for Astronomy, University of Hawaii, 2680 Woodlawn Drive, Honolulu, HI 96822, USA
(January 15, 2021; to be submitted to MNRAS)
Abstract
A reliable model of galaxy bias is necessary for interpreting data from future dense galaxy surveys. Conventional bias models are inaccurate, in that they can yield unphysical results ($\delta_{g}<-1$) for voids that might contain half of the available cosmological information. For this reason, we present a physically-motivated bias model based on an analogy with the Ising model. With only two free parameters, the model produces sensible results for both high- and low-density regions. We also test the model using a catalog of Millennium Simulation galaxies in cubical survey pixels with side lengths from $2h^{-1}$–$31h^{-1}$Mpc, at redshifts from 0 to 2. We find the Ising model markedly superior to linear and quadratic bias models on scales smaller than $10h^{-1}$Mpc, while those conventional models fare better on scales larger than $30h^{-1}$Mpc. While the largest scale where the Ising model is applicable might vary for a specific galaxy catalog, it should be superior on any scale with a non-negligible fraction of cells devoid of galaxies.
††pagerange: An Ising model for galaxy bias–37
1 Introduction
Galaxy surveys represent an important observational constraint on cosmology; indeed, one of the main science drivers for planned surveys such as Euclid (Laureijs
et al. 2011) and WFIRST (Green et al. 2012) is the expectation that their data will encode large amounts of information on the properties of dark energy. Furthermore, voids – comprising roughly half the initial volume of the universe – contain up to half of the cosmological information borne by matter-clustering statistics (see, e.g., Wolk
et al. 2015a). Thus, in order to extract cosmological information from survey data, we require techniques applicable to both clusters and voids.
However, galaxies are biased tracers of matter (see the theoretical treatment in Kaiser 1984; Bardeen et al. 1986), and thus cosmological inference from galaxy surveys requires modeling of the relationship between the matter and galaxy overdensities ($\delta=\rho/\overline{\rho}-1$ and $\delta_{g}=N_{\mathrm{gal}}/\overline{N}-1$, respectively). Perhaps the most common approach (e.g., Hoffmann et al. 2017) is to expand $\delta_{g}$ either linearly ($\delta_{g}=b\delta$) or quadratically ($\delta_{g}=b_{1}\delta+b_{2}\delta^{2}/2$). Another approach (e.g., de la Torre &
Peacock 2013) performs the expansion in log space so that $\ln(1+\delta_{g})=b\ln(1+\delta)$.
One could also attempt to derive a galaxy-dark matter relationship from full-scale halo models and/or hydrodynamical simulations. However, halo modeling is nontrivial, and simulations rely on assumptions about galaxy formation and evolution, many aspects of which remain uncertain. Furthermore, the vast range of scales involved requires simplifications, of unknown impact, in order to render the computations tractable.
Hence, considering the bias models listed above, the first (linear bias) emerges from linear perturbation theory (e.g., Desjacques et al. 2018) and thus provides a reasonable description on large scales. However, future surveys (such as Euclid and WFIRST) will require models that are accurate on smaller, non-linear scales, and on such scales the standard bias models become problematic. For instance, Neyrinck et al. (2014) note an exponential decline of dark matter haloes at low densities, a decline not captured by a linear bias. In particular, linear and quadratic models easily yield non-physical results ($\delta_{g}<-1$) in voids, and thus these models fail in regions that potentially constitute half of a survey’s information on dark energy. Thus these models fail in regions that potentially constitute half of a survey’s information on dark energy. Voids are expected to play an increasingly significant role in cosmological constraints (Pisani
et al. 2019). Likewise, in order to detect any screening effects of modified gravity, one must analyze both ends of the density spectrum, again requiring a model of galaxy bias that yields reasonable results at both density extremes.
In this work we present a model (inspired by the Ising model of ferromagnetism) which meets these conditions (Section 2). We then analyze its accuracy compared to simulation results (Section 3); discussion and conclusion follow in Sections 4 and 5.
2 The Ising Model
In formulating this model, we focus on dark matter subhaloes, which can host individual galaxies, rather than on the larger haloes (which potentially host many galaxies).
We also make a few simple assumptions about galaxy formation. First, we assume that whether or not a galaxy forms in a particular subhalo depends (to first order) only on initial densities and local physics – thus ignoring any tidal influences. Since $\ln(1+\delta)$ captures the approximate initial conditions (Neyrinck
et al. 2009; Carron &
Szapudi 2013), we express our model in terms of $A\equiv\ln(1+\delta)$.
Second, for the galaxies under consideration in a given survey, we assume it is legitimate to treat the subhaloes in which they form as roughly equivalent. Hence we assume that we can, to first order, characterize these subhaloes as identical entities, each of which is in one of two possible states – namely, either hosting a galaxy or not.
Third, we note that the release of gravitational potential energy during galaxy formation results in energetic favorability for the “hosting” state. And since clustered galaxies collectively occupy deeper potential wells than isolated galaxies, we can extend this assumption to argue that galaxy formation is increasingly favorable in survey cells of higher overall density.
It is now straightforward to map these assumptions on to the Ising model of ferromagnetism, with subhaloes playing the role of atoms. First, an Ising model typically allows interaction between neighboring atoms (facilitating the formation of ferromagnetic domains); however, pursuant to our locality assumption we set the interaction term to zero. Second, an atom in the Ising model has two possible states (spin-up or spin-down); replacing the atoms with subhaloes, the two spin states are analogous to two occupation states (i.e., either hosting a galaxy or not). Third, an Ising model allows external fields to render one of the spin states energetically favorable; analogously, we can treat one of the subhalo occupation states (namely, the galaxy-hosting state) as preferable on energetic grounds.
We thus propose an interactionless Ising model of galaxy incidence in which occupation states replace spin states. These assumptions then yield (e.g., Pathria 1972) the following expression for the fraction of subhaloes in a favorable (galaxy-hosting) state:
$$f_{\mathrm{gal}}=\frac{1}{2}e^{\beta E}\operatorname{sech}\beta E=\frac{1}{1+e%
^{-2\beta E}},$$
(1)
where $E$ is energy difference between the two states and $\beta$ is analogous to inverse temperature. (The thermodynamics of gravitational collapse suggest that this temperature analogue will be negative.) The unitless quantity $\beta E$ depends (pursuant to our first and third assumptions) on the initial density, for which we use the log density $A\equiv\ln(1+\delta)$ as a proxy (Carron &
Szapudi 2013). We thus substitute a linear function of $A$ for $\beta E$ and find that Equation 1 reduces to a Fermi-Dirac distribution:
$$f_{\mathrm{gal}}=\frac{1}{1+\exp\left(\frac{A-A_{t}}{-T}\right)}\hskip 28.4527%
56pt(T>0).$$
(2)
In this expression, $A_{t}$ marks the transition density between empty and occupied subhaloes. The subhaloes in higher-density cells fill up first, and $T$ (the absolute value of the negative “temperature”) parametrizes the sharpness of the transition from the empty to the occupied state: as $T$ approaches zero, Equation 2 approaches a step function.
Note that the Fermi-Dirac form of Equation 2 suggests other analogies: one could argue that galaxies observe an exclusion principle, with at most one galaxy occupying a given subhalo. One could also treat the transition density $A_{t}$ as analogous to chemical potential since it characterizes the (log) density at which the next galaxy would form.
Continuing, our goal is to describe the expected number of galaxies per survey cell as a function of the underlying dark matter density. We would expect (ceteris paribus) the number of subhaloes in a survey cell to be proportional to the cell’s matter density ($\langle N_{\mathrm{sh}}\rangle_{A}\propto 1+\delta\equiv e^{A}$). Hence, given a cell of log density $A$, we express the occupied fraction $f_{\mathrm{gal}}$ of subhaloes as follows:
$$f_{\mathrm{gal}}=\frac{\langle N_{\mathrm{gal}}\rangle_{A}}{\langle N_{\mathrm%
{sh}}\rangle_{A}}=\frac{\langle N_{\mathrm{gal}}\rangle_{A}}{b\overline{N}(1+%
\delta)},$$
(3)
where $\overline{N}$ is the global mean number of galaxies per cell. (Here, as in the rest of this paper, subscripted $A$ indicates the underlying log dark matter density; thus $\langle N_{\mathrm{gal}}\rangle_{A}$ denotes the expected number of galaxies in a cell with log density $A$.) Thus it is convenient to write Equation 2 in terms of $M$, the expected number of galaxies per dark matter mass:
$$M\equiv\langle N_{\mathrm{gal}}\rangle_{A}\cdot(1+\delta)^{-1}=\frac{b%
\overline{N}}{1+\exp\left(\frac{A_{t}-A}{T}\right)}\hskip 28.452756pt(T>0),$$
(4)
where, again, $A\equiv\ln(1+\delta)$.
In high-density regions $A\gg A_{t}$, so that $M$ approaches $b\overline{N}$, and $\langle N_{\mathrm{gal}}\rangle_{A}$ approaches $b\overline{N}(1+\delta)$; thus at high densities the number of galaxies is proportional to the amount of underlying matter. For low-density regions ($A\ll A_{t}$), the number of galaxies drops exponentially to zero, as Neyrinck et al. (2014) observe.
Note that the parameter $b$ represents the overall bias in high-density regions; in these regions, $(1+\delta_{g})=\langle N_{\mathrm{gal}}\rangle_{A}/\overline{N}=b(1+\delta)$, so that $\delta_{g}\sim b\delta$, as in the linear bias model. Since high-density regions are the predominant influence on the matter power spectrum, it is not surprising that linear bias models seem to fit the relationship between matter and galaxy spectra. However, both of these spectra – and the linear bias model, as noted above – discard much of the (substantial) information inherent in voids (Neyrinck
et al. 2009; Carron &
Szapudi 2013, 2014; Wolk
et al. 2015b; Repp
et al. 2015).
Finally, we can derive an additional constraint – and thus reduce the number of free model parameters to two – by considering the (global) mean number of galaxies. We obtain this mean by integrating the expected galaxy counts from Equation 4 against the probability distribution $\mathcal{P}(A)$ of the underlying matter density:
$$\overline{N}=\int dA\,\mathcal{P}(A)\langle N_{\mathrm{gal}}\rangle_{A}=\int dA%
\,\mathcal{P}(A)\frac{b\overline{N}e^{A}}{1+\exp\left(\frac{A_{t}-A}{T}\right)}.$$
(5)
It follows that
$$b=\left(\int dA\,\,\mathcal{P}(A)\frac{e^{A}}{1+\exp\left(\frac{A_{t}-A}{T}%
\right)}\right)^{-1}.$$
(6)
To impose this constraint we must know the dark matter probability distribution. Since in this work we consider simulation results – in which we have access to both the dark matter and the galaxy contents of each cell – we shall for this paper use the empirical dark matter distribution given by the simulation itself.
Before proceding to comparison with simulation results, we state explicitly the following analogues of Equation 4 for the linear, quadratic, and logarithmic bias models, where again we define $M\equiv\langle N_{\mathrm{gal}}\rangle_{A}\cdot(1+\delta)^{-1}$:
$$\displaystyle M$$
$$\displaystyle=\overline{N}\left(b+(1-b)e^{-A}\right)\hskip 14.226378pt\mbox{(%
linear bias)}$$
(7)
$$\displaystyle M$$
$$\displaystyle=\overline{N}\left(\frac{b_{2}}{2}e^{A}+\left(b_{1}-b_{2}\right)+%
e^{-A}\left(\frac{b_{2}}{2}-b_{1}+1\right)\right)$$
(8)
$$\displaystyle \mbox{ (quadratic bias)}$$
$$\displaystyle M$$
$$\displaystyle=\overline{N}e^{A(b-1)}\hskip 14.226378pt\mbox{(logarithmic bias)}$$
(9)
We fit Equations 4 and 7–9 to simulation results in the following section.
3 Comparison to Simulation Results
To validate the Ising model of Equation 4, we must compare it to simulations and/or observations. This section accomplishes the former by reporting a series of comparisons to the Millennium Simulation (Springel et al. 2005). In a subsequent paper (Repp & Szapudi, in prep.) we perform comparisons to observational data.
The Millennium Simulation111http://gavo.mpa-garching.mpg.de/Millennium/ provides dark matter densities in a cubical volume with sides of $500h^{-1}$ Mpc. We consider simulation snapshots 32, 41, 48, and 63, corresponding in the original Millennium Simulation cosmology to $z=2.07$, 0.99, 0.51, and 0.00, respectively.
For our galaxy catalog, we employ the results described in Bertone
et al. (2007), in which the L-Galaxies semi-analytic model (also known as the Munich model, outlined in Croton
et al. 2006 and De Lucia et al. 2006) is applied to the Millennium Simulation outputs. To obtain a dense sample, we impose the stellar mass cut $M_{\star}\geq 10^{9}h^{-1}M_{\odot}$. We then determine counts-in-cells (for both the dark matter and galaxy distributions) using cubical cells with side lengths ranging from $1.95h^{-1}$ Mpc to $31.25h^{-1}$ Mpc; in this way we obtain results for five different smoothing scales. (The smallest scale yields $256^{3}$ cells; the largest scale, $16^{3}$.) In this manner, we construct galaxy catalogs for each of the above four redshifts. At $z=0$, the mean number $\overline{N}$ of galaxies per cell ranges from 0.352 on the smallest scale to almost 1450 on the largest; at $z=2.1$, $\overline{N}$ ranges from 0.275 to approximately 1120.
3.1 Fits to Simulation Data
Figure 1 displays in light gray (for each of the five cell sizes) the counts-in-cells values for $A$ and $N_{\mathrm{gal}}$ at $z=0$; Figures 2–4 do the same for redshifts 0.5, 1.0, and 2.1. Note that, as in Equation 4, we normalize the galaxy counts by $e^{A}\equiv 1+\delta$. In the first three panels of the figures, the loci corresponding to $N_{\mathrm{gal}}=0,1,\ldots$ are evident, with the curvature of the $N_{\mathrm{gal}}\geq 1$ loci being the result of the aforementioned normalization. These plots show clearly the large degree of stochasticity on scales smaller than $\sim 10h^{-1}$Mpc, in that for a given dark matter density $A$ there is a wide range of resulting galaxy counts $N_{\mathrm{gal}}$. Thus our first two assumptions in Section 2 are true only in an average sense – a nuance reflected in Equation 4 by the use of $\langle N_{\mathrm{gal}}\rangle_{A}$, the mean number of galaxies for a given matter density.
We proceed to consider these mean values by calculating $\langle N_{\mathrm{gal}}\rangle_{A}$ in a set of bins in $A$. To achieve sufficient resolution for estimating the probability distribution $\mathcal{P}(A)$ (without unduly manipulating the bin placement), we begin with a bin width $\Delta A=0.2$ and then reduce the bin size if necessary to ensure that at least twenty bins contain survey cells. The resulting bin widths are $\Delta A=0.2$ for all panels of Figures 1–4, with the following exceptions: for $31.2h^{-1}$-Mpc cells at $z=0$, 0.5, and 1.0, we have $\Delta A=0.1$; for $15.6h^{-1}$-Mpc cells at $z=2.1$ we have $\Delta A=0.1$; and for $31.2h^{-1}$-Mpc cells at $z=2.1$ we have $\Delta A=0.06$.
In each bin we then calculate the values of $\langle M\rangle_{\mathrm{bin}}=\langle N_{\mathrm{gal}}\cdot e^{-A}\rangle_{%
\mathrm{bin}}$ from Equation 4; we plot these values in red on Figures 1–4. The horizontal error bars for these points depict the standard deviation of the $A$-values in each bin, and the vertical error bars depict the estimated uncertainties of the measured $M$-values. (See Appendix A for details, where we explain our use of a subsample to reduce the effect of covariance on the error estimates.) It is clear from the plots that a sigmoid function, such as that provided by the Ising model, is a reasonable approximation on all five scales.
To quantify the fit of the various bias models, we determine for each one the best-fitting parameter values and the corresponding values of $\chi^{2}_{\nu}$ (chi-squared per degree of freedom), for each scale and redshift. To impose the net-galaxy constraint (Equation 6) on the Ising model, we use the empirical probability distribution $\mathcal{P}(A)$ derived from counting cells in the $A$-bins described above. The resulting best-fitting Ising bias models appear as the thick blue curves on Figures 1–4, and the best-fitting linear, quadratic, and logarithmic bias models appear in green. The best-fitting parameter values and their corresponding $\chi^{2}_{\nu}$-values appear in Table 1, and Figure 5 displays the $\chi^{2}_{\nu}$-values for the various models.
Considering the reduced chi-squared values, we conclude that though the Ising model is not a perfect fit to the data, its fit is superior to the others – in many cases, vastly superior – at scales smaller than $\sim 10h^{-1}$ Mpc, specifically because it avoids unphysical predictions for low densities. In particular, at scales $\la 5h^{-1}$ the linear and quadratic models significantly underestimate the galaxy bias in the high-density regime. At intermediate scales (around $15h^{-1}$ Mpc), none of the models seems to provide a particularly good fit to the simulation results. However, at the largest scale analyzed (around $30h^{-1}$ Mpc) the linear and quadratic models fit the data better than the Ising model.
3.2 A Modified Ising model
Figures 1–4 (especially the panels displaying the 3.9–15.6$h^{-1}$-Mpc scales) seem to show that the Ising model parameters which give the best fit at moderate densities do not necessarily reflect the shape of the data at high densities; indeed, at high densities the linear bias seems to best reflect the curvature of the data points (albeit with a significant horizontal offset). Given this pattern, it is worthwhile to explore, in a preliminary fashion, the utility of modifiying the Ising model to provide better asymptotic behavior at high densities.
Our strategy is to expand the models about $e^{-A}=0$. We note that the linear bias model (Equation 7) is (naturally) strictly first-order in this expansion:
$$M=\overline{N}b+\overline{N}(1-b)e^{-A}.$$
(10)
Expanding the Ising model (Equation 4) in the same way, we obtain
$$M=\overline{N}b+O(e^{-2A}),$$
(11)
thus converging in the limit $A\rightarrow\infty$ to the same constant value as Equation 10, but lacking the correct first-order approach to that limit.
We thus desire a simple function $f(e^{-A})$ which approaches $\overline{N}(1-b)e^{-A}$ in the limit $A\rightarrow\infty$ while approaching zero in the limit $A\rightarrow-\infty$. A double exponential yields the correct behavior, so we let
$$\displaystyle f(e^{-A})$$
$$\displaystyle=\overline{N}(1-b)e^{-A}\exp\left(-ke^{-A}\right)$$
(12)
$$\displaystyle=\overline{N}(1-b)e^{-A}+O(e^{-2A})$$
(13)
serve as our correction term, where $k$ is a free parameter controlling the location of the transition between the high- and low-density regimes. Hence, we can now investigate whether it is advantageous to consider a modified Ising model
$$M=\frac{b\overline{N}}{1+\exp\left(\frac{A_{t}-A}{T}\right)}+(1-b)\overline{N}%
e^{-A}\exp\left(-ke^{-A}\right).$$
(14)
The net-galaxy constraint analogous to Equation 6 now becomes
$$b=\frac{1-I_{2}}{I_{1}-I_{2}},$$
(15)
where $I_{1}$ and $I_{2}$ are the integrals
$$\displaystyle I_{1}$$
$$\displaystyle=\int dA\,\mathcal{P}(A)\frac{e^{A}}{1+\exp\left(\frac{A_{t}-A}{T%
}\right)}$$
(16)
$$\displaystyle I_{2}$$
$$\displaystyle=\int dA\,\mathcal{P}(A)\exp\left(-ke^{-A}\right).$$
(17)
In the limit of $k\rightarrow\infty$, we have $I_{2}=0$, and Equations 14 and 15 reduce to Equations 4 and 6, respectively.
We thus repeat the fitting procedure of Section 3.1 to investigate the improvement (if any) achieved by the addition of the extra term. The resulting best fits appear as thin blue lines in Figures 1–4, and the best-fitting parameters for the modified model appear, along with their reduced $\chi^{2}$-values, in Table 2. The $\chi^{2}_{\nu}$ values for this model also appear in Figure 5 as points connected by thin blue lines.
The utility of including the modification term (with its extra free parameter $k$) seems to be mixed. At the smallest scale (around $2h^{-1}$ Mpc) and the largest scale (around $30h^{-1}$ Mpc), it provides essentially no improvement; in particular, at $30h^{-1}$-Mpc scales the quadratic model still consistently outperforms the Ising model. However, at intermediate scales (roughly $5$–$10h^{-1}$ Mpc) there is significant improvement, with $\chi^{2}_{\nu}$ values in many cases comparable to unity. At scales around $15h^{-1}$ Mpc, the modification does improve the fit – but as before, none of the models seems to yield a particularly good description of the simulation results.
Thus there seems to a non-negligible scale-dependence in the shape of the matter-galaxy relationship. This phenomenon merits further investigation which we, however, leave to future work. In addition, unlike the “plain” Ising model, the modification term seems to have no straightforward physical motivation. Arguably one could invoke linear perturbation theory to explain the asymptotically linear behavior in the high-density regime (Desjacques et al. 2018), but $4h^{-1}$-Mpc scales, at which the modification term still provides a good fit, are well-outside the linear regime. In a subsequent paper (Repp & Szapudi, in prep.) we explore whether or not the modification is useful in characterizing empirical galaxy data.
3.3 Best Fits to Abundance-Matched Data
The procedure in Section 3.1 demonstrates that, for this simulation, the Ising model provides a superior description (compared to standard bias models) of the small-scale galaxy distribution. In addition, the Ising model (without the modification term of Section 3.2) requires no more free parameters than the quadratic model.
However, adapting this procedure to empirical galaxy counts is problematic since dark matter densities are typically unavailable. We thus consider an alternative test relying on our understanding of the dark matter probability distribution $\mathcal{P}(A)$. Here, as in the previous sections, we use the distribution $\mathcal{P}(A)$ derived from the Millennium Simulation results; in dealing with observational data, one could use the log-GEV prescription of Repp &
Szapudi (2018).
The alternative approach explored here is to approximate the relationship between $A=\ln(1+\delta)$ and $N_{\mathrm{gal}}$ using an abundance matching procedure in which we match the cumulative distribution functions of $A$ and $N_{\mathrm{gal}}$. Thus, we define $N_{\mathrm{AM}}(A)$ as the lowest value which satisfies the inequality
$$\mathcal{F}(A)\equiv\int_{-\infty}^{A}dA^{\prime}\,\mathcal{P}(A^{\prime})\leq%
\sum_{N_{\mathrm{gal}}=0}^{N_{\mathrm{AM}}(A)}\mathcal{P}(N_{\mathrm{gal}})%
\equiv\mathcal{F}(N_{\mathrm{AM}}(A)).$$
(18)
We then assume that for an underlying dark matter density $A$, the mean number of galaxies $\langle N_{\mathrm{gal}}\rangle_{A}$ equals $N_{\mathrm{AM}}(A)$; in essence this approximation ignores the scatter of $N_{\mathrm{gal}}$ about its mean.
To gauge the accuracy of this procedure, we obtain 15 evenly-spaced points (in $A$-space) between the 0.5th and 99.5th percentiles of the $A$-distribution. For these values of $A$, we then use Equation 18 to determine $N_{\mathrm{AM}}(A)$, which we then set equal to $\langle N_{\mathrm{gal}}\rangle_{A}$. These abundance-matched values appear as light magenta points in Figures 1–4, thus facilitating comparison with the true values (in red) of $\langle N_{\mathrm{gal}}\rangle_{A}$.
Two facts are evident: first, at small scales ($\la 5h^{-1}$ Mpc) and high values of $A$, the abundance-matched points are systematically higher than the direct-comparison points. This systematic offset is unsurprising, given that the distribution produced by Poisson scatter skews to the right. Second, we see that the abundance-matched points drop more sharply at low values of $A$ than do the true values. This fact is a direct consequence of discreteness in the distribution of $N_{\mathrm{gal}}$: namely, $N_{\mathrm{AM}}(A)$ vanishes for all values of $A$ such that $\int_{-\infty}^{A}dA^{\prime}\,\mathcal{P}(A^{\prime})\,\leq\,\mathcal{P}(N_{%
\mathrm{gal}}=0)$. It is also for this reason that this second effect disappears at larger scales ($\ga 15h^{-1}$ Mpc) where discreteness is less pronounced.
The abundance-matched points still display sigmoid behavior. In general, the transition slope (parametrized by $T$) between the low and high asymptotic values is sharper for the abundance-matched points, precisely because of the aforementioned cutoff. A fit to these abundance-matched points, though crude, would still avoid the unphysical behaviors endemic to the linear and quadratic models at small scales.
We thus conclude that the abundance-matching procedure gives us a reasonable qualitative understanding of the relationship between $A$ and $N_{\mathrm{gal}}$, with the advantage that we need not actually determine the dark matter density in each survey cell. However, reliance on abundance-matching will typically exaggerate the sharpness of the transition between the low- and high-$N_{\mathrm{gal}}$ regimes; thus, while abundance-matching seems capable of discriminating between classes of models (e.g., Ising vs. linear) at small scales, it is nevertheless suboptimal for actual model-fitting. We present a better alternative in future work (Repp & Szapudi, in prep.).
4 Discussion
It is first worthwhile to re-emphasize the stochastic nature of this model, in that it predicts mean values only. This aspect in fact compensates for the approximate nature of the assumptions outlined at the start of Section 2. The Poisson distribution is perhaps the most natural assumption for the distribution of $N_{\mathrm{gal}}$ given $A$; however, the Ising model is compatible with other prescriptions for $\mathcal{P}(N_{\mathrm{gal}}|A)$ as well.
Second, it seems that, at the smallest scale ($2.0h^{-1}$ Mpc) in Figures 1–4, we observe a “bump” in $M$ just to the right of the sharp rise (although the bump may not be present for $z=2.07$). The feature seems to be a real effect not captured by any of the models here considered. It is tempting to speculate that there exists an “optimal” density for galaxy formation, above which the early formation of galaxies in multiple subhaloes suppresses subsequent formation in neighboring sites. (In constrast, we assumed that galaxy formation is strictly local.) It is also possible that this effect is peculiar to the particular semi-analytic model behind our galaxy catalog; it is even possible that the effect might depend on the stellar mass cutoff employed in selecting galaxies for the catalog.
Next, it is instructive to consider the trends in the best-fitting Ising bias parameters ($b$ in Equations 4 and 14), shown in Figure 6. In almost every case, the bias values from the modified model are higher than those from the original model; nevertheless, they show similar trends. First, the bias increases monotonically with redshift (as expected, given that the common motion of both dark matter and galaxies should reduce the bias over time, Tegmark &
Peebles 1998). Second, at scales larger than $5h^{-1}$ Mpc, the best-fitting values vary with scale by only a few per cent (with the exception of $z=2.1$). A similar behavior appears in fig. 2 of Contarini
et al. (2019), where the values of $\delta$ and $\delta_{g}$ from different scales lie on the same curve. Contarini
et al. (2019) consider scales larger than those in this work; it appears that at much smaller scales, the bias is not scale-independent but rises sharply.
Finally, let us consider the physical origin of the pseudo-temperature parameter $T$. In a true Fermi-Dirac situation, this value would parametrize the ability of (say) electrons to scatter from one state to another. In our case, $T$ parametrizes the various possibilities for the distribution of mass within a survey pixel: recall from Section 2 that we assume greater energetic favorability for clustered subhaloes. Since a cell of log density $A\equiv\ln(1+\delta)$ can host a variety of matter configurations (number of subhaloes, degree of clustering, etc.), a given value of $A$ corresponds to a range of energetic favorabilities. Just as $T$ (or, strictly speaking, $-T$) is analogous to temperature, the range of matter distributions internal to a given cell is analogous to entropy. In other words, $T$ parametrizes the thermodynamic effect of coarse-graining.
We show in Appendix B that, for a given redshift, $T^{2}$ should be a decreasing function of the log variance $\sigma_{A}^{2}=\langle A^{2}\rangle-\langle A\rangle^{2}$. Figure 7 displays the actual best-fitting values of $T$ (from the unmodified Ising model), and we see that this expectation is justified. At large scales (low variances) $T$ rises, producing the shallower transitions seen in the lower panels of Figures 1–4. Physically, this behavior reflects that fact that larger cells accommodate a wider range of internal matter configurations. At small scales (high variances) $T$ approaches zero, producing the sharper transitions in the earlier panels of those figures. Again, the physical interpretation of this trend is that the survey cell size is closer to the typical subhalo scale. In general, of course, the value of $T$ depends on both the type of galaxy under consideration and the specific survey parameters.
It is also apparent in Figure 7 that the relationship between $T^{2}$ and $\sigma_{A}^{2}$ is approximately exponential, and that a change in redshift simply offsets the relationship by a multiplicative factor (to first order) without significantly changing the shape. It turns out for a given $\sigma_{A}^{2}$, the height of the curve varies linearly with the growth function $D^{2}(z)$. For instance, taking as fiducial the values at $\sigma_{A}^{2}=0.50$ (since that point is in the domain of all four plotted curves), we find that
$$T_{z}^{2}(\sigma_{A}^{2}=0.5)=0.378D^{2}(z)+0.039,$$
(19)
where $D^{2}(z)$ is the amplitude of the linear power spectrum normalized such that $D^{2}(0)=1$. If we approximate the $\sigma_{A}^{2}$-$T^{2}$ relationship (at each redshift) by an exponential function and scale using Equation 19, then we obtain the dashed lines in Figure 7. In Appendix B we provide theoretical justification for both the exponential dependence of $T^{2}$ on $\sigma_{A}^{2}$ and for the form of the $z$-dependence in Equation 19.
5 Conclusions
A reliable model of galaxy bias is necessary in order to interpret the data from dense galaxy surveys. Conventional bias models (such as the linear or quadratic) work well in high-density regions, but they yield unphysical results for voids, which contain significant cosmological information.
We have here presented an Ising model that avoids unphysical predictions. This model follows from a small set of simple physical assumptions about galaxy formation. We have tested the model using Millennium Simulation galaxy catalogs and have found it vastly superior to conventional models on scales approximately $2$–$10h^{-1}$Mpc, although the linear and quadratic models are preferable at scales above $30h^{-1}$Mpc. (In the intermediate regime we find that none of the models seems to describe the bias well, although the Ising model is still typically preferable to the others.) At all scales considered, however, the Ising model provides reasonable results in both low- and high-density regions and furthermore requires only two free parameters (as does the quadratic model).
At high densities, the Ising model yields the correct zeroth-order (constant) asymptotic behavior. We also considered (in Section 3.2) the possibility of modifying the Ising model to guarantee the same first-order asymptotics as the linear model. The additional term produces a lower reduced $\chi^{2}$ at intermediate scales but does not seem to improve the fit at the smallest ($2h^{-1}$ Mpc) and largest ($30h^{-1}$ Mpc) scales. This behavior – as well as a possible physical motivation for the modification – merits further investigation.
In this work we have restricted our tests of the Ising model to simulation data alone. We are currently (Repp & Szapudi, in prep.) testing the model against empirical galaxy survey data. Nevertheless, the results presented here already seem to demonstrate the superiority of the Ising model for the analysis of galaxy surveys at non-linear scales.
Acknowledgements
The Millennium Simulation data bases used in this work and the web application providing online access to them were constructed as part of the activities of the German Astrophysical Virtual Observatory (GAVO). This work was supported by NASA Headquarters under the NASA Earth and Space Science Fellowship program – “Grant 80NSSC18K1081” – and AR gratefully acknowledges the support. IS acknowledges support from National Science Foundation (NSF) award 1616974.
References
Bardeen et al. (1986)
Bardeen J. M., Bond J. R., Kaiser N., Szalay A. S., 1986, ApJ, 304, 15
Bertone
et al. (2007)
Bertone S., De Lucia G., Thomas P. A., 2007, MNRAS, 379, 1143
Carron &
Szapudi (2013)
Carron J., Szapudi I., 2013, MNRAS,
434, 2961
Carron &
Szapudi (2014)
Carron J., Szapudi I., 2014, MNRAS,
439, L11
Contarini
et al. (2019)
Contarini S., Ronconi T., Marulli F., Moscardini L., Veropalumbo
A., Baldi M., 2019, arXiv e-prints, p. arXiv:1904.01022
Croton
et al. (2006)
Croton D. J., et al., 2006, MNRAS, 365, 11
De Lucia et al. (2006)
De Lucia G., Springel V., White S. D. M., Croton D., Kauffmann
G., 2006, MNRAS, 366, 499
Desjacques et al. (2018)
Desjacques V., Jeong D., Schmidt F., 2018, Phys. Rep., 733, 1
Green et al. (2012)
Green J., Schechter P., Baltay C., Bean R., Bennett D., Brown
R., Conselice C., et al., 2012, preprint, (arXiv:1208.4012)
Hoffmann et al. (2017)
Hoffmann K., Bel J., Gaztañaga E., 2017, MNRAS, 465, 2225
Kaiser (1984)
Kaiser N., 1984, ApJ, 284, L9
Laureijs
et al. (2011)
Laureijs R., Amiaux J., Arduini S., Auguères J. ., Brinchmann
J., Cole R., Cropper M., et al., 2011, preprint, (arXiv:1110.3193)
Neyrinck
et al. (2009)
Neyrinck M. C., Szapudi I., Szalay A. S., 2009, ApJ, 698, L90
Neyrinck et al. (2014)
Neyrinck M. C., Aragón-Calvo M. A., Jeong D., Wang X., 2014,
MNRAS, 441, 646
Pathria (1972)
Pathria R., 1972, Statistical Mechanics.
Pergamon, Oxford
Pisani
et al. (2019)
Pisani A., et al., 2019, BAAS, 51, 40
Repp &
Szapudi (2017)
Repp A., Szapudi I., 2017, MNRAS,
464, L21
Repp &
Szapudi (2018)
Repp A., Szapudi I., 2018, MNRAS,
473, 3598
Repp
et al. (2015)
Repp A., Szapudi I., Carron J., Wolk M., 2015, MNRAS, 454, 3533
Springel et al. (2005)
Springel V., White S. D. M., Jenkins A., Frenk C. S., Yoshida N.,
Gao L., Navarro J., et al., 2005, Nature, 435, 629
Tegmark &
Peebles (1998)
Tegmark M., Peebles P. J. E., 1998, ApJ,
500, L79
Wolk
et al. (2015a)
Wolk M., Carron J., Szapudi I., 2015a, MNRAS, 451, 1682
Wolk
et al. (2015b)
Wolk M., Carron J., Szapudi I., 2015b, MNRAS, 454, 560
de la Torre &
Peacock (2013)
de la Torre S., Peacock J. A., 2013, MNRAS, 435, 743
Appendix A Determining Error on $M$
The vertical axes of Figures 1–4 display the values of $M(A)=\langle N_{\mathrm{gal}}\rangle_{A}\cdot e^{-A}$. (For the remainder of this appendix, we write $N$ for $N_{\mathrm{gal}}$.)
In a bin $\mathcal{A}$ of values of $A$, it is straightforward to show that the average value of $M$ in the bin is
$$\langle M\rangle_{\mathcal{A}}=\sum_{N}\int_{\mathcal{A}}dA\,\,\mathcal{P}_{%
\mathcal{A}}(N,A)\,Ne^{-A},$$
(20)
where $\mathcal{P}_{\mathcal{A}}(N,A)$ denotes the joint probability distribution within the bin $\mathcal{A}$. Hence, given
a set of measurements $N_{i}$, $A_{i}$ within an $A$-bin, the average $\hat{M}\equiv\langle N_{i}e^{-A_{i}}\rangle$ is the proper estimator for $\langle M\rangle_{\mathcal{A}}$, the expected value of $M$ within the bin $\mathcal{A}$; likewise, $\langle N_{i}^{2}e^{-2A_{i}}\rangle$ correctly estimates $\langle M^{2}\rangle_{\mathcal{A}}$. Thus the variance $\sigma_{M_{i}}^{2}$ of $N_{i}e^{-A_{i}}$ properly estimates $\sigma_{M}^{2}$; and thus the uncertainty of our estimator $\hat{M}$ is $\sigma_{\hat{M}}=\sigma_{M_{i}}/\sqrt{N_{\mathrm{cells}}}$, where $N_{\mathrm{cells}}$ is the number of survey cells in the bin $\mathcal{A}$. $\hat{M}$ and $\sigma_{\hat{M}}$ provide the vertical positions and vertical error bars in Figures 1–4.
The preceding analysis assumes that the measurements $N_{i}$, $A_{i}$ within a bin $\mathcal{A}$ are independent and thus uncorrelated; however, significant correlation does exist between cells on small scales. Proper accounting for this correlation would require a model of the two-point probability distribution as well as a model of galaxy bias – leading to circularity, since a model of galaxy bias is precisely what we attempt to validate in this work.
Thus, in this work we use only a subset of the dark matter and galaxy catalogs, chosen such that all cells are at least $\sim 10h^{-1}$ Mpc distant from each other; in this way we seek to minimize the correlations among cells and thus to ensure that they do not significantly influence the size of $\sigma_{\hat{M}}$. In particular, from the $1.95h^{-1}$-Mpc catalogs, we use only every fifth cell (in each dimension); from the $3.91h^{-1}$-Mpc catalogs, every third cell; and from the $7.81h^{-1}$-Mpc catalogs, every other cell. For the larger-scale catalogs, the cell centers are already at least $10h^{-1}$ Mpc apart, and so we use the full catalogs.
Appendix B The Relationship of $T$ and $\sigma_{A}$
The Ising-model parameter $T$ (analogous to negative temperature) is the result of coarse-graining (see Section 4): for a survey cell of (log) density $A$, there exist multiple possible internal matter configurations yielding that density, and each matter configuration could potentially result in a different number of galaxies. Thus, there is an inherent scatter in the relationship $A\longmapsto N$, and $T$ parametrizes this scatter. This rationale for the role of $T$ suggests that we consider the relationship between fine-grain and coarse-grain variability.
Therefore, as in Figure 8, consider a survey cell $V$ of side length $R$. Subdivide $V$ into many smaller cells of side length $r$, where $r$ approximates the scale of an individual subhalo. Choose a specific small cell $v$, and let $V^{\prime}$ be the complement of $v$ in $V$ (i.e., $V^{\prime}=V\setminus v$).
Let $\delta_{R}$ be the density of $V$ (i.e., smoothed with length $R$), and $\delta_{r}$ be the density of $v$ (smoothed with length $r$). Then the density in the region $V^{\prime}$ is
$$\delta_{R^{\prime}}=\frac{V\delta_{R}-v\delta_{r}}{V-v},$$
(21)
so that
$$\displaystyle\delta_{R^{\prime}}\delta_{r}$$
$$\displaystyle=\frac{V\delta_{R}\delta_{r}-v\delta_{r}^{2}}{V-v}$$
(22)
$$\displaystyle\langle\delta_{R}\delta_{r}\rangle$$
$$\displaystyle=\frac{V-v}{V}\langle\delta_{R^{\prime}}\delta_{r}\rangle+\frac{v%
}{V}\langle\delta_{r}^{2}\rangle,$$
(23)
where the average is taken over the entire survey volume (i.e., all cells $V$ containing cells $v$). We also have
$$\displaystyle\delta_{R^{\prime}}\delta_{r}$$
$$\displaystyle=\frac{1}{(V-v)v}\int_{V^{\prime}}d^{3}r^{\prime}\,\delta(r^{%
\prime})\int_{v}d^{3}r\,\delta(r)$$
(24)
$$\displaystyle\langle\delta_{R^{\prime}}\delta_{r}\rangle$$
$$\displaystyle=\frac{1}{(V-v)v}\int_{V^{\prime}}d^{3}r^{\prime}\int_{v}d^{3}r\,%
\langle\delta(r^{\prime})\delta(r)\rangle$$
(25)
$$\displaystyle=\frac{1}{(V-v)v}\int_{V^{\prime}}d^{3}r^{\prime}\int_{v}d^{3}r\,%
\xi(r^{\prime}-r),$$
(26)
where $\xi(r)$ is the two-point correlation function. Making some approximations in the limit of small $v$, Equation 26 implies
$$\displaystyle\langle\delta_{R^{\prime}}\delta_{r}\rangle$$
$$\displaystyle\approx\frac{1}{V-v}\int_{V^{\prime}}d^{3}r^{\prime}\,\xi(r^{%
\prime})$$
(27)
$$\displaystyle\approx\frac{V}{V-v}\sigma_{R}^{2}-\frac{v}{V-v}\sigma_{r}^{2}.$$
(28)
It follows from Equations 23 and 28 that the correlation $\rho_{Rr}$ between large- and small-scale density is simply
$$\rho_{Rr}=\frac{\xi_{Rr}}{\sigma_{R}\sigma_{r}}=\frac{\langle\delta_{R}\delta_%
{r}\rangle}{\sigma_{R}\sigma_{r}}\approx\frac{\sigma_{R}^{2}}{\sigma_{R}\sigma%
_{r}}=\frac{\sigma_{R}}{\sigma_{r}}.$$
(29)
(Note that the implication $\xi_{Rr}=\sigma_{R}^{2}$ is as expected in the limit of small $v$.) As a result of 29, the fine-grain variability within a given coarse-grain cell is
$$\left.\sigma_{r}^{2}\right|_{R}=\sigma_{r}^{2}\left(1-\rho_{Rr}^{2}\right)%
\approx\sigma_{r}^{2}-\sigma_{R}^{2}.$$
(30)
The quantity $\left.\sigma_{r}^{2}\right|_{R}$ is the variance of the fine-grained matter density within a coarse-grained cell, and we thus identify it with $T^{2}$ and note that it is a decreasing function of $\sigma_{R}^{2}$.
Now, our first assumption in constructing the Ising model is that galaxy formation is determined (to first order) by the initial conditions; thus, for the variance $\sigma_{R}^{2}$ we should, strictly speaking, use the initial variance or, as evolved forward in time, the linear variance $\sigma^{2}_{\mathrm{lin}}$. However, the log variance $\sigma_{A}^{2}$ is an increasing function of $\sigma^{2}_{\mathrm{lin}}$ (Repp &
Szapudi 2017), and thus we can conclude that $T^{2}$ is a decreasing function of $\sigma_{A}^{2}$.
Indeed, we can go further to explain the approximately exponential relationship observed in Figure 7, in which $\ln T^{2}$ is roughly a linear function of $\sigma_{A}^{2}$. The key to doing so is the relationship between $\sigma_{A}^{2}$ and the linear variance, $\sigma_{\mathrm{lin}}^{2}$, developed in Repp &
Szapudi (2017), where we show that
$$\sigma_{\mathrm{lin}}^{2}=\mu\exp\frac{\sigma_{A}^{2}}{\mu}-\mu,$$
(31)
with $\mu=0.73$. So, identifying $\left.\sigma_{r}^{2}\right|_{R}$ in Equation 30 as $T^{2}$ and explicitly specifying the linear variance, we can write
$$T^{2}=\sigma_{\mathrm{lin},r}^{2}-\sigma_{\mathrm{lin},R}^{2},$$
(32)
where again $r$ and $R$ specify the fine- and coarse-graining scales, respectively. Since we are measuring $\sigma_{A}^{2}$ at the coarse-grain scale, we now write
$$T^{2}=\left(\sigma_{\mathrm{lin},r}^{2}+\mu\right)-\mu\exp\frac{\sigma_{A}^{2}%
}{\mu},$$
(33)
so that $T^{2}$ has an exponential dependence on $\sigma_{A}^{2}$, as seen in the figure. (The first term of Equation 33 introduces the slight non-linearity observed in the curves of Figure 7.)
Next, let us examine the effect of changing the redshift. We note that the linear variance scales by $D^{2}(z)$, where $D(z)$ is the growth function normalized to unity at $z=0$. So if $\sigma_{r,0}$ is the linear variance at the fine-grain scale at $z=0$, we have from Equation 33
$$T^{2}_{z}(\sigma_{A}^{2})=\left(D^{2}(z)\cdot\sigma_{r,0}^{2}+\mu\right)-\mu%
\exp\frac{\sigma_{A}^{2}}{\mu},$$
(34)
which is linear in $D^{2}(z)$, as in Equation 19.
Finally, Figure 7 indicates that the effect of changing the redshift is, to first approximation, a simple multiplicative scaling independent of $\sigma_{A}^{2}$. If we let $c(z)\equiv D^{2}(z)\cdot\sigma_{r,0}^{2}+\mu$, we can from Equation 34 write
$$\displaystyle\ln T^{2}_{z}(\sigma_{A}^{2})$$
$$\displaystyle=\ln c(z)+\ln\left(1-\frac{\mu}{c(z)}\exp\frac{\sigma_{A}^{2}}{%
\mu}\right)$$
(35)
$$\displaystyle\approx\ln c(z)-\frac{\mu}{c(z)}\exp\frac{\sigma_{A}^{2}}{\mu},$$
(36)
since we take the fine-graining scale to be small enough that $\sigma^{2}_{r,0}$ is quite large, so that $c(z)\gg\mu=0.73$. Then
$$\ln\frac{T_{z_{1}}^{2}}{T_{z_{2}}^{2}}=\ln\frac{c(z_{1})}{c(z_{2})}+\mu\left(%
\frac{1}{c(z_{2})}-\frac{1}{c(z_{1})}\right)\exp\frac{\sigma_{A}^{2}}{\mu}.$$
(37)
However, the large size of $\sigma^{2}_{r,0}$, to which we already appealed, means that $1/c(z)$ will be small, and $1/c(z_{2})-1/c(z_{1})$ will be smaller yet. It follows that the dependence on $\sigma_{A}^{2}$ in 37 is quite weak, so that, to a good approximation, the ratio $T_{z_{1}}^{2}/T_{z_{2}}^{2}$ depends only on the redshifts involved and not on $\sigma_{A}^{2}$, which is precisely what we see in Figure 7. In particular, it is the effect of redshift evolution on the fine-grain scale, as reflected in $c(z)$, which shifts the curves in Figure 7 down at higher redshifts. |
Sound propagation in a uniform superfluid two-dimensional Bose gas
J.L. Ville
R. Saint-Jalm
É. Le Cerf
M. Aidelsburger${}^{\dagger}$
S. Nascimbène
J. Dalibard
J. Beugnon
beugnon@lkb.ens.fr
[
${}^{1}$Laboratoire Kastler Brossel, Collège de France, CNRS, ENS-PSL University, Sorbonne Université, 11 Place Marcelin Berthelot, 75005 Paris, France
(November 23, 2020)
Abstract
In superfluid systems several sound modes can be excited, as for example first and second sound in liquid helium. Here, we excite propagating and standing waves in a uniform two-dimensional Bose gas and we characterize the propagation of sound in both the superfluid and normal regime. In the superfluid phase, the measured speed of sound is well described by a two-fluid hydrodynamic model, and the weak damping rate is well explained by the scattering with thermal excitations. In the normal phase the sound becomes strongly damped due to a departure from hydrodynamic behavior.
${}^{\dagger}$Present address: ]Fakultät für Physik, Ludwig-Maximilians-Universität München, Schellingstr. 4, 80799 Munich, Germany.
Propagation of sound waves is at the heart of our understanding of quantum fluids. In liquid helium, the celebrated two-fluid model was confirmed by the observation of first and second sound modes Donnelly (2009). There, first sound stands for the usual sound appellation, namely a density wave for which normal and superfluid fractions oscillate in phase. Second sound corresponds to a pure entropy wave with no perturbation in density (normal and superfluid components oscillating out of phase), and is generally considered as a smoking gun of superfluidity.
Sound wave propagation is also central to the study of dilute quantum gases, providing information on thermodynamic properties, relaxation mechanisms and superfluid behavior. In ultracold strongly interacting Fermi gases, the existence of first and second sound modes in the superfluid phase was predicted Taylor et al. (2009) and observed in experiments Joseph et al. (2007); Sidorenkov et al. (2013), with a behavior similar to liquid helium. In weakly interacting Bose-Einstein condensates (BECs), one still expects two branches of sound with speeds $c^{(1)}>c^{(2)}$ but the nature of first and second sound is strongly modified because of their large compressibility Griffin and Zaremba (1997). While at zero temperature density perturbations propagate as Bogoliubov sound waves, at finite temperature we expect them to couple mostly to second sound – a behavior contrasting with the case of liquid helium – with a sound speed proportional to the square root of the superfluid fraction Griffin and Zaremba (1997); Ota and Stringari (2018). Sound waves in an elongated three-dimensional (3D) BEC were observed in Refs. Andrews et al. (1997); Meppelink et al. (2009a) in a regime where the sound speed remains close to the Bogoliubov sound speed.
Propagation of sound in weakly interacting two-dimensional (2D) Bose gases was recently discussed in Ref. Ozawa and Stringari (2014) using a hydrodynamic two-fluid model, predicting the existence of first and second sound modes of associated speeds $c^{(1)}_{\rm HD}$ and $c^{(2)}_{\rm HD}$, respectively. In 2D Bose gases, superfluidity occurs via the Berezinskii-Kosterlitz-Thouless (BKT) mechanism Hadzibabic and Dalibard (2011). The superfluid to normal transition is associated with a jump of the superfluid density that cannot be revealed from the thermodynamic properties of the gas. As the second sound speed is related to the superfluid fraction, one expects $c^{(2)}_{\rm HD}$ to remain non-zero just below the critical point of the superfluid to normal transition and to disappear just above the transition.
In this Letter, we study the propagation of sound in a 2D uniform Bose gas. We observe a single density sound mode both in the superfluid and normal phases. Deep in the superfluid regime, the measured sound speed agrees well with the Bogoliubov prediction. We measure a weak damping rate compatible with Landau damping, a fundamental mechanism for the understanding of collective modes of superfluids at finite temperature Pitaevskii and Stringari (1997). For higher temperatures, we observe a decrease of the sound velocity consistent with the second sound speed variation predicted in Ref. Ozawa and Stringari (2014) from two-fluid hydrodynamics. The damping of sound increases with temperature, and the sound propagation becomes marginal for temperatures close to the superfluid to normal transition. Above the critical point, we still observe strongly damped density waves, with no discernable discontinuity at the critical point. The discrepancy with the two-fluid model predictions could be due to a departure from hydrodynamic behavior, that manifests in our experiments as a strong damping of sound around the critical point.
Our experimental setup has been described in Refs Ville et al. (2017); Aidelsburger et al. (2017). Briefly, we confine ${}^{87}$Rb atoms in the $|F=1,m=0\rangle$ ground state into a 2D rectangular box potential of size $L_{x}\times L_{y}=30(1)\times 38(1)\,\,$\mathrm{\SIUnitSymbolMicro}\mathrm{m}$$ (see Fig. 1a). The trapping potential is made by a combination of far-detuned repulsive optical dipole traps. The confinement along the vertical $z$ direction can be approximated by a harmonic potential of frequency $\omega_{z}/(2\pi)=4.59(4)\,$kHz. We always operate in the quasi-2D regime where interaction and thermal energies are smaller than $\hbar\omega_{z}$. Collisions in our weakly-interacting Bose gas are characterized by the effective coupling constant $g=\hbar^{2}\tilde{g}/m=(\hbar^{2}/m)\sqrt{8\pi}\,a_{\rm s}/\ell_{z}$, where $a_{s}$ is the s-wave scattering length, $\ell_{z}=\sqrt{\hbar/(m\omega_{z})}$ and $m$ the atomic mass Hadzibabic and Dalibard (2011). With our confinement we have $\tilde{g}=0.16(1)$ 111The value of $\tilde{g}$ is slightly modified by the effect of interactions. We estimate that $\tilde{g}$ varies by about 10% for the range of surface densities explored in this work..
We control the temperature $T$ thanks to evaporative cooling by varying the height of the potential barrier providing the in-plane confinement. The surface density $n_{\rm 2D}$ of the cloud is varied from 10 to 80 $$\mathrm{\SIUnitSymbolMicro}\mathrm{m}$^{-2}$ by removing a controlled fraction of the atoms from our densest configuration 222This removal is realized by a partial transfer of the atoms to the $|F=2,m=0\rangle$ state with a microwave resonant field and a subsequent blasting of the transferred fraction with a resonant laser beam. In the quasi-2D regime and for a given $\tilde{g}$, the equilibrium state of the cloud is only characterized by a dimensionless combination of $T$ and $n_{\rm 2D}$, thanks to an approximate scale-invariance Hadzibabic and Dalibard (2011). In the following we use the ratio $T/T_{\rm c}$, where $T_{\rm c}=2\pi n_{\rm 2D}\hbar^{2}/[mk_{\rm B}\ln(380/\tilde{g})]$ is the calculated critical temperature for the BKT phase transition Prokof’ev et al. (2001). We determine the ratio $T/T_{\rm c}$ by a method inspired from Ref. Hueck et al. (2018) and based on a measurement of the equation of state of the system (see REF for more details). In this work, we study Bose gases from the highly degenerate regime ($T/T_{\rm c}\approx 0.2$) to the normal regime ($T/T_{\rm c}\approx 1.4$).
We first investigate propagating waves which we excite by a density perturbation. Prior to evaporative cooling in the box potential, we apply to the cloud a repulsive potential, which creates a density dip on one side of the rectangle (see Fig. 1a). The extension of this dip is about 1/4 of the length of the box and its amplitude is chosen so that the density in this region is decreased by a factor of 1/3. After equilibration, we abruptly remove the additional potential and monitor the propagation of this density dip. We show in Fig. 1b a typical time evolution of the density profile integrated along the transverse direction to the perturbation for a strongly degenerate gas 333We did not observe any excitation along the $x$ direction.. In this regime, the density perturbation propagates at constant speed and bounces several times off the walls of the box. Using the calibrated size of the box, we extract a speed $c=1.49(3)\,$mm/s. This value is slightly lower than the Bogoliubov sound speed $c_{\rm B}=\sqrt{gn_{\rm 2D}/m}=1.6(1)\,$mm/s expected at zero temperature for the measured density $n_{\rm 2D}=29(3)\,\,$\mathrm{\SIUnitSymbolMicro}\mathrm{m}$^{-2}$. The measured speed is also close to the second sound mode velocity $c^{(2)}_{\rm HD}=1.4(1)\,$mm/s, estimated from two-fluid hydrodynamics at our experimental value of $T/T_{\rm c}=0.37(12)$ Ozawa and Stringari (2014). The first sound, expected to propagate at a much higher speed $c^{(1)}_{\rm HD}=3.3(3)\,$mm/s Ozawa and Stringari (2014), does not appear in our measurements that feature a single wavefront only. The absence of first sound in our experiments can be explained by its very small coupling to density excitations in a weakly interacting gas Ozawa and Stringari (2014).
In order to probe the role of the cloud degeneracy on the sound wave propagation, we vary both $n_{\rm 2D}$ and $T$. For each configuration, we excite the cloud with the protocol described above, while adjusting the intensity of the depleting laser beam to keep the density dip around 1/3 of non-perturbed density. At lower degeneracies, sound waves are strongly damped and the aforementioned measurements of the density dip position become inadequate. We thus focus on the time evolution of the lowest-energy mode 444A related experimental study of the evolution of the fundamental mode of a 3D uniform weakly interacting Bose gas can be found in Ref. Navon16.. We decompose the density profiles integrated along $x$ as
$$\displaystyle n(y,t)=\bar{n}+\sum_{j=1}^{\infty}A_{j}(t)\>\cos(j\pi y/L_{y}),$$
(1)
where $\bar{n}$ is the average density along $y$ and the $A_{j}$ are the amplitudes of the modes. The choice of the cosine basis ensures the cancellation of the velocity field on the edges of the box. Our excitation protocol mainly couples to the lowest energy modes. We keep the excitation to a low value to be in the linear regime while still observing a clear signal for the lowest-energy mode, which in return provides a too weak signal for a quantitative analysis of higher modes 555The study of the second spatial mode gives oscillation frequencies that are in good approximation twice larger than the lowest-energy mode and thus results in very similar speeds of sounds. However, the damping rate of this mode is also larger (see Fig. 4) and we cannot robustly estimate its lifetime for our deliberately weak excitation protocol.. For each duration of the evolution, we compute the overlap of the atomic density profile with the lowest-energy mode. Examples of the time evolution of the normalized amplitude $\tilde{A}_{1}(t)=A_{1}(t)/A_{1}(0)$ for different degrees of degeneracy are shown in Fig. 2. We observe damped oscillations with a damping rate increasing with $T/T_{\rm c}$. We fit the experimental data by an exponentially damped sinusoidal curve $e^{-\Gamma t/2}[\Gamma/2\omega\sin(\omega t)+\cos(\omega t)]$ to determine the energy damping rate $\Gamma$ and the frequency $\omega$ 666The choice of this oscillating function ensures a null derivative of the amplitude of the mode at $t=0$, when the potential creating the density dip is removed. This behavior is expected from the continuity of the wavefunction and of its derivative describing the state of the gas at $t=0$.. We then determine the speed of sound $c=L_{y}\omega/\pi$ and the quality factor of this mode $Q=2\omega/\Gamma.$
We consolidate all our measurements of speed of sound and quality factors in Fig. 3. To facilitate comparison with theory, we show in Fig. 3a the values of $c$ normalized to $c_{\rm B}$. The non-normalized results are reported in Ref. REF for completeness. In the temperature range $T\lesssim 0.9\,T_{\rm c}$, we measure weakly damped density oscillations, corresponding to a well-defined sound mode ($Q\gtrsim 10$). In this regime, we observe a significant decrease by about $\approx 25\%$ of the sound velocity for increasing values of $T/T_{\rm c}$ . The measured velocities agree well with the prediction from two-fluid hydrodynamics Ozawa and Stringari (2014) combined with the equation of state of the 2D Bose gas Prokof’ev and Svistunov (2002). According to the analysis of Ozawa and Stringari (2014) for weakly interacting gases, this variation is mainly due to the variation of the superfluid fraction $f_{\rm s}$ from $\approx 1$ at $T=0$ to $\approx 0.5$ close to $T=T_{\rm c}$ with the approximate scaling $c^{(2)}_{\rm HD}\propto f_{\rm s}^{1/2}$ Ota and Stringari (2018).
The measured quality factors (see Fig. 3b) compare rather well with the predictions of Ref. Chung and Bhattacherjee (2009), which calculates the decay of Bogoliubov quasi-particles via the Landau damping mechanism for a 2D uniform system 777Note that Beliaev damping, another mechanism for the decay of low-lying excitations, is absent for the first spatial mode of the box. Indeed, it corresponds to a decay of a low-lying excitation into two excitations with lower energies and thus does not exist for the lowest energy mode.. Landau damping describes the decay of low-lying collective excitations via scattering on thermal excitations Pitaevskii and Stringari (1997); Meppelink et al. (2009b). It predicts an increase of the quality factor when decreasing temperature due to the reduction of the number of thermal excitations available for scattering with the sound mode. While the agreement between our measurement and the Landau damping theory is fairly good for $T>0.5\,T_{c}$, we measure significantly larger quality factors for lower temperatures. This could be attributed to the collisionless nature of Landau prediction which does not apply well to our experimental situation.
For temperatures above $0.9\,T_{c}$, we still observe sound waves, albeit with higher damping. In this regime, the measured sound speeds no longer match the two-fluid hydrodynamic model. The latter predicts a finite sound speed $c^{(2)}_{\rm HD}\simeq 0.6\,c_{\rm B}$ for temperatures slightly below the critical temperature, a value significantly below our measured values $c\simeq 0.7-0.8\,c_{\rm B}$. More strikingly, it predicts a disappearance of second sound for $T>T_{c}$, while we still observe sound waves with a velocity comparable to $c^{(2)}_{\rm HD}$. The discrepancy between the hydrodynamic prediction and our measurements can be explained by the strong damping of the measured density oscillations. Indeed, for quality factors of order 1, dissipation effects are no longer perturbative, making dissipationless hydrodynamic models less relevant. In the normal phase, we expect hydrodynamics to describe well the propagation of a sound wave when the collision rate $\Gamma_{\mathrm{coll}}$ between particles largely exceeds the oscillation frequency $\omega$. Assuming quasi-2D kinematics, we estimate $\Gamma_{\mathrm{coll}}/\omega$ to be in the range $1.6-3.4$ 888For a thermal gas evolving in a quasi-2D geometry, the collision rate reads $\Gamma_{\mathrm{coll}}=\hbar\tilde{g}^{2}n/(2m)$ Petrov01. We did not include the bosonic enhancement factor 2 as it should not be relevant for our temperature range, where we expect reduced density fluctuations even in the normal phase Prokof’ev and Svistunov (2002).. We conclude that, in the normal phase, the gas dynamics is not expected to follow the hydrodynamics prediction, which could explain our observations 999The existence of a sound mode in an interacting but collisionless cloud is still expected (S. Stringari, private communication). The role of interactions remains indeed important in the normal phase, even if hydrodynamics does not apply, because the healing length of the cloud is much smaller than the size of the box potential..
In the highly degenerate regime, the low damping rate allows us to observe standing waves. To study them, we modulate sinusoidally the amplitude of the potential creating the dip of density on one edge of the box 101010A similar protocol has been used in Wang15 to excite a degenerate Bose gas in a ring geometry.. After $\approx$1 s we extract, for each frequency $\nu$ of the excitation, the amplitude of the (time-dependent) density modulation induced on the cloud (see Ref. REF for details). We show in Fig. 4 the contribution of the three lowest-energy modes to the amplitude of the modulation as a function of the excitation frequency. For each mode $j$ we observe a clear resonance peak centered at a frequency $\nu_{j}$. We display in the insets the resonance frequencies and width of the modes. The $\nu_{j}$’s are equally spaced, as confirmed by the linear fit. In addition, the right inset shows the widths of the peaks. They also increase approximately linearly with $j$ 111111Because of the finite duration of the excitation (1 s), the width of the peaks is Fourier limited at a typical width of 1 Hz, which should be taken into account for a more quantitative analysis., meaning that the quality factor associated to these peaks is almost the same, as expected for Landau damping.
In conclusion, we have reported the first measurement of second sound velocity and the associated damping in a uniform 2D quantum fluid, and we have characterized their variation with temperature. Surprisingly, this sound mode extends to above the critical temperature and may corresponds to a collisionless mode. This work focuses on a weakly interacting Bose gas which features a large compressibility compared to liquid helium or strongly interacting Fermi gases. A natural extension of this work would thus be to investigate second sound propagation for increasing interactions Ota and Stringari (2018). It would also be interesting to investigate first sound, e.g. by applying a localized temperature excitation Sidorenkov et al. (2013). During the completion of this work we were informed that a related study with a homogeneous 3D Fermi gas was currently performed at MIT 121212M. Zwierlein, talk at the BEC 2017 Frontiers in Quantum Gases, Sant Feliu de Guixols.
Acknowledgements.
${}^{\dagger}$Present address: Fakultät für Physik, Ludwig-Maximilians-Universität München, Schellingstr. 4, 80799 Munich, Germany. This work is supported by DIM NanoK and ERC (Synergy UQUAM). This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement N${}^{\circ}$ 703926. We thank S. Stringari, L. Pitaevskii, M. Ota, N. Proukakis, F. Dalfovo, F. Larcher and P.C.M. Castilho for fruitful discussions, M. Villiers for experimental assistance and F. Gerbier, R. Lopes and M. Zwierlein for their reading of the manuscript.
J.L.V. and R.S.J. contributed equally to this work.
References
Donnelly (2009)
R.J. Donnelly, “The two-fluid
theory and second sound in liquid helium,” Phys. Today 62, 34–39 (2009).
Taylor et al. (2009)
E. Taylor, H. Hu, X.-J. Liu, L. P. Pitaevskii, A. Griffin, and S. Stringari, “First and second sound in a strongly interacting Fermi
gas,” Phys. Rev. A 80, 053601 (2009).
Joseph et al. (2007)
J. Joseph, B. Clancy,
L. Luo, J. Kinast, A. Turlapov, and J. E. Thomas, “Measurement of sound velocity in a Fermi gas near a Feshbach
resonance,” Phys. Rev. Lett. 98, 170401 (2007).
Sidorenkov et al. (2013)
L.A. Sidorenkov, M.K. Tey,
R. Grimm, Y.-H. Hou, L. Pitaevskii, and S. Stringari, “Second sound and the superfluid fraction in a Fermi gas
with resonant interactions,” Nature 498, 78–81 (2013).
Griffin and Zaremba (1997)
A. Griffin and E. Zaremba, “First and
second sound in a uniform Bose gas,” Phys.
Rev. A 56, 4839–4844
(1997).
Ota and Stringari (2018)
M. Ota and S. Stringari, “Second sound
in a two-dimensional Bose gas: From the weakly to the strongly interacting
regime,” Phys. Rev. A 97, 033604 (2018).
Andrews et al. (1997)
M. R. Andrews, D. M. Kurn,
H.-J. Miesner, D. S. Durfee, C. G. Townsend, S. Inouye, and W. Ketterle, “Propagation of sound in a Bose-Einstein condensate,” Phys. Rev. Lett. 79, 553–556 (1997).
Meppelink et al. (2009a)
R. Meppelink, S. B. Koller, and P. van der
Straten, “Sound
propagation in a Bose-Einstein condensate at finite temperatures,” Phys. Rev. A 80, 043605 (2009a).
Ozawa and Stringari (2014)
T. Ozawa and S. Stringari, “Discontinuities in the first and second sound velocities at the
Berezinskii-Kosterlitz-Thouless transition,” Phys. Rev. Lett. 112, 025302 (2014).
Hadzibabic and Dalibard (2011)
Z. Hadzibabic and J. Dalibard, “Two-dimensional Bose fluids: An atomic physics perspective,” Rivista del Nuovo Cimento 34, 389 (2011).
Pitaevskii and Stringari (1997)
L.P. Pitaevskii and S. Stringari, “Landau
damping in dilute Bose gases,” Phys. Lett. A 235, 398–402 (1997).
Ville et al. (2017)
J.L. Ville, T. Bienaimé,
R. Saint-Jalm, L. Corman, M. Aidelsburger, L. Chomaz, K. Kleinlein, D. Perconte, S. Nascimbène, J . Dalibard, and J. Beugnon, ‘‘Loading and compression of a single two-dimensional Bose gas in an
optical accordion,” Phys. Rev. A. 95, 013632 (2017).
Aidelsburger et al. (2017)
M. Aidelsburger, J. L. Ville, R. Saint-Jalm,
S. Nascimbène, J. Dalibard, and J. Beugnon, “Relaxation dynamics in the merging of $N$
independent condensates,” Phys. Rev. Lett. 119, 190403 (2017).
(14)
The value of $\mathaccentV{tilde}07Eg$ is slightly
modified by the effect of interactions. We estimate that $\mathaccentV{tilde}07Eg$ varies by about 10% for the range of surface
densities explored in this work.
(15)
This removal is realized by a partial transfer of the atoms
to the $|F=2,m=0\rangle$ state with a microwave resonant field and a
subsequent blasting of the transferred fraction with a resonant laser
beam.
Prokof’ev et al. (2001)
N. Prokof’ev, O. Ruebenacker, and B. Svistunov, “Critical
point of a weakly interacting two-dimensional Bose gas,” Phys. Rev. Lett. 87, 270402 (2001).
Hueck et al. (2018)
K. Hueck, N. Luick,
L. Sobirey, J. Siegl, T. Lompe, and H. Moritz, “Two-dimensional homogeneous fermi gases,” Phys. Rev. Lett. 120, 060402 (2018).
(18)
“See Supplemental Materials,” .
(19)
We did not observe any excitation along the $x$
direction.
(20)
A related experimental study of the evolution of the
fundamental mode of a 3D uniform weakly interacting Bose gas can be found in
Ref. Navon16.
(21)
The study of the second spatial mode gives oscillation
frequencies that are in good approximation twice larger than the
lowest-energy mode and thus results in very similar speeds of sounds.
However, the damping rate of this mode is also larger (see Fig.\tmspace+.1667em4) and we cannot robustly estimate
its lifetime for our deliberately weak excitation protocol.
(22)
The choice of this oscillating function ensures a null
derivative of the amplitude of the mode at $t=0$, when the potential creating
the density dip is removed. This behavior is expected from the continuity of
the wavefunction and of its derivative describing the state of the gas at
$t=0$.
Chung and Bhattacherjee (2009)
M.-C. Chung and A.B. Bhattacherjee, “Damping
in 2D and 3D dilute Bose gases,” New J. Phys. 11, 123012 (2009).
Prokof’ev and Svistunov (2002)
N. Prokof’ev and B. Svistunov, “Two-dimensional weakly interacting Bose gas in the fluctuation region,” Phys. Rev. A 66, 043608 (2002).
(25)
Note that Beliaev damping, another mechanism for the decay
of low-lying excitations, is absent for the first spatial mode of the box.
Indeed, it corresponds to a decay of a low-lying excitation into two
excitations with lower energies and thus does not exist for the lowest energy
mode.
Meppelink et al. (2009b)
R. Meppelink, S. B. Koller, J. M. Vogels,
H. T. C. Stoof, and P. van der Straten, “Damping of superfluid flow
by a thermal cloud,” Phys. Rev. Lett. 103, 265301 (2009b).
(27)
For a thermal gas evolving in a quasi-2D geometry, the
collision rate reads $\Gamma_{\mathrm{coll}}=\hbar\mathaccentV{tilde}07Eg^{2}n/(2m)$ Petrov01. We did not include the
bosonic enhancement factor 2 as it should not be relevant for our temperature
range, where we expect reduced density fluctuations even in the normal phase
Prokof’ev and Svistunov (2002).
(28)
The existence of a sound mode in an interacting but
collisionless cloud is still expected (S. Stringari, private communication).
The role of interactions remains indeed important in the normal phase, even
if hydrodynamics does not apply, because the healing length of the cloud is
much smaller than the size of the box potential.
(29)
A similar protocol has been used in Wang15 to excite
a degenerate Bose gas in a ring geometry.
(30)
Because of the finite duration of the excitation (1\tmspace+.1667ems), the width of the peaks is Fourier limited
at a typical width of 1\tmspace+.1667emHz, which
should be taken into account for a more quantitative analysis.
(31)
M. Zwierlein, talk at the BEC 2017 Frontiers in Quantum
Gases, Sant Feliu de Guixols. |
An Historical View: The Discovery of Voids in the Galaxy Distribution
Laird A. Thompson
Astronomy Department, University of Illinois Urbana-Champaign, 1002 W. Green
Street, Urbana, IL 61801
Stephen A. Gregory
Physics and Astronomy Department, University of New Mexico, 800 Yale Blvd. NE, Albuquerque, NM 87131, and Boeing LTS Inc., Kirkland AFB, NM 87185
thompson@astro.illinois.edu, sgregory58@gmail.com
Abstract
Voids in the large scale distribution of galaxies were first
recognized and discussed as an astrophysical phenomenon in two
papers published in 1978. We published the first (Gregory and
Thompson 1978) and Joeveer, Einasto and Tago (1978) published the second.
The discovery of voids altered the accepted view of the large scale
structure of the universe. In the old picture, the universe
was filled with field galaxies, and occasional density enhancements
could be found at the locations of rich galaxy clusters or
superclusters. In the new picture, voids are interspersed between
complex filamentary supercluster structure that forms the so-called
cosmic web. The key observational
prerequisite for the discovery of voids was a wide-angle redshift survey
displayed as a cone diagram that extended far enough in distance to
show a fair sample of the universe (i.e. well beyond the Local
Supercluster). The initial impact of the 1978 discovery of voids was
stunted for several years by theoretical cosmologists in the West who
were not quite sure how filaments and voids could emerge from what
otherwise appeared to be a homogeneous universe. After it became clear
several years later that theoretical models of structure formation could
explain the phenomenon, the new void and supercluster paradigm became
widely accepted.
Subject headings:history and philosophy of astronomy — large scale structure
of universe
1. Introduction
In the mid-1970’s two independent research programs revealed the
beautiful void and supercluster structure of the universe, what is
now referred to as the cosmic web. We initiated the first of these
research programs (Gregory and Thompson 1978). The second was an
entirely separate effort by observational astronomers in Estonia and
theoreticians in Russia (Joeveer, Einasto and Tago 1978). These two
programs could not have been more different, but they both led to
the conclusion that there are large volumes in the local universe
with diameters $\sim$20 h${}^{-}$${}^{1}$ Mpc that contain no galaxies
whatsoever111h = Hubble constant/100 km s${}^{-}$${}^{1}$
Mpc${}^{-}$${}^{1}$. We called these empty regions “voids” while Joeveer
et al. at first simply called them “big holes”.
Our work was empirically driven. The existence of supercluster
structure was widely known in this era, and Abell (1961) had called
attention to “second order clusters”, i.e. congregations of rich
cluster cores. Our aim was to understand the galaxy distribution in
and around these collections of Abell clusters. For example, we
asked whether adjacent Abell cluster cores are isolated density
enhancements or whether they are interconnected by “bridges” of
galaxies. Because the nearest rich Abell clusters are distributed
within a volume that extends to distances of $\sim$100 h${}^{-}$${}^{1}$
Mpc, we designed our redshift surveys to sample the appropriate
volume. Our first target was the Coma and A1367 cluster pair, and it
was in the large foreground volume between the Milky Way and the
Coma/A1367 supercluster that we identified the first voids. Our work
was not related to nor driven by any theoretical model of structure
formation.
The Estonian work was aimed at testing a specific model of structure
formation championed in the 1970’s by the prominent Russian
cosmologist Yakov Zeldovich. Zeldovich (1970) postulated the
so-called top-down model of galaxy formation. In his model, large
supercluster-sized structures are the first to form, and individual
galaxies fragment out of the larger collapsing structures. These
models have not stood the test of time, but in the mid-1970’s
Joeveer, Einasto and Tago (1978) collaborated with Zeldovich and
mapped the galaxy and cluster distribution (using catalogues
containing known galaxy redshifts) over volumes similar to those we
were studying. Further historical details about this research program
can be found in Einasto (2009).
In the 1970’s the hierarchical clustering model (so-called bottom-up
model) was most popular in the West, but in its original form it could not
explain, ab initio, the existence of voids as large as those
reported in Gregory and Thompson (1978). At first, those who held
fast to the early hierarchical model acknowledged that the observed
“holes” exist in the galaxy distribution, but they postulated that
they appeared naturally as a result of random statistical processes.
When we presented our evidence for voids as discrete astrophysical
entities with borders defined by filamentary structures, we
encountered skepticism that is documented most clearly in Soneira
and Peebles (1978) where they said the following: ”We know that the
eye does tend to judge in a biased way–for example, one readily
picks out “chains” of points in a uniform random distribution”. In
other words, the filamentary structures that we identified as the
walls of voids were, according to their interpretation, nothing more
than false visual constructs. Not everyone held this extreme view,
of course, but the immediate impact of the early redshift survey
work was diminished despite the fact that the voids that we
identified in the Coma/A1367 foreground (and their filamentary
walls) have stood the test of time and are just as dramatic today as
they were when we first saw them in the mid-1970’s.
Fortunately, in this same era advances were being made in computer
simulations aimed at modeling the growth and evolution of structure
in the large scale distribution of galaxies. These models generally
used as a starting point a simple galaxy distribution but
eventually included dark matter (either hot or cold).
Even these early evolutionary models showed how dramatic
filamentary structure can develop in the galaxy distribution over
time, and by doing so, they removed the theoretical prejudice
against the existence of voids. At first the numerical simulations
were relatively simple (Aarseth et al. 1979 and Doroskevich et al.
1980), but the level of sophistication rapidly increased so that by
1983 it was possible for the first time to begin realistically
testing key features of the large scale galaxy distribution (Melott
1983, Frenk et al. 1983, to mention just two).
This brief summary provides a context for understanding how the
discoveries that emerged from the early galaxy redshift surveys
influenced theoretical models of structure formation. Our aim in
what follows is to document the redshift surveys published in the
first decade after this research began.
2. Early Redshift Surveys
The redshift survey revolution of the 1970’s began when high-voltage
image intensifier devices came into wide use and displaced the
photographic plate as the primary detector in astronomical
spectroscopy. To measure a redshift accurate to $\pm$100 km/s, a
photographic exposure of $\sim$2.5 hours had previously been
required for an m$\sim$15 galaxy. An image intensifier system could
produce a similar result in 10 to 15 minutes. Significant numbers of
new redshifts began to be published soon after the Kitt Peak
National Observatory 2.1-m telescope and the Steward Observatory
2.3-m telescope were both equipped with image tube spectrographs. It
is at this point that the Redshift Survey Timeline begins (Table 1).
Key participants in the early work included the late Herb Rood as
well as Guido Chincarini, William Tifft, Stephen Gregory, Laird
Thompson, and Massimo Tarenghi. Gregory and Thompson had been
graduate students of Tifft at the University of Arizona. Tarenghi
was Tifft’s postdoctoral researcher. Herb Rood (then at Michigan State
University) and Guido Chincarini (then at the University of Oklahoma) formed a
separate team.
The earliest redshift surveys have been called pencil beams because
target galaxies were selected from cluster cores that span a small
solid angle. Table 1 contains a complete list of the galaxy
redshift surveys (including the pencil beam surveys) published in the
period 1971-1981.
Hints of the large scale structure first began to
appear as irregularities in the redshift distribution in the pencil
beam foregrounds. But to see such structure with clarity, a
redshift survey had to probe to m$\sim$15. Surveys to m$\sim$14
showed no significant structure because their distance range was
inadequate, and they could not sample foreground structure in any
detail. With perfect hindsight, any observer who pushed to m$\sim$15
in a pencil beam survey might have said that they saw initial hints
of the structure, but no one immediately grasped its meaning.
A complete description of the large scale structure required
formulating a
new three-dimensional picture of the galaxy distribution, one that
includes voids interspersed between supercluster structure. The
concept of a galaxy supercluster was well developed by the 1970’s
(c.f., Oort 1983), and yet those who accepted the existence of
superclusters assumed that they were simple density
enhancements embedded in a uniform field of background galaxies.
Others suggested that superclusters had a core-halo structure with a
halo that slowly merged into the uniformly distributed field
population of galaxies. So it was the discovery of voids – and not
superclusters – that provided the basis for the paradigm change.
The primary catalyst for this discovery was the cone diagram: a
polar plot with redshift used for the radial coordinate and a
galaxy’s angular position on the sky used as the azimuth (the third
dimension being projected into the plane of the plot). Of course, all
modern redshift surveys like CfA, SDSS, 2dF, etc. rely on this diagram to
display their data. Those who used the cone diagram for the first
time (Tifft and Gregory 1976; Gregory and Thompson 1978; Joeveer,
Einasto and Tago 1978) were the first to visualize the void and
supercluster structure. Table I lists how all authors plotted
their redshift data.
In Chinicarini and Rood’s first redshift survey paper, they completed
a pencil beam survey for the Perseus cluster and then
determined its virial mass. Their second and
third papers list data from the KPNO 2.1-m telescope, and then for
three years they published no redshifts. In the mean time Tifft
began his work at the Steward Observatory 2.3-m telescope. When
Chincarini and Rood resumed their work with a study of the Coma
cluster, Tifft and graduate student Gregory were doing same. Tifft
and Gregory worked in the central regions of Coma (to r = 6${}^{o}$)
while Chincarini and Rood aimed to trace the Coma cluster to a
radial distance of 14${}^{o}$ by surveying primarily to the west of the
cluster core. Both groups aimed to collect complete samples to m =
15, and they began the transition away from the narrow pencil beam surveys.
Fig. 1 shows the Tifft and Gregory (1976) Coma cone diagram. This
is the first redshift survey shown as a cone diagram. In their
discussion Tifft and Gregory (1976) note the lack of field galaxies,
and in the caption to their cone diagram, they say that the
foreground is “devoid” of galaxies. The Tifft and Gregory survey
of Coma covered too narrow an angular span to include a complete
void (from one wall through the empty region and on to the other
wall). An average void spans 20 h${}^{-}$${}^{1}$ Mpc and at the most distant extent
they surveyed 12.6 h${}^{-}$${}^{1}$ Mpc.
Chincarini and Rood published papers in 1975 and 1976 (just before
and just after Tifft and Gregory 1976), but they always plotted
their redshifts as a function of radial distance from the core of
Coma, and their plots were square. By making a radial analysis,
they essentially destroyed 3D information. In fact, Chincarini
and Rood never discussed 3D structure in any of the papers they
published prior to 1978. They talked about “redshift segregation”
whenever their survey intercepted first a void and then a
supercluster filament in the foreground, but they never gave a
physical interpretation as to what it meant, i.e. they never
mentioned concepts like “holes” or “voids”.
Rood and Chincarini discussed their views of the large
scale galaxy distribution most clearly at the end of their last
observational paper on the Coma cluster (Chincarini and Rood 1976a).
They used the following words: “The large sizes of clusters and
their fading into low-density supercluster backgrounds leave little
if any space between them. On the other hand, Figure 3 clearly
shows also a pronounced effect of segregation of redshifts.”
In May, 1975, we submitted
an observing proposal to Kitt Peak National
Observatory to use the 2.1-m telescope for a new redshift survey. In
our proposal, we posed the hypothesis that the two rich Abell
clusters Coma and A1367 are enveloped in a common supercluster.
Since they are separated by 21${}^{o}$ in the plane of the sky, our aim
was to survey a slice in the intercluster region between Coma and
A1367 with dimensions 4${}^{o}$ wide and 19${}^{o}$ long to a depth of m =
15.0. We predicted that
we would detect an over-density of galaxies (a bridge at their
common redshift) in the region between the two clusters. This proposal
was accepted, and we collected the new redshifts in April, 1976.
The redshifts in our Coma/A1367 survey were measured and plotted
in a cone diagram by the early summer, 1976 (reproduced here as Fig.
2). This plot displays the entire galaxy sample in the region that
surrounds and includes Coma and A1367 (11${}^{o}$ x 23${}^{o}$) and not the
smaller area (4${}^{o}$ x 19${}^{o}$) mentioned in our observing proposal.
Fig. 2 is the first wide-angle cone diagram that displays a complete
magnitude-limited sample to m = 15.0. It is not a pencil beam
survey: at the deepest extent it spans 36 h${}^{-}$${}^{1}$ Mpc across the sky.
Upon seeing this plot we immediately realized the
significance of the irregular distribution of galaxies that had
appeared in the foregrounds of the early pencil beam surveys.
Our Coma/A1367 paper (Gregory and Thompson 1978) arrived at the
Astrophysical Journal on September 7, 1977. It discusses for the
first time the large scale structure using the new paradigm. We
list here the key points that are unique to this paper.
•
We recognized huge empty regions in the 3D plot of our survey
volume and for the first time used the word “void” to describe them.
•
We outlined possible hypotheses that might
explain the void phenomenon by using the
following words: “It is an important challenge for any
cosmological model to explain the origin of these vast, apparently
empty regions of space. There are two possibilities: (1) the
regions are truly empty, or (2) the mass in these regions is in some
form other than bright galaxies. In the first case, severe
constraints will be placed on theories of galaxy formation because
it requires a careful (and perhaps impossible) choice of both
$\Omega$ (the present mass density/closure density) and the spectrum
of initial irregularities in order to grow such large density
irregularities…” It seemed impossible to us at the time
because cold dark matter had not yet been proposed.
•
The abstract to our paper states: “there are large
regions of space with radii r $>$ 20h${}^{-}$${}^{1}$ Mpc where there appear
to be no galaxies whatever.”
•
Before the observations were made, we had hypothesized
the existence of a bridge of galaxies between Coma
and A1367. Our 3D cone diagram confirmed it. Today this bridge of galaxies
is a small segment of what is often called “The Great Wall”.
•
The general structure shown in our cone diagram includes
the body, the right leg and right arm of what some call the “Coma stickman”.
The sharp contrast between the description of the large
scale distribution of galaxies as given by Gregory and Thompson
(1978) and that of Chincarini and Rood (1976a) explains why a
paradigm change occurred.
Table 1 includes only papers that appeared in refereed journals and
omits papers that were not refereed: observatory publications,
presentations made at meetings like the American Astronomical
Society, and at conferences like the Tallin conference (IAU Symposium
No. 79) held in Tallin, Estonian SSR, September 12-16, 1977222Neither
Gregory nor Thompson attended the Tallin conference. Gregory submitted
a request to attend, but his request was denied..
The papers presented at this conference were published in
“The Large Scale Structure of the Universe” (Longair and Einasto
1978). This conference was an important event in the early
study of the large scale distribution of galaxies. Milikel Joeveer
and Jaan Einasto discussed preliminary
results that they later published in a refereed journal as Joever,
Einasto, and Tago (1978). Our Coma/A1367 supercluster manuscript arrived
at the Astrophysical Journal five days before the conference began, and we
had no prior knowledge of any presentations that would be made at IAU Symposium
No. 79 except for one by Tifft and Gregory that
briefly mentions Gregory and Thompson (1978).
The Joeveer, Einasto and Tago (1978) paper is an interesting study
of the 3D distribution of both galaxies and clusters of galaxies in
the south galactic hemisphere. They present several cone diagrams
that display the large scale structure in a region of sky
70${}^{o}$x70${}^{o}$. Their cone diagrams show large empty regions, regions
that they call “big holes”. Because their galaxy redshift data
were taken from previously published sources (the major source being
the Second Reference Catalogue of Bright Galaxies by de Vaucouleurs
et al. 1976), Joeveer et al. (1978) do not claim to present a
complete magnitude-limited survey.
Our surveys always aimed to be magnitude-limited because critics in
the 1970’s were quick to raise the possibility that empty
regions appeared empty due to incomplete sampling.
The Joeveer et al. (1978) paper was published
in the November issue of Monthly Notices of the Royal Astronomical
Society, five months after our paper on
Coma/A1367. Their paper contains four cone
diagrams (four different cuts through the same survey area) as well as
a reference to Gregory and Thompson (1978). Figure
3 below displays one of these four cuts.
We continued our program to study the galaxy distribution in and around
the nearest Abell clusters. In Gregory, Thompson and Tifft (1981), we investigated the properties of the long filamentary chain of rich clusters in Perseus, and in Gregory and Thompson (1984) we studied another double cluster, A2197 and A2199. A larger collection of the early redshift survey workers analyzed the properties of the Hercules supercluster region (Tarenghi et al. 1979), and in Chincarini, Thompson and Rood (1981) an extended filament or bridge of galaxies was found to stretch
44h${}^{-}$${}^{1}$ Mpc from the A2197/A2199 supercluster to the Hercules
supercluster. This publication
is significant because
it demonstrates the filamentary nature of the large scale structure
over an extensive length scale. In this case, the filament extends
primarily in depth rather than in an angular span across the sky. We note that none of these studies from the early 1980’s are so-called pencil beam surveys. The survey areas were large enough to include multiple Abell cluster cores and revealed other voids and supercluster structure.
It is a common misconception that the early phases of the Center for
Astrophysics (CfA) redshift survey contributed to the paradigm
change. In Table 1 it is easy to see how the CfA work progressed
relative to the work of the Arizona groups. The first CfA study by
Davis, Geller and Huchra (1978) had a limiting magnitude of m =
13.0. It was shallow and showed no evidence of any structure.
Their stated aim in undertaking this first survey was to measure
the mean mass density of the universe. The second CfA milestone
paper was by Davis, Huchra, Latham and Tonry (1982). With a survey
limit at m = 14.5, it shows hints of the large scale structure. But
by the time this paper was published in 1982, the nature of voids
and supercluster structure was widely known and widely discussed.
For example, Zeldovich, Einasto and Shandarin (1982) published a review
article for Nature entitled “Giant Voids in the Universe”.
In addition to technical references cited in Table 1, popular
accounts were also being published (Chincarini and Rood 1980;
Gregory and Thompson 1982). The later article for Scientific
American is entitled “Superclusters and Voids in the
Distribution of Galaxies”.
3. The Second Wave Redshift Survey Work
Other research groups began to substantiate the new void and
supercluster structural features in the galaxy distribution. The
field grew so rapidly in the mid-1980s – a time when CCD cameras
began to replace image intensifiers – that it is virtually
impossible to make the last half of Table 1 complete. One important
group that contributed to the second-wave included Robert Kirshner,
Augustus Oemler,Jr., Paul Schechter, and Stephen Shectman. During
the late 1970’s they measured galaxy redshifts in three small survey
fields, each separated by 35${}^{o}$ on the sky (in the direction of the
constellations Bootes and Corona Borealis) with the aim of
determining the galaxy luminosity function. After learning about
voids from the early redshift survey papers, they noticed a
deficiency of galaxies in the redshift interval between 12,000 km/s
and 18,000 km/s in all three of their narrow survey fields. On this
basis they speculated that the entire region between their three
small survey fields could be “A million cubic megaparsec void”
(Kirshner 1981). In the end (Kirshner et al. 1987), the volume of
the Bootes void was remeasured to be 1/3 the value quoted in the
title of their 1981 paper. Its radius is $\sim$34h${}^{-}$${}^{1}$ Mpc.
Martha Haynes, Ricardo Giovanelli and their friend Guido Chincarini began
redshift surveys in 1982 based on observations of the 21 cm emission line
of neutral hydrogen (Chincarini et al. 1983). Giovanelli and Haynes continued
in subsequent years to make many seminal contributions to our knowledge of
the neutral hydrogen content of galaxies and to the structure and nature of
superclusters. One of the better examples of their redshift survey work is
paper V in their series of eight papers on the Pisces-Perseus supercluster
(Wegner, Haynes and Giovanelli 1993).
The CfA galaxy redshift survey reached
full stride by the mid-1980’s when this group began to push deeper than m = 15.
The first dramatic CfA results came with the publication of de Lapparent,
Geller and Huchra (1986), a wide angle survey to m = 15.5.
Figure 4a is a plot of the data used by de Lapparent et al. 1986. For comparison sake,
Figure 4b is the same cone diagram with an overlay added showing the
Gregory and Thompson (1978) survey area.
4. Summary Statement on the Early Redshift Survey Work
IAU Symposium No. 124 entitled “Observational Cosmology” was held
in Beijing, August 25-30, 1986. Allan Sandage gave the invited
opening presentation. In his presentation,
Sandage (1987) says the following about the early redshift survey work:
Gregory and Thompson (1978) “marks the discovery of voids, which
have become central to the subject [of the large scale structure]. Prior
work by Einasto et al. (1980 with earlier references), Tifft and Gregory (1976),
and Chincarini and Rood (1976) foreshadowed the development, but the
Gregory and Thompson discovery is generally recognized as the most
convincing early demonstration. Rapid developments in the mapping of
various filaments and voids include the studies of Tarenghi et al. (1979),
Gregory et al. (1981), Kirshner et al. (1981), Gregory and Thompson (1984),
Chincarini et al. (1983), and Huchra et al. (1983). A general review is given
by Oort (1983).”
IAU Symposium No. 124 was held immediately after the the March 1,
1986, publication of the paper by de Lapparent, Geller, and Huchra
(1986) showing the “Coma stickman” in the center of the CfA cone
diagram (Fig. 4).
Acknowledgments
This research has made use of NASA’s Astrophysics Data System
Bibliographic Services.
References
Abell (1961)
Abell, G.O. 1961, AJ, 66, 607
Aarseth et al. (1979)
Aarseth, S.J., Turner, E.L. &
Gott, J.R. III 1979, ApJ, 228, 664
Chincarini et al. (1993)
Chincarini, G.L, Giovanelli, R.,
& Haynes, M.P. 1983, ApJ, 269, 13
Chincarini & Martins
(1975)
Chincarini, G. & Martins, D. 1975, ApJ, 196, 335
Chincarini & Rood (1971)
Chincarini, G. & Rood, H.J.
1971, ApJ, 168, 321
Chincarini & Rood
(1972a)
Chincarini, G. & Rood, H.J. 1972a, AJ, 77, 4
Chincarini & Rood (1972b)
Chincarini, G. & Rood, H.J.
1972b, AJ, 77, 448
Chincarini & Rood (1975)
Chincarini,
G. & Rood, H.J. 1975, Nature, 257, 294
Chincarini & Rood
(1976a)
Chincarini, G. & Rood, H.J. 1976a, ApJ, 206, 30
Chincarini & Rood (1976b)
Chincarini, G. & Rood, H.J
1976b, PASP, 88, 388
Chincarini & Rood
(1980)
Chincarini, G. & Rood, H.J. 1980, Sky & Tel., 59, 364
Chincarini et al. (1981)
Chincarinit, G., Thompson, L.A. &
Rood, H.J.1981, ApJ, 249, L47
Davis et al. (1978)
Davis,
M., Geller, M.J. & Huchra, J. 1978, ApJ, 221, 1
Davis et
al. (1982)
Davis, M., Huchra, J., Latham, D.W. & Tonry, J. 1982,
ApJ, 253, 423
de Lapparent et al. (1986)
de Lapparent, Geller, M.J. &
Huchra, J.P. 1986, ApJ, 302, L1-L5
de Vaucouleurs et al.
(1976)
de Vaucouleurs, G., de Vaucouleurs, A. & Crowin, H.G., Jr.
1976, Second Reference Catalogue of Bright Galaxies (Austin: Univ.
Texas Press)
Doroshkevich et al. (1980)
Doroshkevich, A.G., Kotok, E.V.,
Poliudov, A.N., Shandarin, S.F., Sigov, I.S., & Novikov, I.D. 1980,
MNRAS, 192, 321
Einasto (2009)
Einasto, J. 2009, Dark Matter. UNESCO EOLSS Encyclopedia
(Eolss Publishers: Paris)
Einasto, J. et al. (1980)
Einasto, J., Joeveer,
M. & Saar, E. 1980, MNRAS, 193, 353
Frenk, C.S. et al. (1983)
Frenck, C.S., White, S.D.M. & Davis, M. 1983,
ApJ, 271, 417
Gregory, S.A. &
Thompson, L.A. (1978)
Gregory, S.A. & Thompson, L.A. 1978, ApJ,
222, 784
Gregory, S.A. & Thompson, L.A. (1982)
Gregory, S.A. &
Thompson, L.A. 1982, Sci. Am., 246, 106
Gregory, S.A. &
Thompson, L.A. (1984)
Gregory, S.A. & Thompson, L.A. 1984,
ApJ, 286, 422
Gregory, S.A. et al. (1981)
Gregory, S.A.,
Thompson, L.A. & Tifft, W.G. 1981, ApJ, 243, 411
Joeveer, M. et al. (1978)
Joeveer, M., Einasto, J &
Tago, E. 1978, MNRAS, 185, 357
Kirshner, R.P. et al.
(1981)
Kirshner, R.P., Oemler, A., Jr., Schechter, P.L. &
Shectman, S.A. 1981, ApJ, 248, L57
Kirshner, R.P. et al.
(1987)
Kirshner, R.P., Oemler, A., Jr., Schechter, P.L. &
Shectman, S.A. 1987, ApJ, 314, 493
Longair, M.S. &
Einasto, J. (1978)
Longair, M.S. & Einasto, J. 1978, The Large
Scale Structure of the Universe, IAU Symp. No. 79 (Dordrecht: D.
Reidel Publishing)
Melott, A.L. (1983)
Melott, A.L. 1983, MNRAS, 202, 595
Oort, J.H. (1983)
Oort, J.H. 1983,
ARA&A, 21, 373
Sandage, A. (1987)
Sandage, A. 1987,
Observational Cosmology, IAU Symp. No. 124, eds. A. Hewitt, G.
Burbidge, & L.Z. Feng (Dordrecht: D. Reidel Publishing), 1
Soneira, R.M. & Peebles, P.J.E. (1978)
Soneira, R.M. & Peebles,
P.J.E. 1978, AJ, 83, 845
Tarenghi et al. (1979)
Tarenghi, M., Tifft, W.G.,
Chincarini, G., Rood, H.J. & Thompson, L.A. 1979, ApJ, 234, 793
Tarenghi et al. (1980)
Tarenghi, M., Chincarini, G., Rood,
H.J. & Thompson, L.A. 1980, ApJ, 235, 724
Tifft, W.G. &
Gregory, S.A. (1973)
Tifft, W.G. & Gregory, S.A. 1973, ApJ, 181,
15
Tifft, W.G. & Gregory, S.A. (1976)
Tifft, W.G. &
Gregory, S.A. 1976, ApJ, 205, 696
Tifft, W.G. et al.
(1975)
Tifft, W.G., Hillsman, K.A. & Corrado, L.C. 1975, ApJ,
199, 16
Tifft, W.G. et al. (1973)
Tifft, W.G., Jewsbury,
C.P. & Sargent, T.A. 1973, ApJ, 185, 115
Tifft, W.G. &
Tarenghi, M. (1975a)
Tifft, W.G. & Tarenghi, M. 1975a, ApJ, 198,
L7 bibitem[Tifft, W.G. & Tarenghi, M. (1975b)]Tifft, W.G. &
Tarenghi, M. 1975b, ApJ, 199, 10
Tifft, W.G. & Tarenghi,
M. (1977)
Tifft, W.G. & Tarenghi, M. 1977, ApJ, 217, 944
Wegner, G. et al. (1993)
Wegner, G., Haynes, M.P. &
Giovanelli, R. 1993, AJ, 105, 1251
Zeldovich (1970)
Zeldovich, Y.B. 1970, A&A, 5, 84 |
July 2, 2022
On Current-Squared Flows and ModMax Theories
Christian Ferko,${}^{a}$
Liam Smith,${}^{b}$ and
Gabriele Tartaglino-Mazzucchelli${}^{b}$
${}^{a}$
Center for Quantum Mathematics and Physics (QMAP),
Department of Physics & Astronomy, University of California, Davis, CA 95616, USA
${}^{b}$
School of Mathematics and Physics, University of Queensland,
St Lucia, Brisbane, Queensland 4072, Australia
caferko@ucdavis.edu, liam.smith1@uq.net.au, g.tartaglino-mazzucchelli@uq.edu.au
We show that the recently introduced ModMax theory of electrodynamics and its Born-Infeld-like generalization are related by a flow equation driven by a quadratic combination of stress-energy tensors. The operator associated to this flow is a $4d$ analogue of the $T\overline{T}$ deformation in two dimensions. This result generalizes the observation that the ordinary Born-Infeld Lagrangian is related to the free Maxwell theory by a current-squared flow. As in that case, we show that no analogous relationship holds in any other dimension besides $d=4$. We also demonstrate that the $\mathcal{N}=1$ supersymmetric version of the ModMax-Born-Infeld theory obeys a related supercurrent-squared flow which is formulated directly in $\mathcal{N}=1$ superspace.
Contents
1 Introduction
2 ModMax-BI is a $T^{2}$ flow
3 Derivation from $T^{2}$ Master Flow Equation
3.1 Review of General Flow Equation
3.2 Application to ModMax-BI Theory
3.3 No ModMax-BI Solutions to $T^{2}$ Flows in $d>4$
4 Supersymmetric ModMax-BI is a supercurrent-squared flow
4.1 Review of the Bagger-Galperin Lagrangian flow and the $d=4$, ${\cal N}=1$ super-current squared operator
4.2 The Born-Infeld-ModMax case
5 Conclusion
A Proof of Determinant Condition
B General $T^{2}$ Flows for Scalar Theories
B.1 Master Flow Equation for Scalars
B.2 General Analysis of $O_{T^{2}}^{[r]}$ Flows
B.3 Other Stress Tensor Flows in $d=4$
1 Introduction
Since 2016
there has been a wide range of activities surrounding the study
of $d=2$ quantum field theories deformed by the irrelevant $T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ operator
[1, 2, 3].
Such operator is defined as the determinant of the stress-energy tensor,
$O_{T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}\propto\det{T_{\mu\nu}}$,
which in $d=2$ is equivalent to the following quadratic combination
$$\displaystyle O_{T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}\propto\Big{(}T^{\mu\nu}T_{\mu\nu}-\Theta^{2}\Big{)}~{},~{}~{}~{}~{}~{}~{}\Theta:=T^{\mu}_{\mu}~{}.$$
(1.1)
Despite being irrelevant, the local operator
$O_{T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ proves to be quantum mechanically well defined
[1, 2]
and to preserve many of the symmetries of the seed theory, including integrability [4, 2, 5],
and supersymmetry [6, 7, 8, 9].
By now the field of research surrounding $T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$-like deformations
contains a large body of literature which we will not attempt to review in detail here. Instead we refer the reader to [10] for a pedagogical
introduction to the subject.
Within this context,
the main focus of this paper concerns classical Lagrangian flows triggered by current-squared $T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$-like operators.
The $T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ deformation of a two-dimensional theory
leads to a classical flow equation for the deformed Lagrangian ${\mathcal{L}}_{\lambda}$ of the form
$$\frac{\partial}{\partial\lambda}\mathcal{L}_{\lambda}=-\frac{1}{8}O_{T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}\propto\det\big{(}T_{\mu\nu}[\mathcal{L}_{\lambda}]\big{)}~{},$$
(1.2)
where $T_{\mu\nu}[\mathcal{L}_{\lambda}]$ is the stress-energy tensor for the deformed theory at value $\lambda$ of the flow parameter.
Solving this type of flow equation proves, on the one hand,
to be a fairly involved task even for classical systems,
and in the last few years
various direct, geometric, and string theory inspired techniques have been developed to tackle this problem [3, 11, 12, 13, 6, 7, 14, 15, 16, 17, 18, 19].
On the other hand, the solutions to such flows
lead to surprising and remarkable results.
The simplest example, that was considered for the first time in
[3], is the deformation of the Lagrangian
of a free real scalar field in $d=2$ dimensions.
The undeformed Lagrangian is
$${\mathcal{L}}_{0}=\frac{1}{2}\partial^{\mu}\phi\partial_{\mu}\phi~{}.$$
(1.3)
The deformed Lagrangian ${\mathcal{L}}_{\Lambda}$ satisfying (1.2)
was shown to be
[3] (see also [11])
$$\displaystyle{\mathcal{L}}_{\lambda}$$
$$\displaystyle=-\frac{1}{2\lambda}+\frac{1}{2\lambda}\sqrt{1+2\lambda\partial^{\mu}\phi\partial_{\mu}\phi}~{},$$
(1.4)
which is the gauge-fixed Nambu-Goto Lagrangian for a string
with tension determined by $\lambda$.
This simple result is one of the many links that $T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ deformations
have found with string theory,
see e.g. [20, 21, 6, 22, 23, 14, 18, 24],
and shows how these deformations
can be used to shed new light on the realm of non-local quantum field theories.
The result (1.4) was also extended to the supersymmetric case, where
$\mathcal{N}=(0,1),\,(1,1),\,(0,2)$ and $(2,2)$ supersymmetric extensions
of the Nambu-Goto string were proven to be $T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$-flows
[6, 7, 9, 15, 8, 25].
Such proofs made use of manifestly supersymmetric forms of $T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$
formulated in superspace in terms of supercurrent-squared operators
[6, 7, 8, 9].
Extensions of the Lagrangian flow (1.2)
in terms of operators defined by squared combinations of the stress-energy tensor
have been considered also in $d>2$.111Unlike the $d=2$ case, it is not known whether such classical flows correspond to well-defined operators at the quantum mechanical level. Understanding the quantum properties of $T^{2}$ flows in $d>2$, perhaps with additional assumptions such as maximal supersymmetry, remains an important open question. Related interesting developments
in this direction were obtained for $d=4$, ${\cal N}=4$ SYM
in [26].
One notable example is the
proposal of [27, 28]
that arises from an holographic interpretation of $T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$-like deformations in $d\geq 2$.
Another very surprising and inspiring example is the flow equation
that has been discovered in [13]
for the Maxwell-Born-Infeld theory.
Its Lagrangian
$$\displaystyle{\mathcal{L}}_{\text{BI}}=\frac{1}{\alpha^{2}}\Bigg{\{}~{}1-\sqrt{1+\frac{\alpha^{2}}{2}F^{2}-\frac{\alpha^{4}}{16}(F\tilde{F})^{2}}~{}\Bigg{\}}~{},$$
(1.5a)
was in fact shown to be
a deformation of the free Maxwell Lagrangian as follows:
$$\displaystyle\frac{\partial\mathcal{{\mathcal{L}}}_{\text{BI}}}{\partial\alpha^{2}}=\frac{1}{8}\Big{(}T^{\mu\nu}T_{\mu\nu}-\frac{1}{2}\Theta^{2}\Big{)}~{},~{}~{}~{}~{}~{}~{}{\mathcal{L}}_{\text{BI}}\big{|}_{\alpha^{2}=0}=-\frac{1}{4}F^{2}={\mathcal{L}}_{\text{Maxwell}}~{}.$$
(1.5b)
Extensions of this result were considered in
[25, 29, 30].
For instance,
a manifestly supersymmetric extension of the flow equation (1.5b)
was proven in [25]
to hold for the $4d$, ${\cal N}=1$ supersymmetric
Maxwell-Born-Infeld theory proposed by Bagger and Galperin in
[31].
The (supersymmetric) Born-Infeld theory is of great importance due to its role in the low-energy, effective description of brane systems in string theory.
From this point of view, the flow (1.5b) reads as a $4d$ extension
of the $2d$ Nambu-Goto case and raises questions about whether
current-squared flows might be a universal feature of string theory yet to be
uncovered.
One of the well-known features that characterises Maxwell theory,
together with its Born-Infeld extension,
is invariance under electro-magnetic duality, a property which is also shared by its Bagger-Galperin supersymmetric extension.
This U(1) duality symmetry can be thought of as a phase rotation of a complex combination of the field strength $F_{\mu\nu}$ and its dual $\widetilde{F}^{\mu\nu}=\frac{1}{2}\epsilon^{\mu\nu\rho\sigma}F_{\rho\sigma}$:
$$\displaystyle F_{\mu\nu}+{\rm i}\widetilde{F}_{\mu\nu}\longrightarrow e^{{\rm i}\theta}\left(F_{\mu\nu}+{\rm i}\widetilde{F}_{\mu\nu}\right)\,.$$
(1.6)
It is natural to ask whether there are other theories of electromagnetism which also exhibit such eletro-magnetic symmetry (1.6).
In this context, recently it was discovered in [32]
(see also [33])
that there is a unique
one-parameter family of
Lorentz invariant modifications of the Maxwell Lagrangian in $d=4$ which preserve both duality invariance and conformal symmetry.
This unique deformation is called the Modified Maxwell (or ModMax) theory and is described by the Lagrangian
$$\displaystyle\mathcal{L}_{\text{ModMax}}=-\frac{1}{4}\cosh(\gamma)F^{2}+\frac{1}{4}\sinh(\gamma)\sqrt{(F^{2})^{2}+(F\widetilde{F})^{2}}\,.$$
(1.7)
Here $\gamma$ is a dimensionless real parameter that controls the deformation; when $\gamma=0$, the Lagrangian (1.7) reduces to the usual Maxwell theory. Since the equations of motion for
Maxwell theory are duality invariant, and because the combination under the square root in (1.7) is proportional to $z_{\mu\nu}z^{\mu\nu}\overline{z}^{\rho\sigma}\overline{z}_{\rho\sigma}$ where
$z_{\mu\nu}=F_{\mu\nu}+{\rm i}\widetilde{F}_{\mu\nu}$, the ModMax theory is also invariant under U(1) duality rotations (1.6).
Note that study of duality invariant models, with and without supersymmetry, has a very long history.
We refer the reader to the following
(incomplete) list of papers and references therein
[34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 32, 33, 46, 47, 48, 49]. For a pedagogical introduction to theories of non-linear electrodynamics such as ModMax, see [50].
Due to the presence of the square root in (1.7),
one is tempted to compare ModMax with the Maxwell-Born-Infeld theory
(1.5a).
However, it is clear that the ModMax theory of electrodynamics is qualitatively quite different from the Born-Infeld theory
and not only because one is conformal and the other is not. Although both Lagrangians involve square roots, the Born-Infeld Lagrangian (1.5a) can be Taylor expanded around small field strength to yield the Maxwell Lagrangian plus an infinite series of higher-derivative corrections. But since the square root appearing in the ModMax Lagrangian (1.7) is not of the form $\sqrt{1+x}$ for some quantity $x$ involving field strengths
(in fact it is non-analytic at $z=0$),
ModMax does not admit a derivative expansion of the same form.
Of course, the Born-Infeld-like extension of ModMax
[46], which we will define shortly, does possess an $\alpha^{2}$-type expansion, and this theory will be the main focus
of our discussion.
There are other interesting properties of the ModMax theory
that have recently been investigated — we will mention a few. Although the theory exhibits superluminal propagation when the deformation parameter $\gamma$ is negative, for $\gamma\geq 0$ it has well-behaved plane wave solutions. In particular, small-amplitude waves in the ModMax theory obey a polarization-dependent dispersion relation (birefringence), unlike the Born-Infeld theory. The ModMax theory has been shown to descend, via dimensional reduction, from a $6d$ theory of a chiral $2$-form which can be described by a modified version of the Pasti-Sorokin-Tonin (PST) action [46]; for details on the original PST theory see [51, 52, 53]. Black hole solutions which are the analogues of the Reissner–Nordström black hole, but which are electrically charged under a gauge field described by ModMax, have been studied in [54, 55, 56, 57, 58, 59, 60].
Directly relevant for this paper are
the Born-Infeld-like extensions
of the $4d$ ModMax theory that
have been constructed in [46]
and then supersymmetrised in [48, 47],
see [61] for the ${\cal N}=2$ case,
obtaining an explicit example of the infinite class of supersymmetric duality invariant models defined in
[39, 40, 41, 42, 43, 44].
Written in terms of the $\gamma$ and $\alpha^{2}$ parameters,
the Lagrangian for the Born-Infeld-ModMax theory takes the following form
$$\displaystyle\mathcal{L}_{\gamma{\rm BI}}=\frac{1}{\alpha^{2}}\Bigg{\{}~{}1-\sqrt{1+\frac{\alpha^{2}}{2}\left[\cosh(\gamma)F^{2}-\sinh(\gamma)\sqrt{(F^{2})^{2}+(F\widetilde{F})^{2}}\right]-\frac{\alpha^{4}}{16}(F\widetilde{F})^{2}}~{}\Bigg{\}}~{}.~{}~{}~{}~{}~{}~{}$$
(1.8)
Considering the flow equations described in eq. (1.5b)
for the Maxwell-Born-Infeld theory,
together with its supersymmetric extension of [25],
it is natural to wonder whether
the whole one parameter family of
Born-Infeld-like ModMax theories
satisfies a $T^{2}$-like flow equation both in the
non-supersymmetric and supersymmetric cases.
The main purpose of this paper is in fact to analyse this query
and to provide an affirmative answer
to the following question: is the
(supersymmetric) ModMax-BI Lagrangian satisfying a $T^{2}$-like flow
for any $\gamma$?
An intuition that this might be the case
comes from the the auxiliary field
formulation of duality invariant theories
[62, 38],
see [44] for the supersymmetric case.
In this framework, Maxwell theory with $\gamma=0$
does not seem to have any special property compared to the ModMax case
with $\gamma\neq 0$ [48, 61].
This suggests that, if a Lagrangian flow exists for $\gamma=0$
it should then exist for any $\gamma$,
as we will indeed prove explicitly in our paper
for the non-supersymmetric and ${\cal N}=1$ supersymmetric cases.
This paper is organized as follows. In Section 2, we verify by direct computation that the Born-Infeld extension of the ModMax theory satisfies a $T^{2}$ flow for any value of the parameter $\gamma$. Section 3 then provides a different proof of this fact which begins from a general equation that applies to $T^{2}$ flows for Abelian gauge theories in any spacetime dimension. In Section 4, we extend this analysis to the case with $\mathcal{N}=1$ supersymmetry, demonstrating that the supersymmetric extension of the ModMax-BI theory satisfies a supercurrent-squared flow equation which is the superspace analogue of the ordinary $T^{2}$ deformation. Section 5 summarizes these results and identifies some directions for future research.
We also include two Appendices; in the first we elaborate on
the equivalence of $T^{2}$ operators and
$\sqrt{\det(T_{\mu\nu})}$ in $d=4$, and in the second we derive a general flow equation for $T^{2}$ flows in scalar theories for any spacetime dimension.
Note Added: During the preparation of this work
the interesting paper [63]
appeared with the overlapping result for the
non-supersymmetric Born-Infeld like deformation
of ModMax as a stress-tensor squared deformation.
Interestingly, the authors of [63] also identified
an operator which is a functional of the stress-energy tensor of ModMax-BI that
triggers classically marginal (though non-analytic) deformations
associated with the parameter $\gamma$. The supersymmetric extension of
this result is an interesting venue for future research.
2 ModMax-BI is a $T^{2}$ flow
In [13] it was proven that
the Maxwell-Born-Infeld theory
with Lagrangian ${\mathcal{L}}_{\rm BI}$,
$$\displaystyle S_{\rm BI}=\int d^{4}x\;{\mathcal{L}}_{\text{BI}}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{\alpha^{2}}\int d^{4}x\;\Big{[}1-\sqrt{-\det(\eta_{\mu\nu}+\alpha F_{\mu\nu})}\Big{]}~{}$$
(2.1)
$$\displaystyle=$$
$$\displaystyle\frac{1}{\alpha^{2}}\int d^{4}x\;\Big{[}1-\sqrt{1+\frac{\alpha^{2}}{2}F^{2}-\frac{\alpha^{4}}{16}(F\tilde{F})^{2}}\Big{]}~{}$$
$$\displaystyle=$$
$$\displaystyle-\frac{1}{4}\int d^{4}x\;F^{2}+{\rm higher~{}derivative~{}terms}~{},$$
satisfies the following flow equation with respect to
the $\alpha^{2}$ parameter:
$$\frac{\partial\mathcal{{\mathcal{L}}}_{\text{BI}}}{\partial\alpha^{2}}=\frac{1}{8}O_{T^{2}}~{}.$$
(2.2)
The operator $O_{T^{2}}$
is defined as
$$O_{T^{2}}\equiv T^{\mu\nu}T_{\mu\nu}-\frac{1}{2}\Theta^{2}~{},\qquad\Theta\equiv T_{\mu}^{\mu}~{},$$
(2.3)
where $T^{\mu\nu}$ is the symmetric and conserved stress-energy tensor of the Maxwell-Born-Infeld theory.
The previous composite operator
is one of the representatives of an infinite family of
stress-tensor squared operators of the following form:
$$O_{T^{2}}^{[r]}=T^{\mu\nu}T_{\mu\nu}-r\,\Theta^{2}~{}.$$
(2.4)
These are defined for any real constant parameter $r$
and stress-energy tensor of a relativistic QFT in
$d$-dimensions. We will discuss some more examples in the
next section. However, it is worth reminding that
for $d=2$ and $r=1$, $O_{T^{2}}^{[1]}$
is the $T\bar{T}$ operator,
which is proportional to
$\det[T_{\mu\nu}]$, see [1, 2, 3].
In $d>2$ it is still an open question whether there are
operators that play the same role as $T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$, and share the same remarkable
properties.
Notable proposals for extensions of $T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ in $d>2$ are
$O_{T^{2}}^{[1/(d-1)]}$ in $d$-dimensions, that
were motivated from bulk cut-off holography
[27, 28].
Rather than deforming a QFT by a flow triggered by generic
$O_{T^{2}}^{[r]}$ (we will elaborate on general flows in the next section),
in this section we are interested to check whether
the Born-Infeld-like extension of the ModMax theory
[32] satisfies a stress-tensor squared
flow for a specific value of $r$. We will explicitly show that this is the
case for $r=1/2$, exactly as for the Born-Infeld Lagrangian (2.1).
The Born-Infeld-like extension of ModMax
is defined by the following Lagrangian222The parameter $t$ is the same as $T$ of [32].
$$\displaystyle\mathcal{L}_{\gamma{\rm BI}}=t-\sqrt{t^{2}-2t\left[\cosh(\gamma)S+\sinh(\gamma)\sqrt{S^{2}+P^{2}}\right]-P^{2}}~{},$$
(2.5)
where
$$\displaystyle S=-\frac{1}{4}F^{\mu\nu}F_{\mu\nu},\quad P=-\frac{1}{4}F_{\mu\nu}\tilde{F}^{\mu\nu},\quad\tilde{F}^{\mu\nu}=\frac{1}{2}\epsilon^{\mu\nu\lambda\tau}F_{\lambda\tau}~{},$$
(2.6)
and $F_{\mu\nu}=(\partial_{\mu}v_{\nu}-\partial_{\nu}v_{\mu})$ is the field strength for an Abelian gauge field $v_{\mu}$.
The $t\to+\infty$ limit leads to the ModMax Lagrangian
$$\displaystyle\mathcal{L}_{{\rm ModMax}}=\cosh(\gamma)S+\sinh(\gamma)\sqrt{S^{2}+P^{2}}~{}.$$
(2.7)
For $\gamma=0$, and after identifying
$t=1/\alpha^{2}$, the Lagrangian ${\mathcal{L}}_{\gamma{\rm BI}}$ in eq. (2.5)
turns into the Maxwell-Born-Infeld Lagrangian (2.1).
After minimally coupling the Born-Infeld-ModMax Lagrangian
to a metric $g^{\mu\nu}$,
it is a straightforward exercise
to derive the Hilbert stress-energy tensor, given by333In this subsection
we work in a $d=4$ Lorentzian space-time with mostly plus metric signature.
$$\displaystyle T^{\gamma{\rm BI}}_{\mu\nu}=-\frac{2}{\sqrt{-g}}\frac{\delta S_{\gamma{\rm BI}}}{\delta g^{\mu\nu}}~{},$$
(2.8)
for (2.5).
The result can be written as
$$\displaystyle T^{\gamma{\rm BI}}_{\mu\nu}=\eta_{\mu\nu}f_{1}(S,P)+f_{2}(S,P)F_{\mu}{}^{\lambda}F_{\nu\lambda}~{},$$
(2.9)
where the two functions
$f_{1}(S,P)$
and
$f_{2}(S,P)$
are defined as:
$$\displaystyle f_{1}(S,P)$$
$$\displaystyle=$$
$$\displaystyle\frac{t\left(-t+2\cosh{(\gamma)}S+\sinh{(\gamma)}\frac{P^{2}+2S^{2}}{\sqrt{S^{2}+P^{2}}}\right)}{\sqrt{t^{2}-2t\Big{[}\cosh(\gamma)S+\sinh(\gamma)\sqrt{S^{2}+P^{2}}\Big{]}-P^{2}}}+t~{},$$
(2.10)
$$\displaystyle f_{2}(S,P)$$
$$\displaystyle=$$
$$\displaystyle\frac{t\Big{(}\cosh(\gamma)+\sinh(\gamma)\frac{S}{\sqrt{S^{2}+P^{2}}}\Big{)}}{\sqrt{t^{2}-2t\Big{[}\cosh(\gamma)S+\sinh(\gamma)\sqrt{S^{2}+P^{2}}\Big{]}-P^{2}}}~{}.$$
(2.11)
The trace of the stress-energy tensor is
$$\Theta=\Theta(S,P)=\frac{4t\left(\cosh(\gamma)S+\sinh(\gamma)\sqrt{S^{2}+P^{2}}-t\right)}{\sqrt{t^{2}-2t\Big{[}\cosh(\gamma)S+\sinh(\gamma)\sqrt{S^{2}+P^{2}}\Big{]}-P^{2}}}+4t~{}.$$
(2.12)
It is worth underlining
that the stress-energy tensor for the
Born-Infeld-ModMax theory presented above is invariant under U$(1)$
electro-magnetic duality transformations. This indicates that any
deformation triggered by composite operators defined only in terms of
the stress-energy tensor should remain electro-magnetic invariant.
Let us compute explicitly the $O_{T^{2}}$ operator, eq. (2.3).
Thanks to the identity
$$(F\tilde{F})^{2}=\frac{1}{4}(\epsilon_{\mu\nu\rho\sigma}F^{\mu\nu}F^{\rho\sigma})^{2}=4F_{\mu\nu}F^{\nu\rho}F_{\rho\sigma}F^{\sigma\mu}-2(F^{2})^{2}~{},$$
(2.13)
which implies
$$F_{\mu\nu}F^{\nu\rho}F_{\rho\sigma}F^{\sigma\mu}=8S^{2}+4P^{2}~{},$$
(2.14)
the $O_{T^{2}}$ operator takes the form
$$\displaystyle O_{T^{2}}=4f_{1}^{2}+4f_{2}^{2}\left(2S^{2}+P^{2}\right)-8f_{1}f_{2}S-\frac{1}{2}\Theta^{2}~{},$$
(2.15)
which, after plugging in the explicit expressions for
$f_{1}(S,P)$, $f_{2}(S,P)$, and $\Theta(S,P)$,
simplifies to
$$\displaystyle O_{T^{2}}$$
$$\displaystyle=$$
$$\displaystyle\frac{8t^{2}}{(t-{\mathcal{L}}_{\gamma{\rm BI}})^{3}}\,\Big{\{}\,t^{3}-2P^{2}t+t\cosh(2\gamma)(P^{2}+2S^{2})+\cosh(\gamma)S(P^{2}-3t^{2})$$
(2.16)
$$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\,+\sqrt{S^{2}+P^{2}}\left[2\sinh(2\gamma)St+\sinh(\gamma)(P^{2}-3t^{2})\right]\Big{\}}$$
$$\displaystyle+\frac{8t^{2}}{(t-{\mathcal{L}}_{\gamma{\rm BI}})^{2}}\left\{P^{2}-t^{2}+2t\sinh(\gamma)\sqrt{S^{2}+P^{2}}+2t\cosh(\gamma)S\right\}~{},$$
where
$$\displaystyle t-{\mathcal{L}}_{\gamma{\rm BI}}=\sqrt{t^{2}-2t\left[\cosh(\gamma)S+\sinh(\gamma)\sqrt{S^{2}+P^{2}}\right]-P^{2}}~{}.$$
(2.17)
Despite the seemingly involved expression,
it is straightforward to directly check that (2.16)
is the same as a derivative with respect to $t$, or equivalently
with respect to $\alpha^{2}=1/t$,
of the Born-Infeld-ModMax Lagrangian.
More specifically, it holds
$$\displaystyle\frac{\partial{\mathcal{L}}_{\gamma{\rm BI}}}{\partial\alpha^{2}}=\frac{1}{8}O_{T^{2}}~{},~{}~{}~{}~{}~{}~{}\Longleftrightarrow~{}~{}~{}~{}~{}~{}\frac{\partial{\mathcal{L}}_{\gamma{\rm BI}}}{\partial t}=-\frac{1}{8t^{2}}O_{T^{2}}~{}.$$
(2.18)
This remarkably shows that the stress-tensor squared operator
leading the flow is the same for any value of $\gamma$.
3 Derivation from $T^{2}$ Master Flow Equation
We have seen in Section 2 that the
Born-Infeld-like extension of the ModMax Lagrangian can be shown, via direct computation, to satisfy a ${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$-like flow. In this section, we will present a complementary derivation of this result which begins from a general differential equation for $T^{2}$ flows involving an Abelian gauge theory in arbitrary spacetime dimension. We now briefly review this general flow equation, which first appeared in [30].
3.1 Review of General Flow Equation
For simplicity, in this section we will work in Euclidean signature.444Our conventions follow those in Section 7.2.1 of [30], to which we refer the reader for more details. In particular, the choice of Euclidian signature
in this section is made merely for convenience and does not substantively affect any of the results.
In $d$ spacetime dimensions
the field strength $F_{\mu\nu}$ associated with an Abelian gauge field $v_{\mu}$ can be thought of as a $d\times d$ matrix whose indices are raised or lowered with $\delta_{\mu\nu}$. By the Cayley-Hamilton theorem, every such matrix obeys its characteristic equation
$$\displaystyle p(M)=M^{d}+c_{d-1}M^{d-1}+\cdots+c_{1}M+(-1)^{d}\det(M)\mathbb{I}_{d}=0\,,$$
(3.1)
where $\mathbb{I}_{d}$ is the $d\times d$ identity matrix. The constants $c_{i}$ are given by
$$\displaystyle c_{i}=\sum_{\{k_{l}\}}\,\prod_{l=1}^{d}\,\frac{(-1)^{k_{l}+1}}{l^{k_{l}}k_{l}!}\left[\,\mathrm{tr}\,\big{(}M^{l}\big{)}\,\right]^{k_{l}}\,,$$
(3.2)
where the sum runs over all sets of non-negative integers $k_{l}$ which satisfy
$$\displaystyle\sum_{l=1}^{d}lk_{l}=d-i\,.$$
(3.3)
Because these $c_{i}$ are determined in terms of the lower traces $\operatorname{tr}(M^{j})$ for $j=1,\cdots,d$, equation (3.1) places a limit on the number of independent trace structures that a $d\times d$ matrix may have. In particular, given all of the traces
$$\displaystyle\operatorname{tr}(M)\,,\,\operatorname{tr}\big{(}M^{2}\big{)}\,,\,\cdots\,,\,\operatorname{tr}\big{(}M^{d}\big{)}\,,$$
(3.4)
it follows that all higher traces $\operatorname{tr}\big{(}M^{n}\big{)}$ for $n>d$ can then be expressed in terms of the lower traces. We now restrict to the case of an antisymmetric matrix, appropriate for a field strength $F_{\mu\nu}$. The trace of any odd power of such a matrix vanishes, so a general scalar quantity built from $F_{\mu\nu}$ in $d$ dimensions can be expressed in terms of the independent traces $\operatorname{tr}(F^{2})$, $\operatorname{tr}(F^{4})$, $\cdots$, $\operatorname{tr}(F^{2k})$ where $k=\lfloor\frac{d}{2}\rfloor$. To ease notation, we define
$$\displaystyle x_{i}=\operatorname{tr}(F^{2i})\,,$$
(3.5)
for $i=1,\cdots,k$.
Now consider a general Lagrangian for an Abelian gauge field with field strength $F_{\mu\nu}$ in $d$ Euclidean spacetime dimensions.
Because the Lagrangian is a gauge-invariant scalar, it may therefore be written as a function of the $x_{i}$:555In this paper we only consider manifestly gauge invariant Lagrangians of the form (3.6). In odd dimension it might be
interesting to extend this ansatz and study flows involving terms that depend on both $F$ and Chern-Simons-like couplings. Note however that an ordinary Chern-Simons term is purely topological and would not contribute to the stress tensor.
$$\displaystyle\mathcal{L}(F)=\mathcal{L}(x_{1},\cdots,x_{k})\,.$$
(3.6)
The Hilbert stress-energy tensor associated with this Lagrangian is
$$\displaystyle T_{\mu\nu}=\delta_{\mu\nu}\mathcal{L}-2\sum_{i=1}^{k}\frac{\partial\mathcal{L}}{\partial x_{i}}\cdot\frac{\delta x_{i}}{\delta g^{\mu\nu}}\Bigg{|}_{g=\delta}\,.$$
(3.7)
Computing the metric derivative of one of the $x_{j}$ gives
$$\displaystyle\frac{\delta x_{j}}{\delta g^{\mu\nu}}=2jF^{2j}_{\mu\nu}\,,$$
(3.8)
where we have introduced the notation
$$\displaystyle F^{2j}_{\mu\nu}=g^{\alpha_{1}\beta_{1}}\cdots g^{\alpha_{2j-1}\beta_{2j-1}}F_{\mu\alpha_{1}}F_{\beta_{1}\alpha_{2}}\cdots F_{\beta_{2j-2}\alpha_{2j-1}}F_{\beta_{2j-1}\nu}\,.$$
(3.9)
That is, $F^{2j}_{\mu\nu}$ is a product of $2j$ copies of $F_{\mu\nu}$ with all adjacent indices contracted except for the first and last. Using this in (3.7) yields a general expression for the stress-energy tensor,
$$\displaystyle T_{\mu\nu}=\delta_{\mu\nu}\mathcal{L}-4\sum_{i=1}^{k}i\frac{\partial\mathcal{L}}{\partial x_{i}}F^{2i}_{\mu\nu}\,.$$
(3.10)
The bilinears which appear in general $T^{2}$ flows are then
$$\displaystyle\begin{split}T^{\mu\nu}T_{\mu\nu}&=\mathcal{L}^{2}d-8\mathcal{L}\sum_{i=1}^{k}i\frac{\partial\mathcal{L}}{\partial x_{i}}\operatorname{tr}(F^{2i})+16\sum_{i,j=1}^{k}ij\frac{\partial\mathcal{L}}{\partial x_{i}}\frac{\partial\mathcal{L}}{\partial x_{j}}F^{2i,\mu\nu}F_{\mu\nu}^{2j}\,,\\
\left(T^{\mu}_{\;\;\,\mu}\right)^{2}&=\mathcal{L}^{2}d^{2}-8\mathcal{L}d\sum_{i=1}^{k}i\frac{\partial\mathcal{L}}{\partial x_{i}}\operatorname{tr}(F^{2i})+16\sum_{i,j=1}^{k}ij\frac{\partial\mathcal{L}}{\partial x_{i}}\frac{\partial\mathcal{L}}{\partial x_{j}}\operatorname{tr}(F^{2i})\operatorname{tr}(F^{2j})\,.\end{split}$$
A general flow by the operator $O_{T^{2}}^{[r]}$ defined in (2.4), described by the differential equation
$$\displaystyle\frac{\partial\mathcal{L}}{\partial\lambda}=T^{\mu\nu}T_{\mu\nu}-r\left(T^{\mu}_{\;\;\,\mu}\right)^{2}\,,$$
(3.11)
can therefore be written in terms of the $x_{i}$ as
$$\displaystyle\frac{\partial\mathcal{L}}{\partial\lambda}$$
$$\displaystyle=(1-rd)d\mathcal{L}^{2}-8\mathcal{L}(1-rd)\sum_{i=1}^{k}i\frac{\partial\mathcal{L}}{\partial x_{i}}x_{i}+16\sum_{i,j=1}^{k}ij\frac{\partial\mathcal{L}}{\partial x_{i}}\frac{\partial\mathcal{L}}{\partial x_{j}}\left(x_{i+j}-rx_{i}x_{j}\right)\,.$$
(3.12)
We will refer to (3.12) as the master flow equation, since it describes a general $T^{2}$ flow for a theory of a single Abelian field strength in $d$ dimensions. However, we note that (3.12) is not expressed in terms of the $k$ independent trace structures $x_{1}$, $\cdots$, $x_{k}$, since the quantity involving $x_{i+j}$ will introduce dependence on the higher traces. In applications of the master flow equations we must eliminate these variables in favor of the lower traces using the Cayley-Hamilton theorem, which produces dimension-dependent numerical factors.
We end this subsection by commenting on a condition regarding the stress-energy tensors for Abelian gauge theories, which is easy to understand using the formalism we have just reviewed. It was pointed out in [13] that the stress-energy tensor $T^{(\mathrm{BI})}$ for the Born-Infeld theory in four dimensions satisfies the condition
$$\displaystyle\sqrt{\det\left(T^{(\mathrm{BI})}\right)}=\frac{1}{4}\left(\frac{1}{2}\operatorname{tr}\left(T^{(\mathrm{BI})}\right)^{2}-\operatorname{tr}\left(\left(T^{(\mathrm{BI})}\right)^{2}\right)\right)\,.$$
(3.13)
Note that deformations driven by the determinant
of the stress-energy tensor in higher dimensions appeared also in
[64, 11]
were the operator $\left[\det(T)\right]^{1/(d-1)}$
was proposed as a $T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$-like deformations in $d$ dimensions.
At first sight, equation (3.13)
seems like a fairly special constraint which might be related to the fact that the Born-Infeld Lagrangian satisfies a $T^{2}$ flow in $d=4$.
Indeed, the combination of stress-energy tensor bilinears appearing on the right side of (3.13) is proportional to $O_{T^{2}}$ and the determinant of the energy-momentum tensor is what defines the usual ${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ operator in two dimensions, so this constraint naively appears connected to this family of stress-energy tensor deformations.
However, the condition (3.13) in fact holds for the stress-energy tensor of any theory of an Abelian field strength in four spacetime dimensions (including, of course, the ModMax Lagrangian and its ModMax-BI generalization). The proof of this fact is a simple linear algebra exercise, which we have relegated to Appendix A, and relies only on the form of the Hilbert stress-energy tensor and the fact that the field strength $F_{\mu\nu}$ is antisymmetric. The upshot of this result is that we are free to think of our $T^{2}$ flow as being driven by the operator $O_{T^{2}}$ defined in (2.3), or by the operator $\sqrt{\det(T)}$, up to an overall constant scaling, when we are considering deformations of four-dimensional gauge theories.
3.2 Application to ModMax-BI Theory
We now specialize to the case of $d=4$ spacetime dimensions, appropriate for the ModMax theory and its Born-Infeld-like extension. In this case, the two independent scalars that can be constructed from the field strength $F_{\mu\nu}$ are
$$\displaystyle x_{1}=F_{\mu\nu}F^{\nu\mu}=\operatorname{tr}(F^{2})\,,\qquad x_{2}=F^{\mu\sigma}F_{\sigma}^{\;\;\,\nu}F_{\nu}^{\;\;\,\rho}F_{\rho\mu}=\operatorname{tr}(F^{4})\,.$$
(3.14)
As we mentioned above, the master flow equation (3.12) will introduce dependence on the two higher traces $x_{3}=\operatorname{tr}(F^{6})$ and $x_{4}=\operatorname{tr}(F^{8})$. We must therefore eliminate these in terms of $x_{1}$ and $x_{2}$. The constraint implied by the Cayley-Hamilton theorem (3.1) for a $4\times 4$ matrix $M$ is
$$\displaystyle 0$$
$$\displaystyle=M^{4}-\left(\operatorname{tr}(M)\right)M^{3}+\frac{1}{2}\left(\left(\operatorname{tr}(M)\right)^{2}-\operatorname{tr}(M^{2})\right)M^{2}$$
$$\displaystyle\quad-\frac{1}{6}\left(\left(\operatorname{tr}(M)\right)^{3}-3\operatorname{tr}(M^{2})\operatorname{tr}(M)+2\operatorname{tr}(M^{3})\right)M+\det(M)\mathbb{I}_{4}\,.$$
(3.15)
We first take the trace of equation (3.2) and solve for the determinant to find
$$\displaystyle\det(M)=\frac{1}{24}\left(\left(\operatorname{tr}M\right)^{4}-6\operatorname{tr}(M^{2})\left(\operatorname{tr}M\right)^{2}+3\left(\operatorname{tr}M^{2}\right)^{2}+8\operatorname{tr}(M)\operatorname{tr}(M^{3})-6\operatorname{tr}(M^{4})\right)\,.$$
(3.16)
Replacing $M$ with the antisymmetric matrix $F$, so that traces of odd powers vanish, gives
$$\displaystyle\det(F)=\frac{1}{8}\left(\operatorname{tr}\big{(}F^{2}\big{)}\right)^{2}-\frac{1}{4}\operatorname{tr}(F^{4})\,.$$
(3.17)
If we had first multiplied equation (3.2) by $M^{2}$ or by $M^{4}$ before taking the trace, we would have obtained the conditions
$$\displaystyle\operatorname{tr}(F^{6})$$
$$\displaystyle=\frac{1}{2}\operatorname{tr}(F^{2})\operatorname{tr}(F^{4})-\det(F)\operatorname{tr}(F^{2})\,,$$
(3.18a)
$$\displaystyle\operatorname{tr}(F^{8})$$
$$\displaystyle=\frac{1}{2}\operatorname{tr}(F^{2})\operatorname{tr}(F^{6})-\det(F)\operatorname{tr}(F^{4})\,.$$
(3.18b)
This system of equations can be solved and written in terms of the variables $x_{i}=\operatorname{tr}(F^{2i})$, which yields
$$\displaystyle x_{3}=-\frac{1}{8}x_{1}\left(x_{1}^{2}-6x_{2}\right)\,,\qquad x_{4}=-\frac{1}{16}\left(x_{1}^{4}-4x_{1}^{2}x_{2}-4x_{2}^{2}\right)\,.$$
(3.19)
Substituting the expressions (3.19) into the master flow equation (3.12) and setting $d=4$, we obtain the general differential equation
$$\displaystyle\frac{\partial\mathcal{L}}{\partial\lambda}$$
$$\displaystyle=\left(4-16r\right)\mathcal{L}^{2}-8\left(1-4r\right)\mathcal{L}\left(x_{1}\frac{\partial\mathcal{L}}{\partial x_{1}}+2x_{2}\frac{\partial\mathcal{L}}{\partial x_{2}}\right)+16\left(\frac{\partial\mathcal{L}}{\partial x_{1}}\right)^{2}\left(x_{2}-rx_{1}^{2}\right)$$
$$\displaystyle\quad+16\left(-\frac{1}{2}\frac{\partial\mathcal{L}}{\partial x_{1}}\frac{\partial\mathcal{L}}{\partial x_{2}}x_{1}\left(x_{1}^{2}-6x_{2}+8rx_{2}\right)+\left(\frac{\partial\mathcal{L}}{\partial x_{2}}\right)^{2}\left(-\frac{1}{4}x_{1}^{4}+x_{1}^{2}x_{2}+(1-4r)x_{2}^{2}\right)\right)\,.$$
(3.20)
We now wish to show that equation (3.2) admits a solution which reduces to the ModMax theory when $\lambda=0$, and that this solution exists only for the value $r=\frac{1}{2}$ of the relative coefficient in the deformation. We do this by first making a slightly more general ansatz and then demonstrating that consistency with the flow equation requires the ansatz to take exactly the form of the ModMax-BI Lagrangian.
More precisely, we begin by making an ansatz of the form666Do not confuse the numerical constant $\alpha$
used in this section with the dimensionful coupling constant $\alpha^{2}$ in sections 1, 2 and 4 of the paper, as, for example, in equations (1.5a)
and (1.8).
$$\displaystyle\mathcal{L}(\lambda)=\frac{1}{\alpha\lambda}\left(\sqrt{1+2\alpha\lambda\left(e^{-\gamma}u(x_{1},x_{2})+e^{\gamma}v(x_{1},x_{2})\right)+4\alpha^{2}\lambda^{2}u(x_{1},x_{2})v(x_{1},x_{2})}-1\right)\,,$$
(3.21)
and we further assume that the functions $u,v$ can be written as a sum and difference as
$$\displaystyle u(x_{1},x_{2})=\beta x_{1}+w(x_{1},x_{2})\,,\qquad v(x_{1},x_{2})=\beta x_{1}-w(x_{1},x_{2})\,.$$
(3.22)
At this stage, $w(x_{1},x_{2})$ is an arbitrary function of the two scalars that can be constructed from $F_{\mu\nu}$, while $\alpha,\beta$ are undetermined numerical constants. We now obtain constraints on these quantities from consistency with the master flow equation in $d=4$. From demanding that the ansatz (3.21) be consistent with the differential equation (3.2) at zeroth order in both $\lambda$ and $\gamma$, one finds that the function $w$ must be
$$\displaystyle w(x_{1},x_{2})=\frac{2\sqrt{2}\beta}{\sqrt{\alpha}}\sqrt{x_{1}^{2}-4x_{2}}\,.$$
(3.23)
Substituting this expression for $w$ into the flow equation and then expanding to first order in $\lambda$ but zeroth order in $\gamma$ then produces the constraint
$$\displaystyle\alpha=-8\,.$$
(3.24)
Finally, we expand the flow equation to second order in $\lambda$ and to first order in $\gamma$, after using the above results for $w$ and $\alpha$, and find that the differential equation at this order is satisfied only if
$$\displaystyle r=\frac{1}{2}\,.$$
(3.25)
After imposing these various conditions, we arrive at
$$\displaystyle\mathcal{L}(\lambda)=\frac{1}{8\lambda}\left(1-\sqrt{1-32\beta\lambda\left(x_{1}\cosh(\gamma)+\sinh(\gamma)\sqrt{4x_{2}-x_{1}^{2}}\right)+512\beta^{2}\lambda^{2}(x_{1}^{2}-2x_{2})}\right)\,.$$
(3.26)
This Lagrangian is an exact solution to the $4d$ master flow equation (3.2) to all orders in $\lambda$ and $\gamma$, and with any choice of the arbitrary constant $\beta$. However, to make contact with the preceding section, it is convenient to make a few changes of conventions. First, since (3.26) is a solution for any choice of the normalization $\beta$, we are free to choose $\beta=\frac{1}{32}$. Further, we can eliminate the variables $x_{1}=\operatorname{tr}(F^{2})$ and $x_{2}=\operatorname{tr}(F^{4})$ in terms of the variables $S,P$ defined in equation (2.6). The dictionary which translates between these variables is
$$\displaystyle x_{1}=4S\,,\qquad x_{2}=4P^{2}+8S^{2}\,.$$
(3.27)
After making these replacements, we find
$$\displaystyle\mathcal{L}(\lambda)=\frac{1}{8\lambda}\left(1-\sqrt{1-4\lambda\left(\cosh(\gamma)S+\sinh(\gamma)\sqrt{S^{2}+P^{2}}\right)-4P^{2}\lambda^{2}}\right)\,.$$
(3.28)
This is exactly the form of the Born-Infeld extension of the ModMax Lagrangian which we first defined in equation (2.5) after making the identification $t=\frac{1}{8\lambda}$.
3.3 No ModMax-BI Solutions to $T^{2}$ Flows in $d>4$
The properties of the ModMax Lagrangian (1.7), and its Born-Infeld extension, are special to four dimensions because they are written in terms of the dual field strength $\tilde{F}_{\mu\nu}$, and the Hodge dual of $F_{\mu\nu}$ would be a higher $p$-form in $d>4$ spacetime dimensions. Thus a ModMax-like theory in $d>4$ would, of course, not exhibit any analogue of duality invariance. However, as a pure statement about $T^{2}$ flows, one could ask whether the square-root structure appearing in the ModMax-BI theory is a generic feature of deformations by stress-energy tensor bilinears, or whether it is also special to flows in $d=4$. We saw in equation (3.26) that the ModMax-BI Lagrangian can be written in the form
$$\displaystyle\mathcal{L}(\lambda)=\frac{1}{8\lambda}\left(1-\sqrt{1-\lambda\left(x_{1}\cosh(\gamma)+\sinh(\gamma)\sqrt{4x_{2}-x_{1}^{2}}\right)+\frac{1}{2}\lambda^{2}(x_{1}^{2}-2x_{2})}\right)\,,$$
(3.29)
where for convenience we repeat the definitions of the two indpendent scalars $x_{1},x_{2}$ that can be constructed from a field strength in four dimensions:
$$\displaystyle x_{1}=F_{\mu\nu}F^{\nu\mu}=\operatorname{tr}(F^{2})\,,\qquad x_{2}=F^{\mu\sigma}F_{\sigma}^{\;\;\,\nu}F_{\nu}^{\;\;\,\rho}F_{\rho\mu}=\operatorname{tr}(F^{4})\,.$$
(3.30)
Although (3.29) is equivalent to the ModMax-BI Lagrangian, it is not written in terms of $\widetilde{F}_{\mu\nu}$ and therefore makes sense in any number of spacetime dimensions $d\geq 4$ (the cases for $d<4$ are trivial because $x_{1}$ and $x_{2}$ are no longer independent). Thus one might ask whether a Lagrangian of the form (3.29) satisfies a $T^{2}$ flow in any higher spacetime dimension.777The analogous question of whether the ordinary Born-Infeld Lagrangian arises from a $T^{2}$ flow in $d>4$ was answered in the negative in [30]. However, the naive generalization (3.31) of ModMax-BI does not reduce to Born-Infeld when $\gamma=0$ except in $d=4$. Therefore the absence of Born-Infeld solutions to $T^{2}$ flows in $d>4$ does not imply anything about the absence of ModMax-BI solutions.
We will now show that the answer to this question is no. Suppose that we make an ansatz for a $d$-dimensional Lagrangian which is inspired by the ModMax-BI theory and thus only depends on the first two traces $x_{1},x_{2}$:
$$\displaystyle\mathcal{L}(\lambda)=\frac{1}{\alpha\lambda}\left(\sqrt{1+2\alpha\lambda\left(e^{-\gamma}\,u(x_{1},x_{2})+e^{\gamma}\,v(x_{1},x_{2})\right)+4\alpha^{2}\lambda^{2}\,u(x_{1},x_{2})\,v(x_{1},x_{2})}-1\right)\,.$$
(3.31)
Furthermore, we would like our ansatz to reduce to the free Maxwell action when we take both $\gamma=0$ and $\lambda=0$, so we should make the same refinement to our ansatz as in (3.22) for the $d=4$ case:
$$\displaystyle u(x_{1},x_{2})=\beta x_{1}+w(x_{1},x_{2})\,,\qquad v(x_{1},x_{2})=\beta x_{1}-w(x_{1},x_{2})\,.$$
(3.32)
When $\lambda=0$, this reduces to an undeformed theory of the form
$$\displaystyle\mathcal{L}(0)=e^{\gamma}\left(\beta x_{1}-w\right)+e^{-\gamma}\left(\beta x_{1}+w\right)\,.$$
(3.33)
On dimensional grounds, the function $w$ must be proportional either to $x_{1}$, or to $\sqrt{x_{2}}$, or to a general combination $\sqrt{c_{1}x_{1}^{2}+c_{2}x_{2}}$ which has the same dimension as $x_{1}$. Therefore we will assume that $w$ can be written as
$$\displaystyle w(x_{1},x_{2})=\sqrt{c_{1}x_{1}^{2}+c_{2}x_{2}}\,,$$
(3.34)
which is the same form as in the usual $d=4$ ModMax-BI Lagrangian.
We must check whether any choice of the constants $\alpha,\beta,c_{1},c_{2}$ makes this ansatz consistent with the master flow equation (3.12), which we repeat:
$$\displaystyle\frac{\partial\mathcal{L}}{\partial\lambda}$$
$$\displaystyle=(1-rd)d\mathcal{L}^{2}-8\mathcal{L}(1-rd)\sum_{i=1}^{k}i\frac{\partial\mathcal{L}}{\partial x_{i}}x_{i}+16\sum_{i,j=1}^{k}ij\frac{\partial\mathcal{L}}{\partial x_{i}}\frac{\partial\mathcal{L}}{\partial x_{j}}\left(x_{i+j}-rx_{i}x_{j}\right)\,.$$
(3.35)
There are two cases to consider.
1.
$d\geq 6$. In this case, there is at least one additional independent trace structure $x_{3}$. Because the ansatz (3.31) is a function only of $x_{1}$ and $x_{2}$, the left side of the master flow equation is independent of $x_{3}$. However, the right side of the flow equation contains a term $\frac{\partial\mathcal{L}}{\partial x_{1}}\frac{\partial\mathcal{L}}{\partial x_{2}}x_{3}$ which is non-zero and depends on $x_{3}$. There is no constraint relating $x_{3}$ to other traces in $d\geq 6$, so the two sides cannot be equal and therefore the ansatz does not solve the flow equation.
2.
$d=5$. In this case, $x_{3}$ is not independent of $x_{1}$ and $x_{2}$, but rather satisfies
$$\displaystyle x_{3}=\frac{3}{4}x_{1}x_{2}-\frac{1}{8}x_{1}^{3}\,,$$
(3.36)
by the Cayley-Hamilton theorem. Similarly,
$$\displaystyle x_{4}=\frac{1}{16}\left(4x_{2}^{2}+4x_{1}^{2}x_{2}-x_{1}^{4}\right)\,.$$
(3.37)
One can substitute these relations into the master flow equation and then impose consistency order-by-order in $\lambda$. The constraint which is first order in $\lambda$ will be satisfied so long as
$$\displaystyle\alpha=8\,,\quad r=1\,\quad c_{1}=-\beta^{2}\,\quad c_{2}=4\beta^{2}\,.$$
(3.38)
However, upon expanding to second order in $\lambda$, one finds that the ansatz cannot be made consistent with the flow equation for any non-zero choice of the remaining parameter $\beta$.
Therefore we see that the naive generalization (3.31) of the ModMax-BI theory only satisfies a $T^{2}$ flow in $d=4$, similar to the Born-Infeld action.
Rather than the question that we have addressed above, one could also ask a slightly more general question which exploits the fact that in $d>4$ there are more independent scalars than $x_{1}$ and $x_{2}$. Thus one might wonder whether a different Lagrangian, which still possesses a square root of the form appearing in (3.29) but whose argument depends on $x_{1},x_{2},x_{3}$ and higher trace structures, satisfies a $T^{2}$ flow in higher dimension.
For instance, one could make another ansatz of the form
$$\displaystyle\mathcal{L}(\lambda)=\frac{1}{\alpha\lambda}\left(\sqrt{1+2\alpha\lambda\left(e^{-\gamma}\,u(x_{i})+e^{\gamma}\,v(x_{i})\right)+4\alpha^{2}\lambda^{2}\,u(x_{i})\,v(x_{i})}-1\right)\,.$$
(3.39)
where the functions $u(x_{i})$, $v(x_{i})$ now depend on all trace structures $x_{1},\cdots,x_{k}$. We still assume that
$$\displaystyle u(x_{1},\cdots,x_{k})=\beta x_{1}+w(x_{1},\cdots,x_{k})\,,\qquad v(x_{1},\cdots,x_{k})=\beta x_{1}-w(x_{1},\cdots,x_{k})\,,$$
(3.40)
but now allow a more general form of the function $w$, such as
$$\displaystyle w(x_{i})=\sqrt[k]{c_{1}x_{1}^{k}+c_{2}x_{2}x_{1}^{k-1}+\cdots+c_{N}x_{k}}\,,$$
(3.41)
which reduces to our previous ansatz when $k=2$ and which is again required on dimensional grounds since $w$ must have the same dimension as $x_{1}$. For instance, one could consider a six-dimensional analogue of the ModMax-BI theory where
$$\displaystyle w(x_{1},x_{2},x_{3})=\sqrt[3]{c_{1}x_{1}^{3}+c_{2}x_{1}x_{2}+c_{3}x_{3}}\,.$$
(3.42)
As an example, we will explicitly check whether the six-dimensional ansatz using (3.42) can solve the master flow equation for any choice of parameters. This will require one additional use of the Cayley-Hamilton theorem, with coefficients appropriate for $6\times 6$ matrices, in order to eliminate higher traces in the master flow equation. In this case, the equation obeyed by the field strength $F$ is
$$\displaystyle 0$$
$$\displaystyle=F^{6}-\frac{1}{2}x_{1}F^{4}+\frac{1}{8}\left(x_{1}^{2}-2x_{2}\right)F^{2}+\det(M)\,\mathbb{I}_{6}=0\,,$$
(3.43)
By repeatedly using (3.43) in the same way as above, we can express various higher traces in terms of the three independent structures $x_{1},x_{2},x_{3}$. In particular,
$$\displaystyle x_{4}$$
$$\displaystyle=\frac{1}{48}\left(x_{1}^{4}-12x_{1}^{2}x_{2}+12x_{2}^{2}+32x_{1}x_{3}\right)\,,$$
$$\displaystyle x_{5}$$
$$\displaystyle=\frac{1}{96}\left(x_{1}^{5}-10x_{1}^{3}x_{2}+20x_{1}^{2}x_{3}+40x_{2}x_{3}\right)\,,$$
$$\displaystyle x_{6}$$
$$\displaystyle=\frac{1}{384}\left(x_{1}^{6}-6x_{1}^{4}x_{2}-36x_{1}^{2}x_{2}^{2}+24x_{2}^{3}+16x_{1}^{3}x_{3}+96x_{1}x_{2}x_{3}+64x_{3}^{2}\right)\,.$$
(3.44)
We may now substitute these trace expressions into the master flow equation (3.12) and set $d=6$. The resulting differential equation is rather unwieldy, but we record it here for completeness:
$$\displaystyle\frac{\partial\mathcal{L}}{\partial\lambda}$$
$$\displaystyle=6\mathcal{L}^{2}\left(1-6r\right)+16\left(\frac{\partial\mathcal{L}}{\partial x_{1}}\right)^{2}\left(x_{2}-rx_{1}^{2}\right)+8\mathcal{L}(6r-1)\left(\frac{\partial\mathcal{L}}{\partial x_{1}}x_{1}+2\frac{\partial\mathcal{L}}{\partial x_{2}}x_{2}+3\frac{\partial\mathcal{L}}{\partial x_{3}}x_{3}\right)$$
$$\displaystyle\quad+\frac{4}{3}\left(\frac{\partial\mathcal{L}}{\partial x_{2}}\right)^{2}\left(x_{1}^{4}-12x_{1}^{2}x_{2}+12(1-4r)x_{2}^{2}+32x_{1}x_{3}\right)$$
$$\displaystyle\quad+\frac{4}{3}\frac{\partial\mathcal{L}}{\partial x_{2}}\frac{\partial\mathcal{L}}{\partial x_{3}}\left(x_{1}^{5}-10x_{1}^{3}x_{2}+20x_{1}^{2}x_{3}+8(5-12r)x_{2}x_{3}\right)$$
$$\displaystyle\quad+\frac{3}{8}\left(\frac{\partial\mathcal{L}}{\partial x_{3}}\right)^{2}\left(x_{1}^{6}-6x_{1}^{4}x_{2}-36x_{1}^{2}x_{2}^{2}+24x_{2}^{3}+16x_{1}(x_{1}^{2}+6x_{2})x_{3}+64(1-6r)x_{3}^{2}\right)$$
$$\displaystyle\quad+2\frac{\partial\mathcal{L}}{\partial x_{1}}\left(32\frac{\partial\mathcal{L}}{\partial x_{2}}(x_{3}-rx_{1}x_{2})+\frac{\partial\mathcal{L}}{\partial x_{3}}\left(x_{1}^{4}-12x_{1}^{2}x_{2}+12x_{2}^{2}+16(2-3r)x_{1}x_{3}\right)\right)\,.$$
(3.45)
Upon substituting the ansatz involving the expression $w$ in (3.42) into the $6d$ master flow equation (3.3), one finds that no choice of the parameters is consistent with the differential equation even at the lowest order in $\lambda$ and $\gamma$. One way to see this is to consider the $\gamma=0$ limit and note that $\mathcal{L}(\lambda=0,\gamma=0)$ is proportional to the Maxwell Lagrangian $x_{1}=\operatorname{tr}(F^{2})$. Therefore, the leading deformation from ${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ will only introduce terms involving $x_{2}$ and $x_{1}^{2}$. But the $\mathcal{O}(\lambda)$ expansion of (3.39) also includes dependence on $x_{3}$ (if $c_{3}\neq 0$), which cannot be consistent since $x_{3}$ is independent from $x_{1}$ and $x_{2}$. Therefore there is no Lagrangian of this form which satisfies a $T^{2}$ flow in six dimensions. A similar argument can be used to show that no other ansatz involving a function $w(x_{i})$ as in (3.41) can be consistent with a ${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ flow in any higher number of spacetime dimensions.
This concludes the proof that no obvious analogue of the ModMax-BI theory satisfies a ${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ flow in any dimension other than $d=4$. It is possible that such a generalized ModMax-BI Lagrangian might obey a flow equation driven by some other combination of stress tensors, such as an operator of the form $(\det T)^{p}$ for some power $p$ or a scalar built from contractions of three or more copies of $T_{\mu\nu}$. We will not undertake a general analysis of deformations driven by other stress tensor combinations here. However, if it is true that some ModMax-BI-like theory can be viewed as a deformation by some special stress tensor operator other than $O_{T^{2}}^{[r]}$, one could use this as a principle which identifies a preferred higher-dimensional analogue of the ModMax-BI action.
4 Supersymmetric ModMax-BI is a supercurrent-squared flow
Let us now turn to the $d=4$, ${\cal N}=1$ supersymmetric case. We first review the
result of [25] concerning the extension
of the flow eq. (2.2) for the supersymmetric Maxwell-Born-Infeld
theory introducing the associated supercurrent-squared operator.
We will then discuss the extension to the Born-Infeld-like ModMax theory
of [47] and show that the same
operator introduced in [25]
drives a flow equation with respect to the $\alpha^{2}$ parameter.
4.1 Review of the Bagger-Galperin Lagrangian flow and the $d=4$, ${\cal N}=1$ super-current squared operator
The supersymmetric extension of the Maxwell-Born-Infeld
Lagrangian proposed by Bagger and Galperin in [31]
is defined by the following $d=4$, ${\cal N}=1$ Lagrangian in superspace:
$$\displaystyle\mathcal{L}_{{\rm susy}-{\rm BI}}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{4}\Bigg{[}\int d^{2}\theta\,W^{2}+\int d^{2}\bar{\theta}\,\bar{W}^{2}+\int d^{2}\theta d^{2}\bar{\theta}\,\frac{\alpha^{2}\,W^{2}\bar{W}^{2}}{1-\alpha^{2}\,\mathbb{S}+\sqrt{1-2\alpha^{2}\,\mathbb{S}-\alpha^{4}\,\mathbb{P}}}\Bigg{]}~{}.~{}~{}~{}~{}~{}~{}$$
(4.1)
Here the superfields $\mathbb{S}$ and $\mathbb{P}$
are888We use the notation
$D^{2}:=D^{\alpha}D_{\alpha}$, $W^{2}:=W^{\alpha}W_{\alpha}$,
$\bar{D}^{2}:=\bar{D}_{\dot{\alpha}}\bar{D}^{\dot{\alpha}}$, and $\bar{W}^{2}:=\bar{W}_{\dot{\alpha}}\bar{W}^{\dot{\alpha}}$.
For more detail concerning our notation we refer the reader to
[25].
$$\mathbb{S}=-\frac{1}{16}(D^{2}W^{2}+\bar{{D}}^{2}\bar{W}^{2})~{},\quad\mathbb{P}=\frac{{\rm i}}{16}({D}^{2}W^{2}-\bar{{D}}^{2}\bar{W}^{2})~{},$$
(4.2)
with $W_{\alpha}$, and its conjugate $\bar{W}_{\dot{\alpha}}=(W_{\alpha})^{*}$,
being the superfield strength of a $d=4$, ${\cal N}=1$ Abelian vector multiplet obeying:
$$\displaystyle\bar{{D}}_{\dot{\beta}}W_{\alpha}=0,\quad{D}^{\alpha}W_{\alpha}=\bar{{D}}_{\dot{\alpha}}\bar{W}^{\dot{\alpha}}~{}.$$
(4.3)
In components, $W_{\alpha}$ has the following expansion in terms of the
fields describing the vector multiplet
$$\displaystyle W_{\alpha}$$
$$\displaystyle=$$
$$\displaystyle-{\rm i}\lambda_{\alpha}+\theta_{\alpha}{\sf D}-{\rm i}(\sigma^{\mu\nu}\theta)_{\alpha}F_{\mu\nu}+\theta^{2}(\sigma^{\mu}\partial_{\mu}{\bar{\lambda}})_{\alpha}~{},$$
(4.4)
where the complex spinor $\lambda_{\alpha}$ is the gaugino,
${\sf D}$ is the real auxiliary field,
and $F_{\mu\nu}=2\partial_{[\mu}v_{\nu]}$ is the field strength of an Abelian connection $v_{\mu}$.
The superfields $\mathbb{S}$ and $\mathbb{P}$
are such that their $\theta=0$ components give
$S$ and $P$ of eq. (2.6)
$$\displaystyle\mathbb{S}|_{\theta=0}=S+\frac{1}{2}{\sf D}^{2}~{},~{}~{}~{}~{}~{}~{}\mathbb{P}|_{\theta=0}=P~{},$$
(4.5)
up to a ${\sf D}=D^{\alpha}W_{\alpha}|_{\theta=0}=\bar{D}_{\dot{\alpha}}\bar{W}^{\dot{\alpha}}|_{\theta=0}$ term.
In [25] it was shown that the Bagger-Galperin
supersymmetric extension of the Maxwell-Born-Infeld theory
satisfies a flow equation driven by a supercurrent-squared operator.
More precisely, up to an on-shell condition,
that we will review and use also in the general ModMax case,
the Lagrangian (4.1) was shown
to satisfy999In [65] Cecotti and Ferrara were the first
to observe the flow at order $\alpha^{2}$ where
$\mathcal{O}_{T^{2}}=W^{2}\bar{W}^{2}+\cdots$.
$$\frac{\partial\mathcal{L}_{{\rm susy}-\rm BI}}{\partial\alpha^{2}}=\frac{1}{8}\int d^{2}\theta d^{2}\bar{\theta}\,\mathcal{O}_{T^{2}}~{},$$
(4.6)
where $\mathcal{O}_{T^{2}}$ is
$$\mathcal{O}_{T^{2}}=\frac{1}{16}{\mathcal{J}}^{\alpha\dot{\alpha}}{\mathcal{J}}_{\alpha\dot{\alpha}}-\frac{5}{8}\mathcal{X}\bar{\mathcal{X}}~{}.$$
(4.7)
The operator $\mathcal{O}_{T^{2}}$ is defined in terms of the superfields
of the Ferrara-Zumino (FZ) supercurrent multiplet [66].
The explicit form of ${\mathcal{J}}_{\alpha{\dot{\alpha}}}$, $\mathcal{X}$, and $\mathcal{O}_{T^{2}}$
for the Bagger-Galperin model (4.1) were computed
in [25] by using results of [41].
We will extend this analysis in the next subsection.
In general, the vector superfield ${\mathcal{J}}_{\alpha{\dot{\alpha}}}$
and the complex scalar superfield $\mathcal{X}$
of the FZ multiplet
satisfy the following constraints:
$$\bar{D}^{\dot{\alpha}}\mathcal{J}_{\alpha\dot{\alpha}}=D_{\alpha}\mathcal{X}~{},\qquad\bar{D}_{\dot{\alpha}}\mathcal{X}=0~{}.$$
(4.8)
These constraints lead to $12+12$ independent component fields
that appear in
${\mathcal{J}}_{\alpha{\dot{\alpha}}}$ and $\mathcal{X}$ as follows101010The complex
chiral coordinate $y^{\mu}$ is defined as
$y^{\mu}=x^{\mu}+{\rm i}\theta\sigma^{\mu}\bar{\theta}$,
while the slashed derivatives are $\not{\partial}=\sigma^{\mu}\partial_{\mu},\,\bar{\not{\partial}}=\bar{\sigma}^{\mu}\partial_{\mu}$.
$$\displaystyle\mathcal{J}_{\mu}(x,\theta,{\bar{\theta}})$$
$$\displaystyle=$$
$$\displaystyle j_{\mu}+\theta\Big{(}S_{\mu}-\frac{1}{\sqrt{2}}\sigma_{\mu}\bar{\chi}\Big{)}+\bar{\theta}\Big{(}\bar{S}_{\mu}+\frac{1}{\sqrt{2}}\bar{\sigma}_{\mu}\chi\Big{)}+\frac{{\rm i}}{2}\theta^{2}\partial_{\mu}\bar{\sf x}-\frac{{\rm i}}{2}\bar{\theta}^{2}\partial_{\mu}{\sf x}$$
(4.9a)
$$\displaystyle+\theta\sigma^{\nu}\bar{\theta}\Big{(}2T_{\mu\nu}-\frac{2}{3}\eta_{\mu\nu}\Theta-\frac{1}{2}\epsilon_{\nu\mu\rho\sigma}\partial^{\rho}j^{\sigma}\Big{)}-\frac{{\rm i}}{2}\theta^{2}\bar{\theta}\Big{(}\bar{\not{\partial}}S_{\mu}+\frac{1}{\sqrt{2}}\bar{\sigma}_{\mu}\not{\partial}\bar{\chi}\Big{)}$$
$$\displaystyle-\frac{{\rm i}}{2}\bar{\theta}^{2}\theta\Big{(}{\not{\partial}}{\bar{S}}_{\mu}-\frac{1}{\sqrt{2}}\sigma_{\mu}\bar{\not{\partial}}\chi\Big{)}+\frac{1}{2}\theta^{2}\bar{\theta}^{2}\Big{(}\partial_{\mu}\partial^{\nu}j_{\nu}-\frac{1}{2}\partial^{2}j_{\mu}\Big{)}~{},$$
$$\displaystyle\mathcal{X}(y,\theta)$$
$$\displaystyle=$$
$$\displaystyle{\sf x}(y)+\sqrt{2}\theta\chi(y)+\theta^{2}{\sf F}(y)~{},~{}~{}~{}\chi_{\alpha}=\frac{\sqrt{2}}{3}(\sigma^{\mu})_{\alpha\dot{\alpha}}\bar{S}^{\dot{\alpha}}_{\mu}~{},~{}~{}~{}{\sf F}=\frac{2}{3}\Theta+{\rm i}\partial_{\mu}j^{\mu}~{}.~{}~{}~{}~{}~{}~{}~{}~{}~{}$$
(4.9b)
The operators $T_{\mu\nu}$ and $\Theta$ are the conserved stress-energy tensor
and its trace, while $(S_{\mu}{}^{\alpha},\bar{S}_{\mu}{}_{\dot{\alpha}})$ is the conserved
$d=4$, ${\cal N}=1$ supersymmetry current. The other operators in the FZ multiplet
are required by supersymmetry. Note that, in general, $j_{\mu}$ is not a conserved
vector.
The flow equation (4.6) for the Bagger-Galperin theory
leads to a definition of a manifestly supersymmetric extension of the operator
(2.3) and an associated flow
for any $d=4$, ${\cal N}=1$ supersymmetric QFT admitting a Ferrara-Zumino
supercurrent multiplet [25].
In fact, up to total derivatives and
the (on-shell) conservation equations
(4.8), the following result holds:
$$\displaystyle\int d^{4}\theta\,\mathcal{O}_{T^{2}}$$
$$\displaystyle=$$
$$\displaystyle\Big{(}T^{\mu\nu}T_{\mu\nu}-\frac{1}{2}\Theta^{2}\Big{)}+\frac{3}{8}j_{\mu}\partial^{2}j^{\mu}+\frac{3}{8}\partial_{\mu}{\sf x}\partial^{\mu}\bar{\sf x}-\frac{{\rm i}}{2}\Big{(}S_{\mu}{\not{\partial}}{\bar{S}}^{\mu}-\frac{9}{4}\bar{\chi}\bar{\not{\partial}}\chi\Big{)}$$
(4.10)
$$\displaystyle+\text{total derivatives}+{\rm EOM}~{}.$$
The first two terms in the first
bracket are precisely the operator $O_{T^{2}}$ of
eq. (2.3), while the extra terms in
${\mathcal{O}}_{T^{2}}$ are required by $4d$, ${\cal N}=1$ supersymmetry.
4.2 The Born-Infeld-ModMax case
The Lagrangian density for a Born-Infeld-like extension of the supersymmetric
ModMax theory was derived in [47]. It takes the following form
$$\displaystyle\mathcal{L}_{{\rm susy}-\gamma{\rm BI}}$$
$$\displaystyle=$$
$$\displaystyle\frac{\cosh(\gamma)}{4}\left\{\int d^{2}\theta\,W^{2}+\int d^{2}\bar{\theta}\,\bar{W}^{2}+\int d^{2}\theta d^{2}\bar{\theta}\,W^{2}\bar{W}^{2}K(\mathbb{S},\mathbb{P})\right\}~{},~{}~{}~{}~{}~{}~{}$$
(4.11a)
with the superfield $$K(\mathbb{S},\mathbb{P})$$
given by
$$\displaystyle K(\mathbb{S},\mathbb{P})$$
$$\displaystyle=$$
$$\displaystyle\frac{t-\sqrt{t^{2}-2t\left[\cosh(\gamma)\mathbb{S}+\sinh(\gamma)\sqrt{\mathbb{S}^{2}+\mathbb{P}^{2}}\right]-\mathbb{P}^{2}}-\cosh(\gamma)\mathbb{S}}{\cosh(\gamma)(\mathbb{S}^{2}+\mathbb{P}^{2})}~{}.~{}~{}~{}~{}~{}~{}$$
(4.11b)
In the limit of $\gamma=0$ the Lagrangian (4.2) reduces to
the Bagger-Galperin Lagrangian (4.1) upon identifying $t=1/\alpha^{2}$.
The aim of this subsection is to prove that
$\mathcal{L}_{{\rm susy}-\gamma{\rm BI}}$
satisfies the flow equation
$$\frac{\partial\mathcal{L}_{{\rm susy}-\rm BI}}{\partial\alpha^{2}}=\frac{1}{8}\int d^{2}\theta d^{2}\bar{\theta}\,\mathcal{O}_{T^{2}}~{},~{}~{}~{}~{}~{}~{}\Longleftrightarrow~{}~{}~{}~{}~{}~{}\frac{\partial\mathcal{L}_{{\rm susy}-\rm BI}}{\partial t}=-\frac{1}{8t^{2}}\int d^{2}\theta d^{2}\bar{\theta}\,\mathcal{O}_{T^{2}}~{}.$$
(4.12)
A first step towards proving such a flow equation is to compute
the Ferrara-Zumino multiplet, and then the operator ${\mathcal{O}}_{T^{2}}$ of
eq. (4.7), for the $\mathcal{L}_{{\rm susy}-\rm BI}$
Lagrangian.
It is useful to rewrite the Lagrangian (4.2), and in particular
$K(\mathbb{S},\mathbb{P})$, in terms of the following superfields:
$$\displaystyle u=\frac{1}{8}{D}^{2}W^{2}~{},~{}~{}~{}~{}~{}~{}\bar{u}=\frac{1}{8}\bar{{D}}^{2}\bar{W}^{2}~{},$$
(4.13)
such that
$$\displaystyle\mathbb{S}=-\frac{1}{2}(u+\bar{u})~{},~{}~{}~{}~{}~{}~{}\mathbb{P}=\frac{{\rm i}}{2}(u-\bar{u})~{},~{}~{}~{}~{}~{}~{}\mathbb{S}^{2}+\mathbb{P}^{2}=u\bar{u}~{}.$$
(4.14)
The superfield $K$ then satisfies
$$\displaystyle K(u,\bar{u})=\frac{u+\bar{u}-\text{sech}(\gamma)\left[\sqrt{4t^{2}-8t\sinh(\gamma)\sqrt{u\bar{u}}+4t\cosh(\gamma)(u+\bar{u})+(u-\bar{u})^{2}}-2t\right]}{2u\bar{u}}~{}.$$
(4.15)
The Ferrara-Zumino supercurrent for a large class of models of the form
$$\displaystyle{\mathcal{L}}_{\Lambda}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{4}\int d^{2}\theta\,W^{2}+\frac{1}{4}\int d^{2}\bar{\theta}\,\bar{W}^{2}+\frac{1}{4}\int d^{2}\theta d^{2}\bar{\theta}\,W^{2}\bar{W}^{2}\Lambda(u,\bar{u})~{},$$
(4.16)
was computed in [41] by using superspace techniques for
$d=4$, ${\cal N}=1$ old-minimal Poincaré supergravity.111111See [67] for a
review of $4d$, ${\cal N}=1$ old-minimal supergravity in the
notation of [41].
We can readily use the results of
[41] to write
the expressions that we need for the FZ multiplet derived from
$\mathcal{L}_{{\rm susy}-\rm BI}$.
These are
$$\displaystyle\mathcal{X}$$
$$\displaystyle=$$
$$\displaystyle\frac{\cosh(\gamma)}{6}\,W^{2}\bar{D}^{2}\Big{(}\bar{W}^{2}(\Gamma+\bar{\Gamma}-K)\Big{)}~{},$$
(4.17a)
$$\displaystyle\mathcal{J}_{\alpha\dot{\alpha}}$$
$$\displaystyle=$$
$$\displaystyle\cosh(\gamma)\Bigg{\{}-2{\rm i}M_{\alpha}\bar{W}_{\dot{\alpha}}+2{\rm i}W_{\alpha}\bar{M}_{\dot{\alpha}}+\frac{1}{12}[D_{\alpha},\bar{D}_{\dot{\alpha}}]\big{(}W^{2}\bar{W}^{2}\big{)}\cdot\Big{(}\Gamma+\bar{\Gamma}-K\Big{)}\Bigg{\}}$$
(4.17b)
$$\displaystyle+W^{2}\bar{W}(\cdots)+\bar{W}^{2}W(\cdots)~{}.$$
Here $\Gamma=\Gamma(u,\bar{u})$ and $\bar{\Gamma}=\bar{\Gamma}(u,\bar{u})$
are
$$\displaystyle\Gamma(u,\bar{u})=\frac{\partial\left(uK(u,\bar{u})\right)}{\partial u}~{},~{}~{}~{}~{}~{}~{}\bar{\Gamma}(u,\bar{u})=\frac{\partial\left(\bar{u}K(u,\bar{u})\right)}{\partial\bar{u}}~{},$$
(4.18)
while the superfield $M_{\alpha}$
takes the following form:
$$\displaystyle{\rm i}M_{\alpha}$$
$$\displaystyle=$$
$$\displaystyle W_{\alpha}\Bigg{\{}1-\frac{1}{4}\bar{D}^{2}\Bigg{[}\bar{W}^{2}\Big{(}K+\frac{1}{8}D^{2}\Big{(}W^{2}\frac{\partial K}{\partial u}\Big{)}\Big{)}\Bigg{]}\Bigg{\}}~{},$$
(4.19a)
$$\displaystyle=$$
$$\displaystyle W_{\alpha}\Big{(}1-2\bar{u}\Gamma\Big{)}+W\bar{W}(\cdots)+W^{2}(\cdots)~{}.$$
(4.19b)
Note that in (4.17b) and (4.19b) the ellipsis are quite involved terms
that we avoided writing since they will not contribute
to ${\mathcal{O}}_{T^{2}}$
due to the nilpotency conditions $W_{\alpha}W_{\beta}W_{\gamma}=0$
and $\bar{W}_{\dot{\alpha}}\bar{W}_{\dot{\beta}}\bar{W}_{\dot{\gamma}}=0$.
In fact, for our purposes, we can further simplify the expressions of
${\mathcal{J}}_{\alpha{\dot{\alpha}}}$ and $\mathcal{X}$ to
$$\displaystyle\mathcal{X}$$
$$\displaystyle=$$
$$\displaystyle\frac{4\cosh(\gamma)}{3}\,W^{2}\bar{u}\Big{(}\Gamma+\bar{\Gamma}-K\Big{)}+W^{2}\bar{W}(\cdots)~{},$$
(4.20a)
$$\displaystyle\mathcal{J}_{\alpha\dot{\alpha}}$$
$$\displaystyle=$$
$$\displaystyle\cosh(\gamma)\Bigg{\{}-4W_{\alpha}\bar{W}_{\dot{\alpha}}\Big{(}1-\bar{u}\Gamma-u\bar{\Gamma}\Big{)}+\frac{1}{6}(D_{\alpha}W^{2})(\bar{D}_{\dot{\alpha}}\bar{W}^{2})\Big{(}\Gamma+\bar{\Gamma}-K\Big{)}\Bigg{\}}$$
(4.20b)
$$\displaystyle+W^{2}\bar{W}(\cdots)+\bar{W}^{2}W(\cdots)~{}.$$
Before continuing our analysis,
it is worth mentioning that
the condition for the general Lagrangian
${\mathcal{L}}_{\Lambda}$ in (4.16) to be invariant under electro-magnetic
duality transformation is
$$\displaystyle\rm{Im}\big{\{}\Gamma-\bar{u}\Gamma^{2}\big{\}}=0~{}.$$
(4.21)
The previous self-duality condition was introduced for the first time in [39, 40]
where the general theory of ${\cal N}=1$ and ${\cal N}=2$ supersymmetric nonlinear duality invariant systems was developed.
Under the condition (4.21),
the supercurrent multiplet of the theory ${\mathcal{L}}_{\Lambda}$
is also electro-magnetic duality invariant, see [41] for details.
The ${\cal N}=1$ supersymmetric ModMax theory
is a special case of the analysis
of [39, 40, 41]
where the vector multiplet Lagrangian is not required
to be analytic.
It was in fact proven in [47]
that the supersymmetric Born-Infeld-ModMax Lagrangian
$\mathcal{L}_{{\rm susy}-\gamma{\rm BI}}$,
eq. (4.2), indeed satisfies (4.21).
Exactly as in the non-supersymmetric case,
this indicates that any
deformation triggered by composite operators defined only in terms of
the supercurrent should remain electro-magnetic invariant.
As a next step towards proving (4.12),
we use equations (4.20a) and (4.20b)
for the FZ superfields
to compute $\mathcal{X}\bar{\mathcal{X}}$ and ${\mathcal{J}}^{\alpha{\dot{\alpha}}}{\mathcal{J}}_{\alpha{\dot{\alpha}}}$.
The first one can be easily computed to be
$$\displaystyle\mathcal{X}\bar{\mathcal{X}}=\frac{16\cosh^{2}(\gamma)}{9}\,W^{2}\bar{W}^{2}\,u\bar{u}(\Gamma+\bar{\Gamma}-K)^{2}~{}.$$
(4.22)
The expression for ${\mathcal{J}}^{\alpha{\dot{\alpha}}}{\mathcal{J}}_{\alpha{\dot{\alpha}}}$ is more involved.
It proves to be
$$\displaystyle{\mathcal{J}}^{\alpha\dot{\alpha}}{\mathcal{J}}_{\alpha\dot{\alpha}}$$
$$\displaystyle=$$
$$\displaystyle 16\cosh^{2}(\gamma)\,W^{2}\bar{W}^{2}\Bigg{\{}\,\Big{(}1-\bar{u}\Gamma-u\bar{\Gamma}\Big{)}^{2}+\frac{1}{9}\,u\bar{u}\Big{(}\Gamma+\bar{\Gamma}-K\Big{)}^{2}\Bigg{\}}$$
(4.23)
$$\displaystyle-\frac{4\cosh^{2}(\gamma)}{3}\,W^{2}\bar{W}^{2}(D^{\alpha}W_{\alpha})^{2}\Big{(}1-\bar{u}\Gamma-u\bar{\Gamma}\Big{)}\Big{(}\Gamma+\bar{\Gamma}-K\Big{)}~{}.$$
Note that the second line in (4.23)
is in principle problematic to prove the flow equation
(4.12). In fact, it is clear that the derivative
of $\mathcal{L}_{{\rm susy}-\rm BI}$
with respect to $\alpha^{2}$ is
$$\frac{\partial\mathcal{L}_{{\rm susy}-\rm BI}}{\partial\alpha^{2}}=-\frac{t^{2}}{4}\cosh(\gamma)\int d^{2}\theta d^{2}\bar{\theta}\,W^{2}\bar{W}^{2}\,\frac{\partial K(u,\bar{u})}{\partial t}~{},$$
(4.24)
where
$$\displaystyle\cosh(\gamma)\frac{\partial K(u,\bar{u})}{\partial t}=\frac{1}{u\bar{u}}\Bigg{\{}1-\frac{2t-2\sinh(g)\sqrt{u\bar{u}}+\cosh(\gamma)(u+\bar{u})}{\sqrt{+4t^{2}-8t\sinh(\gamma)\sqrt{u\bar{u}}+4t\cosh(\gamma)(u+\bar{u})+(u-\bar{u})^{2}}}\Bigg{\}}~{}.$$
(4.25)
Therefore, for the flow (4.12) to hold
the operator ${\mathcal{O}}_{T^{2}}$ should be a functional of
$u$ and $\bar{u}$ only.
However, the term in (4.23) that includes the $(D^{\alpha}W_{\alpha})^{2}$ factor
is incompatible with this statement.
The solution to this problem is the same as the one given in
[25]
for the $\gamma=0$ case.121212A similar problem and its solution were also described in the
analysis of flow equations of
$2d$, ${\cal N}=(1,1)$ and ${\cal N}=(2,2)$ supersymmetric
theories where
off-shell supersymmetric multiplets
include auxiliary fields
[6, 9, 25].
In fact, it is enough to prove that, for any $\gamma$, the superspace
equations of motion derived from the Lagrangian
$\mathcal{L}_{{\rm susy}-\gamma{\rm BI}}$, eq. (4.2),
imply the following relation
$$\displaystyle W^{2}\bar{W}^{2}(D^{\alpha}W_{\alpha})\equiv 0~{}.$$
(4.26)
We refer the reader to Appendix A of [25]
for a detailed discussion of this result
for a large class of models that include the Lagrangian ${\mathcal{L}}_{\Lambda}$
in eq. (4.16), and,
in particular, also ${\mathcal{L}}_{{\rm susy}-\gamma{\rm BI}}$
in eq. (4.2).
Notably, equation (4.26)
is equivalent to the fact that
the auxiliary field ${\sf D}\propto D^{\alpha}W_{\alpha}|_{\theta=0}$
satisfies an algebraic equation of motion that sets it to zero up to
terms at least linear in gaugino fields $\lambda_{\alpha}\propto W_{\alpha}|_{\theta=0}$.
For the Born-Infeld-like ModMax theory, this fact — directly related to the preservation of supersymmetry on-shell — was also discussed in [47].
Note also that equation (4.26) alone
is a weaker condition
than imposing the whole set of superfield equations of motion. In fact, imposing (4.26) can be interpreted as only
eliminating the auxiliary field ${\sf D}$ from the vector
multiplet and removing possible ambiguities
of the off-shell description of supersymmetric
Born-Infeld-like theories — see e.g.
[68, 69, 70, 71, 25] for related discussions.
Upon imposing the condition (4.26),
it is simple to show that the ${\mathcal{O}}_{T^{2}}$ operator
(4.7) takes the simple form
$$\displaystyle\mathcal{O}_{T^{2}}=\cosh^{2}(\gamma)\,W^{2}\bar{W}^{2}\Big{[}\left(1-\bar{u}\Gamma-u\bar{\Gamma}\right)^{2}-u\bar{u}\left(\Gamma+\bar{\Gamma}-K\right)^{2}\Big{]}~{},$$
(4.27)
and it is only a functional of $u$ and $\bar{u}$.
To conclude our analysis and finally show that
the flow equation (4.12) is satisfied,
it is now enough to compute explicitly (4.27) by using the
expression of $K(u,\bar{u})$ (4.15) and the definitions of
$\Gamma(u,\bar{u})$ and $\bar{\Gamma}(u,\bar{u})$ in (4.18).
A straightforward calculation shows that the right hand side of
(4.27) precisely coincides with (4.25)
up to a multiplicative factor:
$$\displaystyle\mathcal{O}_{T^{2}}=-2t^{2}\cosh(\gamma)\,\bar{W}^{2}W^{2}\,\frac{\partial K(u,\bar{u})}{\partial t}~{}.$$
(4.28)
By comparing (4.28) with (4.24)
it follows that the flow equation (4.12) is satisfied.
Remarkably, as for the bosonic case,
the structure of the supersymmetric flow equation, and its supercurrent-squared
operator, proves to be the same for any value of $\gamma$.
5 Conclusion
In this work, we have seen that the $T^{2}$ deformation of the ModMax theory is exactly the known Born-Infeld-type generalization of the ModMax theory. Much like the $\gamma=0$ case of this statement, which is the fact that the $T^{2}$ deformation of the free Maxwell theory in $d=4$ gives the usual Born-Infeld action, the flow can also be recast in $\mathcal{N}=1$ superspace, so that the supersymmetric extension of the ModMax-BI theory satisfies a supercurrent-squared flow. This ModMax-BI theory therefore belongs to a collection of other interesting theories which satisfy current-squared flows in various numbers of dimensions, such as the Dirac
action in $d=2$
and the usual Born-Infeld action in $d=4$.
There remain many open questions and directions for future research. First, there is the important conceptual question of what is “special” about theories which satisfy $T^{2}$ flows. Many of the previously studied examples of such theories, like the Dirac and Born-Infeld Lagrangians, are related to strings and branes. It would be very interesting to understand whether there was a more fundamental reason why current-squared flows generate theories of this type, and to see whether there are other examples of interesting theories that satisfy $T^{2}$ flows or related differential equations.
One hint which may prove useful in answering this question is the relationship between $T^{2}$ flows and spontaneously broken symmetries. For instance, the Dirac action which describes the scalar transverse fluctuations of a brane is uniquely fixed by the fact that a brane spontaneously breaks a fraction of the Poincaré symmetry of the ambient space in which it is embedded. The fact that the ordinary $2d$ ${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deformation generates the Dirac Lagrangian suggests that this flow has some relationship with spontaneously broken symmetries (which are then non-linearly realized). In the supersymmetric context, it is known the the Bagger-Galperin model which represents the supercurrent-squared deformation of a free super-Maxwell theory also possesses an extra non-linearly realized supersymmetry [31, 72].
It was also shown that (Volkov-Akulov) Goldstino actions
with non-linearly realised supersymmetry
satisfy ${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$-like flows in $d=2$
[11, 73, 9]
and $d=4$
[25].
One would like to sharpen these observations and perhaps understand whether a similar symmetry breaking pattern is relevant for the ModMax-BI theory.
There is a set of related questions concerning scalars (some rudimentary comments concerning ${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$-like flows for scalar theories are collected in Appendix B). For the ordinary Born-Infeld theory, it is clear that one can incorporate scalars $X^{M}$ by promoting $S_{\rm BI}$ to the Dirac-Born-Infeld action $S_{\rm DBI}$:
$$\displaystyle S_{\rm DBI}=-T_{p}\int d^{p+1}\sigma\,\sqrt{-\det(g_{\mu\nu}+\alpha F_{\mu\nu})}\,,$$
(5.1)
where $g_{\mu\nu}$ is the induced metric
$$\displaystyle g_{\mu\nu}=G_{MN}\partial_{\mu}X^{M}\partial_{\nu}X^{N}\,.$$
(5.2)
However, it is less clear how to incorporate scalars into the ModMax-BI theory. One proposal for such a generalization was presented in [74], but it is not obvious that this is the unique modification which includes scalars, nor is it clear how to interpret this Lagrangian from the perspective of string theory or a modification of brane physics. Therefore, one might ask what principle one should use in order to define a “Mod-DBI” theory – that is, a two-parameter family of theories labeled by $\gamma,\lambda$, including both a gauge field and scalars, and which reduces to the ModMax-BI theory when the scalars are set to zero. In particular, given such a family, one could instead ask what happens when the gauge sector is set to zero. The result would be a two-parameter family of “ModDirac” theories which reduces to the Dirac Lagrangian when $\gamma=0$. On the other hand, when $\lambda=0$, this would yield a new $\gamma$-deformed theory of a scalar which is analogous to the ModMax theory.
One way to probe this question about scalars would be to enhance the amount of supersymmetry. In this work, we have focused on the case of $\mathcal{N}=1$ in four dimensions and considered theories of a vector
superfield strength $W_{\alpha}$. However, with extended supersymmetry such as $\mathcal{N}=2$, the supersymmetric completion of the ModMax-BI theory includes additional fields needed to complete the multiplet
– see [61] for the ${\cal N}=2$
supersymmetric extension of ModMax. In particular, there is a scalar sector. Given a suitable $\mathcal{N}=2$ version of the supercurrent-squared deformation, one could attempt to solve the superspace flow equation and then study the dynamics of the scalars in the resulting deformed theory. This gives a potentially different proposal for incorporating scalars into the ModMax-BI theory, which is not obviously related to the proposal of [74].
The study of Volkov-Akulov-Dirac-Born-Infeld actions with extended supersymmetry in various space-time dimensions and their relationship to string theory has received attention in the past.
In particular, the standard $\gamma=0$, Born-Infeld case with extended
supersymetry has been already studied in
[75, 76, 77, 78, 79, 80, 81, 82, 83, 39, 84, 85, 86, 87, 40, 88, 89, 90, 91].
These works might be a starting point to look for ${\cal N}=2$,
ModMax-BI deformations.
Even without scalars, there are at least other two ways in which one could try to generalize these observations relating ModMax-BI to $T^{2}$ flows.
1.
One way is to look for theories of $p$-form field strengths for $p>2$ which satisfy an appropriate flow equation. It was pointed out in [29] that the $T^{2}$ deformation of a free $3$-form field strength in six dimensions, whose undeformed Lagrangian is proportional to $F_{\mu\nu\rho}F^{\mu\nu\rho}$, does not give a duality-invariant $6d$ analogue of the Born-Infeld theory. However, one could ask whether there is any choice of form rank $p$, dimension $d$, and coefficient $r$ in the operator $O_{T^{2}}^{[r]}$
in eq. (2.4) for which the deformation of a free $p$-form yields an interesting theory (for instance, with a square root structure). It would also be interesting to consider deformations involving chiral $p$-forms. Since the ModMax theory lifts to a $6d$ PST-like theory of a chiral tensor [46], it is natural to wonder about the relationship between $T^{2}$ and theories of this kind.
2.
Another direction for generalization is to consider non-Abelian gauge theories.131313Another interesting, but very different, connection between ${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ and non-Abelian gauge theory involves a $4d$ version of Chern-Simons as in [92]. The formalism developed in Section 3 does not apply in the non-Abelian case because, for each fixed spacetime trace structure $x_{j}$, there can be multiple inequivalent ways to perform the traces over gauge indices. An analogue of the master flow equation has not yet been written down in the non-Abelian case, but it is known that the ${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deformation of free Yang-Mills does not agree with the non-Abelian DBI action in any number of spacetime dimensions. For instance, in $d=2$ the solution of the ${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ for free Yang-Mills coupled to scalars was discussed in [93]. Even though these deformed theories are no longer related to Born-Infeld in the non-Abelian case, they may still be interesting theories in their own right. It might therefore be worthwhile to consider the behavior of a non-Abelian ModMax-type theory under $T^{2}$ flows.
A final puzzle, which we have already alluded to before, is the brane interpretation of the ModMax family of theories. For instance, if the ModMax-BI theory exists at the quantum level, then it should have had a string theoretic interpretation as some deformation of the usual Born-Infeld theory on a brane. What deformation does this correspond to? Is there some brane configuration, perhaps with additional fluxes turned on or other string theory ingredients, which would engineer ModMax-BI in the sense that the brane would have a ModMax-BI theory living on its worldvolume?
To our knowledge, stringy constructions of ModMax
have not appeared yet.
We leave this and the preceding
interesting questions to future work.
Acknowledgements
We are grateful to Hongliang Jiang, Sergei Kuzenko, Emmanouil Raptakis, Savdeep Sethi, and Dmitri Sorokin
for discussions and correspondence related to this work. We also thank Stephen Ebert, Hao-Yu Sun, and Zhengdi Sun for comments on an early draft of this manuscript.
C.F. is supported by U.S. Department of Energy grant DE-SC0009999
and by funds from the University of California.
L.S. is supported by a postgraduate scholarship at the University of Queensland.
The work of G.T.-M. is supported by the Australian Research Council (ARC)
Future Fellowship FT180100353,
and by the Capacity Building Package of the University of Queensland.
Appendix A Proof of Determinant Condition
The goal of this Appendix is to prove that the stress-energy
tensor $T_{\mu\nu}$ for any theory of an Abelian gauge field in four spacetime dimensions satisfies
$$\displaystyle\sqrt{\det\left(T\right)}=\frac{1}{4}\left(\frac{1}{2}\left(\operatorname{tr}(T)\right)^{2}-\operatorname{tr}\left(T^{2}\right)\right)\,.$$
(A.1)
To see this, we first recall from Section 3.1 that a general Lagrangian for an Abelian field strength $F_{\mu\nu}$ in four dimensions can be written as $\mathcal{L}(x_{1},x_{2})$ in terms of the two independent scalars $x_{1},x_{2}$, and that the associated stress-energy tensor is (in Euclidean signature)
$$\displaystyle T_{\mu\nu}=\delta_{\mu\nu}\mathcal{L}-4\frac{\partial\mathcal{L}}{\partial x_{1}}F^{2}_{\mu\nu}-8\frac{\partial\mathcal{L}}{\partial x_{2}}F^{4}_{\mu\nu}\,.$$
(A.2)
At each fixed spacetime location $x$, the stress-energy tensor can therefore be written in components as a $4\times 4$ matrix of the form
$$\displaystyle T=c_{0}\mathbb{I}_{4}+c_{1}F^{2}+c_{2}F^{4}\,,$$
(A.3)
where $\mathbb{I}_{4}$ is the identity matrix and the $c_{i}$ are numbers. We claim that any matrix of the form (A.3), where $F$ is antisymmetric, satisfies (A.1). If the eigenvalues of the antisymmetric matrix $F$ are $\lambda_{i}$, then the eigenvalues of $T$ are $\hat{\lambda}_{i}=c_{0}+c_{1}\lambda_{i}^{2}+c_{2}\lambda_{i}^{4}$, for $i=1,\cdots,4$. The eigenvalues of an antisymmetric matrix are purely imaginary and come in complex conjugate pairs, so we can take $\lambda_{3}=\lambda_{1}^{\ast}=-\lambda_{1}$ and $\lambda_{4}=\lambda_{2}^{\ast}=-\lambda_{2}$. It follows that $\hat{\lambda}_{3}=\hat{\lambda}_{1}$ and $\hat{\lambda}_{4}=\hat{\lambda}_{2}$.
Hence, it holds
$$\displaystyle\sqrt{\det(T)}$$
$$\displaystyle=\sqrt{\hat{\lambda}_{1}^{2}\hat{\lambda}_{2}^{2}}=\pm\hat{\lambda}_{1}\hat{\lambda}_{2}\,.$$
(A.4)
We can take the positive sign on the right side of (A.4) if the stress-energy tensor $T$ is positive definite. On the other hand,
$$\displaystyle\frac{1}{2}\left(\operatorname{tr}\left(T\right)\right)^{2}-\operatorname{tr}\left(T^{2}\right)$$
$$\displaystyle=\frac{1}{2}\left(2\hat{\lambda}_{1}+2\hat{\lambda}_{2}\right)^{2}-\left(2\hat{\lambda}_{1}^{2}+2\hat{\lambda}_{2}^{2}\right)$$
$$\displaystyle=4\hat{\lambda}_{1}\hat{\lambda}_{2}\,.$$
(A.5)
Therefore, assuming $T$ is positive definite so that its determinant is positive, one has
$$\displaystyle\sqrt{\det\left(T\right)}=\frac{1}{4}\left(\frac{1}{2}\left(\operatorname{tr}(T)\right)^{2}-\operatorname{tr}\left(T^{2}\right)\right)\,,$$
(A.6)
for the stress-energy tensor $T_{\mu\nu}$ of a general Abelian gauge theory in four spacetime dimensions.141414Since $\det(T)$ also satisfies equation (3.16), which applies to any $4\times 4$ matrix, one could eliminate $\det(T)$ and express this condition as the vanishing of a particular combination of traces of powers of $T$.
A similar result holds in any even number $d=2k$ of spacetime dimensions.151515In the case of odd spacetime dimension, one must account for the fact that the field strength $F_{\mu\nu}$ has an unpaired zero eigenvalue but the others come in complex-conjugate pairs. The stress-energy tensor (3.10) for an Abelian gauge theory in any dimension takes the form
$$\displaystyle T=c_{0}\mathbb{I}_{4}+\sum_{i=1}^{d}c_{i}F^{2i}\,,$$
(A.7)
and if the eigenvalues of the antisymmetric $d\times d$ matrix $F$ are denoted $\lambda_{i}$, then the eigenvalues of $T$ are
$$\displaystyle\hat{\lambda}_{i}$$
$$\displaystyle=c_{0}+\sum_{i=1}^{k}c_{i}\lambda_{i}^{2i}\,.$$
(A.8)
Since the eigenvalues again come in complex conjugate pairs, we can choose the first $k$ eigenvalues to be independent and then impose $\lambda_{k+1}=\lambda_{1}^{\ast}=-\lambda_{1}$, $\cdots$, $\lambda_{d}=\lambda_{k}^{\ast}=-\lambda_{k}$. This means that
$$\displaystyle\hat{\lambda}_{k+1}=\hat{\lambda}_{1}\,,\cdots\,,\hat{\lambda}_{d}=\hat{\lambda}_{k}\,,$$
(A.9)
and if the stress-energy tensor is positive definite, one then has
$$\displaystyle\sqrt{\det(T)}=\sqrt{\hat{\lambda}_{1}^{2}\cdots\hat{\lambda}_{k}^{2}}=\hat{\lambda}_{1}\cdots\hat{\lambda}_{k}\,.$$
(A.10)
It is an elementary result in the theory of symmetric polynomials, which follows from Newton’s identities, that the symmetric polynomial $\hat{\lambda}_{1}\cdots\hat{\lambda}_{k}$ can be expressed in terms of power sums of the form $\hat{\lambda}_{1}^{j}+\cdots+\hat{\lambda}_{k}^{j}$, which in turn means that (A.10) can be expressed in terms of traces of powers of the matrix $T$. Explicitly, one has
$$\displaystyle\sqrt{\det(T)}=(-1)^{k}\sum_{\{m_{j}\}}\left[\prod_{j=1}^{k}\frac{1}{m_{j}!j^{m_{j}}}\left(-\frac{1}{2}\operatorname{tr}(T^{j})\right)^{m_{j}}\right]\,,$$
(A.11)
where the sum runs over all collections $\{m_{j}\}$ of non-negative integers which satisfy the constraint $m_{1}+2m_{2}+\cdots+km_{k}=k$. For instance, in the case of a $6$-dimensional Abelian gauge theory, one finds
$$\displaystyle\sqrt{\det(T)}=\frac{1}{6}\left(\left(\operatorname{tr}(T)\right)^{3}-3\operatorname{tr}(T^{2})\operatorname{tr}(T)+2\operatorname{tr}(T^{3})\right)\,.$$
(A.12)
However, beyond $d=4$ we see that such combinations are not related to bilinears in stress-energy tensors but rather products involving three or more stress-energy tensor factors. Therefore there does not appear to be any relationship between the combination $\sqrt{\det(T)}$ and any analogue of the $O_{T^{2}}^{[r]}$ operator for deformations of gauge theories in $d>4$.
Finally, there has been some interest [64, 11] in ${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$-like deformations in higher dimension which involve other powers of the determinant of the stress-energy tensor, such as $\left[\det(T)\right]^{1/(d-1)}$. From the analysis of this Appendix, we see that such an operator can never be written in terms of traces of integer powers of the stress-energy tensor when the exponent is smaller than $1/2$, at least in the case of a stress-energy tensor for an Abelian gauge theory. This is because $\left[\det(T)\right]^{1/N}$ will involve fractional powers of the eigenvalues $\hat{\lambda}_{i}$ whenever $N>2$, whereas polynomials in traces of integer powers of $T$ can produce only integer powers of the $\hat{\lambda}_{i}$.
Appendix B General $T^{2}$ Flows for Scalar Theories
In the body of this paper, we have focused on theories whose only physical degree of freedom is an Abelian gauge field, such as the ModMax theory and its ModMax-BI extension. However, as was pointed out in the concluding comments of Section 5, it is natural to wonder about families of theories that involve a gauge field coupled to scalars – such as the Dirac-Born-Infeld (DBI) action – and how such theories interact with $T^{2}$ flows. The aim of this Appendix is to make some preliminary observations in this direction.
Any theory of a gauge field coupled to scalars must, of course, reduce to a pure gauge theory when the scalar sector is turned off, and must reduce to a scalar theory when the field strength is set to zero. Therefore, if such a coupled theory is driven by a $T^{2}$ flow, then as a consistency check we know that the pure gauge sector must satisfy a flow equation of the form developed in Section 3 for general gauge theories. A second consistency check is provided by the requirement that the coupled theory satisfy a version of the master flow equation for scalar fields when the gauge sector is turned off. As a first step towards understanding flows for coupled theories, one would like to repeat the general analysis of Section 3 in the case of a scalar field to obtain a second boundary condition for coupled flows. In this Appendix we will complete such a first step.
For simplicity, we will restrict our attention to theories of a single scalar field $\phi$. We first make some general comments which apply in any spacetime dimension $d$.
B.1 Master Flow Equation for Scalars
In this subsection, we would like to obtain a general flow equation for a Lagrangian $\mathcal{L}(\phi)$ for a single scalar field $\phi$ deformed by some Lorentz scalar constructed from the stress tensor $T_{\mu\nu}$. Our discussion will parallel the derivation of the master flow equation for theories involving a gauge field in Section 3.1, although the scalar case is considerably simpler, which will motivate us to consider more general deformations.
We first note that any Lorentz invariant scalar that can be constructed from $\phi$ with one derivative per field is a function of the combination
$$\displaystyle x=\partial^{\mu}\phi\partial_{\mu}\phi\,.$$
(B.1)
Thus a general Lagrangian for a theory of a single scalar field can be written as $\mathcal{L}=\mathcal{L}(x)$. The Hilbert stress tensor corresponding to this Lagrangian is
$$\displaystyle T_{\mu\nu}$$
$$\displaystyle=\delta_{\mu\nu}\mathcal{L}-2\frac{\partial L}{\partial x}\cdot\frac{\delta x}{\delta g^{\mu\nu}}\Big{|}_{g=\delta}$$
$$\displaystyle=\delta_{\mu\nu}\mathcal{L}-2\frac{\partial\mathcal{L}}{\partial x}\cdot\partial_{\mu}\phi\partial_{\nu}\phi\,.$$
(B.2)
First we will describe the independent scalars that can be constructed from this stress tensor. The determinant is especially simple, since in a component basis at a particular spacetime point $p$, $T_{\mu\nu}(p)$ is written as a linear combination of the identity matrix and the matrix $M_{\mu\nu}=\partial_{\mu}\phi\partial_{\nu}\phi$. Because $M_{\mu\nu}$ is the outer product of a vector with itself, it is rank one and has only a single non-zero eigenvalue. To make this very explicit, if $\bm{v}$ is the vector obtained by writing the components of $\partial_{\mu}\phi$ in a given basis at a fixed spacetime location $p$, then
$$\displaystyle M=\bm{v}\otimes\bm{v}\,.$$
(B.3)
It is an elementary fact from linear algebra that a $d\times d$ matrix $M$ which can be written as the outer product $\bm{v}\otimes\bm{v}$ has one eigenvalue equal to $|\bm{v}|^{2}$ and $d-1$ eigenvalues equal to zero. This will allow us to easily evaluate the eigenvalues of the matrix $T$, which can be written in components at a fixed point $p$ as
$$\displaystyle T=c_{0}\mathbb{I}+c_{1}\bm{v}\otimes\bm{v}\,.$$
(B.4)
Here $c_{0}$ and $c_{1}$ are numbers which depend on the value of the Lagrangian and its derivatives at the point $p$, but which can be treated as constants for this local analysis. Owing to the outer product structure of the second term, the matrix $T$ has $d-1$ eigenvalues equal to $c_{0}$ and a single eigenvalue equal to $c_{0}+c_{1}x$, where $x=|\bm{v}|^{2}$ is the value of $\partial^{\mu}\phi\partial_{\mu}\phi$ at the point $p$. This means that, in an arbitrary number $d$ of spacetime dimensions, the determinant of $T$ is
$$\displaystyle\det(T)=\mathcal{L}^{d-1}\cdot\left(\mathcal{L}-2x\frac{\partial\mathcal{L}}{\partial x}\right)\,.$$
(B.5)
The other scalars that can be constructed from the stress tensor are traces of the form $y_{k}=\operatorname{tr}(T^{k})$. Using our result for the eigenvalues of $T$ above, it is easy to write down a general formula for such traces:
$$\displaystyle y_{k}=\operatorname{tr}(T^{k})$$
$$\displaystyle=(d-1)\mathcal{L}^{k}+\left(\mathcal{L}-2x\frac{\partial\mathcal{L}}{\partial x}\right)^{k}\,.$$
(B.6)
From the Cayley-Hamilton theorem, we know that the determinant $\det(T)$ can be written as a polynomial in the traces $y_{k}$. Furthermore, any trace $y_{k}$ for $k>d$ also satisfies a constraint which relates it to the lower traces $y_{j}$ for $j=1,\cdots,k$. Thus a general flow equation for the Lagrangian driven by a deforming operator $O$ which is a scalar built from the stress tensor $T_{\mu\nu}$ is
$$\displaystyle\frac{\partial\mathcal{L}}{\partial\lambda}=O(y_{1},\cdots,y_{d})\,.$$
(B.7)
For instance, the main operator of interest in this manuscript has been $O_{T^{2}}^{[r]}$, which can be written as
$$\displaystyle O_{T^{2}}^{[r]}(y_{1},y_{2})=y_{2}-ry_{1}^{2}\,.$$
(B.8)
We can use the expressions (B.6) for the $y_{k}$ to write a master flow equations for scalar theories in $d$ dimensions. One has
$$\displaystyle y_{1}=\operatorname{tr}(T)=\mathcal{L}d-2x\frac{\partial\mathcal{L}}{\partial x}\,,\qquad y_{2}=\operatorname{tr}(T^{2})=\mathcal{L}^{2}d-4\mathcal{L}x\frac{\partial\mathcal{L}}{\partial x}+4x^{2}\left(\frac{\partial\mathcal{L}}{\partial x}\right)^{2}\,.$$
(B.9)
Therefore, a general flow driven by the operator of $\mathcal{O}_{T^{2}}^{[r]}$ of equation (B.8) is described by the differential equation
$$\displaystyle\frac{\partial\mathcal{L}}{\partial\lambda}=d(1-rd)\mathcal{L}^{2}+4(rd-1)\mathcal{L}x\frac{\partial\mathcal{L}}{\partial x}+4(1-r)x^{2}\left(\frac{\partial\mathcal{L}}{\partial x}\right)^{2}\,.$$
(B.10)
This is the analogue of the master flow equation (3.12), but now for theories involving a single scalar rather than a gauge field. However, we note that in $d>2$ there are more scalar invariants associated with the stress tensor and one may therefore consider more general deforming operators. For instance, we can obtain another operator with the same mass dimension as $O_{T^{2}}^{[r]}$ by taking a square root of traces involving products of four stress tensors. We adopt the notation $O_{\sqrt{T^{4}}}^{[r_{i}]}$ for such an operator, which depends on three coefficients $r_{1},r_{2},r_{3}$ as
$$\displaystyle O_{\sqrt{T^{4}}}^{[r_{i}]}(y_{1},\cdots,y_{4})=\sqrt{y_{1}^{4}+r_{1}y_{1}^{2}y_{2}+r_{2}y_{2}^{2}+r_{3}y_{4}}\,.$$
(B.11)
We will see below that, in $d=4$ spacetime dimensions, deforming the Lagrangian for a scalar theory by $\sqrt{\det(T)}$ is classically equivalent to deforming by an operator $O_{\sqrt{T^{4}}}^{[r_{i}]}$ for some choice of the constants $r_{i}$.
Before returning to the study of these deformations by more general $O(y_{1},\cdots,y_{d})$ operators, we will first undertake an analysis of flows by the usual $O_{T^{2}}^{[r]}$ for scalar theories in arbitrary dimension.
B.2 General Analysis of $O_{T^{2}}^{[r]}$ Flows
Here we will focus on the master flow equation (B.10) for scalar field theories deformed by $O_{T^{2}}^{[r]}$. Since it is known that this flow equation has a solution of Nambu-Goto type in $d=2$, one might ask whether there are other solutions involving such a square root structure in $d>2$. This is the scalar analogue of the question of whether the Born-Infeld action (or its ModMax-BI extension) emerges as a $T^{2}$ flow in any dimension other than $d=4$, to which we have seen that the answer is no.
We first note that, on dimensional grounds, any Lagrangian which depends only on $\lambda$ and $x$ can be written as
$$\displaystyle\mathcal{L}(\lambda,x)=\frac{1}{\lambda}f(\lambda x)\,,$$
(B.12)
where we define $\xi=\lambda x$ as the dimensionless argument of the function $f$. This parameterization reduces the partial differential equation (B.10) to an ordinary differential equation for $f(\xi)$, namely
$$\displaystyle 4(r-1)\xi^{2}\left(f^{\prime}(\xi)\right)^{2}+\xi f^{\prime}(\xi)-\left(1+4(rd-1)\xi f^{\prime}(\xi)\right)f(\xi)+d(rd-1)\left(f(\xi)\right)^{2}=0\,.$$
(B.13)
This is a quadratic equation in the quantity $f^{\prime}(\xi)$ which can be solved to give
$$\displaystyle f^{\prime}(\xi)=\frac{1}{8\xi(r-1)}\left(4(rd-1)f(\xi)+\sqrt{8f(\xi)(2(d-1)(rd-1)f(\xi)-rd+2r-1)+1}-1\right)\,,$$
(B.14)
where we have chosen the root consistent with $f^{\prime}(\xi)$ being finite as $\xi\to 0$ with $f(0)=0$. One can separate this differential equation as
$$\displaystyle\int_{f(\xi_{0})}^{f(\xi)}\frac{df}{4\left(rd-1\right)f+\sqrt{8f(2(d-1)(rd-1)f-rd+2r-1)+1}-1}=\left[\frac{\log\left(\xi^{\prime}\right)}{8(r-1)}\right]_{\xi^{\prime}=\xi_{0}}^{\xi^{\prime}=\xi}\,.$$
(B.15)
The integral on the left side of (B.15) is quite involved but can be evaluated in closed form in terms of inverse hyperbolic trigonometric functions using Mathematica. The result is not especially illuminating so we do not show it here; we include this integral expression for the solution only to emphasize that, for general $r$ and $d$, the solution for a scalar deformed by $O_{T^{2}}^{[r]}$ is a fairly complicated implicitly defined function which is structurally similar to the result of ${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deforming $2d$ Yang-Mills theory coupled to scalars [93] or deforming $2d$ Born-Infeld theory [30].
The implicit expression (B.15) simplifies for particular choices of the coefficient $r$. For instance, when $r=1$, the quadratic equation for $f^{\prime}(\xi)$ has the much simpler solution
$$\displaystyle f^{\prime}(\xi)=\frac{f(\xi)\left(-1+(d-1)f(\xi)d\right)}{\xi(-1+4(d-1))f(\xi)}\,,$$
(B.16)
which can be integrated to yield the implicit equation
$$\displaystyle\log(f)+\frac{4-d}{d}\log\left(1+f(d-d^{2})\right)=\log(\xi)\,.$$
(B.17)
This equation is transcendental for generic $d$, but when $d=2$, we see that the left side collapses to $\log(f)+\log(1-2f)$ and the solution is
$$\displaystyle f(\xi)=\frac{1}{4}\left(1-\sqrt{1-8\xi}\right)\,.$$
(B.18)
This is the familar result that the ordinary ${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deformation applied to a free scalar seed theory in $d=2$ yields the Nambu-Goto Lagrangian.
Another choice for which the implicit expression simplifies is $r=\frac{1}{d}$, which gives
$$\displaystyle f^{\prime}(\xi)=\frac{d-\sqrt{d^{3}-16d^{2}(d-1)f(\xi)}}{8(d-1)\xi}\,.$$
(B.19)
This differential equation has a solution which can be written in terms of the product logarithm (Lambert $W$ function), but no solution of square-root type.
Given the complexity of the general implicit solution (B.15), and the observation that miraculous simplifications were needed in order to obtain the Nambu-Goto action as a solution with $d=2$ and $r=1$, one might suspect that this is the only choice of the parameters $r,d$ for which a square-root solution exists. This is easy to verify; we first make an ansatz of the form
$$\displaystyle f(\xi)=\frac{1}{a}\left(1-\sqrt{1-2a\xi}\right)\,,$$
(B.20)
where $a$ is some constant. Substituting this ansatz into the ordinary differential equation (B.13) and expanding to second order in $\xi$ yields the constraint
$$\displaystyle a=-8r-2rd^{2}+d(2+8r)\,.$$
(B.21)
Using this value of $a$ in the differential equation and expanding to third order in $\xi$ gives the condition
$$\displaystyle(-2+d)(-4r+d^{3}r^{2}-2d^{2}r(1+2r)+d(1+2r)^{2})\,.$$
(B.22)
This equation is satisfied if either $d=2$ or if $r$ takes one of the values
$$\displaystyle r=\frac{1}{d}\,,\qquad r=\frac{d}{(2-d)^{2}}\,.$$
(B.23)
We handle each of these cases separately.
1.
$d=2$. In this case, plugging the results for $a$ and $d$ back into the flow equation gives an equation which is satisfied if and only if $r=1$. This reduces to the known case.
2.
$r=\frac{1}{d}$. Substitution into (B.13) and expansion to order $\xi^{4}$ yields the constraint $d=1$. We reject this since we are interested in ${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deformations of field theories ($d\geq 2$) rather than quantum mechanics; in one spacetime dimension, the expression $\mathcal{O}_{T^{2}}^{[r]}$ trivializes because the only component of the “stress tensor” is the Hamiltonian.161616Although we will not consider this case in the present work, see [94, 95, 96] for observations on ${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$-like deformations in $(0+1)$-dimensional systems.
3.
$r=\frac{d}{(2-d)^{2}}$. Replacement of $r$ with this value in the flow equation then gives two constraints: $d^{2}-4d+4=0$ and $d^{2}-5d+4=0$. The first condition requires $d=2$ but the second requires either $d=1$ or $d=4$. Thus the two equations cannot be simultaneously satisfied and this choice is inconsistent.
This completes our check that a deformation of a free scalar theory by $O_{T^{2}}^{[r]}$ only produces the Nambu-Goto Lagrangian as a solution in the single case $d=2,r=1$.
Given this conclusion, one is tempted to consider deformations by other Lorentz scalars constructed from the stress tensor. For instance, in $d>2$ dimensions one can deform the Lagrangian by a power of the determinant of the stress tensor, or by some function of the higher independent trace structures $y_{k}$ defined in (B.6). We next turn to an investigation of some other deformations of this type in $d=4$.
B.3 Other Stress Tensor Flows in $d=4$
Another deformation constructed from $T_{\mu\nu}$ is $\sqrt{\det(T)}$. We note that this combination agreed with $O_{T^{2}}^{[r]}$ in the case of $4d$ gauge theory (up to overall scaling), but the two objects disagree for a scalar. In particular, it is no longer true that the eigenvalues of $T_{\mu\nu}$ come in pairs of equal $\hat{\lambda}_{i}$ since the symmetric tensor $\partial_{\mu}\phi\partial_{\nu}\phi$ cannot be written as the square of an antisymmetric tensor in the way that $F^{2}_{\mu\nu}$ could. Rather, in $d=4$, the determinant $\det(T)$ is instead equal to $\mathcal{L}^{3}\cdot\left(\mathcal{L}-2x\frac{\partial\mathcal{L}}{\partial x}\right)$ as we saw in (B.5).
The combination $\sqrt{\det(T)}$ is one member of the general class of deformations (B.7) by some function of the traces $y_{i}=\operatorname{tr}(T^{i})$. In particular, since any $4\times 4$ matrix satisfies (3.16) regardless of its symmetry properties, we have
$$\displaystyle\det(T)=\frac{1}{24}\left(\left(\operatorname{tr}T\right)^{4}-6\operatorname{tr}(T^{2})\left(\operatorname{tr}T\right)^{2}+3\left(\operatorname{tr}T^{2}\right)^{2}+8\operatorname{tr}(T)\operatorname{tr}(T^{3})-6\operatorname{tr}(T^{4})\right)$$
(B.24)
To construct other deformations from the $y_{i}$, it will be convenient to record explicit expressions for these traces:
$$\displaystyle\operatorname{tr}(T)$$
$$\displaystyle=4\mathcal{L}-2x\frac{\partial\mathcal{L}}{\partial x}\,,$$
$$\displaystyle\operatorname{tr}(T^{2})$$
$$\displaystyle=4\mathcal{L}^{2}-4x\mathcal{L}\frac{\partial\mathcal{L}}{\partial x}+4x^{2}\left(\frac{\partial\mathcal{L}}{\partial x}\right)^{2}\,,$$
$$\displaystyle\operatorname{tr}(T^{3})$$
$$\displaystyle=4\mathcal{L}^{3}-6x\mathcal{L}^{2}\frac{\partial\mathcal{L}}{\partial x}+12x^{2}\mathcal{L}\left(\frac{\partial\mathcal{L}}{\partial x}\right)^{2}-8x^{3}\left(\frac{\partial\mathcal{L}}{\partial x}\right)^{3}\,,$$
$$\displaystyle\operatorname{tr}(T^{4})$$
$$\displaystyle=4\mathcal{L}^{4}-8x\mathcal{L}^{3}\frac{\partial\mathcal{L}}{\partial x}+24x^{2}\mathcal{L}^{2}\left(\frac{\partial\mathcal{L}}{\partial x}\right)^{2}-32x^{3}\mathcal{L}\left(\frac{\partial\mathcal{L}}{\partial x}\right)^{3}+16x^{4}\left(\frac{\partial\mathcal{L}}{\partial x}\right)^{4}\,.$$
(B.25)
As a check, plugging these trace expressions into (B.24) gives
$$\displaystyle\det(T)=\mathcal{L}^{4}-2x\mathcal{L}^{3}\frac{\partial\mathcal{L}}{\partial x}\,.$$
(B.26)
which matches (B.5). Therefore, a flow equation of the form
$$\displaystyle\frac{\partial\mathcal{L}}{\partial\lambda}=\sqrt{\det(T^{(\lambda})}$$
(B.27)
is equivalent to the differential equation
$$\displaystyle\frac{\partial\mathcal{L}}{\partial\lambda}=\sqrt{\mathcal{L}^{4}-2x\mathcal{L}^{3}\frac{\partial\mathcal{L}}{\partial x}}\,.$$
(B.28)
We also see that (B.28) is one example of the class of deformations driven by the operators $O_{\sqrt{T^{4}}}^{[r_{i}]}$ defined in (B.11), where the coefficients $r_{i}$ are determined by (B.24).
As written, this flow equation is unsuitable because the argument of the square root need not be positive. For instance, consider the leading order correction in $\lambda$ around a free theory of the form $\mathcal{L}_{0}=cx$ where $x$ is some constant. Then the operator on the right side of (B.28) is
$$\displaystyle\sqrt{\mathcal{L}_{0}^{4}-2x\mathcal{L}^{3}_{0}\frac{\partial\mathcal{L}_{0}}{\partial x}}=\sqrt{-c^{4}x^{4}}\,.$$
(B.29)
This is always a purely imaginary correction for any real value of the constant $c$ and the kinetic term $x=\partial^{\mu}\phi\partial_{\mu}\phi$. To obtain a real deformation at leading order, one should instead consider the flow
$$\displaystyle\frac{\partial\mathcal{L}}{\partial\lambda}$$
$$\displaystyle=\sqrt{-\det(T^{(\lambda})}$$
$$\displaystyle=\sqrt{2x\mathcal{L}^{3}\frac{\partial\mathcal{L}}{\partial x}-\mathcal{L}^{4}}\,.$$
(B.30)
This differential equation has the solution
$$\displaystyle\mathcal{L}(\lambda,x)=\frac{x}{\sqrt{1-2\lambda x}}\,.$$
(B.31)
Solutions of this form for flow equations driven by a power of $\det(T)$ were obtained in [11] using a different strategy.
One could ask whether there is a flow by some other $O(y_{i})$ that reproduces the usual Dirac action. Consider the combination
$$\displaystyle O_{T^{4}}(y_{i})$$
$$\displaystyle=(\operatorname{tr}T)^{4}-\frac{1}{3}\operatorname{tr}(T^{2})(\operatorname{tr}T)^{2}+\frac{1}{3}(\operatorname{tr}T^{2})^{2}-\operatorname{tr}(T^{4})\,$$
$$\displaystyle=y_{1}^{4}-\frac{1}{3}y_{1}^{2}y_{2}+\frac{1}{3}y_{2}^{2}-y_{4}\,.$$
(B.32)
The square root of this object again drives a flow by an operator of the form $O_{\sqrt{T^{4}}}^{[r_{i}]}$, but with a different choice of the coefficients $r_{i}$ than the one which gives $\sqrt{\det(T)}$. Plugging in the expressions (B.3) for the traces gives
$$\displaystyle O_{T^{4}}=\left(\mathcal{L}^{2}-2\mathcal{L}x\frac{\partial\mathcal{L}}{\partial x}\right)^{2}$$
(B.33)
and therefore the flow equation
$$\displaystyle\frac{\partial\mathcal{L}}{\partial\lambda}=\sqrt{O_{T^{4}}}\,,$$
(B.34)
where we take the positive root, gives
$$\displaystyle\frac{\partial\mathcal{L}}{\partial\lambda}=\mathcal{L}^{2}-2\mathcal{L}x\frac{\partial\mathcal{L}}{\partial x}\,.$$
(B.35)
This differential equation has the solution
$$\displaystyle\mathcal{L}(\lambda,x)=\frac{1}{2\lambda}\left(1-\sqrt{1+4\lambda x}\right)\,.$$
(B.36)
Therefore, it is possible to obtain the Dirac Lagrangian as the solution to a stress tensor flow in $d=4$, although one must use a different deformation $O_{\sqrt{T^{4}}}^{[r_{i}]}$ with a special choice of coefficients $r_{i}$, and it is not clear how one would motivate this particular combination.
References
[1]
A. B. Zamolodchikov, “Expectation value of composite field T anti-T in
two-dimensional quantum field theory,”
hep-th/0401146.
[2]
F. A. Smirnov and A. B. Zamolodchikov, “On space of integrable quantum field
theories,” Nucl. Phys. B915 (2017) 363–383,
1608.05499.
[3]
A. Cavaglià, S. Negro, I. M. Szécsényi, and R. Tateo, “$T\bar{T}$-deformed 2D Quantum Field Theories,” JHEP 10 (2016)
112,
1608.05534.
[4]
S. Dubovsky, V. Gorbenko, and M. Mirbabayi, “Natural Tuning: Towards A Proof
of Concept,” JHEP 09 (2013) 045,
1305.6939.
[5]
S. Dubovsky, V. Gorbenko, and M. Mirbabayi, “Asymptotic fragility, near
AdS${}_{2}$ holography and $T\overline{T}$,” JHEP 09 (2017)
136,
1706.06604.
[6]
M. Baggio, A. Sfondrini, G. Tartaglino-Mazzucchelli, and H. Walsh, “On $T\overline{T}$ deformations and supersymmetry,” JHEP 06 (2019)
063,
1811.00533.
[7]
C.-K. Chang, C. Ferko, and S. Sethi, “Supersymmetry and $T\overline{T}$
deformations,” JHEP 04 (2019) 131,
1811.01895.
[8]
H. Jiang, A. Sfondrini, and G. Tartaglino-Mazzucchelli, “$T\bar{T}$
deformations with $\mathcal{N}=(0,2)$ supersymmetry,” Phys. Rev. D100 (2019), no. 4, 046017,
1904.04760.
[9]
C.-K. Chang, C. Ferko, S. Sethi, A. Sfondrini, and G. Tartaglino-Mazzucchelli,
“$T\bar{T}$ flows and (2,2) supersymmetry,” Phys. Rev. D 101
(2020), no. 2, 026008, 1906.00467.
[10]
Y. Jiang, “A pedagogical review on solvable irrelevant deformations of 2D
quantum field theory,” Commun. Theor. Phys. 73 (2021), no. 5,
057201, 1904.13376.
[11]
G. Bonelli, N. Doroud, and M. Zhu, “$T\bar{T}$-deformations in closed
form,” JHEP 06 (2018) 149,
1804.10967.
[12]
R. Conti, S. Negro, and R. Tateo, “The $\mathrm{T}\overline{\mathrm{T}}$
perturbation and its geometric interpretation,” JHEP 02 (2019)
085,
1809.09593.
[13]
R. Conti, L. Iannella, S. Negro, and R. Tateo, “Generalised Born-Infeld
models, Lax operators and the $\mathrm{T}\overline{\mathrm{T}}$
perturbation,” JHEP 11 (2018) 007,
1806.11515.
[14]
S. Frolov, “$T\overline{T}$ Deformation and the Light-Cone Gauge,” Proc. Steklov Inst. Math. 309 (2020) 107–126,
1905.07946.
[15]
E. A. Coleman, J. Aguilera-Damia, D. Z. Freedman, and R. M. Soni, “$T\overline{T}$ -deformed actions and (1,1) supersymmetry,” JHEP 10 (2019) 080, 1906.05439.
[16]
S. Frolov, “$T{\overline{T}}$, $\widetilde{J}J$, $JT$ and $\widetilde{J}T$
deformations,” J. Phys. A 53 (2020), no. 2, 025401,
1907.12117.
[17]
A. J. Tolley, “$T\overline{T}$ deformations, massive gravity and
non-critical strings,” JHEP 06 (2020) 050,
1911.06142.
[18]
A. Sfondrini and S. J. van Tongeren, “$T\bar{T}$ deformations as $TsT$
transformations,” Phys. Rev. D 101 (2020), no. 6, 066022,
1908.09299.
[19]
G. Jorjadze and S. Theisen, “Canonical maps and integrability in $T\bar{T}$
deformed 2d CFTs,” 2001.03563.
[20]
L. McGough, M. Mezei, and H. Verlinde, “Moving the CFT into the bulk with $T\overline{T}$,” JHEP 04 (2018) 010,
1611.03470.
[21]
A. Giveon, N. Itzhaki, and D. Kutasov, “$\mathrm{T}\overline{\mathrm{T}}$
and LST,” JHEP 07 (2017) 122,
1701.05576.
[22]
A. Dei and A. Sfondrini, “Integrable spin chain for stringy
Wess-Zumino-Witten models,” JHEP 07 (2018) 109,
1806.00422.
[23]
M. Baggio and A. Sfondrini, “Strings on NS-NS Backgrounds as Integrable
Deformations,” Phys. Rev. D98 (2018), no. 2, 021902,
1804.01998.
[24]
N. Callebaut, J. Kruthoff, and H. Verlinde, “$T\overline{T}$ deformed CFT
as a non-critical string,” JHEP 04 (2020) 084,
1910.13578.
[25]
C. Ferko, H. Jiang, S. Sethi, and G. Tartaglino-Mazzucchelli, “Non-linear
supersymmetry and $T\overline{T}$-like flows,” JHEP 02 (2020)
016, 1910.01599.
[26]
J. Caetano, W. Peelaers, and L. Rastelli, “Maximally supersymmetric RG flows
in 4D and integrability,” JHEP 12 (2021) 119,
2006.04792.
[27]
M. Taylor, “TT deformations in general dimensions,”
1805.10287.
[28]
T. Hartman, J. Kruthoff, E. Shaghoulian, and A. Tajdini, “Holography at
finite cutoff with a $T^{2}$ deformation,” JHEP 03 (2019) 004,
1807.11401.
[29]
H. Babaei-Aghbolagh, K. B. Velni, D. M. Yekta, and H. Mohammadzadeh, “$T\overline{T}$-like flows in non-linear electrodynamic theories and
S-duality,” JHEP 04 (2021) 187,
2012.13636.
[30]
C. Ferko, Supersymmetry and Irrelevant Deformations.
PhD thesis, Chicago U., 2021.
2112.14647.
[31]
J. Bagger and A. Galperin, “A New Goldstone multiplet for partially broken
supersymmetry,” Phys. Rev. D55 (1997) 1091–1098,
hep-th/9608177.
[32]
I. Bandos, K. Lechner, D. Sorokin, and P. K. Townsend, “A non-linear
duality-invariant conformal extension of Maxwell’s equations,” Phys.
Rev. D 102 (2020) 121703,
2007.09092.
[33]
B. P. Kosyakov, “Nonlinear electrodynamics with the maximum allowable
symmetries,” Phys. Lett. B 810 (2020) 135840,
2007.13878.
[34]
M. K. Gaillard and B. Zumino, “Duality Rotations for Interacting Fields,”
Nucl. Phys. B 193 (1981) 221–244.
[35]
G. W. Gibbons and D. A. Rasheed, “Electric - magnetic duality rotations in
nonlinear electrodynamics,” Nucl. Phys. B 454 (1995) 185–206,
hep-th/9506035.
[36]
G. W. Gibbons and D. A. Rasheed, “Sl(2,R) invariance of nonlinear
electrodynamics coupled to an axion and a dilaton,” Phys. Lett. B
365 (1996) 46–50, hep-th/9509141.
[37]
E. A. Ivanov and B. M. Zupnik, “N=3 supersymmetric Born-Infeld theory,”
Nucl. Phys. B 618 (2001) 3–20,
hep-th/0110074.
[38]
E. A. Ivanov and B. M. Zupnik, “New approach to nonlinear electrodynamics:
Dualities as symmetries of interaction,” Phys. Atom. Nucl. 67
(2004) 2188–2199, hep-th/0303192.
[39]
S. M. Kuzenko and S. Theisen, “Supersymmetric duality rotations,” JHEP 03 (2000) 034,
hep-th/0001068.
[40]
S. M. Kuzenko and S. Theisen, “Nonlinear selfduality and supersymmetry,”
Fortsch. Phys. 49 (2001) 273–309,
hep-th/0007231.
[41]
S. M. Kuzenko and S. A. McCarthy, “Nonlinear selfduality and supergravity,”
JHEP 02 (2003) 038,
hep-th/0212039.
[42]
S. M. Kuzenko and S. A. McCarthy, “On the component structure of N=1
supersymmetric nonlinear electrodynamics,” JHEP 05 (2005) 012,
hep-th/0501172.
[43]
S. M. Kuzenko, “Nonlinear self-duality in N = 2 supergravity,” JHEP
06 (2012) 012, 1202.0126.
[44]
S. M. Kuzenko, “Duality rotations in supersymmetric nonlinear electrodynamics
revisited,” JHEP 03 (2013) 153,
1301.5194.
[45]
I. Antoniadis, H. Jiang, and O. Lacombe, “$\mathcal{N}$ = 2 supersymmetry
deformations, electromagnetic duality and Dirac-Born-Infeld actions,” JHEP 07 (2019) 147,
1904.06339.
[46]
I. Bandos, K. Lechner, D. Sorokin, and P. K. Townsend, “On p-form gauge
theories and their conformal limits,” JHEP 03 (2021) 022,
2012.09286.
[47]
I. Bandos, K. Lechner, D. Sorokin, and P. K. Townsend, “ModMax meets Susy,”
JHEP 10 (2021) 031,
2106.07547.
[48]
S. M. Kuzenko, “Superconformal duality-invariant models and $\mathcal{N}$ =
4 SYM effective action,” JHEP 09 (2021) 180,
2106.07173.
[49]
S. I. Kruglov, “On generalized ModMax model of nonlinear electrodynamics,”
Phys. Lett. B 822 (2021) 136633,
2108.08250.
[50]
D. P. Sorokin, “Introductory Notes on Non-linear Electrodynamics and its
Applications,”
12, 2021.
2112.12118.
[51]
P. Pasti, D. P. Sorokin, and M. Tonin, “Duality symmetric actions with
manifest space-time symmetries,” Phys. Rev. D 52 (1995)
R4277–R4281, hep-th/9506109.
[52]
P. Pasti, D. P. Sorokin, and M. Tonin, “On Lorentz invariant actions for
chiral p forms,” Phys. Rev. D 55 (1997) 6292–6298,
hep-th/9611100.
[53]
P. Pasti, D. P. Sorokin, and M. Tonin, “Covariant action for a D = 11
five-brane with the chiral field,” Phys. Lett. B 398 (1997)
41–46, hep-th/9701037.
[54]
S. I. Kruglov, “Magnetic black holes with generalized ModMax model of
nonlinear electrodynamics,” International Journal of Modern Physics D
0 (0), no. 0, 2250025,
https://doi.org/10.1142/S0218271822500250.
[55]
D. Flores-Alfonso, B. A. González-Morales, R. Linares, and M. Maceda,
“Black holes and gravitational waves sourced by non-linear duality
rotation-invariant conformal electromagnetic matter,” Phys. Lett. B
812 (2021) 136011, 2011.10836.
[56]
A. Bokulić, T. Jurić, and I. Smolić, “Black hole thermodynamics in the
presence of nonlinear electromagnetic fields,” Phys. Rev. D 103
(2021), no. 12, 124059, 2102.06213.
[57]
S. H. Mazharimousavi, “A note on Reissner–Nordström black holes
in the inverse electrodynamics model,” Int. J. Geom. Meth. Mod. Phys.
18 (2021), no. 10, 2150155,
2102.13138.
[58]
Y.-S. An, L. Li, and F.-G. Yang, “No Cauchy horizon theorem for nonlinear
electrodynamics black holes with charged scalar hairs,” Phys. Rev. D
104 (2021), no. 2, 024040,
2106.01069.
[59]
K. Nomura and D. Yoshida, “Quasinormal modes of charged black holes with
corrections from nonlinear electrodynamics,” Phys. Rev. D 105
(2022), no. 4, 044006, 2111.06273.
[60]
A. Ali and K. Saifullah, “Charged black holes in 4D
Einstein–Gauss–Bonnet gravity coupled to nonlinear
electrodynamics with maximum allowable symmetries,” Annals Phys. 437 (2022) 168726.
[61]
S. M. Kuzenko and E. S. N. Raptakis, “Duality-invariant superconformal
higher-spin models,” Phys. Rev. D 104 (2021), no. 12, 125003,
2107.02001.
[62]
E. A. Ivanov and B. M. Zupnik, “New representation for Lagrangians of
selfdual nonlinear electrodynamics,” in Supersymmetries and Quantum
Symmetries. Proceedings, 16th Max Born Symposium, SQS’01: Karpacz, Poland,
September 21-25, 2001, pp. 235–250.
2002.
hep-th/0202203.
[63]
H. Babaei-Aghbolagh, K. B. Velni, D. M. Yekta, and H. Mohammadzadeh,
“Emergence of non-linear electrodynamic theories from $T\bar{T}$-like
deformations,” 2202.11156.
[64]
J. Cardy, “The $T\overline{T}$ deformation of quantum field theory as
random geometry,” JHEP 10 (2018) 186,
1801.06895.
[65]
S. Cecotti and S. Ferrara, “Supersymmetric born-infeld lagrangians,”
Physics Letters B 187 (Mar., 1987) 335–339.
[66]
S. Ferrara and B. Zumino, “Transformation Properties of the Supercurrent,”
Nucl. Phys. B87 (1975)
207.
[67]
I. L. Buchbinder and S. M. Kuzenko, Ideas and methods of supersymmetry
and supergravity: Or a walk through superspace.
Bristol, UK: IOP (1998) 656 p,
1998.
[68]
S. Cecotti and S. Ferrara, “Supersymmetric Born-Infeld Lagrangians,” Phys. Lett. B187 (1987)
335–339.
[69]
F. Gonzalez-Rey, I. Y. Park, and M. Rocek, “On dual 3-brane actions with
partially broken N=2 supersymmetry,” Nucl. Phys. B544 (1999)
243–264,
hep-th/9811130.
[70]
S. M. Kuzenko and S. J. Tyler, “On the Goldstino actions and their
symmetries,” JHEP 05 (2011) 055,
1102.3043.
[71]
J. Bagger and A. Galperin, “The Tensor Goldstone multiplet for partially
broken supersymmetry,” Phys. Lett. B412 (1997) 296–300,
hep-th/9707061.
[72]
M. Rocek and A. A. Tseytlin, “Partial breaking of global D = 4 supersymmetry,
constrained superfields, and three-brane actions,” Phys. Rev. D59 (1999) 106001,
hep-th/9811232.
[73]
N. Cribiori, F. Farakos, and R. von Unge, “2D Volkov-Akulov Model as a $T\overline{T}$ Deformation,” Phys. Rev. Lett. 123 (2019),
no. 20, 201601, 1907.08150.
[74]
H. Nastase, “Coupling ModMax theory precursor with scalars, and BIon-type
solutions,” 2112.01234.
[75]
R. R. Metsaev, M. Rakhmanov, and A. A. Tseytlin, “The Born-Infeld Action as
the Effective Action in the Open Superstring Theory,” Phys. Lett.
B193 (1987)
207–212.
[76]
S. Paban, S. Sethi, and M. Stern, “Constraints from extended supersymmetry in
quantum mechanics,” Nucl. Phys. B534 (1998) 137–154,
hep-th/9805018.
[77]
S. Paban, S. Sethi, and M. Stern, “Supersymmetry and higher derivative terms
in the effective action of Yang-Mills theories,” JHEP 06 (1998)
012,
hep-th/9806028.
[78]
Y.-H. Lin, S.-H. Shao, Y. Wang, and X. Yin, “Higher derivative couplings in
theories with sixteen supersymmetries,” Phys. Rev. D92 (2015),
no. 12, 125017,
1503.02077.
[79]
W.-M. Chen, Y.-t. Huang, and C. Wen, “Exact coefficients for higher
dimensional operators with sixteen supersymmetries,” JHEP 09
(2015) 098,
1505.07093.
[80]
E. Bergshoeff, M. Rakowski, and E. Sezgin, “Higher Derivative Super
Yang-Mills Theories,” Phys. Lett. B185 (1987)
371–376.
[81]
A. A. Tseytlin, “Born-Infeld action, supersymmetry and string theory,”
hep-th/9908105.
[82]
E. Bergshoeff, F. Coomans, R. Kallosh, C. S. Shahbazi, and A. Van Proeyen,
“Dirac-Born-Infeld-Volkov-Akulov and Deformation of Supersymmetry,” JHEP 08 (2013) 100,
1303.5662.
[83]
S. V. Ketov, “Many faces of Born-Infeld theory,” in 7th International
Wigner Symposium (Wigsym 7) College Park, Maryland, August 24-29, 2001.
2001.
hep-th/0108189.
[84]
S. F. Kerstan, “Supersymmetric Born-Infeld from the D9-brane,” Class.
Quant. Grav. 19 (2002) 4525–4536,
hep-th/0204225.
[85]
S. V. Ketov, “A Manifestly N=2 supersymmetric Born-Infeld action,” Mod. Phys. Lett. A14 (1999) 501–510,
hep-th/9809121.
[86]
S. V. Ketov, “Born-Infeld-Goldstone superfield actions for gauge fixed D-5
branes and D-3 branes in 6-d,” Nucl. Phys. B553 (1999)
250–282,
hep-th/9812051.
[87]
S. V. Ketov, “N=2 superBorn-Infeld theory revisited,” Class. Quant.
Grav. 17 (2000) L91,
hep-th/0005126.
[88]
S. Bellucci, E. Ivanov, and S. Krivonos, “N=2 and N=4 supersymmetric
Born-Infeld theories from nonlinear realizations,” Phys. Lett. B502 (2001) 279–290,
hep-th/0012236.
[89]
S. Bellucci, E. Ivanov, and S. Krivonos, “Towards the complete N=2 superfield
Born-Infeld action with partially broken N=4 supersymmetry,” Phys.
Rev. D64 (2001) 025014,
hep-th/0101195.
[90]
S. Bellucci, E. Ivanov, and S. Krivonos, “Superbranes and super-Born-Infeld
theories from nonlinear realizations,” Nucl. Phys. Proc. Suppl. 102 (2001) 26–41, hep-th/0103136.
[,26(2000)].
[91]
N. Berkovits and V. Pershin, “Supersymmetric Born-Infeld from the pure spinor
formalism of the open superstring,” JHEP 01 (2003) 023,
hep-th/0205154.
[92]
V. Py, “$T\bar{T}$ Deformations in Curved Space from 4D Chern-Simons
Theory,” 2202.08841.
[93]
T. D. Brennan, C. Ferko, and S. Sethi, “A Non-Abelian Analogue of DBI from $T\overline{T}$,” SciPost Phys. 8 (2020), no. 4, 052,
1912.12389.
[94]
D. J. Gross, J. Kruthoff, A. Rolph, and E. Shaghoulian, “$T\overline{T}$ in
AdS${}_{2}$ and Quantum Mechanics,” Phys. Rev. D 101 (2020), no. 2,
026011, 1907.04873.
[95]
D. J. Gross, J. Kruthoff, A. Rolph, and E. Shaghoulian, “Hamiltonian
deformations in quantum mechanics, $T\bar{T}$, and the SYK model,” Phys. Rev. D 102 (2020), no. 4, 046019,
1912.06132.
[96]
S. Ebert, C. Ferko, H.-Y. Sun, and Z. Sun, “$T\overline{T}$ Deformations of
Supersymmetric Quantum Mechanics,”
2204.05897. |
Cosmological Perturbations in Elastic Dark Energy Models
Richard A. Battye
Jodrell Bank Observatory, Department of Physics and
Astronomy, University of Manchester, Macclesfield, Cheshire SK11
9DL, UK
Adam Moss
Jodrell Bank Observatory, Department of Physics and
Astronomy, University of Manchester, Macclesfield, Cheshire SK11
9DL, UK
School of Physics and
Astronomy, University of British Columbia, Vancouver, British Columbia, V6T 1Z1,
Canada
(28/3/2007)
Abstract
We discuss the general framework for a perfect continuum medium in cosmology and show that an interesting generalization of the fluids normally used is for the medium to have rigidity and, hence, be analogous to an elastic solid. Such models can provide perfect, adiabatic fluids which are stable even when the pressure is negative, if the rigidity is sufficiently large, making them natural candidates to describe the dark energy. In fact, if the medium is adiabatic and isotropic, they provide the most general description of linearized perturbations. We derive the equations of motion and wave propagation speeds in the isotropic case. We point out that anisotropic models can also be incorporated within the same formalism and that they are classified by the standard Bravais Lattices. We identify the adiabatic and isocurvature modes allowed in both the scalar and vector sectors and discuss the predictions they make for CMB and matter power spectra. We comment on the relationship between these models and other fluid-based approaches to dark energy, and discuss a possible microphysical manifestation of this class of models as a continuum description of defect-dominated scenarios.
pacs:
††preprint:
I Introduction
There are four stress-energy components in the presently preferred standard cosmological model Spergel:2006hy ; Hinshaw:2006ia ; Page:2006hz : the baryons which constitute the visible universe, radiation (photons and neutrinos), cold dark matter (CDM) and some unknown component, often called dark energy, which gives rise to the cosmic acceleration Perlmutter:1996ds ; Riess:1998cb ; Perlmutter:1998np ; Riess:2001gk ; Astier:2005qq . Within the codes Seljak:1996is ; Lewis:1999bs used to make predictions for specific models, the CDM is modelled as a pressureless, perfect fluid and the radiation components are modelled as black-body radiation gases. However, many of the qualitative features of the observed power spectra, for example, the acoustic peaks in the angular power spectrum of the cosmic microwave background (CMB), can be derived by ignoring the higher order moments of the photon and neutrino distributions and treating them as a perfect fluid with $P=\rho/3$, where $P$ is the pressure and $\rho$ is the density. Hence, most of the important components of the universe are well approximated by perfect fluids.
The microphysical origin of the cosmic acceleration is still a mystery. The simplest, and probably most popular, explanation is a cosmological constant, although this has some well documented fine-tuning problems Carroll:1991mt ; Carroll:2000fy . Various ideas exist, collectively known as dark energy (for a recent review of dark energy models, see ref. Copeland:2006wr ), whereby the additional stress-energy component is provided by a slowly rolling scalar field Ratra:1987rm ; Wetterich:1987fm ; Caldwell:1997ii , known as as Quintessence, or a lattice of topological defects Vilenkin:1984rt ; Kibble:1985tf ; Bucher:1998mh ; Battye:1999eq . Alternative explanations require the modification of gravity by the inclusion of non-minimal coupling between matter and gravity Amendola:1999er , extra-dimensions (for example, ref. Dvali:2000hr ) and modifications to the Einstein-Hilbert action Capozziello:2003tk ; Carroll:2003wy ; Carroll:2004de .
In this paper we shall consider a generalization of the perfect continuum fluid approach to describe dark energy. We have already pointed out that perfect fluids can be used to describe the essential properties of the radiation and CDM in the universe, and it seems reasonable to consider the possibility that the dark energy can also be understood in the same way. This will require us to go back to the fundamentals of how formulate a generalized continuum medium in General Relativity and derive equations for the perturbations in the medium. These are essential in computing accurate predictions for observed power spectra Caldwell:1997ii ; Battye:1999eq ; Weller:2003hw ; Bean:2003fb . We will show that the most obvious generalization of a perfect fluid is to allow for rigidity of the medium in a way analogous to a continuum elastic solid Bucher:1998mh , and that this can be stable if the rigidity is sufficiently large even if the pressure is negative. For most cosmological observations (for example, the CMB or large-scale structure measurements) we only need to consider the linearized regime. We will argue that the generalization of fluids to include rigidity, is the most general possibility for an adiabatic medium at linearized order.
In many ways the concept of dark energy is similar to the idea of the Aether postulated in the late 19th century: there is something about the laws of physics which appears awry and we postulate a medium to try and solve the problem 111In fact, as we will discuss in the final section the elastic medium approach is more than just qualitatively similar to the idea of the Aether. We hope, of course, that it is more successful at explaining observational facts than the Aether.. The approach we propose is to formulate generalized properties of such a medium, define ways of computing power spectra and then compare them with observations (see ref. Battye:2005mm for the status of this kind of model after the first year WMAP data).
The idea of generalized dark matter/energy has been considered previously by a number of authors Hu:1998kj ; Hu:1998tj ; Bucher:1998mh ; Battye:1999eq ; Weller:2003hw ; Bean:2003fb ; Koivisto:2005mm . We will attempt to discuss how the approaches suggested by others relate to those presented here. The broad difference between our approach and those taken previously is that we will derive the equations describing perturbations in the medium from a set of well defined physical assumptions. These equations are closed and there is no freedom, except the strength of the rigidity, to play with.
The original derivation of the equations presented here Bucher:1998mh , was motivated by the idea that the dark energy could be a lattice of topological defects (cosmic strings or domain walls) formed at a low energy phase transition. We will explain how such a lattice can provide a possible microphysical model for an elastic dark energy model. However, we believe that the formulation of the elastic dark energy models is more general and should not be thought of as being necessarily linked to these specific type of models which predict very specific values for $w=P/\rho$. In particular, the elastic dark energy models include CDM and a cosmological constant as limiting cases allowing them to provide an interesting phenomenology.
The paper is organized as follows. In section II we discuss the formulation of a generalized medium and derive the equations of motion. We then go on to identify the various adiabatic and isocurvature modes in section III and present the cosmological signatures expected in the CMB and matter power spectra in section IV. In section V we explain the basic ideas of defect-dominated scenarios for dark energy and explain how they relate to the elastic dark energy models.
II Cosmological Dynamics of a Generalized Medium
In Section II.1 we review the medium representation concept and the scheme for specifying perturbations in a general medium. In II.2 we apply this formalism to compute the cosmological equations of motion for an isotropic elastic medium and in II.3 evaluate the propagation speeds of perturbations. We then decompose the perturbations in terms of harmonic basis functions in II.4 and II.5 to give the full set of Einstein and energy-momentum conservation equations. In Section II.6 we compare our results to other treatments of generalized fluids, and in II.7 discuss briefly how the formalism can be applied to anisotropic perturbations.
II.1 Medium Representation Concept
This section provides an overview of work on relativistic elastic media in the context of neutrons stars by Carter and others Carter:1972cq ; Carter:1977qf ; Carter:1980c ; Carter:1982xm , and details how this elegant formalism to describe the mechanics of a generalized medium can be applied to cosmology. In Newtonian theory it is natural to define the properties of a medium, such as the density and pressure, in terms of coordinates of a three-dimensional Euclidean space at some given instant of time. In General Relativity, however, there is no general time-slicing of the four-dimensional space-time manifold $\mathcal{M}$ which can be used to specify the material state. Therefore, it is necessary to consider the projection $\mathcal{P}$: $\mathcal{M}\rightarrow\mathcal{H}$ of $\mathcal{M}$ onto a three-dimensional manifold $\mathcal{H}$ whose elements represent particles of the medium Carter:1972cq . The inverse image $\mathcal{P}^{-1}(X)\subset\mathcal{M}$ of a point $X\in\mathcal{H}$ can then interpreted as the worldline of the particle represented by $X$. The inverse image determines a one-to-one mapping between medium tensors defined on $\mathcal{H}$ and the set of tensors on $\mathcal{M}$ which are orthogonal to the congruence of wordlines. This mapping is important as it allows the intrinsic properties of the material medium to be defined on $\mathcal{H}$, whilst the spacetime evolution is described by tensor fields on $\mathcal{M}$. Moreover, it allows the definition of spacetime tensors in terms the tensors defined on $\mathcal{H}$. We will use Greek indices $\mu,\nu...$ to label tensors on $\mathcal{M}$ and Roman indices $A,B...$ to label tensors on $\mathcal{H}$.
We define $\gamma_{AB}$ to be the metric on $\mathcal{H}$ which quantifies the strain of the medium. At each point of $\mathcal{H}$, we will assume that there are functions determining the density $\rho$ and pressure $P^{AB}$ which can be expressed in terms of $\gamma_{AB}$ (one of the properties of a perfect medium) and, most probably, related by an equation of state $P^{AB}(\rho)$. Under these assumptions one can show that
$$P^{AB}=-2|\gamma|^{-1/2}{\partial\over\partial(\gamma_{AB})}(|\gamma|^{1/2}%
\rho)\,,$$
(1)
where $|\gamma|$ is the determinant of $\gamma_{\rm AB}$, and by taking a second derivative of $\rho$ with respect to $\gamma_{\rm AB}$, one can define
$$E^{ABCD}=-2|\gamma|^{-1/2}{\partial\over\partial(\gamma_{AB})}(|\gamma|^{1/2}P%
^{CD})\,,$$
(2)
the classical elasticity tensor which satisfies $E^{ABCD}=E^{(AB)(CD)}=E^{CDAB}$. We will see that defining these two tensors is sufficient to understand linearized perturbations of the medium.
In order to make correspondence with the spacetime $\mathcal{M}$, we define $\mathcal{P}_{\mu}^{A}$ to be a bi-tensorial projection operator which projects a tensor in $\mathcal{H}$ into $\mathcal{M}$, or vice-versa, then one can define the projected metric tensor $\gamma_{\mu\nu}$ by
$$\gamma_{\mu\nu}=\mathcal{P}_{\mu}^{A}\mathcal{P}_{\nu}^{B}\gamma_{AB}\,.$$
(3)
The field of flow vectors $u^{\mu}$ tangent to wordlines in $\mathcal{M}$ are normalized by the condition
$$u^{\mu}u_{\mu}=-1\,,$$
(4)
and the projection is orthogonal to them, that is, $u^{\mu}\mathcal{P}_{\mu}^{A}=0$. This allows us to construct $\gamma_{\mu\nu}$ as an orthogonal projection tensor
$$\gamma_{\mu\nu}=g_{\mu\nu}+u_{\mu}u_{\nu}\,,$$
(5)
providing an alternative, but equivalent, definition for $\gamma_{\mu\nu}$. It acts as the tensor which projects other tensors onto the tangent subspace orthogonal to the wordlines. For example, any vector $v^{\mu}$ can then be decomposed into components orthogonal and parallel to the flow by $v^{\mu}=_{\perp}\!\!\!v^{\mu}+v^{\parallel}u^{\mu}$, where ${}_{\perp}\!v^{\mu}=\gamma^{\mu}_{\nu}v^{\nu}$ and $v^{\parallel}=-u_{\mu}v^{\mu}$. This allows physically relevant spacetime tensors to be decomposed into the parallel component, which does not contribute to the material projection, and the orthogonal part, which does. This framework is summarized schematically in Fig. 1.
One can use the projection tensor to construct the spacetime pressure tensor, $P^{\mu\nu}$, from that on $\mathcal{H}$
$$P_{\mu\nu}={\cal P}_{\mu}^{A}{\cal P}_{\nu}^{B}P_{AB}\,,$$
(6)
which satisfies $u^{\mu}P_{\mu\nu}=0$. The same can be done for the elasticity tensor $E^{\mu\nu\rho\sigma}$ which satisfies
$$E^{\mu\nu\rho\sigma}=E^{(\mu\nu)(\rho\sigma)}=E^{\rho\sigma\mu\nu},\hskip 14.2%
26378ptE^{\mu\nu\rho\sigma}u_{\sigma}=0\,.$$
(7)
One must also specify how perturbations in $\mathcal{H}$ are related to those in $\mathcal{M}$ in order to deal with dynamics. The convected differential $d[\,\,]$ is an important quantity, as it is the spacetime tensor corresponding to the Lagrangian (wordline preserving) material variation on $\mathcal{H}$. In the case of a vector, there is the bijection $d[v^{\mu}]\longleftrightarrow\{\delta_{\rm L}\,{}_{\perp}\!v^{\rm A},\,\delta_%
{\rm L}v^{\parallel}\}$, where $\delta_{\rm L}$ is the Lagrangian variation. The convected differential can be evaluated in terms of Lagrangian differentials on $\mathcal{M}$ by projecting the orthogonal part of the perturbation. This gives the explicit relationship Carter:1980c
$$d[T^{\mu...}_{\nu...}]=\delta_{\rm L}T^{\mu...}_{\nu...}+T^{\mu...}_{\rho...}u%
^{\rho}\delta_{\rm L}u_{\nu}-T^{\rho...}_{\nu...}u^{\mu}\delta_{\rm L}u_{\rho}\,,$$
(8)
where $T^{\mu...}_{\nu...}$ is a general mixed tensor.
The next stage is to relate the material variations in terms of spacetime tensors. The most direct way to do this is to use the convected derivative, which relates Lagrangian and material variations. Applying (8) to (1) and (2) one obtains Carter:1982xm
$$\delta_{\rm L}\,\rho=-\frac{1}{2}(P^{\mu\nu}+\rho\gamma^{\mu\nu})\delta_{\rm L%
}\,g_{\mu\nu}\,,$$
(9)
$$\delta_{\rm L}P^{\mu\nu}=-\frac{1}{2}(E^{\mu\nu\rho\sigma}+P^{\mu\nu}\gamma^{%
\rho\sigma}-4P^{\rho(\mu}u^{\nu)}u^{\sigma})\delta_{\rm L}\,g_{\rho\sigma}\,,$$
(10)
where extra terms are induced in (10) due to (8).
We can now begin to discuss the dynamics of an elastic medium. A perfect elastic medium is defined by the condition that the energy-momentum tensor $T^{\mu\nu}$ is a material function of the metric tensor with respect to the flow field Carter:1982xm , meaning that it takes the form
$$T^{\mu\nu}=\rho u^{\mu}u^{\nu}+P^{\mu\nu}\,.$$
(11)
The variation in the energy-momentum tensor can be obtained by using (11) in conjunction with (9) and (10) to give
$$\delta_{\rm L}T^{\mu\nu}=-\frac{1}{2}(W^{\mu\nu\rho\sigma}+T^{\mu\nu}g^{\rho%
\sigma})\delta_{\rm L}\,g_{\rho\sigma}\,,$$
(12)
where the non-orthogonal elasticity tensor $W^{\mu\nu\rho\sigma}$ can be decomposed as Friedman:1975fs
$$W^{\mu\nu\rho\sigma}=E^{\mu\nu\rho\sigma}+P^{\mu\nu}u^{\rho}u^{\sigma}+P^{\rho%
\sigma}u^{\mu}u^{\nu}-P^{\mu\rho}u^{\sigma}u^{\nu}-P^{\mu\sigma}u^{\nu}u^{\rho%
}-P^{\nu\rho}u^{\mu}u^{\sigma}-P^{\nu\sigma}u^{\mu}u^{\rho}-\rho u^{\mu}u^{\nu%
}u^{\rho}u^{\sigma}\,,$$
(13)
and has the same symmetry properties as the ordinary elasticity tensor. This allows the variations to be conveniently written in terms of functional derivatives of the Lagrangian with respect to the metric via
$$T^{\mu\nu}=-2|g|^{-1/2}{\delta\over\delta_{\rm L}g_{\mu\nu}}(|g|^{1/2}{\cal L}%
)\,,$$
(14)
$$W^{\mu\nu\rho\sigma}=4|g|^{-1/2}{\delta\over\delta_{\rm L}g_{\rho\sigma}}{%
\delta\over\delta_{\rm L}g_{\mu\nu}}(|g|^{1/2}{\cal L})=-2|g|^{-1/2}{\delta%
\over\delta_{\rm L}g_{\rho\sigma}}(|g|^{1/2}T^{\mu\nu})\,,$$
(15)
where $|g|$ is the determinant of the metric.
It is often more convenient to describe perturbations being fixed with respect to some background space. If the vector field $\xi^{\mu}$ is the infinitesimal displacement of the wordlines with respect to the fixed background space, then the difference between Lagrangian $\delta_{\rm L}$ and fixed (Eulerian) variations $\delta_{\rm E}$ is given by
$$\delta_{\rm L}=\delta_{\rm E}+\mathcal{L}_{\xi}\,,$$
(16)
where $\mathcal{L}_{\xi}$ is the Lie derivative. Such a displacement could be removed by using a mapping of the perturbed space onto the background space, but it will be more convenient to set the gauge by some other means. In the following section, for example, we use the synchronous gauge to derive the cosmological equations of motion. For the particular case of the metric tensor then the relationship (16) gives
$$\delta_{\rm L}\,g_{\mu\nu}=\delta_{\rm E}\,g_{\mu\nu}+2\nabla_{(\mu}\xi_{\nu)}\,,$$
(17)
where $\delta_{\rm E}\,g_{\mu\nu}$ is the Eulerian variation of the metric tensor.
The equation of motion for the vector field $\xi^{\mu}$ gives the complete system of equations for the perturbations. Evaluation of the Lagrangian variation $\delta_{\rm L}({\gamma^{\mu}}_{\nu}\nabla_{\mu}T^{\mu\nu})$ gives
$$\displaystyle({{A^{\mu(\nu}}}{{}_{\rho}{{}^{\sigma)}}}-(\rho{\gamma^{\mu}}_{%
\rho}+{P^{\mu}}_{\rho})u^{\nu}u^{\sigma})\delta_{\rm L}{\Gamma^{\rho}}_{\nu%
\sigma}+\frac{1}{2}{\gamma^{\mu}}_{\rho}{\gamma^{\alpha}}_{\nu}{\gamma^{\beta}%
}_{\sigma}(\delta_{\rm L}\,g_{\alpha\beta})\nabla_{\tau}E^{\rho\tau\nu\sigma}=$$
(18)
$$\displaystyle(P^{\mu\nu}\dot{u}^{\sigma}-\frac{1}{2}P^{\nu\sigma}\dot{u}^{\mu}%
-2{{A^{\mu(\nu}}}{{}_{\rho}{{}^{\tau)}}}{v^{\rho}}_{\tau}u^{\sigma}+(\rho{%
\gamma^{\mu}}_{\rho}+{P^{\mu}}_{\rho})\dot{u}^{\rho}u^{\nu}u^{\sigma})\delta_{%
\rm L}\,g_{\nu\sigma}\,,$$
where dots denote covariant differentiation with respect to the flow (that is, $u^{\mu}\nabla_{\mu}$) and
$${{A^{\mu\nu}}}{{}_{\rho}{{}^{\sigma}}}={{E^{\mu\nu}}}{{}_{\rho}{{}^{\sigma}}}-%
{\gamma^{\mu}}_{\rho}P^{\nu\sigma}\,,$$
(19)
is the relativistic Hadamard elasticity tensor which forms the characteristic equation needed to evaluate the sound speeds in the medium Carter:1973 . This tensor obeys the symmetry and orthogonality conditions
$$A^{\mu\nu\rho\sigma}=A^{\rho\sigma\mu\nu},\hskip 14.226378ptA^{\mu\nu\rho%
\sigma}u_{\sigma}=0\,.$$
(20)
Eqn. (18) takes the form of a wave equation for the displacement vector $\xi^{\mu}$. Since all components are orthogonal the the flow then the additional degree of freedom in $\xi^{\mu}$ can be removed by imposing the orthogonality requirement $\xi^{\mu}u_{\mu}=0$. The remaining quantities required to evaluate (18) are the Lagrangian variation of the connection coefficients, given by
$$\delta_{\rm L}{\Gamma^{\mu}}_{\nu\sigma}=\nabla_{(\nu}\delta_{\rm L}\,{g^{\mu}%
}_{\sigma)}-\frac{1}{2}\nabla^{\mu}\delta_{\rm L}\,g_{\nu\sigma}\,,$$
(21)
and the flow gradient tensor given by
$$v_{\mu\nu}=\nabla_{\nu}u_{\mu}+\dot{u}_{\mu}u_{\nu}\,.$$
(22)
II.2 Cosmological Equations of Motion for an Isotropic Medium
The perturbed space-time metric takes the form
$$g_{\mu\nu}=a^{2}(\tau)\left[\eta_{\mu\nu}+h_{\mu\nu}\right],\hskip 14.226378%
pth^{\mu\nu}=\eta^{\alpha\mu}\eta^{\beta\nu}h_{\alpha\beta}\,,$$
(23)
and so the Eulerian component of the metric perturbation is given by $\delta_{\rm E}\,g_{\mu\nu}=a^{2}h_{\mu\nu}$. We make use of the synchronous gauge conditions ($h_{00}=h_{0i}=0$) to remove the remaining degree of gauge freedom in the Einstein field equations. At zeroth order the flow vector $u^{\mu}=a^{-1}(1,0,0,0)$, and so the non-zero components of the displacement vector $\xi^{\mu}$ are confined to the spatial part by the orthogonality requirement $\xi^{\mu}u_{\mu}=0$.
In the synchronous gauge the non-zero components of the Lagrangian variation of the connection are given by
$$\displaystyle\delta_{\rm L}{\Gamma^{i}}_{00}=\ddot{\xi}^{i}+{\cal H}\dot{\xi}^%
{i}\,,\quad\delta_{\rm L}{\Gamma^{0}}_{0i}={\cal H}\dot{\xi}_{i}\,,\quad\delta%
_{\rm L}{\Gamma^{0}}_{ij}=2{\cal H}\partial_{(i}\xi_{j)}+{\cal H}h_{ij}+\dot{h%
}_{ij}\,,$$
(24)
$$\displaystyle\delta_{\rm L}{\Gamma^{i}}_{0j}=\partial_{j}\dot{\xi}^{i}+\frac{1%
}{2}{{\dot{h}}^{i}}_{\,j}\,,\quad\delta_{\rm L}{\Gamma^{i}}_{jk}=\partial_{j}%
\partial_{k}\xi^{i}-\delta_{jk}{\cal H}\dot{\xi^{i}}+\partial_{(j}{h^{i}}_{k)}%
-\frac{1}{2}\partial^{i}h_{jk}\,,$$
where dots are now understood to denote derivatives with respect to the conformal time and ${\cal H}$ is the conformal time Hubble parameter. We can now obtain the equations of motion and perturbed energy-momentum sources for the cosmological fluids by inserting the appropriate pressure and elasticity tensor expressions for each component into (12) and (18).
II.2.1 Isotropic Perfect Fluid
In an isotropic perfect fluid the pressure tensor is isotropic and is given in terms of the pressure scalar $P$ by
$$P^{\mu\nu}=P\gamma^{\mu\nu}\,,$$
(25)
while the elasticity tensor is given in terms of the bulk modulus $\beta$ by Carter:1972cq
$$E^{\mu\nu\rho\sigma}=(\beta-P)\gamma^{\mu\nu}\gamma^{\rho\sigma}+2P\gamma^{\mu%
(\rho}\gamma^{\sigma)\nu}\,,$$
(26)
and the bulk modulus is defined by
$$\beta=(\rho+P)\frac{dP}{d\rho}\,.$$
(27)
Substituting these expressions into (12), we obtain the components of the perturbed energy-momentum tensor
$$\displaystyle\delta_{\rm E}\,{T^{0}}_{0}$$
$$\displaystyle=$$
$$\displaystyle(\rho+P)\left(\partial_{i}\xi^{i}+\frac{1}{2}h\right)\,,$$
(28a)
$$\displaystyle\delta_{\rm E}\,{T^{i}}_{0}$$
$$\displaystyle=$$
$$\displaystyle-(\rho+P)\dot{\xi}^{i}\,,$$
(28b)
$$\displaystyle\delta_{\rm E}\,{T^{i}}_{j}$$
$$\displaystyle=$$
$$\displaystyle-\beta{\delta^{i}}_{j}\left(\partial_{k}\xi^{k}+\frac{1}{2}h%
\right)\,,$$
(28c)
where $h$ is the trace of the metric perturbation. From now onward we do not make the explicit distinction between Eulerian (fixed) and Lagrangian perturbations, as all relevant quantities are evaluated with respect to the fixed coordinates. Substitution of (25) and (26) into (18) gives the equation of motion for the displacement vector $\xi^{i}$ as
$$(\rho+P)(\ddot{\xi}^{i}+{\cal H}\dot{\xi}^{i})-3\beta{\cal H}\dot{\xi}^{i}-%
\beta(\partial^{i}\partial_{j}\xi^{j}+\partial^{i}h/2)=0\,.$$
(29)
II.2.2 Isotropic Perfect Elastic Medium
We have pointed out that perturbations in an elastic medium can be specified by the pressure tensor $P^{\mu\nu}$ and the elasticity tensor $E^{\mu\nu\rho\sigma}$. In the case of isotropy the pressure tensor is also given by (25) since $\gamma^{\mu\nu}$ is the only isotropic tensor of rank two, upto a scaling. The case of rank four, required to describe the elasticity tensor, is more complicated; it is given by
$$E^{\mu\nu\rho\sigma}=A\gamma^{\mu\nu}\gamma^{\rho\sigma}+B\gamma^{\mu(\rho}%
\gamma^{\sigma)\nu}\,,$$
(30)
where $A$ and $B$ are arbitrary parameters. In order to fit in with the definitions used in the previous section, we will define an additional shear contribution by
$$E^{\mu\nu\rho\sigma}=\Sigma^{\mu\nu\rho\sigma}+(\beta-P)\gamma^{\mu\nu}\gamma^%
{\rho\sigma}+2P\gamma^{\mu(\rho}\gamma^{\sigma)\nu}\,,$$
(31)
where the shear tensor obeys the symmetry and orthogonality conditions
$$\Sigma^{\mu\nu\rho\sigma}=\Sigma^{(\mu\nu)(\rho\sigma)}=\Sigma^{\rho\sigma\mu%
\nu},\hskip 14.226378pt\Sigma^{\mu\nu\rho\sigma}u_{\sigma}=0\,.$$
(32)
In an isotropic elastic fluid the shear tensor in terms of a single shear moduli $\mu$ by
$$\Sigma^{\mu\nu\rho\sigma}=2\mu\left(\gamma^{\mu(\rho}\gamma^{\sigma)\nu}-\frac%
{1}{3}\gamma^{\mu\nu}\gamma^{\rho\sigma}\right)\,.$$
(33)
The contribution of the shear tensor to the elasticity tensor is zero in the perfect fluid case ($\mu=0$). We again substitute these expressions into (12) to obtain the components of the perturbed energy-momentum tensor
$$\displaystyle{\delta T^{0}}_{0}$$
$$\displaystyle=$$
$$\displaystyle(\rho+P)\left(\partial_{i}\xi^{i}+\frac{1}{2}h\right)\,,$$
(34a)
$$\displaystyle{\delta T^{i}}_{0}$$
$$\displaystyle=$$
$$\displaystyle-(\rho+P)\dot{\xi}^{i}\,,$$
(34b)
$$\displaystyle{\delta T^{i}}_{j}$$
$$\displaystyle=$$
$$\displaystyle-{\delta^{i}}_{j}(\beta-\frac{2}{3}\mu)\left(\partial_{k}\xi^{k}+%
\frac{1}{2}h\right)-\mu(2\partial_{(j}\xi^{i)}+h^{i}_{j})\,.$$
(34c)
The equation of motion for the displacement vector $\xi^{i}$ is then given by
$$(\rho+P)(\ddot{\xi}^{i}+{\cal H}\dot{\xi}^{i})-3\beta{\cal H}\dot{\xi}^{i}-%
\beta(\partial^{i}\partial_{j}\xi^{j}+\partial^{i}h/2)-\mu(\partial^{j}%
\partial_{j}\xi^{i}+\partial^{i}\partial_{j}\xi^{j}/3+\partial^{j}h^{i}_{j}-%
\partial^{i}h/3)=0\,.$$
(35)
where (34a, 34b, 34c) and (35) agree with the equations for an elastic medium defined in ref. Bucher:1998mh . If the medium forms at some finite time then boundary conditions imply that $h_{ij}\rightarrow h_{ij}-h_{ij}^{I}$, where $I$ refers to quantities defined at the time of formation of the medium. Since only the second-order variation of the Lagrangian is relevant for linearized perturbations we have shown that the pressure tensor, $P^{\mu\nu}$, and elasticity tensor, $E^{\mu\nu\rho\sigma}$, specify the most general parameterization of perturbations in $T^{\mu\nu}$ under the assumptions discussed earlier.
II.3 Evaluation of Sound Speeds
This section provides a brief summary of how to compute the propagation speed and polarization directions of sound waves (small perturbations) in a perfectly elastic medium, which was originally developed in ref. Carter:1973 . A sound wave front is defined as the hypersurface across which the acceleration vector $u^{\nu}\nabla_{\nu}u^{\mu}$ has a jump discontinuity (that is, at some coordinate point $x^{\mu}_{0}$ the acceleration is a well defined function in the limits $x^{\mu}_{-}$ and $x^{\mu}_{+}$, but is not defined at $x^{\mu}_{0}$). The flow vector $u^{\mu}$, the functions of state $\rho$, $P^{\mu\nu}$, $E^{\mu\nu\rho\sigma}$, the space-time tensor $g_{\mu\nu}$ and the projection tensor $\gamma_{\mu\nu}$ are continuous across this hypersurface, but the acceleration induces discontinuities in the first derivatives of $\rho$, $P^{\mu\nu}$, $E^{\mu\nu\rho\sigma}$ and $\gamma_{\mu\nu}$.
At the wave front hypersurface $u^{\nu}\nabla_{\nu}u^{\mu}=\alpha l^{\mu}$, where $\alpha$ is the amplitude of the sound wave and $l^{\mu}$ is the polarization vector of the wave front, which satisfies
$$l^{\mu}l_{\mu}=1,\hskip 14.226378ptl^{\mu}u_{\mu}=0\,.$$
(36)
If one introduces a propagation direction vector $v^{\mu}$ satisfying the same orthonormality conditions as the polarization vector, then a characteristic equation
$$[v^{2}(\rho\gamma^{\mu\nu}+P^{\mu\nu})-Q^{\mu\nu}]\,l_{\nu}=0\,,$$
(37)
can be constructed whose eigenvalues give the squared propagation speeds $v^{2}$ in the direction specified by $v^{\mu}$ and the eigenvectors are the polarization direction(s). The relativistic Fresnel $Q^{\mu\nu}$ tensor is defined by
$$Q^{\mu\nu}=A^{\mu\rho\nu\sigma}v_{\rho}v_{\sigma}\,,$$
(38)
where the Hadamard tensor $A^{\mu\rho\nu\sigma}$ is defined by (19). The Fresnel tensor satisfies the symmetry and orthogonality conditions
$$Q^{\mu\nu}=Q^{(\mu\nu)},\hskip 14.226378ptQ^{\mu\nu}u_{\nu}=0\,.$$
(39)
II.3.1 Isotropic Perfect Fluid
The Fresnel tensor can be obtained by substituting the expression for the pressure tensor (25) and elasticity tensor (26) into (19) and then using (38) to give
$$Q^{\mu\nu}=\beta v^{\mu}v^{\nu}\,,$$
(40)
so the eigenvalue equation becomes
$$[v^{2}(\rho+P)\gamma^{\mu\nu}-\beta v^{\mu}v^{\nu}]l_{\nu}=0\,.$$
(41)
This equation has a single solution in which the propagation direction is parallel to the polarization vector $(l_{\mu}=v_{\mu})$ and is given by
$$v^{2}=c_{\rm s}^{2}=\frac{\beta}{\rho+P}=\frac{dP}{d\rho}\,,$$
(42)
which corresponds to the longitudinal (scalar) sound speed.
II.3.2 Isotropic Perfect Elastic Medium
In the case of an isotropic elastic medium the Fresnel tensor is given by
$$Q^{\mu\nu}=\left(\beta+\frac{1}{3}\mu\right)v^{\mu}v^{\nu}+\mu\gamma^{\mu\nu}\,,$$
(43)
so the eigenvalue equation becomes
$$\left[v^{2}(\rho+P)\gamma^{\mu\nu}-\mu\gamma^{\mu\nu}-\left(\beta+\frac{1}{3}%
\mu\right)v^{\mu}v^{\nu}\right]l_{\nu}=0\,.$$
(44)
In this case there are two solutions - again there is one where the propagation direction is parallel to the polarization vector — but also another where the propagation direction is orthogonal to the polarization vector $(l^{\mu}v_{\mu}=0)$. This additional solution corresponds to a transversely polarized (vector) sound speed. The two solutions are given by
$$v^{2}=c_{\rm s}^{2}=\frac{\beta+4\mu/3}{\rho+P},\hskip 14.226378ptv^{2}=c_{\rm
v%
}^{2}=\frac{\mu}{\rho+P}\,,$$
(45)
so the two sound speeds are related by
$$c_{\rm s}^{2}=\frac{dP}{d\rho}+\frac{4}{3}c_{\rm v}^{2}\,.$$
(46)
For an equation of state where $P=w\rho$ it can be seen from (46) that if $\mu/\rho$ is sufficiently large then $c_{\rm s}^{2}>0$ even if $w$ is negative. This stabilizing property of the shear modulus initially motivated the use of elastic fluids in the framework of dark energy models, where $w<-1/3$ is required to achieve the observed acceleration Bucher:1998mh ; Battye:1999eq . The relationship between the sound speeds and equation of state is shown in Fig. 2, where we plot lines of constant $\mu/\rho$ and
$c_{\rm v}^{2}$ in the $(w,c_{\rm s}^{2})$ plane. The constraint that $\mu/\rho\geq 0$ requires that $w\geq-1$, while the constraint that $0\leq c_{v}^{2}\leq 1$ restricts the allowed value of $c_{\rm s}^{2}$ for a given $w$. We have assumed that the intrinsic properties of the medium are fixed and have ignored time variations in quantities such as $w$ and $\mu/\rho$. Relaxing this assumption could lead to a well-defined phenomenological model for time-varying dark energy. However, such a model would have to respect the stability conditions discussed above.
II.4 Harmonic Decomposition of Perturbations
Constant time hypersurfaces are homogeneous and isotropic, so it is natural to decompose spatial tensor fields on these hypersurfaces in terms of the irreducible representations of the rotation group SO(3). A spatial tensor field can be decomposed in terms of eigenfunctions of the Laplacian Lifshitz:1963ps ; Bardeen:1980kt ; Abbott:1986ct
$$\eta^{ij}Q_{|ij}^{S,V,T}=-k^{2}Q^{S,V,T}\,,$$
(47)
where the $S,V,T$ index represents irreducible scalar, vector and tensorial quantities and ${}^{\prime}|^{\prime}$ represents covariant differentiation with respect to the three-metric $\eta_{ij}$. These eigenfunctions form a complete basis in which to expand the tensor field. In flat space, for example, Fourier plane waves provide a local orthonormal basis. Scalars can be constructed from longitudinal type vectors and tensors via Hu:1997mn
$$Q^{S}_{i}=-k^{-1}Q_{|i}^{S}\,,\hskip 14.226378ptQ^{S}_{ij}=k^{-2}Q_{|ij}^{S}+%
\frac{1}{3}\eta_{ij}Q^{S}\,,$$
(48)
and similarly vectors can be constructed from solenoidal type tensors by
$$Q_{ij}^{V}=-(k)^{-1}Q_{(i|j)}^{V}\,,$$
(49)
subject to $Q^{V|i}_{i}=Q^{T|i}_{ij}=Q^{Ti}_{i}=0$. A vector field can, therefore, be decomposed as
$$\xi_{i}=\xi^{S}Q^{S}_{i}+\xi^{V}Q_{i}^{V}\,,$$
(50)
and the general decomposition of a symmetric tensor field is
$$H_{ij}=H_{L}^{S}Q^{S}\eta_{ij}+H_{T}^{S}Q_{ij}^{S}+H^{V}Q_{ij}^{V}+H^{T}Q_{ij}%
^{T}\,.$$
(51)
The six degrees of freedom of the tensor field are represented by two scalar parts along with vector and tensor parts each with two degrees of freedom.
II.5 Einstein Equations and Conserved Energy-Momentum
The perturbed Einstein equations subject to the synchronous gauge conditions are
$$\displaystyle a^{2}{G^{0}}_{0}$$
$$\displaystyle=$$
$$\displaystyle-3{\cal H}^{2}-{\cal H}\dot{h}+\frac{1}{2}\partial_{i}\partial^{i%
}h-\frac{1}{2}\partial_{i}\partial_{j}h^{ij}\,,$$
(52a)
$$\displaystyle 2a^{2}{G^{0}}_{i}$$
$$\displaystyle=$$
$$\displaystyle\partial_{i}\dot{h}-\partial_{j}{\dot{h}^{j}}_{\,\,i}\,,$$
(52b)
$$\displaystyle 2a^{2}{G^{i}}_{0}$$
$$\displaystyle=$$
$$\displaystyle\partial_{j}\dot{h}^{ij}-\partial^{i}\dot{h}\,,$$
(52c)
$$\displaystyle a^{2}{G^{i}}_{j}$$
$$\displaystyle=$$
$$\displaystyle\left(2\dot{{\cal H}}-{\cal H}^{2}\right){\delta^{i}}_{j}+\frac{1%
}{2}\left({\ddot{h}^{i}}_{\,\,j}-\ddot{h}{\delta^{i}}_{j}\right)+{\cal H}\left%
({\dot{h}^{i}}_{\,\,j}-\dot{h}{\delta^{i}}_{j}\right)+\frac{1}{2}\left({\delta%
^{i}}_{j}\partial_{k}\partial^{k}h-\partial_{k}\partial^{k}{h^{i}}_{j}\right)$$
$$\displaystyle+\frac{1}{2}\delta^{ik}\left(\partial_{k}\partial_{l}{h^{l}}_{j}+%
\partial_{j}\partial_{l}{h^{l}}_{k}-\partial_{k}\partial_{j}h\right)-\frac{1}{%
2}{\delta^{i}}_{j}\partial_{k}\partial_{l}h^{kl}\,.$$
The metric perturbation is parameterized by $h_{ij}=2H_{ij}$, where $H_{ij}$ is decomposed according to (51). In order to compare the elastic fluid with other fluid based models, we use the parameterization of the the energy-momentum tensor used in ref. Hu:1997mn ,
$$\displaystyle{T^{0}}_{0}$$
$$\displaystyle=$$
$$\displaystyle-(\rho+\delta\rho)\,,$$
(53a)
$$\displaystyle{T^{0}}_{i}$$
$$\displaystyle=$$
$$\displaystyle(\rho+P)v_{i}\,,$$
(53b)
$$\displaystyle{T^{i}}_{0}$$
$$\displaystyle=$$
$$\displaystyle-(\rho+P)v^{i}\,,$$
(53c)
$$\displaystyle{T^{i}}_{j}$$
$$\displaystyle=$$
$$\displaystyle(P+\delta P){\delta^{i}}_{j}+P{\Pi^{i}}_{j}\,,$$
(53d)
where $v_{i}$ is the velocity perturbation from the flow and the anisotropic stress, ${\Pi^{i}}_{j}$, is symmetric and traceless. These quantities are also decomposed according to (50) and (51). The scalar-vector-tensor (SVT) split of the perturbed Einstein equations then gives the following constraint and evolution equations.
Constraint:
$$\displaystyle{\cal H}\dot{h}-2k^{2}\eta$$
$$\displaystyle=$$
$$\displaystyle 8\pi Ga^{2}\delta\rho,$$
(54a)
$$\displaystyle k\dot{\eta}$$
$$\displaystyle=$$
$$\displaystyle 4\pi Ga^{2}(\rho+P)v^{S}\,,$$
(54b)
$$\displaystyle k\dot{H}^{V}$$
$$\displaystyle=$$
$$\displaystyle-16\pi Ga^{2}(\rho+P)v^{V}\,.$$
(54c)
Evolution:
$$\displaystyle\ddot{h}+2{\cal H}\dot{h}-2k^{2}\eta$$
$$\displaystyle=$$
$$\displaystyle-24\pi Ga^{2}\delta P\,,$$
(55a)
$$\displaystyle\ddot{h}+6\ddot{\eta}+2{\cal H}(\dot{h}+6\dot{\eta})-2k^{2}\eta$$
$$\displaystyle=$$
$$\displaystyle-16\pi Ga^{2}P\Pi^{S}\,,$$
(55b)
$$\displaystyle\ddot{H}^{V}+2{\cal H}\dot{H}^{V}$$
$$\displaystyle=$$
$$\displaystyle 8\pi Ga^{2}P\Pi^{V}\,,$$
(55c)
$$\displaystyle\ddot{H}^{T}+2{\cal H}\dot{H}^{T}+k^{2}H^{T}$$
$$\displaystyle=$$
$$\displaystyle 8\pi Ga^{2}P\Pi^{T}\,,$$
(55d)
where $h=6H^{S}_{L}$ and $\eta=-\left(H^{S}_{L}+\frac{1}{3}H^{S}_{T}\right)$ are the metric variables defined in ref. Ma:1995ey . The equations of motion for the scalar and vector components of the displacement vector $\xi^{i}$ of the isotropic elastic fluid given by (35) are then
$$\displaystyle\ddot{\xi}^{S}+\left(1-3\frac{dP}{d\rho}\right){\cal H}\dot{\xi}^%
{S}+k^{2}c_{\rm s}^{2}\left[\xi^{S}+\frac{1}{2k}(h-h_{I})\right]+3k\left(c_{%
\rm s}^{2}-\frac{dP}{d\rho}\right)(\eta-\eta_{I})$$
$$\displaystyle=$$
$$\displaystyle 0\,,$$
(56a)
$$\displaystyle\ddot{\xi}^{V}+\left(1-3\frac{dP}{d\rho}\right){\cal H}\dot{\xi}^%
{V}+k^{2}c_{\rm v}^{2}\left[\xi^{V}+\frac{1}{k}(H^{V}_{I}-H^{V})\right]$$
$$\displaystyle=$$
$$\displaystyle 0\,,$$
(56b)
where the subscript $I$ denotes the metric induced on the medium at the time of formation. Assuming an equation of state $P=w\rho$ then the scalar components of the perturbed energy-momentum tensor (34a, 34b, 34c) are, according to the decomposition (53a, 53b, 53c, 53d),
$$\displaystyle\delta\rho$$
$$\displaystyle=$$
$$\displaystyle-\rho(1+w)\left[k\xi^{S}+\frac{1}{2}(h-h_{I})\right]\,,$$
(57a)
$$\displaystyle v^{S}$$
$$\displaystyle=$$
$$\displaystyle\dot{\xi}^{S}\,,$$
(57b)
$$\displaystyle\delta P$$
$$\displaystyle=$$
$$\displaystyle-\rho(1+w)\frac{dP}{d\rho}\left[k\xi^{S}+\frac{1}{2}(h-h_{I})%
\right]\,,$$
(57c)
$$\displaystyle\Pi^{S}$$
$$\displaystyle=$$
$$\displaystyle\frac{3}{2}\left(c_{\rm s}^{2}-\frac{dP}{d\rho}\right)(1+w^{-1})%
\left[k\xi^{S}+\frac{1}{2}(h-h_{I})+3(\eta-\eta_{I})\right]$$
$$\displaystyle=$$
$$\displaystyle\frac{3}{2}\left(c_{\rm s}^{2}-\frac{dP}{d\rho}\right)(1+w^{-1})%
\left[-\frac{\delta}{1+w}+3(\eta-\eta_{I})\right]\,.$$
Note that $\delta P/\delta\rho=\textstyle{dP\over d\rho}$. Eqn. (56a) can then be used in conjunction with (57a, 57b, 57c, 57) to give a closed set of scalar equations
$$\displaystyle\dot{\delta}$$
$$\displaystyle=$$
$$\displaystyle-(1+w)\left(kv^{S}+\frac{1}{2}\dot{h}\right)\,,$$
(58a)
$$\displaystyle\dot{v}^{S}$$
$$\displaystyle=$$
$$\displaystyle-{\cal H}\left(1-3\frac{dP}{d\rho}\right)v^{S}+\frac{dP}{d\rho}%
\frac{1}{1+w}k\delta-\frac{2}{3}\frac{w}{1+w}k\Pi^{S}\,.$$
(58b)
The anisotropic stress is zero in the perfect fluid limit ($\mu=0$). Similarly, the vector sources are
$$\displaystyle v^{V}$$
$$\displaystyle=$$
$$\displaystyle\dot{\xi}^{V}\,,$$
(59a)
$$\displaystyle\Pi^{V}$$
$$\displaystyle=$$
$$\displaystyle 2c_{\rm v}^{2}(1+w^{-1})(k\xi^{V}+H^{V}_{I}-H^{V})\,,$$
(59b)
and the anisotropic tensor source is
$$\Pi^{T}=2c_{\rm v}^{2}(1+w^{-1})(H^{T}_{I}-H^{T})\,.$$
(60)
The vector or tensor contributions due to the elastic medium are absent in the perfect fluid limit. Eqn. (56b) can then be used in conjunction with (59a, 59b) to give the vector equation of motion
$$\dot{v}^{V}=-{\cal H}\left(1-3\frac{dP}{d\rho}\right)v^{V}-\frac{1}{2}\frac{w}%
{1+w}k\Pi^{V}\,.$$
(61)
II.6 Comparison to Generalized Fluid Systems
The equations of motion derived in the preceding sections are based on a very specific set of assumptions. Given that the understanding the evolution of dark energy perturbations is crucial to making precise predictions for the observed power spectra, a variety of phenomenological models have been discussed Weller:2003hw ; Bean:2003fb ; Hu:1998kj ; Hu:1998tj ; Koivisto:2005mm . In this section we attempt to make some contact between our work and these models.
Energy conservation of a generalized fluid energy-momentum tensor gives the scalar equations Ma:1995ey ; Hu:1998kj :
$$\displaystyle\dot{\left(\frac{\delta}{1+w}\right)}$$
$$\displaystyle=$$
$$\displaystyle-\left(kv^{S}+\frac{1}{2}\dot{h}\right)-3{\cal H}\frac{w}{1+w}%
\Gamma\,,$$
(62a)
$$\displaystyle\dot{v}^{S}$$
$$\displaystyle=$$
$$\displaystyle-{\cal H}\left(1-3\frac{dP}{d\rho}\right)v^{S}+\frac{dP}{d\rho}%
\frac{1}{1+w}k\delta+\frac{w}{1+w}k\Gamma-\frac{2}{3}\frac{w}{1+w}k\Pi^{S}\,,$$
(62b)
where $\Gamma$ is the entropy contribution which is given by
$$w\Gamma=\left(\frac{\delta P}{\delta\rho}-\frac{dP}{d\rho}\right)\delta\,.$$
(63)
In general, $P=w\rho$ does not imply $\delta P=w\,\delta\rho$ due to temporal or spatial variations in $w$ which correspond to entropy perturbations. It transpires that both entropy and anisotropic stress can play a role in stabilizing perturbations which is crucial to the viability of any model.
In an elastic medium, we have already pointed out that $\delta\rho/\delta P=\textstyle{dP\over d\rho}$ which implies that $\Gamma=0$, that is, the medium is adiabatic. However, the non-zero rigidity leads to anisotropic stress and it is that which stabilizes perturbations as shown by (45).
A number of authors Hu:1998kj ; Hu:1998tj ; Koivisto:2005mm have suggested phenomenological approaches which involve the inclusion of anisotropic stress. Based on various arguments, they suggested making $\Pi^{S}$ dynamical and constructed a phenomenological equation of motion for its evolution
$$\dot{\Pi}^{S}+{1\over T}\Pi^{S}={4c_{\rm vis}^{2}\over w}\left(kv^{S}+\frac{1}%
{2}\dot{h}+3\dot{\eta}\right)\,,$$
(64)
where $T$ is some decay timescale (which they suggest should be $(3\mathcal{H})^{-1}$) and $c_{\rm vis}^{2}$ is some arbitrary coefficient, whose physical origin is attributed to viscosity within the fluid. If we differentiate (57) and substitute in (58a), then we obtain
$$\dot{\Pi}^{S}=\frac{2\mu}{P}\left(kv^{S}+\frac{1}{2}\dot{h}+3\dot{\eta}\right)\,,$$
(65)
which has a similar form to (64) if $T=\infty$ and $c_{\rm vis}^{2}=\mu/(2\rho)=c_{\rm v}^{2}(1+w)/2$. The construction of these phenomenological models was somewhat ad-hoc, nonetheless they are very close to those described here. However, we see that the physical mechanism which they describe is not viscosity, but rigidity. It might be possible to construct models with finite values of $T$ by modifying our approach to include some kind of dissipation.
We note that the system of equations which we have derived is very similar to that which might come from a truncated Boltzmann hierarchy for a black-body such as the neutrinos (or photons with no Thomson scattering terms). In particular, if we take $w=1/3$ and $\mu/\rho=4/15$, then the elastic model gives
$$\displaystyle\dot{\delta}$$
$$\displaystyle=$$
$$\displaystyle-{4\over 3}\left(kv^{S}+{1\over 2}\dot{h}\right)\,,$$
(66a)
$$\displaystyle\dot{v}^{S}$$
$$\displaystyle=$$
$$\displaystyle k\left({1\over 4}\delta-{1\over 6}\Pi^{S}\right)\,,$$
(66b)
$$\displaystyle\dot{\Pi}^{S}$$
$$\displaystyle=$$
$$\displaystyle{8\over 5}\left(kv^{S}+{1\over 2}\dot{h}+3\dot{\eta}\right)\,,$$
(66c)
which is exactly that given by a Boltzmann hierarchy with a third moment set to zero Hu:1997mn .
A second class of models seek to model dark energy fluids which are non-adiabatic and could come from scalar field models. For a minimally coupled scalar field with the standard kinetic term
$$\delta\rho=\dot{\phi}\dot{\delta\phi}+{dV\over d\phi}\delta\phi\,,\quad\delta P%
=\dot{\phi}\dot{\delta\phi}-{dV\over d\phi}\delta\phi\,,$$
(67)
and hence $\delta P/\delta\rho$ clearly depends on the frame of the dark energy defined by $\delta\phi$ and $\dot{\delta\phi}$. One can choose to work in the rest frame of the dark energy defined by $(\rho+P)v^{i}={\hat{k}}^{i}k\delta\phi/\dot{\phi}=0$, where $c_{\rm s}^{2}=\delta P/\delta\rho=1$. More general models with non-minimal couplings or exotic kinetic terms could have $c_{\rm s}^{2}\neq 1$.
It was suggested in refs. Weller:2003hw ; Bean:2003fb to use a system of equations described by (62a, 62b) with no anisotropic stress but with $\Gamma\neq 0$. In order to take into account the possibility of different frames, the sound speed was deemed to be defined in the rest frame of the dark energy and the equations of motion were modified to apply in an arbitrary frame, that is, one makes the transformation to
$$\delta^{{\rm rest}}=\delta+3{\cal H}(1+w)(v^{S}-B)/k\,.$$
(68)
Here $B$ is the space-time component of the metric perturbation and is zero in both the synchronous and Newtonian gauges. This leads to equations of motion
$$\displaystyle\dot{\delta}$$
$$\displaystyle=$$
$$\displaystyle-(1+w)\left[kv^{S}\left(1+{9\mathcal{H}^{2}\over k^{2}}(c_{\rm s}%
^{2}-w)\right)+{1\over 2}\dot{h}\right]-3\mathcal{H}(c_{\rm s}^{2}-w)\delta\,,$$
(69a)
$$\displaystyle\dot{v}^{S}$$
$$\displaystyle=$$
$$\displaystyle-\mathcal{H}(1-3c_{\rm s}^{2})v^{S}+{kc_{\rm s}^{2}\over 1+w}%
\delta\,,$$
(69b)
which can be computed from (62b) using
$$w\Gamma_{\rm eff}=(c_{\rm s}^{2}-w)\left(\delta+3\mathcal{H}(1+w){v^{S}\over k%
}\right)\,.$$
(70)
When $c_{\rm s}^{2}=w$ these are clearly the same as for the elastic medium, but they are very different in the limit $c_{\rm s}^{2}\rightarrow 0$, as we shall see in the subsequent discussion.
II.7 Anisotropic Generalizations
In the previous sections we have described how one can construct a general perturbed energy-momentum tensor under a set of simple assumptions. The fact that the general isotropic tensor of rank four has a limited number of degrees of freedom has allowed us to construct the most general set of equations of motion for an isotropic medium defined by its density, pressure and rigidity. More generally we have shown that specification of all of the components of the pressure tensor, $P^{\mu\nu}$, and elasticity tensor $E^{\mu\nu\rho\sigma}$ is sufficient to describe the perturbed energy momentum tensor using (12). In this section we will discuss aspects of a generalization which allows for the perturbations to be anisotropic and consider the case of cubic symmetry which is the simplest anisotropic possibility.
In general, the elasticity tensor has a total of 21 independent components landau:1959 . One of these, the bulk modulus, is specified by the pressure and the other 20 are shear moduli, which can provide an anisotropic response to perturbations depending on the particular symmetry of the system. Fortunately the classification of these shear moduli has been studied in the context of classical elasticity theory and it is known that there 14 different types, known as the Bravais lattices. In table 1 we list the number of non-zero shear moduli for each symmetry group.
An additional requirement that we will impose here is that the pressure tensor is isotropic, $P^{\mu\nu}=P\gamma^{\mu\nu}$, so that the unperturbed spacetime has the standard FRW metric, and the anisotropic response is only present at linearized order. This will probably restrict the range of possibilities allowed, but for sure there is at least one possibility, that of cubic symmetry, which is compatible with this.
The case of cubic response to perturbations was considered in ref. Battye:2006mb . There are now two shear moduli, $\mu_{\rm L}$ and $\mu_{\rm T}$, which specify the non-zero components of the elasticity tensor
$$\displaystyle E^{xxxx}$$
$$\displaystyle=$$
$$\displaystyle E^{yyyy}=E^{zzzz}=\beta+P+\frac{4}{3}\mu_{\rm L}\,,$$
(71)
$$\displaystyle E^{xxyy}$$
$$\displaystyle=$$
$$\displaystyle E^{yyzz}=E^{zzxx}=\beta-P-\frac{2}{3}\mu_{\rm L}\,,$$
$$\displaystyle E^{yzyz}$$
$$\displaystyle=$$
$$\displaystyle E^{xzxz}=E^{xyxy}=P+\mu_{\rm T}\,.$$
The energy-momentum sources are modified from those in (34a, 34b, 34c) to
$$\displaystyle\delta T^{0}_{0}$$
$$\displaystyle=$$
$$\displaystyle(\rho+P)\left(\partial_{i}\xi^{i}+\frac{1}{2}h\right)\,,$$
(72a)
$$\displaystyle\delta T^{i}_{0}$$
$$\displaystyle=$$
$$\displaystyle-(\rho+P)\dot{\xi}^{i}\,,$$
(72b)
$$\displaystyle\delta T^{i}_{j}$$
$$\displaystyle=$$
$$\displaystyle-\delta^{i}_{j}(\beta-\frac{2}{3}\mu_{L})\left(\partial_{k}\xi^{k%
}+\frac{1}{2}h\right)-\mu_{L}(2\partial_{(j}\xi^{i)}+h^{i}_{j})-\Delta\mu\,{S^%
{i}}_{j}\,.$$
(72c)
where $\Delta\mu=\mu_{\rm T}-\mu_{\rm L}$ quantifies the degree of anisotropy. The tensor ${S^{i}}_{j}$ represents the cubic source term and is given by
$${S^{i}}_{j}=\left(\begin{array}[]{ccc}0&2\partial_{(y}\xi^{x)}+h^{x}_{y}&2%
\partial_{(z}\xi^{x)}+h^{x}_{z}\\
2\partial_{(y}\xi^{x)}+h^{y}_{x}&0&2\partial_{(y}\xi^{z)}+h^{y}_{z}\\
2\partial_{(z}\xi^{x)}+h^{z}_{x}&2\partial_{(z}\xi^{y)}+h^{z}_{y}&0\end{array}%
\right)\,.$$
(73)
Furthermore, the evolution equations (35) are modified to
$$(\rho+P)(\ddot{\xi}^{i}+{\cal H}\xi^{i})-3\beta{\cal H}\dot{\xi}^{i}-\beta(%
\partial^{i}\partial_{j}\xi^{j}+\partial^{i}h/2)-\mu_{\rm L}(\partial^{j}%
\partial_{j}\xi^{i}+\partial^{i}\partial_{j}\xi^{j}/3+\partial^{j}{h^{i}}_{j}-%
\partial^{i}h/3)=\Delta\mu\,F^{i}\,,$$
(74)
where the cubic source term $F^{i}$ is given by
$$F^{i}=\left(\begin{array}[]{ccc}(\partial_{y}\partial^{y}+\partial_{z}\partial%
^{z})\xi^{x}+\partial^{x}(\partial_{y}\xi^{y}+\partial_{z}\xi^{z})+\partial^{y%
}{h^{x}}_{y}+\partial^{z}{h^{x}}_{z}\\
(\partial_{x}\partial^{x}+\partial_{z}\partial^{z})\xi^{y}+\partial^{y}(%
\partial_{x}\xi^{x}+\partial_{z}\xi^{z})+\partial^{x}{h^{y}}_{x}+\partial^{z}{%
h^{y}}_{z}\\
(\partial_{x}\partial^{x}+\partial_{y}\partial^{y})\xi^{z}+\partial^{z}(%
\partial_{x}\xi^{x}+\partial_{y}\xi^{y})+\partial^{x}{h^{z}}_{x}+\partial^{y}{%
h^{z}}_{y}\end{array}\right)\,.$$
(75)
One of the consequences of the introduction of two shear moduli is that the sound speeds become dependent on the direction of the wave-vector Battye:2005ik . When one performs the same SVT decomposition as for the isotropic case, the different SVT sectors couple and scalar modes can excite vorticity and gravitational waves Battye:2006mb . It is is possible that this kind of anisotropic behaviour is responsible for anomalies seen the spectrum of the CMB on very large-scales; this is presently under investigation.
III Analytic Solutions
In this section, we identify the regular perturbation modes which can arise in the early universe in the presence of an isotropic elastic medium. This task has been performed in the standard scalar case in ref. Bucher:1999re . Observations show that the primordial perturbation was most likely dominated by an adiabatic scalar mode generated from fluctuations in the metric. However, the most general primordial perturbation can also contain scalar isocurvature modes Bucher:1999re generated from variations in the abundance ratios of different particle species. In the vector sector, a regular mode can also be sourced from non-zero initial photon and neutrino vorticity Lewis:2004kg and primordial magnetic fields can also lead to regular modes Lewis:2004ef .
We consider a universe consisting of cold dark matter (c), baryons (b), neutrinos ($\nu$), photons ($\gamma$) and an elastic fluid component (e). We identify specific modes where the elastic medium has an equation of state with $w=0,-1/3,-2/3$. To identify the regular modes in the early universe we assume that the photons and baryons are tightly coupled due to the large Thomson scattering term $\sigma_{\rm T}$. One can then obtain exact equations for the evolution of their velocity. An expansion in opacity $\kappa_{\rm c}^{-1}=an_{\rm e}\sigma_{\rm T}$ which is valid for max $(k\kappa_{\rm c},{\cal H}\kappa_{\rm c})\ll 1$ gives
$$\displaystyle\dot{v}_{\rm\gamma b}^{S}(1+R)+R{\cal H}v_{\rm\gamma b}^{S}$$
$$\displaystyle=$$
$$\displaystyle\frac{k}{4}\delta_{\rm\gamma}\,,$$
(76a)
$$\displaystyle\dot{v}^{V}_{\rm\gamma b}(1+R)+R{\cal H}v^{V}_{\rm\gamma b}$$
$$\displaystyle=$$
$$\displaystyle 0\,,$$
(76b)
where $R=3\rho_{\rm b}/(4\rho_{\rm\gamma})$. In the following we define $\omega_{\rm x}=\Omega_{\rm x}H_{0}^{2}$, $R_{\rm\nu}=\Omega_{\rm\nu}/\Omega_{\rm r}$ and $R_{\rm\gamma}=\Omega_{\rm\gamma}/\Omega_{\rm r}$. The primordial modes are then given by a series expansion of equations in Section II.5 in terms of the conformal time $\tau$.
III.1 Scalar Modes
In the scalar sector the most general primordial perturbation is specified by a mixture of adiabatic and isocurvature modes. Adiabatic coupling between various components in the universe requires that the relative perturbation between species, given by
$$A_{ij}=\frac{\delta_{i}}{1+w_{i}}-\frac{\delta_{j}}{1+w_{j}}\,,$$
(77)
vanishes. If the various density perturbations compensate in such a way that the initial curvature perturbation ($\eta$) is zero on super-horizon scales then these are termed isocurvature initial conditions. The scalar adiabatic and isocurvature modes (for initial perturbations in fluids other than the elastic fluid) are listed in table 2.
The isocurvature modes which can exist due to an elastic fluid are listed in table 3. These modes are interesting as they correspond to both initial non-zero density fluctuations and non-zero anisotropic stress, resulting from fluctuations of the scalar component of the wordline displacement vector $\xi^{S}$. Physically, we expect these modes to arise due to a perturbed state of the elastic medium relative to the background at formation, that is, the medium is not formed in an equilibrium state.
Due to our definition of anisotropic stress, the stress term for an elastic fluid, defined in (57), will diverge when $w=0$. The equations of motion do not suffer any such divergence, as this arises only due to our use of the standard decomposition of the energy-momentum tensor in (53d). As such, we list the non-zero initial term for the quantity $w\,\Pi^{S}_{e}$ for the elastic anisotropic stress in table 3.
III.2 Vector Modes
The regular vector mode has non-zero initial photon vorticity, having equal and opposite neutrino vorticity Lewis:2004kg . We have computed the initial conditions for this mode in the presence of a elastic fluid, which are shown in table 4, neglecting the small contributions from the photon anisotropic stress.
A regular mode also exists due to a non-zero initial vectorial component of the displacement vector $\xi^{V}$ in the elastic fluid, which also results in non-zero anisotropic stress (as in the scalar mode) and this mode is presented in table 5. In the elastic vector isocurvature mode the photon and neutrino vorticity are zero as they have no source term in the tight coupling limit. In both the regular vector mode and elastic isocurvature mode, the elastic fluid decouples from the equations of motion in the perfect fluid limit.
The vector isocurvature solution for a $w=-1/3$ elastic component can be found analytically in the radiation era due to the absence of photon and neutrino vorticity. This solution is given by
$$\xi^{V}=\xi^{V}_{I}\left[\left(1-\frac{4\omega_{\rm e}}{k^{2}+4\omega_{\rm e}}%
\right)\frac{\sin\left[c_{\rm v}(k^{2}+4\omega_{\rm e})^{\frac{1}{2}}\tau%
\right]}{c_{\rm v}(k^{2}+4\omega_{\rm e})^{\frac{1}{2}}\tau}+\frac{4\omega_{%
\rm e}}{k^{2}+4\omega_{\rm e}}\right]\,.$$
(78)
It can be shown that this solution gives that presented in table 5 if $\xi^{V}_{I}=k^{-1}$ when expanded as a power series in $\tau$.
III.3 Tensor Modes
There are no tensor modes other than the standard adiabatic mode. The tensor solution can be found analytically for a $w=-1/3$ elastic component in the radiation era assuming that the anisotropic stress of photons and neutrinos is zero. Using (55d) and (60) then
$$H^{T}=H_{I}^{T}\left[\frac{\sin\left[(k^{2}+4c_{\rm v}^{2}\omega_{\rm e})^{%
\frac{1}{2}}\tau\right]k^{2}}{(k^{2}+4c_{\rm v}^{2}\omega_{\rm e})^{\frac{3}{2%
}}\tau}+\frac{4c_{\rm v}^{2}\omega_{\rm e}}{k^{2}+4c_{\rm v}^{2}\omega_{\rm e}%
}\right]\,.$$
(79)
This reverts to the standard solution $H^{T}=H_{I}^{T}j_{0}(k\tau)$ when $\omega_{\rm e}=0$ or $c_{\rm v}^{2}=0$.
IV Cosmological Signatures
The observable CMB and matter power spectra were computed by modifying the CAMB software Lewis:1999bs to include an elastic fluid. We consider the elastic fluid in two scenarios - in one instance it acts as the dark energy component in an otherwise standard cosmology. The shear modulus $\mu$ stabilizes perturbations when $w<0$. We also consider in models with $\Omega_{\Lambda}\neq 0$ and the elastic fluid as a pressureless component with $w=0$, but with a non-zero shear modulus. The shear modulus introduces a clustering scale, the Jeans length $\lambda_{\rm J}\sim c_{\rm s}\tau_{0}$, where $\tau_{0}$ is the conformal time today, as $c_{\rm s}^{2}=4\mu/(3\rho)$ when $w=0$. This shares some similarities with a hot (or warm) dark matter component as power will be suppressed on small scales, although the shear modulus also allows for vector perturbations in the medium.
IV.1 Scalar Sector
The scalar $C_{\ell}$’s are given by
$$C_{\ell}=4\pi\int d(\log k)\mathcal{P}_{s}(k)|\Delta_{\ell}(k,\tau_{0})|^{2}\,,$$
(80)
where $\Delta_{\ell}(k,\tau_{0})$ is the associated multipole moment for the photon distribution and $\mathcal{P}_{s}(k)$ is the initial power spectrum, parameterized by $\mathcal{P}_{s}(k)=A_{\rm s}k^{n_{\rm s}-1}$, where $A_{\rm s}$ is the initial scalar amplitude and $n_{\rm s}$ is the scalar spectral index.
Note that $\mathcal{P}(k)=k^{3}P(k)/(2\pi^{2})$, where $P(k)$ is the matter power spectrum.
IV.1.1 Adiabatic Mode
The adiabatic mode is likely to have dominated the primordial fluctuation as a good fit to both CMB and galaxy clustering data can be obtained by this mechanism of structure formation and, assuming a flat universe, specifying a total of 6 cosmological parameters - $\Omega_{\rm b}$, $\Omega_{\rm c}$, $A_{\rm s}$, $n_{\rm s}$, $h$ and the optical depth to reionization $\kappa_{\rm R}$. A dark energy component is required in order to reconcile the observed accelerated expansion Perlmutter:1996ds ; Riess:1998cb ; Perlmutter:1998np ; Riess:2001gk ; Astier:2005qq .
The introduction of a dark energy component effects the CMB anisotropies both by its influence on the background expansion rate and its gravitational perturbations. Any component which has the same equation of state $w$ will give an identical overall expansion effect. However, an elastic fluid can potentially be distinguished by the different evolution of perturbations compared with other dark energy models. If $w<0$ then small-scale anisotropies are unaffected by the dark energy component as the perturbations at these scales entered the horizon when the fractional dark energy density was negligible. The primary contribution of the dark energy then arises at large angular scales through the Integrated Sachs-Wolfe (ISW) effect. The temperature-temperature (TT) contribution of the ISW effect to the scalar multipoles is given by
$$C_{\ell}^{TT}=4\pi\int d(\log k)\mathcal{P}_{s}(k)\left[\int_{\tau_{\rm dec}}^%
{\tau_{\rm 0}}d\tau e^{-\kappa}(\dot{\phi}+\dot{\psi})j_{\ell}(k(\tau_{\rm 0}-%
\tau))\right]^{2}\,,$$
(81)
where $\phi$ and $\psi$ are the Newtonian potentials Ma:1995ey and $\tau_{\rm dec}$ is the conformal time at decoupling. The presence of anisotropic stress alters the evolution of the Newtonian potentials.
In Fig. 3 we plot the CMB anisotropies for multipoles $\ell<25$ for the best fitting $\Lambda$CDM model Spergel:2006hy , along with the best estimates of the TT WMAP three year data Hinshaw:2006ia ; Page:2006hz . We also plot the anisotropies for a number of elastic fluid models, along with the corresponding scalar field model, for various values of $w$ and $c_{\rm s}^{2}$. In each case we exploit the degeneracy between $w$ and $h$ to rescale the first acoustic peak in order to isolate the ISW effect. It is noticeable in both dark energy models that there is a reduction in power at large angular scales for $w=-1/3$ - this reduction is slightly more pronounced for the elastic model due to the anisotropic stress. For the scalar field models, the ISW effect is minimised for $c_{\rm s}^{2}=0$ and thereby increases monotonically with $c_{\rm s}^{2}$. The elastic model is consistent with this behaviour for $w=-1/3$, but for $w=-2/3$ $c_{\rm s}^{2}=0$ actually maximizes the ISW effect.
In Fig. 4 we plot the temperature-polarization (TE) power spectrum for the best fitting $\Lambda$CDM model. The signal on large angular scales can be attributed to early reionization. The contribution to the large scale TE power for the polarization generated at reionization is given by
$$\displaystyle C_{\ell}^{TE}$$
$$\displaystyle=$$
$$\displaystyle 4\pi\int d(\log k)\mathcal{P}_{s}(k)\left[\int_{\tau_{\rm re}}^{%
\tau_{\rm 0}}d\tau e^{-\kappa}(\dot{\phi}+\dot{\psi})j_{\ell}(k(\tau_{\rm 0}-%
\tau))\right]$$
$$\displaystyle\times$$
$$\displaystyle\left[\int_{\tau_{\rm re}}^{\tau_{\rm 0}}-d\tau\frac{3}{4k^{2}}(g%
(k^{2}\Pi+\ddot{\Pi})+2\dot{g}\dot{\Pi}+\ddot{g}\Pi)j_{\ell}(k(\tau_{\rm 0}-%
\tau))\right]\,,$$
where $g=\dot{\kappa}\,e^{-\kappa}$ is the visibility function, $\tau_{\rm re}$ is the conformal time at reionization and the polarization source is given in terms of the photon anisotropic stress and the zeroth and second order polarization moments by $\Pi=\Pi_{\gamma}^{S}/3+\Delta_{P0}+\Delta_{P2}$. The photon anisotropic stress is the dominant term coming from the free streaming of the monopole at recombination. We also plot the TE power spectrum for a number of elastic fluid models, along with the corresponding scalar field models, for various values of $w$ and $c_{\rm s}^{2}$. We find that precision measurement of the TE cross correlation at low $\ell$ can provide a potentially powerful discriminator of the dark energy model, if non-zero anisotropic stress is present. The variation of $c_{\rm s}^{2}$ for a scalar field model generating no anisotropic stress has little effect on the TE spectrum. The elastic fluid, however, shows significant differences for scales of $\ell<10$ due to anisotropic stress modifying the decay of gravitational potentials during reionization.
The possibility of distinguishing two competing models whose differences are only significant on large angular scales is limited by the effect of cosmic variance. The probability of distinguishing model A from B, assuming that A is correct, is then given by Battye:1999eq
$$\left\langle\ln\left(\frac{P(\{a_{\ell m}\}|A)}{P(\{a_{\ell m}\}|B)}\right)%
\right\rangle_{A}=-\frac{1}{2}\sum_{\ell}(2\ell+1)\left[1-\frac{C_{\ell}^{(A)}%
}{C_{\ell}^{(B)}}+\ln\left(\frac{C_{\ell}^{(A)}}{C_{\ell}^{(B)}}\right)\right].$$
(83)
In Fig. 5 we plot this quantity for $A=$ a scalar field model and $B=$ an elastic fluid model in the $(w,c_{s}^{2})$ parameter space, and indicate the region where the models can be distinguished at 1$-\sigma$ . We find that this probability is too small in almost all of the region of interest to discriminate between models, based on CMB TT anisotropies. In the right panel of Fig. 5 we plot the analogous quantity based on TE anisotropies, assuming $\kappa_{\rm R}\neq 0$, which shows that accurate measurement at $\ell<10$ vastly increases the area of parameter space that can be differentiated. If $\kappa_{\rm R}=0$ one would be unable to distinguish between scalar field models from elastic models using TE.
The large scale structure of the universe depends on the growth rate of perturbations in the various species. An important aspect of dark energy is that it has a large Jeans length, $\lambda_{\rm J}\sim c_{\rm s}\tau_{0}$, at the present epoch so that it does not cluster on small scales and contribute to measurements of $\Omega_{\rm m}$ in galaxy clusters. We have assumed that the matter power spectrum is modified to
$$P_{\delta}(k)=b^{2}|\delta_{\rm T}|^{2}\,,$$
(84)
where $\delta_{\rm T}=\sum_{i}\Omega_{i}\delta_{i}$. This makes this untested assumption that the biasing of the dark energy component is proportional to its density. In Fig. 6 we plot the matter power spectrum for $w=-2/3$ elastic and scalar field dark energy models, decomposing the power into components due to the various species.
There are noticeable differences in the two cases. The presence of a clustering scale in the dark energy ($c_{\rm s}^{2}<<1$) affects the elastic fluid power spectrum in several ways. At small scales, power increases as the sound speed becomes non-relativistic. If the assumption (84) is correct then an approximate limit of $c_{\rm s}^{2}\gtrsim 10^{-3}$ would be required in order to be compatible with large scale structure (LSS) data. It is also noticeable that power in the CDM/baryonic components is reduced as the sound speed of the elastic component becomes non-relativistic. In the scalar field dark energy model, however, a sound speed as low as $c_{\rm s}^{2}=10^{-5}$ does not change the total matter power spectrum significantly. We have reduced $c_{\rm s}^{2}$ further and find that $c_{\rm s}^{2}=0$ would be compatible with LSS data.
Finally, we also consider whether the introduction of a non-zero shear modulus in a pressureless $w=0$ component is compatible with CMB and LSS data. In Fig. 7 we plot the ratio of the $\Lambda$CDM matter power spectrum against models where there is elastic rigidity in the CDM, which we denote $\Lambda$ECDM. The introduction of a non-zero shear modulus introduces a clustering scale in the CDM and power is suppressed on small scales. When $c_{\rm s}^{2}=10^{-5}$, for example, power is suppressed by an order of magnitude compared to the $\Lambda$CDM model at $k=0.1h{\,\rm Mpc^{-1}}$. For the values of $c_{\rm s}^{2}<10^{-4}$, the CMB anisotropies are affected at less than the $1\%$ level by variations in $c_{\rm s}^{2}.$
IV.1.2 Isocurvature Modes
Although the initial conditions appear to be dominated by the adiabatic mode, a sub-dominant isocurvature contribution cannot be ruled out. The scale-invariant CDM isocurvature mode suffers from the problem that the CMB anisotropy has greatly suppressed power on small scales relative to large scales. The reason for this is that the Sachs-Wolfe effect and the initial temperature fluctuation both add to give a much greater contribution to large-scale power. Recent work has placed an approximate 10$\%$ upper bound on the CDM isocurvature fraction, and similar constraints also apply for the other isocurvature modes Bean:2006qz .
The power spectrum of the scalar isocurvature mode is defined by
$$\langle|\delta|^{2}\rangle=\int d(\log k)\,\mathcal{P}_{\delta}(k).$$
(85)
The scalar isocurvature modes for an elastic fluid are listed in table 3. We list modes for a pressureless $w=0$ component with a non-zero shear modulus, which can be compared to the standard CDM isocurvature mode, in table 2. The effect of the shear modulus, and hence the sound speed $c_{\rm s}^{2}$, enters at first order in $\tau$ for several of the variables, such as the metric perturbation $\eta$. We also list isocurvature modes where the elastic fluid has an equation of state $w=-1/3$ and $-2/3$.
In Fig. 8 we plot the transfer function $T(k)$ of the total density fluctuation $\delta_{\rm T}$, along with the CMB anisotropies for a scale-invariant ($\mathcal{P}_{\delta}\propto k^{0}$, corresponding to $P_{\delta}(k)\propto k^{-3}$) and white noise power spectrum ($\mathcal{P}_{\delta}\propto k^{3}$, corresponding to $P_{\delta}(k)\propto k^{0}$) for $w=0$ and a primordial power amplitude ratio of $\mathcal{P}_{\delta}/\mathcal{P}_{\eta}\sim 1$, where $\mathcal{P}_{\eta}$ is the power in the curvature perturbation. As in the adiabatic case, density fluctuations are suppressed on small scales when $c_{s}^{2}\neq 0$. Since the curvature fluctuation is zero initially, there is the characteristic phase shift of the CMB peak positions typical of isocurvature modes compared to the adiabatic one, with the first angular peak at $\ell\sim 350$. For the values of $c_{\rm s}^{2}<10^{-4}$, we again find that the CMB anisotropies are affected at less than the $1\%$ level. When $w<0$ perturbation growth is suppressed and $T(k)$ is much flatter, with a fall off in power when $c_{\rm s}^{2}\neq 0$. This is because as these modes cross through the horizon the dark energy density is small, and so the growth in the curvature fluctuation is supressed. These are shown in Fig. 9, where we also plot the CMB anisotropies for a white noise power spectrum when $w=-1/3$ and $-2/3$.
IV.2 Vector Sector
Primordial vector modes constitute vortical perturbations in the early universe. The regular vector mode has been discussed in ref. Lewis:2004kg , along with a mode sourced by the anisotropic stress of a primordial magnetic field in ref Lewis:2004ef . We have also shown in section III.2 that a regular solution exists with a non-zero wordline displacement $\xi^{V}$, which corresponds to non-zero initial anisotropic stress in the elastic fluid.
IV.2.1 Regular Mode
In ref. Lewis:2004kg vector perturbations are expressed in the zero vorticity frame, which coincides with the synchronous gauge used here. The Einstein equation was written
$$k(\dot{\sigma}+2\mathcal{H}\sigma)=-8\pi Ga^{2}\Pi^{V},$$
(86)
where $\sigma$ is the harmonic coefficient of the shear tensor. The relationship $\dot{H}^{V}=-k\sigma$ gives the Einstein evolution equation (55c), and substituting $H^{V}_{1}=-k\sigma_{0}$ into the series solution in section III.2 returns the same result as in ref. Lewis:2004kg . The primordial power spectrum is defined by
$$\langle|\sigma|^{2}\rangle=\int d(\log k)\,\mathcal{P}_{\sigma}(k).$$
(87)
In Fig. 10 we plot the CMB TT anisotropies for multipoles $\ell<25$ for a scale invariant spectrum ($\mathcal{P}_{\sigma}\propto k^{0}$) with a primordial power amplitude ratio of $\mathcal{P}_{\sigma}/\mathcal{P}_{\eta}\sim 10^{-3}$.
When the vector sound speed is zero the elastic dark energy does not contribute to the anisotropic stress of the Einstein equation (86). We find that the large scale CMB power is then the same for both the $\Lambda$CDM model and elastic dark energy models with $c_{\rm v}^{2}=0$. However, $w<0$ requires a non-zero vector sound speed otherwise the scalar perturbations will be unstable to collapse, as shown by (45). In the $w=-1/3$ case, for example, $c_{\rm v}^{2}\geq 1/4$ is required for $c_{\rm s}^{2}\geq 0$. However, the effect of the anisotropic stress on the CMB anisotropies at large angular scales is relativity small, as shown in Fig. 10.
IV.2.2 Isocurvature Modes
Isocurvature modes can exist in an elastic fluid due to non-zero anisotropic stress. We quantity these modes by the spectrum
$$\langle|\Pi|^{2}\rangle=\int d(\log k)\,\mathcal{P}_{\Pi}(k).$$
(88)
In Fig. 11 we plot an example of this mode for $w=-1/3$, with a white noise power spectrum and a primordial power amplitude ratio of $\mathcal{P}_{\Pi}/\mathcal{P}_{\eta}\sim 10^{-3}$. The CMB anisotropies are significantly suppressed on small scales relative to large scales, and are similar to the scalar isocurvature mode. On small scales, the dark energy density is negligible at horizon crossing and the resulting growth in $\sigma$ is small. On larger scales, the dark energy density increases at horizon crossing resulting in larger growth in $\sigma$.
IV.3 Tensor Sector
In many inflationary models a nearly scale invariant spectrum of gravitational waves (tensor modes) is produced. These models are parameterized by the tensor spectral index $n_{t}$ and the tensor to scalar ratio $r=A_{t}/A_{s}$, where $A_{t}$ and $A_{s}$ are the primordial amplitude of tensor and scalar fluctuations and the tensor power spectrum is given by $\mathcal{P}_{t}(k)=A_{t}k^{n_{t}}$. The dominant tensor source to the CMB TT anisotropy is
$$C_{\ell}^{TT}=4\pi\int d(\log k)\mathcal{P}_{t}(k)\left[\int_{\tau_{\rm dec}}^%
{\tau_{\rm 0}}d\tau e^{-\kappa}\dot{H}^{T}\frac{j_{\ell}(k(\tau_{0}-\tau))}{(k%
(\tau_{0}-\tau))^{2}}\right]^{2}\frac{(\ell+2)!}{(\ell-2)!}\,.$$
(89)
For a scale invariant spectrum as $k\rightarrow 0$ the power in the photon Boltzmann hierarchy remains in the quadrupole and thus the tensor contributions have power enhanced on large angular scales. This is shown in Fig. 12, where we plot the TT anisotropy for the $\Lambda$CDM model with $n_{t}=0$ and $r=0.1$ at the pivot scale $k_{0}=0.05{\rm Mpc}^{-1}$. The elastic fluid contributes a source of anisotropic stress to the Einstein equation (55d), and has the effect of damping the evolution of $H^{T}$ and reducing the power on large angular scales. This is another potential signature of elastic dark energy models.
V Possible realizations of elastic dark energy models
One possible realization of the elastic fluid dark energy models is a frustrated network of non-Abelian cosmic strings Vilenkin:1984rt ; Kibble:1985tf ; Spergel:1996ai ; McGraw:1997nx or domain walls Battye:1999eq ; Friedland:2002qs . In this section we discuss some basic aspects of these scenarios and make attempts to link them with the earlier sections. In order to act as dark energy component, any model needs to provide significant negative pressure and have a Jeans length which is comparable to the present horizon, which can be achieved by having a relativistic sound speed.
The idea is that a static lattice of topological defects forms at some point after a phase transition in the Early Universe (see ref. vilenkin:1994s for a review of the physics of topological defects and phase transitions). Such a lattice will have microscopic, mesoscopic and macroscopic scales. The microscopic and mesoscopic scales refer to the core width and the lattice cell size respectively; we will not discuss these specifically here and will just assume that the relevant theory allows for the formation of a lattice which is stable to physics on these scales. Most important to the present discussion is the existence of some macroscopic scale, $L$.
On dimensional grounds the density of static cosmic strings scales as $\rho_{\rm str}\propto L^{-2}$ and that for domain walls is given by $\rho_{\rm dw}\propto L^{-1}$. The standard assumption of defect evolution based on simulations of the simplest field theoretic models is that a self-similar scaling regime is achieved (see, for example, ref. Battye:2006pf and references therein) whereby $L\propto t$. However, if a static, stable lattice forms, then the appropriate macroscopic length scale will scale with the expansion of the universe and $L\propto a(t)$. In this case the appropriate values of $w$ are $w=-1/3$ for strings and $w=-2/3$ for walls and these have been the values which we have focused on in our earlier discussions. It is possible that other values of $w$ might be possible if the equation of state of the walls is not Nambu-Goto. For example, the energy-momentum tensor of a string with energy density per unit length $U$ and tension $T$ is
$$T_{\mu\nu}(x)=\int d\sigma\,\delta[x-X(\sigma)]\,\left(U\epsilon\dot{X}_{\mu}%
\dot{X}_{\nu}-{T\over\epsilon}X^{\prime}_{\mu}X^{\prime}_{\nu}\right)\,,$$
(90)
where $\epsilon^{2}={\bf X}^{\prime\,2}/(1-\dot{\bf X}^{2})$ and $X_{\mu}=(t,{\bf X}(\sigma))$ is the position of the string. Using this one can deduce that
$$P_{\rm str}={1\over 3}\rho_{\rm str}\left[\left(1+{T\over U}\right)\langle v^{%
2}\rangle-{T\over U}\right]\,,$$
(91)
where $\langle v^{2}\rangle^{1/2}$ is the rms velocity of the strings. Hence, one finds that $w=-T/(3U)$ in the static limit. Since causality implies that $T/U\leq 1$ we have that $w\geq-1/3$, with $w=0$ in the non-relativistic limit $T<<U$. Similar arguments can be put together to show that $w\geq-2/3$ for walls.
The static defect lattice will behave like an elastic solid since it has rigidity. Recently, it has been shown Battye:2005hw that under the assumption that the lattice is isotropic $\mu/\rho=4/15$ for static Dirac-Nambu-Goto strings and walls. If the rigidity is comparable to this then the lattice will be stable to macroscopic perturbation modes since $c_{\rm s}^{2}$ and $c_{\rm v}^{2}$ are both positive and the Jeans length will be a substantial fraction of the horizon since $c_{\rm s},c_{\rm v}\sim 0.1$. It is clear that an exactly isotropic network would be difficult to form in cosmological phase transition since it will be uncorrelated on large scales. However, it might be possible for it to be approximately isotropic allowing for the treatment focused on in this paper to be applicable. An alternative is that the lattice has approximate point symmetry so the elasticity tensor will also have point symmetry as described in section II.7. The shear moduli for the Bravais lattices with cubic symmetry (the primitive lattices: simple cubic (SC), face-centred cubic (FCC) and body-centred cubic (BCC)) relevant to the domain wall case were discussed in ref. Battye:2005ik and it was shown that if $w=-2/3$ then the BCC lattice is unstable and the SC/FCC lattices have zero modes. It was argued that for the SC lattice this mode corresponds to perturbation of infinite extent, that is, one of the faces of the cubic system moving toward the another. A number of composite lattices with cubic symmetry which are part of the tetrahedral close-packing (TCP) structures (see, for example ref. kraynick:tcp ) are also possible, although none of these appear to be stable if $w=-2/3$. They could, however, be stable if $w=-2/3+\epsilon$ for $\epsilon$ small and positive. The values of shear moduli are tabulated in table 6 for known cubic lattices.
In addition to not being totally isotropic at formation (or even having exact point symmetry) a lattice may have initial macroscopic fluctuations relative to the equilibrium position which correspond to the isocurvature modes discussed in the previous section. Since the formation of the lattice is a causal process then these modes with have an initial white noise spectrum (${\cal P}_{i}(k)\propto k^{3}$).
Two interesting issues are the scale of symmetry breaking of the phase transition and the likely cell size of the lattice at the present day. Both of these rely on us understanding the lattice formation process which is not well-understood. A simple assumption would be that the lattice forms instantaneously after the phase transition with some initial cell size $\xi_{\rm c}=at_{\rm f}$ which is some fixed fraction of the horizon size at the time of formation $t_{\rm f}$. If $\eta$ is the symmetry breaking scale which relates to the temperature of formation $T_{\rm f}=b\eta$, then assuming the lattice forms in the radiation era the correlation size at formation is given by
$$\xi_{\rm c}(t_{\rm f})=\frac{0.3a}{b^{2}}\mathcal{N}^{-1/2}\frac{m_{\rm pl}}{%
\eta^{2}}\,,$$
(92)
where $\mathcal{N}$ is the number of relativistic species at formation. In the case of domain walls, if the wall density at formation is $\rho(t_{\rm f})=c\eta^{3}/\xi_{\rm c}$, and assuming the network is subsequently swept along by the Hubble flow, we can estimate $\eta$ to be
$$\eta=100{\,\rm keV}\left(\frac{a}{c\,b}\right)^{1/4}\left(\frac{\mathcal{N}}{1%
00}\right)^{-1/8}\left(\Omega_{\rm dw}h^{2}\right)^{1/4}\,.$$
(93)
Since the coefficients representing uncertainty in the wall formation process have small exponents, it seems reasonable to estimate $\eta\approx 100\,$keV if $\Omega_{\rm dw}h^{2}\sim 1$. One can also compute the present day cell size in a similar fashion, which turns out to be
$$\xi_{\rm c}(t_{0})=100{\,\rm pc}\left(\frac{a^{3}\,c}{b^{3}}\right)^{1/4}\left%
(\frac{\mathcal{N}}{100}\right)^{-3/8}\left(\Omega_{\rm dw}h^{2}\right)^{-1/4}\,.$$
(94)
Since the smallest wavenumber at which non-linear effects become important in CMB codes such as CAMB is $k\sim 0.2\,{\rm Mpc^{-1}}$, it seems reasonable to assume that the linear response to perturbations in such a defect network can be treated in the continuum elastic medium framework. The computation of the cell size is more sensitive to the uncertainty parameters, but since causality gives the upper limit of $a\lesssim 1$ the continuum medium description seems justified. One can also perform a similar exercise in the case of cosmic strings. If the wall density at formation is $\rho(t_{\rm f})=c\eta^{2}/\xi_{\rm c}^{2}$ then one finds $\eta\approx 2\,$TeV and $\xi_{\rm c}(t_{0})\approx 1.0\,$AU, with similar uncertainties in these parameters as in the domain wall case.
The allowed symmetry breaking scale may be increased further if the defect network forms during inflation. In this scenario the network is inflated outside the horizon and re-enters during a later epoch thus diluting the initial density. If one assumes that $e^{N}=T_{\rm R}/T_{\rm f}$ where $T_{\rm R}$ is the reheat temperature of inflaton and $N$ the number of e-folds remaining when the network leaves the horizon during inflation then the symmetry breaking scale of a domain wall network is given by
$$\eta={5\times 10^{-7}\,\rm GeV}\left(\frac{a}{c}\right)^{1/3}\left(\frac{%
\mathcal{N}}{100}\right)^{-1/6}\left(\frac{T_{\rm R}}{1\,\rm TeV}\right)^{-1/3%
}e^{N/3}\left(\Omega_{\rm dw}h^{2}\right)^{1/3}\,.$$
(95)
and the present day cell size is modified to
$$\xi_{\rm c}(t_{0})=10^{-5}{\,\rm pc}\,\,a\left(\frac{\mathcal{N}}{100}\right)^%
{-1/2}e^{N}\left(\frac{1\,\rm TeV}{T_{\rm R}}\right)\,.$$
(96)
If $N=30$ and $T_{\rm R}=1\,$TeV then $\eta\approx 10\,$MeV and $\xi_{\rm c}(t_{0})\approx 100\,$Mpc. One should note that quantum fluctuations in the defect forming field, $\psi$, should satisfy satisfy $\delta\psi\sim H<\eta$, otherwise no phase transition can occur during inflation. If $V\sim M^{4}$ this restricts the mass scale of inflation to $M<3\times 10^{8}\,$GeV.
If a lattice of domain walls or cosmic strings exists, their local gravitational interactions can give rise to further observable effects. The symmetry breaking scale $\eta$ is limited observationally by the local generation of density induced perturbations. In the case of domain walls, the mass per unit area is $\sigma\sim\eta^{3}$ and so $\eta=100\,$keV gives $\sigma\sim 5\times 10^{-8}\,\rm{kg\,m}^{-2}$, and for $\eta=10\,$MeV the wall has $\sigma\sim 5\times 10^{-2}\,\rm{kg\,m}^{-2}$. If there are only several walls in our current horizon they induce a density fluctuation $\delta\rho/\rho\sim G\sigma t_{0}$, with a similar sized fluctuations in temperature of the CMB. A wall with $\eta=100\,$keV would induce direct temperature fluctuations of $\delta T/T\sim 10^{-9}$, which is well below the observed value of $\delta T/T\sim 10^{-5}$. Using this constraint directly, one find that domain walls with $\eta\gtrsim 1\,$MeV would be ruled out Zeldovich:1974uw . However, this assumption assumes that the walls move relativistically, with only several walls in our horizon. In the lattice structures we have envisaged here the walls are static and only move with Hubble flow, and the CMB distortion in this case can be much smaller nambu:1991cmb . In the case of strings, the mass per unit length is given by $\mu\sim\eta^{2}$ and so $\eta=1\,$TeV gives $\mu\sim 9\times 10^{-6}\,\rm{kg\,m}^{-1}$. The CMB temperature distortion is of the order $\delta T/T\sim 8\pi G\mu$, and so low energy strings discussed here would not produce significant fluctuations in temperature.
VI Discussion and Conclusions
We have studied the cosmological implications of a perfect elastic fluid in the framework of General Relativity. In previous work on this subject, Bucher and Spergel derived the equations of motion by variation of the action assuming that the fluid is isotropic Bucher:1998mh . Here, we take a more general approach using the material representation concept and obtain equations of motion in terms of the pressure and elasticity tensors of the fluid. This allows us to parameterize the most general description of linearized perturbations of the fluid in a compact and transparent manner.
Under the assumption that the pressure and elasticity tensors are isotropic we derive the Einstein and energy-momentum conservation equations using this approach. When the fluid has non-zero rigidity there is a source of anisotropic stress which stabilizes perturbations when $w<0$, making these models candidates to describe the dark energy. The anisotropic stress also interacts with the vector and tensor sectors as the shear modulus induces transverse waves in the fluid. This phenomenon is well known in the laboratory. In a non-relativistic (low pressure) deformed medium with non-zero rigidity the temperature has both temporal and spatial variations. If the transfer of heat occurs slowly then the oscillatory motions in the deformed body are adiabatic. These motions correspond to longitudinal (scalar) and transverse (vector) elastic waves, which are related by $c_{\rm s}^{2}>4c_{\rm v}^{2}/3$ landau:1959 . In the relativistic (high pressure) treatment we enforce adiabicity by assuming that there are no temporal or spatial variations in $w$ and the relationship between the wave propagation modes is then given by $c_{\rm s}^{2}=w+4c_{\rm v}^{2}/3$.
We find that the elastic fluid model is similar to generalized dark energy models Hu:1998tj ; Hu:1998kj . The construction of the fluid energy-momentum tensor in these models was phenomenological, and it was argued that anisotropic stress can be attributed to viscosity in the fluid. We find that the functional form of anisotropic stress suggested in Hu:1998tj ; Hu:1998kj is identical to our own if the decay timescale associated with the stress term is infinite. In this case, the viscous sound speed can be directly related to the vector sound speed of a fluid with rigidity by $c_{\rm vis}^{2}=c_{\rm v}^{2}(1+w)/2.$
There appears to be two mechanisms for stabilizing dark energy perturbations if we wish to model the dark energy as a fluid. Either the fluid can be adiabatic and have rigidity, or it can be non-adiabatic and have entropy perturbations. A microphysical realization of the latter possibility is scalar field dark energy. As the evolution of perturbations is different in each model, we have investigated the potential observational differences in the CMB and matter power spectrum assuming the initial conditions were generated from curvature fluctuations in the metric. We find that the possibilities for distinguishing the two models are low using only CMB TT data. However, the TE power at large scales is reduced in the elastic fluid models relative to the $\Lambda$CDM and scalar field models with $\kappa_{\rm R}\neq 0$. This increases the parameter space in which the two models can be differentiated, up to $w\gtrsim-0.8$. As $w$ becomes closer to -1 the dark energy perturbations become less important, and the two models essentially become the same from an observational point of view.
The matter power spectrum outlines an interesting difference between the two models. An important aspect of the dark energy is that it has a large Jeans length so that it does not cluster on small scales. In principle though, it can cluster on large scales and we have attempted to model this by including dark energy perturbations in $P_{\delta}(k)$. As $c_{\rm s}\rightarrow 0$ we find that the elastic dark energy contribution to $P_{\delta}(k)$ dominates and $c_{\rm s}\gtrsim 10^{-3}$ would be required to be compatible with large scale structure data. In the case of scalar field dark energy the contribution to $P_{\delta}(k)$ from dark energy increases as $c_{\rm s}\rightarrow 0$ but remains sub-dominant even when $c_{\rm s}^{2}=0$.
We have also shown that both scalar and vector isocurvature modes are allowed in an elastic fluid. These modes correspond to initial fluctuations of the fluid relative to the background, and give rise to anisotropic stress on super-horizon scales. A generic feature of these modes is that power is suppressed on small relative to large scales. The dominant mode in these models would have to be the adiabatic curvature mode in order to fit the data but there could be a sub-dominant isocurvature component.
Our approach has shown that we can parameterize the perturbations of the energy momentum tensor in terms of the pressure and elasticity tensors of the fluid. In this work we mainly focused on the isotropic case, but this formalism can be easily extended if the elastic fluid has point symmetry. The symmetry properties of the elasticity tensor can be classified by the Bravais lattices, which are familiar in crystallography. Recently, there has been evidence of large scale anomalies in the CMB, most noticeably the North-South power asymmetry Eriksen:2003db and alignment of low $\ell$ multipoles Tegmark:2003ve ; deOliveira-Costa:2003pu ; Schwarz:2004gk ; Land:2005ad , which appear to persist in the third year WMAP data Copi:2006tu . A possible solution is that the energy momentum tensor of the dark energy is not rotationally invariant at linearized order. We have considered the case where the elasticity tensor has cubic symmetry and find that perturbation growth is dependent on the direction as well as the magnitude of the wave vector. This is work is developed further in ref. Battye:2006mb , and we are currently investigating whether these anisotropic models are compatable with the data.
Acknowledgements.
We would like to thank Martin Bucher, Brandon Carter and Elie Chachoua for useful discussions.
References
(1)
(2)
D.N. Spergel et al, astro-ph/0603449.
(3)
G. Hinshaw et al, astro-ph/0603451.
(4)
L. Page et al, astro-ph/0603450.
(5)
S. Perlmutter et al, Astrophys. J. 483 (1997) 565.
(6)
A. Riess et al, Astron. J 116 (1998) 1009.
(7)
S. Perlmutter et al, Astrophys. J. 517 (1999) 565.
(8)
A. Riess et al, Astrophys. J. 560 (2001) 49.
(9)
P. Astier et al, A$\&$A 447 (2006) 31.
(10)
U. Seljak and M. Zaldarriaga, Astrophys. J. 469 (1996) 437.
(11)
A. Lewis, A. Challinor and A. Lasenby, Astrophys. J. 538 (2000) 473.
(12)
S. M. Carroll, W. H. Press, E. L. Turner, Ann. Rev. Astron. Astrophys 30 (1992) 499.
(13)
S. M. Carroll, Living Rev. Rel. 4 (2001) 1.
(14)
E. J. Copeland, M. Sami and S. Tsujikawa, hep-th/0603057.
(15)
B. Ratra and P. J. E. Peebles, Phys. Rev. D 37 (1988) 3406.
(16)
C. Wetterich, Nucl. Phys. B 302 (1988) 668.
(17)
R. R. Caldwell, R. Dave and P. J. Steinhardt, Phys. Rev. Lett. 80 (1998) 1582.
(18)
A. Vilenkin, Phys. Rev. Lett. 53 (1994) 1016.
(19)
T.W.B. Kibble, Phys. Rev. D 33 (1986) 328.
(20)
M. S. Bucher and D. N. Spergel, Phys. Rev. D 60 (1999) 043505.
(21)
R.A. Battye, M. Bucher and D.N. Spergel, astro-ph/9908047.
(22)
L. Amendola, Phys. Rev. D 62 (2000) 043511.
(23)
G. R. Dvali, G. Gabadadze and M. Porrati, Phys. Lett. B 485 (2000) 208.
(24)
S. Capozziello, S. Carloni and A. Troisi, astro-ph/0303041.
(25)
S. M. Carroll, V. Duvvuri, M. Trodden and M. S. Turner, Phys. Rev. D 70 (2004) 043528.
(26)
S. M. Carroll et al, Phys. Rev. D 71 (2005) 063513.
(27)
J. Weller and A. Lewis, MNRAS 346 (2003) 987.
(28)
R. Bean and O. Dore, Phys. Rev. D 69 (2004) 083503.
(29)
R.A. Battye and A. Moss, JCAP 0506 (2005) 001
(30)
W. Hu, Astrophys. J. 506 (1998) 485.
(31)
W. Hu and D. Eisenstein, Phys. Rev. D 59 (1999) 083509.
(32)
T. Koivisto and D. F. Mota, Phys. Rev. D 73 (2006) 083502.
(33)
B. Carter. and H. Quintana, Proc. Roy. Soc. A331 (1972) 57.
(34)
B. Carter and H. Quintana, Phys. Rev. D 16, (1977) 2928.
(35)
B. Carter, Proc. Roy. Soc. A372 (1980) 169.
(36)
B. Carter, gr-qc/0102113.
(37)
J. L. Friedman and B. Schutz, Astrophys. J. 200 (1975) 204.
(38)
B. Carter, Phys. Rev. D 7 (1973) 1590.
(39)
E. M. Lifshitz and I. M. Khalatnikov, Adv. Phys. 12 (1963) 185.
(40)
J. M. Bardeen , Phys. Rev. D 22 (1980) 1882.
(41)
L. F. Abbott and R. K. Schaefer, Astrophys. J. 308 (1986) 546.
(42)
W. Hu, U. Seljak, M.J. White and M. Zaldarriaga, Phys. Rev. D 57 (1998) 3290.
(43)
C.P Ma and E. Bertschinger, Astrophys. J. 455 (1995) 7.
(44)
L.D. Landau, E.M. Lifshitz, Theory of Elasticity (Pergamon, London, 1959).
(45)
R.A. Battye and A. Moss, Phys. Rev. D 74 (2006) 041301
(46)
R. A. Battye, E. Chachoua and A. Moss, Phys. Rev. D 73 (2006) 123528.
(47)
M. Bucher, K. Moodley and N. Turok, Phys. Rev. D 62 (2000) 083508.
(48)
A. Lewis, Phys. Rev. D 70 (2004) 043518.
(49)
A. Lewis, Phys. Rev. D 70 (2004) 043011.
(50)
R. Bean, J. Dunkley and E. Pierpaoli, astro-ph/0606685.
(51)
D. Spergel and U. Pen, Astrophys. J. 491 (1997) L67.
(52)
P. McGraw, Phys. Rev. D 57 (1998) 3317.
(53)
A. Friedland, H. Murayama and M. Perelstein, Phys. Rev. D 67 (2003) 043519.
(54)
A. Vilenkin and E. P. S. Shellard, Cosmic Strings and other Topological Defects (Cambridge University Press, 1994).
(55)
R. A. Battye and A. Moss, Phys. Rev. D 74 (2006) 023528.
(56)
R.A. Battye, B. Carter, E. Chachoua and A. Moss, Phys. Rev. D 72 (2005) 023503.
(57)
D. A. Reinelt and A. M. Kraynik, J. Rheol. 44 (2000) 453.
(58)
Y. B. Zeldovich, I. Y. Kobzarev and L. B. Okun, Zh. Eksp. Teor. Fiz. 67 (1974) 3.
(59)
Y. Nambu, H. Ishihara, N. Gouda and N. Sugiyama Astrophys. J. 375 (1991) L35.
(60)
H.K. Eriksen et al, ApJ 605 (2004) 14.
(61)
M. Tegmark, A. de Oliveira-Costa and A.J.S. Hamilton, Phys. Rev. D68 (2003) 123523.
(62)
A. de Oliveira-Costa, M. Tegmark, M. Zaldarriaga and A.Hamilton, Phys. Rev. D69 (2004) 063516.
(63)
D.J. Schwarz, G.D. Starkman, D. Huterer and C.J. Copi, Phys. Rev. Lett. 93 (2004) 221301.
(64)
K. Land and J. Magueijo Phys. Rev. Lett. 95 (2005) 071301.
(65)
C.J. Copi, D. Huterer, D.J. Schwarz and G.D. Starkman, astro-ph/0605135. |
Characterization of the Aging and Excess Noise of a Hamamatsu Fine Mesh Photopentode
D. Fujimoto
fujimoto@phas.ubc.ca
C. Hearty
hearty@physics.ubc.ca
University of British Columbia, 6224 Agricultural Road, Vancouver, BC, V6T 1Z1
Abstract
The excess noise factor and the aging characteristics of 16 Hamamatsu R11283 photopentodes have been tested. These fine-mesh phototubes are to be paired with pure CsI scintillation crystals considered for use in the endcap calorimeter of the Belle II detector. The average excess noise factor was found to be $1.9\pm 0.1\pm 0.4$. The electronic noise of a custom preamplifier produced by the University of Montreal was found as a consequence of this measurement and was $1730\pm 33$ electrons, in agreement with previous values. On average, the gain $\times$ quantum efficiency was reduced to $92\pm 3$ % of the initial value after passing an average of $7$ C through the anode. This corresponds to 70 years of standard Belle II operation.
keywords:
Hamamatsu, Photomultiplier, Mesh, CsI, Belle II, Aging
††journal: Nuclear Instruments and Methods in Physics Research A
1 Introduction
One of the upgrade considerations for the Belle II detector, situated at KEK in Tsukuba, Japan, is to replace the current thallium-doped cesium iodide (CsI(Tl)) scintillation crystals in the endcap electromagnetic calorimeter (ECL) with pure CsI (2). A primary goal of this exchange is to reduce pileup due to the increased luminosity of the SuperKEKB accelerator. While pure CsI has a shorter scintillation time constant, it also has a reduced light yield, with the emission spectrum peaking in the UV range rather than in the visible range (3; 4). Therefore, the new crystals will need new photosensors. Under consideration for the new photosensor is the R11283 photomultiplier tube (PMT), developed by Hamamatsu Photonics for this project. This model has five flying leads and for this reason is often referred to as a photopentode. The PMT is a head-on type with three fine mesh dynodes, UV transparent window, and a bialkali photocathode; similar to previously tested PMTs for CsI scintillation crystals, although this model is of much lower gain and has fewer dynodes (5). The photocathode of the R11283 has a minimum effective diameter of $39$ mm, and a wavelength of maximum response of $420$ nm (6). Pure CsI has an emission maximum at $315$ nm. The PMT is shown in Fig.3 1.
Magnetic fields decrease the performance due to changes in the inter-dynode electron path. These fine mesh PMTs will operate in the 1.5 T Belle II axial magnetic field, reducing the nominal gain by a factor of 3.5 (7). The average nominal gain of the 16 PMTs at an operating voltage of $-1000$ V was $255\pm 11$. As summarized in Table 1, Hamamatsu provides a variety of measurements at $-750$ V with purchase.
The PMT readout electronics were designed and produced by the University of Montreal (1) and consists of a preamp (version 4) and a shaper for every PMT. The preamp has a gain of $0.5$ V/pC and the shaper has a shaping time of 50 ns. For a step-function input, the signal produced by the shaper has a peaking time of $200$ ns. The combination of the two components produces a signal whose amplitude is proportional to the charge at the anode (Fig. 2). The board connected directly to the PMT (Fig. 1) houses both the voltage divider to power the PMT dynodes, and the preamp electronics. The shapers house and are powered by a motherboard which also provides the correct voltages to the preamp electronics.
Additional details on the measurements presented can be found in Ref. (8).
2 Methods
The shaper output was fed into a peak-sensing ADC (LeCroy L2259B) and the output was histogrammed using the MIDAS program (9). The histogram was then fitted with the sum of an exponential and a Novosibirsk function. The exponential roughly describes the background, which was primarily due to backscatter. The Novosibirsk function is an asymmetric Gaussian-like function with four parameters: height, peak location, width, and an asymmetry parameter (10). Fig. 3 shows an example of fitting this sum to the spectrum produced by the $662$ keV decay of ${}^{137}$Cs.
The response of the system to energy deposits in pure CsI is linear. Using several calibration sources, the value of the peak location at zero energy deposited was extrapolated and attributed to a DC offset in the peak-sensing ADC. This pedestal was then subtracted from the measurements prior to any further manipulation. The charge at the anode is proportional to the peak location, and the uncertainty to the width of this distribution. To calibrate the charge at the anode, the calibration test pulse feature of the preamp was used. This allows for a known amount of charge to be injected into the preamp, which is then processed as every other signal. From this calibration, the relationship between the ADC binning and the charge at the anode was found.
3 Excess Noise Factor
The excess noise factor is a common index for estimating the performance of photosensors (11). This factor describes the uncertainty introduced into the system as a result of the electron multiplication process:
$$\left(\frac{\sigma_{a}}{N_{a}}\right)^{2}=F\cdot\left(\frac{\sigma_{c}}{N_{c}}\right)^{2},$$
(1)
where the subscripts $c$ and $a$ denote the cathode and anode respectively. Measured at location $x$, $N_{x}$ and $\sigma_{x}$ are the number of electrons and the signal width respectively, as determined from the Novosibrisk fit. $F$ is the excess noise factor. Typically, the excess noise factor is larger for fine mesh PMTs than standard PMTs. Recognizing the Poisson nature of the photoelectrons and that the measured width at the anode contains contributions from both the constant electronic noise and the excess noise factor, Equation 1 can be written as:
$$\sigma^{2}_{m}=F\cdot N_{c}+\sigma_{o}^{2},$$
(2)
where the internal PMT gain has been applied to put all relevant quantities in units of number of electrons at the photocathode (photoelectrons). Here, $\sigma_{m}$ is the measured width at the anode and $\sigma_{o}$ is the contribution of the electronic noise. From this, the excess noise factor can be easily found by varying the light intensity ($N_{c}$) and measuring the resulting distribution width ($\sigma_{m}$).
To this end, a $405$ nm laser was pulsed at $350$ Hz to illuminate the PMT (Fig. 4). The laser was first reflected off of a diffusive white screen to provide uniform light. The light intensity was controlled via the laser voltage and the screen-to-PMT distance. Using a single preamp, the excess noise factor was found for all 16 PMTs.
Fig. 5 shows the results of this analysis for one of the PMTs. The slope is the excess noise factor, and the intercept is the electronic noise in units of photoelectrons. The operating voltage for these measurements was $-1000~{}$V. On average, the excess noise factor was found to be $1.9\pm 0.1\pm 0.4$, where the statistical error of $0.1$ is the standard deviation across the 16 PMTs. The systematic error of $0.4$ is due to the $25\%$ uncertainty in the value of the capacitor used in the test pulse calibration (1). The average electronic noise was $1730\pm 33$ electrons at the anode, which is in good agreement with previous measurements (1). The range of the excess noise factors was $1.8$ – $2.1$, whereas the range of the electronic noise was $1526$ – $1913$ electrons at the anode. Given that the light yield of the CsI crystal is $85$ photoelectrons per MeV deposited (12), this electronic noise corresponds to an equivalent noise energy of about $80$ keV.
To estimate the impact of magnetic fields on the excess noise, the PMT gain was reduced by lowering the operating voltage. Above a gain of 55, the PMT was seen to have a constant excess noise factor. Below this, the excess noise factor increases non-linearly with decreasing gain, rising to $3.5\pm 0.1\pm 0.4$ at a gain of 25. In comparison, two avalanche photodiodes (APD) from the Hamamatsu S8664 series have been measured to have an excess noise factors of 3.4 and 5.1 (13). These APDs are also being studied for pure CsI application as a competing option for the Belle II endcap ECL upgrade.
4 Aging
Of importance to the Belle II experiment is the effects of the PMT aging. Given that the Belle II endcap ECL will be in an axial magnetic field of approximately $1.5$ T, and that the PMTs will be within $12.4^{\circ}$–$31.4^{\circ}$ to this field (14), it is expected that the gain of the PMTs will drop by about a third (7). To simulate this, the aging process was performed with the operating voltage of the PMTs set to $-491$ V, reducing the average gain to $85\pm 3$ or one third of the nominal gain at $-1000~{}$V.
The performance of the PMT was characterized by the gain$\times$quantum efficiency, which was found from the slope of the peak ADC bin as a function of the energy deposited in the CsI. The change in the gain$\times$quantum efficiency relative to the initial value was monitored as a function of the charge passed through the anode, and also as a function of the real lab time elapsed. Light from a UV LED (335 nm) was used to age the PMTs, which were arrayed in a 4$\times$4 array (Fig. 6). The PMTs were encased in an incubator with a UV transparent acrylic window to maintain the temperature at $37\pm 2~{}^{\circ}$C. The relative humidity was kept within $15$ – $20$ % by means of desiccant and a slow influx of N${}_{2}$ gas into the incubator. One of the 16 PMTs was capped with black rubber to prevent aging and act as a control. This PMT was also used to correct for the residual variation in temperature. The relative peak location response with temperature was roughly linear and varied at a rate of $-1.2\pm 0.2~{}\%/^{\circ}$C. Probably due to the poor thermal contact between dynodes and the environment exterior to the PMT, there is about a $8$ – $10$ h delay before the PMTs reached thermal equilibrium.
To track the performance, a constant light source was produced by gluing pure CsI pucks to the faces of the PMTs (Fig. 1) and triggering scintillation light with a ${}^{207}$Bi source. The glue used was TSE3032 silicone rubber produced by Momentive Performance Materials, which has an index of refraction of $1.406$ for $589$ nm light (15). In comparison, the index of refraction of pure CsI is $1.95$ at the emission maximum of $315$ nm (3). The CsI cylindrical pucks were manufactured by AMCRYS and were approximately the same diameter as the photocathode.
The measurement of the current was enabled by a modified preamp from the University of Montreal. This preamp did not perform any of the signal processing of the regular preamps, but rather contained a resistor such that the current could be measured with a Keithley 6485 Picoammeter. Prior to aging, a current baseline was established for each PMT using constant incident light. Using the baseline, only one PMT was needed to track the current through the anode and the current could be estimated for the other PMTs.
The ${}^{207}$Bi source was chosen for its two easily visible decays at $0.570$ MeV and $1.064~{}$MeV. Given the linear response of the PMT with energy, these two peaks were used to establish the gain$\times$quantum efficiency, which can be seen as a function of the integrated current in Fig. 7.
In Fig. 7, the PMT in the third column of the first row (PP03) is the control PMT and was not aged. There are a few different observed behaviors in the PMT aging. Some exhibited a burn-in period where the performance decreased rapidly during the first coulomb of charge, then appeared to stabilize. Some appeared to age continuously throughout, whereas others exhibited little to no aging or even experienced an improvement in the performance. The control PMT did not see any significant change in performance at the end of the aging process.
After 48 days of aging, the LED was turned off and the stability monitored (Fig. 8). Even after aging, some of the PMTs continued to see a decrease in performance. Of note, one of the 14 aged PMTs (PP02) saw nearly 10 % decrease in performance within a week. This degradation is worrying as it is rapid enough to be difficult to track via a physics-based calibration system. The control PMT did not observe any significant change in performance after the end of the aging process. At the end of the aging process an average of $7.4\pm 1.2$ C was passed through the PMT anodes and the average PMT performance was reduced to $92\pm 3$ % of the initial value.
By definition, each PMT has a relative performance of 1 at zero charge. Other than at this point, the PMTs appear to be scattered normally about the mean (Fig. 9). Due to the internal gain of the PMTs, an equal amount of incident light does not produce the same number of electrons at the anode, and as a result there are fewer statistics available at the larger anode charges.
Fig. 10 shows the average gain$\times$quantum efficiency as a function of charge, with an envelope denoting the RMS. These values were found by taking a linear interpolation to account for the gaps in the measurement. As mentioned earlier, not all of the PMTs saw an equal amount of charge. At $5.6$ C, the total cumulative charge passed through the anode of the lowest gain PMT, the relative gain$\times$quantum efficiency was $95\pm 2$ % across all 14 PMTs. As was seen in Fig. 9, the RMS grows steadily with anode charge.
Empirically, the sum of a linear and exponential function was chosen to be fitted to the curve in Fig. 10 in the range of $[0,5.5]$ C, using the error in the mean as the fit weighting. The resulting function is given by
$$[\mbox{Rel. Gain}\times\mbox{QE}]=0.968-0.0037q+0.0383e^{-q/0.23}.$$
(3)
This shows that there exists a burn-in period, modelled by the exponential term, after which the PMT experiences a linear decay in performance of $-0.4$ %/C. The burn-in period lasts for $1.05$ C, after which the exponential factor scales its coefficient by less than $1$ %. By $1.57$ C the exponential contributes less than $0.1$ % to the function.
5 Summary
The initial quantities measured by Hamamatsu prior to shipping can be found in Table 1. Using processing electronics developed by the University of Montreal, the excess noise factor and aging properties of the R11283 photopentode were studied. The average excess noise factor for 16 PMTs was found to be $1.9\pm 0.1\pm 0.4$ and the average electronic noise was measured to be $1730\pm 33$ electrons at the anode. This electronic noise corresponds to an equivalent noise energy of 80 keV. The phototube aging was also studied by passing a large amount of charge through the anode, by exposing the photocathode to a large amount of incident light. At a reduced gain of $85\pm 3$, 14 PMTs were aged with an average of $7.4\pm 1.2$ C passed through the anode, reducing the average performance of the PMTs to $92\pm 3$ % of the initial measurement. Of the 14 aged PMTs, only one showed signs of rapid aging that could be a problem for some calibration systems. The average change in performance is characterized by an exponential burn-in period that lasts approximately $1.05$ C, after which the performance degrades linearly by $-0.4$ %/C.
Acknowledgments
This work was supported by the technical support staff at TRIUMF, in particular P. Amaudruz who developed the pulsed UV laser and aided in the setup of MIDAS. Additionally, J.P. Martin, N.A. Starinski, and P. Taras of the University of Montreal designed and produced the preamp, motherboard, and shaper electronics. D. Jow aided in the gluing of the CsI to the PMTs. Funding for this work was provided by NSERC.
References
References
(1)
J.P. Martin, N. Starinski, P. Taras, Fast charge-sensitive preamplifier for pure CsI crystals, Nucl. Instrum. Methods A 778 (2015) 120.
(2)
Belle II Collaboration, Belle II technical design report, $\langle$http://xxx.lanl.gov/abs/1011.0352$\rangle$, 2011.
(3)
Saint-Gobain Ceramics & Plastics, Inc., CsI(pure) cesium iodide scintillation material, Technical Report, $\langle$http://www.crystals.saint-gobain.com/uploadedFiles/SG-Crystals/Documents/CsI%20Pure%20Data%20Sheet.pdf$\rangle$, 2007 (accessed 03.08.15).
(4)
Saint-Gobain Ceramics & Plastics, Inc., CsI(Tl), CsI(Na) cesium iodide scintillation material, $\langle$http://www.crystals.saint-gobain.com/uploadedFiles/SG-Crystals/Documents/CsI(Tl)%20and%20(Na)%20data%20sheet.pdf$\rangle$, 2007 (accessed 03.08.15).
(5)
T. K. Komatsubara, et al., Performance of fine-mesh photomultiplier tubes designed for an undoped-CsI endcap photon detector, Nucl. Instrum. Methods A 404 (1998) 315.
(6)
Hamamatsu Photonics K.K. Electron Tube Division, Photomultiplier Tube R11283 Technical Data, Personal Communication, 2013.
(7)
A. Kuzmin, Endcap calorimeter for SuperBelle based on pure CsI crystals, Nucl. Instrum. Methods A 623 (1) (2010) 252–254, http://dx.doi.org/10.1016/j.nima.2010.02.212.
(8)
D. Fujimoto, A low gain fine mesh photomultiplier tube for pure CsI (Master’s thesis), University of British Columbia, 2015.
(9)
S. Ritt, et al., Maximum integrated data acquisition system, $\langle$https://midas.triumf.ca$\rangle$, 1993.
(10)
H. Ikeda, et al., A detailed test of the CsI(Tl) calorimeter for BELLE with photon beams of energy between 20 MeV and 5.4 GeV, Nucl. Instrum. Methods A 441 (3) (2000) 401.
(11)
Hamamatsu Photonics K.K., Photomultiplier tubes: basics and applications, $\langle$http://www.hamamatsu.com/resources/pdf/etd/PMT_handbook_v3aE.pdf$\rangle$, 2007.
(12)
C. Hearty, Pure CsI light output and resolution studies, in: 17th Belle II General Meeting, $\langle$https://kds.kek.jp/indico/event/14531/session/73/contribution/251/material/slides/0.pdf$\rangle$, 2014.
(13)
Y. Jin, et al., Study of a pure CsI crystal readout by APD for Belle II end cap ECL upgrade, Nucl. Instrum. Methods A, $\langle$http://dx.doi.org/10.1016/j.nima.2015.07.034$\rangle$.
(14)
Belle Collaboration, The Belle detector, Nucl. Instrum. Methods A 479 (2002) 117–232.
(15)
Momentive Performance Materials, TSE3032 technical data sheet, $\langle$https://www.momentive.com/products/showtechnicaldatasheet.aspx?id=10404$\rangle$, 2015 (accessed 08.07.15). |
Exact Solutions of a (2+1)-Dimensional
Nonlinear
Klein-Gordon Equation
F. Güngör
Department of Mathematics, Faculty of Science
Istanbul Technical University
80626, Istanbul, Turkey
gungorf@itu.edu.tr
()
Abstract
The purpose of this paper is to present a class of particular
solutions of a C(2,1) conformally invariant nonlinear Klein-Gordon
equation by
symmetry reduction. Using the subgroups of similitude group reduced
ordinary differential equations of second order and their solutions by a
singularity analysis are classified. In particular, it has been shown
that whenever they have the
Painlevé property, they can be transformed to standard forms by
Moebius transformations of dependent variable and arbitrary smooth
transformations of independent variable whose solutions, depending on the
values of parameters, are expressible in terms
of either elementary functions or Jacobi elliptic functions.
PACS numbers: 02.20.-a, 02.30.Jr, 02.30.Hq, 03.50.-z
1 Introduction
In a recent paper [1], we constructed second order differential equations
invariant under the Poincaré, similitude and conformal groups in
$(2+1)$-dimensional space-time. For instance, the planar nonlinear
Klein-Gordon (NLKG) equation
$$\square_{3}u=H(u),$$
where $H$ is an arbitrary sufficiently smooth function of its argument and
$\square_{3}=\partial_{t}^{2}-\Delta_{2}$ is the wave operator in the
(2+1)-dimensional Minkowski space,
is invariant under Poincaré group P(2,1) and more specifically, NLKG
equation with a power nonlinearity
$$\square_{3}u=au^{k}$$
(1.1)
is invariant under the Poincaré group extended by dilations, also called
similitude group. Among equations of the form (1.1), the special case $k=5$ plays a
privileged role in classical and quantum field theory.
In this context Eq. (1.1) with $k=5$, namely
$$\square_{3}u=au^{5}$$
(1.2)
where $a$ is an arbitrary constant,
arises as the equation of motion (Euler-Lagrange equation)
obtained minimizing the action corresponding to a Lagrangian density
$${\cal L}=1/2(\nabla u)^{2}-6au^{5}=1/2(u_{t}^{2}-u_{x}^{2}-u_{y}^{2})-6au^{5}.$$
This equation is also called classical $\phi^{6}$-field equation.
For further physical motivation of this equation, the reader is referred to
Ref. [2]
Another remarkable property of Eq. (1.2) is that,
in addition to being similitude invariant, it is also
invariant under the conformal group C(2,1) of space-time.
It will be of interest to find exact solutions of the NKLG equation
(1.2),
not only from mathematical but also from physical point of view.
While there exists an extensive literature on exact solutions of the
NLKG equation they are mainly devoted to translation invariant solutions
in 1+1 dimension. The study of exact solutions in
higher dimensions appears in a few papers only. For example, in
[2] exact solutions are studied in (3+1)-dimensional case.
Another article dedicated to A.O.Barut [3] investigated
translationally
invariant solutions which actually live in a (1+1)-dimensional case and
static spherically symmetric and similarity solutions in 3+1 dimensions.
To our knowledge, there exists no systematic study of exact solutions of
the NKLG equation in 2+1 dimensions and the present paper aims at
obtaining exact solutions of the NLKG equation.
The method to be used for solving (1.2) is the symmetry method.
This method is described in various books [4, 5] and
lectures [6, 7]. Applications of the method to
find exact solutions of numerous equations of nonlinear mathematical
physics are given in [8].
The first step is to reduce the
considered PDE to an ODE expressed in terms of symmetry variables. Next,
we integrate, whenever possible, the obtained ODEs.
For the NLKG equation (1.2), except for
degenerate cases simplifying either to an algebraic or to a first order
equation that can be integrated directly, most of the reduced ODEs will
be second order and nonlinear.
Often, two approaches are adopted for solving these ODEs. The first
is to find the symmetry group of the reduced equation, if one exists, and
then using it to lower the order of the equation.
The second one consists of performing
a singularity analysis in order to establish whether the equation is of
the Painlevé type meaning that the general solution of the
corresponding equation has no movable singularities (branch points,
essential singularities) other than the poles.
Note that the second method gives more satisfactory results than the first
one. while the equation has no nontrivial symmetries, it may well belong to
the class of the Painlevé type equations. Equations of the form
$$w^{\prime\prime}(z)=f(z,w,w^{\prime})$$
where $f$ is analytic in $z$, rational in $w^{\prime}$, and algebraic in $w$,
possessing the
Painlevé property was classified by Painlevé and Gambier. They
showed that such an ODE can be reduced to one of the 50 equations
listed in [9, 10] whose solutions can be expressed in terms of
either elementary or Jacobi elliptic functions or of solutions of linear
equations. Six of them are irreducible and known to be Painlevé
transcendents which often occur in a host of physical problems.
Equations that do not possess the Painlevé property will in general
contain moving (logarithmic) singularities and we have no systematic
method for integrating them.
In order to obtain in a systematic manner symmetry reductions of Eq.
(1.2) we need a classification of subgroups of the symmetry group
into conjugacy classes under the action of the symmetry group.
The subgroups of similitude groups are known only for small dimensions,
while those of Poincaré groups having generic orbits of codimension one are
known for arbitrary dimension.
The subgroups of S(2,1) are classified in [11]. Making
use of this result we obtain all possible reduced ODEs, mostly of second order
nonlinear. In order to facilitate the tedious computations the MATHEMATICA
package has been used.
2 The Symmetry Group and its Lie Algebra
The symmetry group of (1.2) called similitude group or
extended Poincaré group S(2,1) is the group of
transformations leaving the Lorentz metric form invariant. Its structure
is
$$S(2,1)=D\vartriangleright(\operatorname{SL}(2,\mathbb{R})\vartriangleright T_{%
3})$$
(2.1)
where $T_{3}$ are space-time translations, $\operatorname{SL}(2,\mathbb{R})$ is the special
linear group, $D$ dilations and $\vartriangleright$ denotes a
semi-direct product.
A convenient basis for the Lie algebra of the extended Poincaré group
is given by translations $\{P_{0},P_{1},P_{2}\}$, Lorentz boosts
$\{K_{1},K_{2}\}$, rotation $L_{3}$, and dilation $D$. For Eq. (1.2)
the basis is realized as
$$\begin{array}[]{llll}&P_{0}=\partial_{t},\quad P_{1}=\partial_{x},&&P_{2}=%
\partial_{y},\\
&K_{1}=t\partial_{x}+x\partial_{t},&&K_{2}=t\partial_{y}+y\partial_{t}\\
&L_{3}=y\partial_{x}-x\partial_{y},&&D=x\partial_{x}+y\partial_{y}+t\partial_{%
t}-u/2\partial_{u}.\end{array}$$
(2.2a)
In addition to generators (2.2a), Eq. (1.2) has also
conformal symmetries generated by
$$\begin{array}[]{lll}&C_{0}=2xt\partial_{x}+2yt\partial_{y}+(x^{2}+y^{2}+t^{2})%
\partial_{t}-tu\partial_{u}\\
&C_{1}=(t^{2}+x^{2}-y^{2})\partial_{x}+2xy\partial_{y}+2xt\partial_{t}-ux%
\partial_{u}\\
&C_{2}=2xy\partial_{x}+(t^{2}-x^{2}+y^{2})\partial_{y}+2yt\partial_{t}-uy%
\partial_{u}.\end{array}$$
(2.2b)
3 Symmetry Reductions of NLKG Equation
In this section we give a classification of symmetry reductions of
(1.2) with respect to invariance under the similitude group.
Applying the method of symmetry reduction we will derive all the
solutions invariant under subgroups with generic orbits of codimension 1
in the space of independent variables.
However for the sake of completeness, first, we obtain all reductions of
(1.2) to lower dimensional equations with two independent
variables.
3.1 Symmetry Reductions to
PDEs in Two Variables
We restrict ourselves to subgroups with generic orbits of codimension
two in the space-time $(x,y,t)$ and of codimension three in the space $(x,y,t,u)$.
The corresponding three invariants $I_{i}(x,y,t,u),\;i=1,2,3$ of the group action
on $X\otimes U$ must provide an invertible transformation from the space of
dependent variables to that of the invariants. Hence the invariants of
the subgroup
$H$ of the symmetry group can be written in the form
$$\begin{split}\displaystyle I_{1}&\displaystyle=\xi(x,y,t),\qquad I_{2}=\eta(x,%
y,t),\\
\displaystyle I_{3}&\displaystyle=f(x,y,t,u)=u\tilde{\phi}(x,y,t).\end{split}$$
This permits us to reduce (1.2) to a partial differential equation for $f(\xi,\eta)$ which is
a function of the symmetry variables $\xi$ and $\eta$ and write the solution of
(1.2) as
$$u(x,y,t)=f(\xi,\eta)\phi(x,y,t)$$
(3.1)
where $\phi$, $\xi$ and $\eta$ are known functions whose precise forms are to
be determined by the choice of subgroup. Substituting the reduction
formula (3.1) into (1.2) we obtain a partial differential equation for the function $f$.
The classification of all subgroups of the similitude group is well known
[11].
Using these classification results we classify symmetry reductions.
In the following we give reduction formulas and reduced equations for all
possible subgroups:
(1)
Subgroup $\{K_{2}+L_{3}\}:$
$$\begin{split}&\displaystyle u=f(\xi,\eta),\qquad\xi=x+t,\qquad\eta=y^{2}-2t(x+%
t),\\
&\displaystyle 4(\xi^{2}-\eta)f_{\eta\eta}-4\xi f_{\xi\eta}-6f_{\eta}+af^{5}=0%
.\end{split}$$
(3.2)
(2)
Subgroup $\{K_{1}\}:$
$$\begin{split}&\displaystyle u=f(\xi,\eta),\qquad\xi=t^{2}-x^{2},\\
&\displaystyle 4\xi^{2}f_{\xi\xi}-f_{yy}+4f_{\xi}-af^{5}=0.\end{split}$$
(3.3)
(3)
Subgroup $\{D\}:$
$$\begin{split}&\displaystyle u=t^{-1/2}f(\xi,\eta),\qquad\xi=\frac{y}{t},\qquad%
\eta=\frac{x}{t},\\
&\displaystyle 4(\xi^{2}-1)f_{\xi\xi}+4(\eta^{2}-1)f_{\eta\eta}+8\xi\eta f_{%
\xi\eta}\\
&\displaystyle+12\xi f_{\xi}+12\eta f_{\eta}+3f-4af^{5}=0.\end{split}$$
(3.4)
(4)
Subgroup $\{P_{2}\}:$
$$\begin{split}&\displaystyle u=f(x,t),\\
&\displaystyle f_{tt}-f_{xx}-af^{5}=0,\qquad y\text{-independent equation}.%
\end{split}$$
(3.5)
(5)
Subgroup $\{L_{3}\}:$
$$\begin{split}&\displaystyle u=f(\xi,t),\qquad\xi=(x^{2}+y^{2})^{1/2},\\
&\displaystyle f_{tt}-f_{\xi\xi}-1/\xi f_{\xi}-af^{5}=0.\end{split}$$
(3.6)
(6)
Subgroup $\{P_{0}\}:$
$$\begin{split}&\displaystyle u=f(y,t),\\
&\displaystyle f_{tt}-f_{yy}-af^{5}=0,\qquad x\text{-independent equation}.%
\end{split}$$
(3.7)
(7)
Subgroup $\{P_{0}-P_{1}\}:$
$$\begin{split}&\displaystyle u=f(\xi,y),\qquad\xi=x+t,\\
&\displaystyle f_{yy}+bf^{5}=0.\end{split}$$
(3.8)
(8)
Subgroup $\{K_{1}+P_{2}\}:$
$$\begin{split}&\displaystyle u=f(\xi,\eta),\qquad\xi=t^{2}-x^{2},\qquad\eta=(x+%
t)e^{-y},\\
&\displaystyle\eta^{2}f_{\eta\eta}-4\eta f_{\xi\eta}-4\xi^{2}f_{\xi\xi}+\eta f%
_{\eta}-4\xi f_{\xi}+af^{5}=0.\end{split}$$
(3.9)
(9)
Subgroup $\{L_{3}+P_{0}\}:$
$$\begin{split}&\displaystyle u=f(\xi,\eta),\qquad\xi=x^{2}+y^{2},\qquad\eta=%
\arctan\frac{y}{x}-t,\\
&\displaystyle 4\xi^{2}f_{\xi\xi}+(1-\frac{1}{\xi^{2}})f_{\eta\eta}-4f_{\xi}-%
af^{5}=0.\end{split}$$
(3.10)
(10)
Subgroup $\{D+bK_{1}\}\quad 0<b\leq 1:$
$$\begin{split}&\displaystyle u=y^{-1/2}f(\xi,\eta),\qquad\xi=(x+t)y^{b-1},%
\qquad\eta=(x+t)y^{-(b+1)},\\
&\displaystyle(b-1)^{2}\xi^{2}f_{\xi\xi}-2[(1-b^{2})\xi\eta-2]f_{\xi\eta}+(b+1%
)^{2}\eta^{2}f_{\eta\eta}\\
&\displaystyle+(b-1)(b-3)\xi f_{\xi}+(b+1)(b+3)\eta f_{\eta}+3/4f+af^{5}=0.%
\end{split}$$
(3.11)
(11)
Subgroup $\{D+bL_{3}\},\quad b>0:$
$$\begin{split}&\displaystyle u=t^{-1/2}f(\xi,\eta),\qquad\xi=(x^{2}+y^{2})^{-b/%
2}e^{\arctan y/x},\qquad\eta=(x^{2}+y^{2})t^{-2},\\
&\displaystyle(1+b^{2})\xi^{2}f_{\xi\xi}-4b\xi\eta f_{\xi\eta}-4\eta^{2}(\eta-%
1)f_{\eta\eta}\\
&\displaystyle+(1+b^{2})\xi^{2}f_{\xi}-4\eta(2\eta-1)f_{\eta}-(3/4)\eta f+a%
\eta f^{5}=0.\end{split}$$
(3.12)
(12)
Subgroup $\{D+K_{1}+P_{0}+P_{1}\}:$
$$\begin{split}&\displaystyle u=y^{-1/2}f(\xi,\eta),\qquad\xi=x-t,\qquad\eta=(1+%
x+t)y^{-2},\\
&\displaystyle 16\eta^{2}f_{\eta\eta}+16f_{\xi\eta}+32\eta f_{\eta}+3f+4af^{5}%
=0.\end{split}$$
(3.13)
3.2 Symmetry Reductions to Ordinary Differential
Equations
Subgroups of the symmetry group that have generic orbits of codimension one in
the space of independent variables $(x,y,t)$ and of codimension two in
$(x,y,t,u)$ space will provide reductions to ordinary differential equation. The invariants of subgroup
$H$ have the form
$$I_{1}=\xi(x,y,t)\qquad\text{and}\qquad I_{2}=f(x,y,t,u)=\tilde{\phi}(x,y,t)u.$$
In this case the reduction formula will be
$$u(x,y,t)=\phi(x,y,t)f(\xi)$$
where $\phi$ and $\xi$ are again known functions.
If the action of the subgroup $H$, restricted to the time-space variables, is
transitive the ordinary differential equation is of first order, otherwise of second order.
We mention that subgroups with codimension zero in the time-space variables
will reduce the original equation to an algebraic equation that may or may not
admit nontrivial solutions. We run
through the individual subgroups and obtain the following reductions:
(1)
Subgroup $\{P_{1},D+bK_{2}\}:$
$$\begin{split}&\displaystyle u=(t^{2}-y^{2})^{-1/4}f(\xi),\qquad\xi=(t-y)^{b+1}%
(t+y)^{b-1},\\
&\displaystyle 16(b^{2}-1)\xi^{2}f^{\prime\prime}+8(2b^{2}-b-2)\xi f^{\prime}+%
f-4af^{5}=0.\end{split}$$
(3.14)
(2)
Subgroup $\{D+bL_{3},P_{0}\}:$
$$\begin{split}&\displaystyle u=(x^{2}+y^{2})^{-1/4}f(\xi),\qquad\xi=\arctan{y/x%
}-b/2\log(x^{2}+y^{2}),\\
&\displaystyle 4(1+b^{2})f^{\prime\prime}+4bf^{\prime}+4af^{5}+f=0.\end{split}$$
(3.15)
(3)
Subgroup $\{D+bK_{1},P_{0}-P_{1}\},\quad-1<b\leq 1,\quad b\neq 0:$
$$\begin{split}&\displaystyle u=y^{-1/2}f(\xi),\qquad\xi=(x+t)y^{-(b+1)},\\
&\displaystyle b^{2}\xi^{2}f^{\prime\prime}+b(b+2)\xi f^{\prime}+3/4f+af^{5}=0%
.\end{split}$$
(3.16)
(4)
Subgroup $\{D+K_{1}+P_{0}+P_{1},P_{0}-P_{1}\}:$
$$\begin{split}&\displaystyle u=y^{-1/2}f(\xi),\qquad\xi=(x+t+1)y^{-2},\\
&\displaystyle 16\xi^{2}f^{\prime\prime}+32\xi f^{\prime}+3f+4af^{5}=0.\end{split}$$
(3.17)
(5)
Subgroup $\{D+K_{2}+P_{0}+P_{1},P_{1}\}:$
$$\begin{split}&\displaystyle u=(t+y+1)^{-1/4}f(\xi),\qquad\xi=t-y,\\
&\displaystyle f^{\prime}+af^{5}=0.\end{split}$$
(3.18)
(6)
Subgroup $\{D+bK_{1},K_{2}+L_{3}\},\quad b>0:$
$$\begin{split}&\displaystyle u=(t^{2}-x^{2}-y^{2})^{-1/4}f(\xi),\qquad\xi=(t^{2%
}-x^{2}-y^{2})^{b+1}(t+x)^{-2},\\
&\displaystyle 4(b^{2}-1)\xi^{2}f^{\prime\prime}+2(2b^{2}-1)\xi f^{\prime}-1/4%
f-af^{5}=0.\end{split}$$
(3.19)
(7)
Subgroup $\{D-K_{1}+P_{0}-P_{1},K_{2}+L_{3}\}$
$$\begin{split}&\displaystyle u=(x^{2}+y^{2}-t^{2}-x-t)^{-1/4}f(\xi),\qquad\xi=x%
+t,\\
&\displaystyle 4\xi f^{\prime}+f-4af^{5}=0.\end{split}$$
(3.20)
(8)
Subgroup $\{D,P_{0}\}:$
$$\begin{split}&\displaystyle u=t^{-1/2}f(\xi),\qquad\xi=y/t,\\
&\displaystyle 4(1-\xi^{2})f^{\prime\prime}-12\xi f^{\prime}-3f+4af^{5}=0.\end%
{split}$$
(3.21)
(9)
Subgroup $\{P_{0}-P_{1},D\}:$
$$\begin{split}&\displaystyle u=y^{-1/2}f(\xi),\qquad\xi=(x+t)/y,\\
&\displaystyle 4\xi^{2}f^{\prime\prime}+12\xi f^{\prime}+3f+4af^{5}=0.\end{split}$$
(3.22)
(10)
Subgroup $\{D,P_{2}\}:$
$$\begin{split}&\displaystyle u=t^{-1/2}f(\xi),\qquad\xi=x/t,\\
&\displaystyle 4(1-\xi^{2})f^{\prime\prime}-12\xi f^{\prime}-3f+4af^{5}=0.\end%
{split}$$
(3.23)
(11)
Subgroup $\{K_{1},K_{2}+L_{3}\}:$
$$\begin{split}&\displaystyle u=f(\xi),\qquad\xi=x^{2}+y^{2}-t^{2},\\
&\displaystyle 4\xi f^{\prime\prime}+6f^{\prime}+af^{5}=0.\end{split}$$
(3.24)
(12)
Subgroup $\{P_{0}-P_{1},K_{1}+P_{2}\}$
$$\begin{split}&\displaystyle u=f(\xi),\qquad\xi=e^{-y}(x+t),\\
&\displaystyle\xi^{2}f^{\prime\prime}+\xi f^{\prime}+af^{5}=0.\end{split}$$
(3.25)
(13)
Subgroup $\{P_{0}-P_{1},D+K_{2}+L_{3}\}:$
$$\begin{split}&\displaystyle u=(x+t)^{-1/2}f(\xi),\qquad\xi=\frac{y}{x+t}-\log(%
x+t),\\
&\displaystyle f^{\prime\prime}+af^{5}=0.\end{split}$$
(3.26)
(14)
Subgroup $\{K_{1},K_{2},L_{3}\}:$
$$\begin{split}&\displaystyle u=f(\xi),\qquad\xi=t^{2}-x^{2}-y^{2},\\
&\displaystyle 4\xi f^{\prime\prime}+6f^{\prime}-af^{5}=0.\end{split}$$
(3.27)
(15)
Subgroup $\{L_{3},P_{1},P_{2}\}:$
$$u=f(t),\qquad f^{\prime\prime}-af^{5}=0.$$
(3.28)
(16)
Subgroup $\{K_{1},P_{0},P_{1}\}:$
$$u=f(y),\qquad f^{\prime\prime}+af^{5}=0.$$
(3.29)
4 Discussion of the Reduced Ordinary Differential
Equations
Once the reduced ODEs were obtained the remaining task will be to transform
them,
whenever they have the Painlevé property, into one of the standard forms that
can be integrated once with the exception of the Painlevé transcendents.
By a rescaling of independent and dependent variables
all second order ODEs obtained through the symmetry reduction can be
written in a unified manner as
$$A(\xi)f^{\prime\prime}(\xi)+B(\xi)f^{\prime}(\xi)+C(\xi)f(\xi)+D(\xi)f^{5}(\xi%
)=0.$$
(4.1)
We now pick out those having the Painlevé property. To achieve this
task we subject the reduced equations to the Painlevé test which provides
the necessary conditions for having the Painlevé property.
Eq. (4.1) itself does not directly have the Painlevé property.
However, a leading order analysis indicates that if we make the
substitution
$$f(\xi)=\sqrt{h(\xi)},\quad h(\xi)>0$$
then the equation for $h$ may have the Painlevé property.
In the following we run through all the ODEs separately :
Equation (3.14):
$$B\xi^{2}f^{\prime\prime}+A\xi f^{\prime}+f^{5}=0\qquad B=16(b^{2}-1),\;A=8(2b^%
{2}-b-2).$$
(4.2)
By a change of independent variable $\eta=\ln\xi$ it reduces to
$$B\ddot{f}+(A-B)\dot{f}+f-4af^{5}=0$$
(4.3)
where dot denotes derivative with respect to $\eta.$
For $b=\pm 1$ after scaling variables we have
$$\dot{f}=f(f^{4}-1).$$
(4.4)
Its solution with the original variable is
$$f=(1-\xi_{0}\xi)^{-1/4}$$
where $\xi_{0}$ is an integration constant.
For $b=0$, it has the form
$$f^{\prime\prime}+f+f^{5}=0.$$
(4.5)
This equation passes the Painlevé test.
For $b\neq\pm 1$,
Putting
$$z=f\qquad w(z)=\dot{f}$$
transforms equation (4.3) to
$$16(b^{2}-1)ww_{z}-8bw+z-4az^{5}=0$$
which is an Abel equation of the second kind. Some remarks on this equation
are noteworthy. This equation is not
tractable by standard methods, meaning that there is no systematic method
for solving it in closed form, neither a substitution transforming it into
a linear equation.
A list of solvable examples can be found in the collection of
[12]. In a very recent paper
[13], F. Schwarz studied symmetry analysis of Abel equation and
showed that, when the coefficients of the rational normal form of the
equation satisfy some constraint,
Abel equation admits a one-parameter structure-preserving symmetry group
reducing it to a quadrature. Existence of a two-parameter symmetry group
implies that equation is actually equivalent to a Bernoulli equation.
Equation (3.15):
$$f^{\prime\prime}+Af^{\prime}+f+f^{5}=0,\quad A=2b(1+b^{2})^{-1/2}.$$
(4.6)
Eq. (4.6) passes the Painlevé test only for $b=0(A=0)$. For $b\neq 0$,
a transformation from $(\xi,f(\xi))\to(z,w(z))$ by setting $z=f,\quad w(z)=f^{\prime}$
brings (4.6) to an Abel equation of the second kind
$$ww_{z}+Aw+z+z^{5}=0.$$
Equations (3.16), (3.17), (3.19), (3.22):
They are all treated as similar to equation (3.14).
Equation (3.18):
Eq. (3.18) is immediately integrated to give the singular solution
$$f=\{4a(\xi_{0}-\xi)\}^{-1/4}$$
where $\xi_{0}$ is an integration constant.
Equation (3.20):
$$4\xi f^{\prime}+f-4af^{5}=0.$$
(4.7)
Let us transform the independent variable from $\xi$ to $\zeta=\ln\xi$.
Scaling variables lead to
$$f_{\zeta}=f(f^{4}-1)$$
which is equation (4.4) again.
Equation (3.21):
$$(1-\xi^{2})f^{\prime\prime}-3\xi f^{\prime}-\frac{3}{4}f+f^{5}=0.$$
(4.8)
This equation has the Painlevé property. Eq. (3.23) is treated
similarly.
Equation (3.24):
$$\xi f^{\prime\prime}+\frac{3}{2}f^{\prime}+f^{5}=0.$$
(4.9)
This equation has the Painlevé property.
Eq. (3.27) is treated similarly .
Equations (3.25), (3.26), (3.28),
(3.29):
All of these equations can be transformed to
$$f^{\prime\prime}+f^{5}=0$$
(4.10)
which has the first integral
$${f^{\prime}}^{2}+1/3f^{6}=C.$$
where $C$ is an integration constant.
Setting $\phi=f^{2}$ this equation is further transformed to
the elliptic function equation
$${\phi^{\prime}}^{2}=4C\phi-4/3\phi^{4}$$
The form of solutions will depend on the values of $C$. The solutions of
a more general equation of this type will be discussed below.
In summary, among the considered equations the only ones that pass the
Painlevé test are
$$f^{\prime\prime}+f+f^{5}=0,$$
(4.11a)
$$(1-\xi^{2})f^{\prime\prime}-3\xi f^{\prime}-\frac{3}{4}f+f^{5}=0,$$
(4.11b)
$$\xi f^{\prime\prime}+\frac{3}{2}f^{\prime}+f^{5}=0.$$
(4.11c)
The substitution $f=\sqrt{h}$ transforms these equations to
$$h^{\prime\prime}=\frac{{h^{\prime}}^{2}}{2h}-2(h^{3}+h),$$
(4.12a)
$$h^{\prime\prime}=\frac{{h^{\prime}}^{2}}{2h}-\frac{1}{2(1-\xi^{2})}(4h^{3}-3h-%
6\xi h^{\prime}),$$
(4.12b)
$$h^{\prime\prime}=\frac{{h^{\prime}}^{2}}{2h}-\frac{1}{\xi}(2h^{3}+\frac{3}{2}),$$
(4.12c)
respectively.
By a rescaling of variables (4.12a) can be brought to a
standard form [9]
$$h^{\prime\prime}=\frac{{h^{\prime}}^{2}}{2h}+\frac{3}{2}h^{3}-\frac{h}{2}$$
(4.13)
which has the first integral
$${h^{\prime}}^{2}=h^{4}-h^{2}+Ch=h(h^{3}-h+C),$$
(4.14)
where $C$ is an integration constant. Solutions of this equation can be
expressed in terms of Jacobi elliptic functions depending on the value of $C$.
By a linear transformation
$$h(\xi)=\frac{3}{2}(1-\xi^{2})^{-1/2}W(\eta),\quad\eta=\ln(\xi+\sqrt{\xi^{2}+1})$$
the standard form of (4.12b) becomes (4.13). Finally,
transforming (4.12c) by
$$h(\xi)=\sqrt{-\frac{16}{3}}\xi^{-1}W(\eta),\quad\eta=-\frac{16}{3}\xi^{-3/2}$$
we have again (4.13).
We rewrite Eq. (4.14) as
$${h^{\prime}}^{2}=P(h)=h(h-h_{1})(h-h_{2})(h-h_{3})$$
(4.15)
with
$$\begin{array}[]{ll}&h_{1}+h_{2}+h_{3}=0\\
&h_{1}h_{2}+h_{1}h_{3}+h_{2}h_{3}=-1\\
&h_{1}h_{2}h_{3}=-C.\end{array}$$
The above equation can be simplified by a Moebius transformation of the
dependent variable
$$h(\xi)=\frac{\rho Z(\xi)+\sigma}{\mu Z(\xi)+\nu},$$
where $\rho,\sigma,\mu,\nu$ are constants. If all four roots of
$P(h)$ are distinct we choose $\rho,\sigma,\mu,\nu$ so as to to
transform the zeros at $0$, $h_{1}$, $h_{2}$ and $h_{3}$ into zeros at $\pm 1$
and $\pm M$, where $M$ is some constant. In other words, we transform
(4.15) into the standard form
$${Z^{\prime}}^{2}=K(1-Z^{2})(M^{2}-Z^{2}).$$
(4.16)
The form of solution of this
equation
depends on the multiplicity of the roots of the polynomial $P(h)$.
Solutions can be real or complex, finite or singular, periodic or
localized.
If the four roots are distinct, we obtain solutions in terms of Jacobi
elliptic functions.
The general solution of (4.16) is given by
$$\begin{array}[]{ll}Z&=\operatorname{sn}(\sqrt{K}M(\xi-\xi_{0}),M^{-1}),\quad%
\text{for}\quad M^{2}>1,\;M^{2}\in\mathbb{R}\\
Z&=\operatorname{cn}(\sqrt{-K(1-M^{2})}(\xi-\xi_{0}),(1-M^{2})^{-1/2}),\quad%
\text{for}\quad M^{2}<0\\
Z&=\operatorname{dn}(\sqrt{-K}(\xi-\xi_{0}),(1-M^{2})^{1/2}),\quad\text{for}%
\quad 0<M^{2}<1.\\
\end{array}$$
They are hence always periodic rather than
localized, and they may be finite or have periodically spaced
singularities (poles) on the real axis.
A detailed treatment of these functions can be found in any book on
elliptic functions (for example, see Ref. [14]).
If any of the roots have
multiplicity higher than one, we obtain elementary solutions. These can
be localized, namely solitary waves or kinks. They can also be periodic
and hence delocalized. They can have singularities on the real axis.
Some elementary solutions of (4.15) are listed below:
$$\begin{array}[]{ll}a.)&h_{1}=h_{2}=h_{3}=0\\
&h=(\xi-\xi_{0})^{-1}\\
b.)&h_{2}=h_{3}=0,\;h_{1}\neq 0\\
&h=h_{1}\{1-h_{1}^{2}(\xi-\xi_{0})^{2}/4\}^{-1}\\
c.)&h_{1}=h_{2}=h_{3}\neq 0\\
&h=h_{1}(\xi-\xi_{0})^{2}\{(\xi-\xi_{0})^{2}-4h_{1}^{-2}\}^{-1}\\
d.)&h_{3}=0,\;h_{1}=h_{2}\neq 0\\
&h=h_{1}[1-e^{(\xi-\xi_{0})}]^{-1}\\
e.)&h_{3}=0,\;h_{1}\neq h_{2}\neq 0\\
&h=h_{1}h_{2}\{(h-h_{1})\cosh\sqrt{h_{1}h_{2}}(\xi-\xi_{0})+h_{1}+h_{2}\}^{-1}%
\\
f.)&h_{2}=h_{3}\neq 0,\;h_{1}\neq h_{3}\neq 0\\
&h=h_{1}h_{2}\tanh^{2}X\{h_{1}-h_{2}+h_{2}\tanh^{2}X\}^{-1},\quad X=\sqrt{h_{2%
}(h_{2}-1)}/2(\xi-\xi_{0}).\end{array}$$
In addition to similitude invariant exact solutions, conformal
transformations generated by (2.2b) can be used to obtain conformally
invariant solutions. For example, solutions invariant under a combination of conformal
generators $C_{0},C_{1},C_{2}$, namely $K=\alpha_{0}C_{0}+\alpha_{1}C_{1}+\alpha_{2}C_{2}$ are provided by the following reduction formula
$$u=r^{-1}f(\omega),\quad\omega=r^{-2}(\beta_{0}t+\beta_{1}x+\beta_{2}y),\quad r%
^{2}=t^{2}-x^{2}-y^{2}$$
(4.17)
with constants satisfying
$$\beta_{2}\alpha_{2}+\beta_{1}\alpha_{1}-\beta_{0}\alpha_{0}=0.$$
Substitution of (4.17) into (1.2) gives rise to the
second order ODE
$$\beta^{2}f^{\prime\prime}+af^{5}=0,\qquad\beta^{2}=\beta_{2}^{2}+\beta_{1}^{2}%
-\beta_{0}^{2}.$$
(4.18)
Again, by a change of dependent variable this equation can be
transformed to the elliptic function equation.
On the other hand conformal symmetry allows for new solutions to be
produced from Poincaré and similitude invariant solutions.
More precisely, whenever $f(\tilde{r})$ is a solution
to (1.2), so is
$$u({\bf r})=\sigma^{-1/2}f({\bf\tilde{r}}),\quad\sigma=1-{\boldsymbol{\theta}}.%
{\bf r}+\theta^{2}r^{2},\quad\tilde{\bf r}=\sigma^{-1}({\bf r}-{\boldsymbol{%
\theta}}),\quad{\bf r}=(t,x,y)$$
where $\boldsymbol{\theta}=(\theta_{0},\theta_{1},\theta_{1})$ are group parameters.
5 Conclusions
In this paper we combine group theory with singularity analysis to
obtain exact solutions to a conformally invariant nonlinear
Klein-Gordon equation arising in classical and quantum field theory in
(2+1)-dimensions.
We first classify symmetry reductions of the equation
based on a subgroup classification, rather than to
choose intuitively obvious subgroups. In subsection 3.1 using
the subgroups of the similitude group (i.e. the Poincaré group extended
by dilations) we list all
the reduced equations in two variables. In subsection 3.2 all
possible symmetry
reductions to first and second order ordinary differential equations are
given.
In the final section solutions of the reduced ordinary differential equations are discussed. In
particular, it has been shown that, whenever they pass the Painlevé
test, they can be transformed to the equations for elliptic
functions. Hence, in general, we integrate them in terms of Jacobi
elliptic functions. As the limiting cases of these functions we obtain
interesting elementary solutions which may be periodic or localized.
In cases
when the reduced second order equations do not have the Painlevé property
we reduce
them to Abel type equations of the second kind. A well-known
characteristic of them is
that it is exceptionally that they are integrable with quadratures. That
is why it is no surprise that the ones appearing in the present paper
are far from being integrated by standard methods and they often lack
a symmetry which will enable to find suitable coordinates reducing them to
quadratures.
Note that they may have movable branch points other than movable poles.
This means that an Abel equation does not have the Painlevé property.
References
[1]
F. Güngör.
J. Phys. A: Math. Gen., 31:697–706, 1998.
[2]
P. Winternitz, A. M. Grundland, and J. A. Tuszynski.
J. Math. Phys, 28(9):2194–2212, 1987.
[3]
A. M. Grundland, J. A. Tuszynski, and P. Winternitz.
Found. Phys., 23(4):633–665, 1993.
[4]
P.J. Olver.
Applications of Lie Groups to Differential Equations.
Springer, New York, 1991.
[5]
G.W. Bluman and S. Kumei.
Symmetries and Differential Equations.
Springer, New York, 1989.
[6]
P. Winternitz.
Group theory and exact solutions of partially integrable differential
systems.
In R. Conte and N. Boccara, editors, Partially Integrable
Evolution Equations in Physics, Netherlands, 1989. Kluwer Academic
Publishers.
[7]
P. Winternitz.
Lie groups and solutions of nonlinear partial differential equations.
In L. A. Ibort and M. A. Rodriguez, editors, Integrable Systems,
Quantum Fields, and Quantum Field Theories, Netherlands, 1992. Kluwer
Academic Publishers.
[8]
W. I. Fushchich, W. M. Shtelen, and N. I. Serov.
Symmetry Analysis and Exact Solutions of Equations of Nonlinear
Mathematical Physics.
Kluwer Academic Publishers, Dordrecht, 1993.
[9]
E.L. Ince.
Ordinary Differential Equations.
Dover, New York, 1956.
[10]
H.T. Davis.
Introduction to Nonlinear Differential Equations and Integral
Equations.
Dover, New York, 1962.
[11]
J. Patera, P. Winternitz, R. T. Sharp, and H. Zassenhaus.
Can. J. Phys., 54(9):950–961, 1976.
[12]
E. Kamke.
Differentialgleichungen, Lösungsmethoden und Lösungen.
Akademische Verlagsgesellschaft, Leipzig, 1959.
[13]
F. Schwarz.
Stud. in Appl. Math., 100:269–294, 1998.
[14]
P.F. Byrd and M.D. Friedman.
Handbook of Elliptic Integrals for Engineers and Scientists.
Springer Verlag, Berlin, 1971. |
Abstract
It is embarrassing that 95% of the universe is unaccounted for.
Galaxies and larger-scale cosmic structures are composed mainly of ‘dark
matter’ whose nature is still unknown. Favoured candidates are
weakly-interacting particles that have survived from the very early
universe, but more exotic options cannot be excluded. (There are strong
arguments that the dark matter is not composed of baryons). Intensive
experimental searches are being made for the ‘dark’ particles (which
pervade our entire galaxy), but we have indirect clues to their nature too.
Inferences from galactic dynamics and gravitational lensing allow
astronomers to ‘map’ the dark matter distribution; comparison with
numerical simulations of galaxy formation can constrain (eg) the particle
velocities and collision cross sections. And, of course, progress in
understanding the extreme physics of the ultra-early universe could offer
clues to what particle might have existed then, and how many would have
survived.
The mean cosmic density of dark matter (plus baryons) is now pinned
down to be only about 30% of the so-called critical density
corresponding to a ‘flat’ universe. However, other recent evidence –
microwave background anisotropies, complemented by data on distant
supernovae – reveals that our universe actually is ‘flat’, but that its
dominant ingredient (about 70% of the total mass-energy) is
something quite unexpected — ‘dark energy’ pervading all space, with
negative pressure. We now confront two mysteries:
(i) Why does the universe have three quite distinct basic ingredients –
baryons, dark matter and dark energy – in the proportions (roughly) 5%,
25% and 70%?
(ii) What are the (almost certainly profound) implications of the ‘dark
energy’ for fundamental physics?
INTRODUCTION
Martin J Rees
Institute of Astronomy, Madingley Road, Cambridge, CB3 0HA
1 SOME HISTORY
Astronomers have long known that galaxies and clusters would fly
apart unless they were held together by the gravitational pull of much
more material than we actually see.
The strength of the case built up gradually. The argument that clusters
of galaxies would be unbound without dark matter dates back to
Zwicky (1937) and others in the 1930s. Kahn and Woltjer (1959) pointed
out that the motion of Andromeda towards us implied that there must be
dark matter in our Local Group of galaxies. But the dynamical evidence
for massive halos (or ‘coronae’) around individual galaxies firmed up
rather later (e.g. Roberts and Rots 1973, Rubin, Thonnard and Ford 1978).
Two 1974 papers were specially influential in the latter context. Here
is a quote from each:
The mass of galactic coronas exceeds the mass of populations of known
stars by one order of magnitude, as do the effective dimensions. …. The
mass/luminosity ratio rises to f=100 for spiral and $f=120$ for elliptical
galaxies. With $H=50$ km/sec/Mpc this ratio for the Coma cluster is 170
(Einasto, Kaasik and Saar 1974)
Currently-available observations strongly indicate that the mass of
spiral galaxies increases almost linearly with radius to nearly 1 Mpc….
and that the ratio of this mass to the light within the Holmberg radios,
$f$, is 200 ($M/L_{\odot}$). (Ostriker, Peebles and Yahil, 1974).
The amount of dark matter, and how it is distributed, is now far better
established than it was when those papers were written. The immense advances in
delineated dark matter in clusters and in individual galaxies are manifest
in the programme for this meeting. The rapid current progress stems from
the confluence of several new kinds of data within the same few-year
interval: optical surveys of large areas and high redshifts, CMB
fluctuation measurements, sharp X-ray images, and so forth.
The progress has not been solely observational. Over the last 20
years, a compelling theoretical perspective for the emergence of cosmic
structure has been developed. The expanding universe is unstable to the
growth of structure, in the sense that regions that start off
very slightly
overdense have their expansion slowed by their excess gravity, and evolve
into conspicuous density contrasts. According to this ‘cold dark matter’
(CDM) model, the present-day structure of galaxies and clusters is
moulded by the gravitational aggregation of non-baryonic matter, which is
an essential ingredient of the early universe (Pagels and Primack 1982,
Peebles 1982, Blumenthal et al. 1984, Davis et al. 1985). These models
have been firmed up by vastly improved simulations, rendered possible by
burgeoning computer power. And astronomers can now compare these ‘virtual
universes’ with the real one, not just at the present era but (by observing
very distant objects) can probe back towards the formative stages when the
first galaxies emerged.
The following comments are intended to provide a context for the later
papers. (For that reason, I do not give detailed references to the topics
covered by other speakers – just some citations of historical interest).
2 THE CASE FOR DARK MATTER
2.1 Baryons
The inventory of cosmic baryons is readily compiled. Stars and their
remnants, and gas in galaxies, contribute no more than 1% of the
critical density (i.e. they give $\Omega_{b}<0.01$). However several percent more
could be contributed by diffuse material pervading intergalactic space:
warm gas (with $kT\simeq 0.1$ keV) in groups of galaxies and loose clusters, and cooler gas
pervading intergalactic space that manifests itself via the ‘picket fence’
absorption lines in quasar spectra. (Rich clusters are rare, so their
conspicuous gas content, at several KeV, is not directly significant for
the total inventory, despite its importance as a probe)
These baryon estimates are concordant with those inferred by matching
the He and D abundances at the birth of galaxies with the predicted outcome
of nucleosynthesis in the big bang, which is sensitive to the primordial
baryon/photon ratio, and thus to $\Omega_{b}$. The observational estimates have
firmed up, with improved measurements of deuterium in high-$z$
absorbing clouds. The best fit occurs for $\Omega_{b}\simeq 0.02h^{-2}$ where $h$ is the Hubble constant in units of 100 km s${}^{-1}$ Mpc${}^{-1}$. Observations favour $h\simeq 0.7$.
$\Omega_{b}$ is now pinned down by a variety of argument to be
$0.04-0.05$. This corresponds to only $\sim 0.3$ baryons per cubic metre, a value so low that it leaves little scope for dark baryons. (It is
therefore unsurprising that the MACHO/OGLE searches should have found that
compact objects do not make a substantial contribution to the total mass
of our own galactic halo.)
2.2 How much dark matter?
An important recent development is that $\Omega_{DM}$ can now be constrained to a
value around 0.25 by several independent lines of evidence:
(i) One of the most ingenious and convincing arguments comes from
noting that baryonic matter in clusters – in galaxies, and in intracluster
gas – amounts to $0.15-0.2$ of the inferred virial mass (White et al. 1993)
. If clusters were a fair sample of the universe, this would then be
essentially the same as the cosmic ratio of baryonic to total mass. Such
an argument could not be applied to an individual galaxy, because baryons
segregate towards the centre. However, there is no such segregation on the
much larger scale of clusters: only a small correction is necessary to
allow for baryons expelled during the cluster formation process.
(ii) Very distant galaxies appear distorted, owing to gravitational
lensing by intervening galaxies and clusters. Detailed modelling of the
mass-distributions needed to cause the observed distortions yields a
similar estimate. This is a straight measurement of $\Omega_{DM}$ which (unlike (i))
does not involve assumptions about $\Omega_{b}$, though it does depend on having
an accurate measure of the clustering amplitude.
(iii) Another argument is based on the way density contrasts grow
during the cosmic expansion: in a low density universe, the expansion
kinetic energy overwhelms gravity, and the growth of structure
saturates at recent epochs. The existence of conspicuous clusters of
galaxies with redshifts as large as $z=1$ is hard to reconcile with the
rapid recent growth of structure that would be expected if $\Omega_{DM}$ were
unity. More generally, numerical simulations based on the cold dark
matter (CDM) model model are a better fit to the present-day structure
for this value of $\Omega_{DM}$(partly because the initial fluctuation
spectrum has too little long-wavelength power if $\Omega_{DM}$ is unity).
Other methods will soon offer independent estimates. For instance,
$\Omega_{DM}$
can be estimated from the deviations from the Hubble flow induced by
large-scale irregularities in the mass distribution on supercluster scales.
2.3 What could the dark matter be?
The dark matter is not primarily baryonic. The amount of deuterium
calculated to emerge from the big bang would be far lower than observed
if the average baryon density were $\sim 2$ (rather than $\sim 0.3$) per cubic metre.
Extra exotic particles that do not participate in nuclear reactions,
however, would not scupper the concordance.
Beyond the negative statement that it is non-baryonic, the nature of
the dark matter still eludes us. This key question may yield to a
three-pronged attack:
1. $\underline{\hbox{Direct detection}}$. As described by other contributors to this meeting,
several groups are developing cryogenic detectors for supersymmetric
particles and axions This is an exciting quest. Of course, not even
optimists can be confident that the actual dark matter particles have
parameters within the range that these experiments are yet sensitive to.
But the stakes are high: detection of most of the gravitating stuff in
the universe, as well as a new class of elementary particle. So it seems
well worth committing to these experiments funding that is equivalent to
a small fraction of the cost of a major accelerator
2. $\underline{\hbox{Progress in particle physics}}$. Important recent
measurements suggest that neutrinos have non-zero masses; this result
has crucially important implications for physics beyond the standard
model. The inferred neutrino masses seem, however, too low to be cosmologically
important. If the masses and cross-sections of supersymmetric
particles were known, it should be possible to predict how many
survive, and their contribution to $\Omega$, with the same confidence
with which we can compute the nuclear reactions that control
primordial nucleosynthesis. Associated with such progress, we might
expect a better understanding of how the baryon-antibaryon asymmetry
arose, and the consequence for $\Omega_{b}$. Optimists may hope for
progress on still more exotic options.
3. $\underline{\hbox{Simulations of galaxy formation and large-scale
structure}}$. When and how galaxies form, the way they are clustered,
and the density profiles within individual systems, depend on what
their gravitationally-dominant constituent is. A combination of better
data and better simulations is starting to set generic constraints on
the options. The CDM model works well. But there are claimed
discrepancies, though many of us suspect these may ease when the galaxy formation process is better understood. For instance the centre of a halo would, according to the
simulations, have a ‘cusp’ rather than the measured uniform-density core: this
discrepancy has led some authors to explore modifications where the
particles are assumed to have significant collision probabilities, or to
be moving with non-negligible velocities (i.e. ‘warm’ not
cold.). These calculations are in any case offering interesting constraints on the properties of heavy
supersymmetric particles. (Also, straight astronomical observations
can rule out a contribution to $\Omega$ of more than 0.01 from neutrinos
– this is compatible with current experimental estimates.)
3 DARK ENERGY
The inference that our universe is dominated by dark matter is in itself
a discovery of the first magnitude. But the realisation that even more
mass-energy is in some still more mysterious form – dark energy latent in
space itself – came as a surprise, and probably has even greater import
for fundamental physics.
If this meeting had been taking place 3 years ago, the more open-minded
among us would have given equal billing to two options: a hyperbolic
universe with $\Omega$ of 0.3, (in which it would be a coincidence that the
Robertson-Walker curvature radius was comparable with the present Hubble radius),
or a flat universe in which something other than CDM makes up the balance,
equivalent to $\Omega$ of 0.7 (In this case it would be a coincidence that two
quite different invisible substances make comparable contributions).
But it is now clear that only the second option remains in the
running: there is compelling evidence that the universe is flat. This
evidence comes from the slight temperature-differences over the sky in the
background radiation, due to density irregularities which are the
precursors of cosmic structure. Theory tell us that the temperature
fluctuations should be biggest on a particular length scale that is
related to the distance a sound wave can travel in the early universe.
The angular scale corresponding to this length depends, however, on the
geometry of the universe. If dark matter and baryons were all, we wouldn’t
be in a flat universe – the geometry would be hyperbolic. Distant objects
would look smaller than in a flat universe. In 2001-02, measurements from
balloons and from Antarctica pinned down the angular scale of this
‘doppler peak’: the results indicated ‘flatness’ – a result now
confirmed with greater precision by the WMAP satellite.
A value of 0.3 for $\Omega_{DM}$ would imply (were there no other energy in the
universe) an angle smaller by almost a factor of 2 – definitely in
conflict with observations. So what’s the other 70%? It is not
dark matter but something that does not cluster – some energy latent in
space. The simplest form of this idea goes back to 1917 when Einstein
introduced the cosmological constant, or lambda. A positive lambda can be
interpreted, in the context of the ordinary Friedman equations, as a fixed
positive energy density in all space. This leads to a repulsion because,
according to Einstein’s equation, gravity depends on pressure as well as
density, and vacuum energy has such a large negative pressure – tension
– that the net effect is repulsive.
Einstein’s cosmological constant is just one of the options. A
class of more general models is being explored (under names such as
‘quintessence’) where the energy is time-dependent. Any form of dark
energy must have negative pressure to be compatible with observations –
unclustered relativistic particles, for instance, can be ruled out as
candidates. The argument is straightforward: at present, dark energy
dominates the universe – it amounts to around 70% of the total
mass-energy. But had it been equally dominant in the past, it would
have inhibited the growth of the density contrasts in cosmic structures,
which occurred gravitational instability. This is because the growth timescale for gravitational instability is $\sim\left(G\rho_{c}\right)^{-{1\over 2}}$, where $\rho_{c}$ is the density of the component that participates in the clustering, whereas the expansion timescale scales as $\left(G\rho_{total}\right)^{-{1\over 2}}$ when curvature is unimportant. If $\rho_{total}$ exceeds $\rho_{c}$, the expansion is faster, so the growth is impeded. (Meszaros, 1974)
In the standard model, density contrasts in the dark matter grow by
nearly 1000 since recombination. If this growth had been suppressed, the
existence of present-day clusters would therefore require irregularities
that were already of substantial amplitude at the recombination epoch,
contrary to the evidence from CMB fluctuations. For the ‘dark energy’ to be
less dominant in the past, its density must depend on the scale factor $R$ more slowly than
the $R^{-3}$ dependence of pressure-free matter – i.e. its PdV work must be
negative. Cosmologists have introduced a parameter $w$ such that
$p=w\rho c^{2}$.
A more detailed treatment yields the requirement that
$w<-0.5$.
This comes from taking account of baryons and dark matter, and requiring
that dark energy should not have inhibited the growth of structure
so much that it destroyed the concordance between the CMB fluctuations
(which measure the amplitude at recombination) and the present-day
inhomogeneity. Note however that unless its value is -1 (the special case
of a classical cosmological constant) $w$ will generally be time-dependent.
In principle $w(t)$ can be pinned down by measuring the Hubble expansion
rate at different redshifts
This line of argument would in itself have led to a prediction of
accelerating cosmic expansion. However, as it turned out, studies of the
redshift versus the apparent brightness of distant SNIa – strongly
suggestive if not yet completely compelling – had already conditioned
us to the belief that galaxies are indeed dispersing at an accelerating
rate. As often in science, a clear picture gradually builds up, but the
order in which the bits of the jigsaw fall into place is a matter of
accident or contingency.
CMB fluctuations alone can now pin down $\Omega_{DM}$ and the curvature
independent of all the other measurements.
The ‘modern’ interest in the cosmological constant stems from its
interpretation as a vacuum energy. This leads to the reverse problem: Why
is lambda at least 120 powers of 10 smaller than its ‘natural’ value, even
though the effective vacuum density must have been very high in order to
drive inflation. If lambda is fully resurrected, it will be a posthumous
‘coup’ for de Sitter. His model, dating from the 1920s, not only describes
inflation, but would then also describe future aeons of our cosmos with
increasing accuracy. Only for the 50-odd decades of logarithmic time
between the end of inflation and the present would it need modification!.
But of course the dark energy could have a more complicated and
time-dependent nature – though it must have negative pressure, and it
must not participate in gravitational clustering.
4 SUMMARY AND PROSPECTS
Cosmologists can now proclaim with confidence (but with some
surprise too) that, in round numbers, our universe consists of 5%
baryons, 25% dark matter, and 70% dark energy. It is indeed
embarrassing that 95% of the universe is unaccounted for: even the
dark matter is of quite uncertain nature, and the dark energy is a complete
mystery.
The network of key arguments is summarised in Figure 1. Historically,
the supernova evidence came first. But had the order of events been
different, one could have predicted an acceleration on the basis of CDM
evidence alone; the supernovae would then have offered gratifying
corroboration (despite the unease about possible poorly-understood
evolutionary effects).
Our universe is flat, but with a strange mix of ingredients. Why should
these all give comparable contributions (within a modest factor) when they
could have differed by a hundred powers of ten?
In the coming decade, we can expect advances on several fronts.
Physicists may well develop clearer ideas on what determined the
favouritism for matter over antimatter in the early universe, and on the
particles that make up the dark matter. Understanding the dark
energy, and indeed the big bang itself, is perhaps a remoter goal, but ten years from now
theorists may well have replaced the boisterous variety of ideas on the
ultra-early universe by a firmer best buy. They will do this by discovering
internal inconsistencies in some contending theories, and thereby
narrowing down the field. Better still, maybe one theory will earn
credibility by explaining things we can observe, so that we can apply it
confidently even to things we cannot directly observe. In consequence, we
may have a better insight into the origin of the fluctuations, the dark
energy, and perhaps the big bang itself.
Inflation models have two generic expectations; that the universe should
be flat and that the fluctuations should be gaussian and adiabatic (the
latter because baryogenesis would occur at a later stage than inflation).
But other features of the fluctuations are in principle measurable and
would be a diagnostic of the specific physics. One, the ratio of the
tensor and scalar amplitudes of the fluctuations, will have to await the
next generation of CMB experiments, able to probe the polarization on small
angular scales. Another discriminant among different theories is the
extent to which the fluctuations deviate from a Harrison-Zeldovich
scale-independent format ($n=1$ in the usual notation); they could follow a
different power law (i.e. be tilted) , or have a ‘rollover’ so that the
spectral slope is itself a function of scale. Such effects are already
being constrained by WMAP data, in combination with evidence on smaller
scales from present-day clustering, from the statistics of the Lyman alpha
absorption-line ‘forest’ in quasar spectra, and from indirect evidence on
when the first minihalos collapsed, signalling the formation of the first
Population III stars that ended the cosmic dark age.
In parallel, there will be progress in ‘environmental
cosmology’. The new generation of 10-metre class ground based
telescopes will give more data on the universe at earlier cosmic
epochs, as well as better information on gravitational lensing by dark
matter. And there will be progress by theorists too. The behaviour of
the dark matter, if influenced solely by gravity, can already be
simulated with sufficient accuracy. Gas dynamics, including shocks and
radiative cooling, can be included too (though of course the
resolution isn’t adequate to model turbulence, nor the viscosity in
shear layers). Spectacular recent simulations have been able to
follow the formation of the first stars. But the later stages of
galactic evolution, where feedback is important, cannot be modelled
without parametrising such processes in a fashion guided by physical
intuition and observations. Fortunately, we can expect rapid
improvements, from observations in all wavebands, in our knowledge of
galaxies, and the high-redshift universe.
Via a combination of improved observations, and ever more refined
simulations, we can hope to elucidate how our elaborately structured cosmos
emerged from a near-homogeneous early universe.
References
Blumenthal, G, Faber, S, Primack, J.R, and Rees, M.J. 1984 Nature 311, 517
Davis, M, Efstathiou, G.P, Frenk, C.S. and White, S.D.M., 1985 Astrophys. J. 292, 371
Einasto, J , Kaasik, A and Saar, E, 1974 Nature 250, 309
Kahn, F and Woltjer, L, 1959 Astrophys. J. 130, 705
Meszaros, P. 1974 Astr. Astrophys. 37, 225
Ostriker, J. Peebles, P.J.E., and Yahil, A, 1974 Astrophys.J (Lett) 193, L1
Pagels, H. and Primack, J.R. 1982 Phys. Rev. Lett. 48, 223
Peebles, P.J.E. 1982 Astrophys.J. (Lett) 263, L1
Roberts, M.S. and Rots, A.H. 1973 Astr.Astrophys 26, 483.
Rubin, V.C., Thonnard, N., and Ford, W.K., 1978 Astrophys J. (Lett) 225 ,
L107
White, S.D.M., Navarro, J.F., Evrard, A.E. and Frenk, C.S. 1993, Nature 366, 429
Zwicky, F, 1937 Astrophys. J 86, 217 |
Protein mechanical unfolding: a model with binary variables
A. Imparato
alberto.imparato@polito.it
Dipartimento di Fisica and CNISM, Politecnico di Torino,
c. Duca degli Abruzzi 24, Torino, Italy
INFN, Sezione di Torino, Torino, Italy
A. Pelizzola
alessandro.pelizzola@polito.it
Dipartimento di Fisica and CNISM, Politecnico di Torino,
c. Duca degli Abruzzi 24, Torino, Italy
INFN, Sezione di Torino, Torino, Italy
M. Zamparo
marco.zamparo@polito.it
Dipartimento di Fisica and CNISM, Politecnico di Torino,
c. Duca degli Abruzzi 24, Torino, Italy
Abstract
A simple lattice model, recently introduced as a generalization of the
Wako–Saitô model of protein folding, is used to investigate the
properties of widely studied molecules under external forces. The
equilibrium properties of the model proteins, together with their
energy landscape, are studied on the basis of the exact solution of
the model. Afterwards, the kinetic response of the molecules to a force is
considered, discussing both force clamp and dynamic loading protocols
and showing that theoretical expectations are verified. The kinetic
parameters characterizing the protein unfolding are evaluated by using
computer simulations and agree nicely with experimental results, when
these are available. Finally, the extended Jarzynski equality is
exploited to investigate the possibility of reconstructing the free
energy landscape of proteins with pulling experiments.
pacs: 87.15.Aa, 87.15.He, 87.15.-v
I Introduction
The three-dimensional structure of proteins is strictly connected to
the biological functions these molecules perform in living cells
ABL . Among various experimental techniques, an increasingly
important role in the study of protein structures is being played by single–molecule force spectroscopy,
where proteins KSGB ; rgo ; cv1 ; DBBR ; Ober1 ; exp_fc1 ; exp_fc2 (but also nucleic acid fragments NA_exp ) are pulled by applying controlled forces to their ends through an atomic force microscope (AFM) or optical tweezers.
By studying the dynamical response of proteins to constant or varying
loading, much information on their structure has been gathered
KSGB ; rgo ; cv1 ; DBBR ; Ober1 . In particular, the possibility of controlling
the applied force with high precision has allowed to trace the
molecule folding and unfolding pathways DBBR ; exp_fc1 ; exp_fc2 .
Nevertheless, as the size of the molecules increases, force
spectroscopy outcomes cannot be easily related to molecular
properties. Thus theoretical models of biomolecules subject to an
external force have been developed thiru ; ciep ; LKH ; ILT ,
which represent important tools to study the interplay between protein
response to external force and molecular structure.
In most cases, these models are based on a coarse–grained description
of the biomolecule, their dynamical degrees of freedom being related
to the coordinates and velocities of a suitable set of reference
beads, typically one or a few per amino acid thiru ; ciep ; LKH ; ILT . Equilibrium and
nonequilibrium results are then obtained by means of (time
expensive) computer simulations.
In a recent paper IPZ we have approached the problem of protein unzipping from a different point of view,
introducing a simple lattice model with binary degrees of freedom,
based on (and generalizing) the Wako–Saitô (WS) model of protein
folding WS1 ; WS2 . Despite its simplicity, the model turned
out to exhibit the typical response of real proteins to pulling. In
particular, the mechanical unfolding of the 27th immunoglobulin domain
of titin was investigated, considering both force clamp and dynamic
loading protocols. Theoretically expected laws were verified and an
excellent agreement with the experimental values of the characteristic
kinetic parameter was found. The model was also used to investigate
the possibility of reconstructing the protein free energy landscape by
exploiting an extended version of the Jarzynski equality (JE)
jarz ; HumSza ; alb1 .
The present paper has several purposes.
On the methodological
side, we give a full derivation of the relation between our model and
the original WS model, and show in detail how to obtain free energy
landscapes. On the application side, we consider three other molecules
(including BBL, whose thermal behaviour has recently been the subject
of some debate, see ref. SFM and references therein, and ref. dib ) in addition to titin and, together with the properties
already investigated in IPZ , we discuss also the probability
distribution of unfolding times and forces.
The article is organized as follows. In Section II, we
describe the details of our model, its connection with the WS model,
and the simulation method. In Section III we present the
equilibrium properties of the model proteins, as functions of force
and temperature. We then present the results of mechanical unfolding
obtained by numerical simulations, first, in Section IV, for the force clamp manipulation and then, in Section V, for the dynamic loading. In Section VI, the free energy landscape of
the model proteins is reconstructed from unfolding manipulations, by
using an extended Jarzynski Equality. Conclusions are drawn, and future developments
are sketched, in Section VII.
II The model
In the present section, we define our model and show that in the absence of
an external force it reduces to the WS model of protein folding. The
latter was introduced in 1978 by Wako and Saitô, in two
papers WS1 ; WS2 that appear to have been forgotten until recent
years. The same model was independently reintroduced by Muñoz, Eaton
and coworkers at the end of the ’90s ME1 ; ME2 ; ME3 . These authors
used the model to describe and interpret experimental results, and
soon the model became quite popular Amos ; CCBM ; ItohSasai ; AbeWako ; TD1 ; Ap1 ; Ap2 ; ZP0 ; ZP ; BPZ1 ; BPZ2 . This is why it is
often referred to as the Muñoz–Eaton, or
Wako–Saitô–Muñoz–Eaton, model. The first recent reference to
the original work by Wako and Saitô appeared, as far as we know, in
ItohSasai .
Similarly to the WS model, the state of a protein is defined
according to the conformation of its peptide bond backbone, and in
order to reduce the degrees of freedom, we assume that the peptide bonds can exist
only in two conformations: native and non-native. Thus a $N+1$
amino acid protein is represented as a chain of $N$ peptide bonds, and
a binary variable $m_{k}$ is associated to each peptide bond. Bonds
are numbered from 1 to $N$ and amino acids from 1 to $N+1$, bond $k$
connecting amino acids $k$ and $k+1$. The variable $m_{k}$ takes the
value $0,1$ corresponding to a peptide bond in unfolded or native
state respectively.
In order to couple the molecule to an external force we regard
stretches of consecutive native bonds (delimited by unfolded bonds) as
rigid portions of the protein with their own (native) end-to-end
length: in the following $l_{ij}$ will indicate the end-to-end length of
the stretch of consecutive amino acids connected by native bonds and
delimited by unfolded bonds in positions $i$ and $j>i$: $m_{i}=m_{j}=0$, $m_{k}=1$ for $i<k<j$; if we define the quantity
$$S_{ij}(m)\equiv(1-m_{i})(1-m_{j})\prod_{k=i+1}^{j-1}m_{k},$$
(1)
this condition can be expressed
in a more compact form: $S_{ij}(m)=1$. In the limiting case $j-i=1$ the stretch
reduces to amino acid $j=i+1$, which is characterized by its native
length $l_{i,i+1}$. In the following, “stretch” will refer to
stretches of any length, including single amino acids. Boundary
conditions $m_{0}=m_{N+1}=0$ are introduced to define the stretches
at the protein ends. It is easy to verify that the number of stretches
is equal to 1 plus the number of 0’s in the set $\{m_{k},\,k=1,\ldots N\}$.
We want to keep our model as simple as possible, therefore only two
orientations of rigid stretches are
considered: given the direction of the external force, a stretch can
only be oriented parallel or antiparallel to the force direction.
Thus, following IPZ , we introduce the variables $\sigma_{ij}=\pm 1$ which describe the orientation of the
stretch with respect to the external force $f$. Therefore, the
configuration of the molecule is fixed by the set of
$(\{m_{k}\},\{\sigma_{ij}\})$ values, and for each configuration the protein
end–to–end length is given by
$$L(\left\{{m_{k}}\right\},\{\sigma_{ij}\})=\sum_{0\leq i<j\leq N+1}l_{ij}\sigma%
_{ij}S_{ij}(m).$$
(2)
A cartoon of the model protein is plotted in fig.1.
It is worth noting that a given bond configuration $\{m_{k}\}$ dynamically fixes the
set of variables $\{\sigma_{ij}\}$: for each configuration $\{m_{k}\}$ only those variables $\{\sigma_{ij}\}$ such that $S_{ij}(m)=1$ are considered.
As in the original WS model WS1 ; WS2 ; ME1 ; ME2 ; ME3 , in the present model
two amino acids $i$ and $j+1>i$ interact only if all the
peptide bonds connecting them along the chain (that is, bonds from $i$
to $j$) are in the native state, and if they are close enough in the
native configuration. The Hamiltonian of the model reads thus
$$\mathcal{H}(\left\{{m_{k}}\right\},\{\sigma_{ij}\},f)=-\sum_{i=1}^{N-1}\sum_{j%
=i+1}^{N}\epsilon_{ij}\Delta_{ij}\prod_{k=i}^{j}m_{k}-fL(\left\{{m_{k}}\right%
\},\{\sigma_{ij}\})$$
(3)
The quantity $\epsilon_{ij}>0$ represents the interaction energy between the
amino acids $i$ and $j+1$, (defined as in ME3 ; Ap1 ; BPZ1 , see below
for details), while $\Delta_{ij}=0,1$ is the corresponding element of the
contact matrix, taking the value 1 if the distance between any
two atoms of the two amino acids is smaller than some threshold distance,
or the value 0
if none of all the possible atom pairs satisfies this condition, see discussion below.
The microscopic degrees of freedom $\sigma_{ij}$ do not interact among
themselves, hence the partial partition sum over them can be performed
analytically. We obtain the effective Hamiltonian $\mathcal{H}_{\rm eff}$, defined by
$$\sum_{\{\sigma_{ij}\}}\exp\left[{-\beta\mathcal{H}(\left\{{m_{k}}\right\},\{%
\sigma_{ij}\},f)}\right]=\exp\left[{-\beta\mathcal{H}_{\rm eff}(\left\{{m_{k}}%
\right\},f)}\right],$$
that reads
$$\displaystyle\mathcal{H}_{\rm eff}(\left\{{m_{k}}\right\},f)=-\sum_{i=1}^{N-1}%
\sum_{j=i+1}^{N}\epsilon_{ij}\Delta_{ij}\prod_{k=i}^{j}m_{k}$$
$$\displaystyle-k_{B}T\sum_{0\leq i<j\leq N+1}\ln\left[2\,{\rm cosh}\left(\beta
fl%
_{ij}\right)\right]S_{ij}(m).$$
(4)
This effective Hamiltonian is a linear combination of products of
consecutive $m_{k}$’s (including the single peptide bond) and therefore
it has the same mathematical structure of the WS model, whose
Hamiltonian reads
$$\mathcal{H}_{0}(\left\{{m_{k}}\right\})=-\sum_{i=1}^{N-1}\sum_{j=i+1}^{N}%
\epsilon_{ij}\Delta_{ij}\prod_{k=i}^{j}m_{k}-k_{B}T\sum_{i=1}^{N}q_{i}(1-m_{i}),$$
(5)
where $q_{i}$ is the entropic cost of ordering bond $i$.
Given this similarity, the partition function associated to the
Hamiltonian (4) can be summed exactly, and the thermodynamic
quantities can be obtained, as discussed in ref. Ap1 ; Ap2 .
In the case $f=0$, $\mathcal{H}_{\rm eff}$ reduces to (up to an
additive constant)
$$\mathcal{H}_{\rm eff}(\left\{{m_{k}}\right\},0)=-\sum_{i=1}^{N-1}\sum_{j=i+1}^%
{N}\epsilon_{ij}\Delta_{ij}\prod_{k=i}^{j}m_{k}-k_{B}T\ln 2\sum_{i=1}^{N}(1-m_%
{i}),$$
which corresponds to the WS model, eq.(5), with $q_{i}=\ln 2$.
In the absence of force, our new orientational degrees of freedom
correspond to an entropic gain $q_{i}$ associated to the unfolded
peptide bonds. This quantity takes the value $\ln 2$ because we have made the
simplest choice of two possible orientations per stretch. Of course
this is a very rough discretization of the actual orientational
degrees of freedom of the main chain, namely the dihedral angles. In
principle, one could think to orientational variables taking values in
a larger set, and this would yield different, even non–uniform, $q_{i}$
values.
As discussed in ref. IPZ , in the present model the quantities
$\Delta_{ij}\,,\epsilon_{ij}$ and $l_{ij}$ are chosen by analyzing the native structure
of a given molecule taken by the Protein Data Bank (pdb in the
following, http://www.pdb.org/). We briefly review here the choice
criteria for readability’s sake. Two amino acids $i$ and $j+1$ (with
$j+1>i+2$) are in contact ($\Delta_{ij}=1$) if, in the native state of
the protein, at least two atoms from these amino acids are closer than
$4\,\AA$. In this case $\epsilon_{ij}$ is taken to be equal to $k\epsilon$,
where $k$ is an integer such that $5(k-1)<n_{at}\leq 5k$, and $n_{at}$
is the number of atoms of the two amino acids whose distance is not
larger than the threshold distance. The quantity $\epsilon$ is the
protein energy scale, and is determined by imposing that, at zero
force and at the experimental denaturation temperature $T_{m}$, the
fraction of folded molecules is $p(T_{m})=1/2$. In order to estimate
$p$ we introduce the number of native peptide bonds $M=\sum_{k=1}^{N}m_{k}$ and its average density $m=\langle M\rangle/N$, which in the
following will be used as an order parameter. The fraction of folded
molecules is then estimated, assuming a two–state picture, as $p=(m-m_{\infty})/(m_{0}-m_{\infty})$, where $m_{\infty}=1/3$ is the value
of $m$ at infinite temperature, while $m_{0}$ is a good representative
of the folded state BPZ1 . For most molecules we can take $m_{0}=m(T=0)=1$, an exception being the WW domain of PIN1 (pdb code 1I6C), which orders perfectly
only at zero temperature and exhibits a wide plateau in m(T) in the
range 200–300 K BPZ1 . In this case we choose $m_{0}=m(T=273K)<1$.
As far as the parameters $l_{ij}$ are concerned, the generic amino acid
$i$ is represented by its $N_{i}-C_{\alpha,i}-C_{i}$ sequence. Taking
the native state as the reference configuration, $l_{ij}$ is chosen as
the distance between the midpoint of the $C_{i}$ and $N_{i+1}$ atoms
and the midpoint of the $C_{j}$ and $N_{j+1}$ atoms.
In the present paper we shall consider four different molecules of
increasing size: 1BBL (37 amino acids, see BPZ1 for details), 1I6C (39 amino acids), 1COA (64
amino acids), 1TIT (89 amino acids). Here and in the following the
proteins are indicated with their pdb code. Some results on the
mechanical unfolding of 1TIT and a few about 1I6C have already been
reported in IPZ and will be recalled here for comparison with
the other molecules. Results on the thermal unfolding of the other
molecules, based on the WS model, have been reported in Ap1
(1COA) and BPZ1 (1BBL and 1I6C). In particular, in BPZ1 it
was shown that 1BBL differs from a clear two–state behaviour, though
not being a true downhill folder as some authors have claimed. In the
following we shall see that signals of the peculiar behaviour of 1BBL
appear also in the mechanical unfolding.
While the equilibrium properties of the present model can be calculated exactly (see next section), results
on the unfolding kinetics will be instead obtained by performing Monte Carlo (MC) simulations with Metropolis algorithm, using the effective Hamiltonian
(3). In the following $t_{0}$ will indicate the system time
scale, corresponding to a single MC step.
III Equilibrium properties
Once the temperature and the external force have been fixed, the
macroscopic state of the system is defined by the order parameter $m$
and by the molecule length $L$.
In figure 2 the equilibrium values of such quantities are
plotted as functions of the force, at room temperature, for the
different molecules here considered. Inspection of these figures
suggests that the two larger molecules exhibit a sharp transition to
the unfolded state, while for the two smaller molecules the unfolding is
more gradual as the force is increased.
The small force plateaux correspond to a global reorientation of the
molecule, which remains in its native state, under the external
force. When the force increases a more or less sharp transition to an
elongated state occurs. As in the thermal unfolding study BPZ1
we see that 1BBL exhibits a different, more gradual transition with respect
to the other molecules, which exhibit a clearer two–state behaviour.
For each molecule, we can obtain a phase diagram by computing the locus of points in the temperature–force plane where the
fraction of folded molecules $p=1/2$. Such curves are plotted in Fig. 3: inspection of this figure indicates, again, that 1BBL differs qualitatively from the other molecules.
In order to characterize the macroscopic state of the molecules with
the observable quantity $L$, let us define the constrained zero–force partition
function $Z_{0}(L)$ as follows:
$$Z_{0}(L)=\sum_{\{m_{k}\},\{\sigma_{ij}\}}\delta(L-L(\{m_{k}\},\{\sigma_{ij}\})%
)\mathrm{e}^{-\beta\mathcal{H}(\{m_{k}\},\{\sigma_{ij}\},f=0)},$$
(6)
and the corresponding free energy by
$$F_{0}(L)=-k_{B}T\ln Z_{0}(L).$$
(7)
In appendix A we show how $Z_{0}(L)$, as defined by eq. (6), can be calculated exactly for our model.
In figure 4(a), the equilibrium free energy landscape of
the four molecules here considered is plotted as a function of the molecule length,
for $T=300$ K. Inspection of this figure indicates that all the
molecules have a minimum of the free energy at small $L$, the value of
the minimum being compatible with the value of $\sqrt{\left\langle{L^{2}}\right\rangle}$
at zero force, as plotted in fig. 2(b). At larger force, the
free energy function (7) is tilted and reads $F(L,f)=F_{0}(L)-f\cdot L$: this function exhibits new minima for each molecule,
which are compatible with the values of $\sqrt{\left\langle{L^{2}}\right\rangle}$ in the
large force regime, see fig. 4(b). As an example, let
us consider the 1TIT molecule. In fig. 4(b), the
function $F(L,f)$, with $f=10$ pN, exhibits a global minimum at
$L\simeq 225$ Å, which corresponds to the equilibrium value of
$\sqrt{\left\langle{L^{2}}\right\rangle}$ for the same molecule at $f=10$ pN, as shown in
fig. 2(b).
The function $F(L,f)$ exhibits an energy barrier at small $L$,
whose
height $\Delta F$ and width $\Delta L$ depend on the value of $f$.
In order to give an example of the typical values of these quantities,
let us define $f_{1/2}$ as the force where the molecular length is half of its maximum value.
In table 1, we list
the width $\Delta L$ and the height $\Delta F$ of the energy barrier
separating the two minima of the function $F(L)-f\cdot L$, for
$f=f_{1/2}$ and $T=300$ K.
IV Force Clamp
In this section, we investigate the unfolding of the model proteins by
application of a constant external force: such a manipulation scheme
is usually called “force clamp”. The force on the molecule is
suddenly increased from 0 to its final value $f$, and its length is
measured exp_fc1 ; exp_fc2 . Usually the unfolding of a small molecule or
of a portion of a large molecule is viewed as the overcoming of a
kinetic barrier in the molecular energy landscape
ev2 . Such a barrier is characterized by a width $x_{u}$ along the
reaction coordinate, and by a height $\Delta E_{u}$ over the
corresponding minimum. Thus, the mean unfolding time is expected to
follow the Arrhenius law
$$\tau_{u}=\omega_{0}^{-1}\mathrm{e}^{\beta(\Delta E_{u}-fx_{u})}=\tau_{0}%
\mathrm{e}^{-\beta fx_{u}},$$
(8)
where $\omega_{0}$ is a microscopic attempt rate and $\tau_{0}=\omega_{0}^{-1}\exp(\beta\Delta E_{u})$ is the mean unfolding time at
zero force. Note that Eq. (8) is well defined as long as
$f\leq\Delta E_{\mathrm{u}}/x_{\mathrm{u}}$, i.e. as long as the process can be actually
considered as a jump process over an energy barrier. For $f>\Delta E_{\mathrm{u}}/x_{\mathrm{u}}$ one expects that the mean escape time $\tau_{\mathrm{u}}$ is independent of
the external force but rather depends on the microscopic details of
the system. On the other hand for too small forces the system will not
unfold, if the barrier $\Delta E_{\mathrm{u}}$ is sufficiently high ($\beta\Delta E_{\mathrm{u}}\gg 1$).
In order to check whether the unfolding time under constant force
obeys such a law, and to extract the kinetic parameter $x_{u}$, we
run MC simulations to mimic the unfolding of molecules subject to a
force clamp with force $f$, at time $t=0$. We consider a molecule as
unfolded as soon as its length takes the value $L_{u}=L_{max}/2$, where
$L_{max}$ is the molecule length in the completely unfolded state. For
each molecule 1000 independent unfolding trajectories are considered.
In fig. 5, the mean unfolding time of the four molecules
is plotted as a function of the force. In the case of 1TIT, the force
does not extend to the small force range, since we find that the unfolding time
$\tau_{u}$ goes to infinity in the very small force regime, i.e., the molecules
do not unfold at all.
Inspection of this figure suggests that the mean unfolding time as a
function of the force follows eq. (8) in a wide range of
force values, and then saturates in the large force regime, as found in other works ciep ; LKH . The force value separating the two regimes depends on the
particular molecule. From fits to eq. (8) the values
of the kinetic parameter $x_{u}$ can be obtained for each
molecule. In table 2 the values of the unfolding length are listed for the molecules considered
in this paper. In principle, from such a fit procedure one is also able to obtain an estimate of $\tau_{0}$. However the quantity $\tau_{0}$ cannot be expressed in seconds, since this would require to evaluate the molecular time scale $t_{0}$. This, in turn, requires an experimental estimate of $\tau_{0}$ with the force clamp technique.
It is worth noting that, despite the simplicity of our model, the
unfolding length for 1TIT is in remarkable agreement with the experimental value $x_{\mathrm{u}}=2.5\,\AA$ rgo .
The unfolding of the molecule is a stochastic process, and thus the
unfolding time varies for each realization of the process. It is thus
interesting to study the distribution of the unfolding time. In
figure 6, we plot histograms of the unfolding time of the
1TIT molecule at $T=300$ K, for small and large force. In the large
force regime a lognormal distribution was proposed in ciep ,
without a theoretical justification, and shown to fit quite well the
results of molecular dynamics simulations. Our results are also well
fitted by a lognormal distribution. Nevertheless, we find it
interesting to look for a theoretical distribution for the large force
regime.
In the large force regime, one can imagine the chain as made up of a
number $M<N$ of stretches which can be easily turned by the force from
the antiparallel to the parallel (with respect to the force itself)
configuration. At the beginning of the pulling process, these
stretches will be randomly oriented, half of them antiparallel and
half parallel to the force. With a frequency $\tau_{1}^{-1}$, a
stretch will be selected at random by the kinetics and, if
antiparallel to the force, turned parallel with probability
1. Therefore, after a time $\tau_{u}=k\tau_{1}$, the probability that
the chain will be completely elongated in the force direction will be
$p(M,k)=[1-(1-1/M)^{k}]^{M/2}$. The probability distribution of
the unfolding time will therefore be approximated by $f(\tau_{u}=k\tau_{1})=p(M,k)-p(M,k-1)$, which for large $M$ can be approximated
as
$$f(\tau_{u}=k\tau_{1})=1/2\exp[-k/M-M/2\exp(-k/M)].$$
(9)
As can be seen in Fig. 6, this formula fits reasonably well our data, though
not as well as a lognormal. Similar results are obtained for the other
molecules (data not shown).
IV.1 The Kramers problem
We now ask the following question: is the free energy landscape $F(L)$
the “kinetic” potential associated to the diffusive motion of the
system along the coordinate $L$? In other words, in the zero force
regime, is the thermal motion of the molecule a diffusion process
across the potential $F_{0}(L)$? A positive answer to this question would
imply that the mean first passage time of the system to a given value
$L^{*}$ of the elongation should be given by the solution of the Kramers
problem. The Kramers problem amounts to evaluating the mean first
passage time (mfpt) of a Brownian particle, moving in an energy
potential $U(x)$, to a given point $x_{f}$ of the potential. If we let $x_{0}$
be the initial position of the particle, the mfpt from $x_{0}$ to $x_{f}$
reads Zwa
$$\tau(x_{f})=\frac{1}{D}\int_{x_{0}}^{x_{f}}dy\mathrm{e}^{\beta U(y)}\int_{-%
\infty}^{y}\mathrm{e}^{-\beta U(z)},$$
(10)
where $D$ is the diffusion coefficient, which basically
sets the time scale of the process.
In figure 7(a), we plot the mean first passage time of
the 1I6C molecule at $L^{*}=L_{max}/2$, as a function of $f$, for different
temperatures, as obtained by eq. (10), where $U(x)$ has
been replaced by the free energy $F_{0}(L)-f\cdot L$, as given by
eq. (7). Inspection of this figure suggests that in the
small force range, the unfolding kinetics of the protein is actually
described by the equilibrium free energy landscape $F(L,f)$. The
agreement improves as the temperature decreases. The mean first
passage time of the molecule is not recovered by eq. (10)
in the large force regime, because the energy difference
$F(L^{*},f)-F(L=0,f)$ becomes negative, and thus the motion towards the new
minimum at large $L$ becomes purely diffusive, see
fig. 7(b).
The same behaviour is found for the 1BBL molecule (data not shown). Since the unfolding
time grows exponentially with the size of the molecule, for the two
larger molecules we were not able to observe the unfolding in the
small force regime within reasonable computation times.
V Dynamic loading
In this section we consider the following manipulation strategy, which
is often used in experiments KSGB ; rgo ; cv1 ; DBBR ; Ober1 : starting from equilibrium
configurations, a time-dependent force is applied to our model
molecule and the unfolding time is sampled. As in the case of the
force clamp, we define the unfolding time as the first passage time of
the molecule length across the threshold value $L_{\mathrm{u}}$. Here the force
is taken to increase linearly with time, with a rate $r$: this manipulation scheme corresponds to the force–ramp experimental set–up discussed in Ober1 . Thus, the
rupture force $f_{u}$ is given by $f_{u}=r\tau_{\mathrm{u}}$.
As discussed above, the breaking of a molecular linkage is typically
described as a thermally activated escape process from a bound state
over a barrier which dominates the kinetics. It can be shown that, if
the energy barrier $\Delta E_{u}$ is large (compared to the thermal
energy $k_{B}T$) and rebinding is negligible, the typical unbinding
force of a single molecular bond reads ev2 ; denis2
$$f^{*}=\frac{k_{B}T}{x_{u}}\ln\left[{\frac{rx_{u}\tau_{0}}{k_{B}T}}\right].$$
(11)
In
fig. 8(a) the typical unbinding force $f^{*}$ (the most
probable value of $f_{u}$) is plotted as a function of the pulling
velocity for the 1TIT molecule, for three values of the temperature.
Inspection of this figure suggests that the range of values of $r$,
where $f^{*}$ is a linear function of $\ln r$, depends on the
temperature: the smaller is $T$, the wider is this range. We find
that, for this molecule, the value of the unfolding length, $x_{u}\simeq 3\,\AA$, obtained by fitting the data in the linear regime to eq. (11), is
independent of the temperature, as expected (see caption of the figure
for the numerical values). Note that the value of the unfolding
length, $x_{u}$, in the linear regime defined by eq. (11),
agrees with that found with the force clamp manipulation, and with the
experimental value $x_{\mathrm{u}}=2.5\,\AA$ found in ref. rgo .
By repeating the same procedure for the other molecules, we estimate
their unfolding lengths, the results are listed in table
3. Again, $\tau_{0}$
cannot be directly compared with experimental values, but we see that
the unfolding lengths are comparable to the ones obtained by the force
clamp protocol.
However, as discussed above the rupture force is not a linear function
of $r$ in the whole range of the pulling rate: figure 8 rather suggests that the slope of $f^{*}$ increases as $r$ increases.
In refs. denis1 ; denis2 ; alb2 it has been argued that the appearance of
different slopes in the $f^{*}$ vs. $\ln r$ plot could be the signature of the
presence of different escape paths from the folded state. Each of
these different paths would be selected by pulling the molecule with a given rate $r$.
Thus, the different slopes in the $f^{*}$ vs. $\ln r$ plot correspond
to different characteristic lengths $x_{u}$ of the paths. On the other
hand, in a recent work hs_paper , considering particular
choices of the energy landscape which make exact computations
feasible, it has been argued that, the typical unbinding force $f^{*}$
has a more complex expression
$$f^{*}=\frac{\Delta E_{u}}{\nu x_{u}}\left\{{1-\left[{\frac{k_{B}T}{\Delta E_{u%
}}\ln\left({\frac{\omega_{0}\mathrm{e}^{\gamma}}{\beta x_{u}r}}\right)}\right]%
^{\nu}}\right\},$$
(12)
where the exponent $\nu$ depends on the microscopic details of the
energy landscape, and $\gamma$ is the Euler-Mascheroni constant
$\gamma\simeq 0.577$. Equation (12) reduces to
eq. (11) in the limit $\Delta E_{u}\rightarrow\infty$, or
when the exponent $\nu$ takes the value 1 hs_paper .
A similar expression for the rupture force was previously proposed in Dudk , with $\nu=2/3$.
In fig. 8(b), the fits of the typical unbinding force data
to eq. (12) are plotted for the 1TIT molecule. The fits turn out to be
rather good, but statistical errors are quite large. In order to
reduce them, for each molecule, we considered sets of $(r,f^{*})$ data
at different temperatures, and we made joint fits according to
Eq. (12). The values of the unfolding lengths, energy
barriers and exponents obtained by these fits are listed in table
4.
Comparison of the unfolding lengths listed in table 4
and in table 3 indicates that the values of the $x_{u}$
obtained by fitting the typical unbinding force to eq. (12)
are rather different from the values obtained by fitting the same data
to eq. (11), although, as discussed before, the fit to
eq. (11) is restricted to the intermediate range of
values of $r$. This is in agreement with
refs. Dudk ; hs_paper . In those references it was argued that eq. (12) describes the rupture force in the whole regime of $r$
rather than just in the linear regime, as eq. (11)
does.
It is worth noting that the values of the exponent $\nu$ found for the
1BBL, 1I6C and 1COA molecules (table
4) are compatible with the value $\nu=2/3$ found
in refs. Dudk ; hs_paper for a particular choice of the energy landscape.
Furthermore, the value of the kinetic barrier $\Delta E_{u}=18\pm 1\,k_{B}T$ (at $T=300$ K) found for the 1TIT molecule, is of the same order of magnitude of the experimental value $\sim 37.3k_{B}T$ found in rgo .
Similarly to the case of the force clamp, the distribution of the
unbinding force exhibits a nontrivial dependence on the system kinetic
parameters ev1 :
$$P(f)=\frac{1}{\tau_{0}r}\mathrm{e}^{\beta fx_{u}}\exp\left[{-\frac{k_{B}T}{rx_%
{u}\tau_{0}}\left({\mathrm{e}^{\beta fx_{u}}-1}\right)}\right].$$
(13)
The maximum of this distribution corresponds to the typical unbinding
force, eq. (11).
In figure 9, the distribution of the unfolding force is
plotted for the 1TIT molecule and different rate values. Inspection
of fig. 9 suggests that the observed distribution of
unfolding force agrees nicely with the expected one. Similar results
are obtained for the other molecules.
In order to characterize the different unfolding paths occurring for
different values of $r$, one can assume that the unfolding length
$x_{u}$ is a function of the pulling rate $r$, and exploit
eq. (13) to extract the value $x_{u}(r)$, so as to estimate the
unfolding length at any value of $r$. In figure 10 the
unfolding length $x_{u}$ of the 1COA and 1TIT molecules are plotted as a function
of $r$, as obtained from this fitting procedure. As expected we find
that $x_{u}$ is a decreasing function of $r$. Thus, the values
$x_{u}=22.4$ Å(1COA) and $x_{u}=3.05$ Å(1TIT), obtained by fitting the typical unbinding force to
eq. (11) (see, table 3), turn out to be
weighted averages of the values $x_{u}(r)$ as plotted in
fig. 10.
VI Evaluating the free energy landscape from pulling
experiments
Given that we can compute exactly the free energy landscape $F_{0}(L)$ of
our model, let us now address the problem of evaluating this landscape
through the force manipulation experiments. As discussed above,
applying a dynamic loading is equivalent to drive the system out of
equilibrium by coupling it to the external potential $U(L,t)=-f(t)L$.
The free energy landscape can be evaluated
by using a fluctuation relation, which is an extended form of the Jarzynski equality HumSza ; alb1 .
$$\left\langle{\delta(L-L(\{m_{k}\},\{\sigma_{ij}\}))\mathrm{e}^{-\beta W}}%
\right\rangle_{t}=\mathrm{e}^{-\beta\left(F_{0}(L)-f(t)L\right)}/Z_{0},$$
(14)
where $Z_{0}=\int dL\,Z_{0}(L)$, with $Z_{0}(L)$ as given by eq. (6).
Combining the Jarzynski equality jarz , and
the weighted histogram method ferr , it can be shown that, if the
molecule is driven out of equilibrium by a time-dependent external
potential coupled to its length $U(L,t)$, the free energy $F$ as a
function of $L$ is given by HumSza ; alb1
$$F(L)=-k_{B}T\ln\left[{\frac{\sum_{t}\frac{\left\langle{\delta(L-L(\{m_{k}\},\{%
\sigma_{ij}\}))\exp\left({-\beta W_{t}}\right)}\right\rangle_{t}}{\left\langle%
{\exp\left({-\beta W_{t}}\right)}\right\rangle_{t}}}{\sum_{t}\frac{\exp\left({%
-U(L,t)}\right)}{\left\langle{\exp\left({-\beta W_{t}}\right)}\right\rangle_{t%
}}}}\right],$$
(15)
where $W_{t}$ is the thermodynamic work done on the system by the external potential,
up to the time $t$, defined as $W_{t}=\int^{t}_{0}dt^{\prime}\partial\mathcal{H}/\partial t^{\prime}$, and the average $\left\langle{\dots}\right\rangle_{t}$ is over all
the trajectories of fixed duration $t$. In an experimental situation, the work $W_{t}$ is not sampled continuously, but at successive discrete times $0,\Delta t,2\Delta t,\dots,M\Delta t$. Therefore the sum over $t$ in Eq. (15) runs over these discrete values.
The estimated free energy for the smaller molecules are plotted in
fig. 11, as obtained by 10000 independent pulling
trajectories, together with the exact ones. As expected
IPZ ; alb1 , the curves obtained by numerical “experiments”
collapse onto the expected one as the pulling rate $r$ decreases.
Within the present scheme, it was not possible to evaluate the free energy landscape of the two larger molecules:
indeed the work needed to completely unfold the two molecules (1COA, 1TIT) amounts to some hundreds of $k_{B}T$, and thus
the numerical precision in evaluating the average value of $\exp(-\beta W)$ is rather scanty.
Our results suggest a practical procedure to estimate the free energy landscape
of real proteins with dynamic loading experiments.
The work done on the molecule has to be sampled for different pulling rates:
for each rate eq. (15) provides an estimate of the target function $F_{r}(L)$.
As the rate $r$ decreases, the curves are expected to superimpose more
and more: when the difference between the curves is of the order of
few $k_{B}T$, for the whole range of $L$, the estimate of $F(L)$ can be
considered reliable within this small uncertainty.
VII Conclusions
In this paper we presented a comparative study of four widely studied proteins, by exploiting a simple model recently introduced.
Such a model allows one to obtain analytically the equilibrium properties
of the molecules considered.
By using MC simulations we also study the unfolding kinetics of the molecules
as they are pulled through an external force applied to their free ends.
As already discussed in IPZ , the model turns out to exhibit the typical
behaviour of a protein undergoing a mechanical manipulation, both with the
force–clamp scheme and with the dynamic–loading scheme.
We systematically study the equilibrium free energy landscape of the model molecule, as a function of an experimentally accessible coordinate, namely the molecular elongation.
By comparing the unfolding time with the expected values, as given by the Kramers’ formula, we find that, in the limit of small forces, the free energy landscape (7) represents the kinetic energy landscape which governs the unfolding kinetics of the molecules, as discussed in ref. ev1 .
Furthermore, by comparing the typical width of the energy barrier, as found from eq. (7) and listed in table 1, with the unfolding length obtained from out-of-equilibrium unfolding experiments, listed in tables 2–3,
we conclude that the latter parameters are not related to any typical length in the molecules, but are rather effective parameters.
By considering pulling at different velocities, we found that the unfolding length varies with the pulling velocity: thus our results support the point of view that different escape paths exist for a given molecule, each path being selected by the features of the manipulation.
Finally, the possibility of computing the free energy landscape as an exact result, make our model an excellent test bed to check the application of the fluctuation relations to the study of the equilibrium properties of biomolecules.
Our results indicates that the free energy landscape can be recovered by
out-of-equilibrium manipulations, and the collapse of the reconstructed curve
represents an effective criterion to evaluate the reliability of the results.
In conclusion, we believe that our model is an useful tool to investigate the mechanical unfolding of proteins. We plan to further extend our work by studying the single folding and unfolding paths of proteins, by analyzing the unfolding kinetics of single substructures, so as to compare the results
with experimental results.
Acknowledgements.
We thank A. Szabo for interesting discussions and J. Klafter for his interest in our work.
Appendix A Evaluating the free energy landscape
from the equilibrium exact solution
In this appendix we discuss how the summation in the partition
function (6) can be performed exactly, so as to obtain the
free energy landscape as a function of the molecule length
(7). A key point in our approach is to specify a (finite)
discrete set of values for the length of a molecule. This is of course
not a dangerous assumption, since atomic coordinates in the pdb are
given with a finite resolution and it is therefore fair to round the
distances $l_{ij}$ (measured in $\AA$) to rational numbers with a
finite number of digits. In the applications reported in the present
paper we use the resolution $10^{-3}$ Å. For the sake of
simplicity, in the following we will adopt a length unit such that the
distances $l_{ij}$ are integer numbers. Notice also that their
absolute value is not greater than $L_{\rm max}=\sum_{i=0}^{N}l_{ii+1}$,
which corresponds to the length of the molecule in the completely unfolded, fully stretched configuration.
We compute the partition function (6) within a recursive
scheme, considering sequences of sub-chains made of peptide bonds from
1 to $n\leq N$. For the sub-chain with $n$ bonds we define the
interaction energy
$$E_{n}(m)=-\sum_{i=1}^{n-1}\sum_{j=i+1}^{n}\epsilon_{ij}\Delta_{ij}\prod_{k=i}^%
{j}m_{k}$$
(16)
and the length
$$L_{n}(m,\sigma)=\sum_{i=0}^{n}\sum_{j=i+1}^{n+1}l_{ij}\sigma_{ij}S_{ij}(m).$$
(17)
Notice that $E_{N}(m)-fL_{N}(m,\sigma)$ corresponds to the
Hamiltonian (3). We will also need the reduced partition function
$$\Xi_{n}(f)=\sum_{m}\sum_{\sigma}\exp[-\beta E_{n}(m)+\beta fL_{n}(m,\sigma)],$$
(18)
where the first summation is over the first $n$ peptide bond variables
and the second one is over the orientations of the corresponding stretches.
Since the length values are integer by choice, we can
expand $\Xi_{n}(f)$ in powers of $\text{e}^{\beta f}$ as
$$\Xi_{n}(f)=\sum_{L=-L_{\rm max}}^{+L_{\rm max}}{\mathcal{Z}}_{n}(L)\text{e}^{%
\beta fL}.$$
(19)
Our goal is thus the evaluation of ${\mathcal{Z}}_{N}(L)$, which corresponds to the partition function $Z_{0}(L)$ (6) in the main text.
Using the identity
$$1=1-m_{n}+\sum_{i=1}^{n}(1-m_{i-1})\prod_{k=i}^{n}m_{k},$$
(20)
one can verify the recursive relations
$$\Xi_{n}(f)=2\sum_{i=1}^{n+1}\cosh(\beta fl_{i-1n+1})A_{n}^{i}(f),$$
(21)
and
$$A_{n}^{i}(f)=\left\{\begin{array}[]{ll}\exp\left[{\beta\sum_{k=i}^{n}\epsilon_%
{kn}\Delta_{kn}}\right]A_{n-1}^{i}(f)\;\mbox{
if $i\leq n$;}\\
\Xi_{n-1}(f)\;\mbox{ if $i=n+1$.}\end{array}\right.$$
(22)
where, for $i\in\{1,\ldots,n+1\}$,
$$\displaystyle A_{n}^{i}(f)$$
$$\displaystyle\doteq$$
$$\displaystyle\sum_{m}\sum_{\sigma}(1-m_{i-1})\prod_{k=i}^{n}m_{k}$$
(23)
$$\displaystyle\cdot$$
$$\displaystyle\exp[-\beta E_{n}(m)+\beta fL_{n}(m,\sigma)].$$
where the sums over $m$ and $\sigma$ run as in eq. (18).
In the case $n=0$ we set $\Xi_{0}(f)=2\cosh(\beta fl_{11})$ and
$A_{0}^{1}(f)=1$.
Expanding $A_{n}^{i}$ in powers of $\text{e}^{\beta f}$ as
$$A_{n}^{i}(f)=\sum_{L=-L_{\rm max}}^{+L_{\rm max}}a_{n}^{i}(L)\text{e}^{\beta fL},$$
(24)
we see that the coefficients of the above expansion satisfy
$$a_{n}^{i}(L)=\left\{\begin{array}[]{ll}\exp\left[{\beta\sum_{k=i}^{n}\epsilon_%
{kn}\Delta_{kn}}\right]a_{n-1}^{i}(L)\;\mbox{
if $i\leq n$;}\\
{\mathcal{Z}}_{n-1}(L)\;\mbox{ if $i=n+1$,}\end{array}\right.$$
(25)
with the initial condition $a_{0}^{1}(L)=\delta(L)$. In addition we have
$${\mathcal{Z}}_{n}(L)=\sum_{i=1}^{n+1}\left[a_{n}^{i}(L-l_{i-1n+1})+a_{n}^{i}(L%
+l_{i-1n+1})\right].$$
(26)
Notice that the parity with respect to $L$ can be exploited to reduce
the above scheme to positive $L$ values.
Finally, we want to stress the importance of the integer character of
the microscopic lengths $l_{ij}$. We do not know a priori the
values that the length of the molecule can assume, so our algorithm
needs to span the whole interval $[0,L_{max}]$, that has thus to be a
finite set. On the other hand, the above scheme can be viewed as a
polynomial algorithm to find the values that $L$ can assume: there are
no configurations $(m,\sigma)$ with length $L$ when ${\mathcal{Z}}_{N}(L)=0$.
References
(1)
B. Alberts, A. Johnson, J. Lewis, M. Raff, K. Roberts, P. Walter, Molecular Biology of the Cell, (Garland, New York, 2002).
(2)
M. S. Kellermayer, S. B. Smith, H. L. Granzier C. Bustamante
Science 276,1112 (1997).
(3)
M. Carrion-Vasquez et al, PNAS 96, 3694 (1999).
(4)
M. Carrion-Vasquez et al, Nat. Struct. Biol. 10, 738 (2003); H. Dietz, M. Rief PNAS 101, 16192 (2004).
(5)
H. Dietz, F. Berkemeier, M. Bertz, M. Rief, PNAS 103, 12724 (2006).
(6)
A. F. Oberhauser, P. K. Hansma, M. Carrion-Vazquez, J. M. Fernandez, PNAS 98, 468 (2001).
(7)
J.M. Fernandez, H. Li, Science 303, 1674 (2004).
(8)
M. Schlierf, H. Li, J. M. Fernandez PNAS 101, 7299 (2004).
(9)
C. Danilowicz et al PNAS 100, 1694 (2003); J. Liphardt et al, Science 292, 733 (2001) ; B. Onoa et al Science 299, 1892 (2003).
(10)
D. K. Klimov, D. Thirumalai Proc. Natl. Acad. Sci. U.S.A 97 7254 (2000).
(11)
M. S. Li, M. Kouza, C.-K. Hu, Biophysical Journal 92, 1 (2007).
(12)
P. Szymczak, M. Cieplak,
J. Phys.: Condens. Matter 18, L21 (2006).
(13)
A. Imparato, S. Luccioli, A. Torcini, e-print arXiv:0705.3256.
(14)
A. Imparato, A. Pelizzola, M. Zamparo, Phys. Rev. Lett. 98, 148102 (2007).
(15)
H. Wako, N. Saitô, J. Phys. Soc. Jpn 44, 1931 (1978).
(16)
H. Wako, N. Saitô, J. Phys. Soc. Jpn 44, 1939 (1978).
(17)
C. Jarzynski, Phys. Rev. Lett. 78, 2690 (1997); C. Jarzynski, Phys. Rev. E 56, 5018 (1997); G. E. Crooks, J. Stat. Phys. 90, 1481 (1998); G. E. Crooks, Phys. Rev. E 61, 2361 (1999).
(18)
G. Hummer, A. Szabo, Proc. Natl. Acad. Sci. USA 98, 3658 (2001).
(19)
A. Imparato, L. Peliti, J. Stat. Mech.-Theory Exp. P03005 (2006).
(20)
M. Sadqi, D. Fushman, V. Muñoz, Nature (London) 442, 317 (2006); F. Huang, S. Sato, T. D. Sharpe, L. Ying, A. Fersht, Proc. Natl. Acad. Sci. U.S.A. 104, 123 (2007).
(21)
N. Ferguson, T. D. Sharpe, C. M. Johnson, P. J. Schartau, A. R.
Fersht, Nature (London) 445, E14 (2007);
Z. Zhou and Y. Bai, Nature (London) 445, E16 (2007);
M. Sadqi, D. Fushman, V. Muñoz, Nature (London) 445, E17 (2007).
(22)
V. Muñoz, P.A. Thompson, J. Hofrichter,
W.A. Eaton, Nature 390, 196 (1997).
(23)
V. Muñoz,
E.R. Henry, J. Hofrichter, W.A. Eaton, Proc. Natl. Acad. Sci. USA
95, 5872 (1998).
(24)
V. Muñoz, W.A. Eaton, Proc. Natl. Acad. Sci. USA 96, 11311 (1999).
(25)
A. Flammini, J.R. Banavar, A. Maritan,
Europhys. Lett. 58, 623 (2002).
(26)
I. Chang, M. Cieplak, J.R. Banavar, A. Maritan A. Protein Sci. 13, 2446 (2004).
(27)
K. Itoh K. M. Sasai, Proc. Natl. Acad. Sci. U.S.A 101, 14736 (2004);
K. Itoh, M. Sasai, Proc. Natl. Acad. Sci. USA 103, 7298 (2006).
(28)
H. Abe, H. Wako,Phys. Rev. E 74, 011913 (2006).
(29)
V.I. Tokar, H. Dreyssé Phys. Rev. E 68, 011601 (2003);
V. I. Tokar, H. Dreyssé, J. Phys. Cond. Matter
16, S2203 (2004); V.I. Tokar, H. Dreyssé, Phys. Rev. E 71, 031604 (2005).
(30)
P. Bruscolini, A. Pelizzola Phys. Rev. Lett. 88:258101 (2002).
(31)
A. Pelizzola, J. Stat. Mech.-Theory Exp. P11010 (2005).
(32)
M. Zamparo, A. Pelizzola, Phys. Rev. Lett. 97, 068106 (2006).
(33)
M. Zamparo, A. Pelizzola, J. Stat. Mech.-Theory Exp. P12009 (2006).
(34)
P. Bruscolini , A. Pelizzola, M. Zamparo, J. Chem. Phys. 126, 215103 (2007).
(35)
P. Bruscolini , A. Pelizzola, M. Zamparo to appear in Phys. Rev. Lett..
(36)
E. Evans, Annu. Rev. Biophys. Biomol. Struct. 30, 105 (2001).
(37)
R. Zwanzig, Nonequilibrium Statistical mechanics, Oxford University Press, 2001.
(38)
I. Derényi, D. Bartolo, A. Ajdari, Biophys. J. 86, 1263 (2004).
(39)
D. Bartolo, I. Derényi, A. Ajdari, Phys. Rev. E 65, 051910 (2002).
(40)
A. Imparato, L. Peliti, Eur. Phys. J. B 39, 357 (2004).
(41)
E. Evans, K. Ritchie Biophys. J. 72, 1541 (1997).
(42)
O. K. Dudko, G. Hummer, A. Szabo Phys. Rev. Lett. 96, 108101 (2006).
(43)
O. K. Dudko et al., Proc. Natl. Acad. Sci. U.S.A 100, 11378 (2003).
(44)
A. M. Ferrenberg, R. H. Swendsen, Phys. Rev. Lett 63, 1195 (1989). |
Comparing domain wall synapse with other Non Volatile Memory devices for on-chip learning in Analog Hardware Neural Network
Divya Kaushik${}^{*1}$, Utkarsh Singh${}^{*2}$, Upasana Sahu${}^{1}$,
Indu Sreedevi${}^{2}$ and Debanjan Bhowmik${}^{1}$
${}^{*}$These authors contributed equally to the work.
${}^{1}$Department of Electrical Engineering,
Indian Institute of Technology Delhi, New Delhi-110016, India
${}^{2}$Department of Electronics and Communication Engineering,
Delhi Technological University, Delhi-110042, India
E-mail: debanjan@ee.iitd.ac.in
()
Abstract
Resistive Random Access Memory (RRAM) and Phase Change Memory (PCM) devices have been popularly used as synapses in crossbar array based analog Neural Network (NN) circuit to achieve more energy and time efficient data classification compared to conventional computers. Here we demonstrate the advantages of recently proposed spin orbit torque driven Domain Wall (DW) device as synapse compared to the RRAM and PCM devices with respect to on-chip learning (training in hardware) in such NN. Synaptic characteristic of DW synapse, obtained by us from micromagnetic modeling, turns out to be much more linear and symmetric (between positive and negative update) than that of RRAM and PCM synapse. This makes design of peripheral analog circuits for on-chip learning much easier in DW synapse based NN compared to that for RRAM and PCM synapses. We next incorporate the DW synapse as a Verilog-A model in the crossbar array based NN circuit we design on SPICE circuit simulator. Successful on-chip learning is demonstrated through SPICE simulations on the popular Fisher’s Iris dataset. Time and energy required for learning turn out to be orders of magnitude lower for DW synapse based NN circuit compared to that for RRAM and PCM synapse based NN circuits.
1 Introduction
Crossbar array based analog hardware Neural Network (NN) is considered to be extremely time and energy efficient in executing NN algorithms for data classification applications because it computes at the location of memory itself unlike CPU, GPU and even the recent digital neuromorphic chips which all have memory and computing separate at their smallest cores [1, 2, 3, 4, 5, 6]. Such crossbar based NN needs an analog Non Volatile Memory (NVM) device, also known as synapse, at each of the intersection points of the crossbars. Typically a Resistive Random Access Memory (RRAM) or a Phase Change Memory (PCM) device is used as synapse [1, 7, 8, 9, 10]. Training the NN in hardware (on-chip learning) is achieved by modulating the conductances of the synapses, corresponding to weights stored in synapses, with electrical programming pulses at every iteration. Though the conductance of RRAM and PCM synapses changes by orders of magnitude due to programming pulses, conductance response characteristic is highly non-linear and asymmetric (between positive and negative conductance update) [1, 11, 12, 13]. This leads to issues with design of peripheral circuits for on-chip learning. Learning accuracy suffers. Time and energy consumed in the learning process are also very high [1, 11, 12, 13, 9].
Spin orbit torque driven Domain Wall (DW) device based on heavy metal-ferromagnet hetero-structure has been recently proposed and experimentally demonstrated to exhibit synaptic behaviour [3, 14, 15, 16, 17, 18, 19, 20]. In Section II of this paper, we simulate such DW synapse based on experimentally calibrated micromagnetic model. We show that though the range of conductance variation is much smaller for DW synapse than RRAM and PCM synapse, the conductance response of DW synapse to programming current pulse is linear and symmetric unlike RRAM and PCM synapse. In Section III, we design crossbar array of DW synapses in SPICE circuit simulator, with the synapses being Verilog-A models developed from our micromagnetic simulation results. Fully Connected Neural Network (FCNN) algorithm, with Stochastic Gradient Descent (SGD) based weight/ conductance update, has been used here for on-chip learning [17, 21]. Conductance of DW synapse has been quantized here unlike in Bhowmik et al. [17] to take the effect of DW pinning by defects into account [22, 23, 24]. Despite the quantization, high accuracy is obtained on a popular machine learning dataset- Fisher’s Iris [25], in our circuit simulations. We next show that the time taken and energy consumed for on-chip learning of the DW synapse based NN circuit are orders of magnitude lower than RRAM and PCM synapse based NN circuit.
Section IV concludes the paper. To the best of our knowledge, this is the first comparison study between a spintronic synapse and RRAM/ PCM synapse, with respect to on-chip learning in NN hardware.
2 Device Level Comparison
Schematic of our heavy metal/ ferromagnetic metal hetero-structure based domain wall based synapse is shown in Fig. 1. The operating physics of the device has been discussed extensively in [3, 14, 15, 17]. The core physics is that of spin orbit torque driven DW motion, which has been extensively studied through simulations and experiments in the past[26, 27, 28, 29]. When in-plane current (”write” current) flows through the heavy metal layer (”write” path), a DW in the ferromagnetic layer above the it experiences spin orbit torque. If the DW is of Neel type due to presence of Dzyaloshinskii Moriya Interaction (DMI) [26, 27, 30, 31] at the interface, it moves even in the absence of magnetic field,as observed in several experiments[26, 27, 30] and also our micromagnetic simulations (Fig. 2).
In this paper, we consider a device with lateral dimensions 1000 nm $\times$ 50 nm. Thickness of the heavy metal (Pt) layer is taken to be 10 nm, which is greater than the spin diffusion length. Hence, we can consider the vertical spin current density injected by the heavy metal layer on the ferromagnetic layer above it ($J_{s}$) = in-plane charge current density ($J_{c}$) $\times$ spin Hall angle (0.07 here, considering Pt) [37, 38, 39]. Thickness of the ferromagnetic layer above the heavy metal layer is taken to be 1 nm. Dynamics of the moments of this layer under the influence of this spin current is simulated using micromagnetic simulation package ”mumax3” [36] to model such spin current driven DW motion inside it. We choose micro-magnetic simulation parameters for the ferromagnetic layer based on that used for Pt(heavy metal)/ CoFe (ferromagnet)/ MgO devices in the simulation study of Emori et al. [30], which is based on experimentally observed spin orbit torque driven DW motion in the same devices. The parameters can also be found in Supplementary Material (Section 1) accompanying this paper.
Since the DW is of Neel type (DMI = $1.2\times 10^{-3}$ J/m${}^{2}$), average magnetization inside the wall ($\vec{M_{avg}}$) and direction of spin polarization of the electrons at the interface of heavy metal and ferromagnet due to current flowing through heavy metal ($\vec{\sigma}$) form a non-zero cross product (Fig. 2). The effective magnetic field experienced by DW is equal to that cross-product [26, 28, 30]. As a result, DW moves as seen in our micro-magnetic simulations (Fig. 2). Triangular notch regions with Perpendicular Magnetic Anisotropy (PMA) constant = $9\times 10^{5}$ J/m${}^{3}$ are present on the edges of the simulated ferromagnetic layer in our simulations. PMA in rest of the layer = $8\times 10^{5}$ J/m${}^{3}$. These notch regions mimic defects, which pin the domain wall for in-plane charge current lower than a certain threshold value [15, 22, 23, 24, 31]. Hence, our micro-magnetic simulation Fig.( 3(a)) shows that only above a certain threshold value of current density ($\approx 5\times 10^{6}$ A/cm${}^{2}$), velocity of the domain wall is linearly proportional to the current density [32]. Hence in our device we have only moved the domain wall with a current pulse (3 ns long) of fixed magnitude (25 $\mu$A)(Fig. 2),corresponding to a current density of $5\times 10^{6}$ A/cm${}^{2}$ (Fig. 2) so that the domain wall is never pinned by defects [22, 23, 24]. Pinned ferromagnetic regions are present at each edge of the free layer to stabilize the DW at the edge and prevent it from getting destroyed[3, 14, 33].
Following [14] conductance of the ”read” path (vertical tunnel junction structure in Fig. 1) of the synapse is given by,
$$\emph{ $G^{synapse}$ = ($G_{max}$ + $G_{min}$ )/2 - (($G_{max}$ - $G_{min}$ )$%
<m_{z}>$/2})$$
(1)
where $<m_{z}>$ represents the average out of plane magnetization component of the free layer ($<m_{z}>$ = 1 corresponds to up and $<m_{z}>$ = -1 corresponds to down), $G_{max}$ is maximum conductance of MTJ and $G_{min}$ is minimum conductance of MTJ. Taking the Resistance-Area product of the MTJ to be [34] $4.04\times 10^{-12}$ ohm/m${}^{2}$ and TMR ratio of 120 $\%$ [35], $G_{min}$ $\approx 2.9\times 10^{-3}$ mho and $G_{max}$ $\approx 6.1\times 10^{-3}$ mho. Moment of the fixed layer is in down direction (Fig. 1).
As observed from our micromagnetic simulation, ”write” current pulse of magnitude 25 $\mu$A and positive polarity always moves DW to the right by a fixed distance of $\approx$ 20 nm (Fig. 2). Hence $<m_{z}>$ decreases and following equation (1) conductance increases by a step of $0.071\times 10^{-3}$ mho (Fig. 3(b)). Current pulse of same magnitude and negative polarity moves DW to the left, $<m_{z}>$ increases and conductance decreases by the same step of $0.071\times 10^{-3}$ mho (Fig. 3(b)). Hence conductance response to a series of programming ”write” current pulses of equal magnitude (25 $\mu$A) is linear and is also symmetric between positive and negative pulses. Also conductance of DW synapse and hence the corresponding weight of the synapse only takes quantized values and thus we take defect pinning into account. Energy consumed through Joule heating per programming pulse of 25 $\mu$A for conductance increase/ decrease by a single step is calculated to be 0.18 fJ (Table I).
Next we compare the conductance response of this DW synapse with that of typical RRAM and PCM synapse. Verilog-A model provided by [40], experimentally benchmarked against [41], has been used for RRAM modeling. Following the 1T1M (one transistor, one memristor) configuration [42, 43, 44] we connect this RRAM device with a 65 nm technology node transistor (from UMC library) in Cadence Virtuoso circuit simulator (Fig. 4(a)). We observe that when gate voltage pulses of fixed magnitude and duration (200 ns) are applied at the gate of the transistor for conductance increase (voltage of top electrode kept higher than that of bottom electrode for that purpose) (Check Supplementary Material- Section 2), the conductance of the RRAM synapse does not go up linearly unlike the domain wall synapse. In fact the conductance just saturates to a fixed value (Check Supplementary Material- Section 2). To achieve a linear increase in conductance gate voltage pulses of increasing magnitude (SET pulses) need to be applied (Fig. 4(b)). This has been observed experimentally in the RRAM devices of [42, 44, 45]. Thus, though the conductance varies over a much wider range for RRAM synapse than DW synapse (Table I), the conductance response is inherently non-linear in nature. As a result, if a certain value of weight update is needed for any synapse for an iteration during on-chip, different magnitude of voltage pulses may need to be applied to bring about the same weight update, depending on what weight/ conductance value of the RRAM synapse is before that iteration. This makes designing the analog peripheral circuit for weight update very complicated. In fact, the demonstrations of on-chip learning in RRAM based crossbar NN array so far use a digital FPGA unit or an on-chip CMOS based digital processor, connected to the analog crossbar array, for weight update[44, 46]. ADC-s and DAC-s needed as a result, which can potentially consume a lot of energy and slow down the circuit. Energy consumed in the 1T1M circuit of Fig. 4(a) ranges between 12 pJ (minimum gate voltage) and 51 pJ (maximum gate voltage), which is much larger than energy consumed for weight/ conductance update by a single step in a domain wall synapse (Fig. 3(b)) (Table I). Apart from non-linearity, another issue with conductance response of RRAM synapse is asymmetry between positive and negative update of conductance. If we apply the same gate voltage pulses as in Fig. 4(b) in the reverse order in order to decrease the conductance of the synapse (bottom electrode at higher voltage than top for that purpose), we see that the conductance hardly decreases (Check Supplementary Material- Section 2). Rather in order to decrease conductance by a certain step, long duration (6 $\mu$s) and high magnitude (2.5 V) voltage pulse (hence high energy consuming), known as RESET pulse, needs to be applied at the gate of the transistor for abrupt conductance decrease to the smallest value. It is followed by pulses of gradually increasing voltage pulses (SET pulses) to then increase the conductance.
Conductance response characteristic of PCM synapse we simulated, based on model developed in Nandakumar et al. [48] (See Supplementary Material- Section 3 for more details), is more linear than RRAM i.e. programming current pulse of fixed magnitude 90 $\mu A$ and duration 50 ns increase the conductance of the PCM synapse fairly linearly for a larger number of pulses ( $\approx$ 12) (Fig. 4(c)). Energy associated with each such pulse is 5 pJ [48, 2], still much higher than that for domain wall synapse (Table I). Conductance decrease on the other hand is carried out by an abrupt RESET pulse that consumes 30pJ energy each [2], followed by a series of SET pulses much like RRAM synapse. Thus the conductance response characteristic of PCM synapse is still asymmetric like RRAM synapse.
3 Network Level Comparison
Next we design crossbar array based Fully Connected Neural Network (FCNN) with domain wall synapses [17] and compare the energy and speed performance for on-chip learning with that for equivalent FCNN designed with RRAM and PCM synapses. It is to be noted that this NN is of the second generation non-spiking type [49] and uses standard Stochastic Gradient Descent (SGD) algorithm for weight update [21]. Verilog-A model of domain wall synapse is designed, based on its conductance response obtained from micromagnetic physics as shown in Fig. 3(b)) and inserted in crossbar schematic designed on Cadence Virtuoso circuit simulator (Fig. 5).
Fisher’s Iris dataset, a popular machine learning dataset, is used for the training [25]. Since the dataset is not completely linearly separable, in order to carry out accurate classification on it with a FCNN without a hidden layer which we design here, the 4 input features corresponding to each sample are passed through some basic filters first to convert to 16 features [50]. Input voltages, proportional to these 16 input features, are applied on the crossbar as shown in (Fig. 5 ). Read currents, proportional to product of weight of the synapse and each input feature, add up following Kirchhoff’s current law and enter the neuron/ activation function circuit at each output node. Thus the input Vector- weight Matrix Multiplication (VMM) is carried out in the crossbar array. [1, 17].”tanh” neuron/activation function ($f$) acts on the read current at each output node. This function has been designed with transistors in differential amplifier configuration, as shown in Bhowmik et al. [17]. A weight update circuit follows which calculates the common part of weight update at each output node, using the same SGD method and circuit described in Bhowmik et al. [17]. The common part of weight update computed at each output node is next multiplied with the inputs using the multiplier circuit (x) as shown in(Fig. 5). In Bhowmik et al. [17], write current proportional to the output of the multiplier (x) at each synapse acts on the DW synapse and updates its weight. However, since conductance of the DW synapse here takes only quantized values and is updated by write current pulses of fixed magnitude (25 $\mu$A) only (Fig. 3(b)), an additional quantizer circuit (Q) is present after the multiplier circuit here unlike in Bhowmik et al. [17]. Design and typical output of the quantizer circuit can be found in Supplementary Material (Section 4), accompanying this paper. Despite the fact that conductance and hence weight of each synapse takes only quantized value, on-chip learning is achieved with 89 $\%$ train and 92 $\%$ test accuracy on the Fisher’s Iris dataset (Table I). Test accuracy turns out to be slightly higher than train accuracy because the number of samples available in the dataset is low (100 train, 50 test), so a correct or wrong result just with respect to 1 or 2 samples changes the accuracy number by a few percent.
Similar crossbar based FCNN is designed next with RRAM and PCM synapses, with conductance response as shown in Fig. 4. Similar accuracy for on-chip learning is achieved on Fisher’s Iris dataset (Table I). However, net energy consumed in the synapses for on-chip learning is several orders of magnitude higher for RRAM/PCM synapse than DW synapse (Table I). This is expected because we already showed in Section II that energy consumed for each programming pulse that causes increase of conductance by a step (SET pulse) is orders of magnitude higher for RRAM/PCM synapse than DW synapse. Also, high energy consuming RESET pulses are still needed even though the need for decreasing conductance of a synapse is reduced by using 2 RRAM or 2 PCM per synapse [9, 44] (Check Supplementary Material- Section 5). Also, training takes much longer for RRAM/ PCM synapse based FCNN compared to DW synapse based FCNN because of the need of occasional long duration RESET pulses (in microseconds). Since at each iteration (each sample in the training set) weights of all synapses need to be updated simultaneously, even if one synapse needs a RESET pulse of microsecond duration, time needed to carry out that iteration is in microseconds. Since DW synapse does not have this issue, time taken for each iteration during learning is 3 ns only (duration of each programming pulse for DW synapse in Fig. 3(b))
4 Conclusion
Thus in this paper we have shown through device and network level simulations that on-chip learning in DW synapse based NN circuit can consume much less time and energy than RRAM and PCM synapse based NN circuit.
Supplementary Material
1 Simulation of domain wall synapse
The lateral dimensions of the device are taken to be 1000 nm $\times$ 50 nm. Thickness of the ferromagnetic layer above the heavy metal layer is 1 nm.
Thickness of heavy metal layer is 10 nm.
Magnetization dynamics of the moments in this ferromagnetic layer is simulated in micromagnetic package ”mumax3” with the following parameters:
saturation magnetization ($M_{s}$) = $7\times 10^{5}$ A/m, Perpendicular Magnetic Anisotropy (PMA) constant (K) = $8\times 10^{5}$ J/m${}^{3}$, exchange correlation constant (A) =$1\times 10^{-11}$ J/m and damping factor = 0.3 [30]. Also, from [30], Dzyalonshinskii Moriya Interaction (DMI) is taken to be $1.2\times 10^{-3}$ J/m${}^{2}$ and hence DW acquires Neel type chirality.
Triangular notch regions with Perpendicular Magnetic Anisotropy (PMA) constant = $9\times 10^{5}$ J/m${}^{3}$ are present on the edges of the simulated ferromagnetic layer in order to mimic defects, which can pin the domain wall if the driving current pulse magnitude is below a certain threshold.
Verilog-A model provided by [40], experimentally benchmarked against [41], has been used for RRAM modeling.
Following the 1T1M (one transistor,one memristor) configuration [42, 43, 44] we connect this RRAM device with a 65 nm technology node transistor (from UMC library) in Cadence Virtuoso circuit simulator.
2 Simulation of RRAM synapse
Scheme for conductance increase-
1. Voltage at the Top Electrode is 2V and Bottom Electrode is 0V, to enable conductance increase. When gate voltage pulses of fixed magnitude and duration (200 ns) are applied at the gate of the transistor , conductance increases and then saturates to a fixed value. When gate voltage pulses of fixed but larger magnitude are applied, conductance saturates to a higher final value (Fig. 6(a) of Supplementary Material).
2. Voltage at the Top Electrode is again 2V and Bottom Electrode is again 0V, to enable conductance increase. If gate voltage pulses of fixed duration (200 ns) but increasing magnitude are applied at the gate of the transistor, the conductance goes up linearly (Fig. 4(b) of main manuscript).
Scheme for conductance decrease-
1. Voltage at Top Electrode = 0 V and voltage at Bottom Electrode = 4.8 V, to enable conductance decrease.
If we apply the same gate voltage pulses as in Fig. 4(b) of main manuscript in the reverse order in order to decrease the conductance of the synapse, we see that the conductance hardly decreases (Fig. 6(b) of Supplementary Material).
2. With voltage at Top Electrode being 0 V and Bottom Electrode being 3.5 V, when a long duration (6 $\mu$s) and high magnitude (2.5 V) voltage pulse, known as RESET pulse, is applied at the gate of the transistor, conductance abruptly decreases from maximum to minimum value.
3 Simulation of PCM synapse
To model the conductance response of the Phase Change Memory (PCM) synapse, the model developed in [48], based on the experiments conducted on Ge${}_{2}$Sb${}_{2}$Te${}_{5}$ devices, has been used by us. By averaging the different conductance response curves generated by the model due to the stochasticity inherent in it, we have obtained the conductance response of the PCM synapse as shown in Fig. 4(c) of main manuscript.
Scheme for conductance increase-
The conductance increase characteristic is found to be more linear for PCM synapse than the case of RRAM synapse. For RRAM synapse, applying programming pulse of same magnitude led to saturation of conductance within first $\approx$ 5 pulses (Fig. 1(a) of Supplementary Material). Hence programming pulses of linearly increasing magnitude are needed to increase the conductance linearly for a wide range of pulses and hence obtain many more conductance/ weight states (Fig. 4(b) of main manuscript). However, as observed in Fig.4(c) of main manuscript, programming current pulses (SET pulses) of magnitude 90 $\mu A$ and duration 50 ns increase the conductance of the PCM synapse fairly linearly for a larger number of pulses ( $\approx$ 12) [48]. The energy associated with each such pulse is 5 pJ [48, 2].
Scheme for conductance decrease-
Conductance decrease is carried out by an abrupt RESET pulse that consumes 30pJ energy each [2], followed by a series of SET pulses (for conductance increase in small steps) much like RRAM synapse.
4 Quantizer Circuit for Domain Wall based Spintronic NN
To limit our ”write” current to a magnitude of $25\mu$A for either polarity, an additional ”quantizer” circuit is added after the multiplier circuit which multiplies common part of weight update with the input (Fig. 7 of Supplementary Material). The quantizer circuit consists of a couple of op-amps working in ”Comparator” configuration, which compare the input voltage with voltages at two different points in a potential divider circuit, followed by an op-amp in ”Summing amplifier” configuration which adds the output voltages of the two comparator circuits (Fig. 7 of Supplementary Material). The output of the overall quantizer circuit is hence either $\approx 2.5\times 10^{-3}$ V, 0 or $\approx 2.5\times 10^{-3}$ V. When this output voltage is applied on the ”write” terminal/ ”write” path of the domain wall synapse, ”write” current of three possible values ($\approx-25\mu$A, 0 ,$\approx 25\mu$A) flows through the domain wall synapse. As a result conductance of the synapse goes up or down by a fixed step ($\approx 0.071\ \times 10^{-3}$ mho) or stays unchanged (Fig. 8 of Supplementary Material).
5 RRAM/PCM synapse based NN
Fig. 9 of Supplementary Material shows the variation of conductance values of a randomly chosen synapse in the RRAM based FCNN for 50 epochs during on-chip learning on the Fisher’s Iris dataset. Under the two RRAM or PCM devices per synapse scheme [44, 9] used here ,in order to increase the net conductance of the synapse, conductance of one device (positive synapse) is increased. In order to decrease the net conductance of the synapse, conductance of the other device (negative synapse) is increased. But this way, during the course of the learning, conductance of either or both synapses reaches the maximum and then Reset pulses are needed to lower the conductance to the minimum value. Frequent use of Reset pulses increases the overall energy consumption in the scheme.
6 Classification accuracy of designed Domain wall, RRAM and PCM synapse based Neural Network
Fig. 10 of Supplementary Material shows the train and test accuracies of of the RRAM, PCM and spintronic (Domain wall synapse) based neural network circuits as a function of epochs for on-chip learning on the Fisher’s Iris dataset.
References
[1]
H. Tsai, S. Ambrogio, P. Narayanan, R.M. Shelby, and G.W. Burr, Journal of Physics D: Applied Physics 51, 283001 (2018).
[2]
A. Sebastian, M.L. Gallo, G.W. Burr, S. Kim, M. BrightSky et al., Journal of Applied Physics 124, 111101 (2018).
[3]
A. Sengupta, and K. Roy, IEEE Transactions on Circuits and Systems I: Regular Papers 63, 2267 (2016).
[4]
P.A. Merolla, J.V. Arthur, R. Alvarez-Icaza, A.S. Cassidy, J. Sawada et al., Science 345, 668 (2014).
[5]
M. Davies, N. Srinivasa, T.H. Lin, G. Chinya, Y. Cao et al., IEEE Micro 38, 82 (2018).
[6]
G.W. Burr, P. Narayanan, R.M. Shelby, S. Sidler, I. Boybat, IEEE International Electron Devices Meeting, 4.4.1 (2015).
[7]
I. Boybat, M.L. Gallo, S.R. Nandakumar, T. Moraitis, T. Parnell et al., Nature Communications 9, 1 (2018).
[8]
Y. Li, Z. Wang, R. Midya, Q. Xia, and J.J. Yang, Journal of Physics D: Applied Physics 51, 503002 (2018).
[9]
M. Suri, O. Bichler, D. Querlioz, O. Cueto, L. Perniola et al., IEEE International Electron Devices Meeting, 4.4.1 (2011).
[10]
M. Suri ed., Cognitive Systems Monographs, Springer, 2017.
[11]
G.W. Burr, R.M. Shelby, S. Sidler, C.D. Nolfo, J. Jang et al., IEEE Transactions on Electron Devices 62, 3498 (2015).
[12]
P.Y. Chen, X. Peng, and S. Yu, IEEE International Electron Devices Meeting, 6.1.1 (2017).
[13]
T. Gokmen and Y. Vlasov, Frontiers in Neuroscience 10, 333 (2016).
[14]
A. Sengupta, Y. Shim and K. Roy, IEEE Transactions on Biomedical Circuits and Systems 10, 1152 (2016).
[15]
U. Saxena, D. Kaushik, M. Bansal, U. Sahu and D. Bhowmik, IEEE Transactions on Magnetics 54, 1 (2018).
[16]
S. Zhang, S. Luo, N. Xu, Q. Zou, M. Song et al., Advanced Electronic Materials 5, 1800782 (2019).
[17]
D. Bhowmik, U. Saxena, A. Dankar, A. Verma, D. Kaushik et al., Journal of Magnetism and Magnetic Materials 489, 165271 (2019).
[18]
N. Hassan, X. Hu, L. Jiang-Wei, W.H. Brigner, O.G. Akinola et al., Journal of Applied Physics 124, 152127 (2018).
[19]
K. Yue, Y. Liu, R. K. Lake and A. C. Parker, Science Advances 5, eaau8170 (2019).
[20]
S. Li, W. Kang, Y. Huang, X. Zhang, Y. Zhou et al., Nanotechnology 28, 31LT01 (2017).
[21]
Y. LeCun, Y. Bengio, and G. Hinton, Nature 521, 436 (2015).
[22]
S. Dutta, S. A. Siddiqui, J. A. Currivan-Incorvia, C. A. Ross and M. A. Baldo, AIP Advances 5, 127206 (2015).
[23]
Y. Nakatani, A. Thiaville, and J. Miltat, Nature Materials 2, 521 (2003).
[24]
E. Martinez, Advances in Condensed Matter Physics, 1 (2012).
[25]
R. A. Fisher, Annual Eugenics 7, 179 (1936).
[26]
S. Emori, U. Bauer, S.M. Ahn, E. Martinez and G.S.D. Beach, Nature Materials 12, 611 (2013).
[27]
K.S. Ryu, L. Thomas, S.H. Yang and S. Parkin, Nature Nanotechnology 8, 527 (2013).
[28]
D. Bhowmik, M.E. Nowakowski, L. You, O. Lee, D. Keating et al., Scientific Reports 5, 11823, (2015).
[29]
I.M. Miron, T. Moore, H. Szambolics, L.D. Buda-Prejbeanu, S. Auffret et al., Nature materials 10, 419 (2011).
[30]
S. Emori, E. Martinez, K.J. Lee, H.W. Lee, U. Bauer et al., Physical Review B 90, 184427, 2014.
[31]
J. Sampaio, V. Cros, S. Rohart, A. Thiaville and A. Fert, Nature Nanotechnology 8, 839 (2013).
[32]
E. Martinez, S. Emori, N. Perez, L. Torres and G. S. D. Beach, Journal of Applied Physics 115, 213909 (2014).
[33]
D. M. Bromberg, M. T. Moneck, V. M. Sokalski, J. Zhu, L. Pileggi et al., IEEE International Electron Devices Meeting, 33.1.1 (2014).
[34]
J.-G. Zhu and C. Park, Materials Today 9, 36 (2006).
[35]
S. Ikeda, K. Miura, H. Yamamoto, K. Mizunama, H.D. Gan et al., Nature Materials 9, 721 (2010).
[36]
A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen, F. Garcia-Sanchez et al., AIP Advances 4, 107133 (2014).
[37]
L. Liu, C.F. Pai, Y. Li, H.W. Tseng, D.C. Ralph et al., Science 336, 555 (2012).
[38]
L. Q. Liu, T. Moriyama, D. C. Ralph and R. A. Buhrman, Physical Review Letters 106, 036601 (2011).
[39]
L. Q. Liu, O. J. Lee, T. J. Gudmundsen, D. C. Ralph and R. A. Buhrman, Physical Review Letters 109, 096602 (2012).
[40]
X. Guan, S. Yu and H.S.P. Wong, IEEE Electron Device Letters 33, 1405 (2012).
[41]
S. Yu, Y. Wu, and H.S.P. Wong., Applied Physics Letters 98, 103514 (2011).
[42]
F. Miao, W. Yi, I. Goldfarb, J.J. Yang, M.X. Zhang et al., ACS Nano 6, 2312 (2012).
[43]
H. Jiang, L. Han, P. Lin, Z. Wang, M.H. Jang et al., Scientific Reports 6, 28525 (2016).
[44]
C. Li, D. Belkin, Y. Li, P. Yan, M. Hu et al., Nature Communications 9, 2385 (2018).
[45]
J. Woo, K. Moon, J. Song, M. Kwak, J. Park et al., IEEE Transactions on Electron Devices 63, 5064 (2016).
[46]
F. Cai, J.M. Correll, S.H. Lee, Y. Lim, V. Bothra et al., Nature Electronics 2, 290 (2019).
[47]
O. Bichler, M. Suri, D. Querlioz, D. Vuillaume, B. DeSalvo et al., IEEE Transactions on Electron Devices 59, 2206 (2012).
[48]
S. R.Nandakumar, M. Le Gallo, I. Boybat, B. Rajendran, A. Sebastian et al., Journal of Applied Physics 124, 152135 (2018).
[49]
A. Sengupta, and K. Roy, Applied Physics Reviews 4, 041105 (2017).
[50]
A. Biswas, S. Prasad, S. Lashkare and U. Ganguly, arXiv preprint arXiv:1612.02233 (2016). |
Hodograph Method and Numerical Solution of the
Two Hyperbolic Quasilinear Equations.
Part III. Two-Beam Reduction of the Dense
Soliton Gas Equations
E. V. Shiryaeva
shir@math.sfedu.ru
Institute of Mathematics, Mechanics and Computer Science, Southern Federal University, Rostov-on-Don, Russia
M. Yu. Zhukov
zhuk@math.sdedu.ru
Institute of Mathematics, Mechanics and Computer Science, Southern Federal University, Rostov-on-Don, Russia
(November 19, 2020)
Abstract
The paper presents the solutions for the two-beam reduction of the dense soliton gas equations (or Born-Infeld equation) obtained by analytical and numerical methods. The method proposed by the authors is used. This method allows to reduce the Cauchy problem for two hyperbolic quasilinear PDE’s to the Cauchy problem for ODE’s. In some respect, this method is analogous to the method of characteristics for two hyperbolic equations. The method is effectively applicable in all cases when the explicit expression for the Riemann–Green function for some linear second order PDE, resulting from the use of the hodograph method for the original equations, is known.
The numerical results for the two-beam reduction of the dense soliton gas equations, and the shallow water equations (omitting in the previous papers) are presented.
For computing we use the different initial data (periodic, wave packet).
hodograph method, two-beam reduction of the dense soliton gas equations, numerical method, shallow water equations
pacs: 02.30.Jr, 02.30.Hq, 02.60.-xm, 47.15.gm
I Introduction
In previous papers Zhuk_Shir_ArXiv_2014_1 ; Zhuk_Shir_ArXiv_2014_2 the efficient numerical method, allowing to get solutions, including multi-valued solutions111In Zhuk_Shir_ArXiv_2014_1 the solutions of the shallow water equations describing breaking waves are presented., of the Cauchy problem for two hyperbolic quasilinear PDE’s are presented. This method is based on the results of the paper SenashovYakhno in which the hodograph method based on
conservation laws for two hyperbolic quasilinear PDE’s is presented.
The paper SenashovYakhno shows that the solution of the original equations can easily be written in implicit analytical form if
there is an analytical expression of the Riemann–Green function for some linear hyperbolic equation arising as result of the hodograph transformation.
The paper Zhuk_Shir_ArXiv_2014_1 shows that one can not only write the solution in implicit analytical form, but also construct efficient numerical method of the Cauchy problem integration. Using minor modifications of the results of paper SenashovYakhno it is able to reduce the Cauchy problem for two quasilinear PDE’s to the Cauchy problem for ODE’s.
From the authors point of view, solving of the Cauchy problem for ODE’s, in particular, with the help of the numerical methods, is much easier than solving of nonlinear transcendental equations that must be solved when there is an implicit solution of the original problem.
A key role for the proposed method plays the possibility of constructing an explicit expression for the Riemann–Green function of the corresponding linear equation. This, of course, limits the application of the method. However, the number of the equations admitted application of this method is large enough. These include the shallow water equations (see, e.g. RozhdestvenskiiYanenko ; Whithem ), the gas dynamics equations for a polytropic gas RozhdestvenskiiYanenko ; Whithem , the two-beam reduction of the dense soliton gas equations Whithem ; GenaEl (or Born–Infeld equation), the chromatography equations for classical isotherms RozhdestvenskiiYanenko ; FerapontovTsarev_MatModel ; Kuznetsov , the isotachophoresis and zonal electrophoresis equations BabskiiZhukovYudovichRussian ; ZhukovMassTransport ; ZhukovNonSteadyITP ; ElaevaMM ; Elaeva_ZhVM . In particular, the paper SenashovYakhno presents a
large number of equations for which the explicit expressions for the Riemann–Green functions is known.
Classification of equations that allow an explicit expressions for the Riemann–Green functions, is contained in Copson ; Courant ; Ibragimov (see also Chirkunov ; Chirkunov_2 ).
This paper presents analytical and numerical solution of the Cauchy problem for the two-beam reduction of the dense soliton gas equations Whithem ; GenaEl .
The choice of these problem, in particular, due to the fact that the corresponding Riemann–Green function is very simple. We emphasize that the presented results only demonstrate the method effectiveness and do not claim to any physical interpretation.
Pay attention to the fact that in some sense, the proposed method is ‘exact’. Its realization does not require any approximation of the original hyperbolic PDE’s, which use of the finite-difference methods, finite element method, finite volume method, the Riemann solver, etc.
Also there is no need to introduce an artificial viscosity222The effect of the grid viscosity does not occur due to the absence of approximation. In other words, the original problem is solved without any approximation and modification. The accuracy of the solution is determined by only the accuracy of the ODE’s numerical solution method.
The paper is organized as follows. In Sec. II the Cauchy problem for the two-beam reduction of the dense soliton gas equations is formulated. Here we construct the densities and fluxes of some conservation laws (Sec. II.1), the implicit solution of the problem (Sec. II.2),
the solution on the isochrone (Sec. II.3). In Sec. II.4 we show the impossibility of the breaking solution and investigate the properties of the discontinuity solutions. The numerical results are contained in Sec. II.5. In Sec. III we present the some numerical results for shallow water equations omitted in previous paper Zhuk_Shir_ArXiv_2014_1 . Appendix A gives the short description of the numerical methods (more detail see in Zhuk_Shir_ArXiv_2014_1 ; Zhuk_Shir_ArXiv_2014_2 ).
II Two-beam reduction of the dense soliton gas equations
To demonstrate the effectiveness of the hodograph method based on the conservation laws we consider the equation, the so-called two-beam reduction of the
dense soliton gas equations GenaEl (notations are changed)
$$u^{1}_{t}+(u^{1}R^{1})_{x}=0,\quad u^{2}_{t}+(u^{2}R^{2})_{x}=0,$$
(2.1)
$$R^{1}=4\alpha\frac{1-\kappa(u^{1}-u^{2})}{1-\kappa(u^{1}+u^{2})},\quad R^{2}=-%
4\alpha\frac{1+\kappa(u^{1}-u^{2})}{1-\kappa(u^{1}+u^{2})},\quad R^{1}\neq R^{%
2},$$
(2.2)
where $\alpha$, $\kappa$ are the parameters.
Note that these equations after some transformations are also well known as the Born–Infeld equation (see, e.g., Whithem ), which is investigated enough detailed in Menshikh01 ; Menshikh02 ; Menshikh03 ; Menshikh04 .
The equation (2.1) can be rewritten in the Riemann invariants $R^{1}$, $R^{2}$
$$R^{1}_{t}+R^{2}R^{1}_{x}=0,\quad R^{2}_{t}+R^{1}R^{2}_{x}=0,$$
(2.3)
$$\lambda^{1}=R^{2},\quad\lambda^{2}=R^{1}.$$
(2.4)
The connection of the Riemann invariants with the original variables given by (2.2) has the following form
$$u^{1}=\frac{R^{2}+4\alpha}{\kappa(R^{2}-R^{1})},\quad u^{2}=\frac{R^{1}-4%
\alpha}{\kappa(R^{1}-R^{2})}.$$
(2.5)
The original notations of the paper GenaEl have the following form
$$u^{1}=\rho_{1},\quad u^{2}=\rho_{2},\quad R^{1}=s_{1},\quad R^{2}=s_{2},$$
(2.6)
where $\rho_{1}$, $\rho_{2}$ are the densities and $s_{1}$, $s_{2}$ are the velocities.
II.1 Densities and fluxes of the conservation laws.
To obtain the density $\varphi^{t}$ and flux $\psi^{t}$ of a conservation law
$$\varphi^{t}_{t}+\psi^{t}_{x}=0,$$
(2.7)
which satisfy the conditions (A1.8)
$$(\psi^{t}-\lambda^{1}\varphi^{t})\bigr{|}_{{R^{1}=r^{1}}}=1,\quad(\psi^{t}-%
\lambda^{2}\varphi^{t})\bigr{|}_{{R^{2}=r^{2}}}=-1,$$
(2.8)
we use the natural conservation laws (2.1).
We represent functions $\varphi^{t}$, $\psi^{t}$ as a linear combination of the functions $u^{1}$ and $u^{2}$
$$\varphi^{t}=\beta^{1}u^{1}+\beta^{2}u^{2}+\beta^{0},\quad\psi^{t}=\beta^{1}R^{%
1}u^{1}+\beta^{2}R^{2}u^{2}+\beta.$$
(2.9)
Here, $\beta^{1}$, $\beta^{2}$, $\beta^{0}$, $\beta$ are arbitrary functions depended on $r^{1}$, $r^{2}$.
Substitution (2.9) in (2.8) and identical satisfying of (2.8) gives
$$\beta^{1}=-\frac{\kappa}{4\alpha},\quad\beta^{2}=-\frac{\kappa}{4\alpha},\quad%
\beta^{0}=\frac{\kappa}{4\alpha},\quad\beta=0.$$
(2.10)
Using (2.10) and (2.9) we get (see also SenashovYakhno )
$$\varphi^{t}=\frac{2}{R^{1}-R^{2}},\quad\psi^{t}=\frac{R^{1}+R^{2}}{R^{1}-R^{2}}.$$
(2.11)
Another conservation law
$$\varphi^{x}_{t}+\psi^{x}_{x}=0,$$
(2.12)
which satisfies the conditions (A1.9)
$$\left(\frac{\psi^{x}}{\lambda^{1}}-\varphi^{x}\right)_{{R^{1}=r^{1}}}=1,\quad%
\left(\frac{\psi^{x}}{\lambda^{2}}-\varphi^{x}\right)_{{R^{2}=r^{2}}}=-1$$
(2.13)
can be constructed by analogous.
We assume that $\varphi^{x}$, $\psi^{x}$ are the linear combination
$$\varphi^{x}=\beta^{1}u^{1}+\beta^{2}u^{2}+\beta^{0},\quad\psi^{x}=\beta^{1}R^{%
1}u^{1}+\beta^{2}R^{2}u^{2}+\beta.$$
(2.14)
Identical satisfying of the conditions (2.13) gives $\beta^{1}$, $\beta^{2}$, $\beta^{0}$, $\beta$
$$\beta^{1}=-\kappa,\quad\beta^{2}=\kappa,\quad\beta^{0}=0,\quad\beta=-4\alpha,$$
(2.15)
and functions $\varphi^{x}$, $\psi^{x}$ (see also SenashovYakhno )
$$\varphi^{x}=\frac{R^{1}+R^{2}}{R^{1}-R^{2}},\quad\psi^{x}=\frac{2R^{1}R^{2}}{R%
^{1}-R^{2}}.$$
(2.16)
Note that functions $\varphi^{t}$, $\varphi^{x}$, $\psi^{t}$, $\psi^{x}$ depend only on the variables $R^{1}$, $R^{2}$ and do not depend on the variables $r^{1}$, $r^{2}$.
II.2 Implicit solution of the problem
Taking into account the simple form of the functions $\varphi^{t}$, $\varphi^{x}$, $\psi^{t}$, $\psi^{x}$ we present the solution
of the Cauchy problem for equations (2.3), (2.4) with initial data given on arbitrary curve.
We assume that initial data for the equations (2.3), (2.4) are given for some line $\Gamma$ (not a characteristic)
$$\Gamma=\{(x,t)\,:\,x=x(\tau),\quad t=t(\tau)\},$$
(2.17)
$$R^{1}\bigr{|}_{\Gamma}=R^{1}_{0}(\tau),\quad R^{2}\bigr{|}_{\Gamma}=R^{2}_{0}(%
\tau).$$
(2.18)
Here, $R^{1}_{0}(\tau)$, $R^{2}_{0}(\tau)$ are given functions, $\tau$ is the parameter.
Using the hodograph method based on conservation laws (see SenashovYakhno ) we get
$$2t=t(a)+t(b)-\int\limits_{\Gamma}(\psi^{t}dt-\varphi^{t}dx),$$
(2.19)
$$2x=x(a)+x(b)-\int\limits_{\Gamma}(\psi^{x}dt-\varphi^{x}dx),$$
(2.20)
where the functions $\varphi^{t}$, $\psi^{t}$ are determined by relations (2.11), and the functions $\varphi^{x}$, $\psi^{x}$ are determined by relations (2.16).
We restrict the investigation by the easiest and most natural situation, when the initial data is given at $t=t_{0}$. In this case, the contour $\Gamma$ is an interval of axis $t=t_{0}$, and $\tau=x$ (for $x$ more convenient is to keep the previous notation $\tau$). The conditions (2.18) take the form
$$R^{1}\bigr{|}_{t=t_{0}}=R^{1}_{0}(\tau),\quad R^{2}\bigr{|}_{t=t_{0}}=R^{2}_{0%
}(\tau).$$
(2.21)
Then
$$t=t_{0}+\frac{1}{2}\int\limits_{a}^{b}\varphi^{t}\,d\tau,\quad x=\frac{a+b}{2}%
+\frac{1}{2}\int\limits_{a}^{b}\varphi^{x}\,d\tau.$$
(2.22)
We introduce the notations
$$F(a,b)=\frac{1}{2}\int\limits_{a}^{b}\varphi^{t}\,d\tau=\int\limits_{a}^{b}%
\frac{1}{R^{1}_{0}(\tau)-R^{2}_{0}(\tau)}\,d\tau,$$
(2.23)
$$G(a,b)=\frac{1}{2}\int\limits_{a}^{b}\varphi^{x}\,d\tau=\frac{1}{2}\int\limits%
_{a}^{b}\frac{R^{1}_{0}(\tau)+R^{2}_{0}(\tau)}{R^{1}_{0}(\tau)-R^{2}_{0}(\tau)%
}\,d\tau,$$
(2.24)
where $F(a,b)$, $G(a,b)$ are completely determined by the initial data, and they depend only on the parameters $a$, $b$.
The implicit solution of the problem (2.3), (2.4), (2.21) takes the form
$$t=t(a,b)\equiv t_{0}+F(a,b),\quad x=x(a,b)\equiv\frac{a+b}{2}+G(a,b).$$
(2.25)
$$R^{1}(x,t)=r^{1}\equiv R^{1}_{0}(b),\quad R^{2}(x,t)=r^{2}\equiv R^{2}_{0}(a).$$
(2.26)
Also we present the implicit solutions of the original problem (2.1), (2.2).
Taking into account (2.2), (2.5) we obtain
$$R^{1}-R^{2}=\frac{8\alpha}{1-\kappa(u^{1}+u^{2})},\quad R^{1}+R^{2}=\frac{8%
\alpha\kappa(u^{2}-u^{1})}{1-\kappa(u^{1}+u^{2})},$$
(2.27)
$$F(a,b)=\int\limits_{a}^{b}\frac{1-\kappa(u^{1}_{0}(\tau)+u^{2}_{0}(\tau))}{8%
\alpha}\,d\tau,$$
(2.28)
$$G(a,b)=\int\limits_{a}^{b}\frac{\kappa(u^{2}_{0}(\tau)-u^{1}_{0}(\tau))}{2}\,d\tau,$$
(2.29)
$$u^{1}(x,t)=\frac{u^{1}_{0}(a)(1-\kappa(u^{1}_{0}(b)+u^{2}_{0}(b)))}{1-\kappa(u%
^{1}_{0}(b)+u^{2}_{0}(a))-\kappa^{2}(u^{1}_{0}(a)u^{2}_{0}(b)-u^{1}_{0}(b)u^{2%
}_{0}(a))},$$
(2.30)
$$u^{2}(x,t)=\frac{u^{2}_{0}(b)(1-\kappa(u^{1}_{0}(a)+u^{2}_{0}(a)))}{1-\kappa(u%
^{1}_{0}(b)+u^{2}_{0}(a))-\kappa^{2}(u^{1}_{0}(a)u^{2}_{0}(b)-u^{1}_{0}(b)u^{2%
}_{0}(a))}.$$
(2.31)
Here, $u^{1}_{0}$, $u^{2}_{0}$ are the initial data at $t=t_{0}$.
II.3 The solution on isochrone
We describe the solving of the Cauchy problem for the isochrones, i.e. on line level of function $t(a,b)$, which is determined by (2.23)–(2.25). We recall that the function $x(a,b)$ and hence $G(a,b)$ are not required.
Calculating the derivative of $t_{a}(a,b)$, $t_{b}(a,b)$, i. e. the right parts of the differential equations (A1.18), we get with the help of (2.23)
$$t_{a}=t_{a}(a,b)=-f(a),\quad t_{b}=t_{b}(a,b)=f(b),$$
(2.32)
$$f(\tau)=\frac{1}{R^{1}_{0}(\tau)-R^{2}_{0}(\tau)}.$$
(2.33)
We assume that isochrone is given by the parameters $a_{*}$, $b_{*}$
$$t_{*}=t(a_{*},b_{*}).$$
(2.34)
To determine the coordinates $X_{*}=x(a_{*},b_{*})$, corresponding to the parameter $\tau=0$ we have the Cauchy problem (A1.21), (A1.22), which
can be written with the help of (2.3), (2.32), (2.33) in the following form
$$\frac{dY(b)}{db}=\frac{R^{1}_{0}(b)}{R^{1}_{0}(b)-R^{2}_{0}(b)},\quad Y(a_{*})%
=a_{*}.$$
(2.35)
Integrating from $a_{*}$ to $b_{*}$ we have
$$X_{*}=Y(b_{*})=a_{*}+\int\limits_{a_{*}}^{b_{*}}\frac{R^{1}_{0}(b)\,db}{R^{1}_%
{0}(b)-R^{2}_{0}(b)}.$$
(2.36)
To determine the functions $a(\tau)$, $b(\tau)$ and $x=X(\tau)$ we get the Cauchy problem (A1.18), (A1.20), using again
(2.3), (2.32), (2.33)
$$\frac{da}{d\tau}=\frac{1}{R^{2}_{0}(b)-R^{1}_{0}(b)},\quad\frac{db}{d\tau}=%
\frac{1}{R^{2}_{0}(a)-R^{1}_{0}(a)},$$
(2.37)
$$\frac{dX}{d\tau}=\frac{R^{2}_{0}(a)-R^{1}_{0}(b)}{(R^{1}_{0}(a)-R^{2}_{0}(a))(%
R^{1}_{0}(b)-R^{2}_{0}(b))}.$$
(2.38)
$$a\bigr{|}_{\tau=0}=a_{*},\quad b\bigr{|}_{\tau=0}=b_{*},\quad X\bigr{|}_{\tau=%
0}=X_{*}.$$
(2.39)
Integrating the (2.37)–(2.39)
we obtain the solutions on isochrone
$$R^{1}(x,t_{*})=R^{1}_{0}(b(\tau)),\quad R^{2}(x,t_{*})=R^{2}_{0}(a(\tau)),%
\quad x=X(\tau).$$
(2.40)
II.4 The impossibility of profile breaking. Discontinuous solutions
Before further investigation of the problem we note that the parameters $\alpha$ and $\kappa$ can be excluded from equations with the help of the substitutions
$$t\rightarrow\frac{t}{4\alpha},\quad\kappa u^{i}\rightarrow u^{i}.$$
(2.41)
Further, we just assume
$$4\alpha=1,\quad\kappa=1.$$
(2.42)
One of the breaking solution conditions at some time $t$ (i. e. the formation of the multi-valued solutions) is the tending to infinity of the derivatives $R^{1}_{x}(x,t)$, $R^{2}_{x}(x,t)$. For example, calculating $R^{1}_{x}(x,t)$ we get
$$R^{1}_{x}(x,t)=\partial_{x}R^{1}_{0}(x)=r^{1}_{b}(b)b_{x}.$$
(2.43)
Differentiating $t=t(a,b)$, $x=x(a,b)$ with respect to $x$ we have
$$x_{a}a_{x}+x_{b}b_{x}=1,\quad t_{a}a_{x}+t_{b}b_{x}=0.$$
(2.44)
Then
$$a_{x}=\frac{t_{b}}{\Delta},\quad b_{x}=\frac{-t_{b}}{\Delta},\quad\Delta=x_{a}%
t_{b}-x_{b}t_{a}.$$
(2.45)
Hence,
$$R^{1}_{x}(x,t)=\frac{-r^{1}_{b}(b)t_{a}}{\Delta}.$$
(2.46)
Obviously, the breaking solution condition is $R^{1}_{x}(x,t)=\infty$ or $\Delta=0$. Taking into account (2.23)–(2.26) we calculate the dervaties and get
$$R^{1}_{0}(b)=R^{2}_{0}(a).$$
(2.47)
This equality is impossible because it means that for some point $(x,t)$ we have relation
$$R^{1}(x,t)=R^{2}(x,t),$$
(2.48)
which contradicts to the condition (2.2) (at $4\alpha=1$ and $\kappa=1$)
$$R^{1}-R^{2}=\frac{2}{1-u^{1}+u^{2}}\neq 0.$$
(2.49)
The results obtained indicate that the breaking profile of the function $R^{1}(x,t)$, $R^{2}(x,t)$ is impossible. In other words
$R^{1}(x,t)$, $R^{2}(x,t)$ are the one-valued functions. We recall also that it is impossible to construct a self-similar solution, since the system (2.3) is the degeneracy system
$$\lambda^{1}_{R^{1}}=0,\quad\lambda^{2}_{R^{2}}=0.$$
(2.50)
It means that discontinuities of solutions can be set only at the initial moment (can not occur when initial data are smooth). This discontinuity solution is so called contact discontinuity which can move along characteristics only.
The Rankine–Hugoniot conditions for the conservative system (2.1), after a change of variables (2.5), are written in the form
$$D\left\llbracket\frac{2}{R^{1}-R^{2}}\right\rrbracket=\left\llbracket\frac{R^{%
1}+R^{2}}{R^{1}-R^{2}}\right\rrbracket,\quad D\left\llbracket\frac{R^{1}+R^{2}%
}{R^{1}-R^{2}}\right\rrbracket=\left\llbracket\frac{2R^{1}R^{2}}{R^{1}-R^{2}}%
\right\rrbracket,$$
(2.51)
where $\llbracket\,.\,\rrbracket$ is the jump across discontinuity, $D$ is the discontinuity velocity.
It is easy to show that there are only the following solutions of system (2.51)
$$D=R^{2},\quad\left\llbracket R^{1}\right\rrbracket\neq 0,\quad\quad\left%
\llbracket R^{2}\right\rrbracket=0$$
(2.52)
or
$$D=R^{1},\quad\left\llbracket R^{1}\right\rrbracket=0,\quad\quad\left\llbracket
R%
^{2}\right\rrbracket\neq 0.$$
(2.53)
In particular, the simultaneous discontinuities of the Riemann invariants (i.e., $\llbracket R^{1}\rrbracket\neq 0$, $\llbracket R^{2}\rrbracket\neq 0$) are possible either at the initial moment of time (the Riemann problem), or at intersections in the process of its motion. For example, the moving discontinuities of the Riemann invariants can intersect in some point, and then pass through each other without changing its velocities. Of course, the magnitude of the jumps of discontinuities $\llbracket R^{i}\rrbracket$ in the process of evolution can change its values. To avoid misunderstandings, note that the discontinuities of densities $\llbracket u^{1}\rrbracket$, $\llbracket u^{2}\rrbracket$ can exist simultaneously.
II.5 Numerical results
We demonstrate two examples of the initial density distribution evolution. In the first example, the initial distribution of density is periodic in space
$$u^{1}_{0}=0.2(1+0.1\cos x),\quad u^{2}_{0}=0.3(1+0.2\sin x),\quad 4\lambda=1,%
\quad\kappa=1.$$
(2.54)
On Fig. 1 the distribution of the densities and the Riemann invariants at time $t=8.982$ is shown. The red lines correspond to the initial distribution.
The second example demonstrates the Riemann problem solutions. We solve the general Riemann problem when
the initial discontinuities is not piecewise constant.
$$u^{1}_{0}=(0.2-0.1h(x+2))(1+0.1\cos 2x),\quad 4\lambda=1,\quad\kappa=1$$
(2.55)
$$u^{2}_{0}=(0.3+0.1h(x-2))(1+0.1\sin 3x),$$
where $h(\tau)$ is the Heaviside step function.
On Fig. 2 the distribution of the densities at time $t=1.091$, $t=1.590$ are shown. The red lines correspond to the initial distribution.
On Fig. 3 the distribution of the Riemann invariant at time $t=1.091$, $t=1.590$ are shown. The red lines correspond to the initial distribution.
On Figs. 2, 3 the evolution of the discontinuities is well visible. As already mentioned, the simultaneous discontinuities of the invariants $R^{1}$, $R^{2}$ exist only at their the interaction. On the contrary, the discontinuities of the densities $u^{1}$, $u^{2}$ can be simultaneous.
III The shallow water equations
In this Section we present the numerical results for the shallow water equations omitted in previous paper Zhuk_Shir_ArXiv_2014_1 .
The classic version of the shallow water equations without taking into account the slope of the bottom has the form (see for example RozhdestvenskiiYanenko ; Whithem )
$$h_{t}+(hv)_{x}=0,\quad v_{t}+\left(\frac{1}{2}v^{2}+h\right)_{x}=0,$$
(3.1)
where $h>0$ is the elevation of the free surface, $v$ is the velocity.
We rewrite the equations in the form
$$u^{1}_{t}+(u^{1}u^{2})_{x}=0,\quad u^{2}_{t}+\left(\frac{1}{2}u^{2}u^{2}+u^{1}%
\right)_{x}=0,\quad h=u^{1},\quad v=u^{2}.$$
(3.2)
III.1 Interactions of the ‘solitons’
The initial distribution
$$u^{1}_{0}=1+0.2e^{-(x+3)^{2}}+0.2e^{-(x-3)^{2}},\quad u^{2}_{0}=0.2e^{-(x+3)^{%
2}}-0.2e^{-(x-3)^{2}}$$
(3.3)
simulates the interaction of two ‘solitons’.
The initial perturbations
of the free surface and velocity are given in the form of Gaussian distributions. The right perturbation moves to the left, and the left perturbation moves to the right.
The position of the free surface and the distribution of the velocity field for different moments of time are shown on Fig. 4–9.
III.2 Wave packet
We take the perturbation of the free surface in the wave packet form, assuming that the velocity is equal to nought
$$u^{1}_{0}=1.0+0.1\cos(3x)e^{x^{2}},\quad u^{2}_{0}=0.$$
(3.4)
The position of the free surface and the distribution of the velocity field for different moments of time are shown on Fig. 10, 11.
IV Conclusions
The choice to study equations of two-beam reduction of the dense soliton gas is not random selection.
First, these equations are degeneracy and, therefore, does not admit self-similar solutions.
Secondly, the results of the Sec. II.4 show that the braking solution profile is impossible. All strong discontinuities are the so-called contact discontinuities, i.e. the discontinuities are moving along the characteristics. This, in particular, means that the proposed method allows to solve the Cauchy problem for arbitrary initial data, including discontinuous. Thirdly, the Riemann–Green function has a very simple form that allows us to easily analyze the solution and solve the Cauchy problem with initial data on an arbitrary curve. Note that the densities and fluxes of the conservation laws for equations of two-beam reduction of the dense soliton gas (as well as for equations of the zonal electrophoresis Zhuk_Shir_ArXiv_2014_2 ) can be constructed as linear combinations of the original conservation laws (see (2.9)–(2.14)). Unfortunately, we could not use this method in the case of the shallow water equations.
As already mentioned, the numerical method is accurate, as it does not require any approximations of the original problem. The most efficient method operates when there is an explicit expression for the Riemann–Green function. However, this method can be applied in cases when the Riemann–Green function is determined using the approximate solution of linear equations (A1.3)–(A1.7), for example, in the form of an infinite series or by using numerical methods. Of course, in this case, the inevitably there are errors associated with the construction of the Riemann–Green function.
We say a few words about the Cauchy problem (A1.18)–(A1.20). From our point of view, this problem is a generalization of the characteristics method to the case of two hyperbolic equations. Strictly speaking, formally, the method of characteristics for an arbitrary number of equations to construct is not very difficult. It is sufficient to consider the augmented system and construct the solution, for example, in the form of elementary waves RozhdestvenskiiYanenko ; Bressan . However, such equations are not closed. For the two equations the system can be closed, using the hodograph method and the Riemann–Green function. It would be interesting to build a similar scheme for solving the problem, bypassing the procedure to construct the Riemann–Green function of (and possibly hodograph method), at least for two hyperbolic quasilinear equations.
Acknowledgements.
The authors are grateful to N. M. Zhukova for proofreading the manuscript.
Funding statement. This research is partially supported by the Base Part of the Project no. 213.01-11/2014-1,
Ministry of Education and Science of the Russian Federation, Southern Federal University.
Appendix
Appendix A Reduction of the Cauchy problem for two
hyperbolic quasilinear PDE’s to the Cauchy problem for ODE’s
Referring for details to Zhuk_Shir_ArXiv_2014_1 ; Zhuk_Shir_ArXiv_2014_2 ; SenashovYakhno , here we give only a brief description of the method which allows to reduce
the Cauchy problem for two hyperbolic quasilinear PDE’s to the Cauchy problem for ODE’s.
A.1 The Riemann invariants
Let for a system of two hyperbolic PDE’s, written in the Riemann invariants $R^{1}(x,t)$, $R^{2}(x,t)$, we have the Cauchy problem at $t=t_{0}$
$$R^{1}_{t}+\lambda^{1}(R^{1},R^{2})R^{1}_{x}=0,\quad R^{2}_{t}+\lambda^{2}(R^{1%
},R^{2})R^{2}_{x}=0,$$
(A1.1)
$$R^{1}(\tau,t_{0})=R^{1}_{0}(x),\quad R^{2}(x,t_{0})=R^{2}_{0}(x),$$
(A1.2)
where $R^{1}_{0}(x)$, $R^{2}_{0}(x)$ are the functions determined on some interval of the axis $x$ (possibly infinite), $\lambda^{1}(R^{1},R^{2})$, $\lambda^{2}(R^{1},R^{2})$ are the given functions.
We recall that any system of two quasilinear equations can be reduce to the Riemann invariants (see e.g. RozhdestvenskiiYanenko )
A.2 Hodograph method
Using the hodograph method for some conservation law
$\varphi_{t}+\psi_{x}=0$, where $\varphi(R^{1},R^{2})$ is the density, $\psi(R^{1},R^{2})$ is the flux, we write the equation SenashovYakhno
$$\Phi_{R^{1}R^{2}}+A(R^{1},R^{2})\Phi_{R^{1}}+B(R^{1},R^{2})\Phi_{R^{2}}=0,$$
(A1.3)
$$A(R^{1},R^{2})=\frac{\lambda^{1}_{R^{2}}}{\lambda^{1}-\lambda^{2}},\quad B(R^{%
1},R^{2})=-\frac{\lambda^{2}_{R^{1}}}{\lambda^{1}-\lambda^{2}}.$$
(A1.4)
A.3 The Riemann–Green function
Let the function $\Phi(R^{1},R^{2}|r^{1},r^{2})$ is the Riemann–Green function for equation (A1.3).
The function $\Phi(R^{1},R^{2}|r^{1},r^{2})$ of variables $R^{1}$, $R^{2}$ satisfies the given equation, and
the function $\Phi(R^{1},R^{2}|r^{1},r^{2})$ of variables $r^{1}$, $r^{2}$ is the solution of the conjugate equation
$$\Phi_{r^{1}r^{2}}-(A(r^{1},r^{2})\Phi)_{r^{1}}-(B(r^{1},r^{2})\Phi)_{r^{2}}=0,$$
(A1.5)
with conditions
$$(\Phi_{r^{2}}-A\Phi)\bigr{|}_{r^{1}=R^{1}}=0,\quad(\Phi_{r^{1}}-B\Phi)\bigr{|}%
_{r^{2}=R^{2}}=0,$$
(A1.6)
$$\Phi\bigr{|}_{r^{1}=R^{1},r^{2}=R^{2}}=1.$$
(A1.7)
The methods of the Riemann–Green function construction are described, for example, in Copson ; Chirkunov ; Chirkunov_2 ; Courant ; Ibragimov ; SenashovYakhno .
A.4 Implicit solution of the problem
It is convenient, to write the density of a conservation law, i.e. the function $\varphi(R^{1},R^{2})$,
in the form $\varphi(R^{1},R^{2}|r^{1},r^{2})$
$$\varphi(R^{1},R^{2}|r^{1},r^{2})=M(r^{1},r^{2})\Phi(R^{1},R^{2}|r^{1},r^{2}),%
\quad M(r^{1},r^{2})=\frac{2}{\lambda^{2}(r^{1},r^{2})-\lambda^{1}(r^{1},r^{2}%
)}.$$
(A1.8)
The solution of (A1.1), (A1.2) can be represented in implicit form as SenashovYakhno
$$R^{1}(x,t)=r^{1}(b)=R^{1}_{0}(b),\quad R^{2}(x,t)=r^{2}(a)=R^{2}_{0}(a),$$
(A1.9)
where $a$, $b$ are the new variables (Lagrange variables).
The connection between the new variables $a$, $b$ and old variables $x$, $t$ has the form
$$t=t(a,b),\quad x=x(a,b).$$
(A1.10)
Function $t=t(a,b)$ is calculated using the density of the conservation law $\varphi(R^{1},R^{2}|r^{1},r^{2})$ and the initial data $R^{1}_{0}(x)$, $R^{2}_{0}(x)$ SenashovYakhno ; Zhuk_Shir_ArXiv_2014_1 ; Zhuk_Shir_ArXiv_2014_2
$$t(a,b)=t_{0}+\frac{1}{2}\int\limits_{a}^{b}\varphi(R^{1}_{0}(\tau),R^{2}_{0}(%
\tau)|r^{1}(b),r^{2}(a))\,d\tau.$$
(A1.11)
Function $x=x(a,b)$ is calculated by analogously SenashovYakhno , but this function is not required for further. We assume that this function is the given function.
If the equations (A1.10) are solvable explicitly
$$a=a(x,t),\quad b=b(x,t),$$
(A1.12)
then we have explicit solution
$$R^{1}(x,t)=R^{1}_{0}(b(x,t)),\quad R^{2}(x,t)=R^{2}_{0}(a(x,t)).$$
(A1.13)
A.5 Solution on isochrones
To construct the solution in the form (A1.9) in Zhuk_Shir_ArXiv_2014_1 we proposed to solve the Cauchy problem for ODE’s.
Fix some value $t=t_{*}$, specifying the level line (isochrone) of function $t(a,b)$
$$t_{*}=t(a,b).$$
(A1.14)
We assume that the isochrone is determined on the plane $(a,b)$ by the parametrical equations
$$a=a(\tau),\quad b=b(\tau),$$
(A1.15)
where $\tau$ is the parameter.
We choose the values $a_{*}$, $b_{*}$ which indicate some point on isochrone $t=t_{*}$
$$t_{*}=t(a_{*},b_{*}).$$
(A1.16)
In practice, the values of $a_{*}$, $b_{*}$ one can choose using the line levels of function $t(a,b)$ for some ranges of parameters $a$, $b$.
The coordinate $x$ on isochrone, obviously, is determined by the expression
$$x=x(a(\tau),b(\tau))\equiv X(\tau).$$
(A1.17)
To determine the $a(\tau)$, $b(\tau)$, $X(\tau)$ we have the Cauchy problem Zhuk_Shir_ArXiv_2014_1 ; Zhuk_Shir_ArXiv_2014_2
$$\frac{da}{d\tau}=-t_{b}(a,b),\quad\frac{db}{d\tau}=t_{a}(a,b),$$
(A1.18)
$$\frac{dX}{d\tau}=(\lambda^{2}(r^{1}(b),r^{2}(a))-\lambda^{1}(r^{1}(b),r^{2}(a)%
))t_{a}(a,b)t_{b}(a,b),$$
(A1.19)
$$a\bigr{|}_{\tau=0}=a_{*},\quad b\bigr{|}_{\tau=0}=b_{*},\quad X\bigr{|}_{\tau=%
0}=X_{*}.$$
(A1.20)
Here the values $a_{*}$, $b_{*}$ are given. To determine $X_{*}$ we need to solve the problem
$$\frac{dY(b)}{db}=x_{b}(a_{*},b)=\lambda^{2}(r^{1}(b),r^{2}(a_{*}))t_{b}(a_{*},%
b),\quad Y(a_{*})=a_{*}.$$
(A1.21)
Integrating from $a_{*}$ to $b_{*}$ we get
$$X_{*}=Y(b_{*}).$$
(A1.22)
Note, that $X_{*}=x(a_{*},b_{*})$ is the $x$ coordinate corresponding to $\tau=0$.
Solving (A1.18)–(A1.20)
we obtain the solution on isochrone
$$R^{1}(x,t_{*})=R^{1}_{0}(b(\tau)),\quad R^{2}(x,t_{*})=R^{2}_{0}(a(\tau)),%
\quad x=X(\tau).$$
(A1.23)
Changing the parameter $\tau$ we obtain the solution which depends on $x$ as the fixed $t=t_{*}$. Pay attention to the
fact that the right hand sides of differential equations, in particular, $t_{a}(a,b)$, $t_{b}(a,b)$ are easily computed with help of (A1.8), (A1.9), (A1.11).
Appendix B Additional simplification
Using another parameters $\theta$ instead $\tau$
$$d\tau={(R^{1}_{0}(a)-R^{2}_{0}(a))(R^{1}_{0}(b)-R^{2}_{0}(b))}d\theta$$
(B2.1)
one can simplify the equations (2.37), (2.38).
In this case instead of the equations (2.37), (2.38) we get
$$\frac{da}{d\theta}=R^{2}_{0}(a)-R^{1}_{0}(a),\quad\frac{db}{d\theta}=R^{2}_{0}%
(b)-R^{1}_{0}(b),\quad\frac{dX}{d\theta}=R^{2}_{0}(a)-R^{1}_{0}(b).$$
(B2.2)
References
(1)
Shiryaeva E. V., Zhukov M. Yu.
Hodograph Method and Numerical Solution of the Two Hyperbolic Quasilinear Equations. Part I. Shallow water equations //
(2014) arXiv:1410.2832. 19 p.
(2)
Shiryaeva E. V., Zhukov M. Yu.
Hodograph Method and Numerical Solution of the Two Hyperbolic Quasilinear Equations. Part II. Zonal electrophoresis equations //
(2015) arXiv:1503.01762. 23 p.
(3)
Senashov S. I., Yakhno A. Conservation laws, hodograph transformation and boundary value problems of plane plasticity // SIGMA. 2012. Vol. 8, 071. 16 p.
(4)
Rozdestvenskii B.L., Janenko N.N. Systems of Quasilinear Equations and Their Applications to Gas
Dynamics [in Russian], Nauka, Moscow (1978); English transl.: Transl. Math. Monogr., Vol. 55, Amer. Math.
Soc., Providence, R. I. (1983).
(5)
Whitham G. B. Linear and nonlinear wave. A Wiley-Interscience Publication John Willey &
Sons, 1974, New-York–London–Sydney–Toronto.
(6)
El. G.A., Kamchatnov A.M. Kinetic equation for a dense soliton gas // ArXiv:nlin/0507016v2. 2006. 4 p.
(7)
Ferapontov E. V., Tsarev S. P. Systems of hydrodynamic type that arise in gas chromatography. Riemann invariants and exact solutions. (Russian) Mat. Model. 3 (1991), no. 2, 82–-91.
(8)
Kuznetsov N. N.
Some mathematical questions of chromatography // Computation methohods and programming. 1967. No 6,242–258.
(9)
Elaeva M. S. Investigation of zonal elecrophoresis for two component mixture // (Russian) Mat. Model. 22 (2010), no. 9, 146–-160.
(10)
Elaeva M. S. Separation of two component mixture under action an electric field // Comp. Math. and Mat. Phys. 52:6 (2012), 1143–1159.
(11)
Babskii V. G., Zhukov M. Yu., Yudovich V. I. Mathematical Theory of Electrophoresis.
Kluwer Academic / Plenum Publishers (1989).
(12)
Zhukov M. Yu. Non-staionary isotachophoresis model // Comp. Math. and Math. Phys. 1984. V. 24, No 4, 549–565. (in Rissian).
(13)
Zhukov M. Yu. Masstransport by an electric field. Rostov-on-Don: RGU Press, 2005. 215 p.
(14)
Copson E. T. On the Riemann-Green Function.Arch. Ration. Mech. Anal. 1 (1958) 324-348.
(15)
Courant R., Hilbert D. Methods of Mathematical Physics: Partial Differential Equations, Volume II. New York – London, 1964.
(16)
Ibragimov N. Kh. Group analysis of ordinary differential equations and the invariance principle in mathematical physics (for the 150th anniversary of Sophus Lie) // Russian Mathematical Surveys(1992),47(4):89. P. 83–144.
(17)
Chirkunov Yu. A. On the symmetry classification and conservation laws for quasilinear differential equations of second order // Mathematical Notes. 2010, Volume 87, Issue 1-2, pp 115-121.
(18)
Chirkunov Yu. A. Generalized equivalence transformations and group classification of systems of differential equations // Journal of Applied Mechanics and Technical Physics. 2012, Volume 53, Issue 2, pp 147-155.
(19)
O. F. Menshikh Conservation Laws of Second Order for the Born–Infeld Equation and Other Related Equations //
Mat. Zametki, 84:6 (2008), 874–881.
(20)
Menshikh O. F. Conservation laws and Backlund transformations associated with the Born–Infeld equation //
Mat. Zametki, 77:4 (2005), 551–565.
(21)
Menshikh O. F. Interaction of Born–Infeld solitons //
Teoret. Mat. Fiz., 84:2 (1990), 181–194.
(22)
Menshikh O. F. Interaction of finite solitons for equations of Born–Infeld type //
Teoret. Mat. Fiz., 79:1 (1989), 16–29.
(23)
Bressan A. Hyperbolic Conservation Laws.
An Illustrated Tutorial // 2009, 81 p. http://www.math.psu.edu/bressan/PSPDF/clawtut09.pdf. |
Exploring semiconductor substrates for Silicene epitaxy
Amrita Bhattacharya${}^{\dagger}$, Saswata Bhattacharya${}^{\dagger}$, Gour P. Das${}^{*}$
Dept. of Materials Science, Indian Association for the Cultivation of Science, Kolkata-32, India
(January 18, 2021)
Abstract
We have carried out first-principles based DFT calculation on electronic properties of
silicene monolayer on various (111) semi-conducting surfaces. We find that the relative stability and other properties of the silicene overlayer depends sensitively on whether the interacting top layer of the substrate is metal or non-metal terminated. The nature of silicene-monolayer on the metal terminated surface can be metallic or even magnetic, depending upon the choice of the substrate. The silicene overlayer undergoes n-type doping on
metal terminated surface while it undergoes p-type doping on non metal terminated surfaces of the semiconductor substrates.
Silicene, Planar Nanostructures, Density Functional Theory, Electronic Structure
From first principles
calculations, the 2D mono-layers of Silicon, known as silicene, has also been predicted to be
stabler1 ; r2 ; r3 with graphene like
semi-metallic characteristics viz. a linear dispersion relation at
the K pointr4 ; r5 ; r6 . The epitaxial growth of Ag thin film on Si(111) substrate has been
explored by experimentalists in the past. The limited intermixing between Ag thin
film and Si(111) substrate, make the choice of substrate ideal, for
the growth of Ag thin films, so, it was natural to conjecture a reverse case. Infact, the experimental evidence of formation of silicene on
Ag(001), Ag(110), Ag(111) substrates has been reported by various groups r7 ; r8 ; r9 ; r10 ; r11 ; r12 . The formation of silicene on Ag(110) substrate has also been studied using first principles
methods by A. Kara et al r13 ; r14 . More recently, evidence of epitaxial Silicene on non silver ZrB${}_{2}$-0001 substrate r15 and Ir(111) substrate r16 has also been reported. Moreover, silicene monolayer on Ag(111) was found to undergo a phase
transition to two types of mirror-symmetric boundary-separated rhombic phases at temperatures below
40 K by scanning tunneling microscopy r17 . However, the possibility of existence of other suitable substrates for formation of silicene monolayer can not be ruled out and has not been exhaustively explored till date.
In this communication, we report from our first principles based calculations, the
bonding, stability and electronic structure of silicene monolayer,
when epitaxially grown on various Group II-VI and Group III-V
semiconductor substrates viz.
AlAs(111), AlP(111), GaAs(111), GaP(111), ZnS(111) and ZnSe(111).
The nature of silicene monolayer on the metal
terminated surface of these substrates can be metallic or even magnetic,
depending upon the choice of the substrate. The silicene overlayer undergoes
n-type doping on metal terminated surface while it undergoes p-type doping on
non metal terminated surfaces of the semiconductor substrates.
Our calculations have been carried out using first-principles
density functional theory (DFT)r23 ; r24 based total energy calculations. We have used
VASP r25 code with projected augmented wave
(PAW) potentialr26 and Perdew-Burke-Ernzerhof (PBE) exchange correlation
functionalr27 within generalized gradient
approximation (GGA). An energy cut off of 600 eV has been used. The
k-mesh was generated by Monkhorst-Pack
method and the results were tested for convergence with respect to
mesh size [8$\times$8$\times$1]. However, for the generation of the
electronic density of states (DOS) and band
structure plots, higher values of k-points [16$\times$16$\times$1] were used. In
all our calculations, self-consistency has been achieved with a
$0.0001$ eV convergence in total energy. For optimizing the ground
state geometry, forces on the atoms were
converged to less than $0.001$ eV/Å via conjugate gradient
minimizationr30 .
The silicene unit cell has a buckled hexagonal planar geometry,
with a lattice parameter of 3.84 Å and a homogeneous Si-Si bond
distance of 2.29 Å throughout the sheet. The semiconductor
substrates viz. AlAs(111), AlP(111), GaAs(111), GaP(111), ZnS(111) and
ZnSe(111), that we have considered for epitaxial growth
of silicene monolayer, have resonably good lattice matching with that of
silicene, as given in TABLE-1. In this work, we have only considered
the direct lattice matched non polar-(111) substrates. However, other substrate
orientations may also be relevant and will be addressed in our future study.
Therefore, the unit cell of the
composite system consists of the silicene monolayer (having two Si
atoms) placed on top of six layers of the substrate (having 12 atoms),
with a vacuum slab of 20Å above it.
It is to be noted here that in these substrates, the contact layer of the substrate that interacts
with the silicene overlayer could either be non metal terminated (NMT) or metal
terminated (MT), e.g. in ZnS(111), the interacting layer could be either
S terminated or Zn terminated (FIG.1). If the contact layer is NMT, then the MT
bottom layer in the unit cell does not need H-passivation (FIG.1a). However,
if the contact is MT, then the
lowest layer of the six layer substrate in the unit cell is NMT that
needs to be passivated by hydrogen (FIG.1b), in order to avoid the effect of
dangling bonds. The nomenclature used for designating an unit of NMT
substrate is NM6M6, while the MT substrate is denoted as M6NM6H. For calculating the
binding energy (BE) and magnetic interaction of the silicene monolayer with the semiconductor substrates
we have used a 4$\times$4 supercell.
The BE of silicene monolayer to the MT and NMT surfaces of these
substrates are summarized in TABLE-1. For MT surface the BE values
lie in the range of 0.56 $\pm$ 0.12
eV/atom (TABLE-1) which is comparable to that of silicene monolayer
on Ag(111) substrate, for which our estimated value (0.52 eV/atom) is
in good agreement with first principle result of P. Vogt et.al r17 .
Thus, on comparing the binding energies of silicene to semiconductor MT/ NMT substrates with Ag(111),
we find that they are close or comparable
for metal terminated surfaces of GaAs(111), GaP(111), ZnS(111), ZnSe(111) substrates and non metal terminated surfaces of AlAs(111), GaP(111) and ZnSe(111) substrates (TABLE-1).
Our calculations show that when the bulk substrate (both MT and NMT)
is cleaved to certain number of layers, the top layer contains
uncompensated dangling bonds and therefore, give rise to magnetic
moment in the substrate (TABLE-2). We have estimated the magnetic moment of the
substrate, before and after introduction of silicene monolayer which are given in TABLE-2. We would first discuss the behaviour of
silicene with the NMT semiconductor substrates. When silicene monolayer is
introduced to the NMT surface of these substrates, the magnetic moment of the composite
system gets enhanced. For the sake of simplicity, this magnetism can be
thought upon as the effect of uncompensated dangling bonds at the surface of the
substrate. However, the
nature of charge transfer between the silicene layer and substrate may play a vital role in
enhancement of magnetism of the composite system in all NMT cases. We
have performed Bader charge analysis of NMT substrates before and after
inclusion of silicene layer. In each case, the charge gets transferred from the silicene sheet
to the contact atoms (viz. As, P, S, Se, Ge) of the NMT substrates
(see TABLE-3), because of the later having higher
electronegavity as compared to Si. Therefore, in all NMT cases the silicene overlayer undergoes
p-type doping due to its interaction with the substrate.
In order to understand the interaction of silicene with these substrates and to help their characterization in the laboratory it is necessary to study the magnetic behavior of silicene with
substrate. Therefore, we have plotted the layer projected DOS and the fattening of silicon bands near the Fermi level in order to study the interaction of the sheet with these substrates. Unlike the NMT cases, the behaviour of silicene monolayer on the MT surface can be
divided into three categories. The introduction of silicene monolayer
on MT surface, may quench/
enhance the magnetism or may not have any effect at all, on the magnetic moment of the composite (silicene + substrate) system as shown in TABLE-2. In case
of AlAs(111), AlP(111) and GaAs(111) [Case-I], the surface magnetism of the MT surface
of the substrate is
quenched after introduction of silicene monolayer and as a result of this, the magnetic moment
of the composite system is found to vanish.
Whereas, in GaP(111) and ZnSe(111) [Case-II], the introduction
of silicene monolayer enhances the magnetic moment of the composite system.
However, only in case of ZnS(111) substrate [Case-III],
the magnetic moment of the MT surface remains ‘zero’ before as well as after
inclusion of the silicene monolayer.
The behaviour of silicene on MT surface of semiconductor substrates
can also be analyzed from the layer projected DOS plots of silicene monolayer on these
substrates. The LP-DOS plot of silicene monolayer on MT surface show
that it can be metallic or magnetic behaviour, depending
upon the choice of substrate. In FIG.2, we have shown the LP-DOS and band dispersion plots
of some representative cases [viz. silicene on GaP(111), GaAs(111), ZnS(111) and ZnSe(111)],
while for the same for AlP(111) and AlAs(111) are given as supplementary information [SI-FIG.1].
Silicene on MT surface of AlAs(111),
AlP(111), GaAs(111) and ZnS(111) show metallic behavior with finite density of states
at the Fermi level. FIG.2(b) and (c) show the LP-DOS and band dispersion plots
of silicene on Ga terminated GaAs(111) and Zn terminated ZnS(111) substrate respectively. The
LP-DOS of silicene on these substrates
resemble the overall DOS of free standing silicene monolayer with
their Dirac cone shifted below the Fermi level by 0.7 eV approximately.
Silicene monolayer on MT surface of GaP(111),
show magnetic behavior, with sharp spin up and spin down
splitting near the Fermi level. The spin polarized LP-DOS and band dispersion plots
of silicene on Ga terminated GaP(111) substrate is shown in FIG.2(a).
It is to be noted here that in case of MT III-V substrates, the silicene bands
comprises the valence band and conduction bands near the Fermi level. Therefore,
the behaviour of the composite system is dictated by Silicene itself.
The spin polarized LP-DOS plot of silicene monolayer on the MT surface of ZnSe(111) show half metallic behavior, with the spin down channel completely vanishing at the Fermi level FIG.2(d). However, in the band dispersion plot of the silicene-substrate composite system for substrates involving Zn [ie. ZnSe(111) and ZnS(111)], it is the Zn $\alpha$-band predominantly constitutes the conduction band bottom and it protrudes below the Fermi level at $\Gamma$ point.
Results of the Bader charge analysis of Silicene monolayer on the MT semiconductor substrates is shown in TABLE-3. In case of most of the MT substrates, electrons are transferred from the substrate to the silicene overlayer because of higher electronegativity of the Si atoms as compared to the contact metal atoms of the substrate. As a result of this the Silicene overlayer undergoes a substrate induced n-type dopingr32 ; r33 .
While only in case of MT GaP(111) substrate, electrons are transferred from silicene to the substrate [ie. in the opposite direction] which suggest a
p-type doping in the silicene overlayer similar to NMT cases r34 ; r35 .
For a better understanding of the p-type/ n-type doping of the silicene overlayer, we have estimated the work function (WF) of MT/NMT ($\phi_{s}$) surfaces of semiconductor substrates and compared it with that of free standing
silicene monolayer ($\phi_{fl}$). For $\phi_{s}$ $>$ $\phi_{fl}$, electrons are expected to transfer from silicene overlayer to the substrate, thereby leading to a p-type doping of the silicene overlayer. On the contrary, if $\phi_{fl}$ $>$ $\phi_{s}$, then electrons should be transferred from the substrate to the silicene overlayer leading to n-type doping. The results of our calculations shown in FIG.3, reveals that for
MT substrates the WFs are consistently lower than that of free standing silicene (4.5 eV) with the exception of MT GaP(111) substrate (which is reverse to the other MT cases as also seen from the Bader analysis of TABLE-3). However, for NMT surfaces the trend is just reverse leading to the p-type doping. This is just reverse leading to the p-type doping.
From the layer projected DOS plot of FIG.2, the difference between the substrate induced p-type and n-type doping of the silicene overlayer can be observed.
In case of GaP(111), the p-type doping of the silicene overlayer is accompanied by shift in Fermi level towards the valence band (FIG.2a) while in the case of n-type doping (FIG.2b, c and d) the Fermi level shifts towards the conduction band. This type of p-type or n-type doping in silicene-substrate composite system could be useful for band gap engineering of bilayer silicene following the case of bilayer graphene r36 ; r37 .
In conclusion, we report from our first principles based calculations, the
bonding, stability and electronic structure of silicene monolayer,
when epitaxially grown on various Group II-VI, Group III-V and Group IV
semiconductor substrates viz.
AlAs(111), AlP(111), GaAs(111), GaP(111), ZnS(111) and ZnSe(111). We find that the relative stability and other
properties of the silicene overlayer depends sensitively on whether
the interacting top layer of the substrate is metal or non-metal terminated.
The binding energy of silicene monolayer to the metal terminated
surfaces of these substrates are estimated to be range in the 0.56 $\pm$ 0.12 eV/atom.
The introduction of silicene monolayer on the NMT surface of
all these semiconductor substrates, leads to enhancement in the magnetic moment
of the composite system.
However, the behavior of silicene on MT surface of semiconductor substrates can be metallic,
magnetic or semi-metallic depending on the choice of substrate. It undergoes
substrate induced p-type doping on NMT substrates while n-type doping on MT substrates [with an exception of GaP(111)] depending upon its charge transfer with the substrates.
${}^{\dagger}$ presently affiliated to Fritz-Haber-Institut der Max Planck Gesellschaft, Faradayweg 4-6, Berlin-14195, Germany.
${}^{*}$Corresponding Author Email id: msgpd@iacs.res.in
References
(1)
K. Takeda and K. Shiraishi, Phys. Rev. B 50, 14916 (1994).
(2)
S. B. Fagan, R. J. Baierle, R. Mota, Z. J. R. da
Silva and A. Fazzio, Phys. Rev. B 61, 9994 (2000).
(3)
G. G. Guzmán-Verri and L. C. Lew Yan Voon, Phys. Rev. B 76, 075131 (2007).
(4)
A. K. Geim and K. S. Novoselov, Nat. Mater. 6, 183 (2007); K. S. Novoselov, A. K. Geim,
S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva,
and A. A. Firsov, Science 306, 666 (2004).
(5)
D. C. Elias, R. R. Nair, T. M. G. Mohiuddin, S. V. Morozov, P. Blake, M. P. Halsall, A. C. Ferrari,
D. W. Boukhvalov, M. I. Katsnelson, A. K. Geim and K. S. Novoselov,
Science 323, 610 (2009).
(6)
K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva,
S. V. Dbonos and A. A. Firsov, Nature (London) 438, 197
(2005).
(7)
B. Aufray, A. Kara, S. Vizzini, H. Oughaddou, C. Léandri, B. Ealet, G. Le Lay, Appl.
Phys. Lett. 96 183102 (2010).
(8)
B. Lalmi, H. Oughaddou, H. Enriquez, A. Kara, S. Vizzini, B. Ealet and B. Aufray,
Appl. Phys. Lett 97,223109 (2010).
(9)
P. De Padova, C. Quaresima, C. Ottaviani, P.M. Sheverdyaeva, P. Moras,
C. Carbone, D. Topwal, B. Olivieri, A. Kara, H. Oughaddou, B. Aufray, G. Le Lay,
Appl. Phys. Lett. 96 261905 (2010).
(10)
A. Kara, C. Léandri, M.E. Dévila, P. De Padova, B. Ealet, H. Oughaddou, B. Aufray and G. Le Lay, J. Supercond. Novel Magn. 22 259 (2009).
(11)
P. Vogt, P. D. Padova, C. Quaresima, J. Avila, E. Frantzeskakis,
M. C. Asensio, A. Resta, B. Ealet and G. L. Lay, Phys. Rev. Lett. 108, 155501 (2012).
(12)
B. Feng, Z. Ding, S. Meng, Yugui Yao, X. He, P. Cheng, L. Chen and K. Wu, Nano Lett., 12, 3507 (2012).
(13)
A. Kara, S. Vizzini, C. Leandri, B. Ealet, H. Oughaddou, B. Aufray and G. Le Lay, J. Phys.: Condens. Matter, 22, 045004 (2010).
(14)
A. Kara, H. Enriquez, Ari P. Seitsonen, L.C. Lew Yan Voone, S. Vizzini , B. Aufray , H. Oughaddoub, Surface Science Reports 67, 1 (2012).
(15)
A. Fleurence, R. Friedlein, T. Ozaki, H. Kawai, Y. Wang, and Y. Yamada-Takamura, Phys. Rev. Lett. 108, 245501 (2012).
(16)
L. Meng, Y. Wang, L. Zhang, S. Du, R. Wu, L. Li, Y. Zhang, G. Li, H. Zhou, W. A. Hofer and H. Gao, Nano Lett. 13, 685 (2013).
(17)
L. Chen, H. Li, B. Feng, Z. Ding, J. Qiu, P. Cheng, K. Wu and S. Meng, Phys. Rev. Lett. 110, 085504 (2013).
(18)
P. Hohenberg and W. Kohn, Phys. Rev. B 136, 864 (1964).
(19)
W. Kohn and L. Sham, Phys. Rev. A 140, 1133 (1965).
(20)
G. Kresse and J. Hafner, Phys. Rev. B 49, 14251 (1994); G. Kresse
and J. J. F$\ddot{u}$rthmller, Comput. Mater. Sci. 6, 15 (1996).
(21)
P. E. Bl$\ddot{o}$chl, Phys. Rev. B 50, 17953 (1994).
(22)
J. P. Perdew, K. Burke and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996).
(23)
P. Pulay, Chem. Phys. Lett. 73, 393 (1980).
(24)
W. Chen, S. Chen, D. C. Qi, X. Y. Gao and A. T. S. Wee, J. Am.
Chem. Soc., 129, 10418 (2007).
(25)
N. Jung, N. Kim, S. Jockusch, N. J. Turro, P. Kim and L. Brus,
Nano Lett., 9, 4133 (2009).
(26)
J. H. Chen, C. Jang, S. Adam, M. S. Fuhrer, E. D. Williams and
M. Ishigami, Nat. Phys., 4, 377 (2008).
(27)
A. Bostwick, T. Ohta, T. Seyller, K. Horn and E. Rotenberg, Nat.
Phys. 3, 36 (2007).
(28)
Y. Zhang, T. Tang, C. Girit, Z. Hao, M. C. Martin, A. Zettl, F. Michael, C. Y. R. Shen and F.
Wang, Nature 459, 820 (2009).
(29)
T. Ohta, A. Bostwick, T. Seyller, K. Horn and E. Rotenberg, Science 313, 951 (2006). |
NO CURRENT WITHOUT HEAT
Christian Maes,
Frank Redig and
Michel Verschuere
Instituut voor
Theoretische Fysica, K.U. Leuven, Celestijnenlaan 200D, B-3001
Leuven, Belgium - email:
Christian.Maes@fys.kuleuven.ac.be T.U.Eindhoven. On leave from Instituut
voor Theoretische Fysica, K.U. Leuven, Celestijnenlaan 200D, B-3001
Leuven, Belgium - email: f.h.j.redig@tue.nl Instituut voor
Theoretische Fysica, K.U. Leuven, Celestijnenlaan 200D, B-3001
Leuven, Belgium - email:
Michel.Verschuere@fys.kuleuven.ac.be
Abstract: We show for a large class of interacting particle systems
that whenever the stationary measure is not reversible for the dynamics,
then the mean entropy production in the steady state is strictly positive.
This extends to the thermodynamic limit the equivalence between
microscopic reversibility and zero mean entropy production: time-reversal
invariance cannot be spontaneously broken.
Keywords: stochastic interacting particle systems, entropy
production, (generalized) detailed balance.
1 Introduction
Reversibility and entropy are words with many meanings even within the
context of nonequilibrium statistical mechanics. One class of models that
has often been considered for learning about nonequilibrium behavior is
that of interacting particle systems. These are stochastic dynamics for
spatially extended systems in which particles locally interact. They are
mostly toy-models remaining far from realistic in their microscopic
details. Yet, it is believed
that for some good purposes, the details do not
matter so much and one should be concerned more with the symmetries,
possible conservation laws, locality of the interaction etc.
to hope to understand something about real nature.
This paper is about the relation between time-reversal invariance and the
positivity of entropy production. We do this in the context of
interacting particle systems following the work in
[10, 5, 3, 6, 7]. The physics background will be discussed in
Section 3. The main question is to understand why there cannot exist a
‘superconducting’ interacting particle system in the sense of the title
of this paper to be specified below. To understand the mathematical
problem, let us look first at a finite Markov chain. Suppose that $K$ is
a finite set on which we have an involution $\pi:K\rightarrow K,\pi^{2}=id$, called time-reversal. Often, the most natural choice for
interacting particle systems is $\pi=id$ because we think of the state
space as consisting of occupation variables or of classical spins but our
mathematical set-up will be more general.
Let $(X_{t},t\in[-T,T])$ be a
stationary Markov process (steady state) on $K$ with law $\mathbb{P}_{\rho}$. The
subscript refers to the unique stationary probability measure $\rho$ on
$K$; we assume that $\rho(a)>0,a\in K$.
The rate to go from $a$ to $b$ is denoted by $k(a,b),a,b\in K$
and we assume that $k(\pi b,\pi a)=0$
iff $k(a,b)=0$ (dynamic reversibility). The generator is
$$Lf(a)=\sum_{b}k(a,b)[f(b)-f(a)]$$
(1.1)
The time-reversed process of $(X_{t})$ is the stationary Markov process
$(Y_{t},t\in[-T,T])$ on $K$ with $Y_{t}\equiv\pi X_{-t}$ having transition
rates
$$\tilde{k}(a,b)\equiv k(\pi b,\pi a)\frac{\rho(\pi b)}{\rho(\pi a)}$$
(1.2)
We denote
its law by $\tilde{\mathbb{P}}_{\rho\pi}$ ($\rho\pi$ is stationary for
$(Y_{t})$). Of course, it easily happens that $\rho=\rho\pi$ and yet,
$\mathbb{P}_{\rho}\neq\tilde{\mathbb{P}}_{\rho\pi}$. The corresponding generator for
the time-reversed process is $\tilde{L}=\pi L^{\ast}\pi$ where the $\ast$
refers to the adjoint with respect to the stationary measure
$\rho$.
We say that the process $(X_{t})^{T}_{-T}$ is $\pi$-reversible if
$\mathbb{P}_{\rho}=\tilde{\mathbb{P}}_{\rho\pi}$. This implies that the stationary measure
$\rho$ satisfies $\rho=\rho\pi$ and
$$\rho(a)k(a,b)=k(\pi b,\pi a)\rho(b),a,b\in K$$
(1.3)
which is generalized (or extended) detailed balance (microscopic
reversibility). For the generators, we then have $\tilde{L}=L$.
Observe that (1.3) by itself implies that $\rho(a)=\rho\pi(a)\rho(b)/\rho\pi(b)$ whenever $k(a,b)\neq 0$. Applying this successively
with $b_{1},\ldots,b_{n}\in K$ for which
$k(a,b_{1}),k(b_{1},b_{2}),\ldots,k(b_{n},\pi a)\neq 0$, we find that
$\rho(a)=\rho\pi(a)$. On the other hand, $\pi$-reversibility implies that
$\rho=\rho\pi$
is stationary.
The entropy production is the random variable obtained from taking the
relative action on pathspace with respect to time-reversal, see [4]
for a recent review. Let $\rho_{-T}$ be a probability measure on $K$
which we use to sample the initial data at time $-T$ for the stochastic
time-evolution generated by $L$. The law of this process is denoted by
$\mathbb{P}_{\rho_{-T}}$. Suppose now that for this process the state at time $T$
is described by the probability measure $\rho_{T}$. We could as well start
our process (at time $-T$) from $\rho_{T}\pi$ and then obtain the process
$\mathbb{P}_{\rho_{T}\pi}$. For a particular realization $\omega=(\omega(t),t\in[-T,T])$ of this process we let $\Theta_{\pi}\omega\equiv(\pi\omega(-t),t\in[-T,T])$ be its time-reversal. The entropy production $R_{\pi}(L,\rho_{-T},T)$ is a function of the realization over the
time-interval $[-T,T]$ and is then obtained as
$$R_{\pi}(L,\rho_{-T},T)(\omega)=\log\frac{d\mathbb{P}_{\rho_{-T}}}{d\mathbb{P}_%
{\rho_{T}\pi}\Theta_{\pi}}(\omega)$$
(1.4)
Here we are only interested in its steady state expectation value, that
is the mean entropy production rate, which in fact can be written as
$$\textrm{MEP}_{\pi}(L,\rho)=\lim_{T\uparrow\infty}\frac{1}{2T}{\mbox{\rm$\mbox{%
I}\!\mbox{E}$}}_{\rho}[\log\frac{d\mathbb{P}_{\rho}}{d\tilde{\mathbb{P}}_{\rho%
\pi}}]$$
(1.5)
where ${\mbox{\rm$\mbox{I}\!\mbox{E}$}}_{\rho}$ denotes expectation
with respect to $\mathbb{P}_{\rho}$.
The notation
MEP${}_{\pi}(L,\rho)$ reminds us that this
number depends on the transformation $\pi$, the dynamics
(generated via $L$) and the stationary measure $\rho$. The mean entropy
production thus measures the degree to which $\mathbb{P}_{\rho}$ can be
distinguished from $\tilde{\mathbb{P}}_{\rho\pi}$.
The main property of the mean entropy production is then:
Proposition 1: Consider the stationary process
$(X_{t})$ above with $\rho=\rho\pi$. Then,
$\textrm{MEP}_{\pi}(L,\rho)=\textrm{MEP}_{\pi}(\tilde{L},\rho)\geq 0$
with equality if and only if the process $(X_{t})$ is
$\pi$-reversible.
This says that for finite state space Markov chains
there can be no current without heat, meaning that detailed balance is
equivalent with zero mean entropy production. The problem we address
here is whether the same remains true in the thermodynamic limit, that is
for spatially extended interacting particle systems. In this case we
really should be speaking about the mean entropy production density, i.e., per unit volume, but we will not use this extension. Note
that in this case and from now on we will not and we
cannot assume in general that $\rho=\rho\pi$ even if both are stationary.
We discuss the general physics set-up and further interpretations in
Section 3, after stating our mathematical results in Section 2. We start
however with three examples illustrating some aspects.
1.1 Examples
Example A: We consider particles hopping on the one-dimensional
lattice with a preferred direction that is itself subject to independent
flips. The state space is $\{-1,+1\}\times\{0,1\}^{\mathord{\!{
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is
determined by choosing a constant rate $c(E,\eta)=1$
for changes from a configuration $(E,\eta)$ to $(-E,\eta)$ and
taking rates
$$c(x,E,\eta)=e^{E}\eta_{x}(1-\eta_{x+1})+e^{-E}\eta_{x+1}(1-\eta_{x})$$
for changes to $(E,\eta^{x,x+1})$ where $(\eta^{x,x+1})_{y}=\eta_{y}$ if
$x\neq y\neq x+1$, and $(\eta^{x,x+1})_{y}=\eta_{x}$ when $y=x+1$ and $=\eta_{x+1}$ when $y=x$. The resulting Markov process has generator
$$\displaystyle Lf(E,\eta)$$
$$\displaystyle=$$
$$\displaystyle\sum_{x}[e^{E}\eta_{x}(1-\eta_{x+1})+e^{-E}\eta_{x+1}(1-\eta_{x})%
][f(E,\eta^{x,x+1})-f(E,\eta)]$$
(1.6)
$$\displaystyle+$$
$$\displaystyle f(-E,\eta)-f(E,\eta)$$
For invariant measure $\rho$ we take
$$\rho(E,d\eta)\equiv\frac{1}{2}(\delta_{E,+1}+\delta_{E,-1})\times\nu_{u}(d\eta)$$
where $\nu_{u}$ is the Bernoulli measure with specified density $u\in(0,1)$. For
time-reversal we take $\pi(E,\eta)=(-E,\eta)$ so that $\rho=\rho\pi$.
It is easy to see that the process satisfies generalized detailed balance,
like (1.3), in the sense that
$\rho(E,d\eta)=\rho(-E,d\eta)=\rho(E,d\eta^{x,x+1})$ and both
$$c(E,\eta)=c(-E,\eta)\mbox{ and }c(x,E,\eta)=c(x,-E,\eta^{x,x+1})$$
The last identity depends of course crucially on the fact that $\pi$ is
not the identity and reverses left and right as preferred direction. At
the same time, as can be computed explicitly, the mean entropy production
is zero. The same remains true for $\pi$ a particle-hole transformation,
$(\pi\eta)_{x}=1-\eta_{x}$, leaving the field $E$ unchanged.
Then, $\rho\neq\rho\pi$ for $u\neq 1/2$
but still generalized detailed balance holds. Finally if, instead, we
were to take $\pi=$ identity as time-reversal, then we break the detailed
balance condition and we obtain a strictly positive mean entropy
production.
Example B: We take the simplest example of a spinflip
dynamics for which the one-dimensional Ising model is stationary but not
reversible (for $\pi=$id). Exactly the same can be done in two
dimensions, see [2]. Spinflips are transformations
$U_{x}:\sigma\rightarrow U_{x}(\sigma)=\sigma^{x},x\in\mathord{\!{
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$x$.
Consider the one-dimensional spinflip dynamics with the following
asymmetric rates:
$$c(x,\sigma)=\exp(-2\beta\sigma_{x}\sigma_{x+1})$$
(1.7)
The invariant measure
$\rho$ is the one-dimensional Ising model at inverse temperature $\beta$.
The process starting from $\rho$ is not time-reversal invariant and the
entropy production is equal to MEP$(L,\rho)=4\beta\tanh\beta$ (that is
with time-reversal $\pi=$identity). On the other hand, this time-reversed
process is easy to find; it is a spinflip process with generator
$$L^{\ast}f(\sigma)=\sum_{x}e^{-2\beta\sigma_{x}\sigma_{x-1}}[f(\sigma^{x})-f(%
\sigma)]$$
Let us now take for time-reversal $\pi$ the reflection:
$(\pi\sigma)_{x}=\sigma_{-x}$ which leaves $\rho$ invariant. Since
$$(\pi\sigma)^{x}=\pi(\sigma^{-x})$$
$L^{\ast}=\pi L\pi$ and we have in fact generalized detailed balance
(1.3):
$$\frac{c(x,\sigma)}{c(-x,(\pi\sigma)^{-x})}=\frac{d\rho\circ U_{x}}{d\rho}(%
\sigma)=e^{-2\beta\sigma_{x}(\sigma_{x-1}+\sigma_{x+1})}$$
The denominator in the left hand side is the rate in the original process
by which $\pi U_{x}\sigma=\pi(\sigma^{x})=(\pi\sigma)^{-x}$ is changed to
$\pi\sigma$.
As a result, MEP${}_{\pi}(L,\rho)=$MEP${}_{\pi}(L^{\ast},\rho)=0$.
Example C: Instead of driving the system in the bulk and
breaking detailed balance via some external fields that act on each
component of the system, we may also consider boundary driven processes.
For this we need to start with finite volume. The simplest interesting
case is that of a symmetric exclusion process on a lattice interval that
is driven by independent birth and death processes at its boundaries
corresponding to different chemical potentials. Take
$\Lambda_{n}=\{-n,-n+1,\ldots,n-1,n\}$ and $\eta\in\{0,1\}^{\Lambda_{n}}$ a
particle configuration evolving with generator
$$\displaystyle G_{n}f(\eta)$$
$$\displaystyle=$$
$$\displaystyle\sum_{x=-n}^{n-1}[f(\eta^{x,x+1})-f(\eta)]$$
(1.8)
$$\displaystyle+$$
$$\displaystyle\lambda[e^{h_{1}\eta_{-n}}(f(\eta^{-n})-f(\eta))+e^{h_{2}\eta_{n}%
}(f(\eta^{n})-f(\eta))]$$
The first term corresponds to symmetric hopping with exclusion; the two
last terms are giving birth and death to particles at the ends of the
interval with parameters $h_{1},h_{2}$. One can think here of particle
reservoirs, to the left of the
system with density $1/(1+e^{h_{1}})$ and
to the right with density $1/(1+e^{h_{2}})$. For $\lambda=0$ the system is
uncoupled from the reservoirs and it has all uniform product measures as
reversible measures with vanishing mean entropy production. For
$\lambda\neq 0,h_{1}\neq h_{2}$ this detailed balance
is lost and we have positive mean entropy production.
Yet, it remains of order unity, uniformly in the size $n$ meaning that the
mean entropy production density vanishes in the thermodynamic limit. This
is an instance of a more general fact for interacting particle systems
that will also be treated in the next section: you cannot by driving the
system at its boundaries break the time-reversal invariance in the
limiting infinite volume process, see Proposition 2 below.
We do not
know whether there exists a time-reversal $\pi$ for which (1.8)
would give rise to generalized detailed balance.
In this paper we show more generally how breaking of detailed balance is
strictly equivalent with non-zero mean entropy production. There is no
way to get a current and at the same time to have no dissipation (zero
mean entropy production).
In the next section we describe our class of models and we state our main
result. In section 3, we discuss this result and we give some more
background information concerning entropy production, reversibility and
time-reversal. Section 4 is devoted to the proofs.
2 Models and main result
2.1 Dynamics
This subsection describes the assumptions and introduces the necessary
notation.
The configuration space is $\Omega\equiv S^{\mathord{\!{
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\rule{0.3pt}{0.3pt}}\put(0.1375,0.275){\rule{0.3pt}{0.3pt}}\put(0.15,0.3){%
\rule{0.3pt}{0.3pt}}\put(0.1625,0.325){\rule{0.3pt}{0.3pt}}\put(0.175,0.35){%
\rule{0.3pt}{0.3pt}}\put(0.1875,0.375){\rule{0.3pt}{0.3pt}}\put(0.2,0.4){\rule%
{0.3pt}{0.3pt}}\put(0.2125,0.425){\rule{0.3pt}{0.3pt}}\put(0.225,0.45){\rule{0%
.3pt}{0.3pt}}\put(0.2375,0.475){\rule{0.3pt}{0.3pt}}\put(0.25,0.5){\rule{0.3pt%
}{0.3pt}}\put(0.2625,0.525){\rule{0.3pt}{0.3pt}}\put(0.275,0.55){\rule{0.3pt}{%
0.3pt}}\put(0.2875,0.575){\rule{0.3pt}{0.3pt}}\put(0.3,0.6){\rule{0.3pt}{0.3pt%
}}\put(0.3125,0.625){\rule{0.3pt}{0.3pt}}\put(0.325,0.65){\rule{0.3pt}{0.3pt}}%
\put(0.3375,0.675){\rule{0.3pt}{0.3pt}}\put(0.35,0.7){\rule{0.3pt}{0.3pt}}\put%
(0.3625,0.725){\rule{0.3pt}{0.3pt}}
\put(0.2,0.0){\rule{0.3pt}{0.3pt}}\put(0.2125,0.025){\rule{0.3pt}{0.3pt}}\put(%
0.225,0.05){\rule{0.3pt}{0.3pt}}\put(0.2375,0.075){\rule{0.3pt}{0.3pt}}\put(0.%
25,0.1){\rule{0.3pt}{0.3pt}}\put(0.2625,0.125){\rule{0.3pt}{0.3pt}}\put(0.275,%
0.15){\rule{0.3pt}{0.3pt}}\put(0.2875,0.175){\rule{0.3pt}{0.3pt}}\put(0.3,0.2)%
{\rule{0.3pt}{0.3pt}}\put(0.3125,0.225){\rule{0.3pt}{0.3pt}}\put(0.325,0.25){%
\rule{0.3pt}{0.3pt}}\put(0.3375,0.275){\rule{0.3pt}{0.3pt}}\put(0.35,0.3){%
\rule{0.3pt}{0.3pt}}\put(0.3625,0.325){\rule{0.3pt}{0.3pt}}\put(0.375,0.35){%
\rule{0.3pt}{0.3pt}}\put(0.3875,0.375){\rule{0.3pt}{0.3pt}}\put(0.4,0.4){\rule%
{0.3pt}{0.3pt}}\put(0.4125,0.425){\rule{0.3pt}{0.3pt}}\put(0.425,0.45){\rule{0%
.3pt}{0.3pt}}\put(0.4375,0.475){\rule{0.3pt}{0.3pt}}\put(0.45,0.5){\rule{0.3pt%
}{0.3pt}}\put(0.4625,0.525){\rule{0.3pt}{0.3pt}}\put(0.475,0.55){\rule{0.3pt}{%
0.3pt}}\put(0.4875,0.575){\rule{0.3pt}{0.3pt}}\put(0.5,0.6){\rule{0.3pt}{0.3pt%
}}\put(0.5125,0.625){\rule{0.3pt}{0.3pt}}\put(0.525,0.65){\rule{0.3pt}{0.3pt}}%
\put(0.5375,0.675){\rule{0.3pt}{0.3pt}}\put(0.55,0.7){\rule{0.3pt}{0.3pt}}\put%
(0.5625,0.725){\rule{0.3pt}{0.3pt}}
\put(0.0,0.75){\line(0,-1){0.15}}
\put(0.015,0.75){\line(0,-1){0.1}}
\put(0.03,0.75){\line(0,-1){0.075}}
\put(0.045,0.75){\line(0,-1){0.05}}
\put(0.05,0.75){\line(0,-1){0.025}}
\put(0.6,0.0){\line(0,1){0.15}}
\put(0.585,0.0){\line(0,1){0.1}}
\put(0.57,0.0){\line(0,1){0.075}}
\put(0.555,0.0){\line(0,1){0.05}}
\put(0.55,0.0){\line(0,1){0.025}}
\end{picture}}}^{d}}$ where $S$ is a finite
set and $\mathord{\!{
\begin{picture}(0.6,0.7)\put(0.0,0.0){\line(1,0){0.6}}
\put(0.0,0.75){\line(1,0){0.575}}
\put(0.0,0.0){\rule{0.3pt}{0.3pt}}\put(0.0125,0.025){\rule{0.3pt}{0.3pt}}\put(%
0.025,0.05){\rule{0.3pt}{0.3pt}}\put(0.0375,0.075){\rule{0.3pt}{0.3pt}}\put(0.%
05,0.1){\rule{0.3pt}{0.3pt}}\put(0.0625,0.125){\rule{0.3pt}{0.3pt}}\put(0.075,%
0.15){\rule{0.3pt}{0.3pt}}\put(0.0875,0.175){\rule{0.3pt}{0.3pt}}\put(0.1,0.2)%
{\rule{0.3pt}{0.3pt}}\put(0.1125,0.225){\rule{0.3pt}{0.3pt}}\put(0.125,0.25){%
\rule{0.3pt}{0.3pt}}\put(0.1375,0.275){\rule{0.3pt}{0.3pt}}\put(0.15,0.3){%
\rule{0.3pt}{0.3pt}}\put(0.1625,0.325){\rule{0.3pt}{0.3pt}}\put(0.175,0.35){%
\rule{0.3pt}{0.3pt}}\put(0.1875,0.375){\rule{0.3pt}{0.3pt}}\put(0.2,0.4){\rule%
{0.3pt}{0.3pt}}\put(0.2125,0.425){\rule{0.3pt}{0.3pt}}\put(0.225,0.45){\rule{0%
.3pt}{0.3pt}}\put(0.2375,0.475){\rule{0.3pt}{0.3pt}}\put(0.25,0.5){\rule{0.3pt%
}{0.3pt}}\put(0.2625,0.525){\rule{0.3pt}{0.3pt}}\put(0.275,0.55){\rule{0.3pt}{%
0.3pt}}\put(0.2875,0.575){\rule{0.3pt}{0.3pt}}\put(0.3,0.6){\rule{0.3pt}{0.3pt%
}}\put(0.3125,0.625){\rule{0.3pt}{0.3pt}}\put(0.325,0.65){\rule{0.3pt}{0.3pt}}%
\put(0.3375,0.675){\rule{0.3pt}{0.3pt}}\put(0.35,0.7){\rule{0.3pt}{0.3pt}}\put%
(0.3625,0.725){\rule{0.3pt}{0.3pt}}
\put(0.2,0.0){\rule{0.3pt}{0.3pt}}\put(0.2125,0.025){\rule{0.3pt}{0.3pt}}\put(%
0.225,0.05){\rule{0.3pt}{0.3pt}}\put(0.2375,0.075){\rule{0.3pt}{0.3pt}}\put(0.%
25,0.1){\rule{0.3pt}{0.3pt}}\put(0.2625,0.125){\rule{0.3pt}{0.3pt}}\put(0.275,%
0.15){\rule{0.3pt}{0.3pt}}\put(0.2875,0.175){\rule{0.3pt}{0.3pt}}\put(0.3,0.2)%
{\rule{0.3pt}{0.3pt}}\put(0.3125,0.225){\rule{0.3pt}{0.3pt}}\put(0.325,0.25){%
\rule{0.3pt}{0.3pt}}\put(0.3375,0.275){\rule{0.3pt}{0.3pt}}\put(0.35,0.3){%
\rule{0.3pt}{0.3pt}}\put(0.3625,0.325){\rule{0.3pt}{0.3pt}}\put(0.375,0.35){%
\rule{0.3pt}{0.3pt}}\put(0.3875,0.375){\rule{0.3pt}{0.3pt}}\put(0.4,0.4){\rule%
{0.3pt}{0.3pt}}\put(0.4125,0.425){\rule{0.3pt}{0.3pt}}\put(0.425,0.45){\rule{0%
.3pt}{0.3pt}}\put(0.4375,0.475){\rule{0.3pt}{0.3pt}}\put(0.45,0.5){\rule{0.3pt%
}{0.3pt}}\put(0.4625,0.525){\rule{0.3pt}{0.3pt}}\put(0.475,0.55){\rule{0.3pt}{%
0.3pt}}\put(0.4875,0.575){\rule{0.3pt}{0.3pt}}\put(0.5,0.6){\rule{0.3pt}{0.3pt%
}}\put(0.5125,0.625){\rule{0.3pt}{0.3pt}}\put(0.525,0.65){\rule{0.3pt}{0.3pt}}%
\put(0.5375,0.675){\rule{0.3pt}{0.3pt}}\put(0.55,0.7){\rule{0.3pt}{0.3pt}}\put%
(0.5625,0.725){\rule{0.3pt}{0.3pt}}
\put(0.0,0.75){\line(0,-1){0.15}}
\put(0.015,0.75){\line(0,-1){0.1}}
\put(0.03,0.75){\line(0,-1){0.075}}
\put(0.045,0.75){\line(0,-1){0.05}}
\put(0.05,0.75){\line(0,-1){0.025}}
\put(0.6,0.0){\line(0,1){0.15}}
\put(0.585,0.0){\line(0,1){0.1}}
\put(0.57,0.0){\line(0,1){0.075}}
\put(0.555,0.0){\line(0,1){0.05}}
\put(0.55,0.0){\line(0,1){0.025}}
\end{picture}}}^{d}$ is the regular $d-$dimensional lattice.
Let $\pi$ be an
involution on $\Omega$. A special but important case is when
$\pi=$identity. We assume here that $\pi$ commutes with lattice
translations
$\tau_{x},x\in\mathord{\!{
\begin{picture}(0.6,0.7)\put(0.0,0.0){\line(1,0){0.6}}
\put(0.0,0.75){\line(1,0){0.575}}
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0.025,0.05){\rule{0.3pt}{0.3pt}}\put(0.0375,0.075){\rule{0.3pt}{0.3pt}}\put(0.%
05,0.1){\rule{0.3pt}{0.3pt}}\put(0.0625,0.125){\rule{0.3pt}{0.3pt}}\put(0.075,%
0.15){\rule{0.3pt}{0.3pt}}\put(0.0875,0.175){\rule{0.3pt}{0.3pt}}\put(0.1,0.2)%
{\rule{0.3pt}{0.3pt}}\put(0.1125,0.225){\rule{0.3pt}{0.3pt}}\put(0.125,0.25){%
\rule{0.3pt}{0.3pt}}\put(0.1375,0.275){\rule{0.3pt}{0.3pt}}\put(0.15,0.3){%
\rule{0.3pt}{0.3pt}}\put(0.1625,0.325){\rule{0.3pt}{0.3pt}}\put(0.175,0.35){%
\rule{0.3pt}{0.3pt}}\put(0.1875,0.375){\rule{0.3pt}{0.3pt}}\put(0.2,0.4){\rule%
{0.3pt}{0.3pt}}\put(0.2125,0.425){\rule{0.3pt}{0.3pt}}\put(0.225,0.45){\rule{0%
.3pt}{0.3pt}}\put(0.2375,0.475){\rule{0.3pt}{0.3pt}}\put(0.25,0.5){\rule{0.3pt%
}{0.3pt}}\put(0.2625,0.525){\rule{0.3pt}{0.3pt}}\put(0.275,0.55){\rule{0.3pt}{%
0.3pt}}\put(0.2875,0.575){\rule{0.3pt}{0.3pt}}\put(0.3,0.6){\rule{0.3pt}{0.3pt%
}}\put(0.3125,0.625){\rule{0.3pt}{0.3pt}}\put(0.325,0.65){\rule{0.3pt}{0.3pt}}%
\put(0.3375,0.675){\rule{0.3pt}{0.3pt}}\put(0.35,0.7){\rule{0.3pt}{0.3pt}}\put%
(0.3625,0.725){\rule{0.3pt}{0.3pt}}
\put(0.2,0.0){\rule{0.3pt}{0.3pt}}\put(0.2125,0.025){\rule{0.3pt}{0.3pt}}\put(%
0.225,0.05){\rule{0.3pt}{0.3pt}}\put(0.2375,0.075){\rule{0.3pt}{0.3pt}}\put(0.%
25,0.1){\rule{0.3pt}{0.3pt}}\put(0.2625,0.125){\rule{0.3pt}{0.3pt}}\put(0.275,%
0.15){\rule{0.3pt}{0.3pt}}\put(0.2875,0.175){\rule{0.3pt}{0.3pt}}\put(0.3,0.2)%
{\rule{0.3pt}{0.3pt}}\put(0.3125,0.225){\rule{0.3pt}{0.3pt}}\put(0.325,0.25){%
\rule{0.3pt}{0.3pt}}\put(0.3375,0.275){\rule{0.3pt}{0.3pt}}\put(0.35,0.3){%
\rule{0.3pt}{0.3pt}}\put(0.3625,0.325){\rule{0.3pt}{0.3pt}}\put(0.375,0.35){%
\rule{0.3pt}{0.3pt}}\put(0.3875,0.375){\rule{0.3pt}{0.3pt}}\put(0.4,0.4){\rule%
{0.3pt}{0.3pt}}\put(0.4125,0.425){\rule{0.3pt}{0.3pt}}\put(0.425,0.45){\rule{0%
.3pt}{0.3pt}}\put(0.4375,0.475){\rule{0.3pt}{0.3pt}}\put(0.45,0.5){\rule{0.3pt%
}{0.3pt}}\put(0.4625,0.525){\rule{0.3pt}{0.3pt}}\put(0.475,0.55){\rule{0.3pt}{%
0.3pt}}\put(0.4875,0.575){\rule{0.3pt}{0.3pt}}\put(0.5,0.6){\rule{0.3pt}{0.3pt%
}}\put(0.5125,0.625){\rule{0.3pt}{0.3pt}}\put(0.525,0.65){\rule{0.3pt}{0.3pt}}%
\put(0.5375,0.675){\rule{0.3pt}{0.3pt}}\put(0.55,0.7){\rule{0.3pt}{0.3pt}}\put%
(0.5625,0.725){\rule{0.3pt}{0.3pt}}
\put(0.0,0.75){\line(0,-1){0.15}}
\put(0.015,0.75){\line(0,-1){0.1}}
\put(0.03,0.75){\line(0,-1){0.075}}
\put(0.045,0.75){\line(0,-1){0.05}}
\put(0.05,0.75){\line(0,-1){0.025}}
\put(0.6,0.0){\line(0,1){0.15}}
\put(0.585,0.0){\line(0,1){0.1}}
\put(0.57,0.0){\line(0,1){0.075}}
\put(0.555,0.0){\line(0,1){0.05}}
\put(0.55,0.0){\line(0,1){0.025}}
\end{picture}}}^{d}$.
Let $V_{0}\subset\mathord{\!{
\begin{picture}(0.6,0.7)\put(0.0,0.0){\line(1,0){0.6}}
\put(0.0,0.75){\line(1,0){0.575}}
\put(0.0,0.0){\rule{0.3pt}{0.3pt}}\put(0.0125,0.025){\rule{0.3pt}{0.3pt}}\put(%
0.025,0.05){\rule{0.3pt}{0.3pt}}\put(0.0375,0.075){\rule{0.3pt}{0.3pt}}\put(0.%
05,0.1){\rule{0.3pt}{0.3pt}}\put(0.0625,0.125){\rule{0.3pt}{0.3pt}}\put(0.075,%
0.15){\rule{0.3pt}{0.3pt}}\put(0.0875,0.175){\rule{0.3pt}{0.3pt}}\put(0.1,0.2)%
{\rule{0.3pt}{0.3pt}}\put(0.1125,0.225){\rule{0.3pt}{0.3pt}}\put(0.125,0.25){%
\rule{0.3pt}{0.3pt}}\put(0.1375,0.275){\rule{0.3pt}{0.3pt}}\put(0.15,0.3){%
\rule{0.3pt}{0.3pt}}\put(0.1625,0.325){\rule{0.3pt}{0.3pt}}\put(0.175,0.35){%
\rule{0.3pt}{0.3pt}}\put(0.1875,0.375){\rule{0.3pt}{0.3pt}}\put(0.2,0.4){\rule%
{0.3pt}{0.3pt}}\put(0.2125,0.425){\rule{0.3pt}{0.3pt}}\put(0.225,0.45){\rule{0%
.3pt}{0.3pt}}\put(0.2375,0.475){\rule{0.3pt}{0.3pt}}\put(0.25,0.5){\rule{0.3pt%
}{0.3pt}}\put(0.2625,0.525){\rule{0.3pt}{0.3pt}}\put(0.275,0.55){\rule{0.3pt}{%
0.3pt}}\put(0.2875,0.575){\rule{0.3pt}{0.3pt}}\put(0.3,0.6){\rule{0.3pt}{0.3pt%
}}\put(0.3125,0.625){\rule{0.3pt}{0.3pt}}\put(0.325,0.65){\rule{0.3pt}{0.3pt}}%
\put(0.3375,0.675){\rule{0.3pt}{0.3pt}}\put(0.35,0.7){\rule{0.3pt}{0.3pt}}\put%
(0.3625,0.725){\rule{0.3pt}{0.3pt}}
\put(0.2,0.0){\rule{0.3pt}{0.3pt}}\put(0.2125,0.025){\rule{0.3pt}{0.3pt}}\put(%
0.225,0.05){\rule{0.3pt}{0.3pt}}\put(0.2375,0.075){\rule{0.3pt}{0.3pt}}\put(0.%
25,0.1){\rule{0.3pt}{0.3pt}}\put(0.2625,0.125){\rule{0.3pt}{0.3pt}}\put(0.275,%
0.15){\rule{0.3pt}{0.3pt}}\put(0.2875,0.175){\rule{0.3pt}{0.3pt}}\put(0.3,0.2)%
{\rule{0.3pt}{0.3pt}}\put(0.3125,0.225){\rule{0.3pt}{0.3pt}}\put(0.325,0.25){%
\rule{0.3pt}{0.3pt}}\put(0.3375,0.275){\rule{0.3pt}{0.3pt}}\put(0.35,0.3){%
\rule{0.3pt}{0.3pt}}\put(0.3625,0.325){\rule{0.3pt}{0.3pt}}\put(0.375,0.35){%
\rule{0.3pt}{0.3pt}}\put(0.3875,0.375){\rule{0.3pt}{0.3pt}}\put(0.4,0.4){\rule%
{0.3pt}{0.3pt}}\put(0.4125,0.425){\rule{0.3pt}{0.3pt}}\put(0.425,0.45){\rule{0%
.3pt}{0.3pt}}\put(0.4375,0.475){\rule{0.3pt}{0.3pt}}\put(0.45,0.5){\rule{0.3pt%
}{0.3pt}}\put(0.4625,0.525){\rule{0.3pt}{0.3pt}}\put(0.475,0.55){\rule{0.3pt}{%
0.3pt}}\put(0.4875,0.575){\rule{0.3pt}{0.3pt}}\put(0.5,0.6){\rule{0.3pt}{0.3pt%
}}\put(0.5125,0.625){\rule{0.3pt}{0.3pt}}\put(0.525,0.65){\rule{0.3pt}{0.3pt}}%
\put(0.5375,0.675){\rule{0.3pt}{0.3pt}}\put(0.55,0.7){\rule{0.3pt}{0.3pt}}\put%
(0.5625,0.725){\rule{0.3pt}{0.3pt}}
\put(0.0,0.75){\line(0,-1){0.15}}
\put(0.015,0.75){\line(0,-1){0.1}}
\put(0.03,0.75){\line(0,-1){0.075}}
\put(0.045,0.75){\line(0,-1){0.05}}
\put(0.05,0.75){\line(0,-1){0.025}}
\put(0.6,0.0){\line(0,1){0.15}}
\put(0.585,0.0){\line(0,1){0.1}}
\put(0.57,0.0){\line(0,1){0.075}}
\put(0.555,0.0){\line(0,1){0.05}}
\put(0.55,0.0){\line(0,1){0.025}}
\end{picture}}}^{d}$ be a finite cube
containing the origin and write $\mathcal{P}_{0}$ for any specific
non-empty set of transformations $U_{0}$ on $\Omega$ satisfying, for every
$U_{0}\in\mathcal{P}_{0}$, and for every $\sigma\in\Omega$:
i.
$\left(U_{0}\sigma\right)(y)=\sigma(y)$,
for $y\in V^{c}_{0}$,
ii.
$U_{0}^{-1}\in\mathcal{P}_{0}$,
iii.
$\pi\mathcal{P}_{0}\pi=\mathcal{P}_{0}$,
iv.
If $U_{0}\not=U_{0}^{\prime}$ and $U_{0}\sigma\not=\sigma,U_{0}^{\prime}\sigma\not=\sigma$
then
$U_{0}\sigma\not=U^{\prime}_{0}\sigma$ (for convenience only.)
We consider the translations $V_{x}\equiv\{y+x:y\in V_{0}\}$ and $U_{x}\equiv\tau_{x}U_{0}\tau_{-x}$ to generate a dynamics via local translation
invariant rates $c(U_{x},\sigma)$ for the transition
$\sigma\to\,U_{x}\sigma$. We assume:
v.
Positivity: $c(U_{0},\sigma)=0$ when $U_{0}\sigma=\sigma$ and if not,
$c(U_{0},\sigma)>0$,
vi.
Finite range: there is a finite $\bar{\Lambda}\subset\mathord{\!{
\begin{picture}(0.6,0.7)\put(0.0,0.0){\line(1,0){0.6}}
\put(0.0,0.75){\line(1,0){0.575}}
\put(0.0,0.0){\rule{0.3pt}{0.3pt}}\put(0.0125,0.025){\rule{0.3pt}{0.3pt}}\put(%
0.025,0.05){\rule{0.3pt}{0.3pt}}\put(0.0375,0.075){\rule{0.3pt}{0.3pt}}\put(0.%
05,0.1){\rule{0.3pt}{0.3pt}}\put(0.0625,0.125){\rule{0.3pt}{0.3pt}}\put(0.075,%
0.15){\rule{0.3pt}{0.3pt}}\put(0.0875,0.175){\rule{0.3pt}{0.3pt}}\put(0.1,0.2)%
{\rule{0.3pt}{0.3pt}}\put(0.1125,0.225){\rule{0.3pt}{0.3pt}}\put(0.125,0.25){%
\rule{0.3pt}{0.3pt}}\put(0.1375,0.275){\rule{0.3pt}{0.3pt}}\put(0.15,0.3){%
\rule{0.3pt}{0.3pt}}\put(0.1625,0.325){\rule{0.3pt}{0.3pt}}\put(0.175,0.35){%
\rule{0.3pt}{0.3pt}}\put(0.1875,0.375){\rule{0.3pt}{0.3pt}}\put(0.2,0.4){\rule%
{0.3pt}{0.3pt}}\put(0.2125,0.425){\rule{0.3pt}{0.3pt}}\put(0.225,0.45){\rule{0%
.3pt}{0.3pt}}\put(0.2375,0.475){\rule{0.3pt}{0.3pt}}\put(0.25,0.5){\rule{0.3pt%
}{0.3pt}}\put(0.2625,0.525){\rule{0.3pt}{0.3pt}}\put(0.275,0.55){\rule{0.3pt}{%
0.3pt}}\put(0.2875,0.575){\rule{0.3pt}{0.3pt}}\put(0.3,0.6){\rule{0.3pt}{0.3pt%
}}\put(0.3125,0.625){\rule{0.3pt}{0.3pt}}\put(0.325,0.65){\rule{0.3pt}{0.3pt}}%
\put(0.3375,0.675){\rule{0.3pt}{0.3pt}}\put(0.35,0.7){\rule{0.3pt}{0.3pt}}\put%
(0.3625,0.725){\rule{0.3pt}{0.3pt}}
\put(0.2,0.0){\rule{0.3pt}{0.3pt}}\put(0.2125,0.025){\rule{0.3pt}{0.3pt}}\put(%
0.225,0.05){\rule{0.3pt}{0.3pt}}\put(0.2375,0.075){\rule{0.3pt}{0.3pt}}\put(0.%
25,0.1){\rule{0.3pt}{0.3pt}}\put(0.2625,0.125){\rule{0.3pt}{0.3pt}}\put(0.275,%
0.15){\rule{0.3pt}{0.3pt}}\put(0.2875,0.175){\rule{0.3pt}{0.3pt}}\put(0.3,0.2)%
{\rule{0.3pt}{0.3pt}}\put(0.3125,0.225){\rule{0.3pt}{0.3pt}}\put(0.325,0.25){%
\rule{0.3pt}{0.3pt}}\put(0.3375,0.275){\rule{0.3pt}{0.3pt}}\put(0.35,0.3){%
\rule{0.3pt}{0.3pt}}\put(0.3625,0.325){\rule{0.3pt}{0.3pt}}\put(0.375,0.35){%
\rule{0.3pt}{0.3pt}}\put(0.3875,0.375){\rule{0.3pt}{0.3pt}}\put(0.4,0.4){\rule%
{0.3pt}{0.3pt}}\put(0.4125,0.425){\rule{0.3pt}{0.3pt}}\put(0.425,0.45){\rule{0%
.3pt}{0.3pt}}\put(0.4375,0.475){\rule{0.3pt}{0.3pt}}\put(0.45,0.5){\rule{0.3pt%
}{0.3pt}}\put(0.4625,0.525){\rule{0.3pt}{0.3pt}}\put(0.475,0.55){\rule{0.3pt}{%
0.3pt}}\put(0.4875,0.575){\rule{0.3pt}{0.3pt}}\put(0.5,0.6){\rule{0.3pt}{0.3pt%
}}\put(0.5125,0.625){\rule{0.3pt}{0.3pt}}\put(0.525,0.65){\rule{0.3pt}{0.3pt}}%
\put(0.5375,0.675){\rule{0.3pt}{0.3pt}}\put(0.55,0.7){\rule{0.3pt}{0.3pt}}\put%
(0.5625,0.725){\rule{0.3pt}{0.3pt}}
\put(0.0,0.75){\line(0,-1){0.15}}
\put(0.015,0.75){\line(0,-1){0.1}}
\put(0.03,0.75){\line(0,-1){0.075}}
\put(0.045,0.75){\line(0,-1){0.05}}
\put(0.05,0.75){\line(0,-1){0.025}}
\put(0.6,0.0){\line(0,1){0.15}}
\put(0.585,0.0){\line(0,1){0.1}}
\put(0.57,0.0){\line(0,1){0.075}}
\put(0.555,0.0){\line(0,1){0.05}}
\put(0.55,0.0){\line(0,1){0.025}}
\end{picture}}}^{d}$
such that for all $\sigma,\eta\in\Omega$, and $U_{0}\in\mathcal{P}_{0}$:
$c(U_{0},\sigma)=c(U_{0},\sigma_{\bar{\Lambda}}\eta_{\bar{\Lambda}^{c}})$,
vii.
Translation invariance: for all $x\in\mathord{\!{
\begin{picture}(0.6,0.7)\put(0.0,0.0){\line(1,0){0.6}}
\put(0.0,0.75){\line(1,0){0.575}}
\put(0.0,0.0){\rule{0.3pt}{0.3pt}}\put(0.0125,0.025){\rule{0.3pt}{0.3pt}}\put(%
0.025,0.05){\rule{0.3pt}{0.3pt}}\put(0.0375,0.075){\rule{0.3pt}{0.3pt}}\put(0.%
05,0.1){\rule{0.3pt}{0.3pt}}\put(0.0625,0.125){\rule{0.3pt}{0.3pt}}\put(0.075,%
0.15){\rule{0.3pt}{0.3pt}}\put(0.0875,0.175){\rule{0.3pt}{0.3pt}}\put(0.1,0.2)%
{\rule{0.3pt}{0.3pt}}\put(0.1125,0.225){\rule{0.3pt}{0.3pt}}\put(0.125,0.25){%
\rule{0.3pt}{0.3pt}}\put(0.1375,0.275){\rule{0.3pt}{0.3pt}}\put(0.15,0.3){%
\rule{0.3pt}{0.3pt}}\put(0.1625,0.325){\rule{0.3pt}{0.3pt}}\put(0.175,0.35){%
\rule{0.3pt}{0.3pt}}\put(0.1875,0.375){\rule{0.3pt}{0.3pt}}\put(0.2,0.4){\rule%
{0.3pt}{0.3pt}}\put(0.2125,0.425){\rule{0.3pt}{0.3pt}}\put(0.225,0.45){\rule{0%
.3pt}{0.3pt}}\put(0.2375,0.475){\rule{0.3pt}{0.3pt}}\put(0.25,0.5){\rule{0.3pt%
}{0.3pt}}\put(0.2625,0.525){\rule{0.3pt}{0.3pt}}\put(0.275,0.55){\rule{0.3pt}{%
0.3pt}}\put(0.2875,0.575){\rule{0.3pt}{0.3pt}}\put(0.3,0.6){\rule{0.3pt}{0.3pt%
}}\put(0.3125,0.625){\rule{0.3pt}{0.3pt}}\put(0.325,0.65){\rule{0.3pt}{0.3pt}}%
\put(0.3375,0.675){\rule{0.3pt}{0.3pt}}\put(0.35,0.7){\rule{0.3pt}{0.3pt}}\put%
(0.3625,0.725){\rule{0.3pt}{0.3pt}}
\put(0.2,0.0){\rule{0.3pt}{0.3pt}}\put(0.2125,0.025){\rule{0.3pt}{0.3pt}}\put(%
0.225,0.05){\rule{0.3pt}{0.3pt}}\put(0.2375,0.075){\rule{0.3pt}{0.3pt}}\put(0.%
25,0.1){\rule{0.3pt}{0.3pt}}\put(0.2625,0.125){\rule{0.3pt}{0.3pt}}\put(0.275,%
0.15){\rule{0.3pt}{0.3pt}}\put(0.2875,0.175){\rule{0.3pt}{0.3pt}}\put(0.3,0.2)%
{\rule{0.3pt}{0.3pt}}\put(0.3125,0.225){\rule{0.3pt}{0.3pt}}\put(0.325,0.25){%
\rule{0.3pt}{0.3pt}}\put(0.3375,0.275){\rule{0.3pt}{0.3pt}}\put(0.35,0.3){%
\rule{0.3pt}{0.3pt}}\put(0.3625,0.325){\rule{0.3pt}{0.3pt}}\put(0.375,0.35){%
\rule{0.3pt}{0.3pt}}\put(0.3875,0.375){\rule{0.3pt}{0.3pt}}\put(0.4,0.4){\rule%
{0.3pt}{0.3pt}}\put(0.4125,0.425){\rule{0.3pt}{0.3pt}}\put(0.425,0.45){\rule{0%
.3pt}{0.3pt}}\put(0.4375,0.475){\rule{0.3pt}{0.3pt}}\put(0.45,0.5){\rule{0.3pt%
}{0.3pt}}\put(0.4625,0.525){\rule{0.3pt}{0.3pt}}\put(0.475,0.55){\rule{0.3pt}{%
0.3pt}}\put(0.4875,0.575){\rule{0.3pt}{0.3pt}}\put(0.5,0.6){\rule{0.3pt}{0.3pt%
}}\put(0.5125,0.625){\rule{0.3pt}{0.3pt}}\put(0.525,0.65){\rule{0.3pt}{0.3pt}}%
\put(0.5375,0.675){\rule{0.3pt}{0.3pt}}\put(0.55,0.7){\rule{0.3pt}{0.3pt}}\put%
(0.5625,0.725){\rule{0.3pt}{0.3pt}}
\put(0.0,0.75){\line(0,-1){0.15}}
\put(0.015,0.75){\line(0,-1){0.1}}
\put(0.03,0.75){\line(0,-1){0.075}}
\put(0.045,0.75){\line(0,-1){0.05}}
\put(0.05,0.75){\line(0,-1){0.025}}
\put(0.6,0.0){\line(0,1){0.15}}
\put(0.585,0.0){\line(0,1){0.1}}
\put(0.57,0.0){\line(0,1){0.075}}
\put(0.555,0.0){\line(0,1){0.05}}
\put(0.55,0.0){\line(0,1){0.025}}
\end{picture}}}^{d}$, $U_{x}\in\mathcal{P}_{x}$,
$\sigma\in\Omega$: $c(U_{x},\sigma)=c(U_{0},\tau_{-x}\sigma)$
The generator $L$ corresponding to the given rates is now defined on
local functions $f$ as
$$Lf(\sigma)\equiv\sum_{x\in\mathord{\!{
\begin{picture}(0.6,0.7)\put(0.0,0.0){\line(1,0){0.6}}
\put(0.0,0.75){\line(1,0){0.575}}
\put(0.0,0.0){\rule{0.3pt}{0.3pt}}\put(0.0125,0.025){\rule{0.3pt}{0.3pt}}\put(%
0.025,0.05){\rule{0.3pt}{0.3pt}}\put(0.0375,0.075){\rule{0.3pt}{0.3pt}}\put(0.%
05,0.1){\rule{0.3pt}{0.3pt}}\put(0.0625,0.125){\rule{0.3pt}{0.3pt}}\put(0.075,%
0.15){\rule{0.3pt}{0.3pt}}\put(0.0875,0.175){\rule{0.3pt}{0.3pt}}\put(0.1,0.2)%
{\rule{0.3pt}{0.3pt}}\put(0.1125,0.225){\rule{0.3pt}{0.3pt}}\put(0.125,0.25){%
\rule{0.3pt}{0.3pt}}\put(0.1375,0.275){\rule{0.3pt}{0.3pt}}\put(0.15,0.3){%
\rule{0.3pt}{0.3pt}}\put(0.1625,0.325){\rule{0.3pt}{0.3pt}}\put(0.175,0.35){%
\rule{0.3pt}{0.3pt}}\put(0.1875,0.375){\rule{0.3pt}{0.3pt}}\put(0.2,0.4){\rule%
{0.3pt}{0.3pt}}\put(0.2125,0.425){\rule{0.3pt}{0.3pt}}\put(0.225,0.45){\rule{0%
.3pt}{0.3pt}}\put(0.2375,0.475){\rule{0.3pt}{0.3pt}}\put(0.25,0.5){\rule{0.3pt%
}{0.3pt}}\put(0.2625,0.525){\rule{0.3pt}{0.3pt}}\put(0.275,0.55){\rule{0.3pt}{%
0.3pt}}\put(0.2875,0.575){\rule{0.3pt}{0.3pt}}\put(0.3,0.6){\rule{0.3pt}{0.3pt%
}}\put(0.3125,0.625){\rule{0.3pt}{0.3pt}}\put(0.325,0.65){\rule{0.3pt}{0.3pt}}%
\put(0.3375,0.675){\rule{0.3pt}{0.3pt}}\put(0.35,0.7){\rule{0.3pt}{0.3pt}}\put%
(0.3625,0.725){\rule{0.3pt}{0.3pt}}
\put(0.2,0.0){\rule{0.3pt}{0.3pt}}\put(0.2125,0.025){\rule{0.3pt}{0.3pt}}\put(%
0.225,0.05){\rule{0.3pt}{0.3pt}}\put(0.2375,0.075){\rule{0.3pt}{0.3pt}}\put(0.%
25,0.1){\rule{0.3pt}{0.3pt}}\put(0.2625,0.125){\rule{0.3pt}{0.3pt}}\put(0.275,%
0.15){\rule{0.3pt}{0.3pt}}\put(0.2875,0.175){\rule{0.3pt}{0.3pt}}\put(0.3,0.2)%
{\rule{0.3pt}{0.3pt}}\put(0.3125,0.225){\rule{0.3pt}{0.3pt}}\put(0.325,0.25){%
\rule{0.3pt}{0.3pt}}\put(0.3375,0.275){\rule{0.3pt}{0.3pt}}\put(0.35,0.3){%
\rule{0.3pt}{0.3pt}}\put(0.3625,0.325){\rule{0.3pt}{0.3pt}}\put(0.375,0.35){%
\rule{0.3pt}{0.3pt}}\put(0.3875,0.375){\rule{0.3pt}{0.3pt}}\put(0.4,0.4){\rule%
{0.3pt}{0.3pt}}\put(0.4125,0.425){\rule{0.3pt}{0.3pt}}\put(0.425,0.45){\rule{0%
.3pt}{0.3pt}}\put(0.4375,0.475){\rule{0.3pt}{0.3pt}}\put(0.45,0.5){\rule{0.3pt%
}{0.3pt}}\put(0.4625,0.525){\rule{0.3pt}{0.3pt}}\put(0.475,0.55){\rule{0.3pt}{%
0.3pt}}\put(0.4875,0.575){\rule{0.3pt}{0.3pt}}\put(0.5,0.6){\rule{0.3pt}{0.3pt%
}}\put(0.5125,0.625){\rule{0.3pt}{0.3pt}}\put(0.525,0.65){\rule{0.3pt}{0.3pt}}%
\put(0.5375,0.675){\rule{0.3pt}{0.3pt}}\put(0.55,0.7){\rule{0.3pt}{0.3pt}}\put%
(0.5625,0.725){\rule{0.3pt}{0.3pt}}
\put(0.0,0.75){\line(0,-1){0.15}}
\put(0.015,0.75){\line(0,-1){0.1}}
\put(0.03,0.75){\line(0,-1){0.075}}
\put(0.045,0.75){\line(0,-1){0.05}}
\put(0.05,0.75){\line(0,-1){0.025}}
\put(0.6,0.0){\line(0,1){0.15}}
\put(0.585,0.0){\line(0,1){0.1}}
\put(0.57,0.0){\line(0,1){0.075}}
\put(0.555,0.0){\line(0,1){0.05}}
\put(0.55,0.0){\line(0,1){0.025}}
\end{picture}}}^{d}}\,\sum_{U_{x}\in\mathcal{P}_{x}}c(U_{x},\sigma)[f(U_{x}%
\sigma)-f(\sigma)]$$
(2.9)
That is, $\sigma$ is changed to $\eta$ at rate $c(U_{x},\sigma)$ if
$\eta=U_{x}\sigma$. We will always write $\rho$ for a translation invariant
stationary measure for this dynamics. It can be different from $\rho\pi$
but we assume that also $\rho\pi$ is stationary. Finally, $\rho$ and
$\rho\pi$ give positive weight to all cylinders and writing $\rho^{U}=U\rho$, we always assume that $d\rho^{U_{0}}/d\rho(\sigma)\geq c>0$, which,
even in the present rather general set-up, can be expected quite
generally.
For $V_{0}=\{0\}$ and $S=\{+1,-1\}$, the choice $U_{x}\sigma=\sigma^{x}$
corresponds to a spinflip process. Taking $V_{0}=\{0,e_{1},e_{2},\ldots,e_{d}\}$
with $e_{\alpha}$ the lattice unit vectors, we can make a spin exchange
process or hopping dynamics. We refer to [9] for further details
on constructing the infinite volume process.
2.2 Mean Entropy Production
Put $\Lambda_{n}=[-n,n]^{d}\cap\mathord{\!{
\begin{picture}(0.6,0.7)\put(0.0,0.0){\line(1,0){0.6}}
\put(0.0,0.75){\line(1,0){0.575}}
\put(0.0,0.0){\rule{0.3pt}{0.3pt}}\put(0.0125,0.025){\rule{0.3pt}{0.3pt}}\put(%
0.025,0.05){\rule{0.3pt}{0.3pt}}\put(0.0375,0.075){\rule{0.3pt}{0.3pt}}\put(0.%
05,0.1){\rule{0.3pt}{0.3pt}}\put(0.0625,0.125){\rule{0.3pt}{0.3pt}}\put(0.075,%
0.15){\rule{0.3pt}{0.3pt}}\put(0.0875,0.175){\rule{0.3pt}{0.3pt}}\put(0.1,0.2)%
{\rule{0.3pt}{0.3pt}}\put(0.1125,0.225){\rule{0.3pt}{0.3pt}}\put(0.125,0.25){%
\rule{0.3pt}{0.3pt}}\put(0.1375,0.275){\rule{0.3pt}{0.3pt}}\put(0.15,0.3){%
\rule{0.3pt}{0.3pt}}\put(0.1625,0.325){\rule{0.3pt}{0.3pt}}\put(0.175,0.35){%
\rule{0.3pt}{0.3pt}}\put(0.1875,0.375){\rule{0.3pt}{0.3pt}}\put(0.2,0.4){\rule%
{0.3pt}{0.3pt}}\put(0.2125,0.425){\rule{0.3pt}{0.3pt}}\put(0.225,0.45){\rule{0%
.3pt}{0.3pt}}\put(0.2375,0.475){\rule{0.3pt}{0.3pt}}\put(0.25,0.5){\rule{0.3pt%
}{0.3pt}}\put(0.2625,0.525){\rule{0.3pt}{0.3pt}}\put(0.275,0.55){\rule{0.3pt}{%
0.3pt}}\put(0.2875,0.575){\rule{0.3pt}{0.3pt}}\put(0.3,0.6){\rule{0.3pt}{0.3pt%
}}\put(0.3125,0.625){\rule{0.3pt}{0.3pt}}\put(0.325,0.65){\rule{0.3pt}{0.3pt}}%
\put(0.3375,0.675){\rule{0.3pt}{0.3pt}}\put(0.35,0.7){\rule{0.3pt}{0.3pt}}\put%
(0.3625,0.725){\rule{0.3pt}{0.3pt}}
\put(0.2,0.0){\rule{0.3pt}{0.3pt}}\put(0.2125,0.025){\rule{0.3pt}{0.3pt}}\put(%
0.225,0.05){\rule{0.3pt}{0.3pt}}\put(0.2375,0.075){\rule{0.3pt}{0.3pt}}\put(0.%
25,0.1){\rule{0.3pt}{0.3pt}}\put(0.2625,0.125){\rule{0.3pt}{0.3pt}}\put(0.275,%
0.15){\rule{0.3pt}{0.3pt}}\put(0.2875,0.175){\rule{0.3pt}{0.3pt}}\put(0.3,0.2)%
{\rule{0.3pt}{0.3pt}}\put(0.3125,0.225){\rule{0.3pt}{0.3pt}}\put(0.325,0.25){%
\rule{0.3pt}{0.3pt}}\put(0.3375,0.275){\rule{0.3pt}{0.3pt}}\put(0.35,0.3){%
\rule{0.3pt}{0.3pt}}\put(0.3625,0.325){\rule{0.3pt}{0.3pt}}\put(0.375,0.35){%
\rule{0.3pt}{0.3pt}}\put(0.3875,0.375){\rule{0.3pt}{0.3pt}}\put(0.4,0.4){\rule%
{0.3pt}{0.3pt}}\put(0.4125,0.425){\rule{0.3pt}{0.3pt}}\put(0.425,0.45){\rule{0%
.3pt}{0.3pt}}\put(0.4375,0.475){\rule{0.3pt}{0.3pt}}\put(0.45,0.5){\rule{0.3pt%
}{0.3pt}}\put(0.4625,0.525){\rule{0.3pt}{0.3pt}}\put(0.475,0.55){\rule{0.3pt}{%
0.3pt}}\put(0.4875,0.575){\rule{0.3pt}{0.3pt}}\put(0.5,0.6){\rule{0.3pt}{0.3pt%
}}\put(0.5125,0.625){\rule{0.3pt}{0.3pt}}\put(0.525,0.65){\rule{0.3pt}{0.3pt}}%
\put(0.5375,0.675){\rule{0.3pt}{0.3pt}}\put(0.55,0.7){\rule{0.3pt}{0.3pt}}\put%
(0.5625,0.725){\rule{0.3pt}{0.3pt}}
\put(0.0,0.75){\line(0,-1){0.15}}
\put(0.015,0.75){\line(0,-1){0.1}}
\put(0.03,0.75){\line(0,-1){0.075}}
\put(0.045,0.75){\line(0,-1){0.05}}
\put(0.05,0.75){\line(0,-1){0.025}}
\put(0.6,0.0){\line(0,1){0.15}}
\put(0.585,0.0){\line(0,1){0.1}}
\put(0.57,0.0){\line(0,1){0.075}}
\put(0.555,0.0){\line(0,1){0.05}}
\put(0.55,0.0){\line(0,1){0.025}}
\end{picture}}}^{d}$ for large $n$ and define
$\Lambda^{\sharp}_{n}$ as the maximal subset of $\Lambda_{n}$, such that for all
$x\in\Lambda^{\sharp}_{n}$ and $U_{x}\in\mathcal{P}_{x}$, $c(U_{x},\sigma)$ depends
only on coordinates inside $\Lambda_{n}$, and $V_{x}\subset\Lambda$.
Consider now
the Markov chain on $S^{\Lambda_{n}}$ with generator
$$L_{n}f(\sigma)\equiv\sum_{x\in\Lambda^{\sharp}_{n}}\sum_{U_{x}\in\mathcal{P}_{%
x}}c(U_{x},\sigma)[f(U_{x}\sigma)-f(\sigma)]$$
(2.10)
and started from a probability measure $\rho_{-T}$ on $S^{\Lambda_{n}}$ at
time $-T$. The measure at time $T$ is denoted by $\rho_{T}$. Via a Girsanov
formula this dynamics gives rise to a Hamiltonian (or action functional)
on space-time trajectories $\omega$ (as in [4, 5]),
with corresponding relative energy
with respect to time-reversal given by the entropy production
(1.4) and here equal to
$$R_{\pi}(L_{n},\rho_{-T},T)(\omega)=\ln\rho_{-T}(\omega(-T))-\ln\rho_{T}(\omega%
(T))+\Delta S_{e}(\omega)$$
(2.11)
with
$$\displaystyle\Delta S_{e}(\omega)$$
$$\displaystyle=$$
$$\displaystyle\sum_{x\in\Lambda^{\sharp}_{n}}\sum_{U_{x}\in\mathcal{P}_{x}}\int%
_{-T}^{T}\log\frac{c(U_{x},\omega(s^{-}))}{c(\pi U^{-1}_{x}\pi,\pi U_{x}\omega%
(s^{-}))}dN^{U_{x}}_{s}(\omega)$$
(2.12)
$$\displaystyle+$$
$$\displaystyle\int^{T}_{-T}[c(U_{x},\pi\omega(s))-c(U_{x},\omega(s))]ds$$
where $N^{U_{x}}_{t}(\omega)\equiv\sum_{-T\leq s\leq t}I\left(\omega(s)=U_{x}(\omega(%
s^{-}))\not=\omega(s^{-})\right)$ is the number of times the
transformation $U_{x}$ appeared in the realization $\omega$ up to time
$t\in[-T,T]$. The expression (2.12) must be interpreted as the
variable entropy produced in the reservoirs (environment) when the
microscopic system configuration moves from $\omega(-T)$ to $\omega(T)$:
To get the total variable entropy production (2.11)
one should add to (2.12) the corresponding change in the system’s
entropy, that are the first two
terms in (2.11).
However, when taking steady state averages, this part vanishes (the
entropy of the stationary system does not change on average). We can
therefore define the mean entropy production for the interacting particle
system as
$$\textrm{MEP}_{\pi}(L,\rho)\equiv\lim_{n}\lim_{T\uparrow\infty}\frac{1}{2|%
\Lambda_{n}|T}{\mbox{\rm$\mbox{I}\!\mbox{E}$}}^{n,T}_{\rho}(\Delta S_{e})$$
(2.13)
${\mbox{\rm$\mbox{I}\!\mbox{E}$}}^{n,T}_{\rho}$ denotes the expectation with respect to the path space
measure, in the stationary distribution $\rho$, restricted to trajectories
within $S^{\Lambda_{n}}$. In other words, the mean entropy production is
the expectation of the time-reversal breaking part in the space-time
action functional governing the dynamics. We refer to [5] for a
mathematical discussion on the existence of the limit (2.13) and for a
proof of its non-negativity. We refer to [3, 6, 4] and Section 3
for further background.
2.3 Results
The main question is to see whether for a dynamics where the time-reversal
symmetry is explictly broken (in the sense that there is no detailed
balance), there still can be zero mean entropy production
(dissipationless steady state). Our main result says that this is
impossible.
Main Theorem: Under the conditions above,
$\mbox{MEP}_{\pi}(L,\rho)=\mbox{MEP}_{\pi}(L,\rho\pi)=0$ implies that the
dynamics satisfies (generalized) detailed balance in the sense that for
all $U_{0}$
$$c(\pi U^{-1}_{0}\pi,\pi U_{0}\sigma)\frac{d\rho^{U_{0}}}{d\rho}(\sigma)=c(U_{0%
},\sigma)\quad\rho-a.s.$$
(2.14)
Note that (2.14) is really the analogue of (1.3).
Observe also here that (2.14) implies that the densities
$d\rho^{U_{0}}/d\rho$ are invariant under replacing $\rho$ by $\rho\pi$.
This follows from rewriting (2.14) from right to left with $\sigma\rightarrow\pi U_{0}\sigma$ and $U_{0}\rightarrow\pi U^{-1}_{0}\pi$:
$$\displaystyle c(\pi U^{-1}_{0}\pi,\pi U_{0}\sigma)$$
$$\displaystyle=$$
$$\displaystyle c(\pi\pi U_{0}\pi\pi,\pi\pi U^{-1}_{0}\pi\pi U_{0}\sigma)\frac{d%
\rho^{\pi U^{-1}_{0}\pi}}{d\rho}(\pi U_{0}\sigma)$$
$$\displaystyle=$$
$$\displaystyle c(U_{0},\sigma)\frac{d\rho\pi}{d\rho\pi^{U_{0}}}(\sigma)$$
and comparing it with the original (2.14).
We call $\mathcal{P}_{0}$ complete if every local transformation $h:\Omega\to\Omega$
can be written as
a composition of $U_{x}$: i.e.,
if
$h=U_{x_{1}}\ldots U_{x_{n}}$ for some $x_{1},\ldots,x_{n}\in\mathord{\!{
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\end{picture}}}^{d}$.
Corollary 1: If $\mbox{MEP}_{\pi}(L,\rho)=\mbox{MEP}_{\pi}(L,\rho\pi)=0$ and if $\mathcal{P}_{0}$ is complete and $\pi$
is continuous, then
$\rho$ is a reversible Gibbs measure for the dynamics defined above.
In [5] the converse to these results was already shown: Suppose
that the rates satisfy
$$c(U_{x},\sigma)=c(\pi U^{-1}_{x}\pi,\pi U_{x}\sigma)\exp(-H(U_{x}\sigma)+H(%
\sigma)).$$
(2.15)
This is again the analogue of (1.3). The energy difference in
(2.15) should be interpreted in terms of an absolutely convergent
sum of potentials:
$$H(\sigma_{\Lambda}\eta_{\Lambda^{c}})-H(\xi_{\Lambda}\eta_{\Lambda^{c}})=\sum_%
{A\cap\Lambda\not=\emptyset}\left(V(A,\sigma_{\Lambda}\eta_{\Lambda^{c}})-V(A,%
\xi_{\Lambda}\eta_{\Lambda^{c}})\right),$$
(2.16)
where $(V(A,\cdot):S^{A}\rightarrow(-\infty,+\infty),A$ finite subsets
of $\mathord{\!{
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}{0.3pt}}\put(0.4625,0.525){\rule{0.3pt}{0.3pt}}\put(0.475,0.55){\rule{0.3pt}{%
0.3pt}}\put(0.4875,0.575){\rule{0.3pt}{0.3pt}}\put(0.5,0.6){\rule{0.3pt}{0.3pt%
}}\put(0.5125,0.625){\rule{0.3pt}{0.3pt}}\put(0.525,0.65){\rule{0.3pt}{0.3pt}}%
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(0.5625,0.725){\rule{0.3pt}{0.3pt}}
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\put(0.045,0.75){\line(0,-1){0.05}}
\put(0.05,0.75){\line(0,-1){0.025}}
\put(0.6,0.0){\line(0,1){0.15}}
\put(0.585,0.0){\line(0,1){0.1}}
\put(0.57,0.0){\line(0,1){0.075}}
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\end{picture}}}^{d})$, is a translation invariant (uniformly) absolutely summable
potential:
$$\sum_{A\ni 0}\max_{\sigma\in S^{A}}|V(A,\sigma)|<+\infty$$
(2.17)
Then,
$$\rm{MEP}_{\pi}(L,\rho)=\rm{MEP}_{\pi}(L,\rho\pi)=0$$
When we combine the above we obtain a final
Corollary 2: Under the conditions of Corollary 1,
if there is one translation invariant stationary measure $\rho$ for
which $\rho=\rho\pi$ and MEP${}_{\pi}(L,\rho)=0$, then also
MEP${}_{\pi}(L,\nu)=0$ for all translation invariant stationary
measures $\nu$ and they are all Gibbsian for the same potential.
A caveat
in the above main result is to understand better the relation between
MEP${}_{\pi}(L,\rho)$ and MEP${}_{\pi}(L,\rho\pi)$.
To this we can only add that
MEP${}_{\pi}(\pi L\pi,\rho)=$ MEP${}_{\pi}(L,\rho\pi)$,
as can be verified from a
direct computation starting with (4.24).
The simplest illustration of all this was already obtained in [7]
for a spinflip process. Example B, (1.7), deals with a spinflip
process but there the time-reversal $\pi$ does not commute with
translations. As will be seen from the proof, that is indeed not
essential as long as the dynamics and the stationary measure are
translation invariant. Of course, one should then be extra careful with
condition iii. but also this can be modified accordingly.
It will also be clear that more general lattice
structures and configuration spaces can be employed (e.g. already in Example A).
Finally, for completeness we come back to the situation of Example C in
Section 1.1. For this we must leave the translation invariant infinite
volume context and ask whether boundary driven interacting particle
systems can give rise to non-vanishing mean entropy production density in
the thermodynamic limit. The question can be formalized as follows. We
consider a process on $S^{\Lambda_{n}}$ with generator $G_{n}$ generalizing
(1.8)
$$G_{n}f(\sigma)\equiv L_{n}f(\sigma)+\sum_{\begin{array}[]{c}A\subset\Lambda_{n%
}\setminus\Lambda^{\sharp}_{n}\\
\mbox{\tt diam}A\leq r\end{array}}\sum_{\eta\in S^{A}}k^{(n)}_{A}(\sigma,\eta)%
[f(\sigma^{A,\eta})-f(\sigma)]$$
where $\sigma^{A,\eta}\equiv\sigma_{A^{c}}\eta_{A}$ equals $\sigma$ outside
the set $A$ which has a diameter (maximal lattice
distance within) less than a given constant $r$.
Here the generator $L_{n}$ is given by (2.10) but with rates verifying
condition (2.15) for a finite range potential, and rates
$k^{(n)}_{A}(\sigma,\eta)$ as in (1.8) inducing configurational
changes at the boundary of $\Lambda_{n}$. We further assume that the
$k^{(n)}_{A}(\sigma,\eta)$ are uniformly bounded from below and from above.
In other words, we have a bulk dynamics generated by $L_{n}$ with rates
satisfying (generalized) detailed balance, and at the boundary the
configuration can change quite arbitrarily (but in a local and bounded
way). We suppose that $\rho_{n}$ is the unique stationary measure of this
dynamics and for simplicity we only treat the case $\pi=id$. We are
interested in the mean entropy production MEP$(G_{n},\rho_{n})$ defined in
(1.5) (with $\pi=id$).
Proposition 2: There is a constant $K$ so that
MEP${}_{\pi}(G_{n},\rho_{n})\leq Kn^{d-1}$
The proofs of the above results are postponed to Section 4.
3 Discussion
We briefly discuss some concepts that are important for our result.
3.1 Time-reversal
By this we usually mean a transformation on phase space $\Omega$ which,
for a many-particle system, is defined particle-wise or, for spatially
extended systems, is sufficiently local. Physically speaking, its precise
nature follows from kinematical considerations on the dynamical
variables. In classical mechanics, it reverses the momenta of all the
particles but in the presence of say an electromagnetic potential,
considered part of the system,
one can add an extra transformation
reversing also the magnetic field and thus making the Lorentz force
time-reversal invariant.
In our case, we have a configuration space $\Omega=S^{\mathord{\!{
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with $\mathord{\!{
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0.3pt}}\put(0.4875,0.575){\rule{0.3pt}{0.3pt}}\put(0.5,0.6){\rule{0.3pt}{0.3pt%
}}\put(0.5125,0.625){\rule{0.3pt}{0.3pt}}\put(0.525,0.65){\rule{0.3pt}{0.3pt}}%
\put(0.5375,0.675){\rule{0.3pt}{0.3pt}}\put(0.55,0.7){\rule{0.3pt}{0.3pt}}\put%
(0.5625,0.725){\rule{0.3pt}{0.3pt}}
\put(0.0,0.75){\line(0,-1){0.15}}
\put(0.015,0.75){\line(0,-1){0.1}}
\put(0.03,0.75){\line(0,-1){0.075}}
\put(0.045,0.75){\line(0,-1){0.05}}
\put(0.05,0.75){\line(0,-1){0.025}}
\put(0.6,0.0){\line(0,1){0.15}}
\put(0.585,0.0){\line(0,1){0.1}}
\put(0.57,0.0){\line(0,1){0.075}}
\put(0.555,0.0){\line(0,1){0.05}}
\put(0.55,0.0){\line(0,1){0.025}}
\end{picture}}}^{d}$ the $d$-dimensional lattice and $S$ a finite set.
Time-reversal is an involution $\pi$ on $\Omega$, $\pi^{2}=id$.
Time-reversal extends to a transformation $\Theta_{\pi}$ on path-space, as
introduced for (1.4), by reversing the trajectories. That is, if
we have a trajectory $(\omega_{t},t\in[-T,T])$ then the time-reversed
trajectory $\theta_{\pi}(\omega)$ is given by $(\theta_{\pi}(\omega))_{t}\equiv\pi\omega_{-t}$.
3.2 Reversibility
Dynamic reversibility is a property of the dynamics itself under
time-reversal. It says that if one trajectory $\omega$ of the system is
possible, so is its time-reversed $\theta_{\pi}(\omega)$. For a
deterministic system where $\omega_{t}=\phi(t)\omega_{0}$ with $\phi(t)$ an
invertible flow on phase space, it says that $\phi(t)^{-1}=\pi\phi(t)\pi$, that is a symmetry that anticommutes with the time evolution.
For a stochastic dynamics
this is implied by assuming that if a transition $\sigma\rightarrow U\sigma$ is possible (positive transition rate), then also the same is
true for its time-reversal $\pi U\sigma\rightarrow\pi\sigma$.
Microscopic reversibility is a consequence of dynamic reversibility in
case of an equilibrium dynamics. For our purposes here we do not make a
distinction with the condition of detailed balance. When the dynamics is
driven away from equilibrium, the resulting stochastic model will not
satisfy detailed balance. Usually this produces a current in the system
(but that need not be true in general, see an example in [6]). On
the other hand, a net current signifies the breaking of the detailed
balance condition. In general we like to distinguish between two classes
of finite volume dynamics where microscopic reversibility is explicitly
broken. These are boundary driven versus bulk driven dynamics depending
on the extensivity of the perturbation from an equilibrium dynamics. In
the bulk driven case, one usually verifies so called local detailed
balance, i.e., (2.15) is changed into
$$c(U_{x},\sigma)=c(\pi U^{-1}_{x}\pi,\pi U_{x}\sigma)\exp(-H(U_{x}\sigma)+H(%
\sigma))e^{E\Phi(U_{x}\sigma,\sigma)}$$
where $E$ is some amplitude of an external field and $\Phi$ is
antisymmetric, $\Phi(\pi\eta,\pi\sigma)=-\Phi(\sigma,\eta)$, see e.g.
[10]. Note also that then, necessarily, the relative energies
$H(U_{x}\sigma)-H(\sigma)$ are invariant under exchanging $H$ with $H\pi$.
In boundary driven systems, the process becomes non-translation invariant
and the rates remain of the form (2.15) in the bulk (that is for
$x$ well inside the considered finite volume) while more or less
arbitrary on the boundary. This was the case for Example C in Section
1.1 and was formalized for Proposition 2. Note that there is in fact an
example of a boundary driven system where uniformly in the size of the
system a bulk current can be maintained. This is the nonequilibrium
harmonic crystal treated in [13, 8] where the heat flux is
proportional to the boundary temperature difference rather than to the
temperature gradient (infinite heat conductivity in the thermodynamic
limit). Such ‘superconductors’ do not exist in the context of interacting
particle systems as discussed in the present paper.
3.3 Entropy production
In phenomenological thermodynamics, entropy production appears in open
driven systems as the product of thermodynamic fluxes and forces. The
forces are gradients of intensive quantities (like chemical potential)
generating the currents. The entropy production is identified from a
balance equation for the time-derivative of an entropy density
which is defined in systems close to equilibrium. The definition of
entropy production as we use it here in statistical mechanics comes from
[3, 5, 6, 7, 11, 12, 14] and we refer to the review [4].
The mean entropy production appears there and in
(1.4)-(1.5) as a relative entropy (density) for the
process with respect to its time-reversal. That immediately invites the
following thought (we are grateful to Senya Shlosman for pointing to
this): In equilibrium statistical mechanics, if two translation invariant
Gibbs measures have zero relative entropy density, then they must both be
Gibbsian for the same interaction potential (but not necessarily equal
e.g. because of spontaneous symmetry breaking). Apply this to the
space-time measures obtained for the process $\mathbb{P}_{\rho}$ and the
time-reversed process $\mathbb{P}_{\rho}\Theta$ as introduced for (1.4).
Here we take $\pi=$ identity to avoid extra complications. In some sense,
both processes
are Gibbs measures. Thus, if
the mean entropy production is zero, then the process itself and its
time-reversal have the same (space-time) action functional. Because they
also have the same marginals $\rho$, they must in fact coincide (hence no
spontaneous time-reversal breaking). Hence, zero mean entropy production
implies microscopic reversibility. While convincing on a superficial
level, unfortunately the details of this argument are technically
cumbersome and a direct sufficiently general proof along this line has
not been found.
The only more recent paper that we know of concerning time-reversal
symmetry and the relation with entropy production is [1]. The set-up
there is however quite different from ours. Time-reversal symmetry is
there associated with the anticommutation of an involution with the time
evolution, what we have called dynamic reversibility in the above. In
our discussions here, we deal with spatially extended stochastic dynamics
and the breaking of microscopic reversibility.
4 Proofs
Lemma 1: Under the conditions of Section 2.1, for a
translation invariant stationary measure $\nu$,
$$\sum_{U_{0}\in\mathcal{P}_{0}}\int d\nu(\sigma)c(U_{0},\sigma)\log\frac{d\nu^{%
U_{0}}}{d\nu}(\sigma)=0$$
(4.18)
Proof: Let $\mathcal{F}_{\Lambda}$ be the $\sigma$- field
generated by $\sigma_{x},x\in\Lambda$. Denote by $\nu_{\Lambda}$,
respectively $\nu^{U_{0}}_{\Lambda}$ the $\mathcal{F}_{\Lambda}$- restrictions
of $\nu$ and $\nu^{U_{0}}$. Then we have
$$\frac{d\nu^{U_{0}}_{\Lambda}}{d\nu_{\Lambda}}={\mbox{\rm$\mbox{I}\!\mbox{E}$}}%
_{\nu}\left[\frac{d\nu^{U_{0}}}{d\nu}|\mathcal{F}_{\Lambda}\right]$$
Since $d\nu^{U_{0}}/d\nu\in L^{1}(d\nu)$ for all $U_{0}$, we find using the
martingale convergence theorem that
$$\lim_{\Lambda\uparrow\mathord{\!{
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\rule{0.3pt}{0.3pt}}\put(0.1875,0.375){\rule{0.3pt}{0.3pt}}\put(0.2,0.4){\rule%
{0.3pt}{0.3pt}}\put(0.2125,0.425){\rule{0.3pt}{0.3pt}}\put(0.225,0.45){\rule{0%
.3pt}{0.3pt}}\put(0.2375,0.475){\rule{0.3pt}{0.3pt}}\put(0.25,0.5){\rule{0.3pt%
}{0.3pt}}\put(0.2625,0.525){\rule{0.3pt}{0.3pt}}\put(0.275,0.55){\rule{0.3pt}{%
0.3pt}}\put(0.2875,0.575){\rule{0.3pt}{0.3pt}}\put(0.3,0.6){\rule{0.3pt}{0.3pt%
}}\put(0.3125,0.625){\rule{0.3pt}{0.3pt}}\put(0.325,0.65){\rule{0.3pt}{0.3pt}}%
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25,0.1){\rule{0.3pt}{0.3pt}}\put(0.2625,0.125){\rule{0.3pt}{0.3pt}}\put(0.275,%
0.15){\rule{0.3pt}{0.3pt}}\put(0.2875,0.175){\rule{0.3pt}{0.3pt}}\put(0.3,0.2)%
{\rule{0.3pt}{0.3pt}}\put(0.3125,0.225){\rule{0.3pt}{0.3pt}}\put(0.325,0.25){%
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}{0.3pt}}\put(0.4625,0.525){\rule{0.3pt}{0.3pt}}\put(0.475,0.55){\rule{0.3pt}{%
0.3pt}}\put(0.4875,0.575){\rule{0.3pt}{0.3pt}}\put(0.5,0.6){\rule{0.3pt}{0.3pt%
}}\put(0.5125,0.625){\rule{0.3pt}{0.3pt}}\put(0.525,0.65){\rule{0.3pt}{0.3pt}}%
\put(0.5375,0.675){\rule{0.3pt}{0.3pt}}\put(0.55,0.7){\rule{0.3pt}{0.3pt}}\put%
(0.5625,0.725){\rule{0.3pt}{0.3pt}}
\put(0.0,0.75){\line(0,-1){0.15}}
\put(0.015,0.75){\line(0,-1){0.1}}
\put(0.03,0.75){\line(0,-1){0.075}}
\put(0.045,0.75){\line(0,-1){0.05}}
\put(0.05,0.75){\line(0,-1){0.025}}
\put(0.6,0.0){\line(0,1){0.15}}
\put(0.585,0.0){\line(0,1){0.1}}
\put(0.57,0.0){\line(0,1){0.075}}
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\put(0.55,0.0){\line(0,1){0.025}}
\end{picture}}}^{d}}\frac{d\nu^{U_{0}}_{\Lambda}}{d\nu_{\Lambda}}=\frac{d\nu^{%
U_{0}}}{d\nu},$$
(4.19)
in $L^{1}(d\nu)$. Let $\tilde{\nu}$ be the product measure on $\Omega$
having
as marginals the uniform measure on $S$. From stationarity applied to the
local function $f_{\Lambda}=d\nu_{\Lambda}/d\tilde{\nu}_{\Lambda}$ we find
$$\displaystyle 0$$
$$\displaystyle=$$
$$\displaystyle\sum_{x\in\Lambda^{\prime}}\sum_{U_{x}\in\mathcal{P}_{x}}\int d%
\nu(\sigma)c(U_{x},\sigma)[\log\frac{d\nu^{U_{x}}_{\Lambda}}{d\tilde{\nu}_{%
\Lambda}}-\log\frac{d\nu_{\Lambda}}{d\tilde{\nu}_{\Lambda}}]$$
$$\displaystyle=$$
$$\displaystyle\sum_{x\in\Lambda^{\prime}}\sum_{U_{x}\in\mathcal{P}_{x}}\int d%
\nu(\sigma)c(U_{x},\sigma)\log\frac{d\nu^{U_{x}}_{\Lambda}}{d\nu_{\Lambda}}$$
$$\displaystyle=$$
$$\displaystyle\sum_{x\in\Lambda^{\prime}}\sum_{U_{x}\in\mathcal{P}_{x}}\int d%
\nu(\sigma)c(U_{x},\sigma)\log\frac{d\nu^{U_{x}}}{d\nu}$$
$$\displaystyle+$$
$$\displaystyle\sum_{x\in\Lambda^{\prime}}\sum_{U_{x}\in\mathcal{P}_{x}}\int d%
\nu(\sigma)c(U_{x},\sigma)\left[\log\frac{d\nu^{U_{x}}_{\Lambda}}{d\nu_{%
\Lambda}}-\log\frac{d\nu^{U_{x}}}{d\nu}\right]$$
$$\displaystyle=$$
$$\displaystyle|\Lambda^{\prime}|\sum_{U_{0}\in\mathcal{P}_{0}}\int d\nu(\sigma)%
c(U_{0},\sigma)\log\frac{d\nu^{U_{0}}}{d\nu}(\sigma)$$
$$\displaystyle+$$
$$\displaystyle\sum_{x\in\Lambda^{\prime}}\sum_{U_{x}\in\mathcal{P}_{x}}\int d%
\nu(\sigma)c(U_{x},\sigma)F^{U_{x}}_{\Lambda}(\sigma).$$
The last equality uses translation invariance. We have used the notation
$\Lambda^{\prime}\equiv\{x\in\mathord{\!{
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25,0.1){\rule{0.3pt}{0.3pt}}\put(0.2625,0.125){\rule{0.3pt}{0.3pt}}\put(0.275,%
0.15){\rule{0.3pt}{0.3pt}}\put(0.2875,0.175){\rule{0.3pt}{0.3pt}}\put(0.3,0.2)%
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0.3pt}}\put(0.4875,0.575){\rule{0.3pt}{0.3pt}}\put(0.5,0.6){\rule{0.3pt}{0.3pt%
}}\put(0.5125,0.625){\rule{0.3pt}{0.3pt}}\put(0.525,0.65){\rule{0.3pt}{0.3pt}}%
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(0.5625,0.725){\rule{0.3pt}{0.3pt}}
\put(0.0,0.75){\line(0,-1){0.15}}
\put(0.015,0.75){\line(0,-1){0.1}}
\put(0.03,0.75){\line(0,-1){0.075}}
\put(0.045,0.75){\line(0,-1){0.05}}
\put(0.05,0.75){\line(0,-1){0.025}}
\put(0.6,0.0){\line(0,1){0.15}}
\put(0.585,0.0){\line(0,1){0.1}}
\put(0.57,0.0){\line(0,1){0.075}}
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\end{picture}}}^{d}|V_{x}\cap\Lambda\neq\emptyset\}$ and the
expression
$$F^{U_{x}}_{\Lambda}(\sigma)\equiv\left(\log\frac{d\nu^{U_{x}}_{\Lambda}}{d\nu_%
{\Lambda}}-\log\frac{d\nu^{U_{x}}}{d\nu}\right)$$
We thus have
$$\displaystyle|\sum_{U_{0}\in\mathcal{P}_{0}}\int d\nu(\sigma)c(U_{0},\sigma)%
\log\frac{d\nu^{U_{0}}}{d\nu}(\sigma)|$$
$$\displaystyle\leq$$
$$\displaystyle\frac{1}{|\Lambda^{\prime}|}\sum_{x\in\Lambda^{\prime}}\sum_{U_{x%
}\in\mathcal{P}_{x}}|\int d\nu(\sigma)c(U_{x},\sigma)F^{U_{x}}_{\Lambda}(%
\sigma)|$$
(4.20)
$$\displaystyle\leq$$
$$\displaystyle M\frac{1}{|\Lambda^{\prime}|}\sum_{x\in\Lambda^{\prime}}\sum_{U_%
{0}\in\mathcal{P}_{0}}\int d\nu|F^{U_{0}}_{\Lambda-x}|,$$
by the translation invariance of $\nu$, and $M$ bounds the rates. Now we
use the general fact that if $f_{n}$ converges to $f$ in $L^{1}(d\nu)$ and
both $f_{n},f$ are bounded from below by some constant $c>0$, then $\log f_{n}$ converges to $\log f$ in $L^{1}(d\nu)$. This fact implies that for any
given $\varepsilon>0$, we can choose $\Delta\subset\mathord{\!{
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\put(0.045,0.75){\line(0,-1){0.05}}
\put(0.05,0.75){\line(0,-1){0.025}}
\put(0.6,0.0){\line(0,1){0.15}}
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\end{picture}}}^{d}$ such that for
all $\Delta^{\prime}\supset\Delta$:
$$\max_{U_{0}\in\mathcal{P}_{0}}\int d\nu|F^{U_{0}}_{\Delta^{\prime}}|\leq\frac{%
\varepsilon}{2MN},\quad\mbox{with}\quad|\mathcal{P}_{0}|\equiv N$$
Choose now
$\Lambda\subset\mathord{\!{
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\end{picture}}}^{d}$ so large that
$$\frac{|\{x\in\Lambda^{\prime}:\Delta+x\cap\Lambda^{c}\neq\emptyset\}|}{|%
\Lambda^{\prime}|}\leq\frac{\varepsilon}{2MN\sup_{W,U_{0}}||F^{U_{0}}_{W}||_{L%
^{1}(d\nu)}}$$
We then conclude that
$$\displaystyle|\sum_{U_{0}\in\mathcal{P}_{0}}\int d\nu(\sigma)$$
$$\displaystyle c(U_{0},\sigma)$$
$$\displaystyle\log\frac{d\nu^{U_{0}}}{d\nu}(\sigma)|\leq\frac{1}{|\Lambda^{%
\prime}|}\sum_{x\in\Lambda^{\prime},\Delta+x\subset\Lambda}\frac{\epsilon}{2}+$$
(4.21)
$$\displaystyle MN$$
$$\displaystyle\frac{|\{x\in\Lambda^{\prime}:\Delta+x\cap\Lambda^{c}\neq%
\emptyset\}|}{|\Lambda^{\prime}|}\sup_{W,U_{0}}||F^{U_{0}}_{W}||_{L^{1}(d\nu)}$$
$$\displaystyle\leq$$
$$\displaystyle\varepsilon$$
Proof of Main Theorem: Define
$$\bar{c}(U_{0},\sigma)\equiv\bar{c}(\pi,\rho;U_{0},\sigma)\equiv c(\pi U^{-1}_{%
0}\pi,\pi U_{0}\sigma)\frac{d\rho^{U_{0}}}{d\rho}(\sigma)$$
and substitute it in
$$\sum_{U_{0}\in\mathcal{P}_{0}}\int d\rho(\sigma)[c(U_{0},\sigma)-\bar{c}(U_{0}%
,\sigma)]\log\frac{c(U_{0},\sigma)}{\bar{c}(U_{0},\sigma)}$$
(4.22)
We get four terms, (4.22) =
$$\displaystyle\sum_{U_{0}\in\mathcal{P}_{0}}[\int d\rho(\sigma)\log\frac{c(U_{0%
},\sigma)}{c(\pi\,U_{0}^{-1}\pi,\pi\,U_{0}\sigma)}+\int d\rho(\sigma)c(U_{0},%
\sigma)\log\frac{d\rho}{d\rho^{U_{0}}}$$
(4.23)
$$\displaystyle+$$
$$\displaystyle\int d\rho^{U_{0}}(\sigma)c(\pi U_{0}^{-1}\pi,\pi U_{0}\sigma)%
\log\frac{c(\pi\,U_{0}^{-1}\pi,\pi\,U_{0}\sigma)}{c(U_{0},\sigma)}+\int d\rho^%
{U_{0}}(\sigma)c(\pi U_{0}^{-1}\pi,\pi U_{0}\sigma)\log\frac{d\rho^{U_{0}}}{d%
\rho}]$$
The second term is zero by Lemma 1. The fourth term is also zero because,
using condition iii, we can change $\pi U_{0}^{-1}\pi\rightarrow U_{0}$ in
the sum over $\mathcal{P}_{0}$ getting it equal to
$$\sum_{U_{0}\in\mathcal{P}_{0}}\int d\rho\pi(\sigma)c(U_{0},\sigma)\log\frac{d%
\rho\pi}{d(\rho\pi)^{U_{0}}}$$
which is zero, again by Lemma 1 applied to the stationary measure
$\rho\pi$. Again using iii, we can also
rewrite the third term as
$$\sum_{U_{0}\in\mathcal{P}_{0}}\int d\rho\pi(\sigma)\log\frac{c(U_{0},\sigma)}{%
c(\pi\,U_{0}^{-1}\pi,\pi\,U_{0}\sigma)}$$
Therefore, what remains of (4.23)
is the sum of the first and the third term so that (4.22) equals
$$\sum_{U_{0}\in\mathcal{P}_{0}}[\int d\rho(\sigma)\log\frac{c(U_{0},\sigma)}{c(%
\pi\,U_{0}^{-1}\pi,\pi\,U_{0}\sigma)}+\int d\rho\pi(\sigma)\log\frac{c(U_{0},%
\sigma)}{c(\pi\,U_{0}^{-1}\pi,\pi\,U_{0}\sigma)}]$$
We now recall that the mean
entropy production (2.13) equals
$$\displaystyle\textrm{MEP}_{\pi}(L,\rho)$$
$$\displaystyle=$$
$$\displaystyle\sum_{U_{0}\in\mathcal{P}_{0}}\,(\int d\rho(\sigma)\,c(U_{0},%
\sigma)\log\frac{c(U_{0},\sigma)}{c(\pi\,U_{0}^{-1}\pi,\pi\,U_{0}\sigma)}$$
(4.24)
$$\displaystyle+$$
$$\displaystyle\int d\rho(\sigma)\,[c(U_{0},\pi\sigma)-c(U_{0},\sigma)])$$
This was derived from (2.12) in [5]. We conclude therefore
that (4.22) equals $\textrm{MEP}_{\pi}(L,\rho)+\textrm{MEP}_{\pi}(L,\rho\pi)$ which is zero by hypothesis. This implies
the statement of the Theorem.
Proof of Corollary 1 Since the Radon-Nikodym derivative of
$\rho^{U_{0}}$ with respect to $\rho$ is a local function for all $U_{0}$ and
since by assumption, we can generate with the $U_{0}$ all local excitations
$\sigma^{\prime}$ from $\sigma$, it means that $\rho$ has a continuous version
for its local conditional distributions.
Proof of Corollary 2 From the main result and Corollary 1
it follows that $\rho$ is a translation invariant stationary Gibbs measure
and (2.15) must be satisfied. All other translation invariant
stationary measures must be Gibbsian and for the same potential, see e.g.
[2].
From the
results in [5] as cited above the statement of Corollary 2, it
follows that every other stationary translation invariant measure must
have zero mean entropy production.
Proof of Proposition 2 From the definition
(1.5) we must first compute the relative action under time
reversal, that is
$$R_{n}\equiv\log\frac{d\mathbb{P}_{\rho_{n}}}{d\tilde{\mathbb{P}}_{\rho_{n}}}$$
This can be done via a Girsanov formula and we obtain the analogue of
(2.12):
$$\displaystyle R_{n}(\omega)$$
$$\displaystyle=$$
$$\displaystyle\sum_{x\in\Lambda^{\sharp}_{n}}\sum_{U_{x}\in\mathcal{P}_{x}}\int%
_{-T}^{T}\log\frac{c(U_{x},\omega(s^{-}))}{c(U^{-1}_{x},U_{x}\omega(s^{-}))}dN%
^{U_{x}}_{s}(\omega)$$
$$\displaystyle+$$
$$\displaystyle\sum_{\begin{array}[]{c}A\subset\Lambda_{n}\setminus\Lambda^{%
\sharp}_{n}\\
{\mbox{\tt diam}}(A)\leq r\end{array}}\sum_{\sigma\in S^{A}}\int^{T}_{-T}\log%
\frac{k^{(n)}_{A}(\omega(s^{-}),\omega(s^{-})^{A,\sigma})}{k^{(n)}_{A}(\omega(%
s^{-})^{A,\sigma},\omega(s^{-}))}dN^{A,\sigma,n}_{s}(\omega)$$
The first integral is really a sum over all the times when the trajectory
makes a jump from the action of one of the $U_{x}$; the second integral is a
sum over all times when a configuration $\sigma$ is replacing
$\omega(s^{-})$ in a set $A$ on the boundary. In order to further clarify
this formula, let us first look at trajectories where no boundary
transitions take place (or, what amounts to the same, take $k\equiv 0$
for the moment). Then, we only keep the first term, that is just
(2.12), in case $\pi=id$:
$$\sum_{x\in\Lambda^{\sharp}_{n}}\sum_{U_{x}\in\mathcal{P}_{x}}\int_{-T}^{T}\log%
\frac{c(U_{x},\omega(s^{-}))}{c(U^{-1}_{x},U_{x}\omega(s^{-}))}dN^{U_{x}}_{s}$$
But if we insert the detailed balance condition (2.15), the
above expression telescopes to
$$H(\omega(-T))-H(\omega(T))$$
and the mean entropy production is zero by stationarity.
Turning to the general case we let $\{s_{i}\}_{i=1}^{q}$ be the set of
times at which boundary transitions occur in the sets $A_{i},i=1,\ldots,q,$
for the trajectory $\omega$. These are random but we fix them as $-T\leq s_{1}<s_{2}<..<s_{q}\leq T$. The important thing to realize now is that while
the perfect telescoping of above is broken at each of these times, it can
be restored by adding and subtracting. More precisely, we have
$$\displaystyle R_{n}(\omega)$$
$$\displaystyle=$$
$$\displaystyle H(\omega(-T))-H(\omega(s_{1}^{-}))+H(\omega(s_{1}))-H(\omega(s_{%
2}^{-}))+\ldots$$
$$\displaystyle+$$
$$\displaystyle H(\omega(s_{q}))-H(\omega(T))+\log\frac{k^{(n)}_{A_{1}}(\omega(s%
_{1}^{-}),\omega(s_{1}))}{k^{(n)}_{A_{1}}(\omega(s_{1}),\omega(s_{1}^{-}))}+$$
$$\displaystyle+$$
$$\displaystyle\log\frac{k^{(n)}_{A_{2}}(\omega(s^{-}_{2}),\omega(s_{2}))}{k^{(n%
)}_{A_{2}}(\omega(s_{2}),\omega(s^{-}_{2}))}+\ldots+\log\frac{k^{(n)}_{A_{q}}(%
\omega(s^{-}_{q}),\omega(s^{-}_{q}))}{k^{(n)}_{A_{q}}(\omega(s_{q},\omega(s^{-%
}_{q}))}$$
But by the absolute convergence of the interaction potential we have
$$|H(\omega(s^{-}_{i})-H(\omega(s_{i}))|\leq rC$$
for some constant $C$, since $\omega(s_{i}^{-})$ and $\omega(s_{i})$ only
differ in the set $A_{i}$. Therefore the telescoping of the terms involving
energy differences can be restored upon inserting $q$ terms of order
unity.
As for the other terms, we have assumed uniform boundedness so
that we get
$$|R_{n}(\omega)|\leq q(rC+\log\frac{M}{\epsilon})$$
where $M$ and $\epsilon$ are constant upper and lower bounds for the
transition rates $k^{(n)}$. As the expectation of $q=q(\omega)$ under
${\mbox{\rm$\mbox{I}\!\mbox{E}$}}_{\rho_{n}}$ is proportional to $T|\partial\Lambda_{n}|$, the
proposition is proved.
Acknowledgment: C.M. thanks Senya Shlosman for some very
useful discussions.
References
[1]
Gallavotti G., Breakdown and regeneration of time reversal symmetry
in nonequilibrium Statistical Mechanics Physica D 112 250–257
(1998)
[2]
Künsch, H., Non reversible stationary measures for infinite
interacting particle systems Z. Wahrsch. Verw. Gebiete 66, 407
(1984).
[3]
Maes C., The Fluctuation Theorem as a Gibbs Property, J. Stat.
Phys. 95, 367-392 (1999).
[4]
Maes, C., Statistical mechanics of entropy production: Gibbsian
hypothesis and local fluctuations, preprint from cond-mat/0106464.
[5]
Maes, C., Redig, F. and Verschuere, M., Entropy Production for
Interacting Particle Systems Markov Proc. Rel. Fields 7, 119–134 (2001).
[6]
Maes C., Redig F., Van Moffaert A., On the definition of entropy
production via examples J. Math. Phys. 41, 1528–1554 (2000).
[7]
Maes C., Redig F., Positivity of entropy production J. Stat. Phys.
101, 3–16 (2000).
[8]
Nakazawa, H., On the Lattice Thermal Conduction Suppl.
Prog. Theor. Phys., 45, 231–262 (1970)
[9]
Liggett T. M., Interacting particle systems Springer-Verlag, New
York, Heidelberg, Berlin (1985)
[10]
Lebowitz J. L., Spohn H., A Gallavotti-Cohen type symmetry in the
large deviation functional for stochastic dynamics J. Stat. Phys. 95, 333–365 (1999)
[11]
Qian M. P., Qian M., Qian C., Circulations of markov chains with
continuous time and probability interpretation of some determinants Sci.
Sinica 27, 470-481. (1984)
[12]
Qian M. P., Qian M., The entropy production and reversibility of
Markov processes Proceedings of the first world congress Bernoulli soc.
1988, 307-316
[13]
Rieder, Z., Lebowitz, J.L. and Lieb, E., Properties of
a Harmonic Crystal in a Stationary Nonequilibrium State J. Math. Phys.
8, 1073–1078 (1967)
[14]
Schnakenberg J., Network theory of behavior of master equation
systems Rev. Mod. Phys. 48, 4, 571-585. (1976) |
Coding for Distributed Fog Computing
Songze Li,
Mohammad Ali Maddah-Ali,
and A. Salman Avestimehr
S. Li and A.S. Avestimehr are with the Department of Electrical Engineering, University of Southern California, Los Angeles, CA, 90089, USA (e-mail: songzeli@usc.edu; avestimehr@ee.usc.edu).M. A. Maddah-Ali is with Nokia Bell Labs, Holmdel, NJ, 07733, USA (e-mail: mohammad.maddahali@nokia-bell-labs.com).
I abstract
Redundancy is abundant in Fog networks (i.e., many computing and storage points) and grows linearly with network size. We demonstrate the transformational role of coding in Fog computing for leveraging such redundancy to substantially reduce the bandwidth consumption and latency of computing. In particular, we discuss two recently proposed coding concepts, namely Minimum Bandwidth Codes and Minimum Latency Codes, and illustrate their impacts in Fog computing. We also review a unified coding framework that includes the above two coding techniques as special cases, and enables a tradeoff between computation latency and communication load to optimize system performance. At the end, we will discuss several open problems and future research directions.
II Introduction
The Fog architecture (see Fig. 1) has been recently proposed to better satisfy the service requirements of the emerging Internet-of-Things (IoT) (see, e.g., [1]). Unlike the Cloud computing that stores and processes end-users’ data in remote and centralized datacenters, Fog computing brings the provision of services closer to the end-users by pooling the available resources at the edge of the network (e.g., smartphones, tablets, smart cars, base stations and routers) (see, e.g., [2, 3]). As a result, the main driving vision for Fog computing is to leverage the significant amount of dispersed computing resources at the edge of the network to provide much more user-aware, resource-efficient, scalable and low-latency services for IoT.
The main goal of this paper is to demonstrate how coding can be effectively utilized to trade abundant computing resources at network edge for communication bandwidth and latency. In particular, we illustrate two recently proposed novel coding concepts that leverage the available or under-utilized computing resources at various parts of the network to enable coding opportunities that significantly reduce the bandwidth consumption and latency of computing, which are of particular importance in Fog computing applications.
The first coding concept introduced in [4, 5], which we refer to as Minimum Bandwidth Codes, enables a surprising inverse-linear tradeoff between computation load and communication load in distributed computing. Minimum Bandwidth Codes demonstrate that increasing the computation load by a factor of $r$ (i.e., evaluating each computation at $r$ carefully chosen nodes) can create novel coding opportunities that reduce the required communication load for computing by the same factor. Hence, Minimum Bandwidth Codes can be utilized to pool the underutilized computing resources at network edge to slash the communication load of Fog computing.
The second coding concept introduced in [6], which we refer to as Minimum Latency Codes, enables an inverse-linear tradeoff between computation load and computation latency (i.e., the overall job response time). More specifically, Minimum Latency Codes utilize coding to effectively inject redundant computations to alleviate the effects of stragglers and speed up the computations by a multiplicative factor that is proportional to the amount of injected redundancy. Hence, by utilizing more computation resources at network edge, Minimum Latency Codes can significantly speed up distributed Fog computing applications.
In this paper, we give an overview of these two coding concepts, illustrate their key ideas via motivating examples, and demonstrate their impacts on Fog networks.
More generally, noting that redundancy is abundant in Fog networks (i.e., many computing/storage points) and grows linearly with network size, we demonstrate the transformational role of coding in Fog computing for leveraging such redundancy to substantially reduce the bandwidth consumption and latency of computing. We also point out that while these two coding techniques are also applicable to Cloud computing applications, they are expected to play a much more substantial role in improving the system performance of Fog applications, due to the fact that communication bottleneck and straggling nodes are far more severe issues in Fog computing compared with its Cloud counterpart.
We also discuss a recently proposed unified coding framework, in which the above two coding concepts are systematically combined by introducing a tradeoff between “computation latency” and “communication load”. This framework allows a Fog computing system to operate at any point on the tradeoff, on which the Minimum Bandwidth Codes and the Minimum Latency Codes can be viewed as two extreme points that respectively minimizes the communication load and the computation latency.
We finally conclude the paper and highlight some exciting open problems and research directions for utilizing coding in Fog computing architectures.
III Minimum Bandwidth Codes
We illustrate Minimum Bandwidth Codes in a typical Fog computing scenario, in which a Fog client aims to utilize the network edge for its computation task. For instance, a driver wants to find the best route through a navigation application offered by the Fog, in which the map information and traffic condition are distributedly stored in edge nodes (ENs) like roadside monitors, smart traffic lights, or other smart cars that collaborate to find the best route. Another example is object recognition that is the key enabler of many augmented reality applications. To provide an object recognition service over Fog, edge nodes like routers and base stations, each stores parts of the dataset repository, and collaboratively process the images or videos provided by the Fog client.
For the above Fog computing applications, the computation task is over a large dataset that is distributedly stored on the edge nodes (e.g., map/traffic information or dataset repository), and the computations are often decomposed using MapReduce-type frameworks (e.g., [7, 8]), in which a collection of edge nodes distributedly Map a set of input files, generating some intermediate values, from which they Reduce a set of output functions.
We now demonstrate the main concepts of Minimum Bandwidth Codes in a simple problem depicted in Fig. 2. In this case, a client uploads a job of computing $3$ output functions (represented by red/circle, green/square, and blue/triangle respectively) from $6$ input files to the Fog. Three edge nodes in the Fog, i.e., EN $1$, EN $2$ and EN $3$, collaborate to perform the computation. Each EN is responsible for computing a unique output function, e.g., EN $1$ computes the red/circle function, EN $2$ computes the green/square function, and EN $3$ computes the blue/triangle function. When an EN maps a locally stored input file, it computes $3$ intermediate values, one for each output function. To reduce an output function, each EN needs to know the intermediate values of this output for all $6$ input files.
We first consider the case where no redundancy is imposed on the computations, i.e., each file is mapped exactly once. Then as shown in Fig. 2, each EN maps $2$ input files locally, obtaining $2$ out of $6$ required intermediate values. Hence, each EN needs another $4$ intermediate values transferred from the other ENs, yielding a communication load of $4\times 3=12$.
Now, we demonstrate how Minimum Bandwidth Codes can substantially reduce the communication load by injecting redundancy in computation. As shown in Fig. 2, let us double the computation such that each file is mapped on two ENs (files are downloaded to the ENs offline). It is apparent that since more local computations are performed, each EN now only requires $2$ other intermediate values, and an uncoded shuffling scheme would achieve a communication load of $2\times 3=6$. However, we can do better with the Minimum Bandwidth Codes. As shown in Fig. 2, instead of unicasting individual intermediate values, every EN multicasts a bit-wise XOR, denoted by $\oplus$, of $2$ intermediate values to the other two ENs, simultaneously satisfying their data demands. For example, knowing the blue triangle in File $3$, EN $2$ can cancel it from the coded packet multicast by EN $1$, recovering the needed green square in File $1$. In general, the bandwidth consumption of multicasting one packet to two nodes is less than that of unicasting two packets, and here we consider a scenario in which it is as much as that of unicasting one packet (which is the case for wireless networks). Therefore, the above Minimum Bandwidth Code incurs a communication load of $3$, achieving a $4\times$ gain from the case without computation redundancy and a $2\times$ gain from the uncoded shuffling.
More generally, we can consider a Fog computing scenario, in which $K$ edge nodes collaborate to compute $Q$ output functions from $N$ input files that are distributedly stored at the nodes. We define the computation load, $r$, to be the total number of input files that are mapped across the nodes, normalized by $N$. That is, e.g., $r=2$ means that on average each file is mapped on two nodes. We can similarly define the communication load $L$ to be the total (normalized) number of information bits exchanged across nodes during data shuffling, in order to compute the $Q$ output functions.
For this scenario it was shown in [5] that, compared with conventional uncoded strategies, Minimum Bandwidth Codes can surprisingly reduce the communication load by a multiplicative factor that equals to the computation load $r$, when computing $r$ times more sub-tasks than the execution without redundancy (i.e., $r=1$). Or more specifically,
$$\displaystyle L_{\textup{coded}}=\tfrac{1}{r}L_{\textup{uncoded}}=\tfrac{1}{r}%
(1-\tfrac{r}{K})=\Theta(\tfrac{1}{r}).$$
(1)
Minimum Bandwidth Codes employ a specific strategy to assign the computations of the Map and Reduce functions, in order to enable novel coding opportunities for data shuffling. In particular, each data block is repetitively mapped on $r$ distinct nodes according to a specific pattern, in order to create coded multicast messages that deliver useful data simultaneously to $r\geq 1$ nodes. For example, as demonstrated in Fig. 3, the overall communication load can be reduced by more than 50% when each Map task is repeated at only one other node (i.e., $r=2$).
The idea of efficiently creating and exploiting coded multicast opportunities was initially proposed to solve caching problems in [9, 10], and extended to wireless D2D networks in [11], where caches pre-fetch part of the content to enable coding during the content delivery, minimizing the network traffic. Minimum Bandwidth Codes extend such coding opportunities to data shuffling of distributed computing frameworks, significantly reducing the required communication load.
Apart from significantly slashing the bandwidth consumption, Minimum Bandwidth Codes also have the following major impacts on the design of Fog computing architecture.
Reducing Overall Response Time. Let us consider an arbitrary Fog computing application for which the overall response time is composed of the time spent computing the intermediate tasks, denoted by $T_{\textup{Task Computation}}$, and the time spent moving intermediate results, denoted by $T_{\textup{Data Movement}}$. In many applications of interest (e.g., video/image analytics or recommendation services), most of the job execution time is spent for data movement. For example, consider the scenarios in which $T_{\textup{Data Movement}}$ is $10\times\sim 100\times$ of $T_{\textup{Task Computation}}$. Using a Minimum Bandwidth Code with computation load $r$, we can achieve an overall response time of
$$\displaystyle T_{\textup{total, coded}}\approx\mathbb{E}[rT_{\textup{Task %
Computation}}+\tfrac{1}{r}T_{\textup{Data Movement}}].$$
(2)
To minimize the above response time, one would choose the optimum computation load $r^{*}=\sqrt{\frac{T_{\textup{Data Movement}}}{T_{\textup{Task Computation}}}}$.Then in the above example, utilizing Minimum Bandwidth Codes can reduce the overall job response time by approximately $1.5\sim 5$ times.
The impact of Minimum Bandwidth Codes on reducing the response time has been recently demonstrated in [12] through a series of experiments over Amazon EC2 clusters. In particular, the Minimum Bandwidth Codes were incorporated into the well-known distributed sorting algorithm TeraSort [13], to develop a new coded sorting algorithm, namely CodedTeraSort, which allows a flexible selection of the computation load $r$. Here we summarize in Table I, the runtime performance of a particular job of sorting 12 GB of data over 16 EC2 instances.
Theoretically according to (1), with a computation load $r=5$, CodedTeraSort promises to reduce the data shuffling time by a factor of approximately $5$. From Table I, we can see that while computing $r=5$ times more Map functions increased the Map task computation time by $5.83\times$, CodedTeraSort brought down the data shuffling time, which was the limiting component of the runtime of this application, by $4.24\times$. As a result, CodedTeraSort reduced the overall job response time by $3.39\times$.
Scalable Mobile Computation. The Minimum Bandwidth Codes also found their application in a wireless distributed computing platform proposed in [14], which is a fully decentralized Fog computing environment. In this platform, a collection of mobile users, each has a input to process overall a large dataset (e.g., the image repository of an image recognition application), collaborate to store the dataset and perform the computations, using their own storage and computing resources. All participating users communicate the locally computed intermediate results among each other to reduce the final outputs.
Utilizing Minimum Bandwidth Codes in this wireless computing platform leads to a scalable design. More specifically, let us consider a scenario where $K$ users, each processing a fraction of the dataset, denoted by $\mu$ (for some $\frac{1}{K}\leq\mu\leq 1$), collaborate for wireless distributed computing. It is demonstrated in [14] that Minimum Bandwidth Codes can achieve a (normalized) bandwidth consumption of $\frac{1}{\mu}-1$ to shuffle all required intermediate results. This reduces the communication load of the uncoded scheme, i.e. $K(1-\mu)$, by a factor of $\mu K$, which scales linearly with the aggregated storage size of all collaborating users. Also, since the consumed bandwidth is independent of the number of users $K$, Minimum Bandwidth Code allows this platform to simultaneously serve an unlimited number of users with a constant communication load.
IV Minimum Latency Codes
We now move to the second coding concept, named Minimum Latency Codes, and demonstrate it for a class of Fog computing applications, in which a client’s input is processed over a large dataset (possibly over multiple iterations). The application is supported by a group of edge nodes, which have distributedly stored the entire dataset. Each node processes the client’s input using the parts of the dataset it locally has, and returns the computed results to the client. The client reduces the final results after collecting intermediate results from all edge nodes. Many distributed machine learning algorithms fall into this category. For example, a gradient decent algorithm for linear regression requires multiplying the weight vector with the data matrix in each iteration. To do that at network edge, each edge node stores locally a sub-matrix of the data matrix. During computation, each edge node multiplies the weight vector with the stored sub-matrix and returns the results to the client.
To be more specific, let us consider a simple distributed matrix multiplication problem, in which as shown in Fig. 4, a client wants to multiply a data matrix ${\bf A}$ with the input matrix ${\bf X}$ to compute ${\bf A}{\bf X}$. The data matrix ${\bf A}$ is stored distributedly across $3$ nearby ENs, i.e., EN $1$, EN $2$, and EN $3$, on which the matrix multiplication will be executed distributedly.
One natural approach to tackle this problem is to vertically and evenly divide the data matrix ${\bf A}$
into $3$ sub-matrices, each of which is stored on one EN. Then when each EN receives the input ${\bf X}$, it simply multiplies its locally stored sub-matrix with ${\bf X}$ and returns the results, and the client vertically concatenates the returned matrices to obtain the final result. However, we note that since this uncoded approach relies on successfully retrieving the task results from all $3$ ENs, it has a major drawback that once one of the ENs runs slow or gets disconnected, the computation may take very long or even fail to finish. Minimum Latency Codes deal with slow or unreliable edge nodes by optimally creating redundant computations tasks. As shown in Fig. 4, a Minimum Latency Code vertically partitions the data matrix ${\bf A}$ into $2$ sub-matrices ${\bf A}_{1}$ and ${\bf A}_{2}$, and creates one redundant task by summing ${\bf A}_{1}$ and ${\bf A}_{2}$. Then ${\bf A}_{1}$, ${\bf A}_{2}$ and ${\bf A}_{1}+{\bf A}_{2}$ are stored on EN $1$, EN $2$, and EN $3$ respectively. In the case of Fig. 4, the computation is completed when the client has received the task results only from EN $1$ and $3$, from which ${\bf A}_{2}{\bf X}$ can be decoded. In fact, it is obvious that the client can recover the final result once she receives the task results from any 2 out of the 3 ENs, without needing to wait for the slow/unreachable EN (EN $2$ in this case). In summary, Minimum Latency Codes create redundant computation tasks across Fog networks, such that having any set of certain number of task results is sufficient to accomplish the overall computation. Hence, applying Minimum Latency Codes on the abundant edge nodes can effectively alleviate the effect of stragglers and significantly speed up Fog computing.
As illustrated in the above example, the basic idea of Minimum Latency Codes is to apply erasure codes on computation tasks, creating redundant coded tasks that provide robustness to straggling edge nodes. Erasure codes have been widely exploited to combat symbol losses in communication systems and disk failures in distributed storage systems. The simplest form of erasure codes, i.e., the repetition code, repeats each information symbol multiple times, such that a information symbol can be successfully recovered as long as at least one of the repeats survives. For example, modern distributed files systems like Hadoop Distributed File System (HDFS) replicates each data block three times across different storage nodes. Another type of erasure code, known as the Maximum-Distance-Separable (MDS) code, provides better robustness to erasures. An $(n,k)$ MDS code takes $k$ information symbols and encodes them into $n\geq k$ coded symbols, such that obtaining any $k$ out of the $n$ coded symbols is sufficient to decode all $k$ information symbols. This “any $k$ of $n$” property is highly desirable due to the randomness of erasures. A successful application of the MDS code is the Reed-Solomon Code used to protect CDs and DVDs.
As introduced in [6], Minimum Latency Codes are exactly MDS codes that are used to encode computation tasks. For a Fog computing job executed on $n$ edge nodes, a $(n,k)$ Minimum Latency Code first decomposes the overall computation into $k$ smaller tasks, for some $k\leq n$. Then it encodes them into $n$ coded tasks using an $(n,k)$ MDS code, and assigns each of them to a node to compute. By the aforementioned “any $k$ of $n$” property of the MDS code, we can accomplish the overall computation once we have collected the results from the fastest $k$ out of $n$ coded tasks, without worrying the tasks still running on the slow nodes (or stragglers).
Minimum Latency Codes can help to significantly improve the response time of Fog applications. Let’s consider a computation task performed distributedly across $n$ edge nodes. The response time of the uncoded approach is limited by the slowest node. An $(n,k)$ repetition code breaks the computation into $k$ tasks, and repeats each task $\frac{n}{k}$ times across the $n$ nodes, and the computation continues until each task has been computed at least once. On the other hand, for an $(n,k)$ Minimum Latency Code, the response time is limited by the fastest $k$ out of $n$ nodes that have finished their coded tasks. As shown in [6], for a shifted-exponential distribution, the average response times of the uncoded execution and the repetition code are both $\Theta(\frac{\log n}{n})$. The Minimum Latency Codes can reduce the response time by a factor of $\Theta(\log n)$. For example, in a typical Fog computing scenario with $10\sim 100$ nodes, Minimum Latency Codes can theoretically offer a $2.3\times\sim 4.6\times$ speedup. Moreover, experiments on Amazon EC2 clusters were performed in [6], in which for a gradient descent computation for linear regression, Minimum Latency Codes reduce the response time by 35.7% on average. We further envision that in a Fog computing environment where computing nodes are much more heterogeneous and likely to be irresponsive, the performance gain by using Minimum Latency Codes will be much larger.
Other than speeding up the Fog computing applications, Minimum Latency Codes also maximize the survivability of the computation when faced with nodes failure/disconnection, i.e., when the task results may never come back. We note that an $(n,k)$ Minimum Latency Code requires any $k$ out of $n$ tasks to be returned to guarantee a successful computation, and this level of robustness can not be provided by either the uncoded computation or the the repetition code.
V A Unified Coding Framework
We have so far discussed two different coding techniques that aim at minimizing the bandwidth consumption and the computation latency of Fog computing. However, under a MapReduce-type computing model, a unified coded framework has been recently developed in [15] by introducing a tradeoff between “computation latency” in the Map phase and “communication load” in the Shuffle phase.
As an example, in Fig. 5 we have illustrated the tradeoff between “computation latency” and “communication load” that is achieved by the unified framework for running a distributed matrix multiplication over $18$ edge nodes (see [15] Section III for details). We observe that the achieved tradeoff approximately exhibits an inverse-linearly proportional relationship between the latency and the load. In particular, we can see that the Minimum Bandwidth Codes and the Minimum Latency Codes can be viewed as special instances of the proposed coding framework, by considering two extremes of this tradeoff: minimizing either the communication load or the computation latency individually. Next, we further illustrate how to utilize this tradeoff to minimize the total response time that is the sum of the communication time in the Shuffle phase and the computation latency in the Map phase. For the matrix multiplication problem in Fig. 5, we consider real entries each represented using $2$ bytes, a shift-exponential distribution for the Map task execution time, and a wireless network with speed 10 Mbps. Then, the Minimum Bandwidth Codes that wait for all 18 nodes to finish their Map tasks achieve a total response time of 302s,111The communication load in Fig. 5 is normalized by the number of the rows of the matrix, which is $10^{6}$ in this example. and the Minimum Latency Codes that terminate the Map phase when the fastest 3 nodes (minimum required number) finish their Map tasks achieve a total response time of 263s. Using the unified coding framework, we can wait for the optimal number of the fastest 12 nodes to finish, and achieve the minimum total response time of 186s. Hence, this unified coding approach provides a performance gain of 38.4% and 29.3 % over the Minimum Bandwidth Codes and the Minimum Latency Codes respectively.
This unified coding framework, which is essentially a systematic concatenation of the Minimum Bandwidth Codes and the Minimum Latency Codes, takes advantage of both coding techniques in difference stages of the computation. In the Map phase, MDS codes are employed to create coded tasks, which are then assigned to edge nodes in a specific repetitive pattern for local execution. According to the interested computation latency of the Map phase, all running Map tasks are terminated as soon as a certain number of nodes have finished their local computations. Then in the Shuffle phase, coded multicast opportunities specified by Minimum Bandwidth Codes are greedily utilized, until the data demands of all nodes are satisfied. For example, we can consider executing a linear computation consisting of $m=20$ Map tasks using $K=6$ edge nodes, each of which can process $\mu=\frac{1}{2}$ fractions of the tasks. To be able to end the Map phase when only the fastest $q=4$ nodes finish their local tasks, we can first use a $(\frac{K}{q}m,m)=(30,20)$ MDS code to generate $30$ coded tasks, each of which is then assigned to $\mu q=2$ nodes for execution according to the repetitive assignment pattern specified by the Minimum Bandwidth Codes. For more detailed illustrative examples, we refer the interested readers to Section IV of [15].
The unified coding framework allows us to flexibly select the operation point to minimize the overall job execution time. For example, when the network is slow, we can wait for more nodes to finish their Map computations, creating better multicast opportunities to further slash the amount of data movement. On the other hand, when we have detected that some nodes are running slow or becoming irresponsive, we can shift the load to the network by ending the Map phase as soon as enough coded tasks are executed.
VI Conclusions and Future Research Directions
We demonstrated how coding can be effectively utilized to leverage abundant computing resources at the network edge to significantly reduce the bandwidth consumption and computation latency in Fog computing applications. In particular, we illustrated two recently proposed coding concepts, namely Minimum Bandwidth Codes and Minimum Latency Codes, and discussed their impacts on Fog computing. We also discussed a unified coding framework that includes the above two coding techniques as special cases, and enables a tradeoff between computation latency and communication load to optimize the system performance.
We envision codes to play a fundamental role in Fog computing by enabling an efficient utilization of computation, communication, and storage resources at network edge. This area opens up many important and exciting future research directions. Here we list a few:
Heterogeneous computing nodes: In distributed Fog networks, different nodes have different processing and storage capacities. The ideas outlined in this paper can be used to develop heuristic solutions for heterogeneous networks. For example, one simple approach is to break the more powerful nodes into multiple smaller virtual nodes that have homogeneous capability, and then apply the proposed coding techniques for the homogeneous setting. However, systematically developing practical task assignment and coding techniques for these systems, that are provably optimum (approximately), is a challenging open problem.
Networks with multi-layer and structured topology: The current code designs for distributed computing[4, 5, 15] are developed for a basic topology, in which the processing nodes are connected through a shared link. While these results demonstrate the significant gain of coding in distributed Fog computing, we need to extend these ideas to more general network topologies. In such networks, nodes can be connected through multiple switches and links in different layers with different capacities.
Multi-stage computation tasks: Another important direction is to consider more general computing frameworks, in which the computation job is represented by a Directed Acyclic Task Graph (DAG). While we can apply the aforementioned code designs for each stage of computation locally, we expect to achieve a higher reduction in bandwidth consumption and response time by globally designing codes for the entire task graph and accounting for interactions between consecutive stages.
Coded computing overhead: The current Fog computing system under consideration lacks appropriate modeling of the coding overhead, which includes the cost for the encoding and decoding processes, the cost for performing multicast communications, and the cost for maintaining desired data redundancy across Fog nodes. To make the study of coding in practical Fog systems more relevant, it is important to carefully formulate a comprehensive model that systematically accounts for these overhead.
Verifiable distributed computing: Fog architecture facilitates offloading of computational tasks from relatively weak computational devices (clients) to more powerful nodes in the edge network. As a result, there is a critical need for “Verifiable Computing” methods, in which clients can make sure they receive the correct calculations. This is typically achieved by injecting redundancy in computations by the clients. We expect codes to provide much more efficient methods for leveraging computation redundancy in order to provide verified computing in Fog applications.
Exploiting the algebraic structures of computation tasks: Recall that the Minimum Bandwidth Codes can be applied to any general computation task that can be cast in a MapReduce framework. However, we expect to improve the overall performance, if we exploit the specific algebraic properties of the underlying tasks. For example, if the task has some linearity, we may be able to incorporate it in communication and coding design in order to further reduce the bandwidth consumption and latency. On the contrary, Minimum Latency Codes work only for some particular linear functions (e.g., matrix multiplication). It is of great interest to extend these codes to a broader class of computation tasks.
Communication-heavy applications: Recall that by exploiting Minimum Bandwidth Codes we can envision a Fog system that can handle many distributed Fog nodes with a bounded communication load. Such a surprising feature would enormously expand the list of applications that can be offered over Fog networks. One research direction is to re-examine some communication-heavy tasks to see if Minimum Bandwidth Codes allow them to be implemented over distributed Fog networks.
Plug-and-Play Fog nodes: We can finally envision a software package (or App) that can be installed and maintained distributedly on each Fog node. This package should allow a Fog computing node to join the system anytime to work with the rest of the nodes or leave the system asynchronously, still the entire network operates near optimum. Designing codes that guarantee integrity of computations despite such network dynamics is a very interesting and important research direction.
References
[1]
F. Bonomi, R. Milito, J. Zhu, and S. Addepalli, “Fog computing and its role in
the internet of things,” in Proceedings of the First Edition of the
MCC Workshop on Mobile Cloud Computing, ser. MCC ’12. ACM, Aug. 2012, pp. 13–16.
[2]
M. Chiang and T. Zhang, “Fog and IoT: An overview of research
opportunities,” IEEE Internet of Things Journal, vol. 3, no. 6, pp.
854–864, Dec. 2016.
[3]
S. Yi, C. Li, and Q. Li, “A survey of fog computing: Concepts, applications
and issues,” in Proceedings of the 2015 Workshop on Mobile Big Data,
ser. Mobidata ’15. ACM, June 2015, pp.
37–42.
[4]
S. Li, M. A. Maddah-Ali, and A. S. Avestimehr, “Coded MapReduce,” in
Proceedings of the 2015 53rd Annual Allerton Conference on
Communication, Control, and Computing (Allerton), Sept. 2015, pp. 964–971.
[5]
——, “Fundamental tradeoff between computation and communication in
distributed computing,” in Proceedings of the 2016 IEEE International
Symposium on Information Theory (ISIT), July 2016, pp. 1814–1818.
[6]
K. Lee, M. Lam, R. Pedarsani, D. Papailiopoulos, and K. Ramchandran, “Speeding
up distributed machine learning using codes,” in Proceedings of the
2016 IEEE International Symposium on Information Theory (ISIT), July 2016,
pp. 1143–1147.
[7]
J. Dean and S. Ghemawat, “MapReduce: Simplified data processing on large
clusters,” in Proceedings of the 2004 6th USENIX Symposium on
Operating Systems Design and Implementation, ser. OSDI ’04. USENIX, Dec. 2004, pp. 137–150.
[8]
M. Zaharia, M. Chowdhury, M. J. Franklin, S. Shenker, and I. Stoica, “Spark:
cluster computing with working sets,” in Proceedings of the 2010 2nd
USENIX Workshop on Hot Topics in Cloud Computin, ser. HotCloud ’10. USENIX, June 2010, pp. 10–10.
[9]
M. A. Maddah-Ali and U. Niesen, “Fundamental limits of caching,” IEEE
Transactions on Information Theory, vol. 60, no. 5, pp. 2856–2867, May
2014.
[10]
——, “Decentralized coded caching attains order-optimal memory-rate
tradeoff,” IEEE/ACM Transactions on Networking, vol. 23, no. 4, pp.
1029–1040, Aug. 2015.
[11]
M. Ji, G. Caire, and A. F. Molisch, “Fundamental limits of caching in wireless
D2D networks,” IEEE Transactions on Information Theory, vol. 62,
no. 2, pp. 849–869, Feb. 2016.
[12]
“CodedTeraSort Implementations,” available online at:
http://www-bcf.usc.edu/~avestime/CodedTerasort.html, 2016, Accessed
on January 11, 2017.
[13]
O. O’Malley, “TeraByte Sort on Apache Hadoop,” available online
at: http://sortbenchmark.org/YahooHadoop.pdf, 2008, Accessed on
January 11, 2017.
[14]
S. Li, Q. Yu, M. A. Maddah-Ali, and A. S. Avestimehr, “Edge-facilitated
wireless distributed computing,” in Proceedings of the 2016 IEEE
Global Communications Conference (GLOBECOM), Dec. 2016, pp. 1–7.
[15]
S. Li, M. A. Maddah-Ali, and A. S. Avestimehr, “A unified coding framework for
distributed computing with straggling servers,” in Proceedings of the
2016 IEEE Workshop on Network Coding and Applications (NetCod), Dec. 2016. |
Recognizing [h,2,1] graphs
Liliana Alcón Marisa Gutierrez
María Pía Mazzoleni
Departamento de Matemática
Universidad
Nacional de La Plata
C. C. 172, (1900) La Plata, Argentina
liliana,marisa,pia@mate.unlp.edu.ar
CONICETSupported by CONICET
()
Abstract
An $(h,s,t)$-representation of a graph $G$ consists of a
collection of subtrees of a tree $T$, where each subtree
corresponds to a vertex of $G$ such that (i) the maximum degree of
$T$ is at most $h$, (ii) every subtree has maximum degree at mots
$s$, (iii) there is an edge between two vertices in the graph $G$
if and only if the corresponding subtrees have at least $t$
vertices in common in $T$. The class of graphs that have an
$(h,s,t)$-representation is denoted $[h,s,t]$.
An undirected graph $G$ is called a $VPT$ graph if it is the
vertex intersection graph of a family of paths in a tree. In this
paper we characterize $[h,2,1]$ graphs using chromatic number. We
show that the problem of deciding whether a given $VPT$ graph
belongs to $[h,2,1]$ is NP-complete, while the problem of deciding
whether the graph belongs to $[h,2,1]-[h-1,2,1]$ is NP-hard. Both
problems remain hard even when restricted to $Split\cap VPT$.
Additionally, we present a non-trivial subclass of $Split\cap VPT$ in which these problems are polynomial time solvable.
Key words: intersection graphs, VPT graphs, representations on
trees, recognition problems.
1 Introduction
The intersection graph of a set family is a graph whose vertices
are the members of the family, and the adjacency between them is
defined by a non-empty intersection of the corresponding sets.
Classical examples are interval graphs and chordal graphs.
An interval graph is the intersection graph of a family
of closed intervals on the real line, or equivalently the
intersection graph of a family of subpaths of a path. A
chordal graph is a graph without induced cycles of
length at least four. Gravril [4] proved that a graph is
chordal if and only if it is the intersection graph of a family of
subtrees of a tree, considering vertex intersection. Both classes
has been widely studied [1].
In order to allow larger families of graphs to be represented by
subtrees, several graph classes are defined imposing conditions on
trees, subtrees and intersection sizes [11, 12]. An
(h,s,t)-representation of a graph $G$ consists of a
collection of subtrees of a tree $T$, each subtree corresponding
to a vertex of $G$, such that (i) the maximum degree of $T$ is at
most $h$, (ii) every subtree has maximum degree at mots $s$, (iii)
there is an edge between two vertices in the graph $G$ if and only
if the corresponding subtrees have at least $t$ vertices in common
in $T$. The class of graphs that have an $(h,s,t)$-representation
is denoted [h,s,t]. When there is no restriction on the
degree of $T$ or on the degree of the subtrees, we use $h=\infty$
and $s=\infty$ respectively. Notice that $[\infty,\infty,1]$ is
the class of chordal graphs; $[2,2,1]$ is the class of interval
graphs; $[\infty,2,1]$ and $[\infty,2,2]$ are the well known $VPT$
and $EPT$ graphs [14].
In [3], the minimum $t$ such that a given graph belongs
to $[3,3,t]$ is studied. In [9], $[4,4,2]$ graphs are
characterized and a polynomial time algorithm for their
recognition is given. In [8], the class $[4,2,2]$ is
studied. In [6], different aspects of $[\infty,2,t]$
graphs are considered. The relation between the different classes
is analyzed in [7]. In [5], it is shown that the
problem of recognizing $VPT$ graphs is polynomial times solvable,
but the recognition of $EPT$ graphs is an NP-complete problem.
In this work we focuses in the classes $[h,2,1]$, all them are
subclasses of $VPT$. The problem is deciding whether a given
$VPT$ graph can be represented as intersection of paths in a tree
with maximum degree $h$. Since $[2,2,1]=Interval$ and
$[3,2,1]=[3,2,2]=EPT\cap chordal$ [5], we consider $h\geq 4$. We characterize $[h,2,1]$ graphs using chromatic number. We
show that the problem of deciding whether a given $VPT$ graph
belongs to $[h,2,1]$ is NP-complete, while the problem of deciding
whether the graph belongs to $[h,2,1]-[h-1,2,1]$ is NP-hard. Both
problems remain hard even when restricted to $Split\cap VPT$.
Additionally, we present a non-trivial subclass of $Split\cap VPT$ in which these problems are polynomial time solvable. In
Section 2, we provide basics definitions and
known results. In Section 3, we characterize
$[h,2,1]$ graphs for $h\geq 3$. In Section 4, we
present the results about time complexity. Finally, in Section
5 we present some open questions.
2 Preliminaries
In this paper, all graphs are connected,
finite, simple and loopless. Let $G$ be a graph with vertex set
$V(G)$ and edge set $E(G)$, the open neighborhood
$\mathbf{N_{G}(v)}$ of a vertex $v$ is the set of all vertices
adjacent to $v$. The closed neighborhood
$\mathbf{N_{G}[v]}$ is $N_{G}(v)\cup\{v\}$. The degree of
$v$, denoted by $\mathbf{d_{G}(v)}$, is the cardinality of $N_{G}(v)$.
For simplicity, when no confusion arise, we omit the subindex $G$
and simply write $N(v)$, $N[v]$ or $d(v)$.
For $S\subseteq V(G)$, $\mathbf{G[S]}$ is the subgraph of $G$
induced by $S$;
$\mathbf{G-S}$ is a shorthand for $G[V(G)-S]$; and $\mathbf{G-v}$ is used for
$G-\{v\}$ .
A complete set is a subset of vertices inducing a
complete subgraph. A clique is a maximal complete set. The set of cliques of $G$ is denoted by $\mathbf{\mathcal{C}(G)}$.
A stable set is a subset of vertices pairwise
non-adjacent.
The graph $G$ is split if $V(G)$ can be partitioned into
a stable set $S$ and a clique $K$ [1]. The pair
$\mathbf{(S,K)}$ is the split partition of $G$. The
vertices in $S$ are called stable vertices, and $K$ is
called the central clique of $G$. We say that a stable
vertex $s\in S$ is dominated if there exists $s^{\prime}\in S$
such that $N(s)\subseteq N(s^{\prime})$. Notice that if $G$ is split then
$\mathcal{C}(G)=\{K,N[s]\mbox{ for }s\in S\}$.
A VPT-representation of $G$, denoted by $\langle\mathcal{P},T\rangle$, is an $(\infty,2,1)$- representation.
This means that $\mathcal{P}$ is a family $(P_{v})_{v\in V(G)}$ of
subpaths of a host tree $T$ satisfying that two vertices $v$ and
$v^{\prime}$ of $G$ are adjacent if and only if $P_{v}$ and $P_{v^{\prime}}$ have at
least one vertex in common. If $q$ is a vertex of the host tree
$T$, then $\mathbf{P[q]}$ denote the set $\{P\in\mathcal{P}/q\in V(P)\}$ and $\mathbf{C_{q}}$ denote the complete set $\{v\in V(G)/q\in V(P_{v})\}$. Notice that for every clique $C$ of $G$,
there exists $q\in V(T)$ such that $C=C_{q}$.
Definition 2.1
[5] Let $C\in\mathcal{C}(G)$. The branch
graph of $G$ for the clique $C$ denoted by $\mathbf{B(G/C)}$ is
defined as follows: the vertex set $V(B(G/C))$ contains the
vertices of $V(G)\setminus C$ adjacent to some vertex of $C$. Two
vertices $v$ and $w$ are adjacent in $B(G/C)$ if and only if
1.
$vw\notin E(G)$;
2.
there exists a vertex of $C$ adjacent to both $v$ and $w$;
and
3.
there exist vertices $v^{\prime}$ and $w^{\prime}$ of $C$ such that $v^{\prime}$ is
adjacent to $v$ and non-adjacent to $w$, and $w^{\prime}$ is adjacent to
$w$ and non-adjacent to $v$.
Let $q\in V(T)$, with $N_{T}(q)=\{y_{1},y_{2},..,y_{h}\}$. We call
branches of T at q to the connected components of
$V(T)$ - $\{q\}$. Observe that each $y_{i}$ is contained in a
different branch which will be called $T_{i}$.
The graph $G$ is k-colorable if its vertices can be
colored with at most $k$ colors in such a way that no two
adjacent vertices share the same color. The chromatic
number of $G$, denoted by $\chi(G)$, is the smallest number of
colors needed to coloring $G$.
Theorem 2.1
[13]
Let $G$ be a graph and $k\geq 3$. Deciding whether $G$ is
$k$-colorable is an NP-complete problem.
A graph $G$ is perfect if and only if $G$ is
$\{$$C_{2n+1}$, $\bar{C}_{2n+1}$, with $n\geq 2$ $\}$- free
[2].
Theorem 2.2
[10] Let $G$ be a perfect
graph and $k\geq 3$. Deciding whether $G$ is $k$-colorable is a
polynomial time solvable problem.
3 Characterization of [h,2,1], for $h\geq 3$
In this section we present a characterization of the $VPT$ graphs
that can be represented in a tree with maximum degree at most $h$.
The characterization is given in terms of the chromatic number of
the branch graphs. The following three lemmas are fundamental
tools in the proof of the main Theorems 3.1 and
3.2.
Lemma 3.1
Let $\langle\mathcal{P},T\rangle$ be a $VPT$
representation of $G$. Let $C\in\mathcal{C}(G)$ and $q\in V(T)$
such that $C=C_{q}$. If $v\in V(B(G/C))$ then $P_{v}$ is contained
in some branch of $T$ at $q$. If $v$ is adjacent to $w$ in
$B(G/C)$ then $P_{v}$ and $P_{w}$ are not contained in a same branch
of $T$ at $q$.
Proof:
If $v\in V(B(G/C))$ then $v\notin C$. It follows that $q\notin V(P_{v})$, thus $P_{v}$ is contained in some branch $T_{i}$ of $T$ at
$q$. Let $w\in N_{B(G/C)}(v)$ and assume for a contradiction that
$P_{v}$ and $P_{w}$ are contained in the same branch $T_{i}$. Let
$x$ and $y$ be the vertices of $P_{v}$ and $P_{w}$ respectively at
minimum distance from $q$. Since there exists a vertex of $C$
adjacent to $v$ and $w$, there exists a path in $T$ containing
$q$, $x$ and $y$. We can assume, without loss of generality, that
$x$ is between $q$ and $y$ or that $x=y$. In both cases, $N(w)\cap C\subseteq N(v)\cap C$. This contradicts the fact that $v$ and
$w$ are adjacent in the branch graph.
$\Box$
Lemma 3.2
Let $\langle\mathcal{P},T\rangle$ be a $VPT$
representation of $G$. Let $C\in\mathcal{C}(G)$ and $q\in V(T)$
such that $C=C_{q}$. If $d_{T}(q)=h$, then $B(G/C)$ is
$h$-colorable.
Proof:
Let $T_{1},T_{2},..,T_{h}$ be the branches
of T at q. By Lemma 3.1, if we color each vertex $v$ of
$B(G/C)$ with the index $i$ of the branch $T_{i}$ containing $P_{v}$,
then we obtain a proper coloring of $B(G/C)$. Since there are $h$
branches, $B(G/C)$ is $h$-colorable.
$\Box$
Lemma 3.3
Let $\langle\mathcal{P},T\rangle$ be a $VPT$
representation of $G$. Consider $q\in V(T)$ with $d_{T}(q)=h\geq 4$. Assume there exist $y_{1},y_{2}\in N_{T}(q)$ such that for all
$v\in V(G)$, $\{y_{1},y_{2}\}\nsubseteq V(P_{v})$. Then there
exists a $VPT$ representation $\langle\mathcal{P}^{\prime},T^{\prime}\rangle$ of
$G$ with $V(T^{\prime})=V(T)\cup\{a_{q}\}$, $a_{q}\notin V(T)$, and
$$d_{T^{\prime}}(x)=\left\{\begin{array}[]{ll}3,&\mbox{if\ }x=a_{q}\\
\ h-1,&\mbox{if\ }x=q\\
\ d_{T}(x),&\mbox{if\ }x\in V(T^{\prime})\setminus\{q,a_{q}\}.\end{array}\right.$$
Proof:
We obtain the $\langle\mathcal{P}^{\prime},T^{\prime}\rangle$ representation of
$G$ as follows (Please refer to Figure 1): the set of
vertices of
$T^{\prime}$ is $V(T)\cup\{a_{q}\}$, where $a_{q}$ is a new vertex not in $V(T)$. The set
of edges is $(E(T)\setminus\{y_{1}q,y_{2}q\})\cup\{y_{1}a_{q},y_{2}a_{q},qa_{q}\}$. Observe that the degree of
each vertex $x\in V(T^{\prime})$ is the required in the statement of the
present lemma.
Now we define the paths $P^{\prime}_{v}$ for $v\in V(G)$: if $y_{1}$ and
$q$ or $y_{2}$ and $q$ belong to $V(P_{v})$ then
$V(P^{\prime}_{v})=V(P_{v})\cup\{a_{q}\}$. In any other case,
$V(P^{\prime}_{v})=V(P_{v})$. Since $\{y_{1},q,y_{2}\}\subsetneqq V(P_{v})$, we have that each $V(P^{\prime}_{v})$ induces a path in $T^{\prime}$.
Moreover, since all the paths where vertex $a_{q}$ was added had
vertex $q$ in common, it is clear that, for any pair of vertices
$v,w\in V(G)$, $V(P_{v})\cap V(P_{w})\neq\emptyset$ if and only
if $V(P^{\prime}_{v})\cap V(P^{\prime}_{w})\neq\emptyset$. It follows that
$\langle\mathcal{P}^{\prime},T^{\prime}\rangle$ is a $VPT$ representation of $G$
and the implication is proven.
$\Box$
Theorem 3.1
Let $G\in VPT$ and $h\geq 3$. The graph $G$
belongs to $[h,2,1]$ if and only if $B(G/C)$ is $h$-colorable
for every $C\in\mathcal{C}(G)$. The direct implication is true
also for $h=2$.
Proof:
Let $\langle\mathcal{P},T\rangle$ be an $(h,2,1)$-representation
of $G$ with $h\geq 2$. Assume $C\in\mathcal{C}(G)$, then there
exists $q\in V(T)$ such that $C=C_{q}$. Since $d_{T}(q)\leq h$, by
Lemma 3.2, $B(G/C)$ is $h$-colorable.
The reciprocal implication for $h=3$ was proven by Golumbic and
Jamison in [5]; then we assume $h\geq 4$.
Let $\langle\mathcal{P},T\rangle$ be a $VPT$ representation of $G$. We will prove that $G$
admits an $(h,2,1)$-representation.
We proceed by induction on the number $k$ of vertices
of $T$ whose degree exceeds $h$. If $k=0$ we are done.
If $k>0$, there exists a vertex $q$ of $T$ with degree $d>h$.
Say $N_{T}(q)=\{y_{1},y_{2},...,y_{d}\}$ and for every $i$, $1\leq i\leq d$, let $T_{i}$ be the branch of $T$ at $q$ containing the
vertex $y_{i}$.
If by repeatedly applying the Lemma 3.3 we can obtain a
$VPT$ representation $\langle P^{\prime},T^{\prime}\rangle$ of $G$ with
$d_{T^{\prime}}(q)<h$, then the implication is proven by induction since
no vertex of $T^{\prime}$ increases its degree.
In other case, we can assume that for any pair of vertices
$y_{i}$, $y_{i^{\prime}}$ belonging to $N_{T}(q)$, there exists at least
one $v\in V(G)$ such that $\{y_{i},y_{i^{\prime}}\}\subseteq V(P_{v})$.
Notice that this implies that $C_{q}=$$\{v\in V(G)/q\in V(P_{v})\}$ is a clique of $G$.
We will consider two cases.
Case 1: for every $i$, $1\leq i\leq d$, there exists $v_{i}\in V(G)$ such that $P_{v_{i}}$ is totally contained in the branch
$T_{i}$ and $y_{i}\in V(P_{v_{i}})$. Observe that each $v_{i}$ must
be a vertex of $B(G/C_{q})$. Since $B(G/C_{q})$ is $h$-colorable,
we can partitioned the set $\{y_{1},y_{2},...,y_{d}\}$ in $h$
subsets $D_{j}$, $1\leq j\leq h$, each one containing the vertices
$y_{i}$ for which the associated vertex $v_{i}$ has color $j$.
We obtain a new $VPT$ representation $\langle P^{\prime},T^{\prime}\rangle$ of
$G$ as follows. The tree $T^{\prime}$ is obtained from $T$ by means of the
following procedure (in Figure 2 we offer an example):
1) remove the edges $qy_{i}$, $1\leq i\leq d$; 2) add $h$ new
vertices $\mu_{j}$, $1\leq j\leq h$; 3) add the edges $q\mu_{j}$,
$1\leq j\leq h$; and finally, to connect the vertices $\mu_{j}$
with the vertices $y_{i}$, 4) add for every $j$, $1\leq j\leq h$,
a binary tree rooted at the vertex $\mu_{j}$ and with the
vertices of $D_{j}$ as leaves. The rest of the tree T remains
unchanged.
The only paths of $\mathcal{P}$ which are modified to obtain the
paths of $\mathcal{P^{\prime}}$ are those $Q\in P[q]$. If $Q$ has $q$ as
an endpoint, then we obtain $Q^{\prime}$ by replacing in $Q$ the edge
$qy_{i}$ by the unique subpath of $T^{\prime}$ linking $q$ and $y_{i}$. If
$Q$ has $q$ as an internal vertex, then there exist $i$ and $i^{\prime}$
such that $Q$ contains the edges $qy_{i}$ and $qy_{i^{\prime}}$. Notice
that the existence of $Q$ implies that $v_{i}$ and $v_{i^{\prime}}$ are
adjacent in $B(G/C_{q})$; thus they have different colors, say $j$
and $j^{\prime}$. Therefore, we obtain $Q^{\prime}$ by replacing in $Q$ the edges
$qy_{i}$ and $qy_{i^{\prime}}$ by the only subpath of $T^{\prime}$ linking
$y_{i}$, $q$ and $y_{i^{\prime}}$.
It is easy to see that this construction leaves the intersection
graph of paths unchanged while reducing the number of tree
vertices of degree greater than h. So, by induction, the
implication is proven.
Case 2: there exists $i$, $1\leq i\leq d$, such that every path
$P\in\mathcal{P}$ containing $y_{i}$ is not contained in the
branch $T_{i}$. Thus, every path $P\in\mathcal{P}$ containing
$y_{i}$ contains also $q$. Therefore, we can contract the edge
$qy_{i}$ to obtain a new $VPT$ representation of $G$ and repeat
this as many times as needed to get a representation which is in
Case 1. Notice that in this procedure some vertices of $T$
disappear, and that the degree of $q$ may increase, but the
number of vertices whose degree exceeds $k$ does not grow, thus
the proof follows by induction as in the previous case.
$\Box$
Observe that the reciprocal implication of Theorem
3.1 is false for $h=2$; consider, by instance, the
graph $T^{3}_{2}$ which consists of one central vertex and 3 edge
disjoint paths of 2 edges each intersecting only on the central
vertex. It is easy to see that $T^{3}_{2}\in VPT$ and $B(G/C)$ is
$2$-colorable for all $C\in\mathcal{C}(G)$, but $T^{3}_{2}\notin[2,2,1]$.
Theorem 3.2
Let $G\in VPT$ and $h\geq 4$. The graph $G$ belongs to
$[h,2,1]-[h-1,2,1]$ if and only if $Max_{C\in\mathcal{C}(G)}(\chi(B(G/C)))=h$. The reciprocal implication is
also true for $h=3$.
Proof:
By Theorem 3.1, $G\in[h,2,1]$ if and only if
$Max_{C\in\mathcal{C}(G)}(\chi(B(G/C)))\leq h$. On the other
hand, by the same Theorem 3.1, $G\notin[h-1,2,1]$ if and only if $Max_{C\in\mathcal{C}(G)}(\chi(B(G/C)))>h-1$. Therefore, the proof follows.
$\Box$
4 Complexity
In this Section we prove that the problem
of deciding whether a given graph belongs to $[h,2,1]$ for $h\geq 3$ is NP-complete. We also show that recognizing
$[h,2,1]-[h-1,2,1]$ for $h\geq 4$ is NP-hard. Our results prove
that both problems remain difficult even when restricted to the
class $VPT\cap Split$ without dominated stable vertices.
First we state the following fundamental property of $VPT\cap Split$ graphs which is used in the proof of Theorems 4.1
and 4.2.
Lemma 4.1
Let $s$ be a stable vertex of a $VPT\cap Split$ graph
$G$. The branch graph $B(G/N[s])$ is $1$-colorable.
Proof:
Let $\langle P,T\rangle$ be a $VPT$ representation of $G$ such
that $P_{s}$ is a one vertex path in a leaf $y$ of $T$, in other
words $V(P_{s})=\{y\}$ where $y$ is a leaf of $T$. Thus $N[s]$ is
the clique $C_{y}$. Since $d_{T}(y)=1$, by Lemma 3.2,
$B(G/N[s])$ is $1$-colorable.
$\Box$
For the NP-completeness proof, we use a reduction from the
chromatic number problem [13].
Given a graph $G$ we construct in polynomial time
a graph $\widehat{G}\in VPT\cap Split$ without dominated stable vertices,
in such a way that $\chi(G)=h$
if and only if $\widehat{G}\in[h,2,1]-[h-1,2,1]$.
Let $V(G)=\{v_{1},v_{2},...,v_{n}\}$, we define the graph
$\widehat{G}$ by means of its $VPT$ representation $\langle\mathcal{P},T\rangle$ as follows: the tree $T$ is a star with
central vertex $q$ and leaves $y_{i}$ for $1\leq i\leq n$.
The path family $\mathcal{P}$ contains: a one vertex path $P_{i}$
with $V(P_{i})=\{y_{i}\}$, for each $1\leq i\leq n$; a path
$P_{ij}$ with $V(P_{ij})=\{y_{i},q,y_{j}\}$, for each $1\leq i<j\leq n$ such that $v_{i}v_{j}\in E(G)$; a path $P_{iq}$ with
$V(P_{iq})=\{q,y_{i}\}$, for each $1\leq i\leq n$ such that
$d_{G}(v_{i})=1$.
We call each vertex of $\widehat{G}$ as the corresponding path of
$\mathcal{P}$.
In Figure 3 we offer an example of a graph $G$, the
$VPT$ representation of $\widehat{G}$ and the graph $\widehat{G}$
obtained.
Notice that $\widehat{G}$ is a split graph with the vertex set
partitioned in a stable set of size $n=\mid V(G)\mid$
corresponding to the one vertex paths $P_{i}$; and a central clique
of size $\mid E(G)\mid+\mid\{v\in V(G)/d_{G}(v)=1\}\mid$
corresponding to the remaining paths, all of which contain the
vertex $q$ of $T$, thus this clique is $C_{q}$. The other cliques of
$\widehat{G}$ are the cliques $C_{y_{i}}$ for $1\leq i\leq n$ each
one corresponding to the paths containing the vertex $y_{i}$ of $T$
respectively. The graph $\widehat{G}$ has no more cliques. In
addition, every stable vertex $P_{i}$ of $\widehat{G}$ is
non-dominated.
Observe that the branch graphs $B(\widehat{G}/C_{y_{i}})$ are
described in Lemma 4.1, the following claim does for
$B(\widehat{G}/C_{q})$.
Claim 4.1
If $\widehat{G}$ is the graph obtained from $G$ as
above, then $B(\widehat{G}/C_{q})=G$.
Proof:
Notice that $B(\widehat{G}/C_{q})$ has exactly $n$ vertices: $P_{i}$
for $1\leq i\leq n$.
We will see that $P_{i}$ and $P_{j}$ are adjacent in
$B(\widehat{G}/C_{q})$ if and only if $v_{i}$ and $v_{j}$ are adjacent
in $G$. If $P_{i}P_{j}\in E(B(\widehat{G}/C_{q}))$ then there
exists a vertex of $C_{q}$ adjacent to both $P_{i}$ and $P_{j}$.
Then, there exists a path $P_{ij}\in\mathcal{P}$, thus
$v_{i}v_{j}\in E(G)$. Reciprocally, let $v_{i}v_{j}\in E(G)$.
Notice that $P_{i}$ and $P_{j}$ are non-adjacent in $\widehat{G}$; and
$P_{ij}$ is a vertex of $C_{q}$ adjacent to $P_{i}$ and to $P_{j}$ in
$\widehat{G}$. Let us see that there exists a vertex of $C_{q}$
adjacent to $P_{i}$ and non-adjacent to $P_{j}$. Indeed, if
$d_{G}(v_{i})=1$ then the wanted vertex of $C_{q}$ is $P_{iq}$. If
$d_{G}(v_{i})>1$ then $v_{i}$ must have a neighbor $v_{l}$ with $l\neq j$,
thus the wanted vertex of $C_{q}$ is $P_{il}$. In an analogous
way, there exists a vertex of $C_{q}$ adjacent to $P_{j}$ and
non-adjacent to $P_{i}$. We have proved that $P_{i}$ and $P_{j}$ are
adjacent in $B(\widehat{G}/C_{q})$. We conclude that
$B(\widehat{G}/C_{q})=G$.
$\Box$
The reduction from chromatic number is complete using the next
claim.
Claim 4.2
Let $\widehat{G}$ be the graph obtained from $G$ as
above and $h\geq 4$. The graph $\widehat{G}$ belongs to
$[h,2,1]-[h-1,2,1]$ if and only if
$\chi(G)=h$.
Proof:
By Lemma 4.1 and Claim 4.1, $Max_{C\in\mathcal{C}(\widehat{G})}\chi(B(\widehat{G}/C))=\chi(B(\widehat{G}/C_%
{q}))=\chi(G)$. Hence, by Theorem
3.2, $\widehat{G}$ belongs to $[h,2,1]-[h-1,2,1]$ if
and only if
$\chi(G)=h$.
$\Box$
We have proved the following theorem.
Theorem 4.1
Let $G\in VPT\cap Split$ without dominated stable
vertices, and $h\geq 4$. Decide whether $G\in[h,2,1]-[h-1,2,1]$
is an NP-hard problem.
In addition, since an $(h,2,1)$-representation is a polynomial
certificate of belonging to $[h,2,1]$; using Theorem
3.1 and the construction above, we have proved the
following result.
Theorem 4.2
Let $G\in VPT\cap Split$ without dominated
stable vertices, and $h\geq 3$. Decide whether $G\in[h,2,1]$ is
an NP-complete problem.
We notice that Theorem 4.2 for $h=3$ has been
previously proved in [5].
4.1 A polynomial time solvable subclass
We have proved that deciding whether a given $VPT\cap Split$ graph
without dominated stable vertices admits a representation as
intersection of paths of a tree with maximum degree $h$ is an
NP-complete problem. In what follows we describe a non-trivial
subclass of $VPT\cap Split$ without dominated stable vertices
where the problem is polynomial time solvable.
For $n\geq 4$, a n-sun, denoted by
S${}_{\textbf{n}}$, is a split graph with stable set
$\{s_{1},s_{2},..,s_{n}\}$, central clique
$\{v_{1},v_{2},..,v_{n}\}$, $N(s_{i})=\{v_{i},v_{i+1}\}$ for $1\leq i\leq n-1$, and $N(s_{n})=\{v_{n},v_{1}\}$. See Figure
4.
Let $G$ be a split graph with partition $(S,K)$.
We say that $G$ belongs to $SVS$ (special $VPT$ subclass)
whenever
•
$G\in VPT$,
•
for all $v\in K$, $|N(v)\cap S|\leq 2$, and
•
if $S_{k}$, with $k\in\{4,2n+1$ for $n\geq 2\}$, is induced in $G$
then there exists $v\in K$ such that
$v$ is adjacent to two non-consecutive vertices of the stable set of $S_{k}$.
The class $SVS$ is not trivial, in the sense that it includes
graphs in $[h,2,1]$ for all $h\geq 4$.
For example, let $n\geq 4$ and let $A_{n}$ (see [7]) be a
split graph with partition $(S,K)$, where $S=\{s_{1},..,s_{n}\}$,
$K=\{v_{ij},1\leq i<j\leq n\}$ and $N(v_{ij})=\{s_{i},s_{j}\}$,
for all $1\leq i<j\leq n$. It is clear that $A_{n}$ belongs to
$SVS$, and $B(A_{n}/K)=K_{n}$ with $V(K_{n})=\{s_{1},..,s_{n}\}$.
Hence, by Theorem 3.2, $A_{n}\in[n,2,1]-[n-1,2,1]$.
(As an example see Figure 5).
The following two lemmas are used in the proof of the main Theorem
4.3 which proves that in the class $SVS$ the graphs
belonging to $[h,2,1]$ can be recognized efficiently.
Lemma 4.2
Let $G\in VPT\cap Split$ with partition $(S,K)$ such that
for all $v\in K$, $|N(v)\cap S|\leq 2$, and let $n\geq 4$. If
$B(G/K)$ has an induced $C_{n}$ then $G$ has an induced $S_{n}$.
Proof:
Let $\langle\mathcal{P},T\rangle$ be a $VPT$ representation of
$G$ and $q\in V(T)$ such that $K=C_{q}$. Let $C_{n}$ be an induced
cycle of $B(G/K)$ with vertices $s_{1}$, $s_{2}$,…,$s_{n}$. It is
clear that every $s_{i}\in S$. Since $s_{i}$ is adjacent to $s_{i+1}$
in $B(G/K)$, there exists $v_{i}\in K$ such that $v_{i}$ is
adjacent to $s_{i}$ and to $s_{i+1}$ in $G$. Since, for all $v\in K$, $|N(v)\cap S|\leq 2$, if $i\neq i^{\prime}$ then $v_{i}\neq v_{i^{\prime}}$,
thus $s_{1}$, $s_{2}$,…,$s_{n}$, $v_{1}$, $v_{2}$,…,$v_{n}$ induce a
$n$-sun in $G$ and the proof is completed.
$\Box$
Lemma 4.3
If $G\in SVS$ then every branch graph of $G$ is
perfect.
Proof:
Let $(S,K)$ be a split partition of $G$. By Lemma 4.1,
if $s\in S$ then $B(G/N[s])$ is perfect. Assume for a
contradiction that $B(G/K)$ is not perfect, then $B(G/K)$ has
induced an odd cycle or the complement of an odd cycle. Since the
complement of $C_{5}$ is $C_{5}$; and the complement of any odd cycle
of size 7 or more has an induced $C_{4}$, it follows that $B(G/K)$
has an induced $C_{k}$, for some $k\in\{4,2n+1$ for $n\geq 2\}$.
Therefore, by Lemma 4.2, $G$ has an induced $S_{k}$. Since
$G\in SVS$, there exists $v\in K$ such that $v$ is adjacent to two
non-consecutive vertices $s$ and $s^{\prime}$ of the stable set of
$S_{k}$. Notice that the existence of $v$ implies that the
vertices $s$ and $s^{\prime}$ are adjacent in $B(G/K)$. This contradicts
the fact that $C_{k}$ is an induced cycle of $B(G/K)$. $\Box$
Theorem 4.3
Let $G\in SVS$ and $h\geq 4$. Decide whether $G$
belongs to $[h,2,1]-[h-1,2,1]$ is polynomial time solvable.
Proof:
Given $G\in SVS$, in order to determinate if $G\in[h,2,1]-[h-1,2,1]$, by Theorem 3.1, it is enough to
calculate the chromatic number of $B(G/K)$, where $K$ is the
central clique of $G$. Notice that the branch graph $B(G/K)$ can
be obtained in polynomial time. On the other hand, by Lemma
4.3, $B(G/K)$ is perfect. Thus, by Theorem 2.2,
its chromatic number can be calculated in polynomial time.
$\Box$
5 Future work
In this paper we give a characterization of
the $[h,2,1]$ graphs, with $h\geq 3$. In addition, we prove that
recognizing this class is NP-complete and show a family, called
$SVS$, in which this problem is polynomial time solvable. We are
working in describing a larger subclass of $VPT$ graphs where this
problem remains polynomial. On the other hand, we are analyzing
the possibility of extending the techniques used in the present
paper to characterize the classes $[h,2,2]$.
References
[1]
A. Brandstädt, V. B. Le, J. P. Spinrad, Graph Classes: A
Survey, SIAM Monographs on Discrete Mathematics and Applications.
(1999).
[2]
M. Chudnovsky, N. Robertson, P. Seymour, R. Thomas,
The strong perfect graph theorem, Ann. of Math. 164 (2006)
51-229.
[3]
N. Eaton, Z. Füredi, A. V. Kostochka, J. Skokan,
Tree representations of graphs, Europen Journal of
Combinatorics. 28 (2007) 1087-1098.
[4]
F. Gavril, The intersection graphs of
subtrees in a tree are exactly the chordal graphs, Journal
of Combinatorial Theory. 16 (1974) 47-56.
[5]
M. C. Golumbic, R. E. Jamison, Edge and
vertex intersection of paths in a tree, Discrete Mathematics. 38 (1985) 151-159.
[6]
M. C. Golumbic, M. Lipshteyn, M. Stern,
The k-edge intersection graphs of paths in a tree, Discrete
Applied Mathematics. 156 (2008) 451-461.
[7]
M. C. Golumbic, M. Lipshteyn, M. Stern,
Equivalences and the complete hierarchy of intersection
graphs of paths in a tree, Discrete Applied Mathematics. 156
(2008) 3203-3215.
[8]
M. C. Golumbic, M. Lipshteyn, M. Stern,
Representing edge intersection graphs of paths on degree 4 trees,
Discrete Mathematics. 308 (2008) 1381-1387.
[9]
M. C. Golumbic, M. Lipshteyn, M. Stern,
Intersection models of weakly chordal graphs,
Discrete Applied Mathematics. 157 (2009) 2031-2047.
[10]
M. Grötschel, L. Lovász, A. Schrijver, The ellipsoid method and its
consequences in combinatorial optimization, Combinatorica 1.
(1981) 169-197.
[11]
R. E. Jamison, H. M. Mulder,
Tolerance intersection graphs on binary trees with constant tolerance
3, Discrete Mathematics. 215 (2000) 115-131.
[12]
R. E. Jamison, H. M. Mulder, Constant tolerance
intersection graphs of subtress of a tree, Discrete Mathematics. 290 (2005) 27-46.
[13]
R. Karp, Reducibility among combinatorial problems, In: R. E. Miller
and J. W. Thatcher, editors, Complexity of Computer Computations.
(1972) 85-104.
[14]
C. L. Monma, V. K. Wei, Intersection graphs
of paths in a tree, Journal of Combinatorial Theory. (1986)
140-181. |
Resonant cancellation of off-resonant effects in
a multilevel qubit
Lin Tian${}^{1}$
Seth Lloyd${}^{2}$
${}^{1}$Department of Physics, Massachusetts Institute of
Technology, Cambridge, MA, 02139
${}^{2}$d’Arbeloff Laboratory for Information Systems and
Technology,
Department of Mechanical Engineering, Massachusetts Institute of
Technology, Cambridge, MA, 02139
(November 26, 2020)
Abstract
Off-resonant effects are a significant source of error
in quantum computation. This paper presents a
group theoretic proof that off-resonant transitions to
the higher levels of a multilevel qubit can be completely
prevented in principle. This result can be generalized to
prevent unwanted transitions due to qubit-qubit interactions.
A simple scheme exploiting dynamic pulse control techniques
is presented that can cancel transitions to higher states
to arbitrary accuracy.
pacs: PACS number(s): 03.67.Lx, 03.65.Bz, 89.70.+c
Successful quantum computation depends on the accurate manipulation of
the quantum states of the qubits[1].
In practice, qubits are subject to many sources of
quantum errors including thermal fluctuations of
the environment[2],
qubit-qubit interactions[3],
and intrinsic redundant degrees of freedom
within a qubit such as the quasiparticle
conduction in the superconducting
qubits[4, 5],
and the effect of the higher levels in many practical
qubit designs[5, 6].
This paper proposes a dynamic pulse control technique that
efficiently eliminates unwanted off-resonance transitions.
Various schemes to protect the qubit from qubit errors have been
proposed that can be divided into two categories. The first one is
the quantum error correction technique
[7, 8, 9, 10, 11, 12]
where the qubit state is encoded by
redundant qubits. Different errors project
the qubit-extra-qubit system into different subspaces
that can be determined by measuring the state of the extra qubits.
By applying a transformation according to the measurement, the
correct qubit state can be restored. This approach relies
on large numbers of extra qubits to keep the errors from propagating.
The second approach exploits ‘bang-bang’
control techniques[13]
where the dynamics of the qubit and the environment
is manipulated by fast pulses that flip the qubit.
With the influence of the environment averaged out,
the qubit evolves in the error-free subspace.
This method relies on the ability to apply pulses rapidly
compared with the correlation time of the environment.
This is an open loop control method.
A particularly important form of intrinsic qubit error comes from
the off-resonant transitions to the
higher levels of a qubit when the qubit is being operated.
Real qubits are not $S=1/2$ spins that are perfect
two level systems; redundant levels always exist
that affect the information
content of the qubit.
The additional interaction that is introduced
to achieve qubit operation by
coupling the lowest two states of the qubit
almost always includes unwanted couplings
between the lowest two states and the higher levels.
When the interaction is applied
with frequency
$\omega=\omega_{2}-\omega_{1}$,
resonant transition occurs between the lowest two states;
meanwhile
off-resonant transitions to the higher states are also switched on.
These transitions deviate the phase and amplitude of the qubit state
from perfect Rabi oscillation.
Numerical simulations on the superconducting
persistent current qubit (pc-qubit)[4, 14]
show that this deviation
can be severe when
the unwanted couplings
are of the same order as the Rabi frequency.
In this letter, we study the effect of the higher levels
on qubit dynamics during qubit operation
by a group theoretic approach. We prove
that the errors can be completely avoided by applying
a time varying operation Hamiltonian.
Then we generalize this result
to the qubit-qubit interaction problem which
can be mapped exactly onto the first one.
Using the idea of
dynamic pulse control[13],
we design a pulse sequence
that cancels the leakage to the higher levels
to arbitrary accuracy with $O(N)$ number of pulses,
$N$ being the number of higher levels.
The leakage to higher levels has two significant characteristics.
First, unlike the environmental fluctuations that affect the
qubit only slightly (less than $10^{-4}$) within one operation,
the leakage changes the qubit dynamics on a time scale $1/\omega_{0}$ that
is much shorter than the qubit operation time (about $1/\omega_{Rabi}$).
Conventional quantum error correction corrects errors that
occur with small probability and is not a suitable strategy to
cancel the fast off-resonant transitions.
Neither can we use the bang-bang method to average
out[15]
the leakage simply by manipulating the
lowest two states,
as the pulse induces these unwanted transitions at the same time.
Second, ignoring all interactions with external
variables, the leakage is coherent, although
the coherent oscillation will collapse since the revival time
is too long to be observed due to the large number of
transitions of different frequencies[16].
As will now be shown, the coherent nature of the leakage
implies that this type of error can be corrected by applying
a control sequence that coherently modifies the qubit dynamics.
Consider a $N$ level quantum system with the Hamiltonian ${\cal H}_{0}$,
the lowest two states of which are chosen as the qubit states
$|\uparrow\rangle$ and $|\downarrow\rangle$.
The unitary transformations on this $N$ dimensional
Hilbert space form the $N^{2}$ dimensional compact
Lie group ${\rm U}(N)$. Without
other interaction, the trajectory of the qubit follows
the Abelian subgroup $\{e^{-i{\cal H}_{0}t},t\in R\}$.
Now apply to the qubit the perturbation ${\cal H}_{I}$,
$[{\cal H}_{0},{\cal H}_{I}]\neq 0$,
to induce a desired transformation of the qubit.
In most physical systems, unwanted
transitions to the higher levels are simultaneously
induced. For example,
in the pc-qubit[4] operation,
$({\cal H_{I}})_{mn}=2\pi\delta f\langle m|\sin{(2\phi_{m}+2\pi f)}|n\rangle%
\cos{\omega t}$,
when the bias flux is modulated with rf components of
amplitude $\delta f$ and frequency $\omega$.
This perturbation has couplings between all the levels.
By successive commutation of
${\cal H}_{0}$, ${\cal H}_{I}$, and their commutators
until no independent operator appears,
a Lie algebra ${\cal A}_{I}$ is created.
In almost all cases, ${\cal A}_{\rm I}={\rm u}(N)$[17],
${\rm u}(N)$ being the Lie algebra of ${\rm U}(N)$. The only exception occurs
in a zero measure subspace of ${\rm u}(N)$
when ${\cal H}_{I}$ and ${\cal H}_{0}$ are both in the same
subalgebra of ${\rm u}(N)$. Thus, with almost
all perturbations, the evolution operator
can be any element in ${\rm U}(N)$; and
transitions to higher levels are unavoidable
with an initial state that only occupies the lowest two levels.
To prevent the transitions to the
higher states at time $t$ means to restrict the evolution
operator ${\cal U}(t)$ to the submanifold of
${\rm U}(2)\oplus{\rm U}(N-2)$, ${\rm U}(2)$
being the unitary group on $\{|\uparrow\rangle,|\downarrow\rangle\}$
and ${\rm U}(N-2)$
on the remaining $N-2$ states.
This applies $4(N-2)$ real domain restrictions on ${\cal U}(t)$:
${\cal U}(t)_{1k},{\cal U}(t)_{2k}=0,k=3\,\ldots\,N$.
In contrast to a perfect qubit operation during
which ${\cal U}(t)$ remains
in the subspace ${\rm U}(2)\oplus{\rm U}(N-2)$ all the time,
the qubit is allowed to
stray away from this subspace if only it
goes back to this subspace at the designated time $t$.
The qubit dynamics can be manipulated by varying the
strength and phase of the perturbation with time.
As the $N^{2}$ dimensional Lie group ${\rm U}(N)$ is compact,
any transformation can be reached at time $t$ by adjusting
the $N^{2}+1$ parameters in the following process[17]:
$${\cal U}(t)=e^{-i\alpha{\cal H}_{I}t_{N^{2}}}e^{-i\alpha{\cal H}_{0}t_{N^{2}-1%
}}\,\cdots\,e^{-i\alpha{\cal H}_{0}t_{1}}$$
(1)
where $\alpha$ is introduced to ensure that $t=\sum t_{i}$.
By playing with the $N^{2}+1$ real parameters,
the $4(N-2)$ real numbers in ${\cal U}(t)_{1k},{\cal U}(t)_{2k}$
can be set to zero so that the state of the qubit stays
in the $\{\uparrow,\downarrow\}$ space without
leakage.
Hence by turning the perturbation on and off $O(N^{2})$ times,
the lowest two states are completely decoupled
from the higher states. $O(N^{2})$ pulses give a sufficient condition
that is required to achieve arbitrary transformation. As will be shown
later in this letter, with proper arrangement,
we can design a pulse sequence of $O(N)$ pulses
to cancel the transitions to the higher levels.
Unlike that in the quantum
Zeno effect[18]
where measurement is used to prevent the system from evolving,
the dynamics in this process is described
completely by unitary evolutions.
As the unwanted transitions are off-resonant
transitions whose amplitudes decrease roughly as
$\gamma_{ij}/\omega_{i}$ ($\omega_{i}$ is the energy of the
$i$th level, $\gamma_{ij}$ the coupling between level $i$ and $j$),
the influence of the levels with $\omega_{i}/\omega_{0}\gg 1$
can be ignored.
In the pc-qubit[4]
the energies of the lower levels increase fast enough
($\omega_{10}>10\omega_{0}$) that levels beyond $|10\rangle$ can
be ignored. The energy of the $i$th level of
the charge state qubit[5]
increases as $i^{2}$; less levels affect the qubit dynamics
than that in the pc-qubit.
Hence the number $N$ of the higher states
that are involved in the qubit dynamics in real designs
can be reasonably small. As a result, the number of pulses
in the previous analysis is also reasonable.
One question to ask is whether there is
any fundamental difference
between the errors due to transitions to
the higher levels
and those due to the fluctuations of the environmental variables.
Putting it in another way: what is the difference between the intra-qubit
coupling in a multilevel qubit and the qubit-external-system coupling?
In the following we will show that
the $N$-level qubit can be mapped into interacting subsystems,
and vice versa.
Let the initial state of a $N$-level qubit
be $|\Psi_{0}\rangle=\alpha_{1}^{(0)}|1\rangle+\alpha_{2}^{(0)}|2\rangle$,
$|1\rangle$ and $|2\rangle$ being the lowest two states.
To map the qubit into two subsystems, we
divide the $N$ states into two subspaces $SP_{1}$ and
$SP_{2}$ by adding the vacuum states $|V_{1}\rangle$ and $|V_{2}\rangle$ to
the respective subspaces as $SP_{1}=\{|V_{1}\rangle,|1\rangle,|2\rangle\}$
and $SP_{2}=\{|V_{2}\rangle,|3\rangle,\,\ldots\,,|N\rangle\}$.
Now the $N$ dimensional
Hilbert space of the original qubit is embedded in
the $3(N-1)$ dimensional direct product space $SP_{1}\otimes SP_{2}$.
The states in the expanded space are
$|\overline{\Psi}\rangle=\sum_{i,j}\beta_{i,j}|b_{i}^{(1)}\rangle|b_{j}^{(2)}\rangle$,
where $b_{i}^{(1)}$ and $b_{j}^{(2)}$ are basis of
the two subspaces respectively.
The initial state is
$|\overline{\Psi}_{0}\rangle=(\alpha_{1}^{(0)}|1\rangle+\alpha_{2}^{(0)}|2%
\rangle)|V_{2}\rangle$ in the expanded form.
The unitary transformations in this expanded space
forms the group ${\rm U}(3(N-1))$.
Perturbation introduces coupling between different states.
When mapped to the expanded space, the perturbation $\overline{{\cal H}}_{I}$
connects states in the $N$ dimensional subspace
spanned by $\{|1\rangle|V_{2}\rangle,|2\rangle|V_{2}\rangle,|V_{1}\rangle|3\rangle,\,%
\ldots\,,|V_{1}\rangle|N\rangle\}$.
So $\overline{{\cal H}}_{I}$ and $\overline{{\cal H}}_{0}$
create
$N^{2}$ dimensional subalgebra ${\rm u}(N)$ in the expanded space.
Under the perturbation, the wave function in the expanded space
can be described as
$|\overline{\Psi}\rangle=(\alpha_{1}|1\rangle+\alpha_{2}|2\rangle)|V_{2}\rangle%
+\sum_{i=3}^{N}\alpha_{i}|V_{1}\rangle|i\rangle$,
where $\alpha_{i}$ are time dependent parameters
evolving with the perturbation.
From this analysis, the higher levels
in the qubit form an effective environment that interferes
strongly with the lowest two levels. Interaction
strength
is the major difference between this effective environment and
a real one[4].
The couplings between $SP_{1}$ and $SP_{2}$ are strong and
comparable to the Rabi
coupling that realizes qubit operation. In contrast,
the interactions between the environmental oscillators
and the qubit are weak due to the $O(1/\sqrt{V})$ factor
that originates from the normalization of the extented
modes[2]. So
the thermal fluctuations are not enslaved to the qubit dynamics
and can be
treated classically.
The strong interaction with the higher levels also
explains why the error due to leakage
occurs at such a short time that a particular strategy is required to
correct the error.
Another thing to mention is that this effective environment
only comes with qubit operation,
while the real environment affects the qubit all the time.
Hence we worry about the leakage only during qubit operation
and choose to correct the leakage by controlling the operation process.
By reversing the mapping, interacting qubits
can be modeled as one multilevel quantum system.
One example is two interacting qubits with basis
$|b_{i}^{(1)}\rangle,i=1\,\ldots\,N_{1}$, and
$|b_{j}^{(2)}\rangle,j=1\,\ldots\,N_{2}$, respectively.
Labeling the state $|b_{i}^{(1)}\rangle|b_{j}^{(2)}\rangle$
as $|(i-1)N_{2}+j\rangle$, we have
the $N_{1}N_{2}$ basis for the equivalent multilevel qubit.
In the same way, $n$ two-level qubits
form a quantum system of $2^{n}$
levels. Here the number of states grows exponentially with
the number of qubits as the
entanglement between qubits has to be included
in a single system now[19].
Perturbation applied to one of the qubits
can cause unwanted couplings within the
$2^{n}$ levels,
and induce off-resonant transitions
that affect the qubit performance. Similar to the couplings
of the multilevel qubit,
these couplings are also strong and cause fast errors.
Taking the pc-qubit as an example, the
interaction between the two qubits is[4]:
${\cal H}_{2}=J_{z}\sigma_{z}^{(1)}\sigma_{z}^{(2)}+J_{x}(\sigma_{z}^{(1)}%
\sigma_{x}^{(2)}+\sigma_{x}^{(1)}\sigma_{z}^{(2)})$, where
$J_{z}$ and $J_{x}$ terms are due to
the inductance coupling between qubit circuits.
When a $\sigma_{x}^{(1)}$ term is applied to rotate the first qubit,
the second qubit will be involved and qubit dynamics will
be changed.
Although the mapping of the multilevel qubit and the multiqubit system
into each other
is just another way of looking at the same problem, it
shows that the errors due to the qubit-qubit
interactions[3]
can be treated by the same approach as is used in
cancelling the interference of the higher levels.
Again we turn to the idea of coherent pulse control that is exploited
in the higher state problem.
Now the number of pulses increases exponentially with the number
of interacting qubits,
but it doesn’t cause a disaster in real designs where
only the nearest neighbour qubit interactions are important
and $n$ can be made small in the qubit layout geometry.
To illustrate the general idea of exploiting dynamic pulse control
to cancel the errors
due to the higher states,
we give an example of pulse sequence that completely
cancels the transitions
to the higher levels with $O(N)$ pulses.
Let us start from a three level system
with eigenvalues $\omega_{i},i=1,2,3$.
The energy
difference between level $i$ and $j$ is shorthanded as
$\omega_{ij}$.
An interaction ${\cal H}_{I}$
that couples level $i$ and $j$
by $\gamma_{ij}$
is applied to operate the qubit. When the third level is not
present, $\gamma_{12}$ is the Rabi
frequency of the lowest two states.
For simplicity, we ignore the diagonal couplings $\gamma_{ii}$
as $\gamma_{ii}\ll\omega_{i}$.
As will become clear,
the effectiveness of the designed pulse sequence depends on
the condition $|\gamma_{ij}/\omega_{ij}|\ll 1$ which
is satisfied in most qubits.
The Hamiltonian in the interaction picture is
${\cal H}_{int}=e^{i{\cal H}_{0}t}{\cal H}_{I}e^{-i{\cal H}_{0}t}\cos{(\omega t%
+\phi)}$,
$\omega$ being the pulse frequency.
The wave function
$\Psi(t)=[u\,v\,w]^{T}$ evolves according to the equation
$i\frac{\partial\Psi(t)}{\partial t}={\cal H}_{int}\Psi(t)$. When the perturbation is weak,
this equation is integrated order by order as:
$$\begin{array}[]{lcl}\Psi(t)&=&\Psi(0)+\int_{0}^{t}dt^{\prime}{\cal H}_{int}(t^%
{\prime})\Psi(0)\\
&+&\int_{0}^{t}dt^{\prime}\int_{0}^{t^{\prime}}dt^{\prime\prime}{\cal H}_{int}%
(t^{\prime}){\cal H}_{int}(t^{\prime\prime})\Psi(0)+\cdots\,,\\
\end{array}$$
(2)
The cosine function is used in the rf pulse
instead of the single frequency
rotating wave.
In many systems,
no physical correspondence
of the rotating frame exists.
For example, the circuit of the pc-qubit is biased by
$z$ direction magnetic flux and the perturbation
is high frequency modulation of the $z$ flux.
No rotating frame of transverse
field can be defined for the oscillating
flux.
Our strategy to reduce the unwanted transitions
is to divide the qubit operation into short intervals of $t_{0}$
and attach additional pulses to each operation pulse to
correct errors from this short interval.
The operation pulse
is in resonant with $\omega_{21}$
of the lowest two states. Besides rotating the qubit between
the level $1$ and $2$, it brings up
off-resonant transitions
between the third level and these two levels
through the couplings $\gamma_{13}$ and $\gamma_{23}$.
Then the same perturbation is applied in two
other pulses with different
frequencies, amplitudes and phases as:
$\alpha_{31}{\cal H}_{I}\cos{(\omega_{31}t+\phi_{31})}$ and
$\alpha_{32}{\cal H}_{I}\cos{(\omega_{32}t+\phi_{32})}$, both
for time $t_{0}$, to cancel the unwanted transitions to the
third level.
This three-piece sequence is repeated
$\tau_{op}/t_{0}$ times to finish the qubit operation.
The time $t_{0}$ satisfies
$1/\omega\ll t_{0}\ll 1/\gamma_{ij},i,j=1,2,3$ with both
$1/\omega_{21}t_{0}$ and $\gamma_{ij}t_{0}$ being small parameters of
the same order.
Thus we have two small parameters in this process.
This is crucial for this
simple pulse sequence to work.
Starting with an initial wave function $\Psi(0)=[u_{0}\,v_{0}\,w_{0}]^{T}$,
$w_{0}=0$,
after the $\omega_{21}$ pulse,
the third level has the component:
$$\begin{array}[]{rl}w=&u_{0}(\frac{\gamma_{13}^{*}(e^{-i(\omega_{21}-\omega_{31%
})t_{0}}-1)}{\omega_{21}-\omega_{31}}-\frac{\gamma_{13}^{*}(e^{i(\omega_{21}+%
\omega_{31})t_{0}}-1)}{\omega_{21}+\omega_{31}})\\
+&v_{0}(\frac{\gamma_{23}^{*}(e^{-i(\omega_{21}-\omega_{32})t_{0}}-1)}{\omega_%
{21}-\omega_{32}}-\frac{\gamma_{23}^{*}(e^{i(\omega_{21}+\omega_{32})t_{0}}-1)%
}{\omega_{21}+\omega_{32}})\\
+&u_{0}\theta_{u}+v_{0}\theta_{v}\\
\end{array},$$
(3)
where
$\theta_{u}$ and $\theta_{v}$ are of third order.
The main components in $w$ are second order terms
that depend on the initial wave function $u_{0}$ and $v_{0}$
linearly. With
$t_{0}$ satisfying $e^{2i\omega_{21}t_{0}}=1$,
$u$ and $v$ have third order deviations from the
correct two-level rotation.
The other two pulses are then applied to
cancel the
$w$ component.
The $\omega_{31}$ pulse induces a resonant transition between level
one and level three to
cancel the $u_{0}$ term in $w$; the
$\omega_{32}$ pulse induces a resonant transition between level
two and level three to
cancel the $v_{0}$ term in $w$.
The amplitudes and phase shifts of these two pulses can be expanded
in ascending order as:
$$\begin{array}[]{lcl}\alpha_{31}e^{i\phi_{31}}&=&\alpha_{31}^{(1)}e^{i\phi_{31}%
^{(1)}}+\alpha_{31}^{(2)}e^{i\phi_{31}^{(2)}}+\cdots\\
\alpha_{32}e^{i\phi_{32}}&=&\alpha_{32}^{(1)}e^{i\phi_{32}^{(1)}}+\alpha_{32}^%
{(2)}e^{i\phi_{32}^{(2)}}+\cdots\\
\end{array}\ ,$$
(4)
The first order coefficients cancel the second order terms in $w$ and
modify the higher order terms $\theta_{u}$ and $\theta_{v}$ when:
$$\begin{array}[]{lcl}\alpha_{31}^{(1)}e^{i\phi_{31}^{(1)}}&=&\frac{e^{-i(\omega%
_{21}-\omega_{31})t_{0}}-1}{i(\omega_{21}-\omega_{31})t_{0}}-\frac{e^{i(\omega%
_{21}+\omega_{31})t_{0}}-1}{i(\omega_{21}+\omega_{31})t_{0}}\\
\alpha_{32}^{(1)}e^{i\phi_{32}^{(1)}}&=&\frac{e^{-i(\omega_{21}-\omega_{32})t_%
{0}}-1}{i(\omega_{21}-\omega_{32})t_{0}}-\frac{e^{i(\omega_{21}+\omega_{32})t_%
{0}}-1}{i(\omega_{21}+\omega_{32})t_{0}}\\
\end{array}\ ,$$
(5)
It turns out that the nth order terms of $w$ after the
two correction pulses include
linear terms of $\alpha_{31}^{(n-1)}$ and $\alpha_{32}^{(n-1)}$,
and complicated terms that depend on
$\alpha_{3i}^{(k)}e^{i\phi_{3i}^{(k)}}$ ($k=1\,\ldots\,(n-2)$).
So, for any $n$, $\alpha_{31}^{(n-1)}$ and $\alpha_{32}^{(n-1)}$
can be determined by
the lower order components of $\alpha_{31}$ and $\alpha_{32}$
to cancel the $n$th order of $w$. As a result,
the transitions to the third level can be
completely erased. The parameters $\alpha_{31}$ and $\alpha_{32}$
do not depend on the initial wave function $u_{0}$ and $v_{0}$.
This is similar to solving the wave function
in the perturbation theory where
the higher order terms are derived after the lower order ones.
After the $k$th pulse sequences, with $w=0$,
the wave function of $u$ and $v$ is:
$$\left[\begin{array}[]{l}u_{k+1}\\
v_{k+1}\\
\end{array}\right]=\left[\begin{array}[]{ cc }\cos{\bar{\varphi}}+\bar{s}_{u}&%
-i\sin{\bar{\varphi}}+\bar{t}_{u}\\
-i\sin{\bar{\varphi}}+\bar{t}_{v}&\cos{\bar{\varphi}}+\bar{s}_{v}\\
\end{array}\right]\left[\begin{array}[]{l}u_{k}\\
v_{k}\\
\end{array}\right],$$
(6)
where $\bar{\varphi}=\gamma_{12}t_{0}$ is the phase rotation
of the two-level qubit; the $\bar{s}$
and $\bar{t}$ terms are of third order.
As $w=0$, this is a unitary
transformation that
deviates from the original Rabi oscillation by third order corrections.
The matrix can be written as
${\cal U}(t_{0})=\exp{(-i(\gamma_{12}\sigma_{x}+\delta_{0}+\sum_{i}\delta_{i}%
\sigma_{i})t_{0})}$,
where $\delta_{i}$ are third order small numbers that
can be determined by known parameters and
do not depend on the index $k$. This
is a renormalization of the qubit operation $\gamma_{12}$ with the third
level decoupled.
This correction strategy is easily generalized
to $N$($N\geq 3$) level system.
By applying rf pulses with frequencies $\omega_{i1},\omega_{i2},i=3...N$,
the transitions to the higher levels are completely erased.
Assuming no particular symmetry
between the states, $2(N-2)$ pulses are required in this process.
One may wonder why this simple pulse sequence works so well
to correct the transitions to the higher states.
For $N-2$ higher levels, to decouple these levels
is to exert $4(N-2)$ real domain restrictions on the
transformation matrix:
${\cal U}_{1i},{\cal U}_{2i}=0,i=3...N$.
Our tools are the Hamiltonians ${\cal H}_{0}$ and ${\cal H}_{I}$
that create the whole ${\rm u}(N)$ algebra by commutation.
Our pulse sequence
${\cal U}(t_{0})=\Pi_{i,\beta}P(\alpha_{i\beta},\phi_{i\beta})e^{-i\int{\cal H}%
_{I}\cos{\omega_{21}t^{\prime}}dt^{\prime}}$
($i=3\,\ldots\,N$ and $\beta=\uparrow,\downarrow$ ),
$P(\alpha_{i\beta},\phi_{i\beta})=e^{-i\int{\cal H}_{I}\alpha_{i\beta}\cos{(%
\omega_{i\beta}t^{\prime}+\phi_{i\beta})}dt^{\prime}}$,
contents $4(N-2)$ free
parameters. By choosing proper pulse sequence, we can achieve
the decoupling with proper pulse
parameters.
In conclusion, we discussed the errors due to
unwanted transitions to the higher states of a qubit
during qubit operation. It was shown by a group theoretic
argument that these errors can be completely prevented
in principle.
Then we generalized the result to the errors due to qubit interactions,
which can also be prevented when
the number of coupled qubits is not large.
A simple pulse sequence that
modifies the qubit dynamics and cancels off-resonant transitions
to arbitrary accuracy with $O(N)$ pulses
was proposed to illustrate the general analysis.
Our results showed that the idea of
dynamic pulse control[13]
also works for the fast quantum errors due to the higher states
of a qubit.
These results suggest that dynamic pulse control,
together with conventional quantum error correction, can function
as a powerful tool for performing accurate quantum computation in the
presence of errors.
This work is supported by ARO grant DAAG55-98-1-0369
and DARPA/ARO under the QUIC program.
${}^{\dagger}$ tianl@mit.edu; slloyd@mit.edu
References
[1]
S. Lloyd, Science 261, 1569 (1993);
263, 695 (1994);
D.P. DiVincenzo, ibid. 270, 255 (1995).
[2]
A. Leggett et al., Rev. Mod. Phys. , 1 (1987);
H. Grabert et al., Phys. Rep. 168, 115 (1988);
P.C.E. Stamp, in Tunneling in Complex Systems,
pp.101–197, edited by T. Tomsovic (World Sci. 1998);
S. Schneider & G.J. Milburn, Phys. Rev. A59, 3766 (1999);
57, 3748 (1998);
M. Murao & P.L. Knight, ibid 58, 663 (1998).
[3]
J. Gea-Banacloche, Phys. Rev. A57, R1 (1998);
ibid. 60, 185 (1999).
[4]
J.E. Mooij et al.,
Science 285 , 1036 (1999);
T.P. Orlando et al., Phys. Rev. B60, 15298 (1999;
L. Tian et al., cond-mat/9910062 (1999).
[5]
A. Shnirman et al., Phys. Rev. Lett. 79, 2371
(1997);
Yu. Makhlin et al.,
Nature 398, 305 (1999);
R. Fazio et al., Phys. Rev. Lett. 83, 5385 (1999);
L.B. Ioffe et al., Nature 398, 679 (1999).
[6]
M.B. Plenio and P.L. Knight, Phys. Rev. A 53, 2986 (1996);
Proc. Roy. Soc. A 453, 2017 (1997).
[7]
P. Shor, Phys. Rev. A52, R2493 (1995).
[8]
A. Steane, P. Roy. Soc. Lond. A. Mat 452, 2551 (1996);
A. Steane, Phys. Rev. A54, 4741 (1996);
A. Steane, Phys. Rev. Lett. 77, 793 (1996).
[9]
R. Laflamme et al., Phys. Rev. Lett. 77, 198 (1996);
E. Knill & R. Laflamme, Phys. Rev. A55, 900 (1997).
[10]
D. Gottesman, Phys. Rev. A54, 1862 (1996).
[11]
A.R. Calderbank et al., Phys. Rev. Lett. 78, 405 (1997);
A.R. Calderbank & P. Shor, Phys. Rev. A54, 1098 (1996).
[12]
L.M. Duan & G.C. Guo, Phys. Rev. A54, 737 (1998);
L.M. Duan & G.C. Guo, Phys. Rev. Lett. 79, 1053 (1997).
[13]
L. Viola & S. Lloyd, Phys. Rev. A58, 2733 (1998);
L. Viola et al., Phys. Rev. Lett. 82, 2417 (1999);
L. Viola et al., ibid. 83, 4888 (1999).
[14]
We simulated the dynamic evolution of pc-qubit
under rf magnetic field of $\delta f=0.002$ for $t=T_{Rabi}$.
With an initial state $|0\rangle$, the final state
has components from many higher levels. Details
to be published elsewhere.
[15]
U. Haeberlen, High Resolution NMR in Solids—
Selective Averaging (Academic Press 1976).
[16]
D.F. Walls & G.J. Milburn, Quantum Optics (Springer-Verlag 1994).
[17]
S. Lloyd, Phys. Rev. Lett. 75, 346 (1995).
[18]
C. Presilla, R. Onofrio, and U. Tambini,
Ann. Phys. (NY) 248, 95 (1996); D. Home and M. A. B. Whitaker,
ibid. 258, 237 (1997).
[19]
S. Lloyd, quant-ph/9903057 (1999). |
Maximal subfields of a division algebra
Mai Hoang Bien
Mathematisch Instituut, Leiden Universiteit, Niels Bohrweg 1,2333 CA Leiden,The Netherlands.
Dipartimento di Matematica, Università degli Studi di Padova, Via Trieste 63, 35121 Padova, Italy.
maihoangbien012@yahoo.com
Abstract.
Let $D$ be a division algebra over a field $F$. In this paper, we prove that there exist $a,b,x,y\in D^{*}=D\backslash\{0\}$ such that $F(ab-ba)$ and $F(xyx^{-1}y^{-1})$ are maximal subfields of $D$, which answers questions posted in [5].
Key words and phrases:Maximal subfield, division algebra, commutator, algebraic.
2010 Mathematics Subject Classification. 12F05, 12F10, 12E15, 16K20.
The author would like to thank his supervisor Prof. H.W. Lenstra for the comments.
1. Introduction
Let $F$ be a field. A ring $D$ is called a division algebra over $F$ if the center $Z(D)=\{\,a\in D\mid ab=ba,\forall b\in D\,\}$ of $D$ is equal to $F$, $D$ is a finite dimensional vector space over $F$ and $D$ has neither proper left ideal nor proper right ideal. In other words, $D$ is a division ring with the center $F$ and $\dim_{F}D<\infty$. In some books and papers, $D$ is also called centrally finite [4, Definition 14.1]. A central simple algebra over $F$ is an algebra isomorphic to $M_{n}(D)$ for some positive integer $n$ and division algebra $D$ over $F$. For any central simple algebra $A$ over $F$, $\sqrt{\dim_{F}A}$ is said to be degree of $A$.
For any division algebra $D$ over $F$, it is well known from Kothe’s Theorem that there exists a maximal subfield $K$ of $D$ such that the extension of fields $K/F$ is separable [4, Th. 15.12]. In [1, Theorem 7], authors proved that for any separable extension of fields $K/F$ in $D$, there exists an element $c\in[D,D]$, the group of additive commutators of $(D,+)$, such that $K=F(c)$ unless $\operatorname{Char}(F)=[K:F]=2$ and $4$ does not divide the degree of $D$. Hence, if $K$ is a maximal subfield of $D$ which is separable over $F$ then there exists $c\in[D,D]$ such that $K=F(c)$. In particular, there exists a maximal subfield of $D$ such that it is of the form $F(c)$ for some element $c$ in $[D,D]$. We have a natural question: is it true that there exists a commutator $ab-ba\in[D,D]$ such that $F(ab-ba)$ is a maximal subfield of $D$ (see [5, Problem 28])? Almost similarly, if $K/F$ is a separable extension of fields in $D$ then there exists an element $d\in D^{\prime}=[D^{*},D^{*}]$, the group of multiplicative commutators of $D^{*}=D\backslash\{0\}$, such that $K=F(d)$ (see [5, Theorem 2.26]). Again, the author asked whether $F(xyx^{-1}y^{-1})$ is a maximal subfield of $D$ for some $x,y\in D^{*}$ (see [5, Problem 29]).
The goal of this paper is to answer in the affirmative for both questions. The main tools used in this paper are generalized rational identities over a central simple algebra. Readers can find their definitions and notaions in detail in [3] and [6].
2. Results
Let $R$ be a ring. Recall that an element $a$ of $R$ is called algebraic of degree $n$ over a subring $S$ of $R$ if there exists a polynomial $f(x)$ of degree $n$ over $S$ such that $f(a)=0$ and there is no polynomial of degree less than $n$ vanishing on $a$. In general, $f(x)$ is not necessary unique and irreducible even if $S$ is a field. For example, the matrix $A=\left({\begin{array}[]{*{20}{c}}1&0\\
0&2\end{array}}\right)\in M_{2}(F),$ where $F$ is a field, satisfies the polynomial $f(x)=(x-1)(x-2)$. Since $A\notin F$, $2$ is the smallest degree of all the polynomials vanishing on $A$.
Recall that a generalized rational expression over $R$ is an expression contructed from $R$ and a set of noncommutative inderteminates using addition, substraction, multiplication and division. A generalized rational expression $f$ over $R$ is called a generalized rational identity if it vanishes on all permissible substitutions from $R$. In this case, one says $R$ satisfies $f$. We consider the following example which is important in this paper. Given a positive integer $n$ and $n+1$ noncommutative indeterminates $x,y_{1},\cdots,y_{n}$, put
$$g_{n}(x,y_{1},y_{2},\cdots,y_{n})=\sum\limits_{\delta\in{S_{n+1}}}{%
\operatorname{sign}(\delta){x^{\delta(0)}}{y_{1}}{x^{\delta(1)}}{y_{2}}{x^{%
\delta(2)}}\ldots{y_{n}}{x^{\delta(n)}}},$$
where $S_{n+1}$ is the symmetric group of $\{\,0,1,\cdots,n\,\}$ and $\operatorname{sign}(\delta)$ is the sign of permutation $\delta$. This is a generalized rational expression defined in [3] to connect an algebraic element of degree $n$ and a polynomial of $n+1$ indeterminates.
Lemma 2.1.
Let $F$ be a field and $A$ be a central simple algebra over $F$. For any element $a\in A$, the following conditions are equivalent.
(1)
The element $a$ is algebraic over $F$ of degree less than $n$.
(2)
$g_{n}(a,r_{1},r_{2},\cdots,r_{n})=0$ for any $r_{1},r_{2},\cdots,r_{n}\in A$.
Proof. This is a corollary of [3, Corollary 2.3.8].
In particular, a central simple algebra of degree $m$ satisfies the expression $g_{m}$ since every central simple algebra of degree $m$ over a field $F$ can be considered as a $F$-subalgebra of the ring $M_{m}(F)$ and elements of $M_{m}(F)$ are algebraic of degree less than $m$ over $F$. In other words, $g_{m}$ is a generalized rational indentity of any central simple algebra of degree $m$.
For any central simple algebra $A$, denote ${\mathcal{G}}(A)$ the set of all generalized rational identities of $A$. Then ${\mathcal{G}}(A)\neq\emptyset$ because $g_{m}\in{\mathcal{G}}(A)$. The following theorem gives us a relation between the set of all generalized rational identities of a central simple algebra and the ring of matrices over a field.
Theorem 2.2.
[2, Theorem 11] Let $F$ be an infinite field and $A$ be a central simple algebra of degree $n$ over $F$. Assume that $L$ is an extension field of $F$. Then ${\mathcal{G}}(A)={\mathcal{G}}(M_{n}(F))={\mathcal{G}}(M_{n}(L))$.
Now we are going to prove the main results of this paper. The following lemma is basic.
Lemma 2.3.
Let $D$ be a division algebra of degree $n$ over a field $F$. Assume that $K$ is a subfield of $D$ containing $F$. Then $\dim_{F}K\leq n$. The quality holds if and only if $K$ is a maximal sufield of $D$.
Proof. See [4, Corollary 15.6 and Proposition 15.7]
Lemma 2.4.
Let $F$ be an infinite field and $n\geq 2$ be an integer. There exist two matrices $A,B\in M_{n}(F)$ such that the commutator $ABA^{-1}B^{-1}$ is an algebraic element of degree $n$ over $F$.
Proof. Put
$A=\left({\begin{array}[]{*{20}{c}}0&0&\cdots&0&{{a_{1}}}\\
1&0&\cdots&0&{{a_{2}}}\\
\vdots&\vdots&\ddots&\vdots&\vdots\\
0&0&1&0&{{a_{n-1}}}\\
0&0&0&1&0\end{array}}\right)$ and $B=\left({\begin{array}[]{*{20}{c}}{{b_{1}}}&0&\cdots&0&0\\
0&{{b_{2}}}&\cdots&0&0\\
\vdots&\vdots&\ddots&\vdots&\vdots\\
0&0&0&{{b_{n-1}}}&0\\
0&0&0&0&{{b_{n}}}\end{array}}\right),$ where $a_{i},b_{j}\neq 0$. One has $ABA^{-1}B^{-1}=\left({\begin{array}[]{*{20}{c}}{{b_{n}}b_{1}^{-1}}&0&\cdots&0&%
0\\
*&{{b_{1}}b_{2}^{-1}}&\cdots&0&0\\
\vdots&\vdots&\ddots&\vdots&\vdots\\
*&*&*&{{b_{n-2}}b_{n-1}^{-1}}&0\\
*&*&*&*&{{b_{n-1}}b_{n}^{-1}}\end{array}}\right)$. If we choose $b_{n}b_{1}^{-1},b_{1}b^{-1}_{2},\cdots,b_{n-1}b^{-1}$ all distinct (it is possible since $F$ is infinite), then the characteristic polynomial of $ABA^{-1}B^{-1}$ is a polynomial of smallest degree which vanishes on $ABA^{-1}B^{-1}$. That is, $ABA^{-1}B^{-1}$ is an algebraic element of degree $n$ over $F$.
The following theorem answers Problem 29 in [5, Page 83].
Theorem 2.5.
Let $D$ be a central division algebra over a field $F$. There exist $x,y\in D^{*}$ such that $F(xyx^{-1}y^{-1})$ is a maximal subfield of $D$.
Proof. If $F$ is finite then $D$ is also finite, so that there is nothing to prove. Suppose that $F$ is infinite and $D$ is of degree $n$ over $F$. By Lemma 2.3, it suffices to show that there exist $x,y\in D^{*}$ such that $\dim_{F}F(xyx^{-1}y^{-1})\geq n$. Indeed, put $\ell=\max\{\,\dim_{F}F(xyx^{-1}y^{-1})\mid x,y\in D^{*}\,\}.$ Then from Lemma 2.3,
$$g_{\ell}(rsr^{-1}s^{-1},r_{1},r_{2},\cdots,r_{\ell})=0$$
for any $r_{1},r_{2},\cdots,r_{\ell}\in D$ and $r,s\in D^{*}$. Hence, $g_{\ell}(xyx^{-1}y^{-1},y_{1},y_{2},\cdots,y_{\ell})$ is a generalized rational idenity of $D$, so that, by Lemma 2.2, $g_{\ell}(xy-yx,y_{1},y_{2},\cdots,y_{\ell})$ is also a generalized rational idenity of $M_{n}(F)$. Since $g_{\ell}(ABA^{-1}B^{-1},r_{1},r_{2},\cdots,r_{\ell})=0,$ for any $r_{i}\in M_{n}(F)$ and $A,B$ are chosen in Lemma 2.4. Therefore $n\leq\ell$ because Lemma 2.1 and $AB-BA$ is an algebraic element of degree $n$.
Lemma 2.6.
Let $F$ be an infinite field and $n>2$ be an integer. There exist two matrices $A,B\in M_{n}(F)$ such that $AB-BA$ is an algebraic element of degree $n$ over $F$.
Proof. Put
$A=\left({\begin{array}[]{*{20}{c}}0&0&\cdots&0&{{a_{1}}}\\
1&0&\cdots&0&{{a_{2}}}\\
\vdots&\vdots&\ddots&\vdots&\vdots\\
0&0&1&0&{{a_{n-1}}}\\
0&0&0&1&0\end{array}}\right)$ and $B=\left({\begin{array}[]{*{20}{c}}0&{{b_{1}}}&0&\cdots&0&0\\
0&0&{{b_{2}}}&\cdots&0&0\\
0&0&0&\cdots&0&0\\
\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\
0&0&0&\cdots&0&{{b_{n-1}}}\\
0&0&0&\cdots&0&0\end{array}}\right)$. One has $AB-BA=\left({\begin{array}[]{*{20}{c}}{{b_{1}}}&*&\cdots&*&*\\
0&{{b_{1}}-{b_{2}}}&\cdots&*&*\\
\vdots&\vdots&\ddots&\vdots&\vdots\\
0&0&\cdots&{{b_{n-2}}-{b_{n-1}}}&*\\
0&0&\cdots&0&{{b_{n-1}}}\end{array}}\right)$. Since $F$ is infinite, we can choose $b_{1},b_{2},\cdots,b_{n-1}\in F$ such that $b_{1},b_{1}-b_{2},\cdots,b_{n-2}-b_{n-1},b_{n-1}$ all distinct. Hence, the characteristic polynomial of $AB-BA$ is a polynomial of smallest degree vanishing on $AB-BA$. Therefore, $AB-BA$ is an algebraic element of degree $n$ over $F$.
Almost similar to the proof of Theorem 2.5, we have the following theorem, which answers Problem 28 in [5, Page 83].
Theorem 2.7.
Let $D$ be a central division algebra over a field $F$. There exist $x,y\in D$ such that $F(xy-yx)$ is a maximal subfield of $D$.
Proof. If $F$ is finite then $D$ is also finite, so that there is nothing to prove. Suppose that $F$ is infinite and $D$ is of degree $n$. By Lemma 2.3, it suffices to show that there exist $x,y\in D$ such that $\dim_{F}F(xy-yx)\geq n$. Indeed, if $n=2$, by [4, Corollary 13.5], then there exist $x,y\in D$ such that $xy-yx\notin F$, which implies $F(xy-yx)=2=n$. Assume that $n>2$. Then put $\ell=\max\{\,\dim_{F}F(xy-yx)\mid x,y\in D\,\}.$ By Lemma 2.1,
$$g_{\ell}(rs-sr,r_{1},r_{2},\cdots,r_{\ell})=0$$
for any $r_{1},r_{2},\cdots,r_{\ell}\in D$ and $r,s\in D^{*}$. It follows $g_{\ell}(xy-yx,y_{1},y_{2},\cdots,y_{\ell})$ is a generalized rational idenity of $D$. From Lemma 2.2, $g_{\ell}(xy-yx,y_{1},y_{2},\cdots,y_{\ell})$ is also a generalized rational idenity of $M_{n}(F)$. But because there exist $A,B\in M_{n}(F)$ such that $AB-BA$ is algebraic of degree $n$ (Lemma 2.4), one has
$$g_{\ell}(AB-BA,r_{1},r_{2},\cdots,r_{\ell})=0$$
for any $r_{i}\in M_{n}(F)$. Therefore, by Lemma 2.1, $n\leq\ell$.
References
[1]
S. Akbari, M. Arian-Nejad, M. L. Mehrabadi, On additive commutator groups in division rings, Results Math., 33 (1-2), 9â21, 1998.
[2]
S. A. Amitsur, Rational identities and applications to algebra and geometry, J. Algebra 3, 304–359, 1966.
[3]
K. I. Beidar, W. S. Martindale 3rd and A. V. Mikhalev, Rings with Generalized Identities, Marcel Dekker, Inc., New York- Basel-Hong Kong, 1996.
[4]
T. Y. Lam, A first course in noncommutative rings, MGT 131, Springer, 1991.
[5]
M. Mahdavi-Hezavehi, Commutators in division rings revisited. Bull. Iranian Math. Soc, 26(3): 7–88, 2000.
[6]
L. H. Rowen, Polynomial identities in ring theory, Academic Press, Inc., New York, 1980. |
Local polynomial regression for pooled response data
Dewei Wang label=e1]deweiwang@stat.sc.edu
[
Department of Statistics, University of South Carolina, Columbia, SC 29208, USA
Xichen Mou label=e2]xmou@memphis.edu
[
Division of Epidemiology, Biostatistics, and Environmental Health, University of Memphis, Memphis, Tennessee 38152, USA
Xiang Li label=e3]xl2@email.sc.edu
[
JPMorgan Chase, Jersey City, New Jersey 07310, USA
Xianzheng Huang label=e4]huang@stat.sc.edu
[
Department of Statistics, University of South Carolina, Columbia, SC 29208, USA
(#1)
Abstract
We propose local polynomial estimators for the conditional mean of a continuous response when only pooled response data are collected under different pooling designs. Asymptotic properties of these estimators are investigated and compared. Extensive simulation studies are carried out to compare finite sample performance of the proposed estimators under various model settings and pooling strategies. We apply the proposed local polynomial regression methods to two real-life applications to illustrate practical implementation and performance of the estimators for the mean function.
[
62G08,
62G20,
,
homogeneous pooling,
random pooling,
doi: 10.1214/154957804100000000
††volume: #1
0000
0
and
class=MSC]
Cross validation
1 Introduction
Instead of measuring individual specimens to collect data for biomarkers or analytes of interest, collecting such data on pools of specimens has become increasingly common in epidemiological and environmental studies (Kendziorski et al., 2003; Shih et al., 2004). Collecting pooled data can reduce information loss when there is a detecting limit, and offer a more timely manner to gather information, in addition to the obvious benefit of reducing cost of laboratory assays and preserving irreplaceable specimens. In some econometrics applications, pooled data are all that is available to researchers, such as data aggregated by family or by region. In these applications, data of other attributes at the individual level are often also recorded, and researchers are interested in associations between quantities at the individual level even though some data are collected at the pool level. Our study is motivated by these research questions that require methodologies for regression analysis based on pooled continuous response data and individual-level covariate data.
Traditional regression methodology applicable to individual response data cannot be directly used to analyze pooled response data, and there exist some research on regression analysis for pooled continuous responses. Under the parametric framework, Malinovsky et al. (2012) considered Gaussian random effects models for pooled repeated measures, and studied inference for variance components under different pooling strategies. Mitchell et al. (2014) proposed a Monte Carlo expectation maximization algorithm to carry out regression analyses of pooled biomarker assessments assuming that the biomarker follows a log-normal distribution given covariates. McMahan et al. (2016) developed methods to infer receiver-operating characteristic curves using pooled biomarker measurements. Liu et al. (2017) provided a general strategy based on Monte Carlo maximum likelihood for regression analysis of pooled data under generic parametric models assumed for the individual response given covariates. Under the semiparametric framework, Mitchell et al. (2015) proposed a semiparametric method for regression analysis of a right-skewed and positive response when data for the response are taken from pooled specimens. Without imposing parametric assumptions on the biomarker distribution, Lin and Wang (2018) developed a semiparametric approach for analyzing pooled biomarker measurements originating from a single-index model for the individual response. Under the nonparametric framework, Linton and Whang (2002) proposed a kernel-based estimator for regression function for pooled data when covariate data are also aggregated, with both aggregated response data and covariate data subject to additive measurement error.
Among the existing works on regression analysis of pooled response data, many consider various pooling designs. For example, Ma et al. (2011) compared two pooling designs in the context of linear regression analysis for a pooled continuous response and aggregated covariates, one being random pooling where pools are randomly formed without taking into account covariate information, and the other termed as optimal pooling by the authors, where pools are formed by gathering specimens corresponding to similar covariate values. This latter strategy is better known as homogeneous pooling in the pool/group testing literature (Bilder and Tebbs, 2009), and many researchers have shown efficiency gain in prediction and covariate effects estimation when homogeneous pooled data are used than when random pooled data are used (Vansteelandt et al., 2000; Ma et al., 2011). Mitchell et al. (2014) developed a regression methodology for log-normal response data subject to a special form of homogeneous pooling where covariate values within a pool are identical. Like the regression analysis discussed in Ma et al. (2011), Mitchell et al. (2014) also regressed the pooled continuous response on aggregated covariates to infer the association between the response and covariates at the individual level.
In this article, we propose local polynomial estimators for the mean of a continuous response given covariates using pooled response data and individual-level covariate data. More specifically, the proposed estimators are for the mean function $m(x)=E(Y|X=x)$, where $Y$ is a continuous response of an experimental unit, $X$ is the covariate that can be vector-valued and relate to attributes of the experimental unit or individual. For ease of exposition, we consider a scalar covariate in this article. Observed data available for inferring $m(x)$ include pooled responses from $J$ groups of individuals, $\mathbf{Z}=(Z_{1},\ldots,\,Z_{J})^{\mathrm{\scriptscriptstyle T}}$, where $Z_{j}=c_{j}^{-1}\sum_{k=1}^{c_{j}}Y_{jk}$, in which $c_{j}$ is the number of individuals in pool $j$, and $Y_{jk}$ is the unobserved response of individual $k$ in that pool, for $j=1,\ldots,J$, $k=1,\ldots,c_{j}$. Also observed are covariate data $\mathbb{X}=\{\tilde{\mathbf{X}}_{j},\,j=1,\ldots,J\}$, where $\tilde{\mathbf{X}}_{j}=(X_{j1},\ldots,X_{j,c_{j}})^{\mathrm{\scriptscriptstyle
T}}$, with $X_{jk}$ being the covariate associated with individual $k$ in pool $j$, for $k=1,\ldots,c_{j}$ and $j=1,\ldots,J$. Three proposed local polynomial estimators for $m(x)$ based on data $(\mathbf{Z},\mathbb{X})$ are presented in Section 2 next, where we assume that data arise from random pooling. Section 3 presents local polynomial estimators based on homogeneous pooled data. Asymptotic properties of these estimators are investigated and compared in Section 4 under each of the two pooling designs. Section 5 describes bandwidth selection methods tailored for the proposed estimators. Section 6 presents a simulation study where we compare finite sample performance of the proposed estimators under different model settings and various pooling designs. We further illustrate the implementation and performance of the proposed methods in two real-life applications in Section 7. Finally, in Section 8, we summarize contributions of our study and discuss follow-up research directions.
2 Local polynomial estimators under random pooling
Local polynomial regression has been a well-received and widely applicable nonparametric strategy for estimating $m(x)$ when individual data are available (Fan and Gijbels, 1996). To estimate the regression function $m(x)$ based on individual data $\{(Y_{jk},X_{jk}),k=1,\dots,c_{j}\}_{j=1}^{J}$, this strategy exploits the weighted least squares method to construct an objective function following a $p$-th order Taylor expansion of $m(s)$ around $x$, $m(s)\approx\sum_{\ell=0}^{p}\{m^{(\ell)}(x)/\ell!\}(s-x)^{\ell}$, with $m^{(\ell)}(x)$ equal to $(\partial^{\ell}/\partial s^{\ell})m(s)$ evaluated at $s=x$. In particular, the objective function is given by
$$\displaystyle Q_{0}(\mbox{\boldmath$\beta$})=\sum_{j=1}^{J}\sum_{k=1}^{c_{j}}%
\left\{Y_{jk}-\sum_{\ell=0}^{p}\beta_{\ell}(X_{jk}-x)^{\ell}\right\}^{2}K_{h}(%
X_{jk}-x),$$
(2.1)
where $K_{h}(t)=K(t/h)/h$, $K(t)$ is a symmetric kernel, $h$ is a bandwidth, $\beta_{\ell}=m^{(\ell)}(x)/\ell!$, for $\ell=0,1,\ldots,p$, and $\mbox{\boldmath$\beta$}=(\beta_{0},\beta_{1},\ldots,\beta_{p})^{\mathrm{%
\scriptscriptstyle T}}$. Minimizing $Q_{0}(\mbox{\boldmath$\beta$})$ with respect to $\beta$ yields an estimate of $m(x)(=\beta_{0})$, along with estimates of $m^{(\ell)}(x)(=\ell!\beta_{\ell})$, for $\ell=1,\ldots,p$. Denote by $\hat{m}_{0}(x)$ the so-obtained estimator for $m(x)$.
In what follows, we revise $Q_{0}(\mbox{\boldmath$\beta$})$ to construct new objective functions to adapt the local polynomial regression strategy to pooled response data from random pooling.
2.1 The average-weighted estimator
Now that individual responses $\{Y_{jk},k=1,\dots,c_{j}\}_{j=1}^{J}$ in (2.1) are unobserved but pooled responses $\{Z_{j}\}_{j=1}^{J}$ are instead, it is natural to switch attention from $E(Y_{i}|X_{i})$ to $E(Z_{j}|\tilde{\mathbf{X}}_{j})=c_{j}^{-1}\sum_{k=1}^{c_{j}}m(X_{jk})$, as if one were regressing $Z$ on the accompanying covariates in a pool collectively. This motivates the following weighted least squares objective function,
$$Q_{1}(\mbox{\boldmath$\beta$})=\sum_{j=1}^{J}\left\{Z_{j}-\sum_{\ell=0}^{p}%
\beta_{\ell}c_{j}^{-1}\sum_{k=1}^{c_{j}}(X_{jk}-x)^{\ell}\right\}^{2}\left\{c_%
{j}^{-1}\sum_{k=1}^{c_{j}}K_{h}(X_{jk}-x)\right\}.$$
(2.2)
In (2.1), the weight function $K_{h}(X_{i}-x)$ quantifies the proximity of the $i$th covariate data point to $x$, producing a larger weight for an individual whose covariate value is closer to $x$. In (2.2), the average of such proximity measures associated with $c_{j}$ covariate data points in pool $j$ is used to assess the overall closeness of this collection of covariate values to $x$.
Minimizing $Q_{1}(\mbox{\boldmath$\beta$})$ with respect to $\beta$ and extracting the first element of the resultant minimizer gives a $p$-th order local polynomial estimator for $m(x)$. This estimator can be explicitly expressed as $\hat{m}_{1}(x)=\mbox{\boldmath$e$}_{1}^{\mathrm{\scriptscriptstyle T}}\mathbf{%
S}_{1}^{-1}(x)\mathbf{T}_{1}(x)$,
where $\mbox{\boldmath$e$}_{1}^{\mathrm{\scriptscriptstyle T}}=(1,0,\ldots,0)_{1%
\times(p+1)}$, $\mathbf{S}_{1}(x)=\mathbf{D}_{1}(x)^{\mathrm{\scriptscriptstyle T}}\mathbf{K}_%
{1}(x)\mathbf{D}_{1}(x)$, and $\mathbf{T}_{1}(x)=\mathbf{D}_{1}(x)^{\mathrm{\scriptscriptstyle T}}\mathbf{K}_%
{1}(x)\mathbf{Z}$, in which, $\mathbf{D}_{1}(x)$ is a $J\times(p+1)$ matrix with $\mathbf{D}_{1}(x)[j,\ell+1]=c_{j}^{-1}\sum_{k=1}^{c_{j}}(X_{jk}-x)^{\ell}$, for $j=1,\ldots,J$, $\ell=0,1,\ldots,p$, and $\mathbf{K}_{1}(x)=\textrm{diag}\{c_{1}^{-1}\sum_{k=1}^{c_{1}}K_{h}(X_{1k}-x),%
\,\ldots,\,c_{J}^{-1}\sum_{k=1}^{c_{J}}K_{h}(X_{Jk}-x)\}$. Elaborated expressions of entries in $\mathbf{S}_{1}(x)$ and $\mathbf{T}_{1}(x)$ are given in Appendix A. To highlight the weight function construction in (2.2), $\hat{m}_{1}(x)$ is referred to as the average-weighted estimator in this article.
2.2 The product-weighted estimator
Instead of averaging individual-level weights to construct a weight function as in $Q_{1}(\mbox{\boldmath$\beta$})$, one may view $\tilde{\mathbf{X}}_{j}$ as a multivariate covariate resulting from stacking the $c_{j}$ individual-level covariates in pool $j$ on top of each other, and an alternative weight function can be formulated to measure the nearness of this multivariate covariate to $x\mbox{\boldmath$1$}_{c_{j}}$, where $\mbox{\boldmath$1$}_{c_{j}}$ denotes the $c_{j}\times 1$ vector of one’s. Mimicking the product kernel used in multivariate kernel density estimation, we propose the following weighted least squares objective function with a different weight function,
$$Q_{2}(\mbox{\boldmath$\beta$})=\sum_{j=1}^{J}\left\{Z_{j}-\sum_{\ell=0}^{p}%
\beta_{\ell}c_{j}^{-1}\sum_{k=1}^{c_{j}}(X_{jk}-x)^{\ell}\right\}^{2}\left\{%
\prod_{k=1}^{c_{j}}K_{h}(X_{jk}-x)\right\}.$$
(2.3)
More succinctly, the estimator for $m(x)$ resulting from minimizing $Q_{2}(\mbox{\boldmath$\beta$})$ is given by $\hat{m}_{2}(x)=\mbox{\boldmath$e$}_{1}^{\mathrm{\scriptscriptstyle T}}\mathbf{%
S}_{2}^{-1}(x)\mathbf{T}_{2}(x)$, where $\mathbf{S}_{2}(x)=\mathbf{D}_{1}(x)^{\mathrm{\scriptscriptstyle T}}\mathbf{K}_%
{2}(x)\mathbf{D}_{1}(x)$ and $\mathbf{T}_{2}(x)=\mathbf{D}_{1}(x)^{\mathrm{\scriptscriptstyle T}}\mathbf{K}_%
{2}(x)\mathbf{Z}$, in which the weight matrix is given by $\mathbf{K}_{2}(x)=\textrm{diag}\{\prod_{k=1}^{c_{1}}K_{h}(X_{1k}-x),\ldots,%
\prod_{k=1}^{c_{J}}K_{h}(X_{Jk}-x)\}$. Detailed expressions of entries in $\mathbf{S}_{2}(x)$ and $\mathbf{T}_{2}(x)$ are provided in Appendix B. Due to the construction of the weight function in (2.3), we call $\hat{m}_{2}(x)$ the product-weighted estimator in the sequel.
2.3 The marginal-integration estimator
The first two estimators are motivated by the mean of $Z_{j}$ given all covariate data in pool $j$. The third estimator is inspired by the mean of $c_{j}Z_{j}$ given one arbitrary individual’s covariate in pool $j$ derived next under the assumption that $Y_{jk^{\prime}}\perp X_{jk}$ for $k^{\prime}\neq k$ and the pools are formed randomly independent of covariate information. By the definition of $Z_{j}$, we have
$$\displaystyle E(c_{j}Z_{j}|X_{jk}=x)$$
$$\displaystyle=\sum_{k^{\prime}=1,k^{\prime}\neq k}^{c_{j}}E(Y_{jk^{\prime}}|X_%
{jk}=x)+E(Y_{jk}|X_{jk}=x)$$
$$\displaystyle=\sum_{k^{\prime}=1,k^{\prime}\neq k}^{c_{j}}E(Y_{jk^{\prime}})+m%
(x)=(c_{j}-1)\mu+m(x),$$
where $\mu=E(Y_{jk^{\prime}})$ for $k^{\prime}=1,\ldots,c_{j}$ and $j=1,\ldots,J$.
Hence,
$$\displaystyle E\{c_{j}Z_{j}-(c_{j}-1)\mu|X_{jk}=x\}$$
$$\displaystyle=m(x).$$
(2.4)
If one views $c_{j}Z_{j}-(c_{j}-1)\mu$ as a pseudo response, (2.4) is reminiscent of the conditional mean model for individual-level data, $E(Y_{i}|X_{i}=x)=m(x)$, except for the dependence of the pseudo response on the unknown parameter $\mu$. Since $\mu$ is the marginal mean of $Y$, one may use the overall sample mean response, $\hat{\mu}=N^{-1}\sum_{j=1}^{J}c_{j}Z_{j}$, to estimate $\mu$, where $N=\sum_{j=1}^{J}c_{j}$. This yields a surrogate of the pseudo response defined by $R_{j}=c_{j}Z_{j}-(c_{j}-1)\hat{\mu}$, for $j=1,\ldots,J$. Heuristically, $R_{j}$ can be viewed as an “estimate” for $Y_{jk}$, writing it as $\hat{Y}_{jk}$ for the meantime as a reminder that one tries to return to the mean model one would use had individual responses been available, $E(Y_{jk}|X_{jk}=x)=m(x)$. Certainly, $E(\hat{Y}_{jk}|X_{jk}=x)\neq m(x)$ due to the estimation of $\mu$ in $\hat{Y}_{jk}$. In fact, one can show that $E(\hat{Y}_{jk}|X_{jk}=x)=m(x)+\{\mu-m(x)\}(c_{j}-1)/N$.
Using the surrogate of the pseudo response and (2.4), we formulate the following weighted least squares objective function,
$$Q_{3}(\mbox{\boldmath$\beta$})=\sum_{j=1}^{J}\sum_{k=1}^{c_{j}}\left\{R_{j}-%
\sum_{\ell=0}^{p}\beta_{\ell}(X_{jk}-x)^{\ell}\right\}^{2}K_{h}(X_{jk}-x).$$
(2.5)
Minimizing $Q_{3}(\mbox{\boldmath$\beta$})$ with respect to $\beta$ yields a $p$-th order local polynomial estimator for $m(x)$ yields our third proposed estimator for $m(x)$, denoted by $\hat{m}_{3}(x)$. As one can see from the elaborated expression of it given in Appendix C that $\hat{m}_{3}(x)$ is simply $\hat{m}_{0}(x)$ with $Y_{jk}$ replaced by $R_{j}$, for $j=1,\ldots,J$, $k=1,\ldots,c_{j}$. The construction of $\hat{m}_{3}(x)$ follows the same strategy of marginal integration used in Lin and Wang (2018). For this reason, we refer to $\hat{m}_{3}(x)$ as the marginal-integration estimator henceforth.
All three estimators reduce to $\hat{m}_{0}(x)$ when $c_{j}=1$ for $j=1,\ldots,J$, but are otherwise typically very different from each other. In-depth comparisons between the three estimators that go beyond their formulations demand more systematic investigation on their theoretical properties. This is the content of Section 4, where we look into the asymptotic bias and variance of these estimators under each of the two considered pooling designs.
3 Local polynomial estimators under homogeneous pooling
When pooled data result from homogeneous pooling, it is no longer sensible to consider the mean of $c_{j}Z_{j}$ given one “arbitrary” covariate data point in pool $j$ as we just did to construct $\hat{m}_{3}(x)$, since individuals’ covariates within a pool are not that “arbitrary” now after all, and $E(Y_{jk^{\prime}}|X_{jk}=x)$ is typically not equal to $E(Y_{jk^{\prime}})$ for $k^{\prime}\neq k$. But it is still meaningful to consider the mean of $Z_{j}$ given all covariate data in pool $j$ as we did under random pooling that leads to $\hat{m}_{1}(x)$ and $\hat{m}_{2}(x)$.
To be more concrete, consider the homogeneous pooling design following which pools of individuals are created according to the sorted covariate data in $\mathbb{X}$. This yields covariate data associated with pool $j$, for $j=1,\ldots,J$, given by $\tilde{\mathbf{X}}_{(j)}=(X_{(j1)},\ldots,X_{(jc_{j})})^{\mathrm{%
\scriptscriptstyle T}}$, where $X_{(11)}\leq X_{(12)}\leq\ldots\leq X_{(1c_{1})}\leq X_{(21)}\leq\ldots\leq X_%
{(2c_{2})}\leq\ldots\leq X_{(J1)}\leq\ldots\leq X_{(Jc_{J})}$. Even though the response data are not sorted, we use $Z_{(j)}=c_{j}^{-1}\sum_{k=1}^{c_{j}}Y_{(jk)}$ to denote the corresponding pooled response, where $Y_{(jk)}$ is the response of the individual whose covariate value is $X_{(jk)}$, for $k=1,\ldots,c_{j}$, and $j=1,\ldots,J$. Evaluating the objective functions in (2.2) and (2.3) at $\{(Z_{(j)},\tilde{\mathbf{X}}_{(j)})\}_{j=1}^{J}$ give the following objective functions one maximizes with respect to $\beta$ in order to obtain the average-weighted estimator, $\hat{m}_{1}(x)$, and the product-weighted estimator, $\hat{m}_{2}(x)$, respectively, under homogeneous pooling,
$$\displaystyle Q_{1}(\mbox{\boldmath$\beta$})$$
$$\displaystyle=\sum_{j=1}^{J}\left\{Z_{(j)}-\sum_{\ell=0}^{p}\beta_{\ell}c_{j}^%
{-1}\sum_{k=1}^{c_{j}}(X_{(jk)}-x)^{\ell}\right\}^{2}\left\{c_{j}^{-1}\sum_{k=%
1}^{c_{j}}K_{h}(X_{(jk)}-x)\right\},$$
$$\displaystyle Q_{2}(\mbox{\boldmath$\beta$})$$
$$\displaystyle=\sum_{j=1}^{J}\left\{Z_{(j)}-\sum_{\ell=0}^{p}\beta_{\ell}c_{j}^%
{-1}\sum_{k=1}^{c_{j}}(X_{(jk)}-x)^{\ell}\right\}^{2}\left\{\prod_{k=1}^{c_{j}%
}K_{h}(X_{(jk)}-x)\right\}.$$
4 Comparisons between different estimators
4.1 Asymptotic bias and variance
Under certain regularity conditions listed in Appendix A, we derive asymptotic means and variances of the proposed estimators for $\beta$ as $J\to\infty$ with $\max_{1\leq j\leq J}c_{j}$ bounded. Conditions listed there relate to $m(x)$, the variance function $\sigma^{2}(x)=\textrm{Var}(Y|X=x)$, the density function of $X$, $f_{\hbox{\tiny$X$}}(x)$, and the kernel $K(t)$, which are mostly common conditions seen in the context of local polynomial regression using individual-level data. In what follows, we summarize findings from these derivations (with details provided in the appendices) in two theorems that highlight some interesting contrasts between different estimators for $m(x)$ when pools are of equal size with $c_{j}=c$, for $j=1,\ldots,J$, with additional conditions imposed in each theorem when needed. Several quantities appearing in these theorems are defined next for ease of reference:
$$\displaystyle\mbox{\boldmath$\mu$}^{*}_{\ell}$$
$$\displaystyle=(\mu_{\ell},\mu_{\ell+1},\ldots,\mu_{\ell+p})^{\mathrm{%
\scriptscriptstyle T}},\quad\tilde{\mbox{\boldmath$\mu$}}_{\ell}=[\mu_{\ell_{1%
}+\ell_{2}+\ell}]_{\ell_{1},\ell_{2}=0,1,\ldots,p},$$
$$\displaystyle\tilde{\mbox{\boldmath$\nu$}}_{0}$$
$$\displaystyle=[\nu_{\ell_{1}+\ell_{2}}]_{\ell_{1},\ell_{2}=0,1,\ldots,p},\quad%
\mathbf{R}^{*}_{p}=(R_{0,p}(x),R_{1,p}(x),\ldots,R_{p,p}(x))^{\mathrm{%
\scriptscriptstyle T}},$$
$$\displaystyle\mbox{\boldmath$\Delta$}^{*}_{0}(x)$$
$$\displaystyle=(1,\delta_{1}(x),\ldots,\delta_{p}(x))^{\mathrm{%
\scriptscriptstyle T}},\quad\tilde{\mbox{\boldmath$\Delta$}}_{0}(x)=[\delta_{%
\ell_{1}+\ell_{2}}(x)]_{\ell_{1},\ell_{2}=0,1,\ldots,p},$$
where $R_{\ell,p}(x)=E[(X-x)^{\ell}\{m(X)-\sum_{\ell=0}^{p}\beta_{\ell}(X-x)^{\ell}\}]$ and $\delta_{\ell}(x)=E\{(X-x)^{\ell}\}$, for $\ell=0,1,\ldots,2p$.
The first theorem concerns the three estimators under random pooling. Appendices A, B, and C provide the proof for the three parts of this theorem that allow unequal pool sizes.
Theorem 4.1
As $J\to\infty$ and $h\to 0$, one has the following results regarding the difference between an estimator for $m(x)$ and $m(x)$.
(i)
If the $\ell$-th moment of $X$ exists, for $\ell=1,\ldots,2p$, then
$$\displaystyle\hat{m}_{1}(x)-m(x)=$$
$$\displaystyle\ \mbox{\boldmath$e$}_{1}^{\mathrm{\scriptscriptstyle T}}\mathbf{%
M}_{0}^{-1}(x)\left\{\mathbf{L}_{0}(x)-hf^{-1}_{\hbox{\tiny$X$}}(x)f^{\prime}_%
{\hbox{\tiny$X$}}(x)\mathbf{M}_{1}(x)\mathbf{M}_{0}^{-1}(x)\mathbf{L}_{0}(x)\right.$$
$$\displaystyle\left.+O\left(h^{2}\right)\right\}+\sqrt{c}\times O_{\hbox{\tiny$%
P$}}\left(\frac{1}{\sqrt{Nh}}\right),$$
(4.1)
where
$$\displaystyle\mathbf{L}_{0}(x)=$$
$$\displaystyle\ \frac{c-1}{c^{2}}\left\{R_{0,p}(x)\mbox{\boldmath$e$}_{1}+%
\mathbf{R}_{p}^{*}(x)\right\}+\frac{(c-1)(c-2)_{+}}{c^{2}}R_{0,p}(x)\mbox{%
\boldmath$\Delta$}_{0}^{*}(x),$$
$$\displaystyle\mathbf{M}_{0}(x)=$$
$$\displaystyle\ \frac{\tilde{\mbox{\boldmath$\mu$}}_{0}}{c^{2}}+\frac{c-1}{c^{2%
}}\left\{\tilde{\mbox{\boldmath$\Delta$}}_{0}(x)+\mbox{\boldmath$\Delta$}^{*}_%
{0}(x)\mbox{\boldmath$\mu$}_{0}^{*{\mathrm{\scriptscriptstyle T}}}+\mbox{%
\boldmath$\mu$}_{0}^{*}\mbox{\boldmath$\Delta$}_{0}^{*{\mathrm{%
\scriptscriptstyle T}}}(x)\right\}$$
$$\displaystyle+\frac{(c-1)(c-2)_{+}}{c^{2}}\mbox{\boldmath$\Delta$}_{0}^{*}(x)%
\mbox{\boldmath$\Delta$}_{0}^{*{\mathrm{\scriptscriptstyle T}}}(x),$$
$$\displaystyle\mathbf{M}_{1}(x)=$$
$$\displaystyle\ \frac{\tilde{\mbox{\boldmath$\mu$}}_{1}}{c^{2}}+\frac{c-1}{c^{2%
}}\left\{\mbox{\boldmath$\Delta$}^{*}_{0}(x)\mbox{\boldmath$\mu$}_{1}^{*{%
\mathrm{\scriptscriptstyle T}}}+\mbox{\boldmath$\mu$}_{1}^{*}\mbox{\boldmath$%
\Delta$}_{0}^{*{\mathrm{\scriptscriptstyle T}}}(x)\right\},$$
in which $(t)_{+}=\max(t,0)$.
(ii)
If $m(x)$ is $(p+3)$th-order continuously differentiable, then
$$\displaystyle\ \hat{m}_{2}(x)-m(x)$$
$$\displaystyle=$$
$$\displaystyle\ \mbox{\boldmath$e$}_{1}^{\mathrm{\scriptscriptstyle T}}h^{p+1}%
\left\{\beta_{p+1}\left\{\tilde{\mbox{\boldmath$\mu$}}_{0}+(c-1)\mbox{%
\boldmath$\mu$}_{0}^{*}\mbox{\boldmath$\mu$}_{0}^{*{\mathrm{\scriptscriptstyle
T%
}}}\right\}^{-1}\left\{\mbox{\boldmath$\mu$}^{*}_{p+1}+(c-1)\mu_{p+1}\mbox{%
\boldmath$\mu$}^{*}_{0}\right\}\right.$$
$$\displaystyle+hf^{-1}_{\hbox{\tiny$X$}}(x)\left[\left\{\beta_{p+2}f_{\hbox{%
\tiny$X$}}(x)+\beta_{p+1}f^{\prime}_{\hbox{\tiny$X$}}(x)\right\}\left\{\tilde{%
\mbox{\boldmath$\mu$}}_{0}+(c-1)\mbox{\boldmath$\mu$}_{0}^{*}\mbox{\boldmath$%
\mu$}_{0}^{*{\mathrm{\scriptscriptstyle T}}}\right\}^{-1}\right.$$
$$\displaystyle\times\left\{\mbox{\boldmath$\mu$}^{*}_{p+2}+(c-1)\mu_{p+2}\mbox{%
\boldmath$\mu$}^{*}_{0}\right\}-\beta_{p+1}f^{\prime}_{\hbox{\tiny$X$}}(x)%
\left\{\tilde{\mbox{\boldmath$\mu$}}_{0}+(c-1)\mbox{\boldmath$\mu$}_{0}^{*}%
\mbox{\boldmath$\mu$}_{0}^{*{\mathrm{\scriptscriptstyle T}}}\right\}^{-1}$$
$$\displaystyle\times\left\{\tilde{\mbox{\boldmath$\mu$}}_{1}+(c-1)\left(\mbox{%
\boldmath$\mu$}_{0}^{*}\mbox{\boldmath$\mu$}_{1}^{*{\mathrm{\scriptscriptstyle
T%
}}}+\mbox{\boldmath$\mu$}_{1}^{*}\mbox{\boldmath$\mu$}_{0}^{*{\mathrm{%
\scriptscriptstyle T}}}\right)\right\}\left\{\tilde{\mbox{\boldmath$\mu$}}_{0}%
+(c-1)\mbox{\boldmath$\mu$}_{0}^{*}\mbox{\boldmath$\mu$}_{0}^{*{\mathrm{%
\scriptscriptstyle T}}}\right\}^{-1}$$
$$\displaystyle\times\left.\left.\left\{\mbox{\boldmath$\mu$}^{*}_{p+1}+(c-1)\mu%
_{p+1}\mbox{\boldmath$\mu$}_{0}^{*}\right\}\right]+O(h^{2})\right\}+\sqrt{c}%
\times O_{\hbox{\tiny$P$}}\left(\frac{1}{\sqrt{Nh^{c}}}\right).$$
(iii)
Let $\bar{\sigma}^{2}=E\{\sigma^{2}(X)\}$. If $\textrm{Var}(Y)$ exists, then
$$\displaystyle\hat{m}_{3}(x)-m(x)=$$
$$\displaystyle\ \mbox{\boldmath$e$}_{1}^{\mathrm{\scriptscriptstyle T}}h^{p+1}%
\left\{\beta_{p+1}\tilde{\mbox{\boldmath$\mu$}}_{0}^{-1}\mbox{\boldmath$\mu$}_%
{p+1}^{*}+hf^{-1}_{\hbox{\tiny$X$}}(x)\left[\left\{\beta_{p+2}f_{\hbox{\tiny$X%
$}}(x)\right.\right.\right.$$
$$\displaystyle\left.\left.\left.+\beta_{p+1}f^{\prime}_{\hbox{\tiny$X$}}(x)%
\right\}\tilde{\mbox{\boldmath$\mu$}}_{0}^{-1}\mbox{\boldmath$\mu$}_{p+2}^{*}-%
\beta_{p+1}f^{\prime}_{\hbox{\tiny$X$}}(x)\tilde{\mbox{\boldmath$\mu$}}_{0}^{-%
1}\tilde{\mbox{\boldmath$\mu$}}_{1}\tilde{\mbox{\boldmath$\mu$}}_{0}^{-1}\mbox%
{\boldmath$\mu$}_{p+1}^{*}\right]\right.$$
$$\displaystyle\left.+O(h^{2})\right\}+\sqrt{\sigma^{2}(x)+(c-1)\bar{\sigma}^{2}%
}\times O_{\hbox{\tiny$P$}}\left(\frac{1}{\sqrt{Nh}}\right).$$
(4.2)
Theorem 4.1-(i) indicates that $\hat{m}_{1}(x)$ is an inconsistent estimator for $m(x)$, with the dominating bias given by $\mbox{\boldmath$e$}_{1}^{\mathrm{\scriptscriptstyle T}}\mathbf{M}^{-1}_{0}(x)%
\mathbf{L}_{0}(x)$ that does not depend on $h$, and thus does not diminish as $h\to 0$, but it does vanish when $c=1$. Considering a local constant estimator by setting $p=0$ in (A.20), we show in Appendix A that
$$\displaystyle\ \hat{m}_{1}(x)-m(x)$$
$$\displaystyle=$$
$$\displaystyle\ \frac{c-1}{c}E\{m(X)-m(x)\}+\frac{h^{2}\mu_{2}}{c}\left\{\beta_%
{1}\frac{f^{\prime}_{\hbox{\tiny$X$}}(x)}{f_{\hbox{\tiny$X$}}(x)}+\beta_{2}%
\right\}+O(h^{4})+O_{\hbox{\tiny$P$}}\left(\frac{1}{\sqrt{Jh}}\right),$$
of which the second term (of order $h^{2}$) is $c^{-1}$ times the dominating bias of the Nadaraya-Watson estimator based on individual-level data.
Theorem 4.1-(ii) suggests that $\hat{m}_{2}(x)$ is a consistent estimator for $m(x)$ with the asymptotic variance of order $O\{1/(Jh^{c})\}$, which inflates quickly as $c$ increases. Comparing (ii) and (iii) of Theorem 4.1 reveals that $\hat{m}_{2}(x)$ and $\hat{m}_{3}(x)$ typically do not share the same dominating bias except when $c=1$, and $\hat{m}_{3}(x)$ exhibits the same asymptotic bias as that of $\hat{m}_{0}(x)$ regardless of the pool size. The variability of $\hat{m}_{3}(x)$ is understandably higher than that of $\hat{m}_{0}(x)$, but it only grows linearly in $c$ and thus is much less inflated than the variance of $\hat{m}_{2}(x)$. More specifically, (4.2) implies that the amount of variance inflation of $\hat{m}_{3}(x)$ depends linearly on the pool size and $\bar{\sigma}^{2}$.
Summarizing these implications of Theorem 4.1, we conclude that the marginal-integration estimator $\hat{m}_{3}(x)$ is the preferred estimator among the three proposed under random pooling. It outperforms the average-weighted estimator $\hat{m}_{1}(x)$ for its consistency, and it surpasses the product-weighted estimator $\hat{m}_{2}(x)$ for its much less inflated variance when compared with $\hat{m}_{0}(x)$. However, $\hat{m}_{3}(x)$ is no longer well justified under homogeneous pooling as pointed out in Section 3. The following theorem is regarding the average-weighted estimator and the product-weighted estimator applied to data from the homogeneous pooling design. Appendix D provides the proof for this theorem.
Theorem 4.2
Assume that $x$ is an interior point of a compact and nondegenerate interval $\mathcal{I}$, the pdf of X, $f_{\hbox{\tiny$X$}}(\cdot)$, is bounded away from zero on an interval $\mathcal{J}$, where $\mathcal{I}\subset\mathcal{J}$, and $K(|t|)=0$ for $|t|>1$, with $K^{\prime}(t)$ bounded. Then, as $J\to\infty$, $h\to 0$, and $Jh^{4}\to\infty$,
$$\displaystyle\ \hat{m}_{1}(x)-m(x)$$
$$\displaystyle=$$
$$\displaystyle\ \mbox{\boldmath$e$}_{1}^{\mathrm{\scriptscriptstyle T}}h^{p+1}%
\left\{\beta_{p+1}\tilde{\mbox{\boldmath$\mu$}}_{0}^{-1}\mbox{\boldmath$\mu$}_%
{p+1}^{*}+hf^{-1}_{\hbox{\tiny$X$}}(x)\left[\left\{\beta_{p+2}f_{\hbox{\tiny$X%
$}}(x)+\beta_{p+1}f^{\prime}_{\hbox{\tiny$X$}}(x)\right\}\tilde{\mbox{%
\boldmath$\mu$}}_{0}^{-1}\mbox{\boldmath$\mu$}_{p+2}^{*}\right.\right.$$
$$\displaystyle\left.\left.-\beta_{p+1}f^{\prime}_{\hbox{\tiny$X$}}(x)\tilde{%
\mbox{\boldmath$\mu$}}_{0}^{-1}\tilde{\mbox{\boldmath$\mu$}}_{1}\tilde{\mbox{%
\boldmath$\mu$}}_{0}^{-1}\mbox{\boldmath$\mu$}_{p+1}^{*}\right]+O(h^{2})\right%
\}+O_{\hbox{\tiny$P$}}\left(\frac{1}{\sqrt{Nh}}\right),$$
(4.3)
and
$$\textrm{Var}\left\{\hat{m}_{1}(x)|\mathbb{X}\right\}=\frac{\sigma^{2}(x)}{Nhf_%
{\hbox{\tiny$X$}}(x)}\mbox{\boldmath$e$}_{1}^{\mathrm{\scriptscriptstyle T}}%
\tilde{\mbox{\boldmath$\mu$}}_{0}^{-1}\tilde{\mbox{\boldmath$\nu$}}_{0}\tilde{%
\mbox{\boldmath$\mu$}}_{0}\left\{1+o_{\hbox{\tiny$P$}}(1)\right\}.$$
(4.4)
If Condition (C5) is satisfied for the kernel defined by $K^{\dagger}(t)=K^{c}(t)$, then
$$\displaystyle\ \hat{m}_{2}(x)-m(x)$$
$$\displaystyle=$$
$$\displaystyle\ \mbox{\boldmath$e$}_{1}^{\mathrm{\scriptscriptstyle T}}h^{p+1}%
\left\{\beta_{p+1}\tilde{\mbox{\boldmath$\mu$}}_{\dagger,0}^{-1}\mbox{%
\boldmath$\mu$}_{\dagger,p+1}^{*}+hf^{-1}_{\hbox{\tiny$X$}}(x)\left[\left\{%
\beta_{p+2}f_{\hbox{\tiny$X$}}(x)+\beta_{p+1}f^{\prime}_{\hbox{\tiny$X$}}(x)%
\right\}\tilde{\mbox{\boldmath$\mu$}}_{\dagger,0}^{-1}\mbox{\boldmath$\mu$}_{%
\dagger,p+2}^{*}\right.\right.$$
$$\displaystyle\left.\left.-\beta_{p+1}f^{\prime}_{\hbox{\tiny$X$}}(x)\tilde{%
\mbox{\boldmath$\mu$}}_{\dagger,0}^{-1}\tilde{\mbox{\boldmath$\mu$}}_{\dagger,%
1}\tilde{\mbox{\boldmath$\mu$}}_{\dagger,0}^{-1}\mbox{\boldmath$\mu$}_{\dagger%
,p+1}^{*}\right]+O(h^{2})\right\}+O_{\hbox{\tiny$P$}}\left(\frac{1}{\sqrt{Nh}}%
\right),$$
(4.5)
and
$$\textrm{Var}\left\{\hat{m}_{2}(x)|\mathbb{X}\right\}=\frac{\sigma^{2}(x)}{Nhf_%
{\hbox{\tiny$X$}}(x)}\mbox{\boldmath$e$}_{1}^{\mathrm{\scriptscriptstyle T}}%
\tilde{\mbox{\boldmath$\mu$}}_{\dagger,0}^{-1}\tilde{\mbox{\boldmath$\nu$}}_{%
\dagger,0}\tilde{\mbox{\boldmath$\mu$}}_{\dagger,0}\left\{1+o_{\hbox{\tiny$P$}%
}(1)\right\},$$
(4.6)
where $\mbox{\boldmath$\mu$}^{*}_{\dagger,\ell}$, $\tilde{\mbox{\boldmath$\mu$}}_{\dagger,\ell}$, and $\tilde{\mbox{\boldmath$\nu$}}_{\dagger,0}$ are the counterparts of $\mbox{\boldmath$\mu$}^{*}_{\ell}$, $\tilde{\mbox{\boldmath$\mu$}}_{\ell}$, and $\tilde{\mbox{\boldmath$\nu$}}_{0}$, respectively, with $K(t)$ replaced by $K^{\dagger}(t)$.
Among the additional assumptions imposed in Theorem 4.2, the one on $x$ and the assumption on $K(t)$ are similar to Conditions (T1) and (T5) in Delaigle and Hall (2012), respectively. Theorem 4.2 indicates that both $\hat{m}_{1}(x)$ and $\hat{m}_{2}(x)$ are consistent estimators for $m(x)$ under homogeneous pooling, with the former sharing the same dominating bias as that of $\hat{m}_{0}(x)$, and the latter exhibiting the same form of dominating bias with a re-defined kernel that depends on $c$. Moreover, the asymptotic variances of both estimators are of the same order as that of $\hat{m}_{0}(x)$ despite the pool size. The practical implication of Theorem 4.2 is that, if one uses homogeneous pooled data to infer $m(x)$ via either one of the two proposed local polynomial estimators, one only needs $J$ assays without losing accuracy or efficiency asymptotically compared with when un-pooled data are used that require $N=cJ$ assays.
4.2 Further remarks
We are now in the position to reflect on the findings in Theorems 4.1 and 4.2 to gain a deeper understanding of the three proposed estimators for $m(x)$ using pooled data.
The stark contrast between properties of the average-weighted estimator under the two pooling designs may seem peculiar at first glance. As natural as it initially appears to be, the use of average weights is the root cause for the persistent bias of $\hat{m}_{1}(x)$ under random pooling. For ease of exposition, assume for the time being $c_{j}=2$, for $j=1,\ldots,J$. The objective function $Q_{1}(\mbox{\boldmath$\beta$})$ in (2.2) associated with $\hat{m}_{1}(x)$ is essentially constructed for estimating $m^{*}(x_{1},x_{2})\triangleq\{m(x_{1})+m(x_{2})\}/2$ evaluated at $(x_{1},x_{2})=x\mbox{\boldmath$1$}_{2}$. The same weight, $\{K_{h}(X_{j1}-x)+K_{h}(X_{j2}-x)\}/2$, is assigned to both individuals in pool $j$ whose covariate values are $\tilde{\mathbf{X}}_{j}=(X_{j1},X_{j2})^{\mathrm{\scriptscriptstyle T}}$. This can yield misleading weight when, for example, $X_{j1}$ is close to $x$ but $X_{j2}$ is far away from $x$, which can often happen under random pooling. In contrast, the product weight in $Q_{2}(\mbox{\boldmath$\beta$})$ in (2.3) associated with $\hat{m}_{2}(x)$ avoids such misleading weighting scheme because $K_{h}(X_{j1}-x)K_{h}(X_{j2}-x)$ is small if either one of the two individual weights is small, and thus $\tilde{\mathbf{X}}_{j}$ will only contribute more in estimating $m^{*}(x,x)=m(x)$ when both $X_{j1}$ and $X_{j2}$ are closer to $x$. In particular, when $K(t)$ is the Gaussian kernel, the product weight function amounts to evaluating the bivariate Gaussian density function at the Euclidean distance between $\tilde{\mathbf{X}}_{j}$ and $x\mbox{\boldmath$1$}_{2}$, whereas the average weight function lacks such connection with a meaningful distance measure between the two points in $\mathbb{R}^{2}$.
Even though $\hat{m}_{2}(x)$ exploits a more sensible weight function when comparing with $\hat{m}_{1}(x)$ under random pooling, downplaying $X_{j1}$ even when it is close to $x$ simply because the covariate value of the other individual in the same pool is far away from $x$ is not an efficient use of data. And such waste of data information is more severe when the pool size is bigger, which is essentially the curse-of-dimensionality when one estimates the multivariate function $m^{*}(x\mbox{\boldmath$1$}_{c})$ based on a response along with a $c$-dimensional covariate. It is such inefficient use of data that causes the much inflated variance concluded in Theorem 4.1 for $\hat{m}_{2}(x)$. Figure 1 illustrates the average weight function and the product weight function (in bottom panels) under random pooling when $c=2$ and $K(t)$ is the Epanechnikov kernel. Also shown in Figure 1 (see the top-left panel) are individual-level data generated according to the model specified in (D1) described in Section 6, overlaid with the pseudo response data from random pooling, which are used for the construction of $\hat{m}_{3}(x)$. From there one can see that the pseudo data, $\{(\hat{Y}_{jk},X_{jk}),k=1,2\}_{j=1}^{J}$, are much more variable than the original data used to obtain $\hat{m}_{0}(x)$, and thus the increased variance of $\hat{m}_{3}(x)$ is expected when compared with $\hat{m}_{0}(x)$. Despite the higher variability, the pseudo data cloud does preserve the overall pattern of the original data cloud, which explains the common dominating bias shared between $\hat{m}_{3}(x)$ and $\hat{m}_{0}(x)$. Unlike $Q_{1}(\mbox{\boldmath$\beta$})$ and $Q_{2}(\mbox{\boldmath$\beta$})$, the construction of $Q_{3}(\mbox{\boldmath$\beta$})$ in (2.5) is directly designed for estimating the univariate function $m(x)$ instead of $m^{*}(x\mbox{\boldmath$1$}_{c})$, and thus $\hat{m}_{3}(x)$ overcomes the pitfall of misleading weight assignment in $\hat{m}_{1}(x)$, as well as the curse-of-dimensionality that $\hat{m}_{2}(x)$ suffers.
Figure 2 is the counterpart of Figure 1 under homogeneous pooling. One can see (in the top-left panel) in Figure 2 that the pseudo data, $\{(\hat{Y}_{(jk)},X_{(jk)}),k=1,2\}_{j=1}^{N}$, clearly distort the original data pattern, and thus are inappropriate for estimating $m(x)$. With individuals sharing similar covariates values gathering in the same pool, the concern relating to $\hat{m}_{1}(x)$ of assigning inadequate weight no longer exists, neither does the concern relating to $\hat{m}_{2}(x)$ of inefficient use of data. The bottom panels of Figure 2 depict the average weight function and the product weight function, both are reminiscent of some symmetric kernel function.
5 Bandwidth selection
The choice of bandwidths in local polynomial estimators plays a key role in the performance of these estimators. Besides the usual challenges encountered in bandwidth selection in local polynomial regression, a unique complication we face here is the lack of individual-level response data, which makes loss functions used for bandwidth selection that are based on individual-level residuals (or prediction errors) inapplicable in our context. Next we develop leave-one-out cross-validation (CV) procedures to choose bandwidths in three proposed local polynomial estimators for $m(x)$ using random pooled data.
For the average-weighted estimator, $\hat{m}_{1}(x)$, we choose the bandwidth $h$ that minimizes the following pool-level residual sum of squares,
$$\displaystyle\textrm{RSS}_{1}(h)$$
$$\displaystyle=\sum_{j=1}^{J}\sum_{k=1}^{c_{j}}\left\{Z_{j}-c_{j}^{-1}\sum_{k=1%
}^{c_{j}}\hat{m}_{1,h}(X_{jk})\right\}^{2},$$
(5.1)
where $\hat{m}_{1,h}(X_{jk})$ is the realization of $\hat{m}_{1}(X_{jk})$ based on the observed data $(\mathbf{Z},\mathbb{X})$ excluding data from pool $j$, $(Z_{j},\tilde{\mathbf{X}}_{j})$, with the bandwidth set at $h$.
The bandwidth in the product-weighted estimator, $\hat{m}_{2}(x)$, is chosen by minimizing a CV criterion similarly defined as (5.1),
$$\displaystyle\textrm{RSS}_{2}(h)$$
$$\displaystyle=\sum_{j=1}^{J}\sum_{k=1}^{c_{j}}\left\{Z_{j}-c_{j}^{-1}\sum_{k=1%
}^{c_{j}}\hat{m}_{2,h}(X_{jk})\right\}^{2}.$$
(5.2)
Admittedly, CV criteria or loss functions constructed based on prediction errors at the pool level may not be sensitive to the influence of $h$ on prediction power at the individual level, and thus may not serve as effective model criteria for the purpose for choosing bandwidths.
Given (5.1) and (5.2), one can easily envision a similar CV criterion, denoted by $\textrm{RSS}_{3}(h)$, defined for choosing $h$ in $\hat{m}_{3}(x)$. We however take into account the close tie between $\hat{m}_{3}(x)$ and local polynomial estimators designed for individual-level data, and propose a new and more effective CV criterion. This new criterion tailored for $\hat{m}_{3}(x)$ is mostly thanks to the pseudo individual-level observations, $\{(\hat{Y}_{jk},X_{jk}),\,k=1,\dots,c\}_{j=1}^{J}$, used in $\hat{m}_{3}(x)$. In particular, we choose $h$ used in $\hat{m}_{3}(x)$ that minimizes the following pseudo (individual-level) residual sum of squares,
$$\displaystyle\textrm{PRSS}_{3}(h)$$
$$\displaystyle={\sum_{j=1}^{J}\sum_{k=1}^{c_{j}}}\{\hat{Y}_{jk}-\hat{m}_{3,h}(X%
_{jk})\}^{2},$$
(5.3)
where $\hat{m}_{3,h}(X_{jk})$ is the realization of $\hat{m}_{3}(X_{jk})$ based on the pseudo individual-level data excluding one pseudo data point, $(\hat{Y}_{jk},X_{jk})$, with the bandwidth set at $h$. Empirical evidence suggest that $\textrm{PRSS}_{3}(h)$ is a more effective CV criterion for bandwidth selection than $\textrm{RSS}_{3}(h)$.
When one is concerned about undesirable boundary effects on the prediction assessment of a fitted model, and in turn on the choice of $h$, one may exclude data near the boundary in the pseudo individual-level prediction in (5.3). For instance, one may only consider computing $\hat{m}_{3,h}(X_{jk})$ when $X_{jk}\in[a,b]$, where $a$ and $b$ are the $2.5$th and $97.5$th sample quantiles of observed covariate data, respectively. Similar data exclusion strategy can be employed in computing (5.1) and (5.2). Lastly, when homogeneous pooled data are used, we employ the same CV criteria defined above, although evaluated at homogeneous pooled data, to choose bandwidths.
6 Simulation study
6.1 Design of simulation experiments
To compare different estimators of $m(x)$ in regard to their finite sample performance, and to explore other factors that may influence the estimation, we carry out an empirical study using synthetic data. More specifically, we adopt the following data generating processes reported in Delaigle et al. (2009) to generate individual-level response data:
(D1)
$m(x)=x^{3}\exp(x^{4}/1000)\cos x$, $\epsilon\sim N(0,0.6^{2})$, $X\sim 0.8X_{1}+0.2X_{2}$, where $X_{1}$ follows a distribution with pdf given by $0.1875x^{2}I(-2\leq x\leq 2)$ and $X_{2}\sim\textrm{uniform}(-1,1)$;
(D2)
$m(x)=2x\exp(-10x^{4}/81)$, $\epsilon\sim(0,0.2^{2})$, $X\sim 0.8X_{1}+0.2X_{2}$, where the distributions of $X_{1}$ and $X_{2}$ are as those specified in (D1);
(D3)
$m(x)=x^{3}$, $\epsilon\sim N(0,1.2^{2})$, $X\sim N(0,1)$;
(D4)
$m(x)=x^{4}$, $\epsilon\sim N(0,4^{2})$, $X\sim N(0,1)$.
Under each data generating process, we generate individual-level data of size $N=600$, $\{(Y_{i},X_{i})\}_{i=1}^{N}$. Given an individual-level data set, we create pooled data, first using random pooling and then using homogeneous pooling, with a common pool size $c=2,3,4,5,6$ across all $J$ pools. Given each pooled data set, we obtain three local linear estimates for the mean function, $\hat{m}_{1}(x)$, $\hat{m}_{2}(x)$, and $\hat{m}_{3}(x)$. In addition, we also compute the local linear estimate using individual-level data, $\hat{m}_{0}(x)$, as a benchmark estimate. In all four estimators, we set $K(t)$ as the Epanechnikov kernel. The empirical integrated squared error (ISE) is the metric we use to assess the quality of an estimated mean function, defined by $\textrm{ISE}=\sum_{j=1}^{J}\sum_{k=1}^{c}\{Y_{jk}-\hat{m}(X_{jk})\}^{2}$ for an estimator $\hat{m}(\cdot)$.
6.2 Simulation results
Figure 3 depicts the three proposed estimators when $c=2$ along with the benchmark estimate $\hat{m}_{0}(x)$ using data generated according to (D2). Appendix E provides parallel results under the other three designs of data generating processes and those when $c=6$.
Under random pooling (see upper panels of Figure 3), the average-weighted estimator $\hat{m}_{1}(x)$ is unable to capture the shape of $m(x)$, and it fails more miserably around regions with more curvature. The product-weighted estimator $\hat{m}_{2}(x)$ is able to recover the overall shape of $m(x)$, although exhibiting a higher variability than $\hat{m}_{0}(x)$, especially around the inflection points of $m(x)$. With $c=2$, the marginal-integration estimator $\hat{m}_{3}(x)$ performs similarly as $\hat{m}_{2}(x)$. When one increases $c$ (see Appendix E), one can see that $\hat{m}_{3}(x)$ shows a much more stable performance in estimating $m(x)$ than $\hat{m}_{2}(x)$ does. This is in line with the implication of Theorem 4.1 that the variance of $\hat{m}_{2}(x)$ inflates faster as the pool size increases than the variance of $\hat{m}_{3}(x)$ does.
Under homogeneous pooling (see lower panels of Figure 3), the marginal-integration estimator $\hat{m}_{3}(x)$ distorts the functional form of $m(x)$, whereas both $\hat{m}_{1}(x)$ and $\hat{m}_{2}(x)$ perform similarly as $\hat{m}_{0}(x)$, in regard to both accuracy and precision.
7 Real-life applications
In this section, we analyze data from two real-life applications to illustrate the proposed local linear estimators for a conditional mean function. The individual-level observations are available in both applications, making it feasible to compute the local linear estimate based on individual-level data, $\hat{m}_{0}(x)$, which we compare our proposed estimates based on pooled data with. In all considered estimators, we set $K(t)$ as the Epanechnikov kernel.
Example 1 (Perfluorinated chemicals): The first data set is from the National Health and Nutrition Examination Survey, relating to a study of the bioaccumulation of perfluorinated chemicals (PFCs) in human bodies. PFCs are widely used in the coating of industrial products, such as food packaging foams and non-stick cookware surfaces, many of which are toxic and accumulate in human bodies. Kärrman et al. (2006) studied the relationship between the concentration levels of PFCs in an individual’s blood and one’s age, gender, and geographic region using pooled serum samples of individuals in Australia. The particular data we entertain here include concentration levels of multiple PFCs in the serum samples of 1,904 residents in the United States between 2011 and 2012, along with their demographic information. The goal of our analysis is to infer the relationship between the concentration level of one particular type of PFCs, perfluorohexane sulfonic acid (PFHxS, $Y$), in an individual’s blood and his/her age ($X$).
To assess the uncertainty of each estimation procedure, we generate 500 bootstrap samples from the raw individual-level data. Based on each bootstrap version of the individual-level data, we compute the local linear estimate, $\hat{m}_{0}(x)$, for the mean concentration level of PFHxS given one’s age. Additionally, using the original data, we randomly create 952 pools, each of size two, producing a set of random pooled data; and we also create 952 pools of equal size based on the sorted data for age, producing a set of homogeneous pooled data. With the pool composition under each pooling design fixed, 500 bootstrap versions of random pooled data, and 500 bootstrap versions of homogeneous pooled data are generated by resampling pools with replacement. Using each pooled data set, we compute $\hat{m}_{1}(x)$, $\hat{m}_{2}(x)$, and $\hat{m}_{3}(x)$, resulting in 500 realizations of each estimator.
Figure 4 depicts the average of each estimate across 500 bootstrap samples and two quantiles of selected estimates. When random pooled data are used, the marginal-integration estimate $\hat{m}_{3}(x)$ matches closely with the benchmark estimate based on individual-level data, $\hat{m}_{0}(x)$, both indicating a relatively stable level of PFHxS with a slight decrease as one approaches age 40, and then a steep increase of the concentration level once one passes around age 50. This pattern can be explained by the fact that PFHxS can be partly eliminated from the human body via, for instance, gastrointestinal activities, menstrual bleeding, and breast feeding (Genuis et al., 2013), but many of these pathways of PFCs elimination become less proactive or are completely lost (such as due to menopause) after one reaches certain age. In contrast, the average-weighted estimate, $\hat{m}_{1}(x)$, and the product-weighted estimate, $\hat{m}_{2}(x)$, suggest a much slower and nearly a constant increase in the concentration level as one gets older across the entire observed age range. We believe that this is one case where $\hat{m}_{1}(x)$ fails to capture the underlying pattern of $m(x)$ due to its inherent inconsistency in estimation, and $\hat{m}_{2}(x)$ also misses this pattern due to its high uncertainty in estimation. In conclusion, when only random pooled data are available, $\hat{m}_{3}(x)$ provides a more reliable estimate for the underlying relationship between one’s PFHxS level in blood and age than the other two proposed estimates, although its variability is slightly higher than that of $\hat{m}_{0}(x)$ according to the bootstrap quantiles of the two estimates.
When homogeneous pooled data are used (see the top-right panel of Figure 4), $\hat{m}_{3}(x)$ appears to exaggerate the curvature of the conditional mean function, resulting in a much faster increase in the concentration level once one passes age 50, compared to the rate of increase indicated by the same estimate under random pooling. Despite the use of pooled data, $\hat{m}_{1}(x)$ and $\hat{m}_{2}(x)$ are nearly indistinguishable from $\hat{m}_{0}(x)$, and these three estimates mostly preserve the earlier estimated pattern of $m(x)$ that can be justified on scientific grounds. Moreover, the variability of $\hat{m}_{1}(x)$ is comparable with that of $\hat{m}_{0}(x)$ according to the comparison of the bootstrap quantiles associated with these two estimates. In conclusion, the marginal-integration estimate $\hat{m}_{3}(x)$ based on homogeneous pooled data leads to misleading inference for the underlying truth, whereas the other two estimates based on pooled data provide inference similar to those from the estimate based on individual-level data without noticeable efficiency loss.
Example 2 (Chemokines): The second data set we use to illustrate local linear estimation using different types of data is from the Collaborative Perinatal Project (CPP), which is a long-standing, collaborative project on maternal and child health in the United States. More specifically, this data include chemokine levels collected from 388 pregnant females recruited in CPP, with measurements taken at the individual level as well as the pool level, with 194 non-overlapping pools of size two randomly formed. Chemokines are a family of small proteins related to the homeostatic and inflammatory process in the human body. Medical researchers have studied extensively the role that chemokines play in the immune system. For example, regarding to two particular chemokines, MCP-3 and GRO-$\alpha$, Dhawan and Richmond (2002) investigated the role of the former in tumorigenesis, and Tsou et al. (2007) studied the latter in monocyte mobilization.
Based on the observed individual-level data and the random pooled data available in CPP, we infer the conditional mean concentration of GRO-$\alpha$ ($Y$) given MCP-3 ($X$). For illustration purpose, we generate another pooled data set, with a common pool size of two, following the homogeneous pooling design based on sorted MCP-3 levels. To assess the uncertainty of each estimation method, we generate 500 bootstrap samples for each of the three data types, individual-level data, random pooled data, and homogeneous pooled data, following the same resampling process described in the first example. Figure 5 shows the average of each considered estimate across 500 bootstrap samples and two quantiles of selected estimates.
Similar to the phenomena in the first example, the marginal-integration estimate $\hat{m}_{3}(x)$ yields a similar estimate for the mean concentration level of GRO-$\alpha$ given the level of MCP-3 as that of $\hat{m}_{0}(x)$ when random pooled data are used; but it grossly deviates from this benchmark estimate when homogeneous pooled data are used. In contrast, the other two proposed local linear estimates based on random pooled data go through an obviously uninteresting region of the observed data, yet both estimates applied to homogeneous pooled data follow closely the benchmark estimate $\hat{m}_{0}(x)$, and they only show slight discrepancy from it around the region where data are relatively scarce.
8 Discussion
We present in this article methods for estimating the mean of a continuous response given covariates via local polynomial regression when only pooled response data are observed along with individual-level covariates. Two commonly adopted pooling designs in practice are considered when formulating the local polynomial estimators, and properties of the proposed estimators are compared under each of the pooling designs. We use two real-life applications to illustrate the implementation and performance of the proposed estimators in comparison with their counterpart estimator when individual response data are available. Findings from the two applications are in line with observations on their finite sample performance using synthetic data from the simulation study, which agree with the theoretical implications of the large-sample properties derived for the proposed estimators.
In summary, the marginal-integration estimator $\hat{m}_{3}(x)$ is the winner among the three proposed when pooled data are from a random pooling design, but it fails when pools are not formed randomly; the average-weighted estimator $\hat{m}_{1}(x)$ performs the best when homogeneous pooled data are used, but it is an inconsistent estimator for the mean function when pools are formed randomly; the product-weighted estimator $\hat{m}_{2}(x)$ is a consistent estimator under both pooling designs, but is subject to high variability under random pooling. Based on our discussions in Section 4.2, we believe that there is still room for improvement by more carefully/selectively incorporating individual covariate information within a pool to relate to the pooled response of that pool, as opposed to either using all covariate information (as in $\hat{m}_{1}(x)$ and $\hat{m}_{2}(x)$) or using one individual’s covariate information (as in $\hat{m}_{3}(x)$). Following this more selective incorporation of covariate information for each pool, an alternative construction of the weight function in the objective function may be needed accordingly to exploit a more sensible measure of distance between selected individuals’ covariate information and $x$, the value at which the mean function is of interest. We are hopeful that this more refined strategy for constructing the objective function can lead to a local polynomial estimator that outperforms all three estimators proposed in the current study despite the pooling design.
Another follow-up research is motivated by the fact that, in many applications, covariates of interest cannot be measured precisely or observed directly. It is of interest then to carry out local polynomial regression to infer $m(x)$ using pooled response data and individual-level covariate data that are prone to measurement error.
Appendix A: Proof of Theorem 1-(i)
A.1 Regularity conditions
Define the variance function as $\sigma^{2}(x)=\textrm{Var}(Y|X=x)$, and denote by $f_{\hbox{\tiny$X$}}(x)$ the probability density function (pdf) of $X$. The following conditions imposed on $m(x)$, $\sigma^{2}(x)$, $f_{\hbox{\tiny$X$}}(x)$, and $K(t)$ are assumed throughout the appendices.
(C1)
The mean function $m(x)$ is $(p+3)$-th order continuously differentiable.
(C2)
The variance function $\sigma^{2}(x)$ is second order continuously differentiable.
(C3)
The density function $f_{\hbox{\tiny$X$}}(x)$ is second order continuously differentiable, and $f(x)>0$.
(C4)
The kernel function $K(t)$ is an even function.
(C5)
$\mu_{\ell}=\int t^{\ell}K(t)dt$ and $\nu_{\ell}=\int t^{\ell}K^{2}(t)dt$ are well-defined for $\ell=0,1,\ldots,2p$.
All integrals are over the entire real line $\mathbb{R}$ in this article.
Conditions (C1)–(C3) relate to certain degree of smoothness of the mean function, the variance function, and the density function of $X$ that are typically required for establishing consistency of local polynomial estimators when individual-level data are available. Conditions (C4) and (C5) are also commonly imposed assumptions for the consistency of kernel density estimators besides local polynomial estimators for a smooth function. Additional conditions associated with a particular estimator under a specific pooling design are stated in the relevant theorem when needed.
The average-weighted $p$-th order local polynomial estimator of $m(x)$, denoted by $\hat{m}_{1}(x)$, results from minimizing the following objective function,
$$Q_{1}(\mbox{\boldmath$\beta$})=\sum_{j=1}^{J}\left\{Z_{j}-\sum_{\ell=0}^{p}%
\beta_{\ell}c_{j}^{-1}\sum_{k=1}^{c_{j}}(X_{jk}-x)^{\ell}\right\}^{2}\left\{c_%
{j}^{-1}\sum_{k=1}^{c_{j}}K_{h}(X_{jk}-x)\right\},$$
where $\beta_{\ell}=m^{(\ell)}(x)/\ell!$, for $\ell=0,1,\ldots,p$. Define $\mbox{\boldmath$\beta$}=(\beta_{0},\beta_{1},\ldots,\beta_{p})^{\mathrm{%
\scriptscriptstyle T}}$, with its dependence on $x$ suppressed. Equivalently, one has $\hat{m}_{1}(x)=\mbox{\boldmath$e$}_{1}^{\mathrm{\scriptscriptstyle T}}\mathbf{%
S}_{1}^{-1}(x)\mathbf{T}_{1}(x)$,
where $\mathbf{S}_{1}(x)=\mathbf{D}_{1}(x)^{\mathrm{\scriptscriptstyle T}}\mathbf{K}_%
{1}(x)\mathbf{D}_{1}(x)=\left[S_{1,\ell_{1},\ell_{2}}(x)\right]_{\ell_{1},\ell%
_{2}=0,1,\ldots,p}$, and $\mathbf{T}_{1}(x)=\mathbf{D}_{1}(x)^{\mathrm{\scriptscriptstyle T}}\mathbf{K}_%
{1}(x)\mathbf{Z}=(T_{1,0}(x),\,T_{1,1}(x),\,\ldots,\,T_{1,p}(x))^{\mathrm{%
\scriptscriptstyle T}}$, in which $\mathbf{Z}=(Z_{1},\ldots,\,Z_{J})^{\mathrm{\scriptscriptstyle T}}$,
$$\displaystyle\mathbf{D}_{1}(x)$$
$$\displaystyle=\begin{bmatrix}1&\bar{X}_{1}-x&c_{1}^{-1}\sum_{k=1}^{c_{1}}(X_{1%
k}-x)^{2}&\ldots&c_{1}^{-1}\sum_{k=1}^{c_{1}}(X_{1k}-x)^{p}\\
\vdots&\vdots&\vdots&\ldots&\vdots\\
1&\bar{X}_{J}-x&c_{J}^{-1}\sum_{k=1}^{c_{J}}(X_{Jk}-x)^{2}&\ldots&c_{J}^{-1}%
\sum_{k=1}^{c_{J}}(X_{Jk}-x)^{p}\end{bmatrix},$$
(A.1)
$$\displaystyle\mathbf{K}_{1}(x)$$
$$\displaystyle=\textrm{diag}\left(c_{1}^{-1}\sum_{k=1}^{c_{1}}K_{h}(X_{jk}-x),%
\,\ldots,\,c_{J}^{-1}\sum_{k=1}^{c_{J}}K_{h}(X_{Jk}-x)\right),$$
and $\bar{X}_{j}=c_{j}^{-1}\sum_{k=1}^{c_{j}}X_{jk}$, for $j=1,\ldots,J$. The entry on the $(\ell_{1}+1)$-th row and $(\ell_{2}+1)$-th column of $\mathbf{S}_{1}(x)$ is, for $\ell_{1},\ell_{2}=0,1,\ldots,p$,
$$\displaystyle\ S_{1,\ell_{1},\ell_{2}}(x)$$
$$\displaystyle=$$
$$\displaystyle\ \sum_{j=1}^{J}\left\{c_{j}^{-1}\sum_{k=1}^{c_{j}}(X_{jk}-x)^{%
\ell_{1}}\right\}\left\{c_{j}^{-1}\sum_{k=1}^{c_{j}}(X_{jk}-x)^{\ell_{2}}%
\right\}\left\{c_{j}^{-1}\sum_{k=1}^{c_{j}}K_{h}(X_{jk}-x)\right\};$$
(A.2)
and the $(\ell+1)$-th entry of $\mathbf{T}_{1}(x)$ is, for $\ell=0,1,\ldots,p$,
$$T_{1,\ell}(x)=\sum_{j=1}^{J}Z_{j}\left\{c_{j}^{-1}\sum_{k=1}^{c_{j}}(X_{jk}-x)%
^{\ell}\right\}\left\{c_{j}^{-1}\sum_{k=1}^{c_{j}}K_{h}(X_{jk}-x)\right\}.$$
(A.3)
In what follows, we study the mean and variance of $J^{-1}\mathbf{S}_{1}(x)$ and $\mathbf{C}^{(1)}_{p}(x)=J^{-1}\{\mathbf{T}_{1}(x)-\mathbf{S}_{1}(x)\mbox{%
\boldmath$\beta$}\}$ in order to reveal dominating terms of $\hat{\mbox{\boldmath$\beta$}}_{1}-\mbox{\boldmath$\beta$}$, where $\hat{\mbox{\boldmath$\beta$}}_{1}=\mathbf{S}_{1}^{-1}(x)\mathbf{T}_{1}(x)$.
A.2 Frequently used notations and results
We first defined some frequently used notations and list some useful results to be referenced later. Throughout this document, $\ell$ and $p$ are non-negative integers. For a kernel $K(t)$, define
$$\displaystyle\mu_{\ell}$$
$$\displaystyle=\int t^{\ell}K(t)dt,\quad\nu_{\ell}=\int t^{\ell}K^{2}(t)dt,$$
$$\displaystyle\mbox{\boldmath$\mu$}^{*}_{\ell}$$
$$\displaystyle=(\mu_{\ell},\mu_{\ell+1},\ldots,\mu_{\ell+p})^{\mathrm{%
\scriptscriptstyle T}},\quad\tilde{\mbox{\boldmath$\mu$}}_{\ell}=[\mu_{\ell_{1%
}+\ell_{2}+\ell}]_{\ell_{1},\ell_{2}=0,1,\ldots,p},$$
$$\displaystyle\mbox{\boldmath$\nu$}^{*}_{\ell}$$
$$\displaystyle=(\nu_{\ell},\nu_{\ell+1},\ldots,\nu_{\ell+p})^{\mathrm{%
\scriptscriptstyle T}},\quad\tilde{\mbox{\boldmath$\nu$}}_{\ell}=[\nu_{\ell_{1%
}+\ell_{2}+\ell}]_{\ell_{1},\ell_{2}=0,1,\ldots,p}.$$
All integrals in this document are over the entire real line unless otherwise stated.
By definition, $\mbox{\boldmath$\mu$}^{*}_{\ell}$ is a $(p+1)\times 1$ vector, and $\tilde{\mbox{\boldmath$\mu$}}_{\ell}$ is a $(p+1)\times(p+1)$ matrix of which the entry on the $(\ell_{1}+1)$th row and $(\ell_{2}+1)$th column is $\mu_{\ell_{1}+\ell_{2}+\ell}$, for $\ell_{1},\ell_{2}=0,1,\ldots,p$. The structures of $\mbox{\boldmath$\nu$}^{*}_{\ell}$ and $\tilde{\mbox{\boldmath$\nu$}}_{\ell}$ are similar to those of $\mbox{\boldmath$\mu$}^{*}_{\ell}$ and $\tilde{\mbox{\boldmath$\mu$}}_{\ell}$, respectively. Moreover, because $K(t)$ is a kernel, $\mu_{0}=1$; and, with $K(t)$ being a symmetric kernel as we assume throughout the study, $\mu_{\ell}=\nu_{\ell}=0$ for an odd integer $\ell$.
Given the bandwidth $h$, define the following $(p+1)\times(p+1)$ matrices, $\mathbf{H}=\mbox{diag}(1,h,\ldots,h^{p})$, and $\mathbf{H}^{*}=[h^{2(\ell_{1}+\ell_{2})}]_{\ell_{1},\ell_{2}=0,1,\ldots,p}$. Given the pool composition, define the following constants that are of order $O(1)$ as $J\to\infty$,
$$\displaystyle t_{k0}$$
$$\displaystyle=\frac{1}{J}\sum_{j=1}^{J}\frac{1}{c^{k}_{j}},\textrm{ for $k\in%
\{0,1,2\}$,}$$
$$\displaystyle t_{k_{1}k_{2}}$$
$$\displaystyle=\frac{1}{J}\sum_{j=1}^{J}\frac{[\prod_{k=1}^{k_{2}}(c_{j}-k)]_{+%
}}{c_{j}^{k_{1}}},\textrm{ for $k_{1},k_{2}\in\{1,2,3\}$},$$
where $[s]_{+}=\max(0,s)$ for a constant $s$. Clearly, when $c_{j}=1$ for all $j=1,\ldots,J$, $t_{k0}=1$ and $t_{k_{1}k_{2}}=0$ for all $k,k_{1},k_{2}$.
For a real number $x$ in the support of $X$, define $\delta_{\ell}(x)=E\{(X-x)^{\ell}\}$ and $R_{\ell,p}=E\{(X-x)^{\ell}r_{p}(X,x)\}$, assuming that the expectations exist, where $r_{p}(X,x)=m(X)-\sum_{\ell=0}^{p}\beta_{\ell}(X-x)^{\ell}$. Then define the following vectors and matrices,
$$\displaystyle\mbox{\boldmath$\Delta$}^{*}_{\ell}(x)$$
$$\displaystyle=(\delta_{\ell}(x),\delta_{\ell+1}(x),\ldots,\delta_{\ell+p}(x))^%
{\mathrm{\scriptscriptstyle T}},\quad\tilde{\mbox{\boldmath$\Delta$}}_{\ell}(x%
)=[\delta_{\ell_{1}+\ell_{2}+\ell}(x)]_{\ell_{1},\ell_{2}=0,1,\ldots,p},$$
$$\displaystyle\mathbf{R}^{*}_{p}(x)$$
$$\displaystyle=(R_{0,p}(x),\ldots,R_{p,p}(x))^{\mathrm{\scriptscriptstyle T}}.$$
Lastly, define the following expectations assuming they exist, for $\ell=0,1,\ldots,2p$,
$$\left\{\begin{aligned} \displaystyle\kappa_{1,\ell}(h,x)&\displaystyle=E\{(X-x%
)^{\ell}K_{h}(X-x)\},\\
\displaystyle\kappa_{2,\ell}(h,x)&\displaystyle=E\{(X-x)^{\ell}K^{2}_{h}(X-x)%
\},\\
\displaystyle\kappa^{*}_{1,\ell}(g,h,x)&\displaystyle=E\{g(X)(X-x)^{\ell}K_{h}%
(X-x)\},\\
\displaystyle\kappa^{*}_{2,\ell}(g,h,x)&\displaystyle=E\{g(X)(X-x)^{\ell}K^{2}%
_{h}(X-x)\},\end{aligned}\right.$$
(A.4)
where $g(x)$ is a generic function that is fourth-order continuously differentiable. When it causes no confusion given the context, we suppress the dependence of these quantities on $(h,x)$ in some derivations later. For example, we sometimes write $\kappa^{*}_{1,\ell}(g)$ in place of $\kappa^{*}_{1,\ell}(g,h,x)$.
Assuming that the density of $X$, $f_{\hbox{\tiny$X$}}(x)$, is fourth-order continuously differentiable, one can show the following regarding (A.4),
$$\displaystyle\kappa_{1,\ell}(h,x)$$
$$\displaystyle=\sum_{q=0}^{3}h^{\ell+q}\mu_{\ell+q}f^{(q)}_{\hbox{\tiny$X$}}(x)%
/q!+O\left(h^{\ell+4}\right)$$
(A.5)
$$\displaystyle\kappa_{2,\ell}(h,x)$$
$$\displaystyle=\sum_{q=0}^{3}\ h^{\ell+q-1}\nu_{\ell+q}f^{(q)}_{\hbox{\tiny$X$}%
}(x)/q!+O\left(h^{\ell+3}\right).$$
(A.6)
With $\ell=0$ , one has, for a positive integer $c$,
$$\displaystyle\kappa^{c}_{1,0}(h,x)$$
$$\displaystyle=f^{c-1}_{\hbox{\tiny$X$}}(x)\left\{f_{\hbox{\tiny$X$}}(x)+ch^{2}%
\mu_{2}f_{\hbox{\tiny$X$}}^{\prime\prime}(x)/2\right\}+O(h^{4}),$$
(A.7)
$$\displaystyle\kappa^{c}_{2,0}(h,x)$$
$$\displaystyle=h^{-c}\nu_{0}f^{c}_{\hbox{\tiny$X$}}(x)+O(h^{2-c}).$$
(A.8)
Moreover,
$$\displaystyle\kappa^{*}_{1,\ell}(g,h,x)$$
$$\displaystyle=\ h^{\ell}\mu_{\ell}g(x)f_{\hbox{\tiny$X$}}(x)+h^{\ell+1}\mu_{%
\ell+1}\left\{g(x)f^{\prime}_{\hbox{\tiny$X$}}(x)+g^{\prime}(x)f_{\hbox{\tiny$%
X$}}(x)\right\}+$$
$$\displaystyle\quad\,h^{\ell+2}\mu_{\ell+2}\left\{g(x)f^{\prime\prime}_{\hbox{%
\tiny$X$}}(x)/2+g^{\prime}(x)f^{\prime}_{\hbox{\tiny$X$}}(x)+g^{\prime\prime}(%
x)f_{\hbox{\tiny$X$}}(x)/2\right\}+$$
$$\displaystyle\quad\,O\left(h^{\ell+3}\right),$$
(A.9)
$$\displaystyle\kappa^{*}_{2,\ell}(g,h,x)$$
$$\displaystyle=h^{\ell-1}\nu_{\ell}g(x)f_{\hbox{\tiny$X$}}(x)+h^{\ell}\nu_{\ell%
+1}\left\{g(x)f^{\prime}_{\hbox{\tiny$X$}}(x)+g^{\prime}(x)f_{\hbox{\tiny$X$}}%
(x)\right\}+$$
$$\displaystyle\quad\,h^{\ell+1}\nu_{\ell+2}\left\{g(x)f^{\prime\prime}_{\hbox{%
\tiny$X$}}(x)/2+g^{\prime}(x)f^{\prime}_{\hbox{\tiny$X$}}(x)+g^{\prime\prime}(%
x)f_{\hbox{\tiny$X$}}(x)/2\right\}+$$
$$\displaystyle\quad\,O\left(h^{\ell+1}\right).$$
(A.10)
A.3 About $J^{-1}\mathbf{S}_{1}(x)$
By (A.2), one has
$$\displaystyle E\{J^{-1}S_{1,\ell_{1},\ell_{2}}(x)\}$$
$$\displaystyle=J^{-1}\sum_{j=1}^{J}E\left[\left\{c_{j}^{-1}\sum_{k=1}^{c_{j}}(X%
_{jk}-x)^{\ell_{1}}\right\}\left\{c_{j}^{-1}\sum_{k=1}^{c_{j}}(X_{jk}-x)^{\ell%
_{2}}\right\}\right.$$
$$\displaystyle\quad\left.\times\left\{c_{j}^{-1}\sum_{k=1}^{c_{j}}K_{h}(X_{jk}-%
x)\right\}\right]$$
$$\displaystyle=J^{-1}\sum_{j=1}^{J}c_{j}^{-3}E\left[\sum_{k=1}^{c_{j}}(X_{jk}-x%
)^{\ell_{1}+\ell_{2}}K_{h}(X_{jk}-x)\right.$$
$$\displaystyle\quad+\sum_{k_{1}\neq k_{2}}\left\{(X_{jk_{1}}-x)^{\ell_{1}+\ell_%
{2}}K_{h}(X_{jk_{2}}-x)\right.$$
$$\displaystyle\quad+(X_{jk_{1}}-x)^{\ell_{1}}(X_{jk_{2}}-x)^{\ell_{2}}K_{h}(X_{%
jk_{2}}-x)$$
$$\displaystyle\quad\left.+(X_{jk_{1}}-x)^{\ell_{1}}(X_{jk_{2}}-x)^{\ell_{2}}K_{%
h}(X_{jk_{1}}-x)\right\}$$
$$\displaystyle\quad\left.+\sum_{k_{1}\neq k_{2}\neq k_{3}}(X_{jk_{1}}-x)^{\ell_%
{1}}(X_{jk_{2}}-x)^{\ell_{2}}K_{h}(X_{jk_{3}}-x)\right].$$
Thus,
$$\displaystyle E\{J^{-1}S_{1,\ell_{1},\ell_{2}}(x)\}$$
$$\displaystyle=J^{-1}\sum_{j=1}^{J}c_{j}^{-3}\left[c_{j}\kappa_{1,\ell_{1}+\ell%
_{2}}(h,x)+c_{j}(c_{j}-1)\left\{\delta_{\ell_{1}+\ell_{2}}(x)\kappa_{1,0}(h,x)%
\right.\right.$$
$$\displaystyle\quad\left.+\delta_{\ell_{1}}(x)\kappa_{1,\ell_{2}}(h,x)+\delta_{%
\ell_{2}}(x)\kappa_{1,\ell_{1}}(h,x)\right\}$$
$$\displaystyle\quad\left.+c_{j}(c_{j}-1)(c_{j}-2)\delta_{\ell_{1}}(x)\delta_{%
\ell_{2}}(x)\kappa_{1,0}(h,x)\right].$$
Using (A.5) in the last expression gives
$$\displaystyle E\{J^{-1}S_{1,\ell_{1},\ell_{2}}(x)\}$$
$$\displaystyle=$$
$$\displaystyle\ t_{20}\left\{h^{\ell_{1}+\ell_{2}}\mu_{\ell_{1}+\ell_{2}}f_{%
\hbox{\tiny$X$}}(x)+h^{\ell_{1}+\ell_{2}+1}\mu_{\ell_{1}+\ell_{2}+1}f^{\prime}%
_{\hbox{\tiny$X$}}(x)+0.5h^{\ell_{1}+\ell_{2}+2}\mu_{\ell_{1}+\ell_{2}+2}f^{%
\prime\prime}_{\hbox{\tiny$X$}}(x)\right.$$
$$\displaystyle\left.+O\left(h^{\ell_{1}+\ell_{2}+3}\right)\right\}+t_{21}\left[%
\delta_{\ell_{1}+\ell_{2}}(x)\left\{\mu_{0}f_{\hbox{\tiny$X$}}(x)+0.5h^{2}\mu_%
{2}f^{\prime\prime}_{\hbox{\tiny$X$}}(x)+O(h^{3})\right\}\right.$$
$$\displaystyle\ +\delta_{\ell_{1}}(x)\left\{h^{\ell_{2}}\mu_{\ell_{2}}f_{\hbox{%
\tiny$X$}}(x)+h^{\ell_{2}+1}\mu_{\ell_{2}+1}f^{\prime}_{\hbox{\tiny$X$}}(x)+0.%
5h^{\ell_{2}+2}\mu_{\ell_{2}+2}f^{\prime\prime}_{\hbox{\tiny$X$}}(x)+O\left(h^%
{\ell_{2}+3}\right)\right\}$$
$$\displaystyle\left.+\delta_{\ell_{2}}(x)\left\{h^{\ell_{1}}\mu_{\ell_{1}}f_{%
\hbox{\tiny$X$}}(x)+h^{\ell_{1}+1}\mu_{\ell_{1}+1}f^{\prime}_{\hbox{\tiny$X$}}%
(x)+0.5h^{\ell_{1}+2}\mu_{\ell_{1}+2}f^{\prime\prime}_{\hbox{\tiny$X$}}(x)+O%
\left(h^{\ell_{1}+3}\right)\right\}\right]$$
$$\displaystyle\ +t_{22}\delta_{\ell_{1}}(x)\delta_{\ell_{2}}(x)\left\{\mu_{0}f_%
{\hbox{\tiny$X$}}(x)+0.5h^{2}\mu_{2}f^{\prime\prime}_{\hbox{\tiny$X$}}(x)+O(h^%
{3})\right\}$$
$$\displaystyle=$$
$$\displaystyle\ \{t_{21}\delta_{\ell_{1}+\ell_{2}}(x)+t_{22}\delta_{\ell_{1}}(x%
)\delta_{\ell_{2}}(x)\}\left\{\mu_{0}f_{\hbox{\tiny$X$}}(x)+0.5h^{2}\mu_{2}f^{%
\prime\prime}_{\hbox{\tiny$X$}}(x)+O(h^{3})\right\}$$
$$\displaystyle\ +t_{21}\left[\delta_{\ell_{1}}(x)\left\{h^{\ell_{2}}\mu_{\ell_{%
2}}f_{\hbox{\tiny$X$}}(x)+h^{\ell_{2}+1}\mu_{\ell_{2}+1}f^{\prime}_{\hbox{%
\tiny$X$}}(x)+0.5h^{\ell_{2}+2}\mu_{\ell_{2}+2}f^{\prime\prime}_{\hbox{\tiny$X%
$}}(x)+O\left(h^{\ell_{2}+3}\right)\right\}\right.$$
$$\displaystyle\left.+\delta_{\ell_{2}}(x)\left\{h^{\ell_{1}}\mu_{\ell_{1}}f_{%
\hbox{\tiny$X$}}(x)+h^{\ell_{1}+1}\mu_{\ell_{1}+1}f^{\prime}_{\hbox{\tiny$X$}}%
(x)+0.5h^{\ell_{1}+2}\mu_{\ell_{1}+2}f^{\prime\prime}_{\hbox{\tiny$X$}}(x)+O%
\left(h^{\ell_{1}+3}\right)\right\}\right]$$
$$\displaystyle\ +t_{20}\left\{h^{\ell_{1}+\ell_{2}}\mu_{\ell_{1}+\ell_{2}}f_{%
\hbox{\tiny$X$}}(x)+h^{\ell_{1}+\ell_{2}+1}\mu_{\ell_{1}+\ell_{2}+1}f^{\prime}%
_{\hbox{\tiny$X$}}(x)+0.5h^{\ell_{1}+\ell_{2}+2}\mu_{\ell_{1}+\ell_{2}+2}f^{%
\prime\prime}_{\hbox{\tiny$X$}}(x)\right.$$
$$\displaystyle\left.+O\left(h^{\ell_{1}+\ell_{2}+3}\right)\right\}.$$
(A.11)
Hence,
$$\displaystyle E\left\{J^{-1}\mathbf{S}_{1}(x)\right\}$$
$$\displaystyle=$$
$$\displaystyle\ \left\{\mu_{0}f_{\hbox{\tiny$X$}}(x)+0.5h^{2}\mu_{2}f^{\prime%
\prime}_{\hbox{\tiny$X$}}(x)+O(h^{4})\right\}\left\{t_{21}\tilde{\mbox{%
\boldmath$\Delta$}}_{0}(x)+t_{22}\mbox{\boldmath$\Delta$}^{*}_{0}(x)\mbox{%
\boldmath$\Delta$}^{*{\mathrm{\scriptscriptstyle T}}}_{0}(x)\right\}+$$
$$\displaystyle\ t_{20}\mathbf{H}\left\{f_{\hbox{\tiny$X$}}(x)\tilde{\mbox{%
\boldmath$\mu$}}_{0}+hf^{\prime}_{\hbox{\tiny$X$}}(x)\tilde{\mbox{\boldmath$%
\mu$}}_{1}+0.5h^{2}f^{\prime\prime}_{\hbox{\tiny$X$}}(x)\tilde{\mbox{\boldmath%
$\mu$}}_{2}+O(h^{4})\right\}\mathbf{H}+$$
$$\displaystyle\ t_{21}\left(\mbox{\boldmath$\Delta$}^{*}_{0}(x)\left\{f_{\hbox{%
\tiny$X$}}(x)\mbox{\boldmath$\mu$}^{*{\mathrm{\scriptscriptstyle T}}}_{0}+hf^{%
\prime}_{\hbox{\tiny$X$}}(x)\mbox{\boldmath$\mu$}^{*{\mathrm{%
\scriptscriptstyle T}}}_{1}+0.5h^{2}f^{\prime\prime}_{\hbox{\tiny$X$}}(x)\mbox%
{\boldmath$\mu$}^{*{\mathrm{\scriptscriptstyle T}}}_{2}\right\}\mathbf{H}+\right.$$
$$\displaystyle\ \mathbf{H}\left\{f_{\hbox{\tiny$X$}}(x)\mbox{\boldmath$\mu$}^{*%
}_{0}+hf^{\prime}_{\hbox{\tiny$X$}}(x)\mbox{\boldmath$\mu$}^{*}_{1}+0.5h^{2}f^%
{\prime\prime}_{\hbox{\tiny$X$}}(x)\mbox{\boldmath$\mu$}^{*}_{2}\right\}\mbox{%
\boldmath$\Delta$}^{*{\mathrm{\scriptscriptstyle T}}}_{0}(x)+$$
$$\displaystyle\left.+O\left[h^{3}\left\{\mbox{\boldmath$\Delta$}_{0}^{*}(x)%
\mbox{\boldmath$\mu$}_{3}^{*{\mathrm{\scriptscriptstyle T}}}\mathbf{H}+\mathbf%
{H}\mbox{\boldmath$\mu$}^{*}_{3}\mbox{\boldmath$\Delta$}_{0}^{*{\mathrm{%
\scriptscriptstyle T}}}(x)\right\}\right]\right).$$
(A.12)
If $c_{j}=1$ for $j=1,\ldots,J$, then $t_{20}=1$ and $t_{21}=t_{22}=0$, and thus (A.12) reduces to
$$\mathbf{H}\left\{f_{\hbox{\tiny$X$}}(x)\tilde{\mbox{\boldmath$\mu$}}_{0}+hf^{%
\prime}_{\hbox{\tiny$X$}}(x)\tilde{\mbox{\boldmath$\mu$}}_{1}+0.5h^{2}f^{%
\prime\prime}_{\hbox{\tiny$X$}}(x)\tilde{\mbox{\boldmath$\mu$}}_{2}+O(h^{4})%
\right\}\mathbf{H},$$
which is exactly the counterpart result in local polynomial regression using individual-level data. If $c_{j}>1$ for some pools, the first term (before $t_{20}$) in (A.12) remains there despite the order of the polynomial, i.e., the value of $p$. In fact, the dominating term in each entry of $E\{J^{-1}\mathbf{S}_{1}(x)\}$, that is, (A.11), depend on $(\ell_{1},\ell_{2})$ only via their dependence on $\delta_{\ell}(x)$, for $\ell=\ell_{1},\ell_{2},\ell_{1}+\ell_{2}$, which are constant functions free of $h$. This suggests that orders of the dominating term in (A.11) remain the same for all $(\ell_{1},\ell_{2})$. This stands in stark contrast with the other two proposed estimators, $\hat{m}_{2}(x)$ and $\hat{m}_{3}(x)$, as to become clear in Appendices B and C.
The order of $\textrm{Var}\{J^{-1}S_{1,\ell_{1},\ell_{2}}(x)\}$ is determined the order of
$$\displaystyle\ J^{-2}\sum_{j=1}^{J}E\left[\left\{c_{j}^{-1}\sum_{k=1}^{c_{j}}(%
X_{jk}-x)^{\ell_{1}}\right\}^{2}\left\{c_{j}^{-1}\sum_{k=1}^{c_{j}}(X_{jk}-x)^%
{\ell_{2}}\right\}^{2}\times\right.$$
$$\displaystyle\left.\left\{c_{j}^{-1}\sum_{k=1}^{c_{j}}K_{h}(X_{jk}-x)\right\}^%
{2}\right].$$
(A.13)
According to (A.5) and (A.6), the expectation as the summand of (A.13) is dominated by $E\{K_{h}^{2}(X-x)\}$ for all $(\ell_{1},\ell_{2})$, which is of order $O(h^{-1})$. It follows that (A.13) is of order $O\{1/(Jh)\}$, and so is $\textrm{Var}\{J^{-1}S_{1,\ell_{1},\ell_{2}}(x)\}$. Along with (A.12), we now have
$$\displaystyle J^{-1}\mathbf{S}_{1}(x)$$
$$\displaystyle=$$
$$\displaystyle\ \left\{\mu_{0}f_{\hbox{\tiny$X$}}(x)+0.5h^{2}\mu_{2}f^{\prime%
\prime}_{\hbox{\tiny$X$}}(x)+O(h^{4})\right\}\left\{t_{21}\tilde{\mbox{%
\boldmath$\Delta$}}_{0}(x)+t_{22}\mbox{\boldmath$\Delta$}^{*}_{0}(x)\mbox{%
\boldmath$\Delta$}^{*{\mathrm{\scriptscriptstyle T}}}_{0}(x)\right\}+$$
$$\displaystyle\ t_{20}\mathbf{H}\left\{f_{\hbox{\tiny$X$}}(x)\tilde{\mbox{%
\boldmath$\mu$}}_{0}+hf^{\prime}_{\hbox{\tiny$X$}}(x)\tilde{\mbox{\boldmath$%
\mu$}}_{1}+0.5h^{2}f^{\prime\prime}_{\hbox{\tiny$X$}}(x)\tilde{\mbox{\boldmath%
$\mu$}}_{2}+O(h^{4})\right\}\mathbf{H}+$$
$$\displaystyle\ t_{21}\left(\mbox{\boldmath$\Delta$}^{*}_{0}(x)\left\{f_{\hbox{%
\tiny$X$}}(x)\mbox{\boldmath$\mu$}^{*{\mathrm{\scriptscriptstyle T}}}_{0}+hf^{%
\prime}_{\hbox{\tiny$X$}}(x)\mbox{\boldmath$\mu$}^{*{\mathrm{%
\scriptscriptstyle T}}}_{1}+0.5h^{2}f^{\prime\prime}_{\hbox{\tiny$X$}}(x)\mbox%
{\boldmath$\mu$}^{*{\mathrm{\scriptscriptstyle T}}}_{2}\right\}\mathbf{H}+\right.$$
$$\displaystyle\ \mathbf{H}\left\{f_{\hbox{\tiny$X$}}(x)\mbox{\boldmath$\mu$}^{*%
}_{0}+hf^{\prime}_{\hbox{\tiny$X$}}(x)\mbox{\boldmath$\mu$}^{*}_{1}+0.5h^{2}f^%
{\prime\prime}_{\hbox{\tiny$X$}}(x)\mbox{\boldmath$\mu$}^{*}_{2}\right\}\mbox{%
\boldmath$\Delta$}^{*{\mathrm{\scriptscriptstyle T}}}_{0}(x)+$$
$$\displaystyle\left.+O\left[h^{3}\left\{\mbox{\boldmath$\Delta$}_{0}^{*}(x)%
\mbox{\boldmath$\mu$}_{3}^{*{\mathrm{\scriptscriptstyle T}}}\mathbf{H}+\mathbf%
{H}\mbox{\boldmath$\mu$}^{*}_{3}\mbox{\boldmath$\Delta$}_{0}^{*{\mathrm{%
\scriptscriptstyle T}}}(x)\right\}\right]\right)+O_{\hbox{\tiny$P$}}\left(%
\frac{1}{\sqrt{Jh}}\mathbf{I}_{p+1}\right).$$
(A.14)
It follows that
$$\displaystyle\left\{J^{-1}\mathbf{S}_{1}(x)\right\}^{-1}$$
$$\displaystyle=$$
$$\displaystyle\ \mathbf{H}^{-1}(\mathbf{A}+h\mathbf{B}+h^{2}\mathbf{C})^{-1}%
\mathbf{H}^{-1}+O\left(h^{4}\mathbf{H}^{-2}\right)+O_{\hbox{\tiny$P$}}\left(%
\frac{1}{\sqrt{Jh}}\right)$$
$$\displaystyle=$$
$$\displaystyle\ \mathbf{H}^{-1}\mathbf{A}^{-1}\left\{\mathbf{A}-h\mathbf{B}+h^{%
2}\left(\mathbf{B}\mathbf{A}^{-1}\mathbf{B}+\mathbf{C}\right)-h^{3}\left(%
\mathbf{B}\mathbf{A}^{-1}\mathbf{B}\mathbf{A}^{-1}\mathbf{B}+\mathbf{C}\mathbf%
{A}^{-1}\mathbf{B}+\right.\right.$$
$$\displaystyle\left.\left.\mathbf{B}\mathbf{A}^{-1}\mathbf{C}\right)\right\}%
\mathbf{A}^{-1}\mathbf{H}^{-1}+O\left(h^{4}\mathbf{H}^{-2}\right)+O_{\hbox{%
\tiny$P$}}\left(\frac{1}{\sqrt{Jh}}\mathbf{I}_{p+1}\right),$$
(A.15)
$$\displaystyle\mathbf{A}=$$
$$\displaystyle\ f_{\hbox{\tiny$X$}}(x)\left[t_{20}\tilde{\mbox{\boldmath$\mu$}}%
_{0}+\mathbf{H}^{-1}\left\{t_{21}\tilde{\mbox{\boldmath$\Delta$}}_{0}(x)+t_{22%
}\mbox{\boldmath$\Delta$}_{0}^{*}(x)\mbox{\boldmath$\Delta$}_{0}^{*{\mathrm{%
\scriptscriptstyle T}}}(x)\right\}\mathbf{H}^{-1}+\right.$$
$$\displaystyle\left.t_{21}\left\{\mathbf{H}^{-1}\mbox{\boldmath$\Delta$}_{0}^{*%
}(x)\mbox{\boldmath$\mu$}_{0}^{*{\mathrm{\scriptscriptstyle T}}}+\mbox{%
\boldmath$\mu$}^{*}_{0}\mbox{\boldmath$\Delta$}_{0}^{*{\mathrm{%
\scriptscriptstyle T}}}(x)\mathbf{H}^{-1}\right\}\right],$$
$$\displaystyle\mathbf{B}=$$
$$\displaystyle\ f^{\prime}_{\hbox{\tiny$X$}}(x)\left[t_{20}\tilde{\mbox{%
\boldmath$\mu$}}_{1}+t_{21}\left\{\mathbf{H}^{-1}\mbox{\boldmath$\Delta$}_{0}^%
{*}(x)\mbox{\boldmath$\mu$}_{1}^{T}+\mbox{\boldmath$\mu$}^{*}_{1}\mbox{%
\boldmath$\Delta$}_{0}^{*{\mathrm{\scriptscriptstyle T}}}(x)\mathbf{H}^{-1}%
\right\}\right],$$
$$\displaystyle\mathbf{C}=$$
$$\displaystyle\ 0.5f^{\prime\prime}_{\hbox{\tiny$X$}}(x)\left[t_{20}\tilde{%
\mbox{\boldmath$\mu$}}_{2}+\mu_{2}\mathbf{H}^{-1}\left\{t_{21}\tilde{\mbox{%
\boldmath$\Delta$}}_{0}(x)+t_{22}\mbox{\boldmath$\Delta$}_{0}^{*}(x)\mbox{%
\boldmath$\Delta$}_{0}^{*{\mathrm{\scriptscriptstyle T}}}(x)\right\}\mathbf{H}%
^{-1}+\right.$$
$$\displaystyle\left.t_{21}\left\{\mathbf{H}^{-1}\mbox{\boldmath$\Delta$}_{0}^{*%
}(x)\mbox{\boldmath$\mu$}_{2}^{*{\mathrm{\scriptscriptstyle T}}}+\mbox{%
\boldmath$\mu$}^{*}_{2}\mbox{\boldmath$\Delta$}_{0}^{*{\mathrm{%
\scriptscriptstyle T}}}(x)\mathbf{H}^{-1}\right\}\right].$$
A.4 About $\mathbf{C}^{(1)}_{p}(x)$
Write $\mathbf{C}^{(1)}_{p}(x)$ as $(C^{(1)}_{p,0}(x),C^{(1)}_{p,1}(x),\ldots,C^{(1)}_{p,p}(x))^{\mathrm{%
\scriptscriptstyle T}}$. By (A.2) and (A.3), and using iterated expectations, one has, for $\ell=0,1,\ldots,p$,
$$\displaystyle E\left\{C^{(1)}_{p,\ell}(x)\right\}$$
$$\displaystyle=$$
$$\displaystyle\ J^{-1}E\left[\sum_{j=1}^{J}Z_{j}\left\{c_{j}^{-1}\sum_{k=1}^{c_%
{j}}(X_{jk}-x)^{\ell}\right\}\left\{c_{j}^{-1}\sum_{k=1}^{c_{j}}K_{h}(X_{jk}-x%
)\right\}-\right.$$
$$\displaystyle\left.\sum_{\ell_{2}=0}^{p}\beta_{\ell_{2}}\sum_{j=1}^{J}\left\{c%
_{j}^{-1}\sum_{k=1}^{c_{j}}(X_{jk}-x)^{\ell}\right\}\left\{c_{j}^{-1}\sum_{k=1%
}^{c_{j}}(X_{jk}-x)^{\ell_{2}}\right\}\left\{c_{j}^{-1}\sum_{k=1}^{c_{j}}K_{h}%
(X_{jk}-x)\right\}\right]$$
$$\displaystyle=$$
$$\displaystyle\ J^{-1}E\left[\sum_{j=1}^{J}\left\{c_{j}^{-1}\sum_{k=1}^{c_{j}}m%
(X_{jk})\right\}\left\{c_{j}^{-1}\sum_{k=1}^{c_{j}}(X_{jk}-x)^{\ell}\right\}%
\left\{c_{j}^{-1}\sum_{k=1}^{c_{j}}K_{h}(X_{jk}-x)\right\}-\right.$$
$$\displaystyle\left.\sum_{j=1}^{J}\left\{c_{j}^{-1}\sum_{k=1}^{c_{j}}(X_{jk}-x)%
^{\ell}\right\}\left\{c_{j}^{-1}\sum_{k=1}^{c_{j}}K_{h}(X_{jk}-x)\right\}\sum_%
{\ell_{2}=0}^{p}\beta_{\ell_{2}}\left\{c_{j}^{-1}\sum_{k=1}^{c_{j}}(X_{jk}-x)^%
{\ell_{2}}\right\}\right]$$
$$\displaystyle=$$
$$\displaystyle\ J^{-1}\sum_{j=1}^{J}E\left[\left\{c_{j}^{-1}\sum_{k=1}^{c_{j}}(%
X_{jk}-x)^{\ell}\right\}\left\{c_{j}^{-1}\sum_{k=1}^{c_{j}}K_{h}(X_{jk}-x)%
\right\}\left\{c_{j}^{-1}\sum_{k=1}^{c_{j}}r_{p}(X_{jk},x)\right\}\right]$$
$$\displaystyle=$$
$$\displaystyle\ J^{-1}\sum_{j=1}^{J}c_{j}^{-3}\left[c_{j}E\left\{(X-x)^{\ell}K_%
{h}(X-x)r_{p}(X,x)\right\}+\right.$$
$$\displaystyle\ c_{j}(c_{j}-1)E\left\{(X-x)^{\ell}K_{h}(X-x)\right\}E\left\{r_{%
p}(X,x)\right\}+$$
$$\displaystyle\ c_{j}(c_{j}-1)E\left\{(X-x)^{\ell}\right\}E\left\{K_{h}(X-x)r_{%
p}(X,x)\right\}+$$
$$\displaystyle\ c_{j}(c_{j}-1)E\left\{(X-x)^{\ell}r_{p}(X,x)\right\}E\left\{K_{%
h}(X-x)\right\}+$$
$$\displaystyle\ \left.c_{j}(c_{j}-1)(c_{j}-2)E\left\{(X-x)^{\ell}\right\}E\left%
\{r_{p}(X,x)\right\}E\left\{K_{h}(X-x)\right\}\right].$$
That is,
$$\displaystyle E\left\{C^{(1)}_{p,\ell}(x)\right\}$$
$$\displaystyle=$$
$$\displaystyle\ t_{20}E\left\{(X-x)^{\ell}r_{p}(X,x)K_{h}(X-x)\right\}+t_{21}R_%
{0,p}(x)E\left\{(X-x)^{\ell}K_{h}(X-x)\right\}+$$
$$\displaystyle\ t_{21}\delta_{\ell}(x)E\left\{r_{p}(X,x)K_{h}(X-x)\right\}+t_{2%
1}E\left\{(X-x)^{\ell}r_{p}(X,x)\right\}E\left\{K_{h}(X-x)\right\}+$$
$$\displaystyle\ t_{22}\delta_{\ell}(x)R_{0,p}(x)E\left\{K_{h}(X-x)\right\}$$
$$\displaystyle=$$
$$\displaystyle\ t_{20}\left\{\kappa^{*}_{1,\ell}(m,h,x)-\sum_{\ell_{1}=0}^{p}%
\beta_{\ell_{1}}\kappa_{1,\ell+\ell_{1}}(h,x)\right\}+t_{21}R_{0,p}(x)\kappa_{%
1,\ell}(h,x)+$$
$$\displaystyle\ t_{21}\delta_{\ell}(x)\left\{\kappa^{*}_{1,0}(m,h,x)-\sum_{\ell%
_{1}=0}^{p}\beta_{\ell_{1}}\kappa_{1,\ell_{1}}(h,x)\right\}+t_{21}R_{\ell,p}(x%
)\kappa_{1,0}(h,x)+$$
$$\displaystyle\ t_{22}\delta_{\ell}(x)R_{0,p}(x)\kappa_{1,0}(h,x).$$
Using (A.5) and (A.9) in the above expression gives
$$\displaystyle E\left\{C^{(1)}_{p,0}(x)\right\}=$$
$$\displaystyle\ t_{11}R_{0,p}(x)\left\{f_{\hbox{\tiny$X$}}(x)+0.5h^{2}\mu_{2}f^%
{(2)}_{\hbox{\tiny$X$}}(x)\right\}+$$
$$\displaystyle\ h^{2}\mu_{2}t_{10}\left\{\beta_{1}f^{\prime}_{\hbox{\tiny$X$}}(%
x)I(p=0)+\beta_{2}f_{\hbox{\tiny$X$}}(x)I(p\leq 1)\right\}+O(h^{4}),$$
(A.16)
and, for $\ell>0$,
$$\displaystyle E\left\{C^{(1)}_{p,\ell}(x)\right\}=$$
$$\displaystyle\ \left\{t_{21}R_{\ell,p}(x)+t_{22}\delta_{\ell}(x)R_{0,p}(x)%
\right\}\left\{f_{\hbox{\tiny$X$}}(x)+0.5h^{2}\mu_{2}f^{\prime\prime}_{\hbox{%
\tiny$X$}}(x)\right\}+$$
$$\displaystyle\ h^{\ell}t_{21}R_{0,p}(x)\left\{\mu_{\ell}f_{\hbox{\tiny$X$}}(x)%
+h\mu_{\ell+1}f^{\prime}_{\hbox{\tiny$X$}}(x)\right\}+O(h^{4}).$$
(A.17)
Putting results in (A.16) and (A.17) in a vector, one has
$$\displaystyle E\{\mathbf{C}^{(1)}_{p}(x)\}=$$
$$\displaystyle\ \left\{f_{\hbox{\tiny$X$}}(x)+0.5h^{2}\mu_{2}f^{\prime\prime}_{%
\hbox{\tiny$X$}}(x)\right\}\begin{bmatrix}t_{11}R_{0,p}(x)\\
t_{21}R_{1,p}(x)+t_{22}\delta_{1}(x)R_{0,p}(x)\\
\vdots\\
t_{21}R_{\ell,p}(x)+t_{22}\delta_{\ell}(x)R_{0,p}(x)\\
\vdots\\
t_{21}R_{p,p}(x)+t_{22}\delta_{p}(x)R_{0,p}(x)\end{bmatrix}+$$
$$\displaystyle\ h^{2}\mu_{2}\begin{bmatrix}t_{10}\left\{\beta_{1}f^{\prime}_{%
\hbox{\tiny$X$}}(x)I(p=0)+\beta_{2}f_{\hbox{\tiny$X$}}(x)I(p\leq 1)\right\}\\
t_{21}R_{0,p}(x)f^{\prime}_{\hbox{\tiny$X$}}(x)\\
t_{21}R_{0,p}(x)f_{\hbox{\tiny$X$}}(x)\\
0\\
\vdots\\
0\end{bmatrix}+O\left(h^{4}\mbox{\boldmath$1$}_{p+1}\right)$$
$$\displaystyle=$$
$$\displaystyle\ f_{\hbox{\tiny$X$}}(x)\left\{t_{21}R_{0,p}(x)\mbox{\boldmath$e$%
}_{1}+t_{21}\mathbf{R}^{*}_{p}(x)+t_{22}R_{0,p}(x)\mbox{\boldmath$\Delta$}^{*}%
_{0}(x)\right\}+$$
$$\displaystyle\ 0.5h^{2}\mu_{2}\left[f^{\prime\prime}_{\hbox{\tiny$X$}}(x)\left%
\{t_{21}\mathbf{R}^{*}_{p}(x)+t_{22}R_{0,p}(x)\mbox{\boldmath$\Delta$}^{*}_{0}%
(x)\right\}+\right.$$
$$\displaystyle\ t_{21}R_{0,p}(x)\left\{f^{\prime\prime}_{\hbox{\tiny$X$}}(x)%
\mbox{\boldmath$e$}_{1}+2f^{\prime}_{\hbox{\tiny$X$}}(x)\mbox{\boldmath$e$}_{2%
}I(p\geq 1)+2f_{\hbox{\tiny$X$}}(x)\mbox{\boldmath$e$}_{3}I(p\geq 2)\right\}+$$
$$\displaystyle\left.2t_{10}\left\{\beta_{1}f^{\prime}_{\hbox{\tiny$X$}}(x)I(p=0%
)+\beta_{2}f_{\hbox{\tiny$X$}}(x)I(p\leq 1)\right\}\mbox{\boldmath$e$}_{1}%
\right]+O\left(h^{4}\mbox{\boldmath$1$}_{p+1}\right).$$
(A.18)
As for the order of $\textrm{Var}\{\mathbf{C}^{(1)}_{p}(x)\}$, by iterated expectations, one has
$$\displaystyle\textrm{Var}\{C^{(1)}_{p,\ell}(x)\}$$
$$\displaystyle=$$
$$\displaystyle\ E\left[\textrm{Var}\left\{\left.C^{(1)}_{p,\ell}(x)\right|%
\mathbb{X}\right\}\right]+\textrm{Var}\left[E\left\{\left.C^{(1)}_{p,\ell}(x)%
\right|\mathbb{X}\right\}\right]$$
$$\displaystyle=$$
$$\displaystyle\ E\left[\textrm{Var}\left\{J^{-1}T_{1,\ell}(x)\left|\mathbb{X}%
\right\}\right.\right]+$$
$$\displaystyle\ \textrm{Var}\left[J^{-1}\sum_{j=1}^{J}\left\{c_{j}^{-1}\sum_{k=%
1}^{c_{j}}(X_{jk}-x)^{\ell}\right\}\left\{c_{j}^{-1}\sum_{k=1}^{c_{j}}K_{h}(X_%
{jk}-x)\right\}\left\{c_{j}^{-1}\sum_{j=1}^{c_{j}}r_{p}(X_{jk},x)\right\}\right]$$
$$\displaystyle=$$
$$\displaystyle\ J^{-2}\sum_{j=1}^{J}E\left[\left\{c_{j}^{-2}\sum_{j=1}^{c_{j}}%
\sigma^{2}(X_{jk})\right\}\left\{c_{j}^{-1}\sum_{k=1}^{c_{j}}(X_{jk}-x)^{\ell}%
\right\}^{2}\left\{c_{j}^{-1}\sum_{k=1}^{c_{j}}K_{h}(X_{jk}-x)\right\}^{2}%
\right]+$$
$$\displaystyle\ J^{-2}\sum_{j=1}^{J}\textrm{Var}\left[\left\{c_{j}^{-1}\sum_{k=%
1}^{c_{j}}(X_{jk}-x)^{\ell}\right\}\left\{c_{j}^{-1}\sum_{k=1}^{c_{j}}K_{h}(X_%
{jk}-x)\right\}\left\{c_{j}^{-1}\sum_{j=1}^{c_{j}}r_{p}(X_{jk},x)\right\}%
\right].$$
Because the dominating terms in the above expectations and variances are of the same order as $\kappa_{2,0}(x)=E\{K^{2}_{h}(X-x)\}=O(1/h)$ according to (A.6), $\textrm{Var}\{C^{(1)}_{p,\ell}(x)\}=O\{1/(Jh)\}$ for $\ell=0,1,\ldots,p$. This is also true for $\textrm{Cov}\{C^{(1)}_{p,\ell_{1}}(x),C^{(1)}_{p,\ell_{2}}(x)\}$, for $\ell_{1}\neq\ell_{2}\in\{0,1,\ldots,p\}$.
In conclusion,
$$\displaystyle\mathbf{C}^{(1)}_{p}(x)$$
$$\displaystyle=$$
$$\displaystyle\ f_{\hbox{\tiny$X$}}(x)\left\{t_{21}R_{0,p}(x)\mbox{\boldmath$e$%
}_{1}+t_{21}\mathbf{R}^{*}_{p}(x)+t_{22}R_{0,p}(x)\mbox{\boldmath$\Delta$}^{*}%
_{0}(x)\right\}+$$
$$\displaystyle\ 0.5h^{2}\mu_{2}\left[f^{\prime\prime}_{\hbox{\tiny$X$}}(x)\left%
\{t_{21}\mathbf{R}^{*}_{p}(x)+t_{22}R_{0,p}(x)\mbox{\boldmath$\Delta$}^{*}_{0}%
(x)\right\}+\right.$$
$$\displaystyle\ t_{21}R_{0,p}(x)\left\{f^{\prime\prime}_{\hbox{\tiny$X$}}(x)%
\mbox{\boldmath$e$}_{1}+2f^{\prime}_{\hbox{\tiny$X$}}(x)\mbox{\boldmath$e$}_{2%
}I(p\geq 1)+2f_{\hbox{\tiny$X$}}(x)\mbox{\boldmath$e$}_{3}I(p\geq 2)\right\}+$$
$$\displaystyle\left.2t_{10}\left\{\beta_{1}f^{\prime}_{\hbox{\tiny$X$}}(x)I(p=0%
)+\beta_{2}f_{\hbox{\tiny$X$}}(x)I(p\leq 1)\right\}\mbox{\boldmath$e$}_{1}%
\right]+O\left(h^{4}\mbox{\boldmath$1$}_{p+1}\right)+$$
$$\displaystyle\ O_{\hbox{\tiny$P$}}\left(\frac{1}{\sqrt{Jh}}\mbox{\boldmath$1$}%
_{p+1}\right).$$
(A.19)
Finally, using (A.15) and (A.19) yields the asymptotic discrepancy
$\hat{\mbox{\boldmath$\beta$}}_{1}-\mbox{\boldmath$\beta$}=\left\{J^{-1}\mathbf%
{S}_{1}(x)\right\}^{-1}\mathbf{C}_{p}^{(1)}(x)$. Extracting the first element of $\hat{\mbox{\boldmath$\beta$}}_{1}-\mbox{\boldmath$\beta$}$ gives
$$\displaystyle\ \hat{m}_{1}(x)-m(x)$$
$$\displaystyle=$$
$$\displaystyle\ \mbox{\boldmath$e$}_{1}^{\mathrm{\scriptscriptstyle T}}\mathbf{%
M}_{0}^{-1}(x)\left\{\mathbf{L}_{0}(x)-hf^{-1}_{\hbox{\tiny$X$}}(x)f^{\prime}_%
{\hbox{\tiny$X$}}(x)\mathbf{M}_{1}(x)\mathbf{M}_{0}^{-1}(x)\mathbf{L}_{0}(x)\right.$$
$$\displaystyle+h^{2}f^{-1}_{\hbox{\tiny$X$}}(x)\left(\mu_{2}\left\{\mathbf{L}_{%
1}(x)+\mathbf{L}_{2}(x)+0.5f^{\prime\prime}_{\hbox{\tiny$X$}}(x)\mathbf{L}_{3}%
(x)\right\}\right.$$
$$\displaystyle\left.\left.+\left[\frac{\left\{f^{\prime\prime}_{\hbox{\tiny$X$}%
}(x)\right\}^{2}}{f_{\hbox{\tiny$X$}}(x)}\mathbf{M}_{1}(x)\mathbf{M}_{0}^{-1}(%
x)\mathbf{M}_{1}(x)+0.5f^{\prime\prime}_{\hbox{\tiny$X$}}(x)\mathbf{M}_{2}(x)%
\right]\mathbf{M}_{0}^{-1}\mathbf{L}_{0}(x)\right)\right\}$$
$$\displaystyle+O\left(h^{4}\right)+O_{\hbox{\tiny$P$}}\left(\frac{1}{\sqrt{Jh}}%
\right),$$
(A.20)
where
$$\displaystyle\mathbf{M}_{0}(x)=$$
$$\displaystyle\ t_{20}\tilde{\mbox{\boldmath$\mu$}}_{0}+t_{21}\left\{\tilde{%
\mbox{\boldmath$\Delta$}}_{0}(x)+\mbox{\boldmath$\Delta$}^{*}_{0}(x)\mbox{%
\boldmath$\mu$}_{0}^{*{\mathrm{\scriptscriptstyle T}}}+\mbox{\boldmath$\mu$}_{%
0}^{*}\mbox{\boldmath$\Delta$}_{0}^{*{\mathrm{\scriptscriptstyle T}}}(x)\right\}$$
$$\displaystyle+t_{22}\mbox{\boldmath$\Delta$}_{0}^{*}(x)\mbox{\boldmath$\Delta$%
}_{0}^{*{\mathrm{\scriptscriptstyle T}}}(x),$$
$$\displaystyle\mathbf{M}_{1}(x)=$$
$$\displaystyle\ t_{20}\tilde{\mbox{\boldmath$\mu$}}_{1}+t_{21}\left\{\mbox{%
\boldmath$\Delta$}^{*}_{0}(x)\mbox{\boldmath$\mu$}_{1}^{*{\mathrm{%
\scriptscriptstyle T}}}+\mbox{\boldmath$\mu$}_{1}^{*}\mbox{\boldmath$\Delta$}_%
{0}^{*{\mathrm{\scriptscriptstyle T}}}(x)\right\},$$
$$\displaystyle\mathbf{M}_{2}(x)=$$
$$\displaystyle\ t_{20}\tilde{\mbox{\boldmath$\mu$}}_{2}+t_{21}\left\{\mu_{2}%
\tilde{\mbox{\boldmath$\Delta$}}_{0}(x)+\mbox{\boldmath$\Delta$}^{*}_{0}(x)%
\mbox{\boldmath$\mu$}_{2}^{*{\mathrm{\scriptscriptstyle T}}}+\mbox{\boldmath$%
\mu$}_{2}^{*}\mbox{\boldmath$\Delta$}_{0}^{*{\mathrm{\scriptscriptstyle T}}}(x%
)\right\}$$
$$\displaystyle+t_{22}\mu_{2}\mbox{\boldmath$\Delta$}_{0}^{*}(x)\mbox{\boldmath$%
\Delta$}_{0}^{*{\mathrm{\scriptscriptstyle T}}}(x),$$
$$\displaystyle\mathbf{L}_{0}(x)=$$
$$\displaystyle\ t_{21}\left\{R_{0,p}(x)\mbox{\boldmath$e$}_{1}+\mathbf{R}_{p}^{%
*}(x)\right\}+t_{22}R_{0,p}(x)\mbox{\boldmath$\Delta$}_{0}^{*}(x),$$
$$\displaystyle\mathbf{L}_{1}(x)=$$
$$\displaystyle\ t_{10}\left\{\beta_{1}f^{\prime}_{\hbox{\tiny$X$}}(x)I(p=0)+%
\beta_{2}f_{\hbox{\tiny$X$}}(x)I(p\leq 0)\right\}\mbox{\boldmath$e$}_{1},$$
$$\displaystyle\mathbf{L}_{2}(x)=$$
$$\displaystyle\ t_{21}R_{0,p}(x)\left\{0.5f^{\prime\prime}_{\hbox{\tiny$X$}}(x)%
\mbox{\boldmath$e$}_{1}+f^{\prime}_{\hbox{\tiny$X$}}(x)\mbox{\boldmath$e$}_{2}%
I(p\geq 1)+f_{\hbox{\tiny$X$}}(x)\mbox{\boldmath$e$}_{3}I(p\geq 2)\right\},$$
$$\displaystyle\mathbf{L}_{3}(x)=$$
$$\displaystyle\ t_{21}\mathbf{R}^{*}_{p}(x)+t_{22}R_{0,p}(x)\mbox{\boldmath$%
\Delta$}^{*}(x).$$
A.5 A special case of $\hat{m}_{1}(x)$ with $p=0$
To gain more insight on the effects of pooling and other factors that influence the performance of the average-weighted estimator, we consider a special case for $\hat{m}_{1}(x)$ by setting $p=0$ in (A.20) to obtain a local constant estimator of $m(x)$. In particular, setting $p=0$ in (A.15) gives
$$\displaystyle\left\{J^{-1}\mathbf{S}_{1}(x)\right\}^{-1}$$
$$\displaystyle=f^{-1}_{\hbox{\tiny$X$}}(x)\left\{1+0.5h^{2}\mu_{2}\frac{f^{%
\prime\prime}_{\hbox{\tiny$X$}}(x)}{f_{\hbox{\tiny$X$}}(x)}(t_{20}+t_{11})%
\right\}+O(h^{4})+O_{\hbox{\tiny$P$}}\left(\frac{1}{\sqrt{Jh}}\right).$$
Setting $p=0$ in (A.19) gives
$$\displaystyle\mathbf{C}_{0}^{(1)}(x)=$$
$$\displaystyle\left\{f_{\hbox{\tiny$X$}}(x)+0.5h^{2}\mu_{2}f^{\prime\prime}_{%
\hbox{\tiny$X$}}(x)\right\}t_{11}R_{0,0}(x)+h^{2}\mu_{2}t_{10}\left\{\beta_{1}%
f^{\prime}_{\hbox{\tiny$X$}}(x)+\beta_{2}f_{\hbox{\tiny$X$}}(x)\right\}+$$
$$\displaystyle\ O(h^{4})+O_{\hbox{\tiny$P$}}\left(\frac{1}{\sqrt{Jh}}\right),$$
where $R_{0,0}(x)=E\{m(X)\}-m(x)$. It follows, for the local constant estimator using the average-weighted estimator,
$$\displaystyle\hat{m}_{1}(x)-m(x)$$
$$\displaystyle=$$
$$\displaystyle\ E\{m(X)-m(x)\}t_{11}+h^{2}\mu_{2}t_{10}\left\{\beta_{1}\frac{f^%
{\prime}_{\hbox{\tiny$X$}}(x)}{f_{\hbox{\tiny$X$}}(x)}+\beta_{2}\right\}+O(h^{%
4})+O_{\hbox{\tiny$P$}}\left(\frac{1}{\sqrt{Jh}}\right)$$
$$\displaystyle=$$
$$\displaystyle\ t_{11}E\{m(X)-m(x)\}+t_{10}\textrm{NW}(x)+O(h^{4})+O_{\hbox{%
\tiny$P$}}\left(\frac{1}{\sqrt{Jh}}\right),$$
(A.21)
where
$$\textrm{NW}(x)=h^{2}\mu_{2}\left\{\beta_{1}\frac{f^{\prime}_{\hbox{\tiny$X$}}(%
x)}{f_{\hbox{\tiny$X$}}(x)}+\beta_{2}\right\}$$
is the dominating bias of the Nadaraya-Watson (NW) estimator.
Clearly, (A.21) reduces to the dominating bias of the NW estimator when $c_{j}=1$ for all $j=1,\ldots,J$. If $c_{j}>1$, the result in (A.21) suggests that the bias of $\hat{m}_{1}(x)$ does not tend to zero as $J\to\infty$ and $h\to 0$, unless $E\{m(X)-m(x)\}=0$.
Appendix B: Proof of Theorem 1-(ii)
The product-weighted estimator $\hat{m}_{2}(x)$ results from minimizing the following weighted objective function,
$$Q_{2}(\mbox{\boldmath$\beta$})=\sum_{j=1}^{J}\left\{Z_{j}-\sum_{\ell=0}^{p}%
\beta_{\ell}c_{j}^{-1}\sum_{k=1}^{c_{j}}(X_{jk}-x)^{\ell}\right\}^{2}\left\{%
\prod_{k=1}^{c_{j}}K_{h}(X_{jk}-x)\right\}.$$
In the matrix form, $\hat{m}_{2}(x)=\mbox{\boldmath$e$}_{1}^{\mathrm{\scriptscriptstyle T}}\mathbf{%
S}_{2}^{-1}(x)\mathbf{T}_{2}(x)$,
where
$$\displaystyle\mathbf{S}_{2}(x)$$
$$\displaystyle=\mathbf{D}_{1}(x)^{\mathrm{\scriptscriptstyle T}}\mathbf{K}_{2}(%
x)\mathbf{D}_{1}(x)=\left[S_{2,\ell_{1},\ell_{2}}(x)\right]_{\ell_{1},\ell_{2}%
=0,1,\ldots,p},$$
$$\displaystyle\mathbf{T}_{2}(x)$$
$$\displaystyle=\mathbf{D}_{1}(x)^{\mathrm{\scriptscriptstyle T}}\mathbf{K}_{2}(%
x)\mathbf{Z}=(T_{2,0}(x),\,T_{2,1}(x),\,\ldots,\,T_{2,p}(x))^{\mathrm{%
\scriptscriptstyle T}},$$
$$\displaystyle\mathbf{K}_{2}(x)$$
$$\displaystyle=\textrm{diag}\left(\prod_{k=1}^{c_{1}}K_{h}(X_{jk}-x),\,\ldots,%
\,\prod_{k=1}^{c_{J}}K_{h}(X_{Jk}-x)\right).$$
From above, one can see that entries in $\mathbf{S}_{2}(x)$ are, for $\ell_{1},\ell_{2}=0,1,\ldots,p$,
$$\displaystyle S_{2,\ell_{1},\ell_{2}}(x)$$
$$\displaystyle=$$
$$\displaystyle\ \sum_{j=1}^{J}\left\{c_{j}^{-1}\sum_{k=1}^{c_{j}}(X_{jk}-x)^{%
\ell_{1}}\right\}\left\{c_{j}^{-1}\sum_{k=1}^{c_{j}}(X_{jk}-x)^{\ell_{2}}%
\right\}\left\{\prod_{k=1}^{c_{j}}K_{h}(X_{jk}-x)\right\};$$
(B.1)
and entries in $\mathbf{T}_{2}(x)$ are, for $\ell=0,1,\ldots,p$.
$$T_{2,\ell}(x)=\sum_{j=1}^{J}Z_{j}\left\{c_{j}^{-1}\sum_{k=1}^{c_{j}}(X_{jk}-x)%
^{\ell}\right\}\left\{\prod_{k=1}^{c_{j}}K_{h}(X_{jk}-x)\right\}.$$
(B.2)
In what follows, we derive the mean and variance of $J^{-1}\mathbf{S}_{2}(x)$ and $\mathbf{C}^{(2)}_{p}(x)=J^{-1}\{\mathbf{T}_{2}(x)-\mathbf{S}_{2}(x)\mbox{%
\boldmath$\beta$}\}$ in order to reveal dominating terms of $\hat{\mbox{\boldmath$\beta$}}_{2}-\mbox{\boldmath$\beta$}$, where $\hat{\mbox{\boldmath$\beta$}}_{2}=\mathbf{S}_{2}^{-1}\mathbf{T}_{2}(x)$. For notational simplicity in the derivations, we define the following constants,
$$\displaystyle d(x)$$
$$\displaystyle=J^{-1}\sum_{j=1}^{J}f^{c_{j}-1}_{\hbox{\tiny$X$}}(x),$$
$$\displaystyle d_{0}(x)$$
$$\displaystyle=J^{-1}\sum_{j=1}^{J}c_{j}^{-1}f^{c_{j}-1}_{\hbox{\tiny$X$}}(x),$$
$$\displaystyle d_{k}(x)$$
$$\displaystyle=J^{-1}\sum_{j=1}^{J}\frac{[\prod_{i=1}^{k}(c_{j}-i)]_{+}}{c_{j}}%
f^{c_{j}-1}_{\hbox{\tiny$X$}}(x),\textrm{ for $k=1,2$,}$$
$$\displaystyle w_{20}(x,h,J)$$
$$\displaystyle=J^{-2}\sum_{j=1}^{J}c_{j}^{-2}f^{c_{j}-1}_{\hbox{\tiny$X$}}(x)h^%
{-c_{j}},$$
$$\displaystyle w_{30}(x,h,J)$$
$$\displaystyle=J^{-2}\sum_{j=1}^{J}c_{j}^{-3}f^{c_{j}-1}_{\hbox{\tiny$X$}}(x)h^%
{-c_{j}},$$
$$\displaystyle w_{3k}(x,h,J)$$
$$\displaystyle=J^{-2}\sum_{j=1}^{J}\frac{[\prod_{i=1}^{k}(c_{j}-i)]_{+}}{c_{j}^%
{3}}f^{c_{j}-1}_{\hbox{\tiny$X$}}(x)h^{-c_{j}},\textrm{ for $k=1,2,3$}.$$
As $J\to\infty$, the first three quantities are of order $O(1)$; and the latter three quantities are of order $O(J^{-1}h^{-c^{*}})$, where $c^{*}=\max_{1\leq j\leq J}c_{j}$.
B.1 Derive $E\{J^{-1}\mathbf{S}_{2}(x)\}$
By (B.1), one has
$$\displaystyle\ E\{J^{-1}S_{2,\ell_{1},\ell_{2}}(x)\}$$
$$\displaystyle=$$
$$\displaystyle\ J^{-1}\sum_{j=1}^{J}E\left[\left\{c_{j}^{-1}\sum_{k=1}^{c_{j}}(%
X_{jk}-x)^{\ell_{1}}\right\}\left\{c_{j}^{-1}\sum_{k=1}^{c_{j}}(X_{jk}-x)^{%
\ell_{2}}\right\}\left\{\prod_{k=1}^{c_{j}}K_{h}(X_{jk}-x)\right\}\right]$$
$$\displaystyle=$$
$$\displaystyle\ J^{-1}\sum_{j=1}^{J}c_{j}c_{j}^{-2}E\left\{(X-x)^{\ell_{1}+\ell%
_{2}}K_{h}(X-x)\right\}\kappa^{c_{j}-1}_{1,0}(h,x)+$$
$$\displaystyle\ J^{-1}\sum_{j=1}^{J}c_{j}(c_{j}-1)c_{j}^{-2}E\left\{(X-x)^{\ell%
_{1}}K_{h}(X-x)\right\}\times$$
$$\displaystyle\ E\left\{(X-x)^{\ell_{2}}K_{h}(X-x)\right\}\kappa^{c_{j}-2}_{1,0%
}(h,x)$$
$$\displaystyle=$$
$$\displaystyle\ \kappa_{1,\ell_{1}+\ell_{2}}(h,x)J^{-1}\sum_{j=1}^{J}\frac{1}{c%
_{j}}\kappa_{1,0}^{c_{j}-1}(h,x)+$$
$$\displaystyle\ \kappa_{1,\ell_{1}}(h,x)\kappa_{1,\ell_{2}}(h,x)J^{-1}\sum_{j=1%
}^{J}\frac{c_{j}-1}{c_{j}}\kappa_{1,0}^{c_{j}-2}(h,x)$$
$$\displaystyle=$$
$$\displaystyle\ h^{\ell_{1}+\ell_{2}}J^{-1}\sum_{j=1}^{J}c_{j}^{-1}f^{c_{j}}_{%
\hbox{\tiny$X$}}(x)\left\{\mu_{\ell_{1}+\ell_{2}}+(c_{j}-1)\mu_{\ell_{1}}\mu_{%
\ell_{2}}\right\}+h^{\ell_{1}+\ell_{2}+1}f^{\prime}_{\hbox{\tiny$X$}}(x)\times$$
$$\displaystyle\ J^{-1}\sum_{j=1}^{J}c_{j}^{-1}f^{c_{j}-1}_{\hbox{\tiny$X$}}(x)%
\left\{\mu_{\ell_{1}+\ell_{2}+1}+(c_{j}-1)(\mu_{\ell_{1}}\mu_{\ell_{2}+1}+\mu_%
{\ell_{1}+1}\mu_{\ell_{2}})\right\}+O(h^{\ell_{1}+\ell_{2}+2}).$$
Written in the matrix form, the above suggests that
$$\displaystyle E\left\{J^{-1}\mathbf{S}_{2}(x)\right\}=$$
$$\displaystyle\ \mathbf{H}\left[d_{0}(x)\left\{f_{\hbox{\tiny$X$}}(x)\tilde{%
\mbox{\boldmath$\mu$}}_{0}+hf^{\prime}_{\hbox{\tiny$X$}}(x)\tilde{\mbox{%
\boldmath$\mu$}}_{1}\right\}+d_{1}(x)\left\{f_{\hbox{\tiny$X$}}(x)\mbox{%
\boldmath$\mu$}^{*}_{0}\mbox{\boldmath$\mu$}^{*{\mathrm{\scriptscriptstyle T}}%
}_{0}\right.\right.$$
$$\displaystyle\left.\left.+hf^{\prime}_{\hbox{\tiny$X$}}(x)\left(\mbox{%
\boldmath$\mu$}^{*}_{0}\mbox{\boldmath$\mu$}_{1}^{*{\mathrm{\scriptscriptstyle
T%
}}}+\mbox{\boldmath$\mu$}^{*}_{1}\mbox{\boldmath$\mu$}_{0}^{*{\mathrm{%
\scriptscriptstyle T}}}\right)\right\}+O(h^{2})\right]\mathbf{H}.$$
(B.3)
Note that, when $c_{j}=1$ for $j=1,\ldots,J$, (B.3) is equal to the familiar counterpart result when individual-level data are available since now $d_{0}(x)=1$ and $d_{1}(x)=0$.
B.2 The order of $\textrm{Var}\{J^{-1}\mathbf{S}_{2}(x)\}$
The order of $\textrm{Var}\{J^{-1}S_{2,\ell_{1},\ell_{2}}(x)\}$ is determined by that of
$$J^{-2}\sum_{j=1}^{J}E\left[\left\{c_{j}^{-1}\sum_{k=1}^{c_{j}}(X_{jk}-x)^{\ell%
_{1}}\right\}^{2}\left\{c_{j}^{-1}\sum_{k=1}^{c_{j}}(X_{jk}-x)^{\ell_{2}}%
\right\}^{2}\prod_{k=1}^{c_{j}}K^{2}_{h}(X_{jk}-x)\right].$$
(B.4)
According to (A.6) and (A.8), (B.4) reaches its highest order (i.e., tends to zero the slowest) when $\ell_{1}=\ell_{2}=0$, in which case (B.4) is equal to
$$\displaystyle J^{-2}\sum_{j=1}^{J}E\left\{\prod_{k=1}^{c_{j}}K^{2}_{h}(X_{jk}-%
x)\right\}=$$
$$\displaystyle\ J^{-2}\sum_{j=1}^{J}\kappa_{2,0}^{c_{j}}(h,x)$$
$$\displaystyle=$$
$$\displaystyle\ J^{-2}\sum_{j=1}^{J}\left\{h^{-c_{j}}\nu_{0}f^{c_{j}}_{\hbox{%
\tiny$X$}}(x)+O\left(h^{2-c_{j}}\right)\right\},\textrm{ by (\ref{eq:EK2c}),}$$
$$\displaystyle\leq$$
$$\displaystyle\ J^{-1}h^{-c^{*}}\nu_{0}f_{\hbox{\tiny$X$}}(x)d(x)+O\left(J^{-1}%
h^{2-c^{*}}\right).$$
Orders of (B.4) with one or both of $\ell_{1}$ and $\ell_{2}$ larger than zero can be similarly derived using (A.6) and (A.8). Putting these orders together reveals that
$$\textrm{Var}\left\{J^{-1}\mathbf{S}_{2}(x)\right\}=O\left(J^{-1}h^{-c^{*}}%
\mathbf{H}^{*}\right).$$
(B.5)
According to (B.3) and (B.5), we conclude that
$$\displaystyle J^{-1}\mathbf{S}_{2}(x)=$$
$$\displaystyle\ E\left\{J^{-1}\mathbf{S}_{2}(x)\right\}+O_{\hbox{\tiny$P$}}%
\left[\textrm{Var}\left\{J^{-1}\mathbf{S}_{2}(x)\right\}\right]$$
$$\displaystyle=$$
$$\displaystyle\ \mathbf{H}\left[d_{0}(x)\left\{f_{\hbox{\tiny$X$}}(x)\tilde{%
\mbox{\boldmath$\mu$}}_{0}+hf^{\prime}_{\hbox{\tiny$X$}}(x)\tilde{\mbox{%
\boldmath$\mu$}}_{1}\right\}+d_{1}(x)\left\{f_{\hbox{\tiny$X$}}(x)\mbox{%
\boldmath$\mu$}^{*}_{0}\mbox{\boldmath$\mu$}^{*{\mathrm{\scriptscriptstyle T}}%
}_{0}+\right.\right.$$
$$\displaystyle\ \left.\left.hf^{\prime}_{\hbox{\tiny$X$}}(x)\left(\mbox{%
\boldmath$\mu$}^{*}_{0}\mbox{\boldmath$\mu$}_{1}^{*{\mathrm{\scriptscriptstyle
T%
}}}+\mbox{\boldmath$\mu$}^{*}_{1}\mbox{\boldmath$\mu$}_{0}^{*{\mathrm{%
\scriptscriptstyle T}}}\right)\right\}+O\left(h^{2}\right)\right]\mathbf{H}+O_%
{\hbox{\tiny$P$}}\left(\mathbf{H}^{*1/2}/\sqrt{Jh^{c^{*}}}\right).$$
It follows that
$$\displaystyle\ \left\{J^{-1}\mathbf{S}_{2}(x)\right\}^{-1}$$
(B.6)
$$\displaystyle=$$
$$\displaystyle\ \mathbf{H}^{-1}\left[f^{-1}_{\hbox{\tiny$X$}}(x)\left\{d_{0}(x)%
\tilde{\mbox{\boldmath$\mu$}}_{0}+d_{1}(x)\mbox{\boldmath$\mu$}^{*}_{0}\mbox{%
\boldmath$\mu$}_{0}^{*{\mathrm{\scriptscriptstyle T}}}\right\}^{-1}-h\frac{f^{%
\prime}_{\hbox{\tiny$X$}}(x)}{f^{2}_{\hbox{\tiny$X$}}(x)}\times\right.$$
$$\displaystyle\ \left\{d_{0}(x)\tilde{\mbox{\boldmath$\mu$}}_{0}+d_{1}(x)\mbox{%
\boldmath$\mu$}^{*}_{0}\mbox{\boldmath$\mu$}_{0}^{*{\mathrm{\scriptscriptstyle
T%
}}}\right\}^{-1}\left\{d_{0}(x)\tilde{\mbox{\boldmath$\mu$}}_{1}+d_{1}(x)\left%
(\mbox{\boldmath$\mu$}^{*}_{0}\mbox{\boldmath$\mu$}_{1}^{*{\mathrm{%
\scriptscriptstyle T}}}+\mbox{\boldmath$\mu$}^{*}_{1}\mbox{\boldmath$\mu$}_{0}%
^{*{\mathrm{\scriptscriptstyle T}}}\right)\right\}\times$$
$$\displaystyle\ \left.\left\{d_{0}(x)\tilde{\mbox{\boldmath$\mu$}}_{0}+d_{1}(x)%
\mbox{\boldmath$\mu$}^{*}_{0}\mbox{\boldmath$\mu$}_{0}^{*{\mathrm{%
\scriptscriptstyle T}}}\right\}^{-1}+O\left(h^{2}\right)+O_{\hbox{\tiny$P$}}%
\left(1/\sqrt{Jh^{c^{*}}}\right)\right]\mathbf{H}^{-1}.$$
B.3 Derive $E\{\mathbf{C}^{(2)}_{p}(x)\}$
View $\mathbf{C}^{(2)}_{p}(x)=(C^{(2)}_{p,0}(x),C^{(2)}_{p,1}(x),\ldots,C^{(2)}_{p,p%
}(x))^{\mathrm{\scriptscriptstyle T}}$.
By (B.1) and (B.2),
$$\displaystyle\ E\{C^{(2)}_{p,\ell}(x)\}$$
$$\displaystyle=$$
$$\displaystyle\ J^{-1}E\left[\sum_{j=1}^{J}Z_{j}\left\{c_{j}^{-1}\sum_{k=1}^{c_%
{j}}(X_{jk}-x)^{\ell}\right\}\left\{\prod_{k=1}^{c_{j}}K_{h}(X_{jk}-x)\right\}%
-\right.$$
$$\displaystyle\left.\sum_{\ell_{2}=0}^{p}\beta_{\ell_{2}}\sum_{j=1}^{J}\left\{c%
_{j}^{-1}\sum_{k=1}^{c_{j}}(X_{jk}-x)^{\ell}\right\}\left\{c_{j}^{-1}\sum_{k=1%
}^{c_{j}}(X_{jk}-x)^{\ell_{2}}\right\}\left\{\prod_{k=1}^{c_{j}}K_{h}(X_{jk}-x%
)\right\}\right].$$
(B.7)
Hence,
$$\displaystyle\ E\{C^{(2)}_{p,\ell}(x)\}$$
$$\displaystyle=$$
$$\displaystyle\ J^{-1}E\left[\sum_{j=1}^{J}\left\{c_{j}^{-1}\sum_{k=1}^{c_{j}}m%
(X_{jk})\right\}\left\{c_{j}^{-1}\sum_{k=1}^{c_{j}}(X_{jk}-x)^{\ell}\right\}%
\left\{\prod_{k=1}^{c_{j}}K_{h}(X_{jk}-x)\right\}-\right.$$
$$\displaystyle\left.\sum_{j=1}^{J}\left\{c_{j}^{-1}\sum_{k=1}^{c_{j}}(X_{jk}-x)%
^{\ell}\right\}\left\{\prod_{k=1}^{c_{j}}K_{h}(X_{jk}-x)\right\}\sum_{\ell_{2}%
=0}^{p}\beta_{\ell_{2}}\left\{c_{j}^{-1}\sum_{k=1}^{c_{j}}(X_{jk}-x)^{\ell_{2}%
}\right\}\right]$$
$$\displaystyle=$$
$$\displaystyle\ J^{-1}\sum_{j=1}^{J}E\left[\left\{c_{j}^{-1}\sum_{k=1}^{c_{j}}(%
X_{jk}-x)^{\ell}\right\}\left\{c_{j}^{-1}\sum_{k=1}^{c_{j}}r_{p}(X_{jk},x)%
\right\}\left\{\prod_{k=1}^{c_{j}}K_{h}(X_{jk}-x)\right\}\right]$$
$$\displaystyle=$$
$$\displaystyle\ J^{-1}\sum_{j=1}^{J}c_{j}c_{j}^{-2}E\left\{(X-x)^{\ell}r_{p}(X,%
x)K_{h}(X-x)\right\}\kappa^{c_{j}-1}_{1,0}(h,x)+$$
$$\displaystyle\ J^{-1}\sum_{j=1}^{J}c_{j}(c_{j}-1)c_{j}^{-2}E\left\{(X-x)^{\ell%
}K_{h}(X-x)\right\}E\left\{r_{p}(X,x)K_{h}(X-x)\right\}\kappa^{c_{j}-2}_{1,0}(%
h,x)$$
$$\displaystyle=$$
$$\displaystyle\ E\left\{(X-x)^{\ell}r_{p}(X,x)K_{h}(X-x)\right\}J^{-1}\sum_{j=1%
}^{J}c_{j}^{-1}\kappa^{c_{j}-1}_{1,0}(h,x)+$$
$$\displaystyle\ \kappa_{1,\ell}(h,x)E\left\{r_{p}(X,x)K_{h}(X-x)\right\}J^{-1}%
\sum_{j=1}^{J}(c_{j}-1)c_{j}^{-1}\kappa^{c_{j}-2}_{1,0}(h,x)$$
$$\displaystyle=$$
$$\displaystyle\ E\left\{(X-x)^{\ell}r_{p}(X,x)K_{h}(X-x)\right\}\left\{d_{0}(x)%
+O\left(h^{2}\right)\right\}+$$
$$\displaystyle\ \kappa_{1,\ell}(h,x)E\left\{r_{p}(X,x)K_{h}(X-x)\right\}\left\{%
f^{-1}_{\hbox{\tiny$X$}}(x)d_{1}(x)+O\left(h^{2}\right)\right\}.$$
(B.8)
Assuming the $(p+3)$-th derivative of $m(\cdot)$ exists, one has
$$r_{p}(X,x)=\beta_{p+1}(X-x)^{p+1}+\beta_{p+2}(X-x)^{p+2}+\beta^{*}_{p+3}(X-x)^%
{p+3},$$
(B.9)
where $\beta^{*}_{p+3}=m^{(p+3)}(x^{*})/(p+3)!$, in which $x^{*}$ lies between $X$ and $x$. Using this expansion of $r_{p}(X,x)$, one has,
$$\displaystyle\ E\left\{(X-x)^{\ell}r_{p}(X,x)K_{h}(X-x)\right\}$$
$$\displaystyle=$$
$$\displaystyle\ h^{\ell+p+1}\mu_{\ell+p+1}\beta_{p+1}f_{\hbox{\tiny$X$}}(x)+h^{%
\ell+p+2}\mu_{\ell+p+2}\left\{\beta_{p+1}f^{\prime}_{\hbox{\tiny$X$}}(x)+\beta%
_{p+2}f_{\hbox{\tiny$X$}}(x)\right\}+$$
$$\displaystyle\ O\left(h^{\ell+p+3}\right).$$
(B.10)
Using (A.5) and (B.10) in (B.8) leads to
$$\displaystyle\ E\left\{C^{(2)}_{p,\ell}(x)\right\}$$
$$\displaystyle=$$
$$\displaystyle\ d_{0}(x)\left[h^{\ell+p+1}\mu_{\ell+p+1}\beta_{p+1}f_{\hbox{%
\tiny$X$}}(x)+h^{\ell+p+2}\mu_{\ell+p+2}\left\{\beta_{p+1}f^{\prime}_{\hbox{%
\tiny$X$}}(x)+\beta_{p+2}f_{\hbox{\tiny$X$}}(x)\right\}\right]+$$
$$\displaystyle\ d_{1}(x)f^{\prime}_{\hbox{\tiny$X$}}(x)\left[h^{p+1}\mu_{p+1}%
\beta_{p+1}f_{\hbox{\tiny$X$}}(x)+h^{p+2}\mu_{p+2}\left\{\beta_{p+1}f^{\prime}%
_{\hbox{\tiny$X$}}(x)+\beta_{p+2}f_{\hbox{\tiny$X$}}(x)\right\}\right]\times$$
$$\displaystyle\ \left\{h^{\ell}\mu_{\ell}f_{\hbox{\tiny$X$}}(x)+h^{\ell+1}\mu_{%
\ell+1}f^{\prime}_{\hbox{\tiny$X$}}(x)\right\}+O\left(h^{\ell+p+3}\right)$$
$$\displaystyle=$$
$$\displaystyle\ h^{\ell+p+1}\beta_{p+1}f_{\hbox{\tiny$X$}}(x)\left\{d_{0}(x)\mu%
_{\ell+p+1}+d_{1}(x)\mu_{\ell}\mu_{p+1}\right\}+$$
$$\displaystyle\ h^{\ell+p+2}\left[\left\{\beta_{p+1}f^{\prime}_{\hbox{\tiny$X$}%
}(x)+\beta_{p+2}f_{\hbox{\tiny$X$}}(x)\right\}\left\{d_{0}(x)\mu_{\ell+p+2}+d_%
{1}(x)\mu_{\ell}\mu_{p+2}\right\}+\right.$$
$$\displaystyle\left.d_{1}(x)\mu_{\ell+1}\mu_{p+1}\beta_{p+1}f^{\prime}_{\hbox{%
\tiny$X$}}(x)\right].$$
This is equivalent to, in the matrix form,
$$\displaystyle\ E\{\mathbf{C}^{(2)}_{p}(x)\}$$
$$\displaystyle=$$
$$\displaystyle\ h^{p+1}\mathbf{H}\left(\beta_{p+1}f_{\hbox{\tiny$X$}}(x)\left\{%
d_{0}(x)\mbox{\boldmath$\mu$}^{*}_{p+1}+d_{1}(x)\mu_{p+1}\mbox{\boldmath$\mu$}%
^{*}_{0}\right\}\right.$$
$$\displaystyle\ +h\left[\left\{\beta_{p+2}f_{\hbox{\tiny$X$}}(x)+\beta_{p+1}f^{%
\prime}_{\hbox{\tiny$X$}}(x)\right\}\left\{d_{0}(x)\mbox{\boldmath$\mu$}^{*}_{%
p+2}+d_{1}(x)\mu_{p+2}\mbox{\boldmath$\mu$}^{*}_{0}\right\}\right.$$
$$\displaystyle\left.\left.+d_{1}(x)\beta_{p+1}f^{\prime}_{\hbox{\tiny$X$}}(x)%
\mu_{p+1}\mbox{\boldmath$\mu$}^{*}_{1}\right]+O\left(h^{2}\mbox{\boldmath$1$}_%
{p+1}\right)\right).$$
(B.11)
B.4 The order of $\textrm{Var}\{\mathbf{C}^{(2)}_{p}(x)\}$
By iterated expectations,
$$\displaystyle\textrm{Var}\{C^{(2)}_{p,\ell}(x)\}$$
$$\displaystyle=$$
$$\displaystyle E\left[\textrm{Var}\left\{C^{(2)}_{p,\ell}(x)\left|\mathbb{X}%
\right\}\right.\right]+\textrm{Var}\left[E\left\{C^{(2)}_{p,\ell}(x)\left|%
\mathbb{X}\right\}\right.\right]$$
$$\displaystyle=$$
$$\displaystyle E\left[\textrm{Var}\left\{J^{-1}T_{2,\ell}(x)\left|\mathbb{X}%
\right\}\right.\right]+$$
$$\displaystyle\textrm{Var}\left[J^{-1}\sum_{j=1}^{J}\left\{c_{j}^{-1}\sum_{k=1}%
^{c_{j}}(X_{jk}-x)^{\ell}\right\}\left\{\prod_{k=1}^{c_{j}}K_{h}(X_{jk}-x)%
\right\}\left\{c_{j}^{-1}\sum_{j=1}^{c_{j}}r_{p}(X_{jk},x)\right\}\right]$$
$$\displaystyle=$$
$$\displaystyle J^{-2}\sum_{j=1}^{J}E\left[\left\{c_{j}^{-2}\sum_{j=1}^{c_{j}}%
\sigma^{2}(X_{jk})\right\}\left\{c_{j}^{-1}\sum_{k=1}^{c_{j}}(X_{jk}-x)^{\ell}%
\right\}^{2}\left\{\prod_{k=1}^{c_{j}}K_{h}(X_{jk}-x)\right\}^{2}\right]+$$
$$\displaystyle J^{-2}\sum_{j=1}^{J}\textrm{Var}\left[\left\{c_{j}^{-1}\sum_{k=1%
}^{c_{j}}(X_{jk}-x)^{\ell}\right\}\left\{\prod_{k=1}^{c_{j}}K_{h}(X_{jk}-x)%
\right\}\left\{c_{j}^{-1}\sum_{j=1}^{c_{j}}r_{p}(X_{jk},x)\right\}\right].$$
Following similar derivations as those in Section B.2, one can show that $\textrm{Var}\{C^{(2)}_{p,\ell}(x)\}=O(J^{-1}h^{2\ell-c^{*}})$. Along with (B.11), now one has
$$\displaystyle\ \mathbf{C}^{(2)}_{p}(x)$$
$$\displaystyle=$$
$$\displaystyle\ h^{p+1}\mathbf{H}\left(\beta_{p+1}f_{\hbox{\tiny$X$}}(x)\left\{%
d_{0}(x)\mbox{\boldmath$\mu$}^{*}_{p+1}+d_{1}(x)\mu_{p+1}\mbox{\boldmath$\mu$}%
^{*}_{0}\right\}\right.$$
$$\displaystyle\ +h\left[\left\{\beta_{p+2}f_{\hbox{\tiny$X$}}(x)+\beta_{p+1}f^{%
\prime}_{\hbox{\tiny$X$}}(x)\right\}\left\{d_{0}(x)\mbox{\boldmath$\mu$}^{*}_{%
p+2}+d_{1}(x)\mu_{p+2}\mbox{\boldmath$\mu$}^{*}_{0}\right\}+\right.$$
$$\displaystyle\left.\left.d_{1}(x)\beta_{p+1}f^{\prime}_{\hbox{\tiny$X$}}(x)\mu%
_{p+1}\mbox{\boldmath$\mu$}^{*}_{1}\right]+O\left(h^{2}\right)\right)+O_{\hbox%
{\tiny$P$}}\left\{\textrm{vecdiag}(\mathbf{H})/\sqrt{Jh^{c^{*}}}\right\}.$$
(B.12)
By (B.6) and (B.12), one has
$$\displaystyle\ \hat{\mbox{\boldmath$\beta$}}_{2}-\mbox{\boldmath$\beta$}$$
$$\displaystyle=$$
$$\displaystyle\ h^{p+1}\mathbf{H}^{-1}\left[f^{-1}_{\hbox{\tiny$X$}}(x)\left\{d%
_{0}(x)\tilde{\mbox{\boldmath$\mu$}}_{0}+d_{1}(x)\mbox{\boldmath$\mu$}^{*}_{0}%
\mbox{\boldmath$\mu$}_{0}^{*{\mathrm{\scriptscriptstyle T}}}\right\}^{-1}-h%
\frac{f^{\prime}_{\hbox{\tiny$X$}}(x)}{f^{2}_{\hbox{\tiny$X$}}(x)}\times\right.$$
$$\displaystyle\ \left\{d_{0}(x)\tilde{\mbox{\boldmath$\mu$}}_{0}+d_{1}(x)\mbox{%
\boldmath$\mu$}^{*}_{0}\mbox{\boldmath$\mu$}_{0}^{*{\mathrm{\scriptscriptstyle
T%
}}}\right\}^{-1}\left\{d_{0}(x)\tilde{\mbox{\boldmath$\mu$}}_{1}+d_{1}(x)\left%
(\mbox{\boldmath$\mu$}^{*}_{0}\mbox{\boldmath$\mu$}_{1}^{*{\mathrm{%
\scriptscriptstyle T}}}+\mbox{\boldmath$\mu$}^{*}_{1}\mbox{\boldmath$\mu$}_{0}%
^{*{\mathrm{\scriptscriptstyle T}}}\right)\right\}\times$$
$$\displaystyle\left.\left\{d_{0}(x)\tilde{\mbox{\boldmath$\mu$}}_{0}+d_{1}(x)%
\mbox{\boldmath$\mu$}^{*}_{0}\mbox{\boldmath$\mu$}_{0}^{*{\mathrm{%
\scriptscriptstyle T}}}\right\}^{-1}\right]\left(\beta_{p+1}f_{\hbox{\tiny$X$}%
}(x)\left\{d_{0}(x)\mbox{\boldmath$\mu$}^{*}_{p+1}+d_{1}(x)\mu_{p+1}\mbox{%
\boldmath$\mu$}^{*}_{0}\right\}+\right.$$
$$\displaystyle\left.h\left[\left\{\beta_{p+2}f_{\hbox{\tiny$X$}}(x)+\beta_{p+1}%
f^{\prime}_{\hbox{\tiny$X$}}(x)\right\}\left\{d_{0}(x)\mbox{\boldmath$\mu$}^{*%
}_{p+2}+d_{1}(x)\mu_{p+2}\mbox{\boldmath$\mu$}^{*}_{0}\right\}+\right.\right.$$
$$\displaystyle\left.\left.d_{1}(x)\beta_{p+1}f^{\prime}_{\hbox{\tiny$X$}}(x)\mu%
_{p+1}\mbox{\boldmath$\mu$}^{*}_{1}\right]\right)+\mathbf{H}^{-1}\mbox{%
\boldmath$1$}_{p+1}\left\{o\left(h^{p+2}\right)+O_{\hbox{\tiny$P$}}\left(\frac%
{1}{\sqrt{Jh^{c^{*}}}}\right)\right\}.$$
(B.13)
When $c_{j}=1$ for $j=1,\ldots,J$, with $d_{0}(x)=1$ and $d_{1}(x)=0$, (B.13) reduces to the a much simpler expression that matches the counterpart result for non-pooling data.
B.5 Special cases of $\hat{m}_{2}(x)$ with $p=0,1$
Setting $p=0$ in (B.13) gives the result regarding the local constant estimator $\hat{m}_{2}(x)$, $\hat{m}_{2}(x)-m(x)=h^{2}\mu_{2}\left\{{\beta_{1}f^{\prime}_{\hbox{\tiny$X$}}(%
x)}/{f_{\hbox{\tiny$X$}}(x)}+\beta_{2}\right\}+O(h^{4})+O_{p}\left(\frac{1}{%
\sqrt{Jh^{c^{*}}}}\right),$
which coincides with the result for the Nadaraya-Watson estimator in regard to the dominating bias, although the order of the asymptotic variance is inflated whenever $c^{*}>1$.
Setting $p=1$ in (B.13) and extracting the first entry gives the result regarding the local linear estimator $\hat{m}_{2}(x)$, $\hat{m}_{2}(x)-m(x)=h^{2}\mu_{2}\beta_{2}+O(h^{4})+O_{p}\left(\frac{1}{\sqrt{%
Jh^{c^{*}}}}\right),$ which suggests the same dominating bias of order $h^{2}$ that is equal to the dominating bias of the same order associated with the local linear estimator for $m(x)$ based on individual-level data.
B.6 Variance of $\hat{\mbox{\boldmath$\beta$}}_{2}$
By iterative expectations,
$$\displaystyle\ \textrm{Var}(\hat{\mbox{\boldmath$\beta$}}_{2})$$
$$\displaystyle=$$
$$\displaystyle\ \textrm{Var}\left\{\mathbf{S}_{2}^{-1}(x)\mathbf{T}_{2}(x)\right\}$$
$$\displaystyle=$$
$$\displaystyle\ E\left[\mathbf{S}_{2}^{-1}(x)\textrm{Var}\left\{\mathbf{T}_{2}(%
x)|\mathbb{X}\right\}\mathbf{S}_{2}^{-{\mathrm{\scriptscriptstyle T}}}(x)%
\right]+\textrm{Var}\left[\mathbf{S}_{2}^{-1}(x)E\left\{\mathbf{T}_{2}(x)|%
\mathbb{X}\right\}\right]$$
$$\displaystyle=$$
$$\displaystyle\ E\left[J\mathbf{S}_{2}^{-1}(x)\textrm{Var}\left\{J^{-1}\mathbf{%
T}_{2}(x)|\mathbb{X}\right\}J\mathbf{S}_{2}^{-{\mathrm{\scriptscriptstyle T}}}%
(x)\right]+\textrm{Var}\left[J\mathbf{S}_{2}^{-1}(x)E\left\{J^{-1}\mathbf{T}_{%
2}(x)|\mathbb{X}\right\}\right].$$
On the other hand, we have the dominating term of $J\mathbf{S}_{2}^{-1}(x)$ given in (B.6) that holds for all $\mathbb{X}$. Hence, after substituting $J\mathbf{S}_{2}^{-1}(x)$ with (B.6), we will essentially deal with
$$\textrm{Var}\{J^{-1}\mathbf{T}_{2}(x)\}=E\left\{J^{-1}\mathbf{T}_{2}(x)J^{-1}%
\mathbf{T}^{\mathrm{\scriptscriptstyle T}}_{2}(x)\right\}-E\left\{J^{-1}%
\mathbf{T}_{2}(x)\right\}E\left\{J^{-1}\mathbf{T}^{\mathrm{\scriptscriptstyle T%
}}_{2}(x)\right\}.$$
(B.14)
We derive the second term in (B.14) first.
By (A.5), (A.7), and (A.9),
$$\displaystyle\ E\left\{J^{-1}T_{2,\ell}(x)\right\}$$
$$\displaystyle=$$
$$\displaystyle\ J^{-1}\sum_{j=1}^{J}E\left[\left\{c_{j}^{-1}\sum_{k=1}^{c_{j}}m%
(X_{jk})\right\}\left\{c_{j}^{-1}\sum_{k=1}^{c_{j}}(X_{jk}-x)^{\ell}\right\}%
\left\{\prod_{k=1}^{c_{j}}K_{h}(X_{jk}-x)\right\}\right]$$
$$\displaystyle=$$
$$\displaystyle\ J^{-1}\sum_{j=1}^{J}c_{j}c_{j}^{-2}\kappa^{*}_{1,\ell}(m)\kappa%
^{c_{j}-1}_{1,0}(h,x)+$$
$$\displaystyle\ J^{-1}\sum_{j=1}^{J}c_{j}(c_{j}-1)c_{j}^{-2}\kappa^{*}_{1,0}(m)%
\kappa_{1,\ell}(h,x)\kappa^{c_{j}-2}_{1,0}(h,x)$$
$$\displaystyle=$$
$$\displaystyle\ \left[h^{\ell}\mu_{\ell}\beta_{0}f_{\hbox{\tiny$X$}}(x)+h^{\ell%
+1}\mu_{\ell+1}\left\{\beta_{0}f^{\prime}_{\hbox{\tiny$X$}}(x)+\beta_{1}f_{%
\hbox{\tiny$X$}}(x)\right\}+O(h^{\ell+2})\right]\times$$
$$\displaystyle\ J^{-1}\sum_{j=1}^{J}c_{j}^{-1}\left\{f^{c_{j}-1}_{\hbox{\tiny$X%
$}}(x)+(c_{j}-1)h^{2}\mu_{2}f^{c_{j}-2}_{\hbox{\tiny$X$}}(x)f^{\prime\prime}_{%
\hbox{\tiny$X$}}(x)/2+O(h^{4})\right\}+$$
$$\displaystyle\ \left[\beta_{0}f_{\hbox{\tiny$X$}}(x)+h^{2}\mu_{2}\left\{\beta_%
{0}f^{\prime\prime}_{\hbox{\tiny$X$}}(x)/2+\beta_{1}f^{\prime}_{\hbox{\tiny$X$%
}}(x)+\beta_{2}f_{\hbox{\tiny$X$}}(x)\right\}+O\left(h^{4}\right)\right]\times$$
$$\displaystyle\ \left\{h^{\ell}\mu_{\ell}f_{\hbox{\tiny$X$}}(x)+h^{\ell+1}\mu_{%
\ell+1}f^{\prime}_{\hbox{\tiny$X$}}(x)+O\left(h^{\ell+2}\right)\right\}\times$$
$$\displaystyle\ J^{-1}\sum_{j=1}^{J}(c_{j}-1)c_{j}^{-1}\left\{f^{c_{j}-2}_{%
\hbox{\tiny$X$}}(x)+(c_{j}-2)h^{2}\mu_{2}f^{c_{j}-3}_{\hbox{\tiny$X$}}(x)f^{%
\prime\prime}_{\hbox{\tiny$X$}}(x)/2+O(h^{4})\right\}$$
$$\displaystyle=$$
$$\displaystyle\ h^{\ell}\mu_{\ell}\beta_{0}f_{\hbox{\tiny$X$}}(x)d(x)+h^{\ell+1%
}\mu_{\ell+1}\left\{d(x)\beta_{0}f^{\prime}_{\hbox{\tiny$X$}}(x)+d_{0}(x)\beta%
_{1}f_{\hbox{\tiny$X$}}(x)\right\}+O\left(h^{\ell+2}\right).$$
Summarizing the above in matrix form, one has
$$\displaystyle\ E\left\{J^{-1}\mathbf{T}_{2}(x)\right\}$$
$$\displaystyle=$$
$$\displaystyle\ \mathbf{H}\left[\beta_{0}f_{\hbox{\tiny$X$}}(x)d(x)\mbox{%
\boldmath$\mu$}_{0}^{*}+h\left\{d(x)\beta_{0}f^{\prime}_{\hbox{\tiny$X$}}(x)+d%
_{0}(x)\beta_{1}f_{\hbox{\tiny$X$}}(x)\right\}\mbox{\boldmath$\mu$}_{1}^{*}+O(%
h^{2})\right].$$
(B.15)
It follows that
$$\displaystyle\ E\left\{J^{-1}\mathbf{T}_{2}(x)\right\}E\left\{J^{-1}\mathbf{T}%
^{\mathrm{\scriptscriptstyle T}}_{2}(x)\right\}$$
$$\displaystyle=$$
$$\displaystyle\ \beta_{0}f_{\hbox{\tiny$X$}}(x)d(x)\mathbf{H}\left[\beta_{0}f_{%
\hbox{\tiny$X$}}(x)d(x)\mbox{\boldmath$\mu$}^{*}_{0}\mbox{\boldmath$\mu$}^{*{%
\mathrm{\scriptscriptstyle T}}}_{0}+h\left\{d(x)\beta_{0}f^{\prime}_{\hbox{%
\tiny$X$}}(x)+d_{0}(x)\beta_{1}f_{\hbox{\tiny$X$}}(x)\right\}\times\right.$$
$$\displaystyle\left.\left(\mbox{\boldmath$\mu$}_{0}^{*}\mbox{\boldmath$\mu$}_{1%
}^{*{\mathrm{\scriptscriptstyle T}}}+\mbox{\boldmath$\mu$}_{1}^{*}\mbox{%
\boldmath$\mu$}_{0}^{*{\mathrm{\scriptscriptstyle T}}}\right)+O(h^{2})\right]%
\mathbf{H}.$$
(B.16)
The first term in (B.14) relates to $J^{-2}E\{T_{2,\ell_{1}}(x)T_{2,\ell_{2}}(x)|\mathbb{X}\}$that we derive next. Because
$$\displaystyle\ T_{2,\ell_{1}}(x)T_{2,\ell_{2}}(x)$$
$$\displaystyle=$$
$$\displaystyle\ \sum_{j=1}^{J}Z^{2}_{j}\left\{c_{j}^{-1}\sum_{k=1}^{c_{j}}(X_{%
jk}-x)^{\ell_{1}}\right\}\left\{c_{j}^{-1}\sum_{k=1}^{c_{j}}(X_{jk}-x)^{\ell_{%
2}}\right\}\prod_{k=1}^{c_{j}}K^{2}_{h}(X_{jk}-x)+$$
$$\displaystyle\ \sum_{j_{1}\neq j_{2}}Z_{j_{1}}Z_{j_{2}}\left\{c_{j_{1}}^{-1}%
\sum_{k=1}^{c_{j_{1}}}(X_{j_{1}k}-x)^{\ell_{1}}\right\}\left\{c_{j_{2}}^{-1}%
\sum_{k=1}^{c_{j_{2}}}(X_{j_{2}k}-x)^{\ell_{2}}\right\}\times$$
$$\displaystyle\ \left\{\prod_{k=1}^{c_{j_{1}}}K_{h}(X_{j_{1}k}-x)\right\}\left%
\{\prod_{k=1}^{c_{j_{2}}}K_{h}(X_{j_{2}k}-x)\right\},$$
hence,
$$\displaystyle\ J^{-2}E\{T_{2,\ell_{1}}(x)T_{2,\ell_{2}}(x)|\mathbb{X}\}$$
$$\displaystyle=$$
$$\displaystyle\ J^{-2}\sum_{j=1}^{J}E(Z^{2}_{j}|\tilde{\mathbf{X}}_{j})\left\{c%
_{j}^{-1}\sum_{k=1}^{c_{j}}(X_{jk}-x)^{\ell_{1}}\right\}\left\{c_{j}^{-1}\sum_%
{k=1}^{c_{j}}(X_{jk}-x)^{\ell_{2}}\right\}\prod_{k=1}^{c_{j}}K^{2}_{h}(X_{jk}-%
x)+$$
$$\displaystyle\ J^{-2}\sum_{j_{1}\neq j_{2}}E(Z_{j_{1}}|\tilde{\mathbf{X}}_{j_{%
1}})E(Z_{j_{2}}|\tilde{\mathbf{X}}_{j_{2}})\left\{c_{j_{1}}^{-1}\sum_{k=1}^{c_%
{j_{1}}}(X_{j_{1}k}-x)^{\ell_{1}}\right\}\left\{c_{j_{2}}^{-1}\sum_{k=1}^{c_{j%
_{2}}}(X_{j_{2}k}-x)^{\ell_{2}}\right\}\times$$
$$\displaystyle\ \left\{\prod_{k=1}^{c_{j_{1}}}K_{h}(X_{j_{1}k}-x)\right\}\left%
\{\prod_{k=1}^{c_{j_{2}}}K_{h}(X_{j_{2}k}-x)\right\}$$
$$\displaystyle=$$
$$\displaystyle\ J^{-2}\sum_{j=1}^{J}\left\{c_{j}^{-2}\sum_{k=1}^{c_{j}}\sigma^{%
2}(X_{jk})\right\}\left\{c_{j}^{-1}\sum_{k=1}^{c_{j}}(X_{jk}-x)^{\ell_{1}}%
\right\}\left\{c_{j}^{-1}\sum_{k=1}^{c_{j}}(X_{jk}-x)^{\ell_{2}}\right\}\times$$
$$\displaystyle\ \prod_{k=1}^{c_{j}}K^{2}_{h}(X_{jk}-x)$$
(B.17)
$$\displaystyle\ +J^{-2}\sum_{j=1}^{J}\left\{c_{j}^{-1}\sum_{k=1}^{c_{j}}m(X_{jk%
})\right\}^{2}\left\{c_{j}^{-1}\sum_{k=1}^{c_{j}}(X_{jk}-x)^{\ell_{1}}\right\}%
\left\{c_{j}^{-1}\sum_{k=1}^{c_{j}}(X_{jk}-x)^{\ell_{2}}\right\}\times$$
$$\displaystyle\ \prod_{k=1}^{c_{j}}K^{2}_{h}(X_{jk}-x)$$
(B.18)
$$\displaystyle\ +J^{-2}\sum_{j_{1}\neq j_{2}}\left\{c_{j_{1}}^{-1}\sum_{k=1}^{c%
_{j_{1}}}m(X_{j_{1}k})\right\}\left\{c_{j_{2}}^{-1}\sum_{k=1}^{c_{j_{2}}}m(X_{%
j_{2}k})\right\}\left\{c_{j_{1}}^{-1}\sum_{k=1}^{c_{j_{1}}}(X_{j_{1}k}-x)^{%
\ell_{1}}\right\}\times$$
$$\displaystyle\ \left\{c_{j_{2}}^{-1}\sum_{k=1}^{c_{j_{2}}}(X_{j_{2}k}-x)^{\ell%
_{2}}\right\}\left\{\prod_{k=1}^{c_{j_{1}}}K_{h}(X_{j_{1}k}-x)\right\}\left\{%
\prod_{k=1}^{c_{j_{2}}}K_{h}(X_{j_{2}k}-x)\right\}.$$
(B.19)
According to our earlier derivation of $E\{J^{-1}T_{2,\ell}(x)\}$, the expectation of (B.19) is of order $O(J^{-1}h^{\ell_{1}+\ell_{2}})$, and thus is dominated by the expectations of (B.17) and (B.18), which are of order $O(J^{-1}h^{\ell_{1}+\ell_{2}-c^{*}})$ as we show next.
By (A.8) and (A.10), the expectation of the summand of (B.17) is equal to $c_{j}^{-4}$ times
$$\displaystyle\ E\left[\left\{\sum_{k=1}^{c_{j}}\sigma^{2}(X_{jk})\right\}\left%
\{\sum_{k=1}^{c_{j}}(X_{jk}-x)^{\ell_{1}}\right\}\left\{\sum_{k=1}^{c_{j}}(X_{%
jk}-x)^{\ell_{2}}\right\}\prod_{k=1}^{c_{j}}K^{2}_{h}(X_{jk}-x)\right]$$
$$\displaystyle=$$
$$\displaystyle\ c_{j}\kappa^{*}_{2,\ell_{1}+\ell_{2}}(\sigma^{2})\kappa^{c_{j}-%
1}_{2,0}(h,x)+c_{j}(c_{j}-1)\left\{\kappa^{*}_{2,0}(\sigma^{2})\kappa_{2,\ell_%
{1}+\ell_{2}}(h,x)+\right.$$
$$\displaystyle\left.\kappa^{*}_{2,\ell_{1}}(\sigma^{2})\kappa_{2,\ell_{2}}(h,x)%
+\kappa^{*}_{2,\ell_{2}}(\sigma^{2})\kappa_{2,\ell_{1}}(h,x)\right\}\kappa^{c_%
{j}-2}_{2,0}(h,x)+$$
$$\displaystyle\ c_{j}(c_{j}-1)(c_{j}-2)\kappa^{*}_{2,0}(\sigma^{2})\kappa_{2,%
\ell_{1}}(h,x)\kappa_{2,\ell_{2}}(h,x)\kappa^{c_{j}-3}_{2,0}(h,x)$$
$$\displaystyle=$$
$$\displaystyle\ h^{\ell_{1}+\ell_{2}-c_{j}}\nu_{0}\sigma^{2}(x)c_{j}f^{c_{j}}_{%
\hbox{\tiny$X$}}(x)\{\nu_{\ell_{1}+\ell_{2}}+(c_{j}-1)(\nu_{0}\nu_{\ell_{1}+%
\ell_{2}}+2\nu_{\ell_{1}}\nu_{\ell_{2}})+$$
$$\displaystyle\ [(c_{j}-1)(c_{j}-2)]_{+}\nu_{0}\nu_{\ell_{1}}\nu_{\ell_{2}}\}+h%
^{\ell_{1}+\ell_{2}-c_{j}+1}\nu_{0}\sigma(x)c_{j}f^{c_{j}-1}_{\hbox{\tiny$X$}}%
(x)\left(\nu_{\ell_{1}+\ell_{2}+1}\times\right.$$
$$\displaystyle\ \left\{\sigma(x)f^{\prime}_{\hbox{\tiny$X$}}(x)+2\sigma^{\prime%
}(x)f_{\hbox{\tiny$X$}}(x)\right\}+(c_{j}-1)\left[\nu_{\ell_{1}+\ell_{2}+1}%
\sigma(x)f^{\prime}_{\hbox{\tiny$X$}}(x)+\right.$$
$$\displaystyle\left.(\nu_{\ell_{1}+1}\nu_{\ell_{2}}+\nu_{\ell_{1}}\nu_{\ell_{2}%
+1})\left\{\sigma(x)f^{\prime}_{\hbox{\tiny$X$}}(x)+2\sigma^{\prime}(x)f_{%
\hbox{\tiny$X$}}(x)\right\}\right]+$$
$$\displaystyle\left.[(c_{j}-1)(c_{j}-2)]_{+}\nu_{0}(\nu_{\ell_{1}+1}\nu_{\ell_{%
2}}+\nu_{\ell_{1}}\nu_{\ell_{2}+1})\sigma(x)f^{\prime}_{\hbox{\tiny$X$}}(x)%
\right)+O\left(h^{\ell_{1}+\ell_{2}+2-c_{j}}\right).$$
Hence,
$$\displaystyle\ J^{-2}\sum_{j=1}^{J}E\left[\left\{c_{j}^{-2}\sum_{k=1}^{c_{j}}%
\sigma^{2}(X_{jk})\right\}\left\{c_{j}^{-1}\sum_{k=1}^{c_{j}}(X_{jk}-x)^{\ell_%
{1}}\right\}\times\right.$$
$$\displaystyle\left.\left\{c_{j}^{-1}\sum_{k=1}^{c_{j}}(X_{jk}-x)^{\ell_{2}}%
\right\}\prod_{k=1}^{c_{j}}K^{2}_{h}(X_{jk}-x)\right]$$
$$\displaystyle=$$
$$\displaystyle\ h^{\ell_{1}+\ell_{2}}\nu_{0}\sigma^{2}(x)f_{\hbox{\tiny$X$}}(x)%
\{w_{30}(x,h,J)\nu_{\ell_{1}+\ell_{2}}+w_{31}(x,h,J)(\nu_{0}\nu_{\ell_{1}+\ell%
_{2}}+2\nu_{\ell_{1}}\nu_{\ell_{2}})+$$
$$\displaystyle\ w_{32}(x,h,J)\nu_{0}\nu_{\ell_{1}}\nu_{\ell_{2}}\}+h^{\ell_{1}+%
\ell_{2}+1}\nu_{0}\sigma^{2}(x)\left[f^{\prime}_{\hbox{\tiny$X$}}(x)w_{20}(x,h%
,J)\nu_{\ell_{1}+\ell_{2}+1}+\right.$$
$$\displaystyle\ 2\sigma^{\prime}(x)f_{\hbox{\tiny$X$}}(x)\{w_{30}(x,h,J)\nu_{%
\ell_{1}+\ell_{2}+1}+w_{31}(x,h,J)(\nu_{\ell_{1}+1}\nu_{\ell_{2}}+\nu_{\ell_{1%
}}\nu_{\ell_{2}+1})\}+$$
$$\displaystyle\left.\sigma(x)f^{\prime}_{\hbox{\tiny$X$}}(x)\{w_{31}(x,h,J)+\nu%
_{0}w_{32}(x,h,J)\}(\nu_{\ell_{1}+1}\nu_{\ell_{2}}+\nu_{\ell_{1}}\nu_{\ell_{2}%
+1})\right]+$$
$$\displaystyle\ O\left(J^{-1}h^{\ell_{1}+\ell_{2}+2-c^{*}}\right).$$
For $\ell_{1},\ell_{2}=0,1,\ldots,p$, the above expression is the $[\ell_{1}+1,\ell_{2}+1]$ entry of the following $(p+1)\times(p+1)$ matrix,
$$\displaystyle\ \nu_{0}\sigma^{2}(x)\mathbf{H}\left\{f_{\hbox{\tiny$X$}}(x)%
\left\{w_{30}(x,h,J)\tilde{\mbox{\boldmath$\nu$}}_{0}+w_{31}(x,h,J)\left(\nu_{%
0}\tilde{\mbox{\boldmath$\nu$}}_{0}+2\mbox{\boldmath$\nu$}_{0}^{*}\mbox{%
\boldmath$\nu$}_{0}^{*{\mathrm{\scriptscriptstyle T}}}\right)+\right.\right.$$
$$\displaystyle\left.w_{32}(x,h,J)\nu_{0}\mbox{\boldmath$\nu$}_{0}^{*}\mbox{%
\boldmath$\nu$}_{0}^{*{\mathrm{\scriptscriptstyle T}}}\right\}+h\sigma^{-1}(x)%
\left(\sigma(x)f^{\prime}_{\hbox{\tiny$X$}}(x)\left[w_{20}(x,h,J)\tilde{\mbox{%
\boldmath$\nu$}}_{1}+\right.\right.$$
$$\displaystyle\left.\left\{w_{31}(x,h,J)+\nu_{0}w_{32}(x,h,J)\right\}(\mbox{%
\boldmath$\nu$}_{1}^{*}\mbox{\boldmath$\nu$}_{0}^{*{\mathrm{\scriptscriptstyle
T%
}}}+\mbox{\boldmath$\nu$}_{0}^{*}\mbox{\boldmath$\nu$}_{1}^{*{\mathrm{%
\scriptscriptstyle T}}})\right]+2\sigma^{\prime}(x)f_{\hbox{\tiny$X$}}(x)\times$$
$$\displaystyle\left.\left.\left\{w_{30}(x,h,J)\tilde{\mbox{\boldmath$\nu$}}_{1}%
+w_{31}(x,h,J)(\mbox{\boldmath$\nu$}^{*}_{1}\mbox{\boldmath$\nu$}^{*{\mathrm{%
\scriptscriptstyle T}}}_{0}+\mbox{\boldmath$\nu$}_{0}^{*}\mbox{\boldmath$\nu$}%
_{1}^{*{\mathrm{\scriptscriptstyle T}}})\right\}\right)+O\left(J^{-1}h^{2-c^{*%
}}\right)\right\}\mathbf{H}.$$
(B.20)
The expectation of the summand of (B.18) is equal to $c_{j}^{-4}$ times
$$\displaystyle\ E\left[\left\{\sum_{k=1}^{c_{j}}m(X_{jk})\right\}^{2}\left\{%
\sum_{k=1}^{c_{j}}(X_{jk}-x)^{\ell_{1}}\right\}\left\{\sum_{k=1}^{c_{j}}(X_{jk%
}-x)^{\ell_{2}}\right\}\prod_{k=1}^{c_{j}}K^{2}_{h}(X_{jk}-x)\right]$$
$$\displaystyle=$$
$$\displaystyle\ c_{j}\kappa^{c_{j}-1}_{2,0}(h,x)\kappa^{*}_{2,\ell_{1}+\ell_{2}%
}(m^{2})+$$
$$\displaystyle\ c_{j}(c_{j}-1)\kappa_{2,0}^{c_{j}-2}(h,x)\left\{\kappa^{*}_{2,0%
}(m^{2})\kappa_{2,\ell_{1}+\ell_{2}}(h,x)+2\kappa^{*}_{2,\ell_{1}}(m)\kappa^{*%
}_{2,\ell_{2}}(m)+\right.$$
$$\displaystyle\left.2\kappa^{*}_{2,0}(m)\kappa^{*}_{2,\ell_{1}+\ell_{2}}(m)+%
\kappa^{*}_{2,\ell_{1}}(m^{2})\kappa_{2,\ell_{2}}(h,x)+\kappa^{*}_{2,\ell_{2}}%
(m^{2})\kappa_{2,\ell_{1}}(h,x)\right\}+$$
$$\displaystyle\ [c_{j}(c_{j}-1)(c_{j}-2)]_{+}\kappa_{2,0}^{c_{j}-3}(h,x)\left\{%
\kappa^{*2}_{2,0}(m)\kappa_{2,\ell_{1}+\ell_{2}}(h,x)+\right.$$
$$\displaystyle\left.\kappa^{*}_{2,0}(m^{2})\kappa_{2,\ell_{1}}(h,x)\kappa_{2,%
\ell_{2}}(h,x)+\kappa^{*}_{2,\ell_{1}}(m)\kappa^{*}_{2,0}(m)\kappa_{2,\ell_{2}%
}(h,x)\right.$$
$$\displaystyle\left.+\kappa^{*}_{2,\ell_{2}}(m)\kappa^{*}_{2,0}(m)\kappa_{2,%
\ell_{1}}(h,x)\right\}+[c_{j}(c_{j}-1)(c_{j}-2)(c_{j}-3)]_{+}\times$$
$$\displaystyle\ \kappa_{2,0}^{c_{j}-4}(h,x)\kappa^{*2}_{2,0}(m)\kappa_{2,\ell_{%
1}}(h,x)\kappa_{2,\ell_{2}}(h,x).$$
Using (A.6) and (A.10) in the above gives
$$\displaystyle\ J^{-2}E\left[\left\{c_{j}^{-1}\sum_{k=1}^{c_{j}}m(X_{jk})\right%
\}^{2}\left\{c_{j}^{-1}\sum_{k=1}^{c_{j}}(X_{jk}-x)^{\ell_{1}}\right\}\times\right.$$
$$\displaystyle\left.\left\{c_{j}^{-1}\sum_{k=1}^{c_{j}}(X_{jk}-x)^{\ell_{2}}%
\right\}\prod_{k=1}^{c_{j}}K^{2}_{h}(X_{jk}-x)\right]$$
$$\displaystyle=$$
$$\displaystyle\ h^{\ell_{1}+\ell_{2}}w_{30}(x,h,J)\nu_{0}\beta_{0}\left[\beta_{%
0}f_{\hbox{\tiny$X$}}(x)\nu_{\ell_{1}+\ell_{2}}+h\left\{\beta_{0}f^{\prime}_{%
\hbox{\tiny$X$}}(x)+2\beta_{1}f_{\hbox{\tiny$X$}}(x)\right\}\nu_{\ell_{1}+\ell%
_{2}+1}\right]+$$
$$\displaystyle\ h^{\ell_{1}+\ell_{2}}w_{31}(x,h,J)\nu_{0}\beta_{0}\left(\beta_{%
0}f_{\hbox{\tiny$X$}}(x)(3\nu_{0}\nu_{\ell_{1}+\ell_{2}}+4\nu_{\ell_{1}}\nu_{%
\ell_{2}})+h\left[\nu_{0}\nu_{\ell_{1}+\ell_{2}+1}\times\right.\right.$$
$$\displaystyle\left.\left.\left\{2\beta_{0}f^{\prime}_{\hbox{\tiny$X$}}(x)+%
\beta_{1}f_{\hbox{\tiny$X$}}(x)\right\}+4(\nu_{\ell_{1}}\nu_{\ell_{2}+1}+\nu_{%
\ell_{1}+1}\nu_{\ell_{2}})\left\{\beta_{0}f^{\prime}_{\hbox{\tiny$X$}}(x)+%
\beta_{1}f_{\hbox{\tiny$X$}}(x)\right\}\right]\right)+$$
$$\displaystyle\ h^{\ell_{1}+\ell_{2}}w_{32}(x,h,J)\nu^{2}_{0}\beta_{0}\left(%
\beta_{0}f_{\hbox{\tiny$X$}}(x)\nu_{0}(\nu_{\ell_{1}+\ell_{2}}+3\nu_{\ell_{1}}%
\nu_{\ell_{2}})+h\left[\left\{3\beta_{0}f^{\prime}_{\hbox{\tiny$X$}}(x)+\beta_%
{1}f_{\hbox{\tiny$X$}}(x)\right\}\times\right.\right.$$
$$\displaystyle\left.\left.(\nu_{\ell_{1}}\nu_{\ell_{2}+1}+\nu_{\ell_{1}+1}\nu_{%
\ell_{2}})+\beta_{0}f^{\prime}_{\hbox{\tiny$X$}}(x)\nu_{0}\nu_{\ell_{1}+\ell_{%
2}+1}\right]\right)+$$
$$\displaystyle\ h^{\ell_{1}+\ell_{2}}w_{33}(x,h,J)\nu_{0}^{3}\beta_{0}\left\{%
\beta_{0}f_{\hbox{\tiny$X$}}(x)\nu_{\ell_{1}}\nu_{\ell_{2}}+h\beta_{0}f^{%
\prime}_{\hbox{\tiny$X$}}(x)(\nu_{\ell_{1}}\nu_{\ell_{2}+1}+\nu_{\ell_{1}+1}%
\nu_{\ell_{2}})\right\}+$$
$$\displaystyle\ O\left(J^{-1}h^{\ell_{1}+\ell_{2}+2-c^{*}}\right)$$
$$\displaystyle=$$
$$\displaystyle\ h^{\ell_{1}+\ell_{2}}\nu_{0}\beta_{0}^{2}f_{\hbox{\tiny$X$}}(x)%
\left\{w_{30}(x,h,J)\nu_{\ell_{1}+\ell_{2}}+w_{31}(x,h,J)(3\nu_{0}\nu_{\ell_{1%
}+\ell_{2}}+4\nu_{\ell_{1}}\nu_{\ell_{2}})+\right.$$
$$\displaystyle\left.w_{32}(x,h,J)\nu_{0}^{2}(\nu_{\ell_{1}+\ell_{2}}+3\nu_{\ell%
_{1}}\nu_{\ell_{2}})+w_{33}(x,h,J)\nu_{0}^{2}\nu_{\ell_{1}}\nu_{\ell_{2}}%
\right\}+$$
$$\displaystyle\ h^{\ell_{1}+\ell_{2}+1}\nu_{0}\beta_{0}\left(w_{30}(x,h,J)\left%
\{\beta_{0}f^{\prime}_{\hbox{\tiny$X$}}(x)+2\beta_{1}f_{\hbox{\tiny$X$}}(x)%
\right\}\nu_{\ell_{1}+\ell_{2}+1}+\right.$$
$$\displaystyle\ w_{31}(x,h,J)\left[\left\{2\beta_{0}f^{\prime}_{\hbox{\tiny$X$}%
}(x)+\beta_{1}f_{\hbox{\tiny$X$}}(x)\right\}\nu_{0}\nu_{\ell_{1}+\ell_{2}+1}+4%
\left\{\beta_{0}f^{\prime}_{\hbox{\tiny$X$}}(x)+\beta_{1}f_{\hbox{\tiny$X$}}(x%
)\right\}\times\right.$$
$$\displaystyle\left.(\nu_{\ell_{1}}\nu_{\ell_{2}+1}+\nu_{\ell_{1}+1}\nu_{\ell_{%
2}})\right]+w_{32}(x,h,J)\nu_{0}\left[\left\{3\beta_{0}f^{\prime}_{\hbox{\tiny%
$X$}}(x)+\beta_{1}f_{\hbox{\tiny$X$}}(x)\right\}\times\right.$$
$$\displaystyle\left.(\nu_{\ell_{1}}\nu_{\ell_{2}+1}+\nu_{\ell_{1}+1}\nu_{\ell_{%
2}})+\beta_{0}f^{\prime}_{\hbox{\tiny$X$}}(x)\nu_{0}\nu_{\ell_{1}+\ell_{2}+1}%
\right]+$$
$$\displaystyle\left.w_{33}(x,h,J)\beta_{0}f^{\prime}_{\hbox{\tiny$X$}}(x)\nu_{0%
}^{2}(\nu_{\ell_{1}}\nu_{\ell_{2}+1}+\nu_{\ell_{1}+1}\nu_{\ell_{2}})\right)+O%
\left(J^{-1}h^{\ell_{1}+\ell_{2}+2-c^{*}}\right)$$
For $\ell_{1},\ell_{2}=0,1,\ldots,p$, the above expression is the $[\ell_{1}+1,\ell_{2}+1]$ entry of the following $(p+1)\times(p+1)$ matrix,
$$\displaystyle\ \nu_{0}\beta_{0}\mathbf{H}\left[\beta_{0}f_{\hbox{\tiny$X$}}(x)%
\left\{w_{30}(x,h,J)\tilde{\mbox{\boldmath$\nu$}}_{0}+w_{31}(x,h,J)\left(3\nu_%
{0}\tilde{\mbox{\boldmath$\nu$}}_{0}+4\mbox{\boldmath$\nu$}_{0}^{*}\mbox{%
\boldmath$\nu$}_{0}^{*{\mathrm{\scriptscriptstyle T}}}\right)+\right.\right.$$
$$\displaystyle\left.w_{32}(x,h,J)\nu^{2}_{0}\left(\tilde{\mbox{\boldmath$\nu$}}%
_{0}+3\mbox{\boldmath$\nu$}^{*}_{0}\mbox{\boldmath$\nu$}^{*{\mathrm{%
\scriptscriptstyle T}}}_{0}\right)+w_{33}(x,h,J)\nu_{0}^{2}\mbox{\boldmath$\nu%
$}_{0}^{*}\mbox{\boldmath$\nu$}_{0}^{*{\mathrm{\scriptscriptstyle T}}}\right\}+$$
$$\displaystyle\ h\left(w_{30}(x,h,J)\left\{\beta_{0}f^{\prime}_{\hbox{\tiny$X$}%
}(x)+2\beta_{1}f_{\hbox{\tiny$X$}}(x)\right\}\tilde{\mbox{\boldmath$\nu$}}_{1}%
+w_{31}(x,h,J)\left[\nu_{0}\tilde{\mbox{\boldmath$\nu$}}_{1}\times\right.\right.$$
$$\displaystyle\left.\left\{2\beta_{0}f^{\prime}_{\hbox{\tiny$X$}}(x)+\beta_{1}f%
_{\hbox{\tiny$X$}}(x)\right\}+4\left\{\beta_{0}f^{\prime}_{\hbox{\tiny$X$}}(x)%
+\beta_{1}f_{\hbox{\tiny$X$}}(x)\right\}\left(\mbox{\boldmath$\nu$}_{0}^{*}%
\mbox{\boldmath$\nu$}_{1}^{*{\mathrm{\scriptscriptstyle T}}}+\mbox{\boldmath$%
\nu$}_{1}^{*}\mbox{\boldmath$\nu$}_{0}^{*{\mathrm{\scriptscriptstyle T}}}%
\right)\right]+$$
$$\displaystyle\ w_{32}(x,h,J)\nu_{0}\left[\left\{3\beta_{0}f^{\prime}_{\hbox{%
\tiny$X$}}(x)+\beta_{1}f_{\hbox{\tiny$X$}}(x)\right\}\left(\mbox{\boldmath$\nu%
$}_{0}^{*}\mbox{\boldmath$\nu$}_{1}^{*{\mathrm{\scriptscriptstyle T}}}+\mbox{%
\boldmath$\nu$}_{1}^{*}\mbox{\boldmath$\nu$}_{0}^{*{\mathrm{\scriptscriptstyle
T%
}}}\right)+\beta_{0}f^{\prime}_{\hbox{\tiny$X$}}(x)\nu_{0}\tilde{\mbox{%
\boldmath$\nu$}}_{1}\right]+$$
$$\displaystyle\left.\left.w_{33}(x,h,J)\beta_{0}f^{\prime}_{\hbox{\tiny$X$}}(x)%
\nu_{0}^{2}\left(\mbox{\boldmath$\nu$}_{0}^{*}\mbox{\boldmath$\nu$}_{1}^{*{%
\mathrm{\scriptscriptstyle T}}}+\mbox{\boldmath$\nu$}_{1}^{*}\mbox{\boldmath$%
\nu$}_{0}^{*{\mathrm{\scriptscriptstyle T}}}\right)\right)+O\left(J^{-1}h^{2-c%
^{*}}\right)\right]\mathbf{H}.$$
(B.21)
Summing (B.20) and (B.21) gives the following $(p+1)\times(p+1)$ matrix as the first term in (B.14),
$$\displaystyle\ E\left\{J^{-1}\mathbf{T}_{2}(x)J^{-1}\mathbf{T}^{\mathrm{%
\scriptscriptstyle T}}_{2}(x)\right\}$$
$$\displaystyle=$$
$$\displaystyle\ E\left[E\left\{J^{-1}\mathbf{T}_{2}(x)J^{-1}\mathbf{T}^{\mathrm%
{\scriptscriptstyle T}}_{2}(x)|\mathbb{X}\right\}\right]$$
$$\displaystyle=$$
$$\displaystyle\ \nu_{0}f_{\hbox{\tiny$X$}}(x)\mathbf{H}\left[w_{30}(x,h,J)\left%
\{\sigma^{2}(x)+\beta_{0}^{2}\right\}\tilde{\mbox{\boldmath$\nu$}}_{0}+w_{31}(%
x,h,J)\left\{\sigma^{2}(x)\left(\nu_{0}\tilde{\mbox{\boldmath$\nu$}}_{0}+2%
\mbox{\boldmath$\nu$}^{*}_{0}\mbox{\boldmath$\nu$}_{0}^{*{\mathrm{%
\scriptscriptstyle T}}}\right)\right.\right.$$
$$\displaystyle\left.+\beta^{2}_{0}\left(3\nu_{0}\tilde{\mbox{\boldmath$\nu$}}_{%
0}+4\mbox{\boldmath$\nu$}_{0}^{*}\mbox{\boldmath$\nu$}_{0}^{*{\mathrm{%
\scriptscriptstyle T}}}\right)\right\}+w_{32}(x,h,J)\nu_{0}\left\{\sigma^{2}(x%
)\mbox{\boldmath$\nu$}^{*}_{0}\mbox{\boldmath$\nu$}^{*{\mathrm{%
\scriptscriptstyle T}}}_{0}+\beta_{0}^{2}\nu_{0}\left(\tilde{\mbox{\boldmath$%
\nu$}}_{0}+3\mbox{\boldmath$\nu$}^{*}_{0}\mbox{\boldmath$\nu$}^{*{\mathrm{%
\scriptscriptstyle T}}}_{0}\right)\right\}$$
$$\displaystyle\left.+w_{33}(x,h,J)\beta_{0}^{2}\nu_{0}^{2}\mbox{\boldmath$\nu$}%
_{0}^{*}\mbox{\boldmath$\nu$}^{*{\mathrm{\scriptscriptstyle T}}}_{0}\right]%
\mathbf{H}+\nu_{0}h\mathbf{H}\left\{w_{30}(x,h,J)\left[\sigma^{2}(x)f^{\prime}%
_{\hbox{\tiny$X$}}(x)+2\sigma(x)\sigma^{\prime}(x)f_{\hbox{\tiny$X$}}(x)\right%
.\right.$$
$$\displaystyle\left.+\beta_{0}\left\{\beta_{0}f^{\prime}_{\hbox{\tiny$X$}}(x)+2%
\beta_{1}f_{\hbox{\tiny$X$}}(x)\right\}\right]\tilde{\mbox{\boldmath$\nu$}}_{1%
}+w_{31}(x,h,J)\left(\sigma^{2}(x)f^{\prime}_{\hbox{\tiny$X$}}(x)\left(\tilde{%
\mbox{\boldmath$\nu$}}_{1}+\mbox{\boldmath$\nu$}_{0}^{*}\mbox{\boldmath$\nu$}_%
{1}^{*{\mathrm{\scriptscriptstyle T}}}+\mbox{\boldmath$\nu$}_{1}^{*}\mbox{%
\boldmath$\nu$}_{0}^{*{\mathrm{\scriptscriptstyle T}}}\right)\right.$$
$$\displaystyle\ +2\sigma(x)\sigma^{\prime}(x)f_{\hbox{\tiny$X$}}(x)\left(\mbox{%
\boldmath$\nu$}_{0}^{*}\mbox{\boldmath$\nu$}_{1}^{*{\mathrm{\scriptscriptstyle
T%
}}}+\mbox{\boldmath$\nu$}_{1}^{*}\mbox{\boldmath$\nu$}_{0}^{*{\mathrm{%
\scriptscriptstyle T}}}\right)+\beta_{0}\left[\left\{2\beta_{0}f^{\prime}_{%
\hbox{\tiny$X$}}(x)+\beta_{1}f_{\hbox{\tiny$X$}}(x)\right\}\nu_{0}\tilde{\mbox%
{\boldmath$\nu$}}_{1}\right.$$
$$\displaystyle\left.\left.+4\left\{\beta_{0}f^{\prime}_{\hbox{\tiny$X$}}(x)+%
\beta_{1}f_{\hbox{\tiny$X$}}(x)\right\}\left(\mbox{\boldmath$\nu$}_{0}^{*}%
\mbox{\boldmath$\nu$}_{1}^{*{\mathrm{\scriptscriptstyle T}}}+\mbox{\boldmath$%
\nu$}_{1}^{*}\mbox{\boldmath$\nu$}_{0}^{*{\mathrm{\scriptscriptstyle T}}}%
\right)\right]\right)+w_{32}(x,h,J)\nu_{0}\left(\sigma^{2}(x)f^{\prime}_{\hbox%
{\tiny$X$}}(x)\times\right.$$
$$\displaystyle\left.\left(\mbox{\boldmath$\nu$}_{0}^{*}\mbox{\boldmath$\nu$}_{1%
}^{*{\mathrm{\scriptscriptstyle T}}}+\mbox{\boldmath$\nu$}_{1}^{*}\mbox{%
\boldmath$\nu$}_{0}^{*{\mathrm{\scriptscriptstyle T}}}\right)+\beta_{0}\left[%
\left\{3\beta_{0}f^{\prime}_{\hbox{\tiny$X$}}(x)+\beta_{1}f_{\hbox{\tiny$X$}}(%
x)\right\}\left(\mbox{\boldmath$\nu$}_{0}^{*}\mbox{\boldmath$\nu$}_{1}^{*{%
\mathrm{\scriptscriptstyle T}}}+\mbox{\boldmath$\nu$}_{1}^{*}\mbox{\boldmath$%
\nu$}_{0}^{*{\mathrm{\scriptscriptstyle T}}}\right)+\beta_{0}f^{\prime}_{\hbox%
{\tiny$X$}}(x)\nu_{0}\tilde{\mbox{\boldmath$\nu$}}_{1}\right]\right)$$
$$\displaystyle\left.+w_{33}(x,h,J)\nu_{0}^{2}\beta_{0}^{2}f^{\prime}_{\hbox{%
\tiny$X$}}(x)\left(\mbox{\boldmath$\nu$}_{0}^{*}\mbox{\boldmath$\nu$}_{1}^{*{%
\mathrm{\scriptscriptstyle T}}}+\mbox{\boldmath$\nu$}_{1}^{*}\mbox{\boldmath$%
\nu$}_{0}^{*{\mathrm{\scriptscriptstyle T}}}\right)+O\left(J^{-1}h^{1-c^{*}}%
\right)\right\}\mathbf{H}.$$
(B.22)
Because (B.22) is of order $O\{\mathbf{H}^{2}/(Jh^{c^{*}})\}$, whereas (B.16) is of order $O(\mathbf{H}^{2})$, the dominating terms of $\textrm{Var}\{J^{-1}\mathbf{T}_{2}(x)\}$ are given by (B.22).
Using (B.6) and (B.22), we now have
$$\displaystyle\mbox{Var}\left(\left.\hat{\mbox{\boldmath$\beta$}}_{2}\right|%
\mathbb{X}\right)$$
$$\displaystyle=$$
$$\displaystyle\ \frac{\nu_{0}}{f_{\hbox{\tiny$X$}}(x)}\mathbf{H}^{-1}\mathbf{D}%
^{-1}_{0}(x)\left[\mathbf{F}(x,h,J)+h\frac{f^{\prime}_{\hbox{\tiny$X$}}(x)}{f_%
{\hbox{\tiny$X$}}(x)}\left\{\mathbf{F}(x,h,J)\mathbf{D}^{-1}_{0}(x)\mathbf{D}_%
{1}(x)-\right.\right.$$
$$\displaystyle\left.f_{\hbox{\tiny$X$}}(x)\mathbf{D}_{1}(x)\mathbf{D}_{0}(x)^{-%
1}\mathbf{F}(x,h,J)\mathbf{D}_{0}(x)+\frac{\mathbf{G}(x,h,J)}{f^{\prime}_{%
\hbox{\tiny$X$}}(x)}\right\}+O(h^{2})+$$
$$\displaystyle\left.O_{\hbox{\tiny$P$}}\left(\frac{1}{Jh^{c^{*}}}\right)\right]%
\mathbf{D}^{-1}_{0}(x)\mathbf{H}^{-1},$$
(B.23)
where
$$\displaystyle\mathbf{D}_{0}(x)=$$
$$\displaystyle\ d_{0}(x)\tilde{\mbox{\boldmath$\mu$}}_{0}+d_{1}(x)\mbox{%
\boldmath$\mu$}_{0}^{*}\mbox{\boldmath$\mu$}_{0}^{*{\mathrm{\scriptscriptstyle
T%
}}},$$
$$\displaystyle\mathbf{D}_{1}(x)=$$
$$\displaystyle\ d_{0}(x)\tilde{\mbox{\boldmath$\mu$}}_{1}+d_{1}(x)\left(\mbox{%
\boldmath$\mu$}_{0}^{*}\mbox{\boldmath$\mu$}_{1}^{*{\mathrm{\scriptscriptstyle
T%
}}}+\mbox{\boldmath$\mu$}_{1}^{*}\mbox{\boldmath$\mu$}_{0}^{*{\mathrm{%
\scriptscriptstyle T}}}\right),$$
$$\displaystyle\mathbf{F}(x,h,J)=$$
$$\displaystyle\ w_{30}(x,h,J)\left\{\sigma^{2}(x)+\beta_{0}^{2}\right\}\tilde{%
\mbox{\boldmath$\nu$}}_{0}+$$
$$\displaystyle\ w_{31}(x,h,J)\left\{\sigma^{2}(x)\left(\nu_{0}\tilde{\mbox{%
\boldmath$\nu$}}_{0}+2\mbox{\boldmath$\nu$}^{*}_{0}\mbox{\boldmath$\nu$}_{0}^{%
*{\mathrm{\scriptscriptstyle T}}}\right)+\beta^{2}_{0}\left(3\nu_{0}\tilde{%
\mbox{\boldmath$\nu$}}_{0}+4\mbox{\boldmath$\nu$}_{0}^{*}\mbox{\boldmath$\nu$}%
_{0}^{*{\mathrm{\scriptscriptstyle T}}}\right)\right\}+$$
$$\displaystyle\ w_{32}(x,h,J)\nu_{0}\left\{\sigma^{2}(x)\mbox{\boldmath$\nu$}^{%
*}_{0}\mbox{\boldmath$\nu$}^{*{\mathrm{\scriptscriptstyle T}}}_{0}+\beta_{0}^{%
2}\nu_{0}\left(\tilde{\mbox{\boldmath$\nu$}}_{0}+3\mbox{\boldmath$\nu$}^{*}_{0%
}\mbox{\boldmath$\nu$}^{*{\mathrm{\scriptscriptstyle T}}}_{0}\right)\right\}+$$
$$\displaystyle\ w_{33}(x,h,J)\beta_{0}^{2}\nu_{0}^{2}\mbox{\boldmath$\nu$}_{0}^%
{*}\mbox{\boldmath$\nu$}^{*{\mathrm{\scriptscriptstyle T}}}_{0},$$
$$\displaystyle\mathbf{G}(x,h,J)=$$
$$\displaystyle\ w_{30}(x,h,J)\left[\sigma^{2}(x)f^{\prime}_{\hbox{\tiny$X$}}(x)%
+2\sigma(x)\sigma^{\prime}(x)f_{\hbox{\tiny$X$}}(x)+\right.$$
$$\displaystyle\left.\beta_{0}\left\{\beta_{0}f^{\prime}_{\hbox{\tiny$X$}}(x)+2%
\beta_{1}f_{\hbox{\tiny$X$}}(x)\right\}\right]\tilde{\mbox{\boldmath$\nu$}}_{1}+$$
$$\displaystyle\ w_{31}(x,h,J)\left(\sigma^{2}(x)f^{\prime}_{\hbox{\tiny$X$}}(x)%
\left(\tilde{\mbox{\boldmath$\nu$}}_{1}+\mbox{\boldmath$\nu$}_{0}^{*}\mbox{%
\boldmath$\nu$}_{1}^{*{\mathrm{\scriptscriptstyle T}}}+\mbox{\boldmath$\nu$}_{%
1}^{*}\mbox{\boldmath$\nu$}_{0}^{*{\mathrm{\scriptscriptstyle T}}}\right)+2%
\sigma(x)\sigma^{\prime}(x)\times\right.$$
$$\displaystyle\ f_{\hbox{\tiny$X$}}(x)\left(\mbox{\boldmath$\nu$}_{0}^{*}\mbox{%
\boldmath$\nu$}_{1}^{*{\mathrm{\scriptscriptstyle T}}}+\mbox{\boldmath$\nu$}_{%
1}^{*}\mbox{\boldmath$\nu$}_{0}^{*{\mathrm{\scriptscriptstyle T}}}\right)+%
\beta_{0}\left[\left\{2\beta_{0}f^{\prime}_{\hbox{\tiny$X$}}(x)+\beta_{1}f_{%
\hbox{\tiny$X$}}(x)\right\}\nu_{0}\tilde{\mbox{\boldmath$\nu$}}_{1}+\right.$$
$$\displaystyle\left.\left.4\left\{\beta_{0}f^{\prime}_{\hbox{\tiny$X$}}(x)+%
\beta_{1}f_{\hbox{\tiny$X$}}(x)\right\}\left(\mbox{\boldmath$\nu$}_{0}^{*}%
\mbox{\boldmath$\nu$}_{1}^{*{\mathrm{\scriptscriptstyle T}}}+\mbox{\boldmath$%
\nu$}_{1}^{*}\mbox{\boldmath$\nu$}_{0}^{*{\mathrm{\scriptscriptstyle T}}}%
\right)\right]\right)+$$
$$\displaystyle\ w_{32}(x,h,J)\nu_{0}\left(\sigma^{2}(x)f^{\prime}_{\hbox{\tiny$%
X$}}(x)\left(\mbox{\boldmath$\nu$}_{0}^{*}\mbox{\boldmath$\nu$}_{1}^{*{\mathrm%
{\scriptscriptstyle T}}}+\mbox{\boldmath$\nu$}_{1}^{*}\mbox{\boldmath$\nu$}_{0%
}^{*{\mathrm{\scriptscriptstyle T}}}\right)+\right.$$
$$\displaystyle\left.\beta_{0}\left[\left\{3\beta_{0}f^{\prime}_{\hbox{\tiny$X$}%
}(x)+\beta_{1}f_{\hbox{\tiny$X$}}(x)\right\}\left(\mbox{\boldmath$\nu$}_{0}^{*%
}\mbox{\boldmath$\nu$}_{1}^{*{\mathrm{\scriptscriptstyle T}}}+\mbox{\boldmath$%
\nu$}_{1}^{*}\mbox{\boldmath$\nu$}_{0}^{*{\mathrm{\scriptscriptstyle T}}}%
\right)+\beta_{0}f^{\prime}_{\hbox{\tiny$X$}}(x)\nu_{0}\tilde{\mbox{\boldmath$%
\nu$}}_{1}\right]\right)+$$
$$\displaystyle\ w_{33}(x,h,J)\nu_{0}^{2}\beta_{0}^{2}f^{\prime}_{\hbox{\tiny$X$%
}}(x)\left(\mbox{\boldmath$\nu$}_{0}^{*}\mbox{\boldmath$\nu$}_{1}^{*{\mathrm{%
\scriptscriptstyle T}}}+\mbox{\boldmath$\nu$}_{1}^{*}\mbox{\boldmath$\nu$}_{0}%
^{*{\mathrm{\scriptscriptstyle T}}}\right).$$
One major distinction from the counterpart result when individual-level data are available lies in the fact that the order of the dominating term inside the square brackets in (B.23) is determined by the order of $\mathbf{F}(x,h,J)=O\{1/(Jh^{c^{*}})\}$. Consequently, the asymptotic variance of $\hat{\mbox{\boldmath$\beta$}}_{2}$ is inflated compared to that of $\hat{\mbox{\boldmath$\beta$}}_{0}$, with more inflation when $c^{*}$ is larger.
Appendix C: Proof of Theorem 1-(iii)
C.1 Bias of the marginal-integration estimator $\hat{m}_{3}(x)$
Consider $\hat{\bm{\beta}}_{3}(x)=\mathbf{S}_{3}^{-1}(x)\mathbf{T}_{3}(x)$. For $\ell=0,1,\ldots,p$, the $(\ell+1)$-th element of $\mathbf{T}_{3}(x)$ is
$$\displaystyle T_{3,\ell}(x)=$$
$$\displaystyle\ \sum_{j=1}^{J}\sum_{k=1}^{c_{j}}\{c_{j}Z_{j}-(c_{j}-1)\widehat{%
\mu}\}(X_{jk}-x)^{\ell}K_{h}(X_{jk}-x)$$
$$\displaystyle=$$
$$\displaystyle\ \sum_{j=1}^{J}\sum_{k=1}^{c_{j}}\left\{\sum_{g=1}^{c_{j}}Y_{jg}%
-(c_{j}-1)\widehat{\mu}\right\}(X_{jk}-x)^{\ell}K_{h}(X_{jk}-x)$$
$$\displaystyle=$$
$$\displaystyle\ \sum_{j=1}^{J}\sum_{k=1}^{c_{j}}Y_{jk}(X_{jk}-x)^{\ell}K_{h}(X_%
{jk}-x)$$
$$\displaystyle\ +\sum_{j=1}^{J}\sum_{k=1}^{c_{j}}\left\{\sum_{g=1,g\neq k}^{c_{%
j}}Y_{jg}-(c_{j}-1)\mu\right\}(X_{jk}-x)^{\ell}K_{h}(X_{jk}-x)$$
$$\displaystyle\ +(\mu-\widehat{\mu})\sum_{j=1}^{J}\sum_{k=1}^{c_{j}}(c_{j}-1)(X%
_{jk}-x)^{\ell}K_{h}(X_{jk}-x)$$
$$\displaystyle\equiv$$
$$\displaystyle\ T_{31,\ell}(x)+T_{32,\ell}(x)+(\mu-\widehat{\mu})T_{33,l}(x).$$
Further, denote $\mathbf{T}_{31}(x)=(T_{31,0}(x),\dots,T_{31,p}(x))^{\mathrm{\scriptscriptstyle
T}}$, $\mathbf{T}_{32}(x)=(T_{32,0}(x),\dots,T_{32,p}(x))^{\mathrm{\scriptscriptstyle
T}}$, and $\mathbf{T}_{33}(x)=(T_{33,0}(x),\dots,T_{33,p}(x))^{\mathrm{\scriptscriptstyle
T}}$. Then
$$\hat{\bm{\beta}}_{3}(x)=\mathbf{S}_{3}^{-1}(x)\mathbf{T}_{31}(x)+\mathbf{S}_{3%
}^{-1}(x)\mathbf{T}_{32}(x)+(\mu-\widehat{\mu})\mathbf{S}_{3}^{-1}(x)\mathbf{T%
}_{33}(x).$$
Note that $\hat{\bm{\beta}}_{0}(x)\equiv\mathbf{S}_{3}^{-1}(x)\mathbf{T}_{31}(x)$ is the local polynomial estimator based on individual-level data, hence,
$$\displaystyle\mbox{Bias}\left\{\left.\hat{\bm{\beta}}_{3}(x)\right|\mathbb{X}\right\}$$
$$\displaystyle=$$
$$\displaystyle\mbox{Bias}\left\{\left.\hat{\bm{\beta}}_{0}(x)\right|\mathbb{X}%
\right\}+E\left\{\left.\mathbf{S}_{3}^{-1}(x)\mathbf{T}_{32}(x)\right|\mathbb{%
X}\right\}+E\left\{\left.(\mu-\widehat{\mu})\mathbf{S}_{3}^{-1}(x)\mathbf{T}_{%
33}(x)\right|\mathbb{X}\right\},$$
in which, by Fan and Gijbels (1996),
$$\displaystyle\mbox{Bias}\left\{\hat{\bm{\beta}}_{0}(x)|\mathbb{X}\right\}=$$
$$\displaystyle\ h^{p+1}\mathbf{H}^{-1}\left\{\beta_{p+1}\tilde{\bm{\mu}}_{0}^{-%
1}\bm{\mu}_{p+1}^{*}+h\frac{\beta_{p+2}f_{\hbox{\tiny$X$}}(x)+\beta_{p+1}f^{%
\prime}_{\hbox{\tiny$X$}}(x)}{f_{\hbox{\tiny$X$}}(x)}\tilde{\bm{\mu}}_{0}^{-1}%
\bm{\mu}_{p+2}^{*}\right.$$
$$\displaystyle\ \left.-h\beta_{p+1}\frac{f^{\prime}_{\hbox{\tiny$X$}}(x)}{f_{%
\hbox{\tiny$X$}}(x)}\tilde{\bm{\mu}}_{0}^{-1}\tilde{\bm{\mu}}_{1}\tilde{\bm{%
\mu}}_{0}^{-1}\bm{\mu}_{p+1}^{*}+O_{\hbox{\tiny$P$}}(h^{2}+1/\sqrt{Nh})\right\}.$$
Also from Fan and Gijbels (1996),
$$\mathbf{S}_{3}^{-1}(x)=N^{-1}\mathbf{H}^{-1}\left\{f_{X}^{-1}(x)\tilde{\bm{\mu%
}}_{0}^{-1}-h\frac{f_{X}^{\prime}(x)}{f_{X}^{2}(x)}\tilde{\bm{\mu}}_{0}^{-1}%
\tilde{\bm{\mu}}_{1}\tilde{\bm{\mu}}_{0}^{-1}+O_{\hbox{\tiny$P$}}(h^{2})\right%
\}\mathbf{H}^{-1},$$
when $h\rightarrow 0$ and $Nh^{3}\rightarrow\infty$ as $N\rightarrow\infty$, provided that $f_{\hbox{\tiny$X$}}(x)>0$, $f_{\hbox{\tiny$X$}}(\cdot)$ and $m^{(p+1)}(\cdot)$ (or $f^{\prime}_{\hbox{\tiny$X$}}(\cdot)$ and $m^{(p+2)}$(x) if $p-\ell$ is even) are continuous in a neighborhood of $x$.
Now we derive $E\{\mathbf{S}_{3}^{-1}(x)\mathbf{T}_{32}(x)|\mathbb{X}\}=\mathbf{S}_{3}^{-1}(x%
)E\{\mathbf{T}_{32}(x)|\mathbb{X}\}$. Note that
$$\displaystyle E\{T_{32,\ell}(x)|\mathbb{X}\}=\sum_{j=1}^{J}\sum_{k=1}^{c_{j}}%
\left\{\sum_{g=1,g\neq k}^{c_{j}}m(X_{jg})-(c_{j}-1)\mu\right\}(X_{jk}-x)^{%
\ell}K_{h}(X_{jk}-x).$$
It is easy to see that the expectation of the right-hand side of the last equation is zero. Furthermore,
$$\displaystyle\ \mbox{Var}\left[\sum_{j=1}^{J}\sum_{k=1}^{c_{j}}\left\{\sum_{g=%
1,g\neq k}^{c_{j}}m(X_{jg})-(c_{j}-1)\mu\right\}(X_{jk}-x)^{\ell}K_{h}(X_{jk}-%
x)\right]$$
$$\displaystyle=$$
$$\displaystyle\ \sum_{j=1}^{J}E\left[\sum_{k=1}^{c_{j}}\left\{\sum_{g=1,g\neq k%
}^{c_{j}}m(X_{jg})-(c_{j}-1)\mu\right\}(X_{jk}-x)^{\ell}K_{h}(X_{jk}-x)\right]%
^{2}$$
$$\displaystyle=$$
$$\displaystyle\ \sum_{j=1}^{J}\sum_{k=1}^{c_{j}}E\left[\left\{\sum_{g=1,g\neq k%
}^{c_{j}}m(X_{jg})-(c_{j}-1)\mu\right\}^{2}(X_{jk}-x)^{2\ell}K_{h}^{2}(X_{jk}-%
x)\right]$$
$$\displaystyle\ +\sum_{j=1}^{J}\sum_{k_{1}\neq k_{2}}^{1,\dots,c_{j}}E\left[%
\left\{\sum_{g=1,g\neq k_{1},k_{2}}^{c_{j}}m(X_{jg})+m(X_{jk_{1}})-(c_{j}-1)%
\mu\right\}(X_{jk_{1}}-x)^{\ell}K_{h}(X_{jk_{1}}-x)\right.$$
$$\displaystyle\ \quad\quad\quad\quad\times\left.\left\{\sum_{g=1,g\neq k_{1},k_%
{2}}^{c_{j}}m(X_{jg})+m(X_{jk_{2}})-(c_{j}-1)\mu\right\}(X_{jk_{2}}-x)^{\ell}K%
_{h}(X_{jk_{2}}-x)\right],$$
which is
$$\displaystyle\ \sum_{j=1}^{J}\sum_{k=1}^{c_{j}}E\left[\left\{\sum_{g=1,g\neq k%
}^{c_{j}}m(X_{jg})-(c_{j}-1)\mu\right\}^{2}(X_{jk}-x)^{2\ell}K_{h}^{2}(X_{jk}-%
x)\right]$$
$$\displaystyle\ +\sum_{j=1}^{J}\sum_{k_{1}\neq k_{2}}^{1,\dots,c_{j}}(c_{j}-2)%
\tilde{\sigma}^{2}E\left[(X_{jk_{1}}-x)^{\ell}K_{h}(X_{jk_{1}}-x_{0})(X_{jk_{2%
}}-x)^{\ell}K_{h}(X_{jk_{2}}-x)\right]$$
$$\displaystyle\ +\sum_{j=1}^{J}\sum_{k_{1}\neq k_{2}}^{1,\dots,c_{j}}E[\{m(X_{%
jk_{1}})-\mu\}\{m(X_{jk_{2}})-\mu\}(X_{jk_{1}}-x)^{\ell}K_{h}(X_{jk_{1}}-x)(X_%
{jk_{2}}-x)^{\ell}K_{h}(X_{jk_{2}}-x)]$$
$$\displaystyle=$$
$$\displaystyle\ \sum_{j=1}^{J}c_{j}(c_{j}-1)h^{2\ell-1}\tilde{\sigma}^{2}f_{%
\hbox{\tiny$X$}}(x)\nu_{2\ell}\{1+o(1)\}$$
$$\displaystyle\ +\sum_{j=1}^{J}c_{j}(c_{j}-1)(c_{j}-2)O(h^{2\ell})+\sum_{j=1}^{%
J}c_{j}(c_{j}-1)O(h^{2\ell}),$$
in which $\tilde{\sigma}^{2}=E[\{m(X)-\mu\}^{2}]=\mbox{Var}\{m(X)\}$.
It follows that
$$\displaystyle E[T_{32,\ell}(x)|\mathbb{X}]=$$
$$\displaystyle\ 0+O_{\hbox{\tiny$P$}}\left(\sqrt{\sum_{j=1}^{J}c_{j}(c_{j}-1)h^%
{2\ell-1}\tilde{\sigma}^{2}f_{\hbox{\tiny$X$}}(x)\nu_{2\ell}}\right)$$
$$\displaystyle=$$
$$\displaystyle\ Nh^{\ell}\tilde{\sigma}\sqrt{\sum_{j=1}^{J}c_{j}(c_{j}-1)/\sum_%
{j=1}^{J}c_{j}}\times O_{\hbox{\tiny$P$}}\left(1/\sqrt{Nh}\right).$$
Thus
$$\displaystyle E\left\{\mathbf{S}_{3}^{-1}(x)\mathbf{T}_{32}(x)|\mathbb{X}\right\}$$
$$\displaystyle=$$
$$\displaystyle\ \mathbf{H}^{-1}\left\{f_{\hbox{\tiny$X$}}^{-1}(x)\tilde{\bm{\mu%
}}_{0}^{-1}-h\frac{f_{\hbox{\tiny$X$}}^{\prime}(x)}{f_{\hbox{\tiny$X$}}^{2}(x)%
}\tilde{\bm{\mu}}_{0}^{-1}\tilde{\bm{\mu}}_{1}\tilde{\bm{\mu}}_{0}^{-1}+O_{%
\hbox{\tiny$P$}}(h^{2})\right\}\bm{1}\times O_{\hbox{\tiny$P$}}(1/\sqrt{Nh})$$
$$\displaystyle=$$
$$\displaystyle\ \mathbf{H}^{-1}\sqrt{\frac{\sum_{j=1}^{J}c_{j}(c_{j}-1)}{\sum_{%
j=1}^{J}c_{j}}}\times O_{\hbox{\tiny$P$}}(1/\sqrt{Nh}).$$
To calculate
$E\{(\mu-\widehat{\mu})\mathbf{S}_{3}^{-1}(x)\mathbf{T}_{33}(x)|\mathbb{X}\}$, where
$$T_{33,l}(x)=\sum_{j=1}^{J}\sum_{k=1}^{c_{j}}(c_{j}-1)(X_{jk}-x)^{\ell}K_{h}(X_%
{jk}-x),$$
we have
$$\displaystyle E\left\{(\mu-\widehat{\mu})\mathbf{S}_{3}^{-1}(x)\mathbf{T}_{33}%
(x)|\mathbb{X}\right\}=E\left(\mu-\widehat{\mu}|\mathbb{X}\right)\mathbf{S}_{3%
}^{-1}(x)\mathbf{T}_{33}(x).$$
It is easy to see that
$$E\left(\mu-\widehat{\mu}|\mathbb{X}\right)=\mu-N^{-1}\sum_{j=1}^{J}\sum_{k=1}^%
{c_{j}}m(X_{ij})=O_{\hbox{\tiny$P$}}(1/\sqrt{N}),$$
and
$$\mathbf{T}_{33}(x)=N\mathbf{H}\{O(1)+O_{\hbox{\tiny$P$}}(1/\sqrt{Nh})\}.$$
Thus
$$E\left\{(\mu-\widehat{\mu})\mathbf{S}_{3}^{-1}(x)\mathbf{T}_{33}(x)\right\}=%
\mathbf{H}^{-1}O_{\hbox{\tiny$P$}}(1/\sqrt{N})=o_{\hbox{\tiny$P$}}\left[E\left%
\{\mathbf{S}_{3}^{-1}(x)\mathbf{T}_{32}(x)|\mathbb{X}\right\}\right]$$
if $h\rightarrow 0$.
Thus, we have the bias term given by
$$\displaystyle\hat{\bm{\beta}}_{3}(x)-\mbox{\boldmath$\beta$}=$$
$$\displaystyle\ h^{p+1}\mathbf{H}^{-1}\left\{\beta_{p+1}\tilde{\bm{\mu}}_{0}^{-%
1}\bm{\mu}_{p+1}^{*}+h\frac{\beta_{p+2}f_{\hbox{\tiny$X$}}(x)+\beta_{p+1}f^{%
\prime}_{X}(x)}{f_{\hbox{\tiny$X$}}(x)}\tilde{\bm{\mu}}_{0}^{-1}\bm{\mu}_{p+2}%
^{*}\right.$$
$$\displaystyle\ \quad\quad\quad\quad\left.-h\beta_{p+1}\frac{f^{\prime}_{\hbox{%
\tiny$X$}}(x)}{f_{\hbox{\tiny$X$}}(x)}\tilde{\bm{\mu}}_{0}^{-1}\tilde{\bm{\mu}%
}_{1}\tilde{\bm{\mu}}_{0}^{-1}\bm{\mu}_{p+1}^{*}+O_{\hbox{\tiny$P$}}(h^{2}+1/%
\sqrt{Nh})\right\}$$
$$\displaystyle\ +\mathbf{H}^{-1}\sqrt{\frac{\sum_{j=1}^{J}c_{j}(c_{j}-1)}{\sum_%
{j=1}^{J}c_{j}}}\bm{1}\times O_{\hbox{\tiny$P$}}(1/\sqrt{Nh})$$
$$\displaystyle=$$
$$\displaystyle\ \hat{\bm{\beta}}_{0}(x)-\mbox{\boldmath$\beta$}+\mathbf{H}^{-1}%
\sqrt{\frac{\sum_{j=1}^{J}c_{j}(c_{j}-1)}{\sum_{j=1}^{J}c_{j}}}\bm{1}\times O_%
{\hbox{\tiny$P$}}(1/\sqrt{Nh}),$$
where $\bm{1}$ is a vector of size $1$ of an appropriate length.
The effect of pooling in terms of bias is reflected by the second term in the last line, which disappears if $c_{j}=1$ for all $j$’s (i.e., no pooling).
When $p=0$, the marginal-integration estimator returns a local constant estimator with dominating bias given by
$$\displaystyle\mbox{Bias}\left\{\hat{m}_{3}(x)|\mathbb{X}\right\}$$
$$\displaystyle=$$
$$\displaystyle\ \left\{\frac{1}{2}m^{{}^{\prime\prime}}(x)+\frac{f_{\hbox{\tiny%
$X$}}^{\prime}(x)}{f_{X}(x)}m^{\prime}(x)\right\}\mu_{2}h^{2}+\sqrt{\frac{\sum%
_{j=1}^{J}c_{j}(c_{j}-1)}{\sum_{j=1}^{J}c_{j}}}\times O_{\hbox{\tiny$P$}}(1/%
\sqrt{Nh})$$
$$\displaystyle\ +o_{\hbox{\tiny$P$}}(h^{2}+1/\sqrt{Nh}).$$
Setting $p=1$, we have the local linear marginal-integration estimator with dominating bias given by
$$\displaystyle\mbox{Bias}\left\{\hat{m}_{3}(x)|\mathbb{X}\right\}=$$
$$\displaystyle\ \frac{1}{2}m^{\prime\prime}(x)\mu_{2}h^{2}+\sqrt{\frac{\sum_{j=%
1}^{J}c_{j}(c_{j}-1)}{\sum_{j=1}^{J}c_{j}}}\times O_{\hbox{\tiny$P$}}(1/\sqrt{%
Nh})$$
$$\displaystyle+o_{\hbox{\tiny$P$}}(h^{2}+1/\sqrt{Nh}).$$
C.2 Variance of $\hat{m}_{3}(x)$
Again, we focus on $\hat{\bm{\beta}}_{3}(x)=\mathbf{S}_{3}^{-1}(x)\mathbf{T}_{3}(x)$, where
$$\displaystyle T_{3,\ell}(x)=$$
$$\displaystyle\ \sum_{j=1}^{J}\sum_{k=1}^{c_{j}}\{c_{j}Z_{j}-(c_{j}-1)\widehat{%
\mu}\}(X_{jk}-x)^{\ell}K_{h}(X_{jk}-x)$$
$$\displaystyle=$$
$$\displaystyle\ \sum_{j=1}^{J}c_{j}Z_{j}\sum_{k=1}^{c_{j}}(X_{jk}-x)^{\ell}K_{h%
}(X_{jk}-x)$$
$$\displaystyle\ -\widehat{\mu}\sum_{j=1}^{J}\sum_{k=1}^{c_{j}}(c_{j}-1)(X_{jk}-%
x)^{\ell}K_{h}(X_{jk}-x)$$
$$\displaystyle\equiv$$
$$\displaystyle\ T_{31,\ell}^{*}(x)+T_{32,\ell}^{*}(x).$$
First, we have
$$\displaystyle\mbox{Cov}\left\{T_{31,\ell_{1}}^{*}(x),T_{31,\ell_{2}}^{*}(x)|%
\mathbb{X}\right\}=$$
$$\displaystyle\ \sum_{j=1}^{J}\sum_{k=1}^{c_{j}}\sigma^{2}(X_{jk})\sum_{k=1}^{c%
_{j}}(X_{jk}-x)^{\ell_{1}}K_{h}(X_{jk}-x)$$
$$\displaystyle\ \times\sum_{k=1}^{c_{j}}(X_{jk}-x)^{\ell_{2}}K_{h}(X_{jk}-x).$$
Straightforward calculation presents, by (A.6) and (A.10),
$$\displaystyle\ E\left\{\mbox{Cov}[T_{31,\ell_{1}}^{*}(x),T_{31,\ell_{2}}^{*}(x%
)|\mathbb{X}]\right\}$$
$$\displaystyle=$$
$$\displaystyle\ \sum_{j=1}^{J}E\left\{\sum_{k=1}^{c_{j}}\sigma^{2}(X_{jk})\sum_%
{k=1}^{c_{j}}(X_{jk}-x)^{\ell_{1}+\ell_{2}}K_{h}^{2}(X_{jk}-x)\right\}+\mbox{%
Lower-order terms}$$
$$\displaystyle=$$
$$\displaystyle\ \sum_{j=1}^{J}\left[c_{j}\kappa^{*}_{2,\ell_{1}+\ell_{2}}(%
\sigma^{2},h,x)+c_{j}(c_{j}-1)E\left\{\sigma^{2}(X)\right\}\kappa_{2,\ell_{1}+%
\ell_{2}}(h,x)\right]+\mbox{Lower-order terms}$$
$$\displaystyle=$$
$$\displaystyle\ Nh^{\ell_{1}+\ell_{2}-1}f_{\hbox{\tiny$X$}}(x)\nu_{\ell_{1}+%
\ell_{2}}\left\{\sigma^{2}(x)+\bar{\sigma}^{2}N^{-1}\sum_{j=1}^{J}\sum_{k=1}^{%
c_{j}}(c_{j}-1)\right\}+O\left(h^{\ell_{1}+\ell_{2}}\right),$$
where $\bar{\sigma}^{2}=E\{\sigma^{2}(X)\}$.
Moreover,
$$\displaystyle\ \mbox{Var}\left[\mbox{Cov}\left\{T_{31,\ell_{1}}^{*}(x),T_{31,%
\ell_{2}}^{*}(x)|\mathbb{X}\right\}\right]$$
$$\displaystyle=$$
$$\displaystyle\ \sum_{j=1}^{J}\mbox{Var}\left[\sum_{k=1}^{c_{j}}\sigma^{2}(X_{%
jk})\sum_{k=1}^{c_{j}}(X_{jk}-x)^{\ell_{1}}K_{h}(X_{jk}-x)\sum_{k=1}^{c_{j}}(X%
_{jk}-x)^{\ell_{2}}K_{h}(X_{jk}-x)\right]$$
$$\displaystyle=$$
$$\displaystyle\ O(Nh^{2\ell_{1}+2\ell_{2}-3}).$$
Thus
$$\mbox{Cov}\left\{\mathbf{T}_{31}^{*}(x)|\mathbb{X}\right\}=h^{-1}N\mathbf{H}%
\left\{\sigma^{2}(x)+\bar{\sigma}^{2}N^{-1}\sum_{j=1}^{J}\sum_{k=1}^{c_{j}}(c_%
{j}-1)\right\}f_{\hbox{\tiny$X$}}(x)\tilde{\mbox{\boldmath$\nu$}}_{0}\{1+o_{p}%
(1)\}\mathbf{H}.$$
Then
$$\displaystyle\mbox{Cov}[\mathbf{S}_{3}^{-1}(x)\mathbf{T}_{31}^{*}(x)|\mathbb{X}]$$
$$\displaystyle=$$
$$\displaystyle\ N^{-1}\mathbf{H}^{-1}\left\{f_{\hbox{\tiny$X$}}^{-1}(x)\tilde{%
\mbox{\boldmath$\mu$}}_{0}^{-1}+O_{\hbox{\tiny$P$}}(h)\right\}\mathbf{H}^{-1}$$
$$\displaystyle\ \times h^{-1}N\mathbf{H}\left\{\sigma^{2}(x)+\bar{\sigma}^{2}N^%
{-1}\sum_{j=1}^{J}\sum_{k=1}^{c_{j}}(c_{j}-1)\right\}f_{\hbox{\tiny$X$}}(x)%
\tilde{\mbox{\boldmath$\nu$}}_{0}\{1+o_{\hbox{\tiny$P$}}(1)\}\mathbf{H}$$
$$\displaystyle\ \times N^{-1}\mathbf{H}^{-1}\left\{f_{\hbox{\tiny$X$}}^{-1}(x)%
\tilde{\mbox{\boldmath$\mu$}}_{0}^{-1}+O_{\hbox{\tiny$P$}}(h)\right\}\mathbf{H%
}^{-1}$$
$$\displaystyle=$$
$$\displaystyle\ \frac{\sigma^{2}(x)+\bar{\sigma}^{2}N^{-1}\sum_{j=1}^{J}\sum_{k%
=1}^{c_{j}}(c_{j}-1)}{hf_{\hbox{\tiny$X$}}(x)}N^{-1}\mathbf{H}^{-1}\tilde{%
\mbox{\boldmath$\mu$}}_{0}^{-1}\tilde{\mbox{\boldmath$\nu$}}_{0}\tilde{\mbox{%
\boldmath$\mu$}}_{0}^{-1}\mathbf{H}^{-1}\{1+o_{\hbox{\tiny$P$}}(1)\},$$
in which $\sigma^{2}(x)\{hf_{X}(x)\}^{-1}N^{-1}\mathbf{H}^{-1}\tilde{\mbox{\boldmath$\mu%
$}}_{0}^{-1}\tilde{\mbox{\boldmath$\nu$}}_{0}\tilde{\mbox{\boldmath$\mu$}}_{0}%
^{-1}\mathbf{H}^{-1}\{1+o_{p}(1)\}$ is the same as that in Fan and Gijbels (1996) and $\bar{\sigma}^{2}N^{-1}\sum_{j=1}^{J}\sum_{k=1}^{c_{j}}(c_{j}-1)$ accounts for the effect of pooling and disappears when $c_{j}=1$ for all $j$’s.
Now we calculate $\mbox{Cov}\{\mathbf{S}_{3}^{-1}(x)\mathbf{T}_{32}^{*}(x)|\mathbb{X}\}$.
$$\displaystyle\mbox{Cov}\left\{T_{32,\ell_{1}}^{*}(x),T_{32,\ell_{2}}^{*}(x)|%
\mathbb{X}\right\}=$$
$$\displaystyle\ \sum_{j=1}^{J}\sum_{k=1}^{c_{j}}(c_{j}-1)(X_{jk}-x)^{\ell_{1}}K%
_{h}(X_{jk}-x)\mbox{Var}(\widehat{\mu}|\mathbb{X})$$
$$\displaystyle\ \times\sum_{j=1}^{J}\sum_{k=1}^{c_{j}}(c_{j}-1)(X_{jk}-x)^{\ell%
_{2}}K_{h}(X_{jk}-x),$$
in which
$$\displaystyle\mbox{Var}(\widehat{\mu}|\mathbb{X})=\frac{\sum_{j=1}^{J}\sum_{k=%
1}^{c_{j}}\sigma^{2}(X_{ij})}{N^{2}}=N^{-1}\{\bar{\sigma}^{2}+O_{\hbox{\tiny$P%
$}}(1/\sqrt{N})\}$$
and
$$\displaystyle\ \sum_{j=1}^{J}\sum_{k=1}^{c_{j}}(c_{j}-1)(X_{jk}-x)^{\ell}K_{h}%
(X_{jk}-x)$$
$$\displaystyle=$$
$$\displaystyle\ f_{X}(x)h^{\ell}\mu_{\ell}\sum_{j=1}^{J}\sum_{k=1}^{c_{j}}(c_{j%
}-1)\{1+o(1)\}$$
$$\displaystyle\ +h^{\ell}O_{\hbox{\tiny$P$}}\left(\sqrt{h^{-1}\sum_{j=1}^{J}%
\sum_{k=1}^{c_{j}}(c_{j}-1)^{2}}\right).$$
Hence
$$\displaystyle\mbox{Cov}\left\{T_{32,\ell_{1}}^{*}(x),T_{32,\ell_{2}}^{*}(x)|%
\mathbb{X}\right\}$$
$$\displaystyle=$$
$$\displaystyle\ Nh^{\ell_{1}}\mu_{\ell_{1}}f_{\hbox{\tiny$X$}}(x)\frac{\sum_{j=%
1}^{J}\sum_{k=1}^{c_{j}}(c_{j}-1)}{N}\{1+o(1)+O_{\hbox{\tiny$P$}}(1/\sqrt{Nh})\}$$
$$\displaystyle\ \times Nh^{\ell_{2}}\mu_{\ell_{2}}f_{\hbox{\tiny$X$}}(x)\frac{%
\sum_{j=1}^{J}\sum_{k=1}^{c_{j}}(c_{j}-1)}{N}\{1+o(1)+O_{\hbox{\tiny$P$}}(1/%
\sqrt{Nh})\}$$
$$\displaystyle\ \times N^{-1}\{\bar{\sigma}^{2}+O_{\hbox{\tiny$P$}}(1/\sqrt{N})\};$$
i.e.,
$$\displaystyle\mbox{Cov}[\mathbf{T}_{33}^{*}(x)|\mathbb{X}]=$$
$$\displaystyle\ \bar{\sigma}^{2}f_{X}^{2}(x)\left(\frac{\sum_{j=1}^{J}\sum_{k=1%
}^{c_{j}}(c_{j}-1)}{N}\right)^{2}N\mathbf{H}\tilde{\bm{\mu}}_{0}\tilde{\bm{\mu%
}}_{0}^{\top}\mathbf{H}\{1+o_{\hbox{\tiny$P$}}(1)\}$$
$$\displaystyle=$$
$$\displaystyle\ \mbox{Cov}[\mathbf{T}_{31}^{*}(x)|\mathbb{X}]\times o_{\hbox{%
\tiny$P$}}(1),$$
which becomes negligible.
The last term is $\mbox{Cov}\{\mathbf{S}_{3}^{-1}(x)\mathbf{T}_{31}^{*}(x),\mathbf{S}_{3}^{-1}(x%
)\mathbf{T}_{32}^{*}(x)|\mathbb{X}\}$, in which $\mbox{Cov}\{T_{31,\ell_{1}}^{*}(x),T_{32,\ell_{2}}^{*}(x)|\mathbb{X}\}$ equals to
$$\displaystyle\ \mbox{Cov}\left\{\sum_{j=1}^{J}c_{j}Z_{j}\sum_{k=1}^{c_{j}}(X_{%
jk}-x)^{\ell_{1}}K_{h}(X_{jk}-x),\widehat{\mu}\sum_{j=1}^{J}\sum_{k=1}^{c_{j}}%
(c_{j}-1)(X_{jk}-x)^{\ell_{2}}K_{h}(X_{jk}-x)|\mathbb{X}\right\}$$
$$\displaystyle=$$
$$\displaystyle\ \sum_{j=1}^{J}\mbox{Cov}\left(c_{j}Z_{j},\widehat{\mu}|\mathbb{%
X}\right)\sum_{k=1}^{c_{j}}(X_{jk}-x)^{\ell_{1}}K_{h}(X_{jk}-x)\times\sum_{j=1%
}^{J}\sum_{k=1}^{c_{j}}(c_{j}-1)(X_{jk}-x)^{\ell_{2}}K_{h}(X_{jk}-x)$$
$$\displaystyle=$$
$$\displaystyle\ N^{-1}\sum_{j=1}^{J}\left\{\sum_{k=1}^{c_{j}}\sigma^{2}(X_{jk})%
\sum_{k=1}^{c_{j}}(X_{jk}-x)^{\ell_{1}}K_{h}(X_{jk}-x)\right\}\times\sum_{j=1}%
^{J}\sum_{k=1}^{c_{j}}(c_{j}-1)(X_{jk}-x)^{\ell_{2}}K_{h}(X_{jk}-x),$$
in which
$$\displaystyle\sum_{j=1}^{J}\left\{\sum_{k=1}^{c_{j}}\sigma^{2}(X_{jk})\sum_{k=%
1}^{c_{j}}(X_{jk}-x)^{\ell_{1}}K_{h}(X_{jk}-x)\right\}=$$
$$\displaystyle\ O(Nh^{\ell_{1}})+O_{\hbox{\tiny$P$}}(\sqrt{Nh^{2\ell_{1}-1}})$$
$$\displaystyle=$$
$$\displaystyle\ Nh^{\ell_{1}}\{O(1)+O_{p}(1/\sqrt{Nh})\},$$
$$\displaystyle\sum_{j=1}^{J}\sum_{k=1}^{c_{j}}(c_{j}-1)(X_{jk}-x)^{\ell}K_{h}(X%
_{jk}-x)=$$
$$\displaystyle\ O(Nh^{\ell_{2}})+O_{\hbox{\tiny$P$}}(\sqrt{Nh^{\ell_{2}-1}})$$
$$\displaystyle=$$
$$\displaystyle\ Nh^{\ell_{2}}\{O(1)+O_{\hbox{\tiny$P$}}(1/\sqrt{Nh})\}.$$
Thus
$$\mbox{Cov}\left\{T_{31,\ell_{1}}^{*}(x),T_{32,\ell_{2}}^{*}(x)|\mathbb{X}%
\right\}=Nh^{\ell_{1}+\ell_{2}}\{O(1)+O_{\hbox{\tiny$P$}}(1/\sqrt{Nh})\},$$
i.e.,
$$\mbox{Cov}\left\{\mathbf{T}_{31}^{*}(x),\mathbf{T}_{32}^{*}(x)|\mathbb{X}%
\right\}=\mbox{Cov}\left\{\mathbf{T}_{31}^{*}(x)|\mathbb{X}\right\}\times o_{%
\hbox{\tiny$P$}}(1),$$
which also becomes negligible.
Finally, we have
$$\displaystyle\mbox{Var}\left\{\left.\hat{\mbox{\boldmath$\beta$}}_{3}(x)\right%
|\mathbb{X}\right\}=\frac{\sigma^{2}(x)+\bar{\sigma}^{2}N^{-1}\sum_{j=1}^{J}%
\sum_{k=1}^{c_{j}}(c_{j}-1)}{hf_{X}(x)}N^{-1}\mathbf{H}^{-1}\tilde{\mbox{%
\boldmath$\mu$}}_{0}^{-1}\tilde{\mbox{\boldmath$\nu$}}_{0}\tilde{\mbox{%
\boldmath$\mu$}}_{0}^{-1}\mathbf{H}^{-1}\{1+o_{\hbox{\tiny$P$}}(1)\}.$$
C.2.1 Variance of $\hat{m}_{3}(x)$ when $p=0,1$
With $p=0$ or 1, we have
$$\displaystyle\ \mbox{Var}\left\{\hat{m}^{(3)}(x)|\mathbb{X}\right\}$$
$$\displaystyle=$$
$$\displaystyle\ \nu_{0}\frac{\sigma^{2}(x)+\bar{\sigma}^{2}N^{-1}\sum_{j=1}^{J}%
\sum_{k=1}^{c_{j}}(c_{j}-1)}{f_{X}(x)Nh}\{1+o_{\hbox{\tiny$P$}}(1)\}.$$
Appendix D: Proof of Theorem 2
It is assumed that $c_{j}=c$, for $j=1,\ldots,J$, in this appendix. Suppose that covariate data are sorted before forming pools, resulting in ordered covariate data, $X_{(11)}\leq X_{(12)}\leq\ldots\leq X_{(1c)}\leq X_{(21)}\leq\ldots\leq X_{(2c%
)}\leq\ldots\leq X_{(Jc)}$, so that $\tilde{\mathbf{X}}_{(j)}=(X_{(j1)},\ldots,X_{(jc)})^{\top}$ are covariate data in the $j$th pool, and $Z_{(j)}$ is the corresponding pooled response, for $j=1,\ldots,J$.
Throughout this section, we consider the kernel function $K(\cdot)$ satisfying that $K(|t|)=0$ if $|t|>1$. Kernels that satisfy this condition include the Epanechnikov, quartic, triweight, and tricube kernel. Along with the condition that $h\rightarrow 0$ as $N\rightarrow\infty$, this condition on the kernel shares the same spirit as Condition (T5) in Delaigle and Hall (2012).
D.1 Bias and variance of $\hat{m}_{1}(x)$
Using pooled data from homogeneous pooling, the weighted least squares objective function $Q_{1}(\mbox{\boldmath$\beta$})$ defined in the main article is
$$Q_{1}(\mbox{\boldmath$\beta$})=\sum_{j=1}^{J}\left\{Z_{(j)}-\sum_{\ell=0}^{p}%
\beta_{\ell}c^{-1}\sum_{k=1}^{c}\left(X_{(jk)}-x\right)^{\ell}\right\}^{2}%
\left\{c^{-1}\sum_{k=1}^{c}K_{h}\left(X_{(jk)}-x\right)\right\}.$$
Then the averaged-weighted local polynomial estimator of order $p$ for $m(x)$ is $\hat{m}_{1}(x)=\mbox{\boldmath$e$}_{1}^{\mathrm{\scriptscriptstyle T}}\mathbf{%
S}_{1}^{-1}(x)\mathbf{T}_{1}(x)$, where entries in
$\mathbf{S}_{1}(x)=\ [S_{1,\ell_{1},\ell_{2}}(x)]_{\ell_{1},\ell_{2}=0,1,\dots,p}$ and $\mathbf{T}_{1}(x)=\ (T_{1,0}(x),T_{1,1}(x),\dots,T_{1,p}(x))^{\mathrm{%
\scriptscriptstyle T}}$ are
$$\displaystyle S_{1,\ell_{1},\ell_{2}}(x)=$$
$$\displaystyle\ J^{-1}\sum_{j=1}^{J}\left\{c^{-1}\sum_{k=1}^{c}\left(X_{(jk)}-x%
\right)^{\ell_{1}}\right\}\left\{c^{-1}\sum_{k=1}^{c}\left(X_{(jk)}-x\right)^{%
\ell_{2}}\right\}\times$$
$$\displaystyle\left\{c^{-1}\sum_{k=1}^{c}K_{h}\left(X_{(jk)}-x\right)\right\},%
\mbox{ for }\ell_{1},\ell_{2}=0,1,\dots,p,$$
$$\displaystyle T_{1,\ell}(x)=$$
$$\displaystyle\ J^{-1}\sum_{j=1}^{J}Z_{(j)}\left\{c^{-1}\sum_{k=1}^{c}\left(X_{%
(jk)}-x\right)^{\ell}\right\}\left\{c^{-1}\sum_{k=1}^{c}K_{h}\left(X_{(jk)}-x%
\right)\right\},$$
$$\displaystyle\mbox{ for }\ell=0,1,\dots,p.$$
We focus on deriving the bias of $\hat{\bm{\beta}}_{1}(x)=\mathbf{S}_{1}^{-1}(x)\mathbf{T}_{1}(x)$ conditioning on the collection of all covariate data $\mathbb{X}$ first,
$$\displaystyle\mbox{Bias}[\hat{\bm{\beta}}_{1}(x)|\mathbb{X}]=\mbox{$\hbox{%
\hbox to 0.25pt{$S$\hss}\hbox to 0.25pt{$S$\hss}\hbox{$S$}}$}_{1}^{-1}(x)E[%
\mbox{$\hbox{\hbox to 0.25pt{$T$\hss}\hbox to 0.25pt{$T$\hss}\hbox{$T$}}$}_{1}%
(x)-\mbox{$\hbox{\hbox to 0.25pt{$S$\hss}\hbox to 0.25pt{$S$\hss}\hbox{$S$}}$}%
_{1}(x)\bm{\beta}(x)|\mathbb{X}].$$
Because, for $\ell=0,1,\dots,p$, the $\ell$th component of $\mathbf{T}_{1}(x)-\mathbf{S}_{1}(x)\mbox{\boldmath$\beta$}(x)$ is
$$\displaystyle T_{1,\ell}(x)-\mathbf{S}_{1}(x)[\ell,\,]\bm{\beta}(x)=$$
$$\displaystyle\ J^{-1}\sum_{j=1}^{J}\left\{Z_{(j)}-\sum_{\ell^{\prime}=0}^{p}%
\beta_{\ell^{\prime}}c^{-1}\sum_{k=1}^{c}\left(X_{(jk)}-x\right)^{\ell^{\prime%
}}\right\}$$
$$\displaystyle\ \times\left\{c^{-1}\sum_{k=1}^{c}\left(X_{(jk)}-x\right)^{\ell}%
\right\}\left\{c^{-1}\sum_{k=1}^{c}K_{h}\left(X_{(jk)}-x\right)\right\},$$
one has
$$\displaystyle E\{T_{1,\ell}(x)-\mathbf{S}_{1}(x)[\ell,\,]\bm{\beta}(x)|\mathbb%
{X}\}$$
$$\displaystyle=$$
$$\displaystyle\ N^{-1}\sum_{j=1}^{J}\left\{\sum_{k=1}^{c}K_{h}\left(X_{(jk)}-x%
\right)\right\}\left\{c^{-1}\sum_{k=1}^{c}\left(X_{(jk)}-x\right)^{\ell}\right\}$$
$$\displaystyle\times\left[c^{-1}\sum_{k=1}^{c}\left\{m(X_{(jk)})-\sum_{\ell^{%
\prime}=0}^{p}\beta_{\ell^{\prime}}\left(X_{(jk)}-x\right)^{\ell^{\prime}}%
\right\}\right].$$
(D.1)
Here, we only consider $x$ being in the interior set of $\mathcal{I}$ that is a compact, nondegenerate interval. Besides continuity, we also assume $f_{X}(\cdot)$ bounded away from zero on an open interval $\mathcal{J}$ such that $\mathcal{I}\subset\mathcal{J}$, which is precisely Condition (T1) in Delaigle and Hall (2012). For any such $x$, we consider the cumulative distribution function (cdf) of $X$ evaluated at $x$, $F_{X}(x)$. Because $x$ is an interior point of $\mathcal{I}$, and $f_{X}(x)$ is bounded away from zero in $\mathcal{J}$, there exist $a$ and $b$ such that $a<x<b$ and $0<F_{X}(a)<F_{X}(x)<F_{X}(b)<1$.
We argue that when $N$ is large, for any $j$, if one of $X_{(j1)},\dots,X_{(jc)}$ is in $[x-h,x+h]$, then all covariate data in pool $j$ fall in $[a,b]$, where we note that $h\rightarrow 0$ as $N\rightarrow\infty$. To signify the dependence on $N$ of this part of the discussions, we write these covariate data as $X_{(j1:N)},\dots,X_{(jc:N)}$ and the bandwidth as $h_{N}$. Now suppose the opposite is true. Then there exists a sequence of sample sizes $c<N_{1}<N_{2}<\cdots<N_{m}<\cdots<\infty$, where $\lim_{m\rightarrow\infty}N_{m}=\infty$, such that for each $m$, $-h_{N_{m}}<X_{(j1:N_{m})}-x<h_{N_{m}}$ and $X_{(jc:N_{m})}>b+\epsilon$ for some $j\in\{1,\dots,J\}$, where $\epsilon$ is a very small constant such that $F_{X}(b+\epsilon)>F_{X}(b)$. We have that $F_{N_{k}}(X_{(j1:N_{m})})=((j-1)c+1)/N_{m}$ and $F_{N_{k}}(X_{(jc:N_{m})})=\{(j-1)c+k\}/N_{m}$, where $F_{N}$ is the empirical function of $F_{X}$ when the sample size is $N$. Because $h_{N_{m}}$ goes to zero as $m$ goes to infinity, $\lim_{m\rightarrow\infty}X_{(j1:N_{m})}=x$. Further more, because $F_{N_{k}}(\cdot)$ converges to $F_{X}(\cdot)$ from the classic uniform convergence of an empirical process, and that $F_{X}$ is continuous, we conclude that $\lim_{m\rightarrow\infty}\{(j-1)c+1\}/N_{m}=F_{X}(x)<F_{X}(b)$. Consequently, $\lim_{m\rightarrow\infty}\{(j-1)c+c\}/N_{m}=F_{X}(x)<F_{X}(b)$; i.e., $\lim_{m\rightarrow\infty}F_{N_{k}}(X_{jc:N_{m}})=F_{X}(x)<F_{X}(b)$. However, $F_{N_{k}}(X_{jc:N_{m}})>F_{N_{k}}(b+\epsilon)$ implies that $\lim_{m\rightarrow\infty}F_{N_{k}}(X_{jc:N_{m}})\geq F_{X}(b+\epsilon)>F_{X}(b)$, which provides a contradiction.
Define a partition of the index set $\{1,\ldots,J\}=\mathcal{J}_{1}\cup\mathcal{J}_{2}$, where $\mathcal{J}_{1}=\{j\in\{1,\ldots,J\}:\,\textrm{at least one of $X_{(j1)},\dots%
,X_{(jc)}$ is in $[x-h,x+h]$}\}$, and $\mathcal{J}_{2}=\{j\in\{1,\ldots,J\}:\,\textrm{none of $X_{(j1)},\dots,X_{(jc)%
}$ are in $[x-h,x+h]$}\}$. Following this partition, (D.3) can be re-expressed as
$$E\left\{\left.T_{1,\ell}(x)-\mathbf{S}_{1,\ell,\cdot}(x)\bm{\beta}(x)\right|%
\mathbb{X}\right\}=\mathcal{L}_{1}+\mathcal{L}_{2},$$
where, for $a=1,2$, $\mathcal{L}_{a}=$
$$N^{-1}\sum_{j\in\mathcal{J}_{a}}\sum_{k=1}^{c}K_{h}\left(X_{(jk)}-x\right)%
\left\{c^{-1}\sum_{k=1}^{c}\left(X_{(jk)}-x\right)^{\ell}\right\}c^{-1}\sum_{k%
=1}^{c}\left\{m(X_{(jk)})-\sum_{\ell=0}^{p}\beta_{\ell}\left(X_{(jk)}-x\right)%
^{\ell}\right\}.$$
When $j\in\mathcal{J}_{2}$, all the $K_{h}\left(X_{(jk)}-x\right)$’s are zero. Hence
$$\displaystyle\mathcal{L}_{2}=0=N^{-1}\sum_{j\in\mathcal{J}_{2}}\sum_{k=1}^{c}K%
_{h}\left(X_{(jk)}-x\right)\left(X_{(jk)}-x\right)^{\ell}\left\{m(X_{(jk)})-%
\sum_{\ell=0}^{p}\beta_{\ell}\left(X_{(jk)}-x\right)^{\ell}\right\}.$$
When $j\in\mathcal{J}_{1}$, we known that when $N$ is large, all the $X_{(j1)},\dots,X_{(jc)}$ will be in $[a,b]$. Using the classical result of an empirical quantile process, we know that
$$\sup_{u\in[F_{X}(a),F_{X}(b)]}\sqrt{N}|F_{N}^{-1}(u)-F_{X}^{-1}(u)|=O_{\hbox{%
\tiny$P$}}(1).$$
where we denote by $F_{N}^{-1}$ the empirical quantile function using the random sample $X_{jk}$’s and $F^{-1}$ the quantile function of the distribution of $X$. Then we have
$X_{(jk)}=F_{N}^{-1}(\{(j-1)c+k\}/N)$,
and
$$\displaystyle\sup_{j\in\mathcal{J}_{1}}\sqrt{N}|X_{(jc)}-X_{(j1)}|$$
$$\displaystyle\leq$$
$$\displaystyle\ \sup_{j\in\mathcal{J}_{1}}\sqrt{N}\left|F_{N}^{-1}\left(\frac{%
\sum_{m=1}^{j-1}c_{m}+c}{N}\right)-F_{X}^{-1}\left(\frac{\sum_{m=1}^{j-1}c_{m}%
+c}{N}\right)\right|$$
$$\displaystyle\ +\sup_{j\in\mathcal{J}_{1}}\sqrt{N}\left|F_{N}^{-1}\left(\frac{%
\sum_{m=1}^{j-1}c_{m}+1}{N}\right)-F_{X}^{-1}\left(\frac{\sum_{m=1}^{j-1}c_{m}%
+1}{N}\right)\right|$$
$$\displaystyle\ +\sup_{j\in\mathcal{J}_{1}}\sqrt{N}\left|F_{X}^{-1}\left(\frac{%
\sum_{m=1}^{j-1}c_{m}+c}{N}\right)-F_{X}^{-1}\left(\frac{\sum_{m=1}^{j-1}c_{m}%
+1}{N}\right)\right|$$
$$\displaystyle\leq$$
$$\displaystyle\ 2\sup_{u\in[F_{X}(a),F_{X}(b)]}\sqrt{N}|F_{N}^{-1}(u)-F_{X}^{-1%
}(u)|$$
$$\displaystyle\ +\sup_{j\in\mathcal{J}_{1}}\sqrt{N}\left|F_{X}^{-1}\left(\frac{%
\sum_{m=1}^{j-1}c_{m}+c}{N}\right)-F_{X}^{-1}\left(\frac{\sum_{m=1}^{j-1}c_{m}%
+1}{N}\right)\right|.$$
(D.2)
Hence, when $j\in\mathcal{J}_{1}$, $\sup_{j}|X_{(jc)}-X_{(j1)}|=O_{\hbox{\tiny$P$}}(1/\sqrt{N})$.
Then, for $j\in\mathcal{J}_{1}$, we have $|X_{(jk)}-x|\leq h$. Consequently,
$$\displaystyle c^{-1}\sum_{k=1}^{c}\left(X_{(jk)}-x\right)^{\ell}=$$
$$\displaystyle\ \left(X_{(jk)}-x\right)^{\ell}+c^{-1}\sum_{k_{1}\neq k}^{c}\{(X%
_{(jk_{1})}-x)^{\ell}-(X_{(jk)}-x)^{\ell}\}$$
$$\displaystyle=$$
$$\displaystyle\ \left(X_{(jk)}-x\right)^{\ell}+O_{p}(h^{\ell-1}/\sqrt{N})\{1-I(%
\ell=0)\}.$$
Assuming that $m(\cdot)$ is $(p+2)$-th order continuously differentiable,
$$\displaystyle\ c^{-1}\sum_{k_{1}=1}^{c}\left\{m(X_{(jk_{1})})-\sum_{\ell=0}^{p%
}\beta_{\ell}\left(X_{(jk_{1})}-x\right)^{\ell}\right\}-\left\{m(X_{(jk)})-%
\sum_{\ell=0}^{p}\beta_{\ell}\left(X_{(jk)}-x\right)^{\ell}\right\}$$
$$\displaystyle=$$
$$\displaystyle\ c^{-1}\sum_{k_{1}\neq k}\left[\left\{m(X_{(jk_{1})})-\sum_{\ell%
=0}^{p}\beta_{\ell}\left(X_{(jk_{1})}-x\right)^{\ell}\right\}-\left\{m(X_{(jk)%
})-\sum_{\ell=0}^{p}\beta_{\ell}\left(X_{(jk)}-x\right)^{\ell}\right\}\right]$$
$$\displaystyle=$$
$$\displaystyle\ c^{-1}\sum_{k_{1}\neq k}\left\{m^{(p+1)}(X_{(jk_{1})}^{*})(X_{(%
jk_{1})}-x)^{p+1}-m^{(p+1)}(X_{(jk)}^{*})(X_{(jk)}-x)^{p+1}\right\}/(p+1)!$$
$$\displaystyle=$$
$$\displaystyle\ O_{\hbox{\tiny$P$}}(h^{p}/\sqrt{N}),$$
where $X_{(jk_{1})}^{*}$ lies between $X_{(jk_{1})}$ and $x$, and $X_{(jk)}^{*}$ lies between $X_{(jk)}$ and $x$.
Consequently, recalling the definition of $\mathcal{J}_{2}$, we can write $E\{T_{1,\ell}(x_{0})-\mbox{$\hbox{\hbox to 0.25pt{$S$\hss}\hbox to 0.25pt{$S$%
\hss}\hbox{$S$}}$}_{1,\ell,\cdot}(x)\bm{\beta}(x)|\mathbb{X}\}=\mathcal{L}_{1}$ as follows,
$$\displaystyle\ N^{-1}\sum_{j=1}^{J}\sum_{k=1}^{c}K_{h}\left(X_{(jk)}-x\right)%
\left(X_{(jk)}-x\right)^{\ell}\left\{m(X_{(jk)})-\sum_{\ell=0}^{p}\beta_{\ell}%
\left(X_{(jk)}-x\right)^{\ell}\right\}$$
$$\displaystyle\ +O_{\hbox{\tiny$P$}}(h^{\ell-1}/\sqrt{N})\{1-I(l=0)\}N^{-1}\sum%
_{j=1}^{J}\sum_{k=1}^{c}K_{h}\left(X_{(jk)}-x\right)\left\{m(X_{(jk)})-\sum_{%
\ell=0}^{p}\beta_{\ell}\left(X_{(jk)}-x\right)^{\ell}\right\}$$
$$\displaystyle\ +O_{\hbox{\tiny$P$}}(h^{p}/\sqrt{N})N^{-1}\sum_{j=1}^{J}\sum_{k%
=1}^{c}K_{h}\left(X_{(jk)}-x\right)\left(X_{(jk)}-x\right)^{\ell}$$
$$\displaystyle\ +O_{\hbox{\tiny$P$}}(h^{\ell-1}/\sqrt{N})\{1-I(\ell=0)\}O_{%
\hbox{\tiny$P$}}(h^{p}/\sqrt{N})N^{-1}\sum_{j=1}^{J}\sum_{k=1}^{c}K_{h}\left(X%
_{jk}-x\right)$$
$$\displaystyle=$$
$$\displaystyle\ N^{-1}\sum_{j=1}^{J}\sum_{k=1}^{c}K_{h}\left(X_{(jk)}-x\right)%
\left(X_{(jk)}-x_{0}\right)^{\ell}\left\{m(X_{(jk)})-\sum_{\ell=0}^{p}\beta_{%
\ell}\left(X_{(jk)}-x\right)^{\ell}\right\}\{1+o_{\hbox{\tiny$P$}}(1)\}$$
provided that $Nh^{4}\rightarrow\infty$.
Similarly, we have
$$\displaystyle S_{1,\ell_{1},\ell_{2}}(x)=$$
$$\displaystyle\ N^{-1}\sum_{j=1}^{J}\sum_{k=1}^{c}K_{h}\left(X_{jk}-x\right)%
\left(X_{jk}-x\right)^{\ell_{1}+\ell_{2}}\{1+o_{\hbox{\tiny$P$}}(1)\}.$$
Hence, $\mbox{Bias}\{\hat{\bm{\beta}}_{1}(x)|\mathbb{X}\}=\mbox{Bias}\{\hat{\bm{\beta}%
}_{0}(x)|\mathbb{X}\}\{1+o_{\hbox{\tiny$P$}}(1)\}$.
Now we consider the variance,
$$\mbox{Cov}\{\mathbf{S}_{1}(x)^{-1}\mathbf{T}_{1}(x)|\mathbb{X}\}=\mathbf{S}_{1%
}(x)^{-1}\mbox{Cov}\{\mathbf{T}_{1}(x)|\mathbb{X}\}\mathbf{S}_{1}(x)^{-1}.$$
We see that
$$\displaystyle\mbox{Cov}\left\{T_{1,\ell_{1}}(x),T_{1,\ell_{2}}(x)|\mathbb{X}%
\right\}=$$
$$\displaystyle\ J^{-2}\sum_{j=1}^{J}c^{-2}\sum_{k=1}^{c}\sigma^{2}(X_{(jk)})%
\left\{c^{-1}\sum_{k=1}^{c}\left(X_{(jk)}-x\right)^{\ell_{1}}\right\}$$
$$\displaystyle\ \times\left\{c^{-1}\sum_{k=1}^{c}\left(X_{(jk)}-x\right)^{\ell_%
{2}}\right\}\left\{c^{-1}\sum_{k=1}^{c}K_{h}\left(X_{(jk)}-x\right)\right\}^{2}.$$
Similarly, we define $\mathcal{J}_{1}$ and $\mathcal{J}_{2}$. For $j\in\mathcal{J}_{2}$,
$$\displaystyle\ J^{-2}\sum_{j\in\mathcal{J}_{2}}c^{-2}\sum_{k=1}^{c}\sigma^{2}(%
X_{(jk)})\left\{c^{-1}\sum_{k=1}^{c}\left(X_{(jk)}-x\right)^{l_{1}}\right\}$$
$$\displaystyle\ \times\left\{c^{-1}\sum_{k=1}^{c}\left(X_{(jk)}-x\right)^{l_{2}%
}\right\}\left\{c^{-1}\sum_{k=1}^{c}K_{h}\left(X_{(jk)}-x\right)\right\}^{2}$$
$$\displaystyle=$$
$$\displaystyle\ (Nh)^{-1}N^{-1}\sum_{j\in\mathcal{J}_{2}}\sum_{k=1}^{c}\sigma^{%
2}(X_{jk})\left(X_{jk}-x\right)^{\ell_{1}+\ell_{2}}K^{\ddagger}_{h}\left(X_{jk%
}-x\right),$$
where $K^{\ddagger}(t)=K^{2}(t)$.
For $j\in\mathcal{J}_{1}$, assuming smoothness of $\sigma^{2}(\cdot)$, we have
$$\displaystyle c^{-1}\sum_{k=1}^{c}\sigma^{2}(X_{(jk)})=$$
$$\displaystyle\ \sigma^{2}(X_{jk})+O_{\hbox{\tiny$P$}}(1/\sqrt{N})$$
$$\displaystyle c^{-1}\sum_{k=1}^{c}\left(X_{(jk)}-x\right)^{\ell}=$$
$$\displaystyle\ \left(X_{jk}-x\right)^{\ell}+O_{\hbox{\tiny$P$}}(h^{l-1}/\sqrt{%
N})\{1-I(\ell=0)\}.$$
Further, provided that $K^{\prime}(t)$ is bounded,
$$\displaystyle\ \left\{c^{-1}\sum_{k=1}^{c}K_{h}\left(X_{(jk)}-x\right)\right\}%
^{2}=c^{-2}\sum_{k=1}^{c}\left\{K_{h}\left(X_{(jk)}-x\right)\right\}^{2}$$
$$\displaystyle\ +c^{-2}\sum_{k_{1}\neq k_{2}}K_{h}\left(X_{(jk_{1})}-x\right)K_%
{h}\left(X_{(jk_{2})}-x\right)$$
$$\displaystyle=$$
$$\displaystyle\ c^{-2}\sum_{k=1}^{c}\left\{K_{h}\left(X_{(jk)}-x\right)\right\}%
^{2}+c^{-2}(c-1)\sum_{k_{1}=1}^{c}K_{h}\left(X_{(jk_{1})}-x\right)\{K_{h}\left%
(X_{(jk_{1})}-x\right)+O_{\hbox{\tiny$P$}}(1/\sqrt{Nh^{4}})\}$$
$$\displaystyle=$$
$$\displaystyle\ c^{-1}h^{-1}\sum_{k=1}^{c}K^{\ddagger}_{h}\left(X_{(jk)}-x%
\right)+\frac{c-1}{c^{2}}h^{-1}\sum_{k=1}^{c}K_{h}(X_{(jk)}-x)O_{\hbox{\tiny$P%
$}}(1/\sqrt{Nh^{2}}).$$
Then
$$\displaystyle\ J^{-2}\sum_{j\in\mathcal{J}_{1}}c^{-2}\sum_{k=1}^{c}\sigma^{2}(%
X_{(jk)})\left\{c^{-1}\sum_{k=1}^{c}\left(X_{(jk)}-x\right)^{l_{1}}\right\}$$
$$\displaystyle\ \times\left\{c^{-1}\sum_{k=1}^{c}\left(X_{(jk)}-x\right)^{l_{2}%
}\right\}\left\{c^{-1}\sum_{k=1}^{c}K_{h}\left(X_{(jk)}-x\right)\right\}^{2}$$
$$\displaystyle=$$
$$\displaystyle\ (Nh)^{-1}N^{-1}\sum_{j\in\mathcal{J}_{1}}\left\{\sigma^{2}(X_{(%
jk)})+O_{\hbox{\tiny$P$}}(1/\sqrt{N})\right\}\left\{\left(X_{(jk)}-x\right)^{%
\ell_{1}}+O_{\hbox{\tiny$P$}}(h^{\ell_{1}-1}/\sqrt{N})\{1-I(\ell_{1}=0)\}\right\}$$
$$\displaystyle\ \times\left\{\left(X_{(jk)}-x\right)^{\ell_{2}}+O_{\hbox{\tiny$%
P$}}(h^{\ell_{2}-1}/\sqrt{N})\{1-I(\ell_{2}=0)\}\right\}$$
$$\displaystyle\ \times\left\{\sum_{k=1}^{c}K^{\ddagger}_{h}\left(X_{(jk)}-x%
\right)+(c-1)\sum_{k=1}^{c}K_{h}(X_{(jk)}-x)O_{\hbox{\tiny$P$}}(1/\sqrt{Nh^{2}%
})\right\}$$
$$\displaystyle=$$
$$\displaystyle\ (Nh)^{-1}N^{-1}\sum_{j=1}^{J}\sum_{k=1}^{c}K^{\ddagger}_{h}%
\left(X_{(jk)}-x\right)\sigma^{2}(X_{(jk)})\left(X_{(jk)}-x_{0}\right)^{\ell_{%
1}+\ell_{2}}\{1+o_{\hbox{\tiny$P$}}(1)\}.$$
provided that $Nh^{4}\rightarrow\infty$.
Thus, we conclude that
$$\displaystyle\ (Nh)\mbox{Cov}\left\{T_{1,\ell_{1}}(x),T_{1,\ell_{2}}(x)|%
\mathbb{X}\right\}$$
$$\displaystyle=$$
$$\displaystyle\ N^{-1}\sum_{j=1}^{J}\sum_{k=1}^{c}K^{\ddagger}_{h}\left(X_{jk}-%
x\right)\sigma^{2}(X_{jk})\left(X_{jk}-x\right)^{\ell_{1}+\ell_{2}}\{1+o_{%
\hbox{\tiny$P$}}(1)\}.$$
Hence, $\mbox{Var}\{\hat{\bm{\beta}}_{1}(x)|\mathbb{X}\}=\mbox{Var}\{\hat{\bm{\beta}}_%
{0}(x)|\mathbb{X}\}\{1+o_{\hbox{\tiny$P$}}(1)\}.$ Note that, homogeneous pooling uses $J$ tests while individual uses $cJ$ tests.
D.2 Bias and variance of $\hat{m}_{2}(x)$
Under homogeneous pooling, the weighted least squares objective function $Q_{2}(\mbox{\boldmath$\beta$})$ is
$$Q_{2}(\mbox{\boldmath$\beta$})=\sum_{j=1}^{J}\left\{Z_{(j)}-\sum_{\ell=0}^{p}%
\beta_{\ell}c^{-1}\sum_{k=1}^{c}\left(X_{(jk)}-x\right)^{\ell}\right\}^{2}%
\left\{\prod_{k=1}^{c}K_{h}\left(X_{(jk)}-x\right)\right\}.$$
Then the product-weighted $p$-th order local polynomial estimator for $m(x)$, is
$$\hat{m}_{2}(x)=\mbox{\boldmath$e$}_{1}^{\top}\mathbf{S}_{2}^{-1}(x)\mathbf{T}_%
{2}(x),$$
where
$$\displaystyle\mathbf{S}_{2}(x)=$$
$$\displaystyle\ [S_{1,\ell_{1},\ell_{2}}(x)]_{\ell_{1},\ell_{2}=0,1,\dots,p},$$
$$\displaystyle\mathbf{T}_{2}(x)=$$
$$\displaystyle\ (T_{1,0}(x),T_{1,1}(x),\dots,T_{1,p}(x))^{\top},\mbox{ in which}$$
$$\displaystyle S_{2,\ell_{1},\ell_{2}}(x)=$$
$$\displaystyle\ J^{-1}\sum_{j=1}^{J}\left\{c^{-1}\sum_{k=1}^{c}\left(X_{(jk)}-x%
\right)^{\ell_{1}}\right\}\left\{c^{-1}\sum_{k=1}^{c}\left(X_{(jk)}-x\right)^{%
\ell_{2}}\right\}\left\{h^{c-1}\prod_{k=1}^{c}K_{h}\left(X_{(jk)}-x\right)%
\right\},$$
$$\displaystyle\mbox{ for }\ell_{1},\ell_{2}=0,1,\dots,p,$$
$$\displaystyle T_{2,\ell}(x)=$$
$$\displaystyle\ J^{-1}\sum_{j=1}^{J}Z_{(j)}\left\{c^{-1}\sum_{k=1}^{c}\left(X_{%
(jk)}-x\right)^{\ell}\right\}\left\{h^{c-1}\prod_{k=1}^{c}K_{h}\left(X_{(jk)}-%
x\right)\right\},\mbox{ for }\ell=0,1,\dots,p.$$
Study the bias term first,
$$\displaystyle\mbox{Bias}\left\{\left.\hat{\bm{\beta}}_{2}(x)\right|\mathbb{X}%
\right\}=\mathbf{S}_{2}^{-1}(x)E\left\{\left.\mathbf{T}_{2}(x)-\mathbf{S}_{2}(%
x)\bm{\beta}(x)\right|\mathbb{X}\right\}$$
Rewrite the $\ell$th component of $\mathbf{T}_{2}(x)-\mathbf{S}_{2}(x)\beta(x)$, where $\ell=0,1,\dots,p$,
$$\displaystyle T_{2,\ell}(x)-\mathbf{S}_{2,l,\cdot}(x)\bm{\beta}(x)=$$
$$\displaystyle\ J^{-1}\sum_{j=1}^{J}\left\{Z_{(j)}-\sum_{\ell=0}^{p}\beta_{\ell%
}c^{-1}\sum_{k=1}^{c}\left(X_{(jk)}-x\right)^{\ell}\right\}$$
$$\displaystyle\ \times\left\{c^{-1}\sum_{k=1}^{c}\left(X_{(jk)}-x\right)^{\ell}%
\right\}\left\{h^{c-1}\prod_{k=1}^{c}K_{h}\left(X_{(jk)}-x\right)\right\}.$$
Then
$$\displaystyle E\left\{\left.T_{1,\ell}(x)-\mathbf{S}_{1,\ell,\cdot}(x)\bm{%
\beta}(x)\right|\mathbb{X}\right\}=$$
$$\displaystyle\ J^{-1}\sum_{j=1}^{J}\left\{h^{c-1}\prod_{k=1}^{c}K_{h}\left(X_{%
(jk)}-x\right)\right\}$$
$$\displaystyle\ \times\left\{c^{-1}\sum_{k=1}^{c}\left(X_{(jk)}-x\right)^{\ell}\right\}$$
$$\displaystyle\ \times c^{-1}\sum_{k=1}^{c}\left\{m(X_{(jk)})-\sum_{\ell=0}^{p}%
\beta_{\ell}\left(X_{(jk)}-x\right)^{\ell}\right\}$$
(D.3)
Similarly, we break the summation $\sum_{j=1}^{J}$ in (D.3) into two parts as $\sum_{j\in\mathcal{J}_{1}}$ and $\sum_{j\in\mathcal{J}_{2}}$ where for $j\in\mathcal{J}_{1}$, at least one of $X_{(j1)},\dots,X_{(jc)}$ is in $[x-h,x+h]$; and for $j\in\mathcal{J}_{2}$, none of $X_{(j1)},\dots,X_{(jc)}$ are in $[x-h,x+h]$. We rewrite (D.3) as
$$E\left\{\left.T_{1,\ell}(x)-\mathbf{S}_{1,\ell,\cdot}(x)\bm{\beta}(x)\right|%
\mathbb{X}\right\}=\mathcal{L}_{1}+\mathcal{L}_{2}.$$
When $j\in\mathcal{J}_{2}$, all the $K_{h}\left(X_{(jk)}-x\right)$’s are zero. Hence
$$\displaystyle\mathcal{L}_{2}=0=N^{-1}\sum_{j\in\mathcal{J}_{2}}\sum_{k=1}^{c}K%
^{\dagger}_{h}\left(X_{(jk)}-x\right)\left(X_{(jk)}-x\right)^{\ell}\left\{m(X_%
{(jk)})-\sum_{\ell=0}^{p}\beta_{\ell}\left(X_{(jk)}-x\right)^{\ell}\right\},$$
where $K^{\dagger}(t)=K^{c}(t)$.
When $j\in\mathcal{J}_{1}$, we have $|X_{(jk)}-x|\leq h$. Consequently,
$$\displaystyle c^{-1}\sum_{k=1}^{c}\left(X_{(jk)}-x\right)^{\ell}=$$
$$\displaystyle\ \left(X_{(jk)}-x\right)^{\ell}+c^{-1}\sum_{k_{1}\neq k}^{c}\{(X%
_{(jk_{1})}-x)^{\ell}-(X_{(jk)}-x)^{\ell}\}$$
$$\displaystyle=$$
$$\displaystyle\ \left(X_{(jk)}-x\right)^{\ell}+O_{\hbox{\tiny$P$}}(h^{\ell-1}/%
\sqrt{N})\{1-I(\ell=0)\}.$$
By the smoothness of $m(\cdot)$,
$$\displaystyle\ c^{-1}\sum_{k=1}^{c}\left\{m(X_{(jk)})-\sum_{\ell=0}^{p}\beta_{%
\ell}\left(X_{(jk)}-x\right)^{\ell}\right\}-\left\{m(X_{(jk)})-\sum_{\ell=0}^{%
p}\beta_{\ell}\left(X_{(jk)}-x\right)^{\ell}\right\}$$
$$\displaystyle=$$
$$\displaystyle\ c^{-1}\sum_{k_{1}\neq k}\left[\left\{m(X_{(jk_{1})})-\sum_{\ell%
=0}^{p}\beta_{\ell}\left(X_{(jk_{1})}-x\right)^{\ell}\right\}-\left\{m(X_{(jk)%
})-\sum_{\ell=0}^{p}\beta_{\ell}\left(X_{(jk)}-x\right)^{\ell}\right\}\right]$$
$$\displaystyle=$$
$$\displaystyle\ c^{-1}\sum_{k_{1}\neq k}\left\{m^{(p+1)}(X_{(jk_{1})}^{*})(X_{(%
jk_{1})}-x)^{p+1}-m^{(p+1)}(X_{(jk)}^{*})(X_{(jk)}-x)^{p+1}\right\}/(p+1)!$$
$$\displaystyle=$$
$$\displaystyle\ O_{\hbox{\tiny$P$}}(h^{p}/\sqrt{N}).$$
And, when $K^{\prime}(t)$ is bounded and $Nh^{2}\rightarrow\infty$,
$$\displaystyle h^{c-1}\prod_{k=1}^{c}K_{h}\left(X_{(jk)}-x\right)=$$
$$\displaystyle\ h^{-1}\prod_{k=1}^{c}K\left(\frac{X_{(jk)-x}}{h}\right)$$
$$\displaystyle=$$
$$\displaystyle\ h^{-1}\prod_{k=1}^{c}\left\{K\left(\frac{X_{(j1)-x}}{h}\right)+%
O_{p}(1/\sqrt{Nh^{2}})\right\}$$
$$\displaystyle=$$
$$\displaystyle\ h^{-1}K^{c}\left(\frac{X_{(j1)-x}}{h}\right)\{1+o_{\hbox{\tiny$%
P$}}(1)\}.$$
Hence, we have
$$\displaystyle h^{c-1}\prod_{k=1}^{c}K_{h}\left(X_{(jk)}-x\right)=$$
$$\displaystyle\ c^{-1}\sum_{k=1}^{c}K^{\dagger}_{h}\left(\frac{X_{(jk)-x}}{h}%
\right)\{1+o_{\hbox{\tiny$P$}}(1)\}.$$
Consequently, recalling the definition of $\mathcal{J}_{2}$, we can write $E[T_{1,l}(x)-\mathbf{S}_{1,l,\cdot}(x)\bm{\beta}(x)|\mathbb{X}]=\mathcal{L}_{1}$ as
$$\displaystyle N^{-1}\sum_{j=1}^{J}\sum_{k=1}^{c}K^{\dagger}_{h}\left(X_{(jk)}-%
x\right)\left(X_{(jk)}-x\right)^{\ell}\left\{m(X_{(jk)})-\sum_{\ell=0}^{p}%
\beta_{\ell}\left(X_{(jk)}-x\right)^{\ell}\right\}\{1+o_{\hbox{\tiny$P$}}(1)\}$$
provided that $Nh^{2}\rightarrow\infty$.
Similarly, we have
$$\displaystyle S_{1,\ell_{1},\ell_{2}}(x)=$$
$$\displaystyle\ N^{-1}\sum_{j=1}^{J}\sum_{k=1}^{c}K^{\dagger}_{h}\left(X_{jk}-x%
\right)\left(X_{jk}-x\right)^{\ell_{1}+\ell_{2}}\{1+o_{\hbox{\tiny$P$}}(1)\}.$$
Hence, $\mbox{Bias}\{\hat{\bm{\beta}}_{1}(x)|\mathbb{X}\}=\mbox{Bias}[\hat{\bm{\beta}}%
_{0}(x)|\mathbb{X}]\{1+o_{\hbox{\tiny$P$}}(1)\}$ when using $K^{\dagger}(t)$ as the kernel function.
Similarly, we can conclude that
$\mbox{Var}\{\hat{\bm{\beta}}_{2}(x)|\mathbb{X}\}=\mbox{Var}\{\hat{\bm{\beta}}_%
{0}(x)|\mathbb{X}\}\{1+o_{\hbox{\tiny$P$}}(1)\}$, again where the kernel function is $K^{\dagger}(t)$ instead of $K(t)$. Note that, homogeneous pooling uses $J$ tests while individual uses $cJ$ tests.
Appendix E: Additional simulation results
Using data generated from models specified in (D1), (D3), and (D4) in Section 6 in the main article, Figures 6, 7, and 8 depict the three proposed estimators based on pooled data when $c=2$ and the benchmark estimate, $\hat{m}_{0}(x)$, based on individual-level data. Figures 9–12 provide parallel results when $c=6$ under (D1), (D2), (D3), and (D4).
Acknowledgments
Wang’s work was supported by the National Institutes of Health (NIH) under the Award Number R03 AI135614.
References
Bilder and Tebbs (2009)
Bilder, C. R. and Tebbs, J. M. (2009).
Bias, efficiency, and agreement for group-testing regression models.
Journal of Statistical Computation and Simulation,
79(1):67–80.
Delaigle et al. (2009)
Delaigle, A., Fan, J., and Carroll, R. (2009).
A design-adaptive local polynomial estimator for the
errors-in-variables problem.
Journal of the American Statistical Association,
104(485):348–359.
Delaigle and Hall (2012)
Delaigle, A. and Hall, P. (2012).
Nonparametric regression with homogeneous group testing data.
The Annals of Statistics, 40(1):131–158.
Dhawan and Richmond (2002)
Dhawan, P. and Richmond, A. (2002).
Role of cxcl1 in tumorigenesis of melanoma.
Journal of Leukocyte Biology, 72(1):9–18.
Fan and Gijbels (1996)
Fan, J. and Gijbels, I. (1996).
Local Polynomial Modelling and Its Applications: Monographs on
Statistics and Applied Probability 66, volume 66.
Chapman & Hall/CRC.
Genuis et al. (2013)
Genuis, S. J., Curtis, L., and Birkholz, D. (2013).
Gastrointestinal elimination of perfluorinated compounds using
cholestyramine and chlorella pyrenoidosa.
ISRN Toxicology, 2013.
Kärrman et al. (2006)
Kärrman, A., Mueller, J. F., Van Bavel, B., Harden, F., Toms, L.-M. L., and
Lindström, G. (2006).
Levels of 12 perfluorinated chemicals in pooled australian serum,
collected 2002- 2003, in relation to age, gender, and region.
Environmental Science & Technology, 40(12):3742–3748.
Kendziorski et al. (2003)
Kendziorski, C., Zhang, Y., Lan, H., and Attie, A. (2003).
The efficiency of pooling mrna in microarray experiments.
Biostatistics, 4(3):465–477.
Lin and Wang (2018)
Lin, J. and Wang, D. (2018).
Single-index regression for pooled biomarker data.
Journal of Nonparametric Statistics, 30(4):813–833.
Linton and Whang (2002)
Linton, O. and Whang, Y.-J. (2002).
Nonparametric estimation with aggregated data.
Econometric Theory, 18(2):420–468.
Liu et al. (2017)
Liu, Y., McMahan, C., and Gallagher, C. (2017).
A general framework for the regression analysis of pooled biomarker
assessments.
Statistics in Medicine, 36(15):2363–2377.
Ma et al. (2011)
Ma, C.-X., Vexler, A., Schisterman, E. F., and Tian, L. (2011).
Cost-efficient designs based on linearly associated biomarkers.
Journal of Applied Statistics, 38(12):2739–2750.
Malinovsky et al. (2012)
Malinovsky, Y., Albert, P. S., and Schisterman, E. F. (2012).
Pooling designs for outcomes under a gaussian random effects model.
Biometrics, 68(1):45–52.
McMahan et al. (2016)
McMahan, C. S., McLain, A. C., Gallagher, C. M., and Schisterman, E. F. (2016).
Estimating covariate-adjusted measures of diagnostic accuracy based
on pooled biomarker assessments.
Biometrical Journal, 58(4):944–961.
Mitchell et al. (2014)
Mitchell, E. M., Lyles, R. H., Manatunga, A. K., Danaher, M., Perkins, N. J.,
and Schisterman, E. F. (2014).
Regression for skewed biomarker outcomes subject to pooling.
Biometrics, 70(1):202–211.
Mitchell et al. (2015)
Mitchell, E. M., Lyles, R. H., Manatunga, A. K., and Schisterman, E. F. (2015).
Semiparametric regression models for a right-skewed outcome subject
to pooling.
American Journal of Epidemiology, 181(7):541–548.
Shih et al. (2004)
Shih, J. H., Michalowska, A. M., Dobbin, K., Ye, Y., Qiu, T. H., and Green,
J. E. (2004).
Effects of pooling mrna in microarray class comparisons.
Bioinformatics, 20(18):3318–3325.
Tsou et al. (2007)
Tsou, C.-L., Peters, W., Si, Y., Slaymaker, S., Aslanian, A. M., Weisberg,
S. P., Mack, M., and Charo, I. F. (2007).
Critical roles for ccr2 and mcp-3 in monocyte mobilization from bone
marrow and recruitment to inflammatory sites.
The Journal of Clinical Investigation, 117(4):902–909.
Vansteelandt et al. (2000)
Vansteelandt, S., Goetghebeur, E., and Verstraeten, T. (2000).
Regression models for disease prevalence with diagnostic tests on
pools of serum samples.
Biometrics, 56(4):1126–1133. |
[
[
Abstract
A pan-tilt camera system has been adopted by a variety of fields since it can cover a wide range of region compared to a single fixated camera setup.
Yet many studies rely on factory-assembled and calibrated platforms and assume an ideal rotation where rotation axes are perfectly aligned with the optical axis of the local camera.
However, in a user-created setup where a pan-tilting mechanism is arbitrarily assembled, the kinematic configurations may be inaccurate or unknown, violating ideal rotation.
These discrepancies in the model with the real physics result in erroneous servo manipulation of the pan-tilting system.
In this paper, we propose an accurate control mechanism for arbitrarily-assembled pan-tilt camera systems.
The proposed method formulates pan-tilt rotations as motion along great circle trajectories and calibrates its model parameters, such as positions and vectors of rotation axes, in 3D space.
Then, one can accurately servo pan-tilt rotations with pose estimation from inverse kinematics of their transformation.
The comparative experiment demonstrates out-performance of the proposed method,
in terms of accurately localizing target points in world coordinates, after being rotated from their captured camera frames.{CCSXML}
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††volume: #1††issue: 2\ShortPresentation\electronicVersion\PrintedOrElectronic
Accurate control of a pan-tilt system based on parameterization of rotational motion]Accurate control of a pan-tilt system based on parameterization of rotational motion
Byun, JungHyun et al.]
JungHyun Byun, SeungHo Chae, and TackDon Han
Media System Lab, Department of Computer Science, Yonsei University, Republic of Korea
1 Introduction
A pan-tilt platform consists of two motors, each rotating in pan and tilt directions. These 2 degrees of rotation freedoms grant the mounted system theoretically 360 degrees field-of-view.
This is particularly beneficial to computer vision applications, for cameras can only obtain data from a field of view that is directed by the optical axis at a time [NLW${}^{*}$17]. To obtain scene information from a larger field of view, the camera had to be translated or rotated to capture a series of images.
Various applications utilizes pan-tilt platforms from obvious video surveillance [DC03] to video conferencing, human-computer interaction, and augmented/mixed reality [WBIH12, BCYH17].
Despite continuous endeavors of the literature on the pan-tilting mechanics, many studies based their ground on fully factory-assembled and calibrated platforms such as in [DC03, WBIH12, LZHT15], which price from several hundreds to thousands of dollars. This may be problematic because the pan-tilting model may work seemingly error-free even if assuming an ideal rotation where rotation axes are aligned with the camera optical axis perfectly orthogonally, due to sound reliable assembly quality.
On the other hand, there is growing interest in user-created robots [Par11], where users create their own version of robots with such as consumer kits on the market [BCYH17]. In this case, it is unlikely that the exact kinematic specifications will be provided, for example in a CAD file format. Moreover, the mechanism may be fabricated and assembled in an do-it-yourself or arbitrary manner.
All these factors attribute to erroneous, if even existent, kinematic equations, rendering
kinematic control methods obsolete.
In this paper, we would like to address the issue of accurate pan-tilting manipulation even the pan-tilting kinematics are loosely coupled, or often unavailable.
More specifically, we propose an operating mechanism of an arbitrarily assembled pan-tilt model with loose kinematics based on rotational motion modeling of the mounted camera.
Our method is based on the pan-tilt model that is general enough to calibrate and compensate for assembly mismatches, such as skewed rotation axes or off-origin rotations.
2 Related Work
Calibrating pan-tilt motion of the camera has been broadly studied in the computer vision field, especially surveillance using pan-tilt-zoom (PTZ) camera. For example,
Davis and Chen in [DC03] presented a method for calibrating pan-tilt cameras that incorporates a more accurate complete model of camera motion. Pan and tilt rotations were modeled as occurring around detached arbitrary axes in space, without the assumption of rotation axes aligned to camera optical axis.
Wu and Radke in [WR13] introduced a camera calibration model for a pan-tilt-zoom camera, which explicitly reflects how focal length and lens distortion vary as a function of zoom scale. Using a nonlinear optimization, authors were able to accurately calibrate multiple parameters in one whole step. They also investigated and analyzed multiple cases of pan-tilt errors to maintain the calibrated state even after extended
continuous operations.
Li et al. in [LZHT15] presented a novel method to online calibrate the rotation angles of a pan-tilt camera by using only one control point. By converting the non-linear pan-tilt camera model into a linear model according to sine and cosine of pan and tilt parameters, a closed-form solution could be derived by solving a quadratic equation of their tangents.
In the optics and measurement literatures, studies regarding calibration methods for a turntable or rotational axis have been proposed.
Chen et al. in [CDCZ14] fixed a checkerboard plate on a turntable and captured multiple views of one pattern. Retrieved 3D corner points in a 360° view were used to form multiple circular trajectory planes, the equation and parameters of which were acquired using constrained global optimization method.
Niu et al. in [NLW${}^{*}$17] proposed a method for calibrating the relative orientation of a camera fixed on a rotation axis, where the camera cannot directly ‘see’ the rotation axis. Utilizing two checkerboards, one for the rotating camera and one for the external observing camera, they were able to calibrate the relative orientation of the two cameras and rotation axis represented in the same coordinate system.
In this paper, we propose
an operating mechanism of an arbitrarily assembled pan-tilt model with loose kinematics
based on rotational motion modeling of the mounted camera.
Our contributions can be summarized as follow:
•
First, the proposed method models and calibrates the rotation of a generic pan-tilt platform by recovering directions and positions of its axes in 3D space, utilizing an RGB-D camera.
•
Second, the proposed method is capable of manipulating servo rotations with respect to the camera, based on the inverse kinematic interpretation of its pan-tilt transformation model.
3 Pan-Tilt Rotation Modeling
Our goal is to model the rotational movement of a pan/tilting platform, so that we can estimate the pose of the RGB-D camera mounted on top of it. The platform rotates about arbitrarily assembled, independent axes [DC03] with loosely coupled kinematics. The structural model of such a setup is illustrated in Figure 1.
3.1 Rotation Parameters Acquisition
To model the movement of the motor rotation, we first calibrate parameters for the pan and tilt rotation. The calibration is a two-step process, where we first estimate the direction vector of the rotation, and then estimate the center of the circular trajectory. The directions and centers of two rotations are all free variables.
When arbitrary points are rotated around a rotation axis, they create a closed circular rotation trajectory on a 3-dimensional plane, where the plane is perpendicular to the rotation axis and the circle center lies on the rotation axis.
From the coordinates of a same point in rotated frames, the circular trajectory can be obtained using the least squares method[Sch06, DCYH13].
During the calibration, the camera captures multiple frames of a large checkerboard in front while it rotates, so that the checkerboard moves from one end of the field of view to another. Since the structure of the checkerboard is pre-known, all the rotation trajectories can be represented with respect to that of the top-leftmost corner. Then, we can parametrize the rotation with every corner of every frame and solve the objective function as a whole.
If the checkerboard comprises $m$ corners in the vertical direction and $n$ corners in the horizontal direction and $l$ frames were taken throughout the calibration, we have total $l\times m\times n$ corners to globally optimize.
For the rotation of the upper-left corner, let us denote its rotation direction vector as $n=[n_{x},\ n_{y},\ n_{z}]^{\intercal},\ ||n||=1$ and rotation circle center as $p=[a,\ b,\ c]^{\intercal}$.
Then the rotation axis equation becomes
$$\frac{x-a}{n_{x}}=\frac{y-b}{n_{y}}=\frac{z-c}{n_{z}},$$
(1)
and the rotation plane, which the upper-left corner is on, is $d$ away from the origin:
$$n_{x}x+n_{y}y+n_{z}z+d=0.$$
(2)
Since all the rotation circles made from checkerboard corners are defined on the same rotation axis, the distance between their planes can be defined with respect to the indices of checkerboard corners.
Let us denote the distances between the planes are $d_{h}$ and $d_{w}$ respectively in vertical and horizontal directions.
Then for the corner at the $i$-th row and $j$-th column of the checkerboard, the rotation circle center becomes $p_{ij}=[a_{ij},\ b_{ij},\ c_{ij}]^{\intercal}$, where
$$\begin{gathered}\displaystyle a_{ij}=a-n_{x}(i\times d_{h}+j\times d_{w}),\\
\displaystyle b_{ij}=b-n_{y}(i\times d_{h}+j\times d_{w}),\\
\displaystyle c_{ij}=c-n_{z}(i\times d_{h}+j\times d_{w}).\\
\end{gathered}$$
(3)
The ideal rotation trajectory will be modeled as a great circle, which is represented as the intersection of a plane and a sphere in 3D space.
Here, the plane can be modeled as
$$n_{x}x+n_{y}y+n_{z}z+d+i\times d_{h}+j\times d_{w}=0$$
(4)
and the sphere, or the intersecting circle can be modeled as
$$(x-a_{ij})^{2}+(y-b_{ij})^{2}+(z-c_{ij})^{2}=r_{ij}^{2}.$$
(5)
Let us denote the 3D vertex of the corner at the $i$th row and $j$th column of the $k$th captured checkerboard frame as $v_{ijk}=[x_{ijk},\ y_{ijk},\ z_{ijk}]^{\intercal}$.
Then, we can setup the objective function where our goal is to find parameters $n_{x},\ n_{y},\ n_{z},\ d,\ d_{h},\ d_{w}$ that minimize the following error for the plane model:
$$\sum\limits_{k=0}^{l-1}\sum\limits_{i=0}^{m-1}\sum\limits_{j=0}^{n-1}(n_{x}x_{%
ijk}+n_{y}y_{ijk}+n_{z}z_{ijk}+d+i\times d_{h}+j\times d_{w})^{2}.$$
(6)
From Equation 3 and Equation 6, one can calculate parameters $a,\ b,\ c,\ r_{ij}$ that minimize the following error for the circle model:
$$\sum\limits_{k=0}^{l-1}\sum\limits_{i=0}^{m-1}\sum\limits_{j=0}^{n-1}((x_{ijk}%
-a_{ij})^{2}+(y_{ijk}-b_{ij})^{2}+(z_{ijk}-c_{ij})^{2}-r_{ij}^{2})^{2}.$$
(7)
The global least squares method is adopted to minimize errors of two objective functions with regard to entire coordinate variations of 70 checkerboard corners in all frames. This yields optimized parameters for the rotation axis calibration [CDCZ14].
3.2 Pan-Tilt Transformation Model
If we denote a 3D point taken in some local camera frame after rotating tilt and pan angles $P_{local}$, and the point before rotations $P_{world}$, the relationship can be written with the rotation model as
$$\begin{gathered}\displaystyle\begin{bmatrix}P_{world}\\
1\end{bmatrix}=T_{pan}\ R_{pan}\ T^{-1}_{pan}\ T_{tilt}\ R_{tilt}\ T^{-1}_{%
tilt}\ \begin{bmatrix}P_{local}\\
1\end{bmatrix},\text{ where }\\
\displaystyle R(\theta)=\resizebox{375.804pt}{}{$\begin{bmatrix}C+n_{x}^{2}(1-%
C)&n_{x}n_{y}(1-C)-n_{z}S&n_{x}n_{z}(1-C)+n_{y}S&0\\
n_{y}n_{x}(1-C)+n_{z}S&C+n_{y}^{2}(1-C)&n_{y}n_{z}(1-C)-n_{x}S&0\\
n_{z}n_{x}(1-C)-n_{y}S&n_{z}n_{y}(1-C)+n_{x}S&C+n_{z}^{2}(1-C)&0\\
0&0&0&1\end{bmatrix}$},\\
\displaystyle C=\cos\theta,\ S=\sin\theta,\text{ and }T=\resizebox{99.7326pt}{%
}{$\begin{bmatrix}1&0&0&a\\
0&1&0&b\\
0&0&1&c\\
0&0&0&1\end{bmatrix}.$}\end{gathered}$$
(8)
Here, $R_{tilt}$ is a 4$\times$4 matrix that rotates around the direction vector $n=[n_{x},\ n_{y},\ n_{z}]^{\intercal}$ of the tilt axis and $T_{tilt}$ is a 4$\times$4 matrix that translates by the coordinates of the pivot point $p=(a,\ b,\ c)$ of the tilt axis. Transformations regarding pan are analogous.
Note that in the model tilt rotation comes before pan rotation. This is due to the kinematic configuration of the pan-tilt system we used. As shown in Figure 2, the tilting servo is installed on the panning arm. Thus to ensure unique rotation in world space, we tilt first, then pan.
3.3 Servo Control with Inverse Kinematics
Our scenario is that the users want to orient the camera so that the target object is located at the image center after the rotation. Specifically, the optical axis of the camera should pass through the target point. Then the task is rotating some point on the optical axis to a target point in world space.
This task can be thought of as rotating the linear-actuated end-effector $P_{local}=[0,\ 0,\ sz]^{\intercal}$ (some point on the optical axis) of a robot arm (pan-tilting platform) to the target point $P_{dst}=[dx,\ dy,\ dz]^{\intercal}$, with unknown $\alpha$ pan and $\ \beta$ tilt angles.
Using inverse kinematics notation on Equation 8, the problem becomes to find the parameter vector $\theta=[\alpha,\ \beta,\ sz]^{\intercal}$ of 2 rotations and 1 translation that minimizes the error $\left\|e\right\|=\left\|P_{dst}-P_{world}\right\|$. We adopt the Jacobian transpose method [Bus04] for Algorithm 1.
In our practice, we initialized $\alpha,\beta$ as 0 and $sz$ as $-\left\|P_{dst}\right\|$ and terminated the optimization if $\left\|e\right\|<1\ mm$. Maximum iteration count was set to 100, though 25 on average was enough to satisfy the condition. One pitfall here is that the magnitude difference can be extensive if the value of $sz$ is set in $mm$, for $\alpha,\ \beta$ range between $-1.57$ and 1.57 radian while $sz$ may vary from 450 to 8000 $mm$. This difference causes too much fluctuations in $\alpha,\beta$ values, leading to far-from-optimal solutions. One solution is to optimize $sz$ in meter unit so that the domain difference to angles becomes minimum.
4 Evaluation
4.1 System Configuration and Calibration
To validate the proposed setup, we set up an experimental setup that comprises an Microsoft Kinect v2 and two pan and tilt HS-785HB servos controlled with Arduino.
For ease of explanation, we assume internal and external parameters of the color and depth cameras are already known.
The checkerboard consists of 8 $\times$ 11 white-and-black checkers, each 100 mm $\times$ 100 mm in size.
We captured 28 frames by rotating the camera end-to-end in pan direction, and 11 frames in tilt rotation. With coordinates of 70 extracted 3D corners, we estimated parameter values in Equation 6 that form a linear function of multiple planes, using the least squares method. Then, we fitted corner trajectories to a circle with parameters in Equation 7.
In Table 1, we show values of 6 key rotation parameters that govern pan-tilt transformation of Equation 8.
4.2 Experiment Design
Here, we examine the performance of the proposed pan-tilt rotation model in terms of the accurate targeting capability.
The system is tasked to adjust its attitude so that the target point is identical to the optical of center of the camera.
We measure errors between the estimated pixels/vertices and actual captured pixels/vertices after the rotation.
The coordinates of the 70 checkerboard corners when the system is at its rest pose, i.e. pan=tilt=0, are used as target points, constituting 70 trials in total.
We compare our result with the result produced using the Single Point Calibration Method (SPCM) of [LZHT15] as the baseline.
To steer the system, we used inverse kinematics Algorithm 1 for the proposed method, while for SPCM pan, tilt values are calculated in closed form based on its geometrical model.
4.3 Results and Discussion
Figure 3 summarizes evaluation results of two models.
In [LZHT15], authors evaluated the model with only Root Mean Squared Errors (RMSE) of L2-norms in XY image pixels.
Since we have depth values available, we also collected RMSEs of real distances ($mm$), and additionally measured Mean Absolute Errors (MAE) in each X, Y (and Z) directions in both pixels and $mm$s for further analysis.
In the graph, the proposed method outperforms SPCM in every metric, especially in Z directions in real distances. This can be explained by the omission of non-origin rotation centers (see Table 1) in SPCM. Also, the rotation axes are assumed to be identical to world XY axes which lead to additional errors. The error differences become small in image planes. We explain this is due to normalizing effects of perspective projections onto the image plane.
Beside the comparative analysis, the results show seemingly high error patterns. We conjecture a number of factors have attributed to this.
First, there is the color-depth coordinate conversion. We detected checkerboard corners in color images, then converted them into depth image coordinates. Due to limitation in Kinect SDK, conversion in sub-pixel resolution was impossible, leading to error increases.
Second, servo rotation manipulation could be inaccurate, maybe due to unreliable kinematics of low-grade servos, such as hysteresis, jitters or inaccurate pulse width-angle mapping.
5 Conclusion and Future Work
In this paper, we have proposed an accurate controlling method for arbitrarily-assembled pan-tilt camera systems based on the rotation transformation model and its inverse kinematics.
The proposed model is capable of recovering rotation model parameters including positions and directions of axes of the pan-tilting platform in 3D space.
The comparative experiment demonstrates outperformance of the proposed method, in terms of accurately localizing target world points in local camera frames captured after rotations.
In following future work, we would like to extend the proposed operating mechanism to a full-scale camera pose-estimation framework that can be used in projection-based augmented reality or point cloud registration and 3D reconstruction. We would also like to delve into hardware limitations discussed in 4.3 and improve the pan-tilt model to compensate errors further.
6 Acknowledgment
This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIP) (No. NRF-2015R1A2A1A10055673).
References
[BCYH17]
Byun J., Chae S., Yang Y., Han T.:
Air: Anywhere immersive reality with user-perspective projection.
In 38th annual Eurographics conference (2017), The
Eurographics Association, pp. 5–8.
[Bus04]
Buss S. R.:
Introduction to inverse kinematics with jacobian transpose,
pseudoinverse and damped least squares methods.
IEEE Journal of Robotics and Automation 17, 1-19 (2004), 16.
[CDCZ14]
Chen P., Dai M., Chen K., Zhang Z.:
Rotation axis calibration of a turntable using constrained global
optimization.
Optik-International Journal for Light and Electron Optics 125,
17 (2014), 4831–4836.
[DC03]
Davis J., Chen X.:
Calibrating pan-tilt cameras in wide-area surveillance networks.
In In IEEE International Conference on Computer Vision (2003),
Citeseer.
[DCYH13]
Dai M., Chen L., Yang F., He X.:
Calibration of revolution axis for 360 deg surface measurement.
Applied optics 52, 22 (2013), 5440–5448.
[LZHT15]
Li Y., Zhang J., Hu W., Tian J.:
Method for pan-tilt camera calibration using single control point.
JOSA A 32, 1 (2015), 156–163.
[NLW${}^{*}$17]
Niu Z., Liu K., Wang Y., Huang S., Deng X., Zhang Z.:
Calibration method for the relative orientation between the rotation
axis and a camera using constrained global optimization.
Measurement Science and Technology 28, 5 (2017), 055001.
[Par11]
Park W.:
Philosophy and strategy of minimalism-based user created robots
(ucrs) for educational robotics-education, technology and business viewpoint.
International Journal of Robots, Education and Art 1, 1
(2011), 26–38.
[Sch06]
Schaffrin B.:
A note on constrained total least-squares estimation.
Linear algebra and its applications 417, 1 (2006), 245–258.
[WBIH12]
Wilson A., Benko H., Izadi S., Hilliges O.:
Steerable augmented reality with the beamatron.
In Proceedings of the 25th annual ACM symposium on User
interface software and technology (2012), ACM, pp. 413–422.
[WR13]
Wu Z., Radke R. J.:
Keeping a pan-tilt-zoom camera calibrated.
IEEE transactions on pattern analysis and machine intelligence
35, 8 (2013), 1994–2007. |
Measurement of the top quark pair production cross section in
$p\bar{p}$ collisions using multijet final states
B. Abbott
${}^{40}$
M. Abolins
${}^{37}$
V. Abramov
${}^{15}$
B.S. Acharya
${}^{8}$
I. Adam
${}^{39}$
D.L. Adams
${}^{48}$
M. Adams
${}^{24}$
S. Ahn
${}^{23}$
H. Aihara
${}^{17}$
G.A. Alves
${}^{2}$
N. Amos
${}^{36}$
E.W. Anderson
${}^{30}$
R. Astur
${}^{42}$
M.M. Baarmand
${}^{42}$
V.V. Babintsev
${}^{15}$
L. Babukhadia
${}^{16}$
A. Baden
${}^{33}$
B. Baldin
${}^{23}$
S. Banerjee
${}^{8}$
J. Bantly
${}^{45}$
E. Barberis
${}^{17}$
P. Baringer
${}^{31}$
J.F. Bartlett
${}^{23}$
A. Belyaev
${}^{14}$
S.B. Beri
${}^{6}$
I. Bertram
${}^{26}$
V.A. Bezzubov
${}^{15}$
P.C. Bhat
${}^{23}$
V. Bhatnagar
${}^{6}$
M. Bhattacharjee
${}^{42}$
N. Biswas
${}^{28}$
G. Blazey
${}^{25}$
S. Blessing
${}^{21}$
P. Bloom
${}^{18}$
A. Boehnlein
${}^{23}$
N.I. Bojko
${}^{15}$
F. Borcherding
${}^{23}$
C. Boswell
${}^{20}$
A. Brandt
${}^{23}$
R. Breedon
${}^{18}$
R. Brock
${}^{37}$
A. Bross
${}^{23}$
D. Buchholz
${}^{26}$
V.S. Burtovoi
${}^{15}$
J.M. Butler
${}^{34}$
W. Carvalho
${}^{2}$
D. Casey
${}^{37}$
Z. Casilum
${}^{42}$
H. Castilla-Valdez
${}^{11}$
D. Chakraborty
${}^{42}$
S.-M. Chang
${}^{35}$
S.V. Chekulaev
${}^{15}$
W. Chen
${}^{42}$
S. Choi
${}^{10}$
S. Chopra
${}^{36}$
B.C. Choudhary
${}^{20}$
J.H. Christenson
${}^{23}$
M. Chung
${}^{24}$
D. Claes
${}^{38}$
A.R. Clark
${}^{17}$
W.G. Cobau
${}^{33}$
J. Cochran
${}^{20}$
L. Coney
${}^{28}$
W.E. Cooper
${}^{23}$
C. Cretsinger
${}^{41}$
D. Cullen-Vidal
${}^{45}$
M.A.C. Cummings
${}^{25}$
D. Cutts
${}^{45}$
O.I. Dahl
${}^{17}$
K. Davis
${}^{16}$
K. De
${}^{46}$
K. Del Signore
${}^{36}$
M. Demarteau
${}^{23}$
D. Denisov
${}^{23}$
S.P. Denisov
${}^{15}$
H.T. Diehl
${}^{23}$
M. Diesburg
${}^{23}$
G. Di Loreto
${}^{37}$
P. Draper
${}^{46}$
Y. Ducros
${}^{5}$
L.V. Dudko
${}^{14}$
S.R. Dugad
${}^{8}$
A. Dyshkant
${}^{15}$
D. Edmunds
${}^{37}$
J. Ellison
${}^{20}$
V.D. Elvira
${}^{42}$
R. Engelmann
${}^{42}$
S. Eno
${}^{33}$
G. Eppley
${}^{48}$
P. Ermolov
${}^{14}$
O.V. Eroshin
${}^{15}$
V.N. Evdokimov
${}^{15}$
T. Fahland
${}^{19}$
M.K. Fatyga
${}^{41}$
S. Feher
${}^{23}$
D. Fein
${}^{16}$
T. Ferbel
${}^{41}$
G. Finocchiaro
${}^{42}$
H.E. Fisk
${}^{23}$
Y. Fisyak
${}^{43}$
E. Flattum
${}^{23}$
G.E. Forden
${}^{16}$
M. Fortner
${}^{25}$
K.C. Frame
${}^{37}$
S. Fuess
${}^{23}$
E. Gallas
${}^{46}$
A.N. Galyaev
${}^{15}$
P. Gartung
${}^{20}$
V. Gavrilov
${}^{13}$
T.L. Geld
${}^{37}$
R.J. Genik II
${}^{37}$
K. Genser
${}^{23}$
C.E. Gerber
${}^{23}$
Y. Gershtein
${}^{13}$
B. Gibbard
${}^{43}$
B. Gobbi
${}^{26}$
B. Gómez
${}^{4}$
G. Gómez
${}^{33}$
P.I. Goncharov
${}^{15}$
J.L. González Solís
${}^{11}$
H. Gordon
${}^{43}$
L.T. Goss
${}^{47}$
K. Gounder
${}^{20}$
A. Goussiou
${}^{42}$
N. Graf
${}^{43}$
P.D. Grannis
${}^{42}$
D.R. Green
${}^{23}$
H. Greenlee
${}^{23}$
S. Grinstein
${}^{1}$
P. Grudberg
${}^{17}$
S. Grünendahl
${}^{23}$
G. Guglielmo
${}^{44}$
J.A. Guida
${}^{16}$
J.M. Guida
${}^{45}$
A. Gupta
${}^{8}$
S.N. Gurzhiev
${}^{15}$
G. Gutierrez
${}^{23}$
P. Gutierrez
${}^{44}$
N.J. Hadley
${}^{33}$
H. Haggerty
${}^{23}$
S. Hagopian
${}^{21}$
V. Hagopian
${}^{21}$
K.S. Hahn
${}^{41}$
R.E. Hall
${}^{19}$
P. Hanlet
${}^{35}$
S. Hansen
${}^{23}$
J.M. Hauptman
${}^{30}$
D. Hedin
${}^{25}$
A.P. Heinson
${}^{20}$
U. Heintz
${}^{23}$
R. Hernández-Montoya
${}^{11}$
T. Heuring
${}^{21}$
R. Hirosky
${}^{24}$
J.D. Hobbs
${}^{42}$
B. Hoeneisen
${}^{4,*}$
J.S. Hoftun
${}^{45}$
F. Hsieh
${}^{36}$
Ting Hu
${}^{42}$
Tong Hu
${}^{27}$
A.S. Ito
${}^{23}$
E. James
${}^{16}$
J. Jaques
${}^{28}$
S.A. Jerger
${}^{37}$
R. Jesik
${}^{27}$
T. Joffe-Minor
${}^{26}$
K. Johns
${}^{16}$
M. Johnson
${}^{23}$
A. Jonckheere
${}^{23}$
M. Jones
${}^{22}$
H. Jöstlein
${}^{23}$
S.Y. Jun
${}^{26}$
C.K. Jung
${}^{42}$
S. Kahn
${}^{43}$
G. Kalbfleisch
${}^{44}$
D. Karmanov
${}^{14}$
D. Karmgard
${}^{21}$
R. Kehoe
${}^{28}$
M.L. Kelly
${}^{28}$
S.K. Kim
${}^{10}$
B. Klima
${}^{23}$
C. Klopfenstein
${}^{18}$
W. Ko
${}^{18}$
J.M. Kohli
${}^{6}$
D. Koltick
${}^{29}$
A.V. Kostritskiy
${}^{15}$
J. Kotcher
${}^{43}$
A.V. Kotwal
${}^{39}$
A.V. Kozelov
${}^{15}$
E.A. Kozlovsky
${}^{15}$
J. Krane
${}^{38}$
M.R. Krishnaswamy
${}^{8}$
S. Krzywdzinski
${}^{23}$
S. Kuleshov
${}^{13}$
Y. Kulik
${}^{42}$
S. Kunori
${}^{33}$
F. Landry
${}^{37}$
G. Landsberg
${}^{45}$
B. Lauer
${}^{30}$
A. Leflat
${}^{14}$
J. Li
${}^{46}$
Q.Z. Li-Demarteau
${}^{23}$
J.G.R. Lima
${}^{3}$
D. Lincoln
${}^{23}$
S.L. Linn
${}^{21}$
J. Linnemann
${}^{37}$
R. Lipton
${}^{23}$
F. Lobkowicz
${}^{41}$
S.C. Loken
${}^{17}$
A. Lucotte
${}^{42}$
L. Lueking
${}^{23}$
A.L. Lyon
${}^{33}$
A.K.A. Maciel
${}^{2}$
R.J. Madaras
${}^{17}$
R. Madden
${}^{21}$
L. Magaña-Mendoza
${}^{11}$
V. Manankov
${}^{14}$
S. Mani
${}^{18}$
H.S. Mao
${}^{23,{\dagger}}$
R. Markeloff
${}^{25}$
T. Marshall
${}^{27}$
M.I. Martin
${}^{23}$
K.M. Mauritz
${}^{30}$
B. May
${}^{26}$
A.A. Mayorov
${}^{15}$
R. McCarthy
${}^{42}$
J. McDonald
${}^{21}$
T. McKibben
${}^{24}$
J. McKinley
${}^{37}$
T. McMahon
${}^{44}$
H.L. Melanson
${}^{23}$
M. Merkin
${}^{14}$
K.W. Merritt
${}^{23}$
C. Miao
${}^{45}$
H. Miettinen
${}^{48}$
A. Mincer
${}^{40}$
C.S. Mishra
${}^{23}$
N. Mokhov
${}^{23}$
N.K. Mondal
${}^{8}$
H.E. Montgomery
${}^{23}$
P. Mooney
${}^{4}$
J. Moromisato
${}^{35}$
M. Mostafa
${}^{1}$
H. da Motta
${}^{2}$
C. Murphy
${}^{24}$
F. Nang
${}^{16}$
M. Narain
${}^{23}$
V.S. Narasimham
${}^{8}$
A. Narayanan
${}^{16}$
H.A. Neal
${}^{36}$
J.P. Negret
${}^{4}$
P. Nemethy
${}^{40}$
D. Norman
${}^{47}$
L. Oesch
${}^{36}$
V. Oguri
${}^{3}$
E. Oliveira
${}^{2}$
E. Oltman
${}^{17}$
N. Oshima
${}^{23}$
D. Owen
${}^{37}$
P. Padley
${}^{48}$
A. Para
${}^{23}$
Y.M. Park
${}^{9}$
R. Partridge
${}^{45}$
N. Parua
${}^{8}$
M. Paterno
${}^{41}$
B. Pawlik
${}^{12}$
J. Perkins
${}^{46}$
M. Peters
${}^{22}$
R. Piegaia
${}^{1}$
H. Piekarz
${}^{21}$
Y. Pischalnikov
${}^{29}$
B.G. Pope
${}^{37}$
H.B. Prosper
${}^{21}$
S. Protopopescu
${}^{43}$
J. Qian
${}^{36}$
P.Z. Quintas
${}^{23}$
R. Raja
${}^{23}$
S. Rajagopalan
${}^{43}$
O. Ramirez
${}^{24}$
S. Reucroft
${}^{35}$
M. Rijssenbeek
${}^{42}$
T. Rockwell
${}^{37}$
M. Roco
${}^{23}$
P. Rubinov
${}^{26}$
R. Ruchti
${}^{28}$
J. Rutherfoord
${}^{16}$
A. Sánchez-Hernández
${}^{11}$
A. Santoro
${}^{2}$
L. Sawyer
${}^{32}$
R.D. Schamberger
${}^{42}$
H. Schellman
${}^{26}$
J. Sculli
${}^{40}$
E. Shabalina
${}^{14}$
C. Shaffer
${}^{21}$
H.C. Shankar
${}^{8}$
R.K. Shivpuri
${}^{7}$
D. Shpakov
${}^{42}$
M. Shupe
${}^{16}$
H. Singh
${}^{20}$
J.B. Singh
${}^{6}$
V. Sirotenko
${}^{25}$
E. Smith
${}^{44}$
R.P. Smith
${}^{23}$
R. Snihur
${}^{26}$
G.R. Snow
${}^{38}$
J. Snow
${}^{44}$
S. Snyder
${}^{43}$
J. Solomon
${}^{24}$
M. Sosebee
${}^{46}$
N. Sotnikova
${}^{14}$
M. Souza
${}^{2}$
G. Steinbrück
${}^{44}$
R.W. Stephens
${}^{46}$
M.L. Stevenson
${}^{17}$
D. Stewart
${}^{36}$
F. Stichelbaut
${}^{42}$
D. Stoker
${}^{19}$
V. Stolin
${}^{13}$
D.A. Stoyanova
${}^{15}$
M. Strauss
${}^{44}$
K. Streets
${}^{40}$
M. Strovink
${}^{17}$
A. Sznajder
${}^{2}$
P. Tamburello
${}^{33}$
J. Tarazi
${}^{19}$
M. Tartaglia
${}^{23}$
T.L.T. Thomas
${}^{26}$
J. Thompson
${}^{33}$
T.G. Trippe
${}^{17}$
P.M. Tuts
${}^{39}$
V. Vaniev
${}^{15}$
N. Varelas
${}^{24}$
E.W. Varnes
${}^{17}$
D. Vititoe
${}^{16}$
A.A. Volkov
${}^{15}$
A.P. Vorobiev
${}^{15}$
H.D. Wahl
${}^{21}$
G. Wang
${}^{21}$
J. Warchol
${}^{28}$
G. Watts
${}^{45}$
M. Wayne
${}^{28}$
H. Weerts
${}^{37}$
A. White
${}^{46}$
J.T. White
${}^{47}$
J.A. Wightman
${}^{30}$
S. Willis
${}^{25}$
S.J. Wimpenny
${}^{20}$
J.V.D. Wirjawan
${}^{47}$
J. Womersley
${}^{23}$
E. Won
${}^{41}$
D.R. Wood
${}^{35}$
Z. Wu
${}^{23,{\dagger}}$
R. Yamada
${}^{23}$
P. Yamin
${}^{43}$
T. Yasuda
${}^{35}$
P. Yepes
${}^{48}$
K. Yip
${}^{23}$
C. Yoshikawa
${}^{22}$
S. Youssef
${}^{21}$
J. Yu
${}^{23}$
Y. Yu
${}^{10}$
B. Zhang
${}^{23,{\dagger}}$
Y. Zhou
${}^{23,{\dagger}}$
Z. Zhou
${}^{30}$
Z.H. Zhu
${}^{41}$
M. Zielinski
${}^{41}$
D. Zieminska
${}^{27}$
A. Zieminski
${}^{27}$
E.G. Zverev
${}^{14}$
and A. Zylberstejn${}^{5}$
(DØ Collaboration)
${}^{1}$Universidad de Buenos Aires, Buenos Aires, Argentina
${}^{2}$LAFEX, Centro Brasileiro de Pesquisas Físicas,
Rio de Janeiro, Brazil
${}^{3}$Universidade do Estado do Rio de Janeiro,
Rio de Janeiro, Brazil
${}^{4}$Universidad de los Andes, Bogotá, Colombia
${}^{5}$DAPNIA/Service de Physique des Particules, CEA, Saclay,
France
${}^{6}$Panjab University, Chandigarh, India
${}^{7}$Delhi University, Delhi, India
${}^{8}$Tata Institute of Fundamental Research, Mumbai, India
${}^{9}$Kyungsung University, Pusan, Korea
${}^{10}$Seoul National University, Seoul, Korea
${}^{11}$CINVESTAV, Mexico City, Mexico
${}^{12}$Institute of Nuclear Physics, Kraków, Poland
${}^{13}$Institute for Theoretical and Experimental Physics,
Moscow, Russia
${}^{14}$Moscow State University, Moscow, Russia
${}^{15}$Institute for High Energy Physics, Protvino, Russia
${}^{16}$University of Arizona, Tucson, Arizona 85721
${}^{17}$Lawrence Berkeley National Laboratory and University of
California, Berkeley, California 94720
${}^{18}$University of California, Davis, California 95616
${}^{19}$University of California, Irvine, California 92697
${}^{20}$University of California, Riverside, California 92521
${}^{21}$Florida State University, Tallahassee, Florida 32306
${}^{22}$University of Hawaii, Honolulu, Hawaii 96822
${}^{23}$Fermi National Accelerator Laboratory, Batavia,
Illinois 60510
${}^{24}$University of Illinois at Chicago, Chicago,
Illinois 60607
${}^{25}$Northern Illinois University, DeKalb, Illinois 60115
${}^{26}$Northwestern University, Evanston, Illinois 60208
${}^{27}$Indiana University, Bloomington, Indiana 47405
${}^{28}$University of Notre Dame, Notre Dame, Indiana 46556
${}^{29}$Purdue University, West Lafayette, Indiana 47907
${}^{30}$Iowa State University, Ames, Iowa 50011
${}^{31}$University of Kansas, Lawrence, Kansas 66045
${}^{32}$Louisiana Tech University, Ruston, Louisiana 71272
${}^{33}$University of Maryland, College Park, Maryland 20742
${}^{34}$Boston University, Boston, Massachusetts 02215
${}^{35}$Northeastern University, Boston, Massachusetts 02115
${}^{36}$University of Michigan, Ann Arbor, Michigan 48109
${}^{37}$Michigan State University, East Lansing, Michigan 48824
${}^{38}$University of Nebraska, Lincoln, Nebraska 68588
${}^{39}$Columbia University, New York, New York 10027
${}^{40}$New York University, New York, New York 10003
${}^{41}$University of Rochester, Rochester, New York 14627
${}^{42}$State University of New York, Stony Brook,
New York 11794
${}^{43}$Brookhaven National Laboratory, Upton, New York 11973
${}^{44}$University of Oklahoma, Norman, Oklahoma 73019
${}^{45}$Brown University, Providence, Rhode Island 02912
${}^{46}$University of Texas, Arlington, Texas 76019
${}^{47}$Texas A&M University, College Station, Texas 77843
${}^{48}$Rice University, Houston, Texas 77005
Abstract
We have studied $t\bar{t}$ production using multijet final states
in $p\bar{p}$ collisions at a center-of-mass energy of 1.8 TeV,
with an integrated luminosity of 110.3 pb${}^{-1}$. Each of the top quarks
with these final states decays exclusively to a bottom quark and a $W$ boson,
with the $W$ bosons decaying into quark-antiquark pairs. The analysis
has been optimized using neural networks to achieve the smallest expected
fractional uncertainty on the $t\bar{t}$ production
cross section, and yields a cross section
of 7.1 $\pm$ 2.8 (stat) $\pm$ 1.5 (syst) pb, assuming a
top quark mass of 172.1 GeV/$c^{2}$.
Combining this result with previous DØ measurements, where one
or both of the $W$ bosons decay leptonically, gives
a $t\bar{t}$ production cross section of 5.9 $\pm$ 1.2 (stat) $\pm$ 1.1 (syst) pb.
Contents
I Introduction
II Outline of Method
III The DØ Detector
III.1 Tracking system
III.2 Calorimeter
III.3 Muon spectrometer
IV Data Sample
IV.1 Initial selection criteria
IV.2 Jet algorithms
IV.3 Jet energy correction
IV.4 Characteristics of jets
IV.5 Simulation of $t\bar{t}$ events
V Kinematic Parameters
V.1 Parameters sensitive to energy scale
V.2 Parameters sensitive to additional radiation
V.3 Aplanarity and sphericity
V.4 Parameters sensitive to rapidity distributions
VI Event Structure Variables
VI.1 $p_{T}$ of tagging muon
VI.2 Widths of jets
VI.3 Mass likelihood parameter
VI.4 Correlations between parameters
VII Analysis
VII.1 Event selection criteria
VII.2 Muon tagging
VII.3 Muon tagging rates
VII.4 Background modeling
VII.5 Neural network analysis
VII.6 Cross section using neural network fits
VII.7 Cross section using counting method
VII.8 Double-tagged events
VII.9 Corrections and uncertainties
VII.10 Measured cross section
VII.11 Significance of signal
VIII Summary
I Introduction
In the standard model, the top quark decays to a $b$ quark and a $W$ boson,
and the dominant decay of the $W$ boson is into a quark-antiquark pair.
Events with a $t\bar{t}$ pair can have both $W$ bosons
decaying to quarks. This is referred to
as the “all-jets” channel, and is expected to
account for 44% of the $t\bar{t}$ production cross section.
The observation of top quark production [3, 4] in the channels
involving one or two leptons motivates us to investigate $t\bar{t}$ decays into
other channels. DØ has measured a top quark mass, $m_{t}$, of
172.1 $\pm$ 5.2 (stat) $\pm$ 4.9 (syst) GeV/$c^{2}$ [5]
and a $t\bar{t}$ production cross section of 5.6 $\pm$ 1.4 (stat) $\pm$ 1.2 (syst)
pb [6], while CDF has measured a mass of
175.9 $\pm$ 4.8 (stat) $\pm$ 4.9 (syst) GeV/$c^{2}$ [7]
and a $t\bar{t}$ production cross section of 7.6 ${}^{+1.8}_{-1.5}$ pb [8].
Recently, CDF has reported on the all-jets channel [9], and
finds the $t\bar{t}$ production cross section to be 10.1 ${}^{+4.5}_{-3.6}$ pb
and a top quark mass of 186 $\pm$ 10 (stat) $\pm$ 12 (syst) GeV/$c^{2}$.
The work presented here is based on 110.3 $\pm$ 5.8 pb${}^{-1}$ of
data recorded between August 1992 and February 1996 at the Fermilab Tevatron
collider, with a $p\bar{p}$ center-of-mass energy of 1.8 TeV.
Assuming the branching ratio and cross section predicted by the standard
model, we expect approximately
200 $t\bar{t}\rightarrow$ all-jets
events in this data sample.
The signature for $t\bar{t}$ production in the all-jets channel is
six or more high transverse momentum jets with kinematic properties consistent
with the top quark decay hypothesis. At least two of these jets originate from
$b$ quarks. The background to this signature
consists of events from other processes that can also produce six
or more jets. The $t\bar{t}$ channel is one of the few examples of multijet
final states that are dominated by quarks rather than gluons.
This fact has motivated us to include the characteristic differences
between quark and gluon jets in separating the top quark to all-jets signal
from background.
Interest in the all-jets decay channel of top quarks also stems from the
fact that, without any unobserved particles in the final-state, the
all-jets mode is the most kinematically constrained of all the top quark decay
channels. Furthermore, since the top quark is quite massive, decays via charged
Higgs may be possible. If channels such as $t\rightarrow H^{+}b$ have
a significant branching fraction, the main effect could be a deficit in
the $t\bar{t}$ final states with energetic electrons or muons, relative
to the all-jets channel.
II Outline of Method
The search for the top quark in the all-jets channel began with the imposition
of preliminary selection criteria at the trigger stage, followed by
more stringent criteria in the offline analysis. As these initial
criteria were not very restrictive, the observed cross section,
primarily from QCD processes, was more than three thousand times larger than
the expected signal. The principal
challenge in the search was to develop a set of selection criteria that could
significantly improve the signal-to-background ratio, and provide an
estimate of the background remaining after imposing any selection requirements.
The data sample consisted of over 600,000 events after the initial selection
criteria. Because of the small number of $t\bar{t}$ events expected in
the presence of this large
background, and with only modest discrimination in any single
kinematic or topological property, traditional methods of analysis
were inadequate. The analysis would have to involve many variables,
which are likely to be highly correlated.
Neural networks were chosen as the
appropriate tool for handling many variables simultaneously.
The analysis relied on Monte Carlo simulations to model the properties
of $t\bar{t}$ events. These simulations were performed for
different top quark masses, and the final results interpolated to the mass
measured by the DØ collaboration.
We note that the $t\bar{t}$ detection efficiency is not
strongly dependent on the assumed mass of the top quark.
In contrast, the background model was determined entirely
from data. An advantage of the overwhelming background-to-signal ratio is that
the data provide an almost pure background sample.
This approach obviates a number of concerns when calculating the
background. The background is predominantly QCD multijet production, which
involves higher-order processes that may not be well modeled in a
Monte Carlo simulation. Furthermore, detector effects are
implicitly included when data are employed for the model of the background.
Soft-lepton tagging, using muons embedded in jets, serves as a possible
signature for the presence of a $b$ quark within the jet, and
is referred to as $b$-tagging. By identifying the muon from the
semileptonic decay of a $b$ quark (or the sequential decay),
$b$-tagging of jets improves the signal-to-background ratio significantly.
The $t\bar{t}$ events are tagged roughly 20% of the time,
whereas the tag rate for QCD multijet events with similar requirements
is about 3%. Requiring the presence of a muon tag in the
event therefore provides nearly a factor of ten in background rejection and
a method to estimate this background.
The background calculation relied on being able to predict the number of
events that are $b$-tagged, based on events without such tags.
To make the untagged data represent the background in this analysis,
a way of estimating the tagging rate in QCD events was needed. This was done
by constructing a “tag rate” function, determined from data, that is applied
to each jet separately. This function is simply the probability
for any individual jet to have a muon tag.
Application of the tag rate function to each jet in untagged events gives
the background model for our final event sample.
The presence of $t\bar{t}$ signal was identified by
an excess observed in the data above this background. This excess should be
small in the regions of the neural network output where background dominates,
but should be enhanced where significant signal is expected.
This analysis employed two neural networks to extract the final $t\bar{t}$
cross section. The first had as its input variables those parameters involving
kinematic and topological properties of the events that were
highly correlated. The output of this neural network was used as an input
variable to a second neural network, along with three other inputs.
These three inputs were the transverse momentum ($p_{T}$)
of the tagging muon, a discriminant based on the widths of the jets,
and a likelihood variable that parameterized the degree to which an event
was consistent with the $t\bar{t}$ decay hypothesis. These three variables
were less correlated than the kinematic variables used in the first neural
network. The $t\bar{t}$ cross section was determined from the output of this
second neural network by fitting the neural network output distributions of
the signal and background outputs to the observed data.
III The DØ Detector
DØ is a multipurpose detector designed to study $p\bar{p}$ collisions
at the Fermilab Tevatron Collider. The detector
was commissioned during the summer of 1992. A full description
of the detector can be found in references [10, 11].
Here we describe the properties of the detector that are most
relevant to the search in the all-jets channel. An isometric
view of the detector is shown in Fig. 1.
III.1 Tracking system
The tracking system consists of a vertex drift chamber,
a transition radiation detector, a central drift chamber, and
two forward drift chambers. The system provides
charged-particle tracking over the pseudorapidity region $|\eta|<$ 3.2,
where $\eta$ = $-\ln({\tan{(\theta/2)}})$; $\theta$ and $\phi$ are,
respectively, the polar and azimuthal angles relative to the proton beam axis.
The resolution for charged particles is 2.5 mrad in $\phi$ and 28 mrad in
$\theta$. The position of the interaction vertex along the beam direction
($z$) is determined typically to an accuracy of 8 mm.
III.2 Calorimeter
The liquid-argon calorimeter, using uranium and stainless steel/copper absorber,
is divided into three parts: a central calorimeter
and two end calorimeters. Each part consists of an inner electromagnetic
section, a fine hadronic section, and a coarse hadronic section, housed in a
stainless steel cryostat. The intercryostat detector
consists of scintillator tiles
inserted in the space between the central and end calorimeter
cryostats.
In addition, “massless gaps”,
installed inside both central and end calorimeters, are
active readout cells, without absorber material, located inside the
cryostat adjacent to the cryostat walls.
The intercryostat detector and massless gaps
improve the energy resolution for jets that straddle two cryostats. The
calorimeter covers the pseudorapidity range $|\eta|<$ 4.2, and has a typical
segmentation of
0.1 $\times$ 0.1 in $\Delta\eta$ $\times$ $\Delta\phi$. The energy
resolution is $\delta(E)/E$ = 15%/$\sqrt{E{\rm(GeV)}}$ $\oplus$ 0.4% for
electrons. For charged pions, the resolution is approximately
50%/$\sqrt{E{\rm(GeV)}}$, and for jets approximately
80%/$\sqrt{E{\rm(GeV)}}$ [10, 11].
As can be seen in Fig. 1, the Main Ring beam pipe penetrates
the outer hadronic section of the calorimeters and the muon spectrometer.
The Main Ring carries protons with energies between 8-150 GeV, and is
used in antiproton production during the Tevatron $p\bar{p}$ running.
Because of this, any losses from the Main Ring can produce
backgrounds in the detector that must be removed.
III.3 Muon spectrometer
The DØ experiment detects muons using proportional drift tubes (PDT) and an
iron toroid. Because muons from top quark decays populate predominantly the
central region, this analysis uses muon detection systems in the region
$|\eta|<$ 1.
The combined material in the calorimeter and iron toroid has between
13 and 19 interaction lengths (the range-out energy for muons is approximately
3.5 GeV), making background from hadronic punchthrough
negligible. Also, the small central tracking volume minimizes background
from in-flight decays of pions and kaons.
A typical muon track is measured in four layers of PDTs before,
and six layers after, the iron toroid. The six layers are constructed
in two super-layers that are separated by about one meter to provide
a good lever arm for measuring the muon momentum, $p$.
The muon momentum is determined from its deflection angle in the magnetic
field of the toroid. The momentum resolution is limited by multiple
scattering in the traversed material, knowledge of the
integrated magnetic field,
and resolution on the measurement of the deflection angle. The resolution
is roughly Gaussian in 1/$p$, and is approximately
$\delta(1/p)$ = 0.18($p$-2)/$p^{2}\oplus$0.003 (with $p$ in GeV/$c$) for
the algorithms that were used in this analysis.
IV Data Sample
This section describes the data sample and the simulated events for the
$t\bar{t}$ signal used in our analysis.
IV.1 Initial selection criteria
The data sample was selected by imposing both hardware
(Level 1) and software (Level 2) trigger requirements.
These requirements were modified slightly over the course of the 1992–1996 run
in order to accommodate the higher instantaneous luminosities later in the run.
Table I indicates the three main running periods, the run numbers
associated with these periods, and the integrated luminosity collected.
The hardware trigger required the presence of at least four
calorimeter trigger towers
(0.2 $\times$ 0.2 in $\Delta\eta$ $\times$ $\Delta\phi$),
each with transverse energy $E_{T}>$ 5 GeV, for the Ia period.
In the Ib and Ic periods, the $E_{T}$ requirement was raised to 7 GeV, and
an additional requirement for at least three large tiles
(0.8 $\times$ 1.6 in $\Delta\eta$ $\times$ $\Delta\phi$)
with $E_{T}>$ 15 GeV was imposed. These were imposed to reduce the
trigger rate and avoid saturating the bandwidth of the trigger system at
high instantaneous luminosities ($\geq$ 10${}^{31}$ cm${}^{-2}$ s${}^{-1}$).
The software filter required five jets, defined by
$\cal{R}$=$\sqrt{(\Delta\eta)^{2}+(\Delta\phi)^{2}}$=0.3 cones,
with $|\eta|<$ 2.5 and $E_{T}>$ 10 GeV. Again, in order to reduce the data
rate at high luminosities during the Ib period, a further condition was added
requiring the scalar sum of the $E_{T}$ of all jets (defined as $H_{T}$)
to be greater than 110 or 115 GeV, depending upon run number.
This $H_{T}$ requirement was raised to 120 GeV during the Ic period.
The effects of these changes on the acceptance for $t\bar{t}$ events were
studied using Monte Carlo simulations, and were found to be negligible.
In addition to imposing trigger and filter requirements, a set of offline
selection criteria was used to reduce the data sample to a manageable size
without greatly affecting the acceptance for the $t\bar{t}$ signal. First, $H_{T}$ was
required to be greater than 115 GeV, where the sum used ${\cal{R}}$=0.5 jets
with $|\eta|<$ 2.5 and $E_{T}>$ 8 GeV. Also, requirements were imposed
in order to eliminate events with spurious jets due to spray from the Main Ring
or effects from noisy cells in the calorimeter [12, 13].
For example, Fig. 2 shows the imbalance in transverse energy, or
missing $E_{T}$ (${\hbox{$E$\kern-6.0pt\lower-0.43pt\hbox{/}}}_{T}$), in the event versus the azimuthal angle ($\phi$) of the
jet, before and after the rejection of Main Ring events. We see that our
requirements have removed the spurious cluster of jets in the region where the
Main Ring pierces the DØ detector (1.6 $<\phi<$ 1.8). Table II summarizes
the impact of the trigger and initial reconstruction criteria on the
$t\bar{t}$ signal for a top quark mass of 180 GeV/$c^{2}$.
IV.2 Jet algorithms
The jet algorithm is the fundamental analysis tool in the search for
$t\bar{t}$ events in the all-jets mode. One of the most important
considerations in choosing a jet algorithm is the efficiency
for reconstructing the six primary $t\bar{t}$ decay products.
The $\eta$ distribution of the jets from $t\bar{t}$ decays tends to be
quite narrow, and therefore the $\cal{R}$ separation between
adjacent jets is frequently small. When two jets are too close together,
they may not be resolved, leading to reconstruction inefficiency.
Figure 3 shows the reconstruction efficiency for the cone
jet algorithm [14] with various cone sizes
for simulated $t\bar{t}$ events in the all-jets
channel, as generated with the herwig Monte Carlo program[15].
Here, the definition of a quark includes any final state gluon radiation
added back to the quark momentum. The matching of reconstructed jets to
quarks relies on using combinations of the two that minimize the distance in
$\cal{R}$ between them. A jet is considered to be matched only if
that distance is less than $\Delta\cal{R}$=0.5, the energy of the jet is
within a factor of two of the quark energy, and the
reconstructed jet $E_{T}$ is greater than 10 GeV.
Figures 3(a) and 3(b) show how the reconstruction
efficiency depends on quark $E_{T}$ and
$\eta$ for the cone algorithm with different cone sizes.
The $\cal{R}$=0.3 cone algorithm shows a higher
jet reconstruction efficiency than the larger cone algorithms.
In the central region, the $\cal{R}$=0.3 cone algorithm has an efficiency
of 94%, while the $\cal{R}$=0.5 and $\cal{R}$=0.7
cone algorithms are 90% and 81% efficient, respectively.
Given an average efficiency $\epsilon$ for reconstructing a single jet, the
reconstruction efficiency for finding $t\bar{t}$ events (with six or more jets)
will be of the order of $\epsilon^{6}$. Therefore, larger cone sizes are
less efficient in the multijet environment.
Figure 3(c) shows the correspondence between parton
and jet energies found for various cone algorithms, after DØ jet energy
corrections are applied (see next section). Linear fits to the quark-jet
correlation in energy are shown in Fig. 3(c) for the three cone
algorithms. Figure 3(d) shows the three-jet invariant mass for the
correct combinations of jets matching top and antitop quarks. The areas of the
mass distributions reflect the event reconstruction efficiencies for different
algorithms.
The shift in the reconstructed mass from the input mass of the top quark
(175 GeV/$c^{2}$) shows that the jet algorithms are not equivalent. The shift
in three-jet mass from the nominal input top quark mass increases as the
cone radius is decreased. The widths of the mass distributions are not very
sensitive to the choice of cone size. The overall root-mean-square, rms,
spread in reconstructed mass for correct combinations of jets is approximately
10% of the mass.
In summary, there are two competing effects when choosing the optimal jet cone
size. Smaller cone sizes are better able to resolve separate jets, but do not
do as well at reconstructing jet energy. However, the ability to resolve
individual jets was deemed of higher importance in the search for a signal.
Hence the $\cal{R}$=0.3 cone algorithm is preferred for analyzing multijet
events. But, due to the relatively large shift in the jet energy
for the $\cal{R}$=0.3 cone algorithm, we chose to use the $\cal{R}$=0.5 cone
algorithm for calculating some quantities that emphasize energy response at the
expense of jet efficiency. Jets with $E_{T}~{}<$ 8 GeV, before application of
energy corrections (see Sec. IV.C), were discarded.
IV.3 Jet energy correction
DØ has developed a correction procedure [16] to calibrate jet energies,
which is applied to both data and Monte Carlo. The underlying assumption is
that the true jet energy, $E_{\rm ptcl}$, is the sum of the energies of all
final state particles entering the cone algorithm applied at the calorimeter
level. $E_{\rm ptcl}$ is obtained from the energy measured in the calorimeter,
$E_{\rm meas}$, as follows:
$$\displaystyle E_{\rm ptcl}=\frac{E_{\rm meas}-E_{\rm O}({\cal R},\eta,{\cal L}%
)}{R(\eta,E,{\sc rms})S({\cal R},\eta,E)}\,,$$
(1)
where:
•
$E_{\rm O}({\cal R},\eta,{\cal L})$
is an offset, which includes the physics of the underlying event, noise from
the radioactive decay of the uranium absorber,
the effect of previous crossings (pile-up), and the
contribution of additional contemporaneous $p\bar{p}$ interactions.
The physics of the underlying
event is defined as the energy contributed by spectators to the hard parton
interaction which resulted in
the high-$p_{T}$ event.
This offset increases as a function of the cone size ${\cal R}$.
It also depends on $\eta$ and on the instantaneous luminosity, ${\cal L}$,
which is related to the contribution from
the additional $p\bar{p}$ interactions.
•
$R(\eta,E,{\sc rms})$ is the energy response of the calorimeter.
It is nearly independent of the jet cone size, ${\cal R}$, but does depend
on the rms width of the jet. The width dependence accounts for
differences in the calorimeter response to narrow jets, which fragmented
into fewer particles (of, on average, higher energy) than broader jets,
with larger particle multiplicities. Because the various detector components
are not identical, $R$ also depends on detector $\eta$. $R$ is typically less
than one, due to energy loss in the uninstrumented regions between modules,
differences between the electromagnetic ($e$) and hadronic response ($h$) of
the detector ($e/h>1$), and module-to-module inhomogeneities.
•
$S({\cal R},\eta,E)$ is the fraction of the jet
energy that is deposited inside the algorithm cone. Since the jet energy is
corrected back to the particle level, the effects of calorimeter showering
must be removed. $S$ is less than one, meaning that the effect of
showering is a net flux of energy from inside to outside the cone.
$S$ depends strongly on the cone size ${\cal R}$, energy, and $\eta$.
IV.4 Characteristics of jets
Comparisons of jet properties (jet multiplicity, inclusive jet $E_{T}$,
$\eta$, and $\phi$, for $\cal{R}$=0.3 cones)
are shown in Fig. 4 for data from the Ia and Ib periods
(see Table I) and for
$t\bar{t}$ Monte Carlo.
Only jets with $E_{T}>$ 10 GeV and $|\eta|<2$ are
included in the comparison. The results from Ia and Ib are in good agreement,
although Ib typically had higher instantaneous luminosity.
Figure 4(a) shows that for events with six jets,
the background (i.e., data) is at least three orders of magnitude
larger than the expected $t\bar{t}$ signal. The peak at five jets is the
result of the initial event selection (see Table II). The inclusive jet $E_{T}$
spectrum in Fig. 4(b) falls exponentially at about the same rate
for signal as for data, and the signal is consistently three orders of
magnitude below the data. In Fig. 4(c), the distributions of
jet $\eta$ are normalized to the same area for signal and data. The signal
is concentrated in the central region, while the data extend to higher $\eta$.
There is a difference of the order of 10% between Ia and Ib in the
intercryostat region ($|\eta|\approx$ 1.2) due to improvements in the Ib
period. Figure 4(d) shows that the $\phi$ distribution of jets is
isotropic, except for a 5% suppression in the region of the Main Ring.
The Monte Carlo does not simulate the effects of the
Main Ring, and consequently has no apparent structure in $\phi$.
IV.5 Simulation of $t\bar{t}$ events
The simulation of $t\bar{t}$ events plays an important role in
extracting a signal in the presence of significant
background. It is necessary, therefore, to have a
good description of the production and decay of $t\bar{t}$ events
in order to calculate detector acceptances accurately and to develop
methods to identify $t\bar{t}$ events in the data.
The $t\bar{t}$ events were generated for top quark masses between 120 and
220 GeV/$c^{2}$ for the reaction $p\bar{p}\rightarrow t\bar{t}+X$
using herwig as a primary model and isajet [18] as a
check. The underlying assumptions in the fragmentation of partons are
different in the two programs. The generated events were put through the
DØ shower library [19], a fast detector simulation package
based on geant [20], which contains the effects of
cracks and other dead material in the DØ calorimeter, and provides
accurate shower simulation. The geant simulation has been tuned
to achieve a good match between generated single-particle characteristics
and observed data [21]. Events were subsequently digitized, passed
through the DØ reconstruction program [10], and subjected
to the same selection criteria as the data (see Table II). Events passing
these criteria served as the model for our studies of $t\bar{t}$ properties.
Generally, acceptances for $t\bar{t}$ production as calculated with
herwig or isajet agree to within 10%, and any differences
between the two are incorporated in the final systematic uncertainties.
As an illustration of the discrepancies, we show in Fig. 5
distributions of jet multiplicity, jet $\eta$, the $E_{T}$ of the leading jet,
and the fifth highest jet $E_{T}$ for herwig and isajet. Except
for jet multiplicity, these distributions are in good agreement. It has been
shown [11] that isajet produces more gluon radiation than
herwig, in accord with our results in Fig. 5(a).
V Kinematic Parameters
The principal background to the $t\bar{t}$ signal is QCD multijet production,
which is dominated by a $2\rightarrow 2$ parton process with additional
jets produced through gluon radiation. Therefore, the background tends to have
jets that are more forward-backward in rapidity. The additional jets are
generally lower in $E_{T}$ (i.e., softer) than the initial outgoing parent
partons. Furthermore, this extra radiation tends to lie in
a plane formed by the incoming beam and the two leading jets.
Because the mass of the top quark is large, the characteristic energy scale
(commonly called $Q^{2}$) of the $t\bar{t}$ event is significantly larger than
that of the average QCD background event. This means that $t\bar{t}$ events
generally have jets with higher $E_{T}$, and have larger multijet invariant
masses.
Extracting a signal from data dominated by background requires
the use of global kinematic parameters based on these differences.
Employing such parameters helps to differentiate between
the $t\bar{t}$ signal and background. We can summarize the salient
features of the background, relative to the $t\bar{t}$ signal, as follows:
•
The overall energy scale is lower; leading jets have lower
$E_{T}$; multijet invariant masses are smaller.
•
The additional radiated jets are softer (have lower $E_{T}$).
•
The event shape is more planar (less spherical).
•
The jets are more forward-backward in rapidity (less central).
We defined
two or more kinematic parameters that quantified aspects of each property.
Only the most effective of these were used and these are discussed below.
We found that, in general, better discrimination was achieved using
$\cal{R}$=0.3 cone jets (with $|\eta|<2.0$ and $E_{T}>15$ GeV) than
$\cal{R}$=0.5 cone jets. However, in some instances, $\cal{R}$=0.5 cone
jets were used, and this is noted where it occurs.
Although correlations exist between many of the kinematic parameters, each
includes useful information not fully contained in any of the others. These
correlations are presented in Sec. VI.D.
V.1 Parameters sensitive to energy scale
Any parameter that depends on the energy scale of the jets
is also sensitive to the mass of the top quark. These “mass sensitive”
parameters usually provide better discrimination against QCD background
than other parameters that provide only a measure of some topological
feature. Three mass sensitive parameters are:
1.
$H_{T}$
The sum of the transverse energies of jets in a given event characterizes
the transverse energy flow, and is defined as:
$$H_{T}=\sum_{j=1}^{N_{\rm jets}}E_{T_{j}}$$
(2)
where $E_{T_{j}}$ is the transverse energy of the $j$th jet, as ordered in
decreasing jet $E_{T}$ rank, and $N_{\rm jets}$ is the number of jets
in the event.
2.
$\sqrt{\hat{s}}$
This parameter is the invariant mass of the $N_{\rm jets}$ system.
3.
$E_{T_{1}}$/$H_{T}$
$E_{T_{1}}$ is the transverse energy of the $\cal{R}$=0.5 cone jet with highest
$E_{T}$. This parameter characterizes the $E_{T}$ fraction carried by the leading
jet, and tends to be high for QCD background. The $t\bar{t}$ events are likely
to have transverse energy roughly equipartitioned among all six jets, and
hence the leading $E_{T}$ jet is, on average, fractionally softer.
Figure 6 shows the distributions of $H_{T}$, $\sqrt{\hat{s}}$, and
$E_{T_{1}}$/$H_{T}$, each of which reveals significant discrimination between
signal and background. This and subsequent figures for the parameters are
shown both normalized to cross section, and normalized to unity.
V.2 Parameters sensitive to additional radiation
As previously noted, the QCD background is primarily a $2\rightarrow 2$
parton process that contains additional radiated gluons. These gluons tend
to be much softer than the leading partons, and therefore the jets associated
with this radiation tend to have smaller $E_{T}$. Three parameters that measure
the hardness of this radiation are:
4.
$H_{T}^{3j}$
This variable is defined as[12, 13]:
$$H_{T}^{3j}=H_{T}-E_{T_{1}}-E_{T_{2}}$$
(3)
where $E_{T_{1}}$ and $E_{T_{2}}$ are the transverse energies of the
two leading (highest $E_{T}$) jets. By subtracting the $E_{T}$ of
the two leading jets, what remains is a better measure of any additional
gluon radiation in QCD events, enhancing the discrimination
between $t\bar{t}$ signal and QCD background.
5.
$N_{\rm jets}^{A}$
An average jet count parameter, $N_{\rm jets}^{A}$, provides a way to
parameterize the number of jets in an event, while taking account of the
hardness of these jets. We define:
$$N_{\rm jets}^{A}={{\displaystyle\int^{55}_{15}}E_{T}^{\rm thr}N(E_{T}^{\rm thr%
})\,dE_{T}^{\rm thr}\over{\displaystyle\int^{55}_{15}}E_{T}^{\rm thr}\,dE_{T}^%
{\rm thr}}$$
(4)
where $N(E_{T}^{\rm thr})$ is the number of jets in a given event with
$|\eta|<2.0$ and $E_{T}$ greater than some threshold, $E_{T}^{\rm thr}$ in GeV.
Therefore, this parameter corresponds to the number of jets, but is
more sensitive to jets of higher $E_{T}$ than just a simple jet count above
some given threshold.
6.
$E_{T_{5,6}}$
The transverse energies of the fifth jet, $E_{T_{5}}$, and sixth jet, $E_{T_{6}}$,
are also useful in discriminating QCD background from $t\bar{t}$ events.
Our final selection (see Sec. VII(a)) requires at least six jets. For
background these usually correspond to soft radiation. The variable chosen is:
$$E_{T_{5,6}}=\sqrt{E_{T_{5}}~{}E_{T_{6}}}.$$
(5)
Figure 7 shows distributions of $H_{T}^{3j}$, $N_{\rm jets}^{A}$ and
$E_{T_{5,6}}$. Again, these variables are effective in differentiating
between signal and background.
V.3 Aplanarity and sphericity
The direction and shape of the momentum flow of jets in
$t\bar{t}$ production are different from those in
QCD background. These differences can be quantified
using event-shape parameters[22]. For each event, we define the
normalized momentum tensor $M_{ab}$:
$$M_{ab}=\sum_{j}^{N_{\rm jets}}p_{ja}p_{jb}/\sum_{j}^{N_{\rm jets}}p_{j}^{2}$$
(6)
where $a$ and $b$ run over the $x,y,z$ components (indices of the tensor), and
$j$ runs over the number of jets in an event. As is clear from its definition,
$M_{ab}$ is a symmetric matrix that is always diagonalizable, and has
positive-definite eigenvalues ($Q_{1},Q_{2},Q_{3}$) satisfying the conditions:
$$Q_{1}+Q_{2}+Q_{3}=1~{}~{}~{}{\rm and}~{}~{}~{}0\leq Q_{1}\leq Q_{2}\leq Q_{3}.$$
(7)
The equation $Q_{1}+Q_{2}+Q_{3}=1$ represents a plane in a space spanned by
$Q_{1},Q_{2}$, and $Q_{3}$, and the inequality restricts
the range of each eigenvalue, as shown in Fig. 8:
$$\displaystyle 0\leq Q_{1}\leq\frac{1}{3},$$
(8)
$$\displaystyle 0\leq Q_{2}\leq\frac{1}{2},$$
$$\displaystyle\frac{1}{3}\leq Q_{3}\leq 1.$$
The magnitude of any $Q_{i}$ represents the portion of momentum flow in the
direction of the $i^{th}$ eigenvector.
Limiting event shapes can therefore be characterized as follows:
•
Linear : $Q_{1}=Q_{2}=0$ and $Q_{3}$ = 1.
•
Planar : $Q_{1}=0$ and $Q_{2}=Q_{3}$ = $\frac{1}{2}$.
•
Spherical : $Q_{1}=Q_{2}=Q_{3}$ = $\frac{1}{3}$.
The aplanarity ($\cal{A}$) and sphericity ($\cal{S}$) parameters that we use
are defined as follows:
7.
${\cal{A}}=\frac{3}{2}Q_{1}$,
8.
${\cal{S}}=\frac{3}{2}\left(Q_{1}+Q_{2}\right)$,
with $0\leq{\cal{A}}\leq 0.5$ and $0\leq{\cal{S}}\leq 1$.
Top quark ($t\bar{t}$) events tend to have higher aplanarity and sphericity
than background events. We calculate $\cal{A}$ and $\cal{S}$ in the $p\bar{p}$
collision frame; little difference is found using the parton center of mass
frame. Distributions of $\cal{A}$ and $\cal{S}$ for herwig $t\bar{t}$
events for $m_{t}$=175 GeV/$c^{2}$ and for data are shown in Fig. 9.
V.4 Parameters sensitive to rapidity distributions
9.
${\cal{C}}$
The centrality (${\cal{C}}$) parameter is defined as:
$${\cal{C}}=\frac{H_{T}}{H_{E}},$$
(9)
where
$$H_{E}=\sum_{j=1}^{N_{\rm jets}}E_{j}.$$
(10)
Centrality is similar to $H_{T}$, characterizing the transverse energy in events,
but is normalized in such a way that it depends only weakly on the mass of the
top quark.
10.
$\left<\eta^{2}\right>$
To good approximation, the $\eta$ distribution for jets in $t\bar{t}$ events is
normally distributed about zero with an rms, $\sigma_{\eta}$, close to
unity. With typically six or more jets in an event, the rms of the jet
$\eta$ distribution can be a useful discriminator. The $\left<\eta^{2}\right>$
variable is defined using only the leading six jets. We use $\cal{R}$=0.5
cone jets for this variable.
We calculate $\left<\eta^{2}\right>$ by taking the square of the difference
between each jet $\eta$ and the $E_{T}$-weighted mean, $\bar{\eta}$, weighted
by a factor ${\cal W}(E_{T})$. ${\cal W}(E_{T})$ depends upon the difference in
rms between $t\bar{t}$ signal ($\sigma_{\eta}^{t\bar{t}}$) and background
($\sigma_{\eta}^{\rm bkg}$), and is larger at those $E_{T}$ values where signal
and background are expected to differ. The $\left<\eta^{2}\right>$ parameter
is given by:
$${\rm\left<\eta^{2}\right>}=\frac{\sum_{j=1}^{6}{\cal{W}}(E_{T_{j}})\left(\eta_%
{j}-\bar{\eta}\right)^{2}}{\sum_{j=1}^{6}{\cal{W}}(E_{T_{j}})},$$
(11)
where
$${\cal{W}}(E_{T})=\frac{\sigma_{\eta}^{t\bar{t}}(E_{T})-\sigma_{\eta}^{\rm bkg}%
(E_{T})}{\sigma_{\eta}^{t\bar{t}}(E_{T})}\quad{\rm and}$$
(12)
$$\bar{\eta}=\frac{1}{H_{T}}\sum_{j=1}^{N_{\rm jets}}E_{T_{j}}~{}\eta_{j}.$$
(13)
Note that both $\sigma_{\eta}^{t\bar{t}}(E_{T})$ and $\sigma_{\eta}^{\rm bkg}(E_{T})$ depend on the $E_{T}$ of the jets in the $\eta$ distribution. Jets with
lower $E_{T}$ tend to be at larger values of $|\eta|$, and consequently
$\sigma_{\eta}$ decreases with increasing $E_{T}$. The QCD multijet background
has a broader distribution in the $\left<\eta^{2}\right>$ variable than the
$t\bar{t}$ signal.
The ${\cal C}$ and $\left<\eta^{2}\right>$ distributions are shown
in Fig. 10, for $m_{t}$ = 175 GeV/$c^{2}$.
The above ten kinematic variables are employed as inputs to the first neural
network. The output of this network is an input to the second (and final)
neural network, whose three other inputs are described in the following section.
VI Event Structure Variables
In addition to the kinematic and topological characteristics examined in
Sec. V, there are other differences between the $t\bar{t}$
signal and the QCD multijet background that we will exploit
in extracting the $t\bar{t}$ signal.
VI.1 $p_{T}$ of tagging muon
The $p_{T}$ of the tagging muon gives further discrimination between $t\bar{t}$
signal and QCD background. Not only does the fragmentation of $b$ quarks
produce higher $p_{T}$ objects, but the $b$ quark is also more energetic in
$t\bar{t}$ events than in background. Thus, the mean muon $p_{T}$, $p_{T}^{\mu}$,
is significantly larger in $t\bar{t}$ events. Figure 11 shows the
muon $p_{T}$ spectra. Figure 11(a) compares the muon $p_{T}$ in
herwig and isajet $t\bar{t}$ events, which shows that the muon
$p_{T}$ spectrum is modeled consistently by Monte Carlo. Figure 11(b)
compares herwig $t\bar{t}$ events and data (predominantly background).
These results show that the $p_{T}$ of the muon can serve as a useful tool in
differentiating between signal and background.
VI.2 Widths of jets
At the simplest level, each $t(\bar{t})$ quark decays into a $b(\bar{b})$
quark and a $W^{+}(W^{-})$ boson, with each $W$ boson decaying into light
quarks. Barring extra gluon bremsstrahlung, this represents six quark-jets
in the final state. The average jet multiplicity for herwig $t\bar{t}$
events ($m_{t}$=175 GeV/$c^{2}$) using our selection criteria is 6.9, implying
that the contribution from gluons is relatively small. Conversely, jets in
the QCD multijet background originate predominantly from gluon radiation.
Although gluon splitting can take place, producing both quark and gluon jets,
it is expected that gluons dominate QCD multijet production.
QCD predicts substantial differences between quark jets and gluon jets and,
in fact, observed differences in quark and gluon jet widths have
been reported by experiments at the KEK $e^{+}e^{-}$ collider (TRISTAN)[23]
and the CERN $e^{+}e^{-}$ collider (LEP)[24]. Parton shower Monte Carlos
such as herwig have been shown to reproduce the widths observed
in data [24], although herwig has been found to
slightly underestimate jet widths at the Fermilab Tevatron [25].
We found that by applying a correction of 3% to the widths, herwig QCD
Monte Carlo reproduces the observed distributions in the width of the jets.
Further studies have shown that the kinematic distributions of the multijet
background are also well modeled using herwig. We have therefore chosen
herwig as the generator for studying jet widths, with a 3% correction
applied to the widths of each jet.
Figure 12(a) shows the mean width of 0.5
cone jets versus jet $E_{T}$ for multijet data and herwig QCD and
Fig. 12(b) compares the data to herwig $t\bar{t}$.
Here, the jet width is:
$$\sigma_{\rm jet}=\sqrt{\sigma_{\eta}^{2}+\sigma_{\phi}^{2}},$$
(14)
where $\sigma_{\eta}$ and $\sigma_{\phi}$ are the transverse energy weighted
rms widths in $\eta$ and $\phi$, respectively, and are calculated using
the ($\eta$,$\phi$) positions of each calorimeter bin
(0.1 $\times$ 0.1 in $\Delta\eta$ $\times$ $\Delta\phi$) weighted by the
transverse energy in that bin. In order to account for the broadening of jets
from additional minimum bias interactions which could overlap an event,
corrections were applied to the widths of each jet in the event. These
corrections were typically a few percent, and depended, among other factors,
upon the instantaneous luminosity during that event. These corrections were
determined by assuming that the energy coming from minimum bias interactions
was uniformly distributed in $\Delta\eta$ and $\Delta\phi$, and therefore the
measured rms of a jet was the sum in quadrature of its true rms
and the rms of a uniform distribution.
It is clear from Fig. 12(a) that herwig QCD describes the
widths observed in the data, and the herwig $t\bar{t}$ has significantly
narrower jets. This suggests that the difference may be due to the different
mix of gluons and quarks in the two processes.
For Monte Carlo it is possible to match initial state quarks to final state
reconstructed jets because the herwig $t\bar{t}$ events are relatively
simple. The mapping between quarks and jets requires a tight match in $\Delta\cal{R}$ between the initial quark and the jet, as well as a reasonable match
in energy. The following criteria were employed to define Monte Carlo
“quark-like jets”:
•
Good quality 0.5 cone jet, reconstructed without merging (not formed
from two or more adjacent jets) and with $|\eta|~{}\leq$ 2.5,
•
Distance between initial quark and its reconstructed jet to be $\Delta\cal{R}$ $\leq$ 0.05,
•
The difference in energy between the quark and the jet
$\Delta E$ $\leq$ $\sqrt{E_{\rm quark}}$ ($E$ in GeV).
Monte Carlo “gluon-like jets” were defined to be good quality jets, without
merging, but where the separation distance to the nearest quark was $\Delta\cal{R}$ $\geq$ 1. Imposing these criteria, the distributions in the jet
rms widths are shown in Fig. 13. To guide the eye, Gaussian
fits have been superimposed on the distributions. With these definitions, it
appears that gluon-like jets are 20-30% wider than quark-like jets.
Figure 13 suggests that the jet rms distributions for these
definitions of quark/gluon jets can be approximated by Gaussians. A Fisher
discriminant can be used to differentiate statistically between any two such
distributions. We defined a Fisher discriminant, ${\cal F}_{\rm jet}$, in
terms of the individual jet width $\sigma_{\rm jet}$ and the width expected
for gluon-like ($\sigma_{\rm gluon}$) and quark-like ($\sigma_{\rm quark}$)
jets, as follows:
$${\cal F}_{\rm jet}=\frac{(\sigma_{\rm jet}-\sigma_{\rm quark}(E_{T}))^{2}}{%
\sigma^{2}_{\rm quark}(E_{T})}-\frac{(\sigma_{\rm jet}-\sigma_{\rm gluon}(E_{T%
}))^{2}}{\sigma^{2}_{\rm gluon}(E_{T})}$$
(15)
We used this single parameter to characterize the quark-like or gluon-like
essence of a jet. This discriminant is summed over the four unmerged jets
with the smallest values of ${\cal F}_{\rm jet}$ in an event to form a
variable ${\cal F}$ which reflects whether the event is more $t\bar{t}$-like
(signal) or more QCD-like (background). Summing only over the four smallest
values of ${\cal F}_{\rm jet}$ (most quark-like jets), according to Monte
Carlo, optimizes the discrimination. This summed discriminant, ${\cal F}$,
will be used in our search for $t\bar{t}$ signal in the all-jets channel.
The distributions of ${\cal F}$ are shown in Fig. 14. It is known
that jet widths are not as well modeled in isajet [26],
and we have, therefore, based this discriminant only on the herwig
generator. Figure 14(a) shows ${\cal F}$ for data and herwig
QCD, and Fig. 14(b) shows ${\cal F}$ for data and herwig
$t\bar{t}\rightarrow$ all-jets. Comparison shows that the jets in data are
significantly wider, and are more consistent with herwig QCD than with
herwig $t\bar{t}$.
VI.3 Mass likelihood parameter
A mass likelihood variable, ${\cal M}$, defined below,
provides good discrimination between signal and background by requiring
two jet pairs that are consistent with the $W$ boson mass, and two $W$ + jet
pairs that are consistent with a single top quark mass of any value.
Since there are no high-$p_{T}$ leptons in the all-jets channel, and hence no
high-$p_{T}$ neutrinos, the event is in principle fully reconstructible. The
presence of two $W$ bosons in $t\bar{t}$ events provides significant rejection
against QCD background. A further requirement that the two reconstructed top
quarks have equal masses provides some additional discrimination.
${\cal M}$ is defined as a $\chi^{2}$-like object:
$${\cal M}=\frac{(M_{W_{1}}-M_{W})^{2}}{\sigma_{W}^{2}}+\frac{(M_{W_{2}}-M_{W})^%
{2}}{\sigma_{W}^{2}}+\frac{(m_{t_{1}}-m_{t_{2}})^{2}}{\sigma_{t}^{2}},$$
(16)
where $M_{W_{1}}$ ($M_{W_{2}}$) is the mass of the two ${\cal{R}}$=0.5 cone jets
corresponding to the $W$ boson from the first (second) top quark, of mass
$m_{t_{1}}$ ($m_{t_{2}}$). The parameters $M_{W}$, $\sigma_{W}$ and $\sigma_{t}$ were
fixed at 80, 16 and 62 GeV/$c^{2}$, respectively. The last two values
approximate the full widths of the two distributions, and taking them to
be constant simplifies the calculation.
The ${\cal M}$ variable is calculated by looping over combinations of jets,
and assigning all jets with $|\eta|~{}\leq$ 2.5 to one of the $W$ bosons or $b$
quarks from the two top quark decays. The smallest value of ${\cal M}$ is
selected as the discriminator. To reduce the number of combinations, two jets
are assigned to each $W$ boson, and one to the $b$ quark from one of the two
top quarks. Jets from the $W$ boson are required to have $E_{T}>10$ GeV, while
those from the $b$ quark must have $E_{T}>15$ GeV. All remaining jets are
assigned to the $b$ quark from the second top quark. To keep $b$-tagged events
on the same footing as untagged events, no a priori assignment is made
between tagged jets and $b$ quarks. Since in the top quark rest frame the $W$
boson and the $b$ quark have equal momenta, the $E_{T}$ of $W$ bosons and
$b$-jets are more similar than for QCD background. The following criterion
helps further reduce combinatorics:
•
$E_{T(W_{1})}~{}+~{}E_{T(W_{2})}~{}\leq~{}0.65~{}H_{T}$,
where $E_{T(W_{1})}$ ($E_{T(W_{2})}$) is the $E_{T}$ from the vector sum of two jet
momenta assigned to the $W$ boson from the first (second) top quark. Although
there are other possible algorithms for assigning jets to the two top quarks,
the discrimination in the ${\cal M}$ variable is not very sensitive to the
choice of reasonable algorithms.
The distributions in the ${\cal M}$ variable are shown in Fig. 15.
Figure 15(a) compares the ${\cal M}$ variable in herwig and
isajet $t\bar{t}$ events ($m_{t}$=175 GeV/$c^{2}$). Figure 15(b)
compares herwig QCD and the data (predominantly background).
Figure 15(c) compares herwig $t\bar{t}$ events and data.
These plots show that this variable is modeled consistently by the two
$t\bar{t}$ Monte Carlos, that herwig QCD models the background well,
and that ${\cal M}$ is useful in discriminating between signal and background.
VI.4 Correlations between parameters
A summary of the 13 parameters used in this analysis is given in
Table 3. The first ten parameters are simple kinematic variables,
and are correlated. To quantify the degree of correlation between any two
variables $x$ and $y$, we define a linear correlation coefficient, $r$
as[27]:
$$r={{\displaystyle N\sum}x_{i}y_{i}-\sum x_{i}\sum y_{i}\over\left[{%
\displaystyle N}\sum x_{i}^{2}-(\sum x_{i})^{2}\right]^{1/2}\left[N\sum y_{i}^%
{2}-(\sum y_{i})^{2}\right]^{1/2}}.$$
(17)
The value of $r$ ranges from 0, when there is no correlation, to $\pm$1, when
there is complete correlation or anticorrelation. Table 4
shows the average correlations among 13 parameters defined in Sec. V and
Sec. VI for data. These are average correlation coefficients; local
correlations can vary significantly, depending upon the region of multivariate
space. Note that the parameters $p_{T}^{\mu}$, ${\cal F}$, and ${\cal M}$
have relatively small correlations with the other kinematic parameters.
VII Analysis
VII.1 Event selection criteria
Before proceeding further with the analysis, basic quality criteria were
applied to the data and to Monte Carlo events:
•
isolated leptons: Events containing an isolated electron or
muon [11, 6] were rejected. This ensured that our event
sample was orthogonal to those used in the $t\bar{t}$ analyses in other decay
channels.
•
$H_{T}^{3j}\geq$ 25 GeV: Removed QCD $2\rightarrow 2$ events with
little additional jet activity.
•
number of jets: Events with fewer than six $\cal{R}$=0.3 cone jets
or more than eight $\cal{R}$=0.5 cone jets were rejected.
–
By eliminating events with fewer than six $\cal{R}$=0.3 cone jets,
the signal-to-background ratio is improved. Only 14% of the signal is
lost, while 36% of the background is rejected. (The $E_{T}$ of the sixth
jet is required in the calculation of several variables.)
–
Removal of events with more than eight $\cal{R}$=0.5 cone jets also
improves signal-to-background, rejecting 13% of the background and only
5% of the signal. The calculation of the ${\cal M}$ variable and Fisher
discriminant are thereby improved because of the reduction in the number
of jet combinations.
Of the roughly 600,000 events passing our initial criteria (see Table II),
approximately 280,000 events survive these selection requirements.
VII.2 Muon tagging
The direct branching fraction of a $b$ quark into a muon plus anything
is 10.7 $\pm$ 0.5%[28]. However, when all contributions from decays of
$b$ and $c$ quarks from the two top quarks are considered, and with a muon
acceptance of about 50%, approximately 20% of the events in the $t\bar{t}$
$\rightarrow$ all-jets mode are expected to yield at least one muon. Muons
in QCD background processes arise mainly from gluon splitting into $c\bar{c}$
or $b\bar{b}$ pairs, but intrinsic $c\bar{c}$ and $b\bar{b}$ production
as well as in-flight pion and kaon decays within jets also contribute. These
sources occur in only a small fraction of the events, and therefore only a few
percent of the QCD multijet background events will have a muon
tag[11].
To take advantage of the difference in the muon tag rate and enhance the
$t\bar{t}$ signal, our analysis requires the presence of at least one muon
near a jet in every event (“$b$-tagging”). This also provides a means of
estimating the background in a given data sample, which can be determined
purely from data. The $b$-tagging requirement should give nearly a factor
of ten improvement in signal/background[11].
Procedures for tagging jets with muons were defined after extensive Monte Carlo
studies of $t\bar{t}$ production in lepton+jets final states [11].
The requirements used to select such muon tags are:
•
The presence of a
fully reconstructed muon track in the central region ($|\eta|<$1.0).
This restriction does not have much impact on the acceptance of muons
from $b$ quark jets from $t\bar{t}$ decay because these $b$ quarks tend
to be produced mainly at central rapidities.
•
The track must be flagged as a high-quality muon. This quality is
based on a $\chi^{2}$ fit to the track in both the bend and non-bend views of
the muon system [29].
•
The signal from the calorimeter in the road defined by the track must
be consistent with the passage of a minimum ionizing particle. The signal
is measured by energy deposited in the calorimeter cells along the track.
•
Because the $p_{T}$ spectrum of muons from pion and kaon decays is
softer than from heavy quarks, an overall $p_{T}>$ 4.0 GeV/$c$
cutoff is imposed to enhance the signal from heavy quarks. Imposing
this cutoff has limited impact on the $t\bar{t}$ acceptance, since
the muon energy must be greater than about 3.5 GeV in order to penetrate
the material of the calorimeter and the iron toroid at $\eta$=0.
•
The muon must be reconstructed near a jet that has $|\eta|<$1.0 and
$E_{T}>$ 10 GeV. The distance $\Delta{\cal R}_{\mu}$ in $\eta$-$\phi$ space
between the muon and the jet axis must be less than 0.5.
If a muon satisfies the above conditions, the jet associated with
the muon is defined as a $b$-tagged jet, and the muon is called a tag.
Of the roughly 280,000 events which survived the initial selection criteria,
3853 have at least one $b$-tagged jet.
VII.3 Muon tagging rates
The probability of tagging QCD background events containing several jets is
observed to be just the sum of the probabilities of tagging individual
jets[11], and is approximately independent of the nature of the
rest of the event. The muon tagging rate is therefore defined in terms of
probability per jet rather than per event. We define the muon tagging rate as
the ratio of tagged to untagged jets, allowing us to multiply this function by
the number of untagged events to obtain an estimate of the tagged background.
Initially, the tagging rate was modeled only as a function of jet
$E_{T}$[3, 11]. However, it was found subsequently that there
was an $\eta$-dependence to the muon tag rate which depended on the date of
the run. This was traced to the fact that the muon chambers experienced
radiation damage, and required that some of the wires be cleaned during the
run. Figure 16 shows the relative muon detection efficiency as a
function of the $\eta$ of the jet for different ranges of runs.
Figures 16(a)–16(c) correspond to the time before
the cleaning and Fig. 16(d) to that after the cleaning
($N_{\rm run}~{}\geq$ 89000). These plots illustrate the need to account
for the dependence on $\eta$ and run number when performing estimates of
tagging rates.
To address this problem, the tag rate for background,
$P_{\rm tag}$($E_{T}$,$\eta$,$N_{\rm run}$), was parameterized as a function
of jet $E_{T}$, jet $\eta$, and the run number, $N_{\rm run}$, and was assumed to
factorize:
$$P_{\rm tag}(E_{T},\eta,N_{\rm run})=f(E_{T})\cdot\epsilon(\eta,N_{\rm run}),$$
(18)
where $f(E_{T})$ is the relative probability that a jet of given $E_{T}$ has a
muon tag, and $\epsilon(\eta,N_{\rm run})$ is the relative muon detection
efficiency. The functions $f(E_{T})$ and $\epsilon(\eta,N_{\rm run})$ are not
normalized individually, but it is the product of the two which is normalized.
Besides the differences in chamber efficiency caused by the deterioration and
cleaning of wires, there were also changes in the gas mixtures used in the
muon chambers between the Ia period and Ib (see Table I), and changes in the
high voltage settings, which were implemented at Run 84000. These required two
additional separations of runs, as shown in Fig. 16. We also found
a small dependence of the tag rate function on $\sqrt{\hat{s}}$ of the entire
event, which is described below.
The jet $E_{T}$ factor in the muon tag rate function ($f(E_{T})$) is shown in
Fig. 17. $f(E_{T})$ was parameterized in two ways, which allowed us to
estimate a systematic error due to the model dependence of this function.
The first parameterization assumed that $f(E_{T})$ saturates at high values of
jet $E_{T}$, and was given by the form:
$$f(E_{T})=A_{0}~{}\aleph(\frac{E_{T}-E_{T_{0}}}{\lambda}),$$
(19)
where $\aleph(x)$ is the normal frequency function (i.e.,
$\aleph(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{x}e^{-z^{2}/2}dz$), which
approaches one at high jet $E_{T}$. The parameters $A_{0},E_{T_{0}}$, and
$\lambda$ are obtained from the fits to the observed tag rates,
shown in Fig. 17.
An alternative parameterization of $f(E_{T})$ assumed a polynomial in ln($E_{T}$),
and was given by:
$$f(E_{T})=a_{0}+a_{1}~{}{\rm ln}(E_{T})+a_{2}~{}{\rm ln}^{2}(E_{T})+a_{3}~{}{%
\rm ln}^{3}(E_{T}).$$
(20)
Here, $f(E_{T})$ continues to increase with jet $E_{T}$, and the constants
$a_{0},~{}a_{1},~{}a_{2}$, and $a_{3}$ are again obtained from fits to the observed
tagged distributions, shown in Fig. 18. The difference in the
background estimate between Eq. 19 and Eq. 20
is discussed in Sec. VII.I. Because the tagging rate in Eq. 20
continues to grow with increasing jet $E_{T}$, it gives a slightly larger
estimate of the background than Eq. 19. Increasing the tag
rate increases the estimated background, thereby decreasing the signal.
Both versions of $f(E_{T})$ give similar $\chi^{2}$ fits, but as our Monte Carlo
studies showed that the tag rate continues to slowly increase with jet
$E_{T}$, even for high $E_{T}$, we chose equation 20 for estimating
the background in this analysis.
Having considered all factors that go into the tag rate function on a
jet-by-jet basis, we looked for dependence on characteristics of the event
as a whole. We observed a small additional dependence, most notable in
variables that are sensitive to the total energy scale of the event.
Figure 19 shows the muon tag rate in two bins of $\sqrt{\hat{s}}$,
which reflects the total energy of the partonic collision. The superimposed
solid curves represent fits to Eq. 20, but where the
coefficients $a_{0},~{}a_{1},~{}a_{2}$, and $a_{3}$ are now second-order polynomials
in $\sqrt{\hat{s}}$. In Fig. 19(b), the dashed curve represents
the fit at 200$<\sqrt{\hat{s}}<$300 GeV/$c^{2}$, and a small shift in the
relative tag rate is apparent. This $\sqrt{\hat{s}}$ dependence was included
through a modification of the principal $E_{T}$-dependent part of the function,
$f(E_{T})$.
As indicated by Eq. 18, the observed tag rate is the product of two
parts. Because of this, the fits of Eqs. 19 or
20 to the observed tag rate are correlated with the muon
detection efficiency. To disentangle the two components, the fit used data only
from central rapidities, where the detection efficiency was a weak function of
$\eta$. The criterion $\epsilon(\eta,N_{\rm run})/\epsilon(0,N_{\rm run})\geq$0.6 defined the region used in the fit, corresponding to the region where
the $\eta$-dependence varied least rapidly. Once this initial $f(E_{T})$ was
determined, it was necessary to use it to re-estimate
$\epsilon(\eta,N_{\rm run})$. This involved taking the ratio of the number of
observed tagged jets to the number predicted using the initial $f(E_{T})$.
This ratio, as a function of $\eta$, is plotted in Fig. 16 for
different run ranges. The process of fitting $f(E_{T})$ and then re-calculating
$\epsilon(\eta,N_{\rm run})$ was iterated several times until stable results
were obtained. The final relative probabilities ($f(E_{T})$) are shown in
Fig. 17 and Fig. 18, and the final relative efficiency
is shown in Fig. 16. These are labeled relative
probabilities/efficiencies because it is not possible to determine the overall
normalizations of $f(E_{T})$ and $\epsilon(\eta,N_{\rm run})$ independently; it
is their product which is well determined.
Using Eq. 18, the number of expected tagged events (from background)
in a given event sample is
$$N_{\rm tag}^{\rm expt}=\sum_{\rm events}~{}~{}\sum_{\rm jets}P_{\rm tag}(E_{T}%
,\eta,N_{\rm run}).$$
(21)
In using Eq. 21 to estimate the tagged background, we assumed that
this relation remains valid for extrapolation from the background region
through to the signal region. These regions will be defined in terms of the
neural network output, in Sec. VII.E. This supposes that there is no
significant correlation between the intrinsic heavy quark ($c\bar{c}$
or $b\bar{b}$) content and the neural network output, apart from any kinematic
correlation through variation in $E_{T}$ and $\eta$, as parametrized by
Eq. 21. Therefore, we attribute any excess of tagged events over
the background predicted by Eq. 21 to $t\bar{t}$
production.
VII.4 Background modeling
Since the kinematic variables are calculated using the jet energies, they are
to some extent sensitive to the small shift in energy due to the presence of
the tagged muon and its associated neutrino. As was described earlier, jets
are measured through the deposition of energy in the calorimeter, and are not
corrected for the muon’s momentum. The neutrino’s energy is, of course, missed
completely, and there is no unique prescription for correcting the jet’s energy
for the neutrino. However, these corrections are typically small
(of the order of the muon momentum).
Previous analyses [5] aimed at determining the top quark mass
have incorporated approximate correction factors for the energies of tagged
jets. For our analysis, such corrections are not strictly needed, and as
we argue below, are disfavored due to the correlations they introduce between
the $E_{T}$ of the tagged jet and the $p_{T}$ of the tagging muon. Our procedure
consists of calculating the muon tag rate function (Eq. 7.1) from jets
containing muon tags and untagged jets as follows: we denote the distribution
of untagged jets as a function of $E_{T}$ by $U(E_{T})$, and the distribution of
the tagged jets by $T(E_{T}^{\prime}$. The distribution $U(E_{T})$ reflects
dominantly QCD background. Here, $E_{T}$ is the transverse energy observed for
jets with no observable muon, and thus is on average the true jet energy;
$E_{T}^{\prime}$ is the observed energy for tagged jets, without corrections,
and thus is missing the contributions to the progenitor jets due to the
transverse energy of the muon and neutrino. We formed the ratio
$T(E_{T}^{\prime})/U(E_{T})$, taking the same numerical values of $E_{T}^{\prime}$
and $E_{T}$. This ratiowas then parameterized, as discussed in Sec. VII.C, to
give the tag rate function, $P_{\rm tag}(E_{T})$. The $E_{T}$ distribution of QCD
background events with a tagged jet, $B(E_{T})$, for our analysis was then
obtained using the untagged jet sample $U(E_{T})$ from the expression $B(E_{T})=P_{\rm tag}(E_{T})\times U(E_{T})$, which, apart from the smoothing applied to
the tag rate function, is equivalent to $B(E_{T})$ = $T(E_{T}^{\prime})$.
Although there is no a priori advantage to using uncorrected $E_{T}^{\prime}$
instead of corrected $E_{T}$ for the tagged jets, it does simplify the background
calculation for the neural network analyses. Our studies show that the $p_{T}$
of the muon is uncorrelated with $E_{T}^{\prime}$, but not with $E_{T}$. This is
illustrated in Fig. 20(a), which shows the mean muon $p_{T}$ as a
function of the tagged jet $E_{T}^{\prime}$ for data. A fit to a straight line
gives a slope consistent with zero. Figure 20(b) shows muon $p_{T}$
distributions for three distinct ranges of tagged jet $E_{T}^{\prime}$ (chosen to
be equally populated); they are indistinguishable. Similar plots are shown in
Fig. 21 for herwig $t\bar{t}$ events. Again, no significant
correlation between muon $p_{T}$ and tagged jet $E_{T}^{\prime}$ is observed.
Since the $p_{T}$ of the muon is not correlated with the uncorrected jet $E_{T}$,
it is largely independent of event kinematics and the probability of finding
a muon of a given $p_{T}$ factorizes from the tag rate function. Tagged
background events can therefore be generated by adding (“fake”) muons to
untagged events by assigning a random $p_{T}$ value from the observed $p_{T}$
spectra. The value of $p_{T}$ enters into the second neural network and must be
generated for the modeled background. The $p_{T}$ distributions for both data
(predominantly background) and herwig $t\bar{t}$ events were fitted
separately to the sum of two exponentials, and the parameterizations from the
fits were used in the random generation of muon $p_{T}$ values for both
background and signal. These spectra and the associated fits are shown in
Fig. 22. As discussed above, correcting the jets for muon and
neutrino $p_{T}$ would introduce correlations that would complicate the
application of the tag rate function; we have consequently not applied such
corrections to the jet energies.
The procedure used for estimating the number of tagged events expected from
background can be checked by comparing the distributions of estimated tags to
those for the observed tags. Figure 23 shows this comparison for
the distributions in each of the 13 parameters used in this analysis, for the
entire multijet tagged data sample. In these distributions the $t\bar{t}$
fraction is negligible, as less than 40 $t\bar{t}$ events are expected. The
predicted rate, absolutely normalized using Eq. 21, is shown for all
distributions, and consistently reproduces the observed number of tagged events.
The values of $\chi^{2}$ per degree of freedom for the plots in
Fig. 23 are given in Table V.
Once the background sample is generated, these events are treated exactly as
the tagged sample (the sample used to extract signal). The neural network is
applied to both sets of events, tagged and modeled background (untagged
events+“fake-tags”), and the difference between the two represents an excess
that is attributed to the $t\bar{t}$ signal. Similarly, “fake-tags” are
applied to the untagged herwig $t\bar{t}$ events, and these events are
used to model the signal. This effectively increases the statistics of the
tagged events in the Monte Carlo $t\bar{t}$ sample.
A correction for the small contamination of the background sample due
to $t\bar{t}$ events is made (see Sec. VII.I).
VII.5 Neural network analysis
Artificial neural networks constitute a powerful extension of conventional
methods of multidimensional data analysis [30], and are well suited
to our search because they handle information from a large number of inputs and
can account for nonlinear correlations between inputs. A neural net is a
multivariate discriminant. Its construction typically consists of input nodes,
output(s), and intermediary “hidden nodes”. The connection between any two
nodes is governed by a sigmoidal function which is characterized by a
“weight” and “threshold”. The neural network is “trained” by
setting weights and thresholds of the nodes through an optimization algorithm.
The output of the neural network is simply a mapping between the
multidimensional space described by our kinematic input variables and a
one-dimensional output space. Setting a threshold on the output of the neural
network corresponds to a set of hypersurface cuts in multidimensional input
space. Consequently, the neural network output may be employed to discriminate
between signal and background as long as the following conditions are observed:
•
The neural network is trained on event samples that are independent of
the sample used for the measurement.
•
There is a reliable method for determining the background level for a
given value of neural network output.
Independence of the training sample and the sample used to extract the
$t\bar{t}$ signal is maintained by considering only $b$-tagged events in the
final extraction of a signal for $t\bar{t}$ production. Events that did not
have a $b$-tagged jet are used for training and for defining the background
sample.
In order to simulate the background, untagged events were made to resemble
tagged events by adding muon tags to one of the jets in the event. With such
“fake” muons, these events were taken to represent the background. The
prescription for adding these muons to the untagged jets was described in
Sec. VII.D. A subset of these events was used to train the neural network
response to background.
The set of 13 parameters (see Table 3) was used as the set of
input nodes in training the neural network. Because training time increases
markedly and quality of convergence decreases with the number of input nodes
and hidden layers, the problem was simplified by first training a neural
network using the first ten kinematic variables. These variables tended to
be more highly correlated than the remaining three (see Sec. VI). Based on
studies using our training samples, we chose to have 20 hidden nodes and one
network output, and used the back-propagation learning algorithm in
jetnet [31]. The output of this neural network and the remaining
three parameters were used as inputs to a second neural network. Here, we
chose eight hidden nodes and one network output.
Events used to train the two neural networks were selected as follows. A
simpler initial network (NN${}_{0}$), using a subset of seven kinematic parameters
(excluding $E_{T_{1}}$/$H_{T}$, $E_{T_{5,6}}$, and $\left<\eta^{2}\right>$), was
trained using all events. The output of this network, for both data and
herwig $t\bar{t}$ Monte Carlo, is shown in Fig. 24.
Figure 24 shows that the $t\bar{t}$ signal tends to peak at values
of neural network output near 1 (the “signal region”), whereas the background
events peak near 0 (the “background region”). For the final training samples,
we selected data and $t\bar{t}$ Monte Carlo events having NN${}_{0}>$ 0.3. This
neural network was used only for choosing the best training samples, and was
not employed in the final analysis (i.e., all events were reanalyzed).
Removing events that were very unlikely $t\bar{t}$ candidates (NN${}_{0}<$ 0.3)
improved the efficiency of the training and increased network sensitivity to
background events that more closely mimic $t\bar{t}$ event characteristics,
thereby improving signal-to-background discrimination in the final analysis.
Training of the two neural networks used in the final analysis proceeded as
follows. The first neural network (NN${}_{1}$) was trained on the ten kinematic
variables using the training sets, as described above. The output of NN${}_{1}$,
and the remaining three variables were then used as inputs to the second neural
network (NN${}_{2}$). NN${}_{2}$ was trained using tagged herwig $t\bar{t}$
Monte Carlo events and “fake” tagged data, also described in Sec. VII.D.
VII.6 Cross section using neural network fits
The $t\bar{t}$ cross section, integrated over all values of neural network
output, is determined from the distributions in the output of the final neural
network. Any excess of the tagged data over the modeled background
distribution is attributed to $t\bar{t}$ production. This excess, integrated
over all values of neural network output, is independent of the neural network,
and depends only on the accuracy of the modeling of the background by the tag
rate function. If the location of any excess appears in the region of
$t\bar{t}$ signal (in neural network output) it would make these events likely
$t\bar{t}$ candidates. The final neural network (NN${}_{2}$) distributions for
the data and the expected background are shown in Fig. 25(a),
and for herwig $t\bar{t}$ events in Fig. 25(b). The
normalization of the $t\bar{t}$ signal is described below. These distributions
demonstrate a strong discrimination between signal and background.
We extract the cross section from a fit to the data of the sum of the neural
network output distributions expected for the $t\bar{t}$ signal and for QCD
multijet background. Because the shapes of the $t\bar{t}$ and QCD network
output distributions differ significantly, the relative amounts of each can be
disentangled. The generated herwig $t\bar{t}$ events were arbitrarily
normalized assuming $\sigma_{t\bar{t}}$ = 6.4 pb at each top quark mass. This
value needs to be factored out in normalizing Fig. 25(b).
The data of Fig. 25(a) are fitted using $\chi^{2}$ minimization
to the hypothesis:
$$N_{\rm expected}={A_{\rm bkg}}~{}N^{i}_{\rm bkg}+\frac{\sigma_{t\bar{t}}}{6.4~%
{}{\rm pb}}~{}N^{i}_{t\bar{t}},$$
(22)
where $N^{i}_{\rm bkg}$ is the expected number of background events in the
$i^{th}$ bin, and $N^{i}_{t\bar{t}}$ is the expected signal in this bin.
Because the full Monte Carlo sample, scaled to the total number of events
(given by 6.4 pb multiplied by the integrated luminosity), is subjected to
exactly the same trigger and selection criteria as the data,
$N^{i}_{t\bar{t}}$ accounts for the luminosity, branching ratio (BR),
and $t\bar{t}$ efficiency of our selection criteria. Both $A_{\rm bkg}$,
the background normalization factor, and $\sigma_{t\bar{t}}$, are obtained
from the fit, along with their respective statistical errors. The results
of this fit are shown in Fig. 26.
By allowing the signal and background normalization factors to be determined
from the fit, this method simultaneously provides the $t\bar{t}$ cross section
and a more sensitive measurement of the background normalization. It
efficiently exploits all information about the $t\bar{t}$ cross section and
background normalization from the entire range of neural network output,
without choosing any particular cutoff on neural network output. The
distributions for signal, background and data are shown separately in
Fig. 26. The error bars are the square root of the number of
data events in each bin.
Events at the lowest values of neural network output ($<0.02$) have been
removed, leaving 2207 events, or slightly more than half of the tagged data
sample. The resulting fits may be checked by varying the region of NN${}_{2}$ used.
(Fig. 26 uses events with NN${}_{2}>$ 0.02). Figure 27 shows results
for $A_{\rm bkg}$ and $\sigma_{t\bar{t}}$ as a function of the lower limit in
NN${}_{2}$ employed in the fit. The results are seen to be quite stable to the
change of this lower limit. We note that the jets in events with NN${}_{2}<$ 0.02
tend to have low $E_{T}$, where the tagging rate may not be as well determined
due to the low tagging probability. Because the background modeling may be
less accurate in the very low NN${}_{2}$ region, where the background so strongly
dominates the data distribution, we impose a cut of NN${}_{2}>$ 0.02 for our fits
to $A_{\rm bkg}$ and $\sigma_{t\bar{t}}$. The stability of the results shown
in Fig. 27 supports this choice.
A similar plot was produced and fitted for several top quark masses, and the
values of the cross section obtained using the output distribution for
herwig $t\bar{t}$ events generated at that mass. The results are shown
in Table VI for several top quark masses. Interpolating to the value for the
top quark mass as measured by DØ [5] ($m_{t}$ = 172.1 $\pm$ 7.1
GeV), we obtain $\sigma_{t\bar{t}}$ = 7.1 $\pm$ 2.8(stat) pb.
Fitting the data in Fig. 26 only to the background
($\sigma_{t\bar{t}}$ forced to zero), changes the normalization to 1.09 $\pm$
0.03, and the total $\chi^{2}$ per degree-of-freedom to 23.1/18. We note that
the change in $\chi^{2}$ comes predominantly from the last three bins of neural
network output (in Fig. 26), and the probability for a change in
$\chi^{2}$ of 6.2 (for $m_{t}$=180 GeV/$c^{2}$) for one additional degree-of-freedom
is consistent with the significance of the extracted cross section, which is
2.5 standard deviations from zero.
VII.7 Cross section using counting method
The traditional method for extracting the $t\bar{t}$ cross section served as a
useful check on the above procedure. We assumed an absolute normalization of
the background as given by the tag rate function. Taking the excess in observed
events (seen in Fig. 26) to be from $t\bar{t}$ production, we
calculate the cross section for the process using the conventional relation:
$$\sigma_{t\bar{t}}=\frac{N_{\rm obs}-N_{\rm bkg}}{\epsilon\times BR\times{\cal{%
L}}}$$
(23)
where N${}_{\rm obs}$ is the number of observed events with neural network
output greater than some threshold, N${}_{\rm bkg}$ is the corresponding
number of expected background events, $\epsilon\times BR$ is the branching
ratio (BR) times the efficiency ($\epsilon$) of the criteria used for selecting
$t\bar{t}$ events, and ${\cal{L}}$ is the total integrated luminosity
(110.3 $\pm$ 5.8 pb${}^{-1}$). We use herwig as the model for calculating
the value of $\epsilon\times BR$.
The number of events, as a function of the threshold placed on the output of
the neural network, is shown in Fig. 28(a). The error bars are
the square root of the number of events in each bin. The upper smooth curve in
Fig. 28(a) represents the sum of the expected signal and
background, and the lower curve is just the expected background. The
statistical error in the cross section depends upon where the threshold is
placed. A plot of the relative statistical error versus the threshold on the
output of the neural network is shown in Fig. 28(b). The fractional
error ${\cal E}$ is approximated by:
$${\cal E}=\frac{\sqrt{(N_{t\bar{t}}+N_{\rm bkg})}}{N_{t\bar{t}}},$$
(24)
where $N_{t\bar{t}}$ and $N_{\rm bkg}$ are the expected number of $t\bar{t}$
and background events above the neural network threshold. We wished to place
the final threshold at or near the minimum error, and chose 0.85, as shown in
Fig. 28(b). The number of events above this threshold, the expected
background, and the expected signal are shown in Table VII.
Using Eq. 23, Table VIII lists the efficiency times branching ratios
for two input top quark mass values, and the extracted $t\bar{t}$ cross
sections. We note that the method in Sec. VII.F gave $t\bar{t}$ cross sections
of 7.2 and 6.3 pb for $m_{t}$ of 170 and 180 GeV/$c^{2}$, respectively, in good
agreement with the values in Table VIII. When interpolated to the measured top
quark mass of 172.1 GeV/c${}^{2}$, this determination yields a cross section of
7.3 $\pm$ 3.0 $\pm$ 1.6 pb. The results from the fit to the neural network are
slightly lower, as one would expect, since the background normalization was
1.07 (instead of being fixed to 1 here). The changes in efficiencies as a
function of top quark mass reflect the sensitivity of the selection criteria
to the input mass $m_{t}$. The statistical and systematic uncertainties in the
cross sections are discussed in Sec. VII.I.
VII.8 Double-tagged events
The requirement of a second $b$-tagged jet in the event further reduces the
background, thereby increasing the signal-to-background ratio. Unfortunately,
the additional requirement significantly reduces the expected yield. However,
the search for these “double-tagged” events serves as a consistency check of
the single-tag analysis, and also as a test of the model for the background.
The number of events that contain two $b$-tagged jets is shown in Table IX for
various NN${}_{2}$ thresholds. The two $b$-tags are required to originate from
separate jets; two tags within the same jet are counted as a single tag. The
higher muon $p_{T}$ is used as the input to the neural network. The background is
again calculated based on Eq. 18, where
$P_{\rm tag}(E_{T},\eta,N_{\rm run})$, summed over all jets, represents the
expected number of tags in the event. The double-tag probability is
obtained via the Poisson distribution, and is the likelihood of observing
at least two tagged jets, given the expected number. This follows since the
tag rate function is a rate per jet, and, within our model, the two tagged
jets are uncorrelated.
We make the assumption that the fraction of double-tagged events from
correlated sources, such as direct heavy-quark pair production ($c\bar{c}$ or
$b\bar{b}$), remains unchanged over the entire range of the neural network
output variable. This assumption is motivated by the fact that the energy
scales in such events are well above the energy thresholds for heavy-quark pair
production, and therefore the fraction of these events should be independent of
the neural network output. The good agreement between the background model and
data in the single-tagged channel supports this assumption.
We determine the normalization of the background by fitting the neural network
output distribution to the expected background and signal contributions as in
Sec. VII.F. The 32 events were binned in neural network output, and the
log-likelihood calculated. The minimum in negative log-likelihood occurs for a
background normalization factor of 0.97${}^{+0.20}_{-0.18}$, where the errors
correspond to a change in log-likelihood of 1/2. In determining this
normalization, the expected $t\bar{t}$ signal was not varied, but the result is
insensitive to this value. Allowing the data to determine the normalization
through this fit accomodates the possibility that the tag rate function for
the second muon in the event is different from that for the first muon. The two
errors on the expected background in Table IX represent the uncertainties due
to the tag rate function, $t\bar{t}$ subtraction and $E_{T}$ scale (see
Sec. VII.I) and the normalization error, respectively.
We note that the fitted normalization is consistent with that for the single
tagged sample indicating that the second muon tag probability is roughly the
same as for the first. The total number of events for NN${}_{2}>0.02$ is in good
agreement with the sum of expected background plus the small contribution from
top. The small excess persists as the NN${}_{2}$ threshold is increased, in
agreement with expectations. The double tag analysis supports our conclusion
that the singly-tagged sample is due to $t\bar{t}$ production.
VII.9 Corrections and uncertainties
In this subsection we discuss the major sources of systematic uncertainty that
affect either the background estimate or signal efficiency. The statistical
errors on the cross section and background normalization come directly from
the fit (Eq. 22) shown in Fig. 26.
•
The statistical error in the calculation of the background is estimated
by the number of untagged events falling in the signal region. This estimate
of 24.8 events, and an approximate mean tagging rate of 2%, implies of the
order of 1240 untagged events for the background, and a consequent 3%
statistical uncertainty in the background estimate. This contributes a 4%
uncertainty in the cross section based on the counting method in
Eq. 23.
•
The error in the normalization of the tagging rate was taken from the
combined fits to the output of the neural networks using Eq. 22.
This error is shown in Fig. 27(a), and was taken to be 5%. It is
used only in the calculation of the error on the background, as it is already
included in the cross section. (The statistical error on the cross section was
obtained from a simultaneous fit to the normalization of both background and
signal, and accounts for the error on the background normalization.)
•
The uncertainty in the parameterization of the tagging rate results in a
5% uncertainty in the predicted number of background events. This was
estimated by comparing the predicted number of tags for two functional forms
(Eq. 19 and Eq. 20) assumed for the tag rate.
Unlike the normalization of the tagging rate, this error accounts for possible
changes in the shape of the background as a function of neural network output.
This results in a 7% uncertainty in the $t\bar{t}$ cross section.
•
The presence of $t\bar{t}$ events in the data used for estimating
background has been taken into account in all results presented thus far. The
procedure used to estimate the correction to the background proceeds as
follows. Calling $N_{t\bar{t}}^{\rm mistag}$ the number of untagged $t\bar{t}$
events wrongly assigned to the background estimate, we can estimate
$N_{t\bar{t}}^{\rm mistag}$ as:
$$N_{t\bar{t}}^{\rm mistag}=\frac{0.8}{0.2}(N_{\rm obs}-N_{\rm bkg})f_{\rm tag}$$
(25)
where the $\frac{0.8}{0.2}$ corrects the $b$-tagged signal back to the untagged
signal (recall that $t\bar{t}$ events are tagged roughly 20% of the time),
$f_{\rm tag}$ is the average tag rate per event, and $N_{\rm obs}$ and
$N_{\rm bkg}$ refer to events in the final tagged data sample. The corrected
background estimation therefore becomes:
$$N_{\rm bkg}(corr)=N_{\rm bkg}-N_{t\bar{t}}^{\rm mistag}$$
(26)
This correction is applied bin by bin in Fig. 26, and is
approximately 4% in the signal region. We therefore assign a systematic
uncertainty of 4% to the background estimate and a corresponding 6%
to the $t\bar{t}$ cross section.
•
Because untagged events, when multiplied by the tag rate function, model
the tagged background, the $E_{T}$ scale of both sets must be the same.
Any mismatch between these can produce subtle differences in the scales of
the kinematic variables. A useful measure of this scale is mean $H_{T}$.
We observe that the difference in mean $H_{T}$ between our data and background
model is 1.5 $\pm$ 1.4 GeV (see Fig. 23(a)), which is consistent
with no mismatch. We take 1.4 GeV to be the uncertainty in the energy scale
of the background model. This 1.4 GeV is added to one of the jets (we
arbitrarily choose the jet with highest $E_{T}$), event-by-event, in the
background calculation and the analysis is redone. The resultant change in
the background is 4.2%, and 9.1% change in the cross section.
•
The statistical error in the $t\bar{t}$ efficiency is 3.2%.
•
Any difference in the turn-on of the trigger efficiency for data and for
$t\bar{t}$ Monte Carlo events can affect the signal efficiency. The difference
can originate, for example, from the modeling of electronic noise or from the
simulation of the underlying event. Furthermore, this efficiency can depend
upon the mass of the top quark. From our trigger simulations, we estimate $<$
5% uncertainty in signal efficiency from such
sources [12, 13].
•
The uncertainty in the integrated luminosity was taken to be
5.3% [32]. This arises mainly from the uncertainty in the absolute
luminosity, and affects all runs systematically.
•
Any difference in the relative energy scale between data and Monte Carlo
affects the efficiency for signal. This uncertainty was determined using the
MPF method[17], as described in Sec. IV.C. Varying the energy scale in
the $t\bar{t}$ Monte Carlo by $\pm$ (4% + 1 GeV) [6] changes the
efficiency for signal by $\pm$ 5.7%.
•
The $t\bar{t}$ tag rate is based on the $t\bar{t}$ Monte Carlo, but
assumes that the performance of all detector components was stable during the
run. The Monte Carlo acceptance was reduced by 7.0% to correct mainly for
muon detection inefficiencies that were not modeled in our simulation. We
estimate a 7.0% uncertainty in the $t\bar{t}$ efficiency from any such
changes in the muon tag rate.
•
Uncertainty in the model for $t\bar{t}$ production is estimated by
comparing $t\bar{t}$ predictions from isajet and herwig generators.
Figure 29 shows the fractional differences in efficiencies
(($\epsilon_{\sc isajet}$ - $\epsilon_{\sc herwig}$)/$\epsilon_{\sc herwig}$)
for different thresholds on $H_{T}$, $H_{T}^{3j}$, Aplanarity and $\cal{C}$ (again, for
$m_{t}$ = 180 GeV/$c^{2}$). Although the two generators differ significantly in the
tails of these distributions, on average they are in reasonable agreement. The
systematic error was estimated by repeating the analysis using events generated
with isajet. In order to remove the effects of the Fisher discriminant
(${\cal F}$), which is not well modeled in isajet, ${\cal F}$ values
were randomly chosen based on a parameterization of the herwig $t\bar{t}$
${\cal F}$ distribution. To further remove the dependence on the tag rate,
randomly generated values of muon $p_{T}$ were taken. The expected distributions
for the two generators, normalized as before, are shown in Fig. 30.
Identical thresholds were placed on the neural network output. The cross
section changed by 6.2%, which we take as the uncertainty in the overall
signal efficiency due to $t\bar{t}$ model dependence.
•
The 6% uncertainty in the $b\rightarrow\mu$ branching fraction [28]
corresponds to an average over the produced $B$-mesons. This 6% enters directly
into the acceptance error in the Monte Carlo.
•
The $p_{T}$ of the tagged muon enters as an input to the neural network.
The mean $p_{T}$ in herwig $t\bar{t}$ events was 14.7 GeV/$c$, while
in isajet it was 15.9 GeV/$c$, an 8% difference. Rescaling the muon
$p_{T}$ in herwig by 8% changes the cross section by 7.0%, which is
taken as a systematic error.
•
The uncertainty resulting from the modeling of the Fisher discriminant
for the jet widths, ${\cal F}$, was estimated by comparing data to our
herwig QCD Monte Carlo. The mean value of ${\cal F}$ in data was 0.0470
$\pm$ 0.0002 and in herwig QCD it was 0.0488 $\pm$ 0.0019. The difference
of 0.0018 $\pm$ 0.0019 indicates that our modeling is reasonable. The
uncertainty on this result, 0.0019, was systematically added to the value
of ${\cal F}$, event-by-event, in the herwig $t\bar{t}$ generator, and
the cross section recalculated. The observed change in the cross section of
2.0% is used as the systematic error from this variable.
The sizes of the above effects are summarized in Table X for the uncertainties
in the background, and in Table XI for the cross section. Adding both
statistical and systematic errors in quadrature, we estimate the background
as 24.8 $\pm$ 2.4 events (see Table VII). Similarly, the uncertainty in
the efficiency of the $t\bar{t}$ signal is calculated from the errors in Table XI.
VII.10 Measured cross section
By fitting the shape of the output in the neural network distribution,
we obtain the $t\bar{t}$ production cross section as a function of the
input mass of the top quark. The $t\bar{t}$ cross sections extracted for
several values of the top quark mass, along with a function used to
interpolate the $t\bar{t}$ cross section (drawn as a smooth curve), are
shown in Fig. 31. Interpolating both the cross section and the
statistical error, we find $\sigma_{t\bar{t}}$ = 7.1 $\pm$ 2.8 $\pm$
1.5 pb for $m_{t}$=172.1 GeV/$c^{2}$ [5].
The all-jets cross section can be combined with previous DØ measurements of
the $t\bar{t}$ production cross section, as extracted from channels where one or
both of the $W$ bosons decay leptonically [6]. This cross section,
averaged over all leptonic channels, was 5.6 $\pm$ 1.4 (stat) $\pm$
1.2 (syst) pb at $m_{t}$=172.1 GeV/$c^{2}$, and is shown superimposed on
Fig. 31. The statistical errors on the all-jets and leptonic
cross section measurements are uncorrelated. The systematic uncertainties in
the following categories were assumed to be correlated with a correlation
coefficient of 1.0.
•
Luminosity.
•
Jet energy scale.
•
Muon tagging efficiency.
•
Non-leptonic trigger efficiency.
•
Top quark generator.
•
$b\rightarrow\mu$ branching ratio and muon $p_{T}$ spectrum.
•
Background tag rate function.
The combined result for the DØ $t\bar{t}$ production cross section is
5.9 $\pm$ 1.2 (stat) $\pm$ 1.1 (syst) pb for $m_{t}$=172.1 GeV/$c^{2}$.
VII.11 Significance of signal
In this section, we estimate the significance of the excess of $t\bar{t}$
signal relative to expected background. We define the probability ($P$) of
seeing at least the number of observed events ($N_{\rm obs}$), when only
background is expected. The significance of a $t\bar{t}$ signal can be
characterized by the likelihood of $P$ being due to a fluctuation. If the
distribution for the expected number of background events, $\mu$, is assumed
to be a Gaussian with mean $b$, and has a systematic uncertainty $\sigma_{b}$,
then $P$ can be calculated as:
$$\displaystyle P$$
$$\displaystyle=$$
$$\displaystyle\sum_{n=N_{\rm obs}}^{\infty}\int^{\infty}_{0}d\mu\frac{e^{-\mu}%
\mu^{n}}{n!}\frac{1}{\sqrt{2\pi}\sigma_{b}}e^{-(\mu-b)^{2}/2\sigma_{b}^{2}}$$
(27)
$$\displaystyle=$$
$$\displaystyle 1-\sum_{n=0}^{N_{\rm obs}-1}\int^{\infty}_{0}d\mu\frac{e^{-\mu}%
\mu^{n}}{n!}\frac{1}{\sqrt{2\pi}\sigma_{b}}e^{-(\mu-b)^{2}/2\sigma_{b}^{2}}$$
The optimal choice of selection criteria can be found by minimizing the
expected value of $P$ and, thereby, maximizing the significance of the excess,
assuming that $N_{\rm obs}$ is composed of $t\bar{t}$ signal and background.
Both the expected value and measured value of the significance are shown, along
with the cutoff for greatest significance, in Fig. 32. The
result of the calculation, optimized for significance, with 18 observed events
and an expected background of 6.9 $\pm$ 0.9, is $P$ = 0.0006, corresponding to
a 3.2 standard deviation effect. This is sufficient to establish the
existence of a $t\bar{t}$ signal in multijet final states.
We consequently observe an excess in the multijet final states which we
attribute to $t\bar{t}$ production. The cross section measured is consistent
with previous measurements in other modes of $t\bar{t}$ decay[6].
VIII Summary
We have performed a measurement of the $t\bar{t}$ production cross section in
multijet final states. As described above, we observe an excess of events that
can be attributed to $t\bar{t}$ production. The level of significance of the
signal, as calculated from a possible upward fluctuation of the background to
produce the observed excess, is sufficiently high to establish independently
the existence of $t\bar{t}$ signal in the all-jets channel.
Using the DØ measured value of 172.1 GeV/$c^{2}$ for the mass of the top
quark, we obtain a cross section of 7.1 $\pm$ 2.8 (stat) $\pm$
1.5 (syst) pb, which agrees with the DØ cross section as measured in the
leptonic channels. Combining this result with previous DØ measurements of the
$t\bar{t}$ production cross section gives 5.9 $\pm$ 1.2 (stat) $\pm$ 1.1 (syst) pb.
Acknowledgement
We thank the staffs at Fermilab and collaborating institutions for their
contributions to this work, and acknowledge support from the
Department of Energy and National Science Foundation (U.S.A.),
Commissariat à L’Energie Atomique (France),
Ministry for Science and Technology and Ministry for Atomic
Energy (Russia),
CAPES and CNPq (Brazil),
Departments of Atomic Energy and Science and Education (India),
Colciencias (Colombia),
CONACyT (Mexico),
Ministry of Education and KOSEF (Korea),
and CONICET and UBACyT (Argentina).
References
[*]
Visitor from Universidad San Francisco de Quito, Quito, Ecuador.
[†]
Visitor from IHEP, Beijing, China.
[3]
DØ Collaboration, S. Abachi et al., Phys. Rev. Lett.
74, 2632 (1995).
[4]
CDF Collaboration, F. Abe et al., Phys. Rev. Lett.
74, 2626 (1995).
[5]
DØ Collaboration, B. Abbott et al., to be published in
Phys. Rev. D, FERMILAB-Pub-98/031-E, (1998).
[6]
DØ Collaboration, S. Abachi et al.,
Phys. Rev. Lett. 79, 1203 (1997).
[7]
CDF Collaboration, F. Abe et al., Phys. Rev. Lett.
80, 2767 (1998).
[8]
CDF Collaboration, F. Abe et al., Phys. Rev. Lett.
80, 2779 (1998).
[9]
CDF Collaboration, F. Abe et al., Phys. Rev. Lett.
79, 1992 (1997).
[10]
DØ Collaboration, S. Abachi et al., Nucl. Instr.
Methods Phys. Res. A 338, 185 (1994).
[11]
DØ Collaboration, S. Abachi et al.,
Phys. Rev. D 52, 4877 (1995).
[12]
C. Cretsinger, Ph.D. thesis, University of Rochester,
1995, (unpublished), available from http://www-d0.fnal.gov/publications_talks/
thesis/cretsinger/thesis_final.ps.
[13]
E. Won, Ph.D. thesis, University of Rochester, 1996,
(unpublished), available from http://www-d0.fnal.gov/publications_talks/
thesis/won/thesis.ps.
[14]
UA1 Collaboration, G. Arnison et al., Phys. Lett.
123B, 115 (1983).
B. Abbott et al., FERMILAB-Pub-97/242-E, unpublished.
[15]
G. Marchesini et al., Comput. Phys. Commun. 67,
465 (1992).
[16]
DØ Collaboration, R. Kehoe, in Proceedings of the
Sixth International Conference on Calorimetry in High
Energy Physics, Frascati, Italy (World Scientific, River
Edge, NJ, 1996), FERMILAB-Conf-96/284-E.
DØ Collaboration, B. Abbott et al.,
submitted to Nucl. Instr. and Methods Res. A,
FERMILAB-Pub-97/330-E, (1998).
[17]
CDF Collaboration, F. Abe et al., Phys. Rev. Lett.
69, 2896 (1992).
[18]
F. Paige and S. Protopopescu, BNL Report no. BNL38034, 1986
(unpublished), release v6.49.
[19]
R. Raja, Proceedings of the Workshop on Detector
Simulation for the SSC, Argonne, 1987, Fermilab-Conf-88/16-E.
[20]
The geant Manual, F. Carminati et al., CERN Geneva
(1996), Unpublished.
[21]
DØ Collaboration, S. Abachi et al., Nucl. Instr.
Methods Phys. Res. A 324, 53 (1993).
[22]
V.D. Barger and R.J.N. Phillips, Collider Physics
(Addison-Wesley, Massachusetts, 1987).
[23]
AMY Collaboration, Y.K. Kim et al., Phys. Rev. Lett.
63, 1772 (1989).
[24]
OPAL Collaboration, G. Alexander et al.,
Phys. Lett. B 265, 462 (1991).
OPAL Collaboration, G. Alexander et al.,
Z.Phys. C68, 179 (1995).
[25]
DØ Collaboration, S. Abachi et al., Phys. Lett. B
357, 500 (1995).
[26]
B. Abbott, Ph.D. thesis, Purdue University, 1994,
(unpublished), available from http://www-d0.fnal.gov/publications_talks
/thesis/abbott/ thesis_abbott.
[27]
P. R. Bevington, Data Reduction and Error Analysis
for the Physical Sciences (McGraw-Hill, New York, 1969).
[28]
Particle Data Group, Phys. Rev. D 54, (1996).
[29]
H. Xu, Ph.D. thesis, Brown University, 1995, (unpublished),
available from http://www-d0.fnal.gov/publications_talks
/thesis/ Xu/thesis.ps.
[30]
P. Bhat, hep-ex/9507007 (1995), unpublished.
R. Beale and T. Jackson, Neural Computing: An
Introduction (Adam Hilger, New York, 1991).
J. Hertz, A. Krogh, and R. G. Palmer,
Introduction to the Theory of Neural Computation,
(Addison-Wesley, Massachusetts, 1990).
[31]
C. Peterson and T. R$\ddot{\rm o}$gnvaldsson,
JETNET 3.0 - A Versatile Artifical
Neural Network Package, CERN-TH.7135/94 (1994).
[32]
J. Bantly et al., FERMILAB-TM-1995, (1997), unpublished. |
Non-linear Realisation of the $\mathcal{N}=2$, $D=6$ Supergravity
Nejat T. Y$\i$lmaz
Department of Mathematics
and Computer Science,
Çankaya University,
Öğretmenler Cad. No:14, 06530,
Balgat, Ankara, Turkey.
ntyilmaz@cankaya.edu.tr
Abstract
We have applied the method of dualisation
to construct the coset realisation of the bosonic sector of the
$\mathcal{N}=2$, $D=6$ supergravity which is coupled to a tensor
multiplet. The bosonic field equations are regained through the
Cartan-Maurer equation which the Cartan form satisfies. The
first-order formulation of the theory is also obtained as a
twisted self-duality condition within the non-linear coset
construction.
Pacs numbers: 04.65.+e, 12.60.Jv, 11.15.-q, 11.10.Lm.
Keywords: Supergravity, Non-linear sigma models,
Non-linear realisations, Coset formulation.
1 Introduction
The coset construction of the bosonic sectors of the maximal
supergravities obtained from the $D=11$ supergravity [1]
by dimensional reduction has been given in [2]. The
non-linear coset construction is based on the doubled formalism of
the theories in which a dual field is introduced for each bosonic
field of the theory. The results of [2] have been
extended to the matter coupled supergravities in
[3, 4, 5]. This has been possible since in
[6, 7] a general coset construction is performed for the
symmetric space sigma models of generic scalar coset manifolds.
Also in [8] a general construction is presented for a
matter coupled scalar coset.
The coset constructions or the non-linear realisations of the
bosonic sectors of the supergravity theories are important for the
understanding of the global symmetries of these theories. The
global symmetry of the bosonic sector; in particular the scalar
sector can be extended over the other fields to be the rigid
global symmetry of the entire theory [9, 10]. Besides a
restriction of the global symmetry group $G$ of a supergravity
theory to the integers is conjectured to be the U-duality symmetry
of the relative string theory whose low energy effective theory
becomes the supergravity theory at hand [11, 12].
In [13] the non-linear coset realisation of the pure
${\mathcal{N}}=4$, $D=5$ supergravity is given. In this work we
construct the coset realisation of the ${\mathcal{N}}=2$, $D=6$
supergravity which is coupled to a tensor multiplet
[14, 15]. The coupling of the tensor multiplet is
necessary to be able to write a Lorentz covariant lagrangian since
for the pure ${\mathcal{N}}=2$, $D=6$ graviton multiplet a
canonical Lorentz covariant lagrangian can not be constructed
owing to the existence of an anti-self-dual three-form field
strength [14, 15]. When one introduces a tensor multiplet
as it will be clear in section two one can lift the constraint of
(anti) self-duality by combining the fields of the graviton and
the tensor multiplets. In section two we will derive the field
equations and then we will give the locally integrated first-order
field equations. In section three following the introduction of
the coset map we will derive the algebra which parameterizes this
map by using the dualisation method of [2]. We will
denote that the first-order field equations can be obtained from
the Cartan form which is induced by the coset map as a twisted
self-duality condition satisfied by it [16].
2 The ${\mathcal{N}}=2$, $D=6$ Supergravity
The field content of the pure ${\mathcal{N}}=2$, $D=6$
supergravity multiplet [14, 15] can be given as
$$(e_{\mu}^{m},\>\psi_{\mu}^{i},\>A^{(-)}_{\mu\nu}),$$
(2.1)
where $e_{\mu}^{m}$ is the vielbein, $A^{(-)}_{\mu\nu}$ is an
anti-self-dual two-form field and $\psi_{\mu}^{i}$ for $i=1,2$ are
the gravitini. Due to the anti-self-duality constraint on the
two-form gauge field $A$ there is no way of constructing a
canonical Lorentz covariant and unconstrained lagrangian for the
pure ${\mathcal{N}}=2$, $D=6$ supergravity
[14, 15, 17, 18, 19]. The minimal coupling which
enables the construction of a Lorentz covariant lagrangian is the
coupling of a matter tensor multiplet whose field content is
$$(\lambda_{i},\>A^{(+)}_{\mu\nu},\>\phi),$$
(2.2)
where $A^{(+)}_{\mu\nu}$ is a self-dual two-form field, $\phi$ is
a scalar and $\lambda_{i}$ for $i=1,2$ are symplectic
Majorana-Weyl spinors. By field redefinitions one may combine the
anti-self-dual two-form $A^{(-)}_{\mu\nu}$ of the supergravity
multiplet and the self-dual two-form $A^{(+)}_{\mu\nu}$ of the
tensor multiplet into a single two-form field $A_{\mu\nu}$ which
is unconstrained and this enables the construction of the Lorentz
covariant and unconstrained lagrangian for the ${\mathcal{N}}=2$,
$D=6$ supergravity coupled to a tensor multiplet. Thus the field
content of the theory can be given as
$$(e_{\mu}^{m},\>\psi_{\mu}^{i},\>A_{\mu\nu},\>\lambda_{i},\>\phi).$$
(2.3)
We assume the signature of the space-time metric as
$$\eta_{AB}=\text{diag}(-,+,+,+,+,+).$$
(2.4)
The bosonic lagrangian of the $\mathcal{N}=2$, $D=6$ supergravity
coupled to a tensor multiplet can be given as [14, 15]
$$\mathcal{L}=-\frac{1}{2}\,R\ast 1-\frac{1}{2}\,\ast d\phi\wedge d\phi-\frac{1}%
{2}\,e^{2\phi}\,\ast F\wedge F,$$
(2.5)
where $F=dA$. In the next section we will workout the coset
construction of the bosonic sector of the theory excluding the
gravity sector; for this reason we are interested in only the
equations of motion for the fields $A$ and $\phi$. If we vary the
lagrangian with respect to the fields $A$ and $\phi$ we find the
corresponding second-order field equations respectively as
$$\displaystyle d(e^{2\phi}\,\ast F)=0,$$
$$\displaystyle d(\ast d\phi)=-e^{2\phi}\,\ast F\wedge F.$$
(2.6)
By using the fact that locally a closed differential form is an
exact one we can locally integrate the second-order field
equations to obtain the local first-order ones. Thus if we
introduce the three-form $\widetilde{A}$ and the four-form
$\widetilde{\phi}$, by eliminating an exterior derivative operator
on both sides of the field equations in (2.6) we can write
down the first-order equations as
$$\displaystyle e^{2\phi}\,\ast F=d\widetilde{A},$$
$$\displaystyle\ast d\phi=d\widetilde{\phi}+d\widetilde{A}\wedge A.$$
(2.7)
If one takes the exterior derivative of the equations in
(2.7) one would obtain the second-order field equations
(2.6) which are free of the lagrange multiplier fields
$\widetilde{A}$ and $\widetilde{\phi}$.
3 The Doubled-formalism and the Coset Structure
In this section we will construct a coset structure for the
bosonic sector of the $\mathcal{N}=2$, $D=6$ supergravity that is
coupled to a tensor multiplet such that the second-order field
equations of (2.6) can be realized by the Cartan form of the
coset map in the Cartan-Maurer equation
[2, 3, 4, 5, 8]. The coset map will be
parameterized by a Lie superalgebra and our aim will be to derive
the structure of this algebra. We first assign the generators
$$(\,Y,\>K\,),$$
(3.1)
to the original bosonic fields $A$ and $\phi$ respectively. We
have already introduced the dual fields $\widetilde{A}$ and
$\widetilde{\phi}$ in (2.7) therefore we also define the dual
generators
$$(\,\widetilde{Y},\,\widetilde{K}\,),$$
(3.2)
which will couple to the dual fields in the construction of the
coset map. Since the Lie superalgebra of the generators of any
coset construction has a $\mathbb{Z}_{2}$ grading the generators
are chosen to be odd if the corresponding coupling field is an odd
degree differential form and otherwise even [2]. For our
case all the generators defined in (3.1) and (3.2) are
even according to the above general scheme. In the construction of
the coset map we will make use of the differential graded algebra
structure of the differential forms and the field generators given
in (3.1) and (3.2). The details of this algebra can be
referred in [2, 6, 7, 8, 13]. Before giving the
definition of the full coset map which includes the dual fields
and the dual generators as well we define
$$\nu=e^{\phi K}e^{AY}.$$
(3.3)
If one assumes a matrix representation for the algebra generated
by the generators $K$ and $Y$, by using the matrix identities
$$\displaystyle de^{X}e^{-X}$$
$$\displaystyle=dX+\frac{1}{2!}[X,dX]+\frac{1}{3!}[X,[X,dX]]+\cdot\cdot\cdot\>,$$
(3.4)
$$\displaystyle e^{X}Ye^{-X}$$
$$\displaystyle=Y+[X,Y]+\frac{1}{2!}[X,[X,Y]]+\cdot\cdot\cdot\>,$$
one can calculate the Cartan form
$$\mathcal{G}=d\nu\nu^{-1},$$
(3.5)
as
$$\mathcal{G}=d\phi K+e^{\phi}dAY,$$
(3.6)
where the only non-vanishing commutator of the algebra of $K$ and
$Y$ is defined as
$$[K,Y]=Y.$$
(3.7)
When we insert the Cartan form (3.6) in the Cartan-Maurer
equation
$$d\mathcal{G}-\mathcal{G}\wedge\mathcal{G}=0,$$
(3.8)
whose validity originates from the definition of $\nu$ in
(3.3) we see that we obtain the trivial Bianchi identities
$$\displaystyle d(e^{\phi}\,dA)=-e^{\phi}dA\wedge d\phi,$$
$$\displaystyle d(d\phi)=0,$$
(3.9)
for the field strengths $e^{\phi}dA$ and $d\phi$ of the potentials
$A$ and $\phi$ respectively. Thus we observe that in order to
obtain the realisation of the field equations (2.6) of the
theory one has to construct the coset map by using the dual fields
and the dual generators together with the original ones. Therefore
next we propose the map
$$\nu^{\prime}=e^{\phi K}e^{AY}e^{\widetilde{A}\widetilde{Y}}e^{\widetilde{\phi}%
\widetilde{K}}.$$
(3.10)
The Cartan form $\mathcal{G}^{\prime}=d\nu^{\prime}\nu^{\prime-1}$
induced by this coset map will also satisfy the Cartan-Maurer
equation
$$d\mathcal{G}^{\prime}-\mathcal{G}^{\prime}\wedge\mathcal{G}^{\prime}=0,$$
(3.11)
canonically. Following the general method of the coset
construction [2, 3, 4, 5, 6, 7, 8, 13] we
will demand that when we calculate the Cartan form
$\mathcal{G}^{\prime}=d\nu^{\prime}\nu^{\prime-1}$ and insert it
in the Cartan-Maurer equation (3.11) we should reach the
second-order field equations (2.6) of the theory. One
immediately observes that the calculation of the Cartan form
$\mathcal{G}^{\prime}$ starting from the definition of the coset
map (3.10) needs the specification of the algebra structure
of the generators $Y,\>K,\>\widetilde{Y},\>\widetilde{K}$. As a
matter of fact this is the mechanism we need to derive the algebra
structure of the non-linear realisation. If one calculates the
Cartan form $\mathcal{G}^{\prime}$ in terms of the unknown
structure constants of the algebra of the generators by using the
identities (3.4) and then inserts it in (3.11); by
comparing the result with the second-order field equations
(2.6) one can read the desired structure constants. We should
remark that to be able to use the identities (3.4) we assume
that we choose a matrix representation for the algebra generated
by the generators of (3.1) and (3.2). Performing the
above mentioned calculation we find that the only non-vanishing
commutators of the algebra of the generators (3.1) and
(3.2) are
$$\displaystyle[K,Y]=Y\quad,\quad[K,\widetilde{Y}]=-\widetilde{Y},$$
$$\displaystyle=\widetilde{K}.$$
(3.12)
Now by using the matrix identities (3.4), also the
commutators of (3.12) we can calculate the Cartan form
$\mathcal{G}^{\prime}=d\nu^{\prime}\nu^{\prime-1}$ of the coset
map (3.10) explicitly as
$$\displaystyle\mathcal{G}^{\prime}$$
$$\displaystyle=d\nu^{\prime}\nu^{\prime-1}$$
(3.13)
$$\displaystyle=d\phi\,K\>+\>e^{\phi}\,dA\,Y\>+\>e^{-\phi}\,d\widetilde{A}\>\,%
\widetilde{Y}$$
$$\displaystyle\quad+(\>d\widetilde{\phi}\>+\>A\,\wedge\,d\widetilde{A}\>)\,%
\widetilde{K}.$$
The coset construction of the supergravities also produces the
first-order formulation of these theories [2, 16].
The locally integrated first-order field equations are encoded in
the doubled Cartan form $\mathcal{G}^{\prime}$ as a twisted
self-duality condition
$$\ast\mathcal{G}^{\prime}=\mathcal{SG}^{\prime},$$
(3.14)
which it satisfies [3, 4, 5, 13]. In (3.14)
$\mathcal{S}$ is a pseudo-involution of the algebra generated by
the generators in (3.1) and (3.2). For our construction
we define its action on the generators as
$$\displaystyle\mathcal{S}Y=\widetilde{Y}\quad,\quad\mathcal{S}K=\widetilde{K},$$
$$\displaystyle\mathcal{S}\widetilde{Y}=Y\quad,\quad\mathcal{S}\widetilde{K}=K.$$
(3.15)
The general construction of $\mathcal{S}$ for a generic coset
formulation can be referred in [2, 6, 7]. Now, by
using the definition of $\mathcal{S}$ given in (3.15);
inserting (3.13) in (3.14) gives
$$\displaystyle e^{\phi}\,\ast dA=e^{-\phi}\,d\widetilde{A},$$
$$\displaystyle\ast d\phi=d\widetilde{\phi}+A\wedge d\widetilde{A}.$$
(3.16)
We see that these are the same equations with (2.7) of
section two which have been obtained from the second-order field
equations (2.6) by differential algebraic integration.
However, here they are obtained through the coset construction
which includes the definition of the coset map (3.10), the
algebra structure of (3.12) whose generators parameterize the
coset map and the matrix representation chosen for this algebra.
As discussed in [14, 15] the ${\mathcal{N}}=2$, $D=6$
supergravity coupled to a tensor multiplet is the minimal
extension of the ${\mathcal{N}}=2$, $D=6$ supergravity multiplet
to write an invariant lagrangian. Accordingly we conclude that
from the coset construction point of view the algebra derived in
(3.12) is a minimal one thus it plays a special role in the
covariant Lagrangian formulation of the theory.
4 Conclusion
We have obtained the coset formulation of the bosonic sector of
the $\mathcal{N}=2$, $D=6$ supergravity which is coupled to a
tensor multiplet [14, 15]. We have derived the algebra
structure which is used to parameterize a coset map such that the
induced Cartan form realizes the second-order field equations in
the Cartan-Maurer equation. Thus the bosonic field equations of
the $\mathcal{N}=2$, $D=6$ supergravity coupled to a tensor
multiplet are obtained within the geometrical construction of the
non-linear sigma model. The first-order formulation of the theory
is also encoded in the coset construction. The locally integrated
first-order field equations can be found through a twisted
self-duality condition satisfied by the Cartan form
[2, 16].
Our main objective in this work was to construct the algebra which
parameterizes the coset map and which generates the field
equations. The group theoretical structure of our coset
realisation can further be studied separately. In section three we
have stated that the algebra we have constructed can be considered
as a minimal one. This fact may be linked to the minimality of the
tensor multiplet coupling to write an invariant lagrangian. One
may inspect the role of the generators of the two-form field and
its dual in the algebra constructed in section three. After
studying the group theoretical construction of the coset one may
question what it means algebraically and geometrically to
introduce the other generators of the algebra to construct a model
which will enable a Lorentz covariant and an unconstrained
lagrangian.
The non-linear coset construction of this work can be extended to
include the gravity and the fermionic sectors. The dualisation of
the $\mathcal{N}=2$, $D=6$ supergravity which is coupled to other
multiplets can be studied to orient the position of the
$\mathcal{N}=2$, $D=6$ supergravity in the general dualisation
scheme of the supergravity theories and to improve our knowledge
of the global symmetries of these theories.
References
[1]
E. Cremmer, B. Julia and J. Scherk, “ Supergravity theory
in eleven-dimensions ”, Phys. Lett. B76 (1978) 409.
[2]
E. Cremmer, B. Julia, H. Lü and C. N. Pope, “ Dualisation of dualities II : Twisted self-duality of doubled fields and
superdualities ”,
Nucl. Phys. B535 (1998) 242, hep-th/9806106.
[3]
T. Dereli and N. T. Y$\i$lmaz, “ Dualisation of the
Salam-Sezgin d=8 supergravity ”, Nucl. Phys. B691 (2004)
223.
[4]
N. T. Y$\i$lmaz, “ Dualisation of the $d=7$ heterotic
string ”, JHEP 0409 (2004) 003.
[5]
N. T. Y$\i$lmaz, “ Dualisation of the $d=9$ matter
coupled supergravity ”, JHEP 0506 (2005) 031.
[6]
N. T. Y$\i$lmaz, “ Dualisation of the general scalar
coset in supergravity theories ”, Nucl. Phys. B664
(2003) 357, hep-th/0301236.
[7]
N. T. Y$\i$lmaz, “ The non-split scalar coset in
supergravity theories ”, Nucl. Phys. B675 (2003) 122.
[8]
T. Dereli and N. T. Y$\i$lmaz, “ Dualisation of the
symmetric space sigma model with couplings ”, Nucl. Phys.
B705 (2005) 60.
[9]
E. Cremmer, B. Julia,
H. Lü and C. N. Pope, “ Dualisation of dualities. I.
”, Nucl. Phys. B523 (1998) 73, hep-th/9710119.
[10]
C. N. Pope, “ Lecture Notes on Kaluza-Klein Theory ”,
(unpublished).
[11]
C. M. Hull and P. K. Townsend, “ Unity of superstring
dualities ”, Nucl. Phys. B438 (1995) 109,
hep-th/9410167.
[12]
E. Witten, “ String theory dynamics in various dimensions
”, Nucl. Phys. B443 (1995) 85, hep-th/9503124.
[13]
N. T. Y$\i$lmaz, “ Non-linear realisation of the pure
$\mathcal{N}=4$, $D=5$ supergravity ”, Phys. Rev. D72
(2005) 105013, hep-th/0511303.
[14]
N. Marcus and J. H. Schwarz, “ Field theories that have
no manifestly Lorentz invariant formulation ”, Phys. Lett.
B115 (1982) 111.
[15]
F. Giani, M. Pernici and P. van Nieuwenhuizen, “ Gauged
N=4 supergravity in six dimensions ”, Phys. Rev. D30
(1984) 1680.
[16]
B. Julia, “ Superdualities: below and beyond U-duality
”, LPTENS 00/02 (2000), hep-th/0002035.
[17]
H. Nishino and E. Sezgin, “ Matter and gauge couplings of
N=2 supergravity in six dimensions ”, Phys. Lett. B144
(1984) 187.
[18]
R. D’Auria, P. Fre and T. Regge, “ Consistent supergravity in
six dimensions without action invariance ”, Phys. Lett. B128 (1983) 44.
[19]
P. K. Townsend, “ A new anomaly free chiral supergravity theory from
compactification on K3 ”, Phys. Lett. B139 (1984) 283. |
DOD-ETL: Distributed On-Demand ETL for Near Real-Time Business Intelligence
Gustavo V. Machado${}^{1}$, Ítalo Cunha${}^{1}$, Adriano C. M. Pereira${}^{1}$, Leonardo B. Oliveira${}^{1,2}$
Abstract
The competitive dynamics of the globalized market demand information on the internal and external reality of corporations. Information is a precious asset and is responsible for establishing key advantages to enable companies to maintain their leadership. However, reliable, rich information is no longer the only goal. The time frame to extract information from data determines its usefulness. This work proposes DOD-ETL, a tool that addresses, in an innovative manner, the main bottleneck in Business Intelligence solutions, the Extract Transform Load process (ETL), providing it in near real-time. DOD-ETL achieves this by combining an on-demand data stream pipeline with a distributed, parallel and technology-independent architecture with in-memory caching and efficient data partitioning. We compared DOD-ETL with other Stream Processing frameworks used to perform near real-time ETL and found DOD-ETL executes workloads up to 10 times faster. We have deployed it in a large steelworks as a replacement for its previous ETL solution, enabling near real-time reports previously unavailable.
1 Introduction
Today, there is a dire need for organizations to find new ways to succeed in an increasingly competitive market. There is no simple answer on how to achieve this goal. One thing is patently true, though: organizations must make use of near real-time and reliable information to thrive in the global market.
Business Intelligence (BI) is a term used to define a variety of analytical tools that provide easy access to information that support decision-making processes [17]. These tools perform collection, consolidation, and analysis of information, enabling analytical capabilities at every level inside and outside a company. Putting it another way, BI allows collected data to be unified, structured, and thus presented in an intuitive and concise manner, assisting organizations in corporate decision-making.
The Extract Transform Load (ETL) pipeline is a vital procedure in the Business Intelligence (BI) workflow. It is the process of structuring data for querying or analysis. ETL is made up of three stages, namely: data extraction, data transformation, and data loading where, respectively, data is extracted from their sources, structured accordingly, and finally loaded into the target data warehouse. Two processing strategies can be used in ETL process: (1) Batch and (2) Stream processing. The difference between them resides in whether the source data is bounded, by known and finite size, or unbounded (arriving gradually over time).
The integration of production systems and BI tools, which is a responsibility of ETL processes, “is the most challenging aspect of BI, requiring about 80 percent of the time and effort and generating more than 50 percent of the unexpected project costs” [34]. For all that, ETL is deemed a mission-critical process in BI and deserves close attention. Getting current, accurate data promptly is essential to the success of BI applications. However, due to the massive amount of data and complex operations, current ETL solutions usually have long run times and therefore are an obstacle to fulfilling BI’s requirements.
The main challenges of ETL lie on performance degradation at data sources during data extraction, and on performing complex operations on large data volumes in short time frames. The ideal solution has two conflicting goals: (1) cause no impact on data sources and (2) process data in near real-time as they are generated or updated. Ideal solutions should handle high-volume input data rates and perform complex operations in short time frames while extracting data with no operational overhead. Both batch and near real-time ETL process and its main characteristics are shown in Figure 1.
Sabtu et al. [27] enumerate several problems related to near real-time ETL and, along with Ellis [8], they provide some directions and possible solutions to each problem. However, due to these problems complexity, ETL solutions do not always address them directly: to avoid affecting efficiency on transaction databases, ETL processes were usually run in batches and off-hours (i.e., after midnight or weekends) and, by avoiding peak hours, the impact on mission-critical applications is mitigated. However, in a context where the delay to extract information from data determines its usefulness, BI tools and decision making are heavily impacted when the ETL process is executed infrequently.
In this paper, we propose DOD-ETL, Distributed On-Demand ETL, a technology-independent tool which combines modern frameworks with custom optimizations and multiple strategies to provide a near real-time ETL. DOD-ETL has minimum impact on the source database during data extraction, delivers a stream of transformed data to the target database at the same speed as data is generated or updated at the source, and provides scalability, being able to respond to data and complexity growth. We achieve all this by synergistically combining multiple strategies and technologies that were once used separately (e.g., in [18, 21, 37, 13]): log-based Change Data Capture (CDC), stream processing, cluster computing, an in-memory data store, a buffer to guarantee join consistency along with efficient data partitioning and an unified programming model. DOD-ETL works in a distributed fashion and on top of a Stream Processing framework, optimizing its performance.
We have developed a DOD-ETL prototype based on Kafka [15], Beam [1] and Spark Streaming [36]. We evaluate DOD-ETL’s performance executing the same workload on a Stream Processing framework with and without DOD-ETL. We have found that our solution, indeed, provides better performance when compared to an unmodified stream processing framework, being able to execute workloads up to 10 times faster without losing its core features: horizontal scalability and fault-tolerance. We have also tested it in a large steelworks as a replacement for its previous ETL solution. DOD-ETL has been able to speed up the ETL process from hours to less than a minute. This, in turn, enabled important near real-time reports that were previously impractical.
Our key contributions are: (1) a robust study and bibliographic review of BI and the ETL process; (2) the design and development of a general-use tool called DOD-ETL, using state-of-the-art messaging, cluster computing tools and in-memory databases; (3) application of DOD-ETL to the steel sector.
The remainder of this work is organized as follows. First, we discuss related work in Section 2. Then, Section 3 presents DOD-ETL and the assumptions under which it has been designed, detailing its implementation and optimization. We evaluate performance, scalability, and fault-tolerance in Section 4. And finally, we summarize our main findings and propose future work in Section 5.
2 Related Work
Due to the increasing pressure on businesses to perform decision-making on increasingly short time frames, data warehouses are required to be updated in real-time or on the shortest possible interval, not waiting for lower workload periods. This requirement leads to the concept of near real-time ETL, a challenging and important topic of research, whose primary definitions, problems and concepts were defined by Vassiliadis and Simitsis [32].
Other emerging concepts related to near real-time ETL are active data warehouses [30], [14] and real-time BI [2], [28], [24]. Both describe a new paradigm for business intelligence, in which data is updated in near real-time and both decision-making process as well as actions are performed automatically.
Both Wibowo [35] and Sabtu et al. [27] identify problems and challenges for developing near real-time ETL systems. Along with Ellis [8], they provide some directions within this topic for researchers and developers, proposing possible solutions to each challenge. According to Wibowo [35], there are problems on each stage of the ETL process that need to be handled to achieve near real-time ETL. Namely, data source overload, integration of heterogeneous sources, handling master data, and need for computational power to perform prompt operations.
Ellis [8] explains that dealing with near real-time ETL requires three key features to be addressed at each ETL phase: high availability, low latency, and horizontal scalability, which ensure that data will flow and be available constantly, providing up-to-date information to the business. They also enable ETL to always improve performance when needed by adding more servers to the cluster. Sabtu et al. [27] also point out the challenges that near real-time solutions faces but focus on multiple and heterogeneous data sources, data backups, inconsistency, performance degradation, data source overload and master data overhead.
Table 1 sums up these solutions and related problems, also described in detail below.
Mesiti et al. [18] take advantage of the Stream Processing paradigm, where processors can work concurrently, to integrate multiple and heterogeneous sensor data sources. This work takes the integration challenge to a more severe domain, where heterogeneity and multiplicity of data sources are accentuated: sensors as data sources. Naeem et al. [21] address near real-time ETL by proposing an event-driven near real-time ETL pipeline based on push technology and with a database queue. They also tried to minimize master data overhead by designing a new approach that manages them during the transformation stage: they divided data into master data, which are more static, and transactional data, which changes more often, and stored that master data on a repository. This strategy made its use more efficient during the transformation step.
Zhang et al. [37] proposes a new framework to process streaming data in healthcare scientific applications. Their framework (1) enables integration between data from multiple data sources and with different arrival rates and (2) estimates workload so it can plan for computational resources to scale appropriately. They extended the Hadoop [7] Map-Reduce framework to address the streaming data varied arrival rate. To estimate the unknown workload characteristics, they propose two methods to predict streaming data workload: smoothing and Kalman filter.
Waas et al. [33] propose a different approach for ETL with streaming. They first import raw records from data sources and only proceed with transformation and data cleaning when requested by reports, turning ETL into ELT. Therefore, data is transformed on-demand processing different data sets at different moments as required. To achieve this capability, they developed a monitoring tool to alert all connected components about new incoming data.
Many works focused on the master data overhead problem, where joins between the data stream and master data can lead to performance bottlenecks. To minimize these bottlenecks, they proposed optimizations strategies to these joins [25, 26, 3, 22].
As for data source overhead, Jain et al. [13] compared CDC proposals using two metrics to identify data source overload. They concluded that log-based CDC was the most efficient, having minimal impact on database performance and minimizing data loss.
The above-mentioned publications propose strategies to overcome the challenges related to the three main features of near real-time ETL solutions (high availability, low latency and horizontal scalability). Each one focusing on a specific problem and not all of them at once. However, the Stream Processing paradigm appears as a common denominator among most solutions, which is consistent with near real-time ETL requirements.
In this stream-based application context, multiple stream processing frameworks facilitate the development of such applications or are used directly to solve near real-time ETL requirements. Most of these frameworks are based on the record-at-a-time processing model, in which each record is processed independently. Examples of stream processors frameworks that use this model are Yahoo!’s S4 [23] and Twitter’s Storm [31].
In contrast to this record-at-a-time processing model, another possibility is to model it as a series of small deterministic batch computations, as proposed by Zaharia et al. [36] as the Spark Streaming framework. This way, among others benefits, integration with a batch system is made easy and performance overhead, due to record-at-a-time operations, is avoided. Flink [4] is currently competing with Spark Streaming as the open source framework for heavyweight data processing. It also merges, in one pipeline, stream and batch processing and it has features such as flexible windowing mechanism and exactly once semantics.
Besides these open source frameworks, there are those offered by cloud providers as services such as Google Dataflow [10] and Azure Stream Analytics [19]. By using these services, it is possible to avoid the installation and configuration overhead and the resources allocation to horizontal scalability gets simpler.
These open source frameworks and stream processing services are both designed for general use. Due to this one size fits all architecture, they lack strategies and optimizations that are used by DOD-ETL. In addition, while the above-mentioned publications propose solutions to a specific problem or challenge, DOD-ETL tries to combine them in a single tool.
3 DOD-ETL
DOD-ETL relies on an on-demand data stream pipeline with a distributed, parallel and technology-independent architecture. It uses Stream Processing along with an in-memory master data cache to increase performance, a buffer to guarantee consistency of join operations on data with different arrival rates, and a unified programming model to allow it to be used on top of a number of Stream Processing frameworks. Besides, our approach takes advantage of (1) data partitioning to optimize parallelism, (2) data filtering, to optimize data storage on DOD-ETL’s In-memory cache and (3) log-based CDC, to process data on-demand and with minimum impact on the source database. Therefore, DOD-ETL has: minimum impact on the source database during extraction, works in near real-time, can scale to respond to data and throughput growth and can work with multiple Stream Processing frameworks.
The insights that enable DOD-ETL to achieve the features mentioned above are: On-demand data stream—as data changes on the source database, they are processed and loaded into the target database, creating a stream of data where the ETL process handles, in a feasible time frame, only the minimum amount of necessary data. Distributed & parallel—perform all steps in a distributed and parallel manner, shortening processing time and enabling a proper use of the computer resources. In-memory cache—perform data transformations with no look-backs on the source database, providing all required data to execute calculations in the In-memory cache which avoids expensive communications with the data source. Unsynchronized consistency—a buffer guarantees that data with different arrival rates can be joined during transformation. Unified programming model—a programming model used to build data pipelines for both batch and streaming in multiple Stream Processing frameworks.
DOD-ETL merges into a single solution multiple strategies and techniques to achieve near real-time ETL that have are have never been integrated to this extent: log-based Change Data Capture (CDC), stream processing, cluster computing, and an in-memory data store along with efficient data partitioning (c.f. Section 2). We note that although these techniques have indeed been used before (e.g., in [18, 21, 37, 13])) they have not been previously integrated into a single solution. By synergistically combining these strategies and techniques DOD-ETL can achieve near real-time ETL.
3.1 Architecture
DOD-ETL has the following workflow (Figure 2): (1) it tracks changes on each source system’s database table and (2) sends these changes as messages to a (3) message queue, following a preconfigured partitioning criteria. Then, (4) these messages are pulled from the message queue and sent to the In-memory cache or (5) transformed to the target reporting technology format, and, finally, (6) loaded into the target database. DOD-ETL’s workflow can be grouped into two central modules: Change Tracker and Stream Processor.
All steps depend on configuration parameters to work properly. Thus, during DOD-ETL’s deployment, it is imperative to go through a configuration process, where decisions are made to set the following parameters: tables to extract—define which tables will have data extracted from; table nature—from the defined tables, detail which ones are operational (constantly updated) and which ones are master data (more static); table row key—column, from each table, that contains each row’s unique identifier; business key—column, from each table, that contains the values used to partition or filter data by a domain-specific key.
3.1.1 Change Tracker
The Change Tracker makes available to the Stream Processor module, at the same time that events occur, any data altered and added to the source database. This module makes use of CDC, a standard database pattern that contains all operations carried out over time and their respective values. Thus, whenever a record is inserted, altered, or removed from a table, the CDC writes that event, together with all of the record’s information.
CDC can take many forms, ranging from log files to structured tables in the database. They can be generated either by the database itself or by an external system. In this work, to be adherent to Jain et al.’s recommendations [13], log-based CDC was used. However, as explained later, the CDC reading step in DOD-ETL is performed by the Listener; thanks to DOD-ETL’s modular architecture, it is possible to create different Listeners to support different CDC implementations.
The Change Tracker works in three stages called Listener, Message Producer, and Message Queue. The Listener stage listens to the CDC changes and, with each new record, extracts its data to be further processed by Message Producer. This stage was built to extract data independently among tables, allowing it to be parallelized. As a prerequisite, then, Listener expects the CDC log to support this feature. The Listener step has minimum impact on the source database’s performance due to two factors: (1) only new and altered records are extracted at each execution, representing a lower data volume, and (2) queries are performed in log files only, which takes the pressure off the database and production tables. As this step has been designed to be decoupled from the others, it can be rebuilt to meet the specific characteristics of the source database, such as different technologies and CDC implementations.
The Message Queue works as a message broker and uses the publish/subscribe pattern with partitioning features, in which each topic is partitioned by its messages key.
On DOD-ETL, a topic contains all insertions, updates and deletions of a table. The topic partitioning criteria vary by the table nature that the topic represents (master or operational). When dealing with master data tables, each topic is partitioned by its respective table unique row identifier and, when dealing with operational data, each topic is partitioned by the business key. This partitioning occurs at the Message Producer, before it publishes each extracted data, based on the aforementioned configuration parameters. Therefore, Message Producer builds messages from data extracted by the Listener and publishes them in topics on the Message Queue according to the preconfigured parameters. These two partitioning strategies (by row key and by business key) have the following objectives:
Row key: since the table unique row identifier is the topic partition key, Message Queue guarantees that the consumption of messages from a specific table row will happen in the right sequence. By calling the last message from each partitioning key (row id), it is possible to reconstruct the most recent table snapshot. In other words and, as shown in Figure LABEL:tts, each topic contains all insertions, updates and deletions from a table and it is partitioned by its id column. The last value from all partitions key represents the current table state.
Business key: the Stream Processor transformation process is parallelized based on the number of partitions defined by operational topics and their business keys. Therefore, the partitioning criteria has a direct impact on DOD-ETL’s performance. In this sense, it is paramount to understand in advance the extracted data and the nature of operations that will be performed by Stream Processor. The goal is to figure out the partitioning criterion and its key, because they may vary depending on the business nature.
3.1.2 Stream Processor
The Stream Processor receives data from the Listener by subscribing to all topics published on Message Queue, receiving all recently changed data as message streams. Stream Processor comprises three steps: (1) In-memory Table Updater, (2) Data Transformer and (3) Target Database Updater.
In-memory Table Updater prevents costly look-backs on the source database by creating and continuously updating distributed in-memory tables that contains supplementary data required to perform a data transformation. Data flowing from topics representing master data tables goes to In-memory Table Updater step. In-memory Table Updater only saves data related to the business keys assigned to its corresponding Stream Processor node, filtering messages by this key. By doing so, only data from keys that will be processed by each node are saved in its in-memory table, taking off pressure from memory resources. In case of node failures, data loss or both, Stream Processor retrieves a snapshot from Message Queue and repopulates in-memory tables that went down. This is possible due to the way each Message Queue master data topic is modeled: it is partitioned by the table’s unique row identifier, allowing In-memory Table Updater to retrieve, from Message Queue, an exact snapshot of this topic table.
Data Transformer receives data and performs operations to transform it into the required BI report format. Data Transformer can run point-in-time queries on the in-memory tables to fetch missing data necessary to carry out its operations, joining streaming and static data efficiently. Each partition is processed in parallel, improving performance. The operations executed in Data Transformer rely on the operators of the cluster computing framework (e.g., map, reduce, filter, group) and are business-dependent. Like In-memory Table Updater, not all messages go through this module, only messages from tables configured as operational. In the event of master data arriving after operational data (master and operational messages are sent to different topics and by different Listener instances), Data Transformer uses a buffer to store this late operational message for late reprocessing. At each new operational message, Data Transformer checks the buffer for late messages and reprocesses them, along with the current one. To optimize performance, Data Transformer only reprocesses buffer messages with transaction dates older than the latest transaction date from the In-memory cache, which avoids reprocessing operational messages that still have no master data As shown in more details in Section 4, DOD-ETL performance is highly dependant on the data model and transformation operations complexity. That is, the data and how it is transformed, in this case by Stream Processor, will determine DOD-ETL processing rate.
Target Database Updater translates the Data Transformer’s results into query statements and then loads the statements into the target database. For performance, the loading process also takes place in parallel and each partition executes its query statements independently.
3.2 Scalability, Fault-tolerance and Consistency
DOD-ETL takes advantage of the adopted cluster computing framework’s and the message queue’s native support for scalability and fault-tolerance. For the former, both DOD-ETL modules can handle node failures, but with different strategies: while Change Tracker focuses on no message loss, Stream Processor concentrates on no processing task loss. As for scalability, both depend on efficient data partitioning to scale properly.
Despite these inherited capabilities, some features had to be implemented on Stream Processor so fault-tolerance and scalability could be adherent to DOD-ETL’s architecture: the In-memory cache and the Operational Message Buffer have to be able to retrieve data from new nodes or from failed nodes. Regarding the In-memory cache, we implemented a trigger that alerts In-memory Table Updater when Data Transformer’s assigned business keys changes. On a change event, In-memory Table Updater resets the In-memory cache, dumps the latest snapshot from Message Queue and filters it by all assigned business keys. By doing so, the In-memory cache keeps up with Stream Processor reassignment process on failure events or even when the cluster scales up or down. As for the Operational Message Buffer, it uses a distributed configuration service, that already comes with Message Queue, to save its values so, in case of a node failure, other Stream Processor instances can keep up with reprocessing. That is, Operational Message Buffer saves all operational messages with late master data and, at each new message, Data Transformer tries to reprocess this buffer by checking at the In-memory cache if its respective master data arrived, solving the out-of-sync message arrival problem.
Since DOD-ETL focuses on delivering data to information systems, not operational mission-critical systems, transactional atomicity was not a requirement and it was left out of the scope of this work: rows of different tables added/altered in the same transaction can arrive at slightly different time frames. However, due to its late operational messages buffer, DOD-ETL can guarantee that only data with referential integrity will be processed and that those that are momentarily inconsistent will be eventually processed. As stated before, Data Transformer reprocesses all messages from the Operational Message Buffer when all needed data arrives on the In-memory cache.
3.3 Implementation
DOD-ETL is a tool with many parts: CDC, Listener, Message Queue, Stream Processor. While the Listener was built from scratch, CDC and Message Queue are simply out of the shelf products, and Stream Processor is a Stream Processing Framework with customizations and optimizations built on top of a unified programming model. Its implementation and used technologies are explained next.
All data exchanged between DOD-ETL modules are serialized and deserialized by the Avro system [6]. The Message Queue role was performed by Kafka [15], an asynchronous real-time message management system, whose architecture is distributed and fault-tolerant. Due to Kafka’s dependency on Zookeeper [11], it was also used on DOD-ETL.
Stream Processor was implemented with Beam [1], a unified programming model, which allows it to be used on top of Stream Processing frameworks such as Spark Streaming and Flink. Its steps, Data Transformer, In-memory Table Updater and Target Database Updater, were all encapsulated together to make communication between them easy. Data Transformer takes advantage of the already deployed Zookeeper to store its late operational messages buffer. It does so to guarantee that, in any failure event, another Stream Processor node could keep processing those late messages.
Regarding the In-memory cache, H2 [20] was used and deployed as an embedded database on the Stream Processor application. To support DOD-ETL’s needs, H2 was configured to work in-memory and embedded so, for each Spark worker, we could have an H2 instance with fast communication. Our prototype and its modules are publicly available.111https://github.com/gustavo-vm/dod-etl
Since DOD-ETL modules were designed to be decoupled, each one can be altered and replaced without impacting the others. Adding to this its technological-independent features, all selected technologies on each module can be replaced, provided that its requirements are satisfied.
4 Evaluation
We used, as a case study, a large steelworks and its BI processes. In this case, a relational database, powered by both its production management system and shop floor level devices, was used as the data source. This steelworks has the following characteristics: total area of 4,529,027 m${}^{2}$, a constructed area of 221,686 m${}^{2}$, 2,238 employees, and it is specialized in manufacturing Special Bar Quality (SBQ) steel.
This steelworks uses OLAP reports [5] as its BI tool. DOD-ETL was used to provide near real-time updates to these reports, which were unavailable; prior to DOD-ETL’s deployment reports were updated twice a day. DOD-ETL’s purpose was to extract data from the source database and transform them into OLAP’s expected model, known as star schema [9]. This transformation involves calculations of Key Process Indicators (KPIs) of this steelworks’ process. For this case study, the Overall Equipment Effectiveness (OEE) [29], developed to support Total Productive Maintenance initiatives (TPM) [16], and its related indicators (Availability, Performance, and Quality) were chosen as the steelworks KPIs.
TPM is a strategy used for equipment maintenance that seeks optimum production by pursuing the following objectives: no equipment breakdowns; no equipment running slower than planned; no production loss.
OEE relates to TPM initiatives by providing an accurate metric to track progress towards optimum production. That is, the following KPIs can quantify the above three objectives: availability—measures productivity losses, tracking the equipment downtime vs. its planned productive time; performance—tracks the equipment actual production speed vs. its maximum theoretical speed; quality—measures losses from manufacturing defects. These three indicators together result in the OEE score: a number that provides a comprehensive dimension of manufacturing effectiveness.
DOD-ETL extracted only the data needed to calculate these indicators. For this industry context, we grouped them into the following categories: production data—information of production parts; equipment data—equipment status information; quality data—produced parts quality information. During the DOD-ETL configuration process, two decisions were made: we defined the nature of each table (operational and/or master data) and decided which table column would be considered as the Stream Processor business partitioning key. Regarding the table nature, we considered the production group as operational and equipment and quality as master data. Due to this decision, all equipment and quality data sent to Stream Processor will be stored on its In-memory cache while production data will go straight to the Data Transformer step of the Stream Processor.
As for the business key, we considered the production equipment unit identifier, since all KPIs are calculated for it. We set, then, the column that represents and stores the related equipment unit code on each table as the business key in the configuration. This column will be used by operational topics for partitioning and by the In-memory cache as filter criteria.
For this industry context, Data Transformer splits data as a requisite to support OLAP multidimensional reports. For the above mentioned KPIs, we defined the splitting criteria as its intersections in the time domain, in which the lowest granularity represents the intersection between all the data obtained for the equipment unit in question: Production, Equipment status and Quality. Figure 3 shows an example of this intersection analysis and data splitting. In this case, Data Transformer searches for intersections between equipment status and production data and breaks them down, generating smaller data groups called fact grain. In the Figure 3 example, these fact grains can be divided in two groups: (1) equipment with status ”off” and (2) equipment with status ”on” with production. As stated before, after the splitting process is completed, Data Transformer performs the calculations.
4.1 Experiments
To evaluate our DOD-ETL prototype’s performance, we used the Spark Streaming framework [36] as the baseline. We generated a synthetic workload, simulating the data sent by the steelworks equipment, and executed Spark with and without DOD-ETL on top of it. We have also performed experiments to check if DOD-ETL achieved Ellis [8] key features (high availability, low latency and horizontal scalability) and, as a differential of our work, we executed DOD-ETL with production workloads from the steelworks to check its behavior in a complex and realistic scenario.
In sum, we evaluated (1) DOD-ETL vs Baseline, checking how Spark performs with and without DOD-ETL on top of it; (2) horizontal scalability, analyzing processing time when computational resources are increased; (3) fault tolerance, evaluating DOD-ETL’s behavior in the event of failure of a compute node; (4) DOD-ETL in production, comparing its performance against real workloads and complex database models from the steelworks.
We used Google Cloud and resources were scaled up as needed, except for the fifth experiment that used the steelworks’s computing infrastructure. The following hardware and configurations were used: database – MySQL (with CDC log activated) deployed on an 8-core 10 GB memory instance; sampler – 20-core 18 GB instance; Change Tracker – 10-core 18 GB instance; Message Queue – seven instances, three for Kafka brokers and three for Zookeeper, with one core and 2 GB of memory each; Stream Processor – we deployed Spark in standalone mode with one master node and multiple worker nodes, which varied from one to twenty instances depending on the experiment (all Stream Processor instances had one core and 2 GB of memory each). All Google Cloud Platform instances had Intel Haswell as their CPU platform and hard disk drives for persistent storage. To represent the three data categories cited before, we used one table per data model group on experiments 1, 2 and 3. Regarding the fourth experiment, we used a more complex data model based on the ISA-95 standard [12].
Also for experiments 1, 2 and 3, as mentioned above, we built a sampler to insert records on each database table. This sampler generates synthetic data, inserting 20,000 records at each table, simulating the steelworks operation. To minimize the impact of external variables in these experiments, the Listener started its data extraction after the sampler finished its execution. To avoid impact on the results, Change Tracker extracted all data before the execution of Stream Processor, so Message Queue could send data at the same pace as requested by Stream Processor. Since Listener depends on the CDC log implementation, its data extraction performance also depends on it. We used MySQL as the database and its binary log as the CDC log.
4.1.1 DOD-ETL vs Baseline
Since DOD-ETL is comprised of out-of-the-shelf components, each module following a decoupled architecture. To evaluate DOD-ETL’s performance, each module needed to be analyzed separately.
As said before, Listener is highly dependant on the used database and CDC implementation and its performance is tied to them. Therefore, Listener has the CDC throughput as its baseline. Since the complete flow from writing to the database to copying it to the binary incurs a lot of I/O, Listener will always be able to read more data than CDC can provide.
Since Message Queue is an out-of-the-shelf product and can be replaced by any other product, provided that its requirements are satisfied, it can keep up with new benchmarks. As for now, Message Producer and Message Queue are instances of Kafka producers and Kafka brokers, respectively. Kreps et al. [15] already demonstrated its performance against other messaging systems.
The Stream Processor includes substantial customizations (Data Transformer, In-memory Table Updater and Target Database Updater) on top of the Spark Streaming framework. We evaluated its performance executing the same workload against Spark Streaming with and without DOD-ETL. We used a fixed number of Spark worker nodes (Table 2): a cluster with ten nodes.
While DOD-ETL was able to process 10,090 records per second, Spark Streaming alone processed ten times fewer records (1,230 records/s). In order to understand in detail this performance difference, we analyzed the Spark job execution log, that shows the processing time in milliseconds of each Spark Worker task (Figure 4). Note that DOD-ETL has an initialization overhead at each Spark worker when it is started and when a new key or partition is assigned. Regarding the used dataset and infrastructure, this initialization overhead costs 40 seconds for each Worker.
Throughout these experiments, we were able to demonstrate that DOD-ETL customizations make Spark significantly faster. Although it has an initialization overhead, due to the In-memory Table Updater data dump from Message Queue, it is minimal and negligible considering the volume of processed messages.
4.1.2 Scalability
We evaluated the scalability of DOD-ETL’s Change Tracker and Stream Processor modules. Listener scalability, as stated before, also depends on the CDC log implementation. We performed two different experiments to evaluate Listener’s performance: (1) Inserted data on extracted tables only, where insertions were made only in databases that we were extracting data, so the number of inserted and extracted tables increased at the same time (from 1 to 16); and Fixed number of inserted tables, where insertions were made in a fixed number of tables (16) and the number of extracted tables was increased (from 1 to 16).
As shown in Figure 5, where number of records inserted per second was plotted against the number of tables, Listener had different results on each experiment: When inserting data on extracted tables only, Listener’s performance increased as a sublinear function and then saturated at 18,600 records per second for eight tables. When using a fixed number of tables, it increased linearly until it also saturated when extracting simultaneously from eight tables, with a throughput of 10,200 records per second. This behavior is directly related to MySQL’s CDC implementation, where it writes changes from all tables on the same log file, so each Listener instance had to read the whole file to extract data from its respective table. Throughput is higher in the first experiment compared to the second experiment because of the difference in log file size: while in the fixed insertions experiment the CDC log file had a fixed size of 320,000 records, the varying insertion experiment log file varied from 20,000 records to 320,000 records. Therefore, going through the whole log file took less time until it matched at 16 tables. Regarding the saturation point, this happened due to MySQL performance and we conjecture it will vary across different databases and CDC implementations.
As already stated, Message Producer and Message Queue are instances of Kafka producers and Kafka brokers, respectively. Kreps et al. [15] already demonstrated that each producer is able to produce orders of magnitude more records per second than Listener can process. Regarding its brokers, their throughput is dictated more by hardware and network restrictions than by the software itself, also enabling it to process more records.
To evaluate Stream Processor’s scalability, we varied the number of available Spark worker nodes from one to twenty and fixed the number of partitions on the operational topic at twenty. To maximize parallelization, the number of equipment units (partition keys) from the sampled data followed the number of topic partitions: sampled data contained 20 equipment unit identifiers, used as partition keys. We used the Spark Streaming Query Progress’ metric average processed rows per second at each of its mini batch tasks. As shown in Figure 6, where the number of processing records per seconds was plotted against the number of Spark Workers.
DOD-ETL modules are scalable: both Change Tracker and Stream Processor can absorb growth by adding more computational resources, despite their difference in throughput and scalability factor. While Change Tracker scales proportionally to the number of tables, Stream Processor scales with the number of partition keys on operational tables.
Regarding Change Tracker, we saw that Listener and Message Producer can process tables independently and that it can scale up as the process incorporates new ones, provided that the database CDC log supports it. As for Message Queue, it also scales linearly but based on multiples variables: the number of available brokers, extracted tables (topics), partitions, and keys.
Stream Processor’s scalability is proportional to the number of partitions at the operational table topics and the number of partitioning keys that, on this steelworks case, are the total number of production equipment units. Data partitioning plays a vital role here, so, it is imperative to understand functionally and in advance all extracted data in order to find partitioning criterion and its key, which varies from business to business. Since Message Queue supports throughput orders of magnitude higher than Listener and Stream Processor, it is possible to integrate multiple database sources and use multiple Stream Processor instances, each performing different transformations.
4.1.3 Fault Tolerance
We have executed DOD-ETL in a cluster with five worker nodes and, midway through the experiment, we shut down two of the worker nodes to evaluate DOD-ETL’s fault tolerance. We measured the rate of processed messages before and after the shutdown and performed a data consistency check on the result. We used the same performance metric as the scalability experiment (Table 2).
Stream Processor went from processing 5,060 messages per second to 2,210, representing a processing rate decrease of 57%. After each execution, we checked the consistency of all messages processed by this module and did not find any error: it processed all messages correctly, albeit at a reduced rate. This result indicates that DOD-ETL is fault tolerant, which significantly increases its robustness.
While the number of available clusters was changed from 5 to 3, a 40% decrease, the performance decrease was more significant (57%). By analyzing the Spark execution log, we found that the In-memory cache also impacts fail-over performance: when a node becomes unavailable and a new one is assigned, the In-memory cache has to dump all data from the newly assigned partition keys, which impacts performance.
Since the Change Tracker and the Stream Processor were built on top of Kafka and Spark, respectively, both modules can resist node failures. Due to their different purposes, each module uses distinct strategies: while Kafka can be configured to minimize message loss, Spark can be configured to minimize interruption of processing tasks.
4.1.4 DOD-ETL in production
We executed DOD-ETL with real workloads from the steelworks. This workload is generated by both a production management system and shop floor level devices and its data model is based on the ISA-95 standard, where multiple tables are used to represent each category (production, equipment and quality). We compared DOD-ETL results on previous experiments (Table 2), where synthetic data was used with a simpler data model (a single table for each category of data), with DOD-ETL executing real and complex data.
Both synthetic and production experiments used the same configuration: a cluster with ten Spark worker nodes. While DOD-ETL can process 10,090 records per second for the data source with simple model complexity, this number decreases to 230 records per second for the complex data model. It is possible to state, then, that data model complexity impacts directly on DOD-ETL performance. Since it depends on the In-memory cache to retrieve missing data, when this retrieval involves complex queries, this complexity impacts on the query execution time and, therefore, on DOD-ETL performance.
This steelworks currently uses an ETL solution to perform the same operation performed by DOD-ETL. It adopts a sequential batch approach, comprises a series of procedures ran within the relational database server, and relies on a twelve core /32 GB memory server. While DOD-ETL takes 0.4 seconds to process 100 records, this solution takes one hour. Although it is not a a fair comparison (a streaming distributed and parallel tool vs a batch and legacy solution), it is important to demonstrate that DOD-ETL can be successfully used in critical conditions and to absorb the steelworks data throughput, providing information in near real-time to its BI tools.
Considering the author’s experience in developing mission critical manufacturing systems and its knowledge in the ISA-95 standard, his opinion regarding these systems data modeling is that the drawbacks of using a standardized and general data model, that seeks a single vision for all types of manufacturing processes, far outweigh the benefits. Manufacturing systems that use generalized data models get way more complex when compared with process-specific models. These systems performance, maintenance and architecture get severely impacted in order to provide a generic model.
Therefore, in this industry context, a more straightforward data model could be used in production management systems and shop-floor without drawbacks. With this, DOD-ETL would perform even better, in addition to the factory systems.
5 Conclusion
DOD-ETL’s novelty relies on synergistically combining multiple strategies and optimizations (that were previously only used separately) with an on-demand data stream pipeline as well as with a distributed, parallel, and technology-independent architecture.
ETL systems need to have three key features to work in near real-time: high availability, low latency, and scalability. DOD-ETL has been able to achieve all three key features and address these challenges by combining log-based Change Data Capture (CDC), stream processing, cluster computing, an in-memory data store, a buffer to guarantee join consistency along with efficient data partitioning, and a unified programming model.
We have been able to demonstrate, by performing independent experiments on each of its main modules, that DOD-ETL strategies and optimizations reduce ETL run time by a factor of ten, outperforming unmodified Stream Processing frameworks. Through these experiments, we showed that DOD-ETL achieves these results without sacrificing scalability and fault-tolerance. We have also found that data source model complexity heavily impacts the transformation stage and that DOD-ETL can be successfully used even for complex models.
Due to its technology-independence, DOD-ETL can use a number of Stream Processor frameworks and messaging systems, provided that its requisites are satisfied. This allows DOD-ETL to adapt and evolve as new technologies arrive and avoid technology lock-ins.
Instantiating DOD-ETL requires customizing the Data Transformer step: each required transformation is translated as Spark operators which, in turn, are compiled as a Java application. This requires DOD-ETL’s users to know how to program, restricting its use. To overcome this, on future work, DOD-ETL will be adapted to integrate a friendly user interface with an intuitive and visual configuration of Data Transformer transformation operations.
Also on future work, we will study the impact on DOD-ETL performance when lightweight Stream Processing frameworks are used, such as Kafka Streams and Samza, by performing new experiments. By doing so, we will be able to compare and evaluate the trade-offs between these two types of frameworks (lightweight vs heavyweight) and its impact on DOD-ETL proposed strategies and optimizations.
In sum, through this work, we were able to achieve near real-time ETL by combining multiple strategies and technologies, to propose a general use tool and to evaluate it in the metals industry context.
References
Apache [2015]
Apache.
Apache Beam.
https://beam.apache.org/, 2015.
Azvine et al. [2006]
B. Azvine, Z. Cui, D. D. Nauck, and B. Majeed.
Real time business intelligence for the adaptive enterprise.
In E-Commerce Technology, 2006. The 8th IEEE International
Conference on and Enterprise Computing, E-Commerce, and E-Services, The 3rd
IEEE International Conference on, pages 29–29. IEEE, 2006.
Bornea et al. [2011]
M. A. Bornea, A. Deligiannakis, Y. Kotidis, and V. Vassalos.
Semi-streamed index join for near-real time execution of etl
transformations.
In Data Engineering (ICDE), 2011 IEEE 27th International
Conference on, pages 159–170. IEEE, 2011.
Carbone et al. [2015]
P. Carbone, A. Katsifodimos, S. Ewen, V. Markl, S. Haridi, and K. Tzoumas.
Apache flink: Stream and batch processing in a single engine.
Bulletin of the IEEE Computer Society Technical Committee on
Data Engineering, 36(4), 2015.
Codd et al. [1993]
E. F. Codd, S. B. Codd, and C. T. Salley.
Providing olap (on-line analytical processing) to user-analysts: An
it mandate.
Codd and Date, 32, 1993.
Cutting [2009]
D. Cutting.
Apache Avro.
https://avro.apache.org/, 2009.
Dean and Ghemawat [2008]
J. Dean and S. Ghemawat.
Mapreduce: simplified data processing on large clusters.
Communications of the ACM, 51(1):107–113,
2008.
Ellis [2014]
B. Ellis.
Real-time analytics: Techniques to analyze and visualize
streaming data.
John Wiley & Sons, 2014.
Giovinazzo [2000]
W. A. Giovinazzo.
Object-oriented data warehouse design: building a star schema.
Prentice Hall PTR, 2000.
Google [2015]
Google.
Google Dataflow.
https://cloud.google.com/dataflow/, 2015.
Hunt et al. [2010]
P. Hunt, M. Konar, F. P. Junqueira, and B. Reed.
Zookeeper: Wait-free coordination for internet-scale systems.
In USENIX annual technical conference, volume 8, page 9.
Boston, MA, USA, 2010.
International Society of
Automation [2001]
International Society of Automation.
Enterprise-Control System Integration Part 2 : Object Model
Attributes.
Isa, 2001.
Jain et al. [2012]
T. Jain, S. Rajasree, and S. Saluja.
Refreshing datawarehouse in near real-time.
International Journal of Computer Applications, 46(18):24–29, 2012.
Karakasidis et al. [2005]
A. Karakasidis, P. Vassiliadis, and E. Pitoura.
Etl queues for active data warehousing.
In Proceedings of the 2nd international workshop on Information
quality in information systems, pages 28–39. ACM, 2005.
Kreps et al. [2011]
J. Kreps, N. Narkhede, J. Rao, et al.
Kafka: A distributed messaging system for log processing.
In Proceedings of the NetDB, pages 1–7, 2011.
Ljungberg [1998]
Õ. Ljungberg.
Measurement of overall equipment effectiveness as a basis for tpm
activities.
International Journal of Operations & Production Management,
18(5):495–507, 1998.
Malhotra [2001]
Y. Malhotra.
From information management to knowledge management. beyond
the’hi-tech hidebound’systems.
Knowledge management and business model innovation, pages
115–134, 2001.
Mesiti et al. [2016]
M. Mesiti, L. Ferrari, S. Valtolina, G. Licari, G. Galliani, M. Dao, K. Zettsu,
et al.
Streamloader: an event-driven etl system for the on-line processing
of heterogeneous sensor data.
In Extending Database Technology, pages 628–631.
OpenProceedings, 2016.
Microsoft [2015]
Microsoft.
Azure Stream Analytics.
https://azure.microsoft.com/en-us/services/stream-analytics/,
2015.
Mueller [2012]
T. Mueller.
H2 Database.
http://www.h2database.com/, 2012.
Naeem et al. [2008]
M. A. Naeem, G. Dobbie, and G. Webber.
An event-based near real-time data integration architecture.
In Enterprise Distributed Object Computing Conference
Workshops, 2008 12th, pages 401–404. IEEE, 2008.
Naeem et al. [2010]
M. A. Naeem, G. Dobbie, G. Weber, and S. Alam.
R-meshjoin for near-real-time data warehousing.
In Proceedings of the ACM 13th international workshop on Data
warehousing and OLAP, pages 53–60. ACM, 2010.
Neumeyer et al. [2010]
L. Neumeyer, B. Robbins, A. Nair, and A. Kesari.
S4: Distributed stream computing platform.
In Data Mining Workshops (ICDMW), 2010 IEEE International
Conference on, pages 170–177. IEEE, 2010.
Nguyen et al. [2005]
T. M. Nguyen, J. Schiefer, and A. M. Tjoa.
Sense & response service architecture (saresa): an approach towards
a real-time business intelligence solution and its use for a fraud detection
application.
In Proceedings of the 8th ACM international workshop on Data
warehousing and OLAP, pages 77–86. ACM, 2005.
Polyzotis et al. [2007]
N. Polyzotis, S. Skiadopoulos, P. Vassiliadis, A. Simitsis, and N.-E.
Frantzell.
Supporting streaming updates in an active data warehouse.
In Data Engineering, 2007. ICDE 2007. IEEE 23rd International
Conference on, pages 476–485. IEEE, 2007.
Polyzotis et al. [2008]
N. Polyzotis, S. Skiadopoulos, P. Vassiliadis, A. Simitsis, and N. Frantzell.
Meshing streaming updates with persistent data in an active data
warehouse.
IEEE Transactions on Knowledge and Data Engineering,
20(7):976–991, 2008.
Sabtu et al. [2017]
A. Sabtu, N. F. M. Azmi, N. N. A. Sjarif, S. A. Ismail, O. M. Yusop, H. Sarkan,
and S. Chuprat.
The challenges of extract, transform and loading (etl) system
implementation for near real-time environment.
In Research and Innovation in Information Systems (ICRIIS),
2017 International Conference on, pages 1–5. IEEE, 2017.
Sahay and Ranjan [2008]
B. Sahay and J. Ranjan.
Real time business intelligence in supply chain analytics.
Information Management & Computer Security, 16(1):28–48, 2008.
Stamatis [2010]
D. Stamatis.
The OEE Primer: Understanding Overall Equipment Effectiveness,
Reliability, and Maintainability.
Productivity Press, 1 pap/cdr edition, 6 2010.
ISBN 9781439814062.
URL http://amazon.com/o/ASIN/1439814066/.
Thalhammer et al. [2001]
T. Thalhammer, M. Schrefl, and M. Mohania.
Active data warehouses: complementing olap with analysis rules.
Data & Knowledge Engineering, 39(3):241–269, 2001.
Toshniwal et al. [2014]
A. Toshniwal, S. Taneja, A. Shukla, K. Ramasamy, J. M. Patel, S. Kulkarni,
J. Jackson, K. Gade, M. Fu, J. Donham, et al.
Storm@ twitter.
In Proceedings of the 2014 ACM SIGMOD international conference
on Management of data, pages 147–156. ACM, 2014.
Vassiliadis and Simitsis [2009]
P. Vassiliadis and A. Simitsis.
Near real time etl.
In New trends in data warehousing and data analysis, pages
1–31. Springer, 2009.
Waas et al. [2013]
F. Waas, R. Wrembel, T. Freudenreich, M. Thiele, C. Koncilia, and P. Furtado.
On-demand elt architecture for right-time bi: extending the vision.
International Journal of Data Warehousing and Mining (IJDWM),
9(2):21–38, 2013.
Watson and Wixom [2007]
H. J. Watson and B. H. Wixom.
The current state of business intelligence.
Computer, 40(9), 2007.
Wibowo [2015]
A. Wibowo.
Problems and available solutions on the stage of extract, transform,
and loading in near real-time data warehousing (a literature study).
In Intelligent Technology and Its Applications (ISITIA), 2015
International Seminar on, pages 345–350. IEEE, 2015.
Zaharia et al. [2012]
M. Zaharia, T. Das, H. Li, S. Shenker, and I. Stoica.
Discretized streams: An efficient and fault-tolerant model for stream
processing on large clusters.
HotCloud, 12:10–10, 2012.
Zhang et al. [2015]
F. Zhang, J. Cao, S. U. Khan, K. Li, and K. Hwang.
A task-level adaptive mapreduce framework for real-time streaming
data in healthcare applications.
Future Generation Computer Systems, 43:149–160,
2015. |
On the Kodaira vanishing theorem for log del Pezzo surfaces in positive characteristic
Emelie Arvidsson
EPFL SB MATH CAG
MA C3 615 (Batiment MA)
Station 8
CH-1015 Lausanne
emelie.arvidsson@epfl.ch
On the Kodaira vanishing theorem for log del Pezzo surfaces in positive characteristic
Emelie Arvidsson
EPFL SB MATH CAG
MA C3 615 (Batiment MA)
Station 8
CH-1015 Lausanne
emelie.arvidsson@epfl.ch
Abstract.
We investigate the vanishing of $H^{1}(X,\mathcal{O}_{X}(-D))$ for a big and nef $\mathbb{Q}$-Cartier $\mathbb{Z}$-divisor $D$ on a log del Pezzo surface $(X,\Delta)$ over an algebraically closed field of positive characteristic $p$.
Key words and phrases:Kodaira vanishing, log del pezzo surfaces, positive characteristic
2010 Mathematics Subject Classification: 14E30, 14F17, 14J45, 13A35
The author was supported by SNF Grant #200021/169639.
Introduction
It has long been known that the (Kodaira and Kawamata–Viehweg) vanishing theorems, so fundamental to birational geometry in characteristic zero, in general fail for surfaces in positive characteristic [Ray78]. Several people have investigated different classes of surfaces over perfect fields of positive characteristic for which the (Kodaira and Kawamata–Viehweg) vanishing theorems may or may not hold. The question is subtle; for example, every smooth rational surface over an algebraically closed field satisfies Kodaira vanishing, however over any algebraically closed field of positive characteristic, there are smooth rational surfaces that violate Kawamata–Viehweg vanishing [CT18]. On the other hand a smooth surface with ample anti-canonical bundle satisfies Kawamata–Viehweg vanishing theorem over any algebraically closed field [CT18, Proposition A.1]. This however is no longer true if (klt) singularities are allowed and counterexamples (even to Kodaira vanishing) have been constructed over algebraically closed fields of characteristic $2$ [CT19] and $3$ [Ber17].
The situation simplifies in large characteristics. In [CTW17] the authors prove the existence of an integer $p_{0}$ such that over an algebraically closed field of characteristic $p>p_{0}$ every log del Pezzo surface satisfies Kawamata–Viehweg vanishing theorem. Finding an effective bound for this $p_{0}$ is a central open question in positive characteristic birational geometry. For example over an algebraically closed field of characteristic $p>\mathrm{max}\{5,p_{0}\}$ a three dimensional klt singularity is rational [HW19, Theorem 1.1], and threefolds satisfy a refined version of the basepoint free theorem [Ber19][Theorem 1.1].
The construction of the integer $p_{0}$ in [CTW17] is implicit. During the course of the proof of [CTW17, Theorem 1.1] the authors consider log del Pezzo surfaces
belonging to a bounded family. The construction of the integer $p_{0}$ does in particular depend on this family in an implicit way.
It is therefore natural to ask if, we, by other methods, may describe vanishing theorems in large explicit characteristics depending on some explicit numerical invariant of a bounded family of log del Pezzo surfaces. In this direction, we show that Kodaira vanishing (for big and nef divisors) holds on a del Pezzo surface of bounded index $I$ in characteristic $p>p_{0}(I)$ where $p_{0}(I)$ is an explicit polynomial in the index $I$. We also determine an explicit bound for $p_{0}$ (for Kodaira vanishing) in terms of $\epsilon>0$ for an $\epsilon$-klt log del Pezzo surface. Our main results in this direction are the following:
Theorem A.
Let $X$ be a projective klt surface over an algebraically closed field $k$ of characteristic $p>0$ with ample anti-canonical divisor $-K_{X}$ of Cartier index $I$. Let $D$ be a big and nef $\mathbb{Q}$-Cartier $\mathbb{Z}$-divisor on $X$.
(1)
If $I=1$, i.e., if $-K_{X}$ is an ample Cartier divisor then:
•
if $p\geq 9221$, then $H^{1}(X,\mathcal{O}_{X}(-D))=0$
•
if $p\geq 5$, then $H^{1}(X,\mathcal{O}_{X}(D))=0.$
(2)
If $I\geq 2$ and $p\geq(13-45I)^{2}(2I^{3}+4I^{2}+2I)+1$, then:
$$H^{1}(X,\mathcal{O}_{X}(D))=H^{1}(X,\mathcal{O}_{X}(-D))=0.$$
Theorem B.
Let $0<\epsilon<3^{-1/2}$ be a real number. Let $(X,\Delta)$ be a projective $\epsilon$-klt surface with $-(K_{X}+\Delta)$ ample over an algebraically closed field $k$ of characteristic
$$p\geq 2^{\frac{128}{\epsilon^{5}}+2}\left(\frac{2}{\epsilon}\right)^{\frac{(12%
8)^{2}}{\epsilon^{25}}}+1.$$
Then Kodaira vanishing holds on $X$, i.e., for all ample $\mathbb{Q}$-Cartier $\mathbb{Z}$-divisors $D$ on $X$ we have that $H^{i}(X,\mathcal{O}_{X}(D+K_{X}))=0$, for all $i>0$.
In this note, we use a technique due to Ekedahl [Eke88]. When trying to prove Kodaira vanishing in explicit characteristics using this technique, the difficulties arise from Weil divisors which are far from being Cartier. For example, on a log del Pezzo surface over an algebraically closed field of characteristic $p\geq 4c+1$ we can prove that $H^{1}(X,\mathcal{O}_{X}(K_{X}+A))=0$ for every ample $\mathbb{Q}$-Cartier Weil divisor $A$ of Cartier index $\leq c$ (see Remark 3.4). Another strategy is therefore, loosely speaking, to, in different ways, control the Cartier index of a Weil divisor $D$ in terms of the Cartier index $I$ of $(K_{X}+\Delta)$. We prove, in this way, that a big and nef $\mathbb{Z}$-divisors $D$ of big volume relative to the index $I$ in characteristic $p\geq 5$ satisfies $H^{1}(X,\mathcal{O}_{X}(-D))=0$.
Theorem C.
Let $(X,\Delta)$ be a projective klt surface over an algebraically closed field $k$ of characteristic $p\geq 5$ with $-(K_{X}+\Delta)$ an ample $\mathbb{Q}$-Cartier $\mathbb{Q}$-divisor of Cartier index $I$. Let $D$ be a big and nef $\mathbb{Q}$-Cartier $\mathbb{Z}$-divisor. Then the following holds true:
(1)
If $I=1$ and $D^{2}\geq 9$, then $H^{1}(X,\mathcal{O}_{X}(-D))=0.$
(2)
If $I\geq 2$ and $D^{2}\geq{2I^{3}+4I^{2}+2I}$, then $H^{1}(X,\mathcal{O}_{X}(-D))=0.$
During the preparation of this note we were informed that J. Lacini has classified all log del Pezzo surfaces of Picard rank one over an algebraically closed field of characteristic $p>5$ [Lac, Theorem 1.1]. From this classification it follows that all log del Pezzo surfaces of Picard rank one over an algebraically closed field of characteristic $p>5$ admit a lift to characteristic zero over a smooth base [Lac, Theorem 7.2]. It therefore follows from his work that Kodaira vanishing holds for log del Pezzo surfaces of Picard rank one over an algebraically closed field of characteristic $p>5$ (see [CTW17, Lemma 6.1]). Combining his result with the techniques exploited in this paper, one can prove the Kodaira vanishing theorem on a log del Pezzo surface over an algebraically closed field of characteristic $p>5$. At the end of this note we illustrate this argument. Building on the liftability of rank one log del Pezzo surfaces in characteristic $p>5$ by J. Lacini we have:
Theorem D.
Let $(X,\Delta)$ be a log del Pezzo surface over an algebraically closed field of characteristic $p>5$. Then Kodaira vanishing holds on $X$, i.e., for all ample $\mathbb{Z}$-divisors $D$ on $X$ we have $H^{i}(X,\mathcal{O}_{X}(D+K_{X}))=0$, for all $i>0$.
We do not know if the results of J. Lacini may be extended to show that every log del Pezzo surface over an algebraically closed field of characteristic $p>5$ admits a log resolution that lifts to characteristic zero over a smooth base. In fact, we do not know if every log del Pezzo surfaces in large characteristic admits such a lift (see also [CTW17, Theorem 1.1]).
Remark 1.
We do not know if the Kodaira vanishing theorem holds on a log del Pezzo surfaces over an algebraically closed field of characteristic $p=5$.
Acknowledgments
I would like to thank my PhD-advisor Zsolt Patakfalvi for the help he has given me during this work. I would like to thank Jakub Witaszek for answering questions about [CTW17]. I also thank Fabio Bernasconi and Maciej Zdanowicz for very useful discussions and for reading earlier drafts of this paper.
1. Preliminaries
By a variety we shall mean a finite type integral separated scheme over a field. We will work exclusively over an algebraically closed field.
A $\mathbb{Q}$-divisor $D$ is said to be $\mathbb{Q}$-Cartier if there exists an integer $m$ such that $mD$ is Cartier. A $\mathbb{Q}$-divisor is ample/nef/big if it is $\mathbb{Q}$- Cartier and an integer multiple is ample/nef/big as a line bundle.
If $X$ is a normal variety, then the reflexive sheaves on $X$ are determined (up to isomorphism) by their restriction to a big open subset $U$ (an open subset $U$ of $X$ is big if $\mathrm{codim}_{X}(X-U)\geq 2$). If $i:U\rightarrow X$ is the inclusion of a big open subset and $\mathcal{M}$ is a reflexive sheaf on $U$ then $i_{*}\mathcal{M}$ is a reflexive sheaf on $X$, moreover for any reflexive sheaf $\mathcal{G}$ on $X$ we have an isomorphism $\mathcal{G}\cong i_{*}\mathcal{G}_{|_{U}}$ [Har80, Proposition 1.6].
Let $U$ denote the smooth locus of $X$. Since $X$ is normal $U$ is big. Therefore, for any Weil divisor $D$ on $X$ the restriction of $D$ to the smooth locus defines a reflexive sheaf $i_{*}\mathcal{O}_{U}(U\cap D)$ of rank one on $X$. Conversely, any reflexive sheaf of rank one $\mathcal{F}$ on $X$ defines a Weil divisor $D$ by setting $D$ to be the closure of a Weil divisor $E$ on $U$ satisfying $\mathcal{F}_{|_{U}}=\mathcal{O}_{U}(E)$. This defines a one to one correspondence between the reflexive sheaves of rank one on $X$ up to isomorphism and the set of Weil divisors on $X$ up to rational equivalence [Har80]. We denote by $\mathcal{O}_{X}(D)$ the reflexive sheaf corresponding to a divisor $D$ on $X$.
We use the notation $K_{X}$ for the class corresponding to the closure of a canonical divisor $K_{U}$ on $U$ in $X$, hence $\mathcal{O}_{X}(K_{X})\cong i_{*}\mathcal{O}_{U}(K_{U})$. If X is projective then $\mathcal{O}_{X}(K_{X})$ can be seen to be a dualizing sheaf for $X$ [KM98, Proposition 5.75].
In general if $D_{1}$ and $D_{2}$ are Weil divisors on a normal variety $X$, then $\mathcal{O}_{X}(D_{1})\otimes\mathcal{O}_{X}(D_{2})$ is not reflexive. However, the dual $\mathcal{M}^{v}$ of a coherent sheaf $\mathcal{M}$ is always reflexive [Har80][Corollary 1.2] and we may define a product structure on the set of reflexive sheaves by defining $\mathcal{O}_{X}(D_{1})\otimes\mathcal{O}_{X}(D_{2})\mathrel{\mathop{:}}=(%
\mathcal{O}_{X}(D_{1})\otimes\mathcal{O}_{X}(D_{2}))^{\vee\vee}$. We have the following identities for Weil divisors $D_{1},D_{2}$ and a Cartier divisor $C$:
$\mathcal{O}_{X}(D_{1}+D_{2})\cong(\mathcal{O}_{X}(D_{1})\otimes\mathcal{O}_{X}%
(D_{2}))^{\vee\vee}$, $\mathcal{O}_{X}(-D_{1})\cong\mathcal{O}_{X}(D_{1})^{\vee}$, $\mathcal{O}_{X}(D_{1}+C)\cong\mathcal{O}_{X}(D_{1})\otimes\mathcal{O}_{X}(C)$ and $Hom_{\mathcal{O}_{X}}(\mathcal{O}_{X}(D_{1}),\mathcal{O}_{X}(D_{2}))\cong%
\mathcal{O}_{X}(D_{2}-D_{1}).$ This product structure makes the set of reflexive sheaves on $X$ (up to isomorphism) into a group such that the correspondence between Weil divisors and reflexive sheaves on $X$, as described above, becomes an isomorphism of groups.
By a log pair $(X,\Delta)$ we shall mean a normal variety $X$ together with an effective $\mathbb{Q}$-divisor $\Delta$ on $X$ such that $K_{X}+\Delta$ is $\mathbb{Q}$-Cartier. By a klt (respectively lc) pair we shall mean a log pair with klt (respectively lc) singularities in the sense of [KM98, Section 2]. By [Tan18, Corollary 4.11] if $X$ is a surface and $(X,\Delta)$ is a klt pair then $X$ is $\mathbb{Q}$-factorial.
A normal surface $X$ is said to be a del Pezzo surface if $-K_{X}$ is $\mathbb{Q}$- Cartier and ample. A log pair $(X,B)$ is said to be a log del Pezzo surface if $-(K_{X}+B)$ is ample.
1.1. Serre vanishing and Serre duality for $\mathbb{Z}$-divisors
One of the most fundamental vanishing theorems in algebraic geometry is that of Serre:
Theorem.
Let $\mathcal{L}$ be an ample line bundle on a proper scheme $X$ and let $\mathcal{F}$ be a coherent sheaf on $X$. Then for all $m$ big enough we have: $H^{i}(X,\mathcal{F}\otimes\mathcal{L}^{\otimes m})=0,$ for all $i>0$.
It naturally implies a Weil divisor version:
Corollary 1.1.
Let $X$ be a normal projective variety. If $D$ is an ample $\mathbb{Q}$-Cartier $\mathbb{Z}$-divisor and $\mathcal{F}$ is a coherent sheaf on $X$ then:
$$H^{i}(X,\mathcal{F}\otimes\mathcal{O}_{X}(lD))=0$$
for all $l$ big enough and all $i>0$.
Proof.
Let $m$ be an integer such that $mD$ is Cartier. For all $1\leq i\leq m$ there exists an $n_{i}$ such that $H^{j}(X,\mathcal{F}\otimes\mathcal{O}_{X}(iD)\otimes\mathcal{O}_{X}(mD)^{%
\otimes n})=0$, for all $n\geq n_{i}$ and all $j>0$. Since $\mathcal{O}_{X}(mD)$ is Cartier we have that $\mathcal{O}_{X}(iD)\otimes\mathcal{O}_{X}(mD)^{\otimes n}=\mathcal{O}_{X}((i+%
mn)D)$. Set $N=\max_{1\leq i\leq m}n_{i}$. For all $l\geq(m+1)N$, we can write $l=i+mk$, for $1\leq i\leq m-1$, where $k\geq N$. Therefore, we have $H^{i}(X,\mathcal{F}\otimes\mathcal{O}_{X}(lD))=H^{i}(X,\mathcal{F}\otimes%
\mathcal{O}_{X}(iD)\otimes\mathcal{O}_{X}(mD)^{\otimes k})=0$, for all $l\geq(m+1)N$.
∎
Theorem 1.2 ([Fuj83, Theorem 10]).
Let $X$ be a normal projective surface and $L$ be a big and nef line bundle on $X$. Then there exists an integer $m$ such that:
$$H^{1}(X,\mathcal{O}_{X}(-nL-E)=0$$
for all nef and effective divisor $E$ and for all $n>m$.
Corollary 1.3.
Let $D$ be a big and nef integral divisor on a normal projective surface $X$, then $H^{1}(X,\mathcal{O}(-mD))=0$ for all $m>>0$.
Proof.
Let $r$ be the Cartier index of $D$. Let $n>0$ be such that $H^{0}(X,\mathcal{O}(lD))$ is non zero for all $l\geq n$, this is possible since $D$ is big. By Theorem 1.2 there exists an integer $m$ such that $H^{1}(X,\mathcal{O}(-(kr+l)D))=0$ for all $l\geq n$ and all $k\geq m$, this proves the Corollary. ∎
We will repeatedly need to use a form of Serre Duality valid for reflexive sheaves on a normal surface:
Theorem 1.4 (Serre Duality for CM-sheaves).
Let $X$ be a projective scheme of pure dimension $n$ over a field $k$. Let $\mathcal{F}$ be a CM sheaf on $X$ such that $\operatorname{Supp}(\mathcal{F})$ is of pure dimension n. Then
$H^{i}(X,\mathcal{F})$ is dual to $H^{n-i}(X,Hom_{\mathcal{O}_{X}}(\mathcal{F},\omega_{X}))$.
Proof.
E.g., [KM98, Theorem 5.71]∎
In particular, by [Har80, Proposition 1.3] for any $\mathbb{Z}$-divisor $D$ on a normal projective surface $X$ we have that $H^{i}(X,\mathcal{O}_{X}(D))$ is dual to $H^{2-i}(X,\mathcal{O}_{X}(K_{X}-D))$ for $i=0,1$.
Remark 1.5.
If $D$ is a big and nef $\mathbb{Z}$-divisor on a normal projective surface $X$ then $H^{0}(X,\mathcal{O}_{X}(-D))=0$ and hence by duality $H^{2}(X,\mathcal{O}_{X}(D+K_{X}))=0$. Therefore, Kodaira vanishing is equivalent to the vanishing of $H^{1}(X,\mathcal{O}_{X}(-D))\cong H^{1}(X,\mathcal{O}_{X}(K_{X}-D)).$
1.2. Frobenius techniques
Let $X$ be a scheme over a positive characteristic base $S$, let $F_{X}$ denote the absolute Frobenius on $X$. $F_{X}$ is not a morphism of $S$-schemes unless $F_{S}$ on $S$ is the identity. In particular we have the following commuting diagram:
$$\tikzcd X\arrow[bendleft]{drr}{F_{X}}\arrow[bendright]{ddr}\arrow[dotted]{dr}[%
description]{F_{X/S}}&&\\
&X\times_{S}S\arrow{r}{\omega}\arrow{d}&X\arrow{d}\\
&S\arrow{r}{F_{S}}&S\makebox[0.0pt][l]{\,.}$$
where $F_{X/S}$ denotes the relative Frobenius. When $S=Spec(k)$ for a perfect field $k$ then the absolute Frobenius on $S$ is an isomorphism and {tikzcd} X ×_S S \arrowrω & X is an isomorphism over $S$ if we consider $X$ as a scheme over $S$ by post-composing the structure morphism to $S$ with $F^{-1}_{S}$ on $S$. We often do this identification, under which the relative and the absolute Frobenius coincide, and denote by $F:X\rightarrow X$ the corresponding morphism.
Let $X$ be a normal variety over an algebraically closed field of positive characteristic. If $D$ is an integral divisor on $X$ then $(F^{*}\mathcal{O}_{X}(D))^{\vee\vee}\cong\mathcal{O}_{X}(pD)$, since they agree on the regular locus. There is a natural map (unit of adjunction) $\mathcal{O}_{X}(D)\rightarrow F_{*}F^{*}\mathcal{O}_{X}(D)$. This induces a morphism:
$$H^{i}(X,\mathcal{O}_{X}(D))\to H^{i}(X,F^{*}\mathcal{O}_{X}(D))\rightarrow H^{%
i}(X,F^{*}\mathcal{O}_{X}(D)^{\vee\vee}).$$
Therefore, the Frobenius on X induces morphisms for all $m$:
$$H^{i}(X,\mathcal{O}_{X}(-p^{m}D))\to H^{i}(X,\mathcal{O}_{X}(-p^{m+1}D)).$$
If $D$ is an ample CM divisor, then these groups eventually vanish for $m>>0$ by Theorem 1.4 and Corollary 1.1. Therefore, Kodaira vanishing theorem for ample CM $\mathbb{Z}$-divisors $D$ holds true on $X$ if and only if the morphisms $H^{i}(X,\mathcal{O}_{X}(-p^{m}D))\rightarrow H^{i}(X,\mathcal{O}_{X}(-p^{m+1}D))$ are injective for all $m$ and all $i<\mathrm{dim}(X)$.
1.3. Non-vanishing and the associated $\alpha_{\mathcal{O}_{X_{0}}(D_{0})}$-torsor.
Let $X$ be a projective normal surface and $D$ be a $\mathbb{Q}$-Cartier $\mathbb{Z}$-divisor on $X$. Let $X_{0}\subset X$ denote the smooth locus of $X$ and $D_{0}\mathrel{\mathop{:}}=D\cap X_{0}$ be the restriction of $D$ to the smooth locus. Let $F_{0}$ denote the Frobenius on $X_{0}$. We will assume that $H^{1}(X,\mathcal{O}_{X}(-D))\neq 0$ and that $H^{1}(X,\mathcal{O}_{X}(-pD))=0$. Let $\alpha\in H^{1}(X,\mathcal{O}_{X}(-D))$ denote a non-trivial element of the kernel of the map $H^{i}(X,\mathcal{O}_{X}(-D))\rightarrow H^{i}(X,\mathcal{O}_{X}(-pD))$ described in subsection 1.2 above. Since $Z\mathrel{\mathop{:}}=X-X_{0}$ is the complement of a big open subset of $X$ and $\mathcal{O}_{X}(nD)$ is reflexive (and so in particular, $H^{0}(X,\mathcal{O}_{X}(nD))\cong H^{0}(X_{0},{\mathcal{O}_{X}}_{0}(nD_{0}))$), we see from the local cohomology long exact sequence that we have an inclusion:
$$H^{1}(X,\mathcal{O}_{X}(-nD))\hookrightarrow H^{1}(X_{0},\mathcal{O}_{X_{0}}(-%
nD_{0}))$$
for all $n$. By our assumptions, $\alpha$ therefore defines a non-trivial element:
$$\alpha_{0}\in H^{1}(X_{0},\mathcal{O}_{X_{0}}(-D_{0})),$$
which belongs to the kernel of the morphism:
$$H^{1}(X_{0},\mathcal{O}_{X_{0}}(-D_{0}))\rightarrow H^{1}(X_{0},\mathcal{O}_{X%
_{0}}(-pD_{0}))$$
induced by $F_{0}$. This kernel has a geometric description on $X_{0}$ as follows:
1.3.1. $\alpha_{L}$-torsors
An invertable sheaf $\mathcal{L}$ on a scheme $X$ naturally defines a sheaf of groups under addition representable by the affine group scheme $L\mathrel{\mathop{:}}=\operatorname{Spec}_{X}(\bigoplus_{i}\mathcal{L}^{-i}).$ Let $X$ be defined over a field $k$ of characteristic $p>0.$ Then there exists a homomorphism of sheaves of additive groups $\mathcal{L}\rightarrow\mathcal{L}^{p}$, defined by raising a local section to its $p$-power. This is a purely characteristic $p>0$ phenomenon, since for any two local sections $s$ and $t$ of $\mathcal{L}$ we have $(s+t)^{p}=s^{p}+t^{p}$.
Relative to the Zariski site of $X$ the above morphism of sheaves is not in general a surjection. However, as a morphism of $X_{\operatorname{fppf}}$-sheaves it is, i.e., the corresponding morphism of sheaves of groups on the flat site of $X$ is surjective [Mil80, II.2.18]. The kernel is a sheaf of groups on $X_{\operatorname{fppf}}$ which we denote by $\alpha_{\mathcal{L}}.$ By construction $\alpha_{\mathcal{L}}$ is representable by an affine group scheme $\alpha_{L}.$ In fact, we have $\alpha_{L}=\operatorname{Spec}_{X}(\bigoplus_{i=o}^{p-1}\mathcal{L}^{-i})$ since it is equal to the relative spectrum over $X$ of the cokernel of the $\mathcal{O}_{X}$-algebra inclusion:
$$\bigoplus_{i}\mathcal{L}^{-pi}\hookrightarrow\bigoplus_{i}\mathcal{L}^{-i}.$$
The short exact sequence of sheaves of groups (relative to the flat- topology on $X$)
$$\tikzcd 0\arrow{r}&\alpha_{\mathcal{L}}\arrow{r}&{\mathcal{L}}\arrow{r}&{%
\mathcal{L}}^{p}\arrow{r}&0$$
induces a long exact sequence on cohomology [Mil80, 4, Prop 4.5]:
$$\dots\rightarrow H^{0}(X,\mathcal{L}^{p})\rightarrow H^{1}_{\operatorname{fppf%
}}(X,\alpha_{\mathcal{L}})\rightarrow H^{1}(X,\mathcal{L})\rightarrow H^{1}(X,%
\mathcal{L}^{p})\rightarrow\dots$$
Therefore, a non-trivial element $\alpha$ in the kernel of $H^{1}(X,\mathcal{L})\rightarrow H^{1}(X,\mathcal{L}^{p})$ defines a non-trivial element $\alpha\in H^{1}_{\operatorname{fppf}}(X,\alpha_{\mathcal{L}}).$ This group has a geometric meaning, i.e., the first cohomology group of a group scheme $G$ over a scheme $X$ corresponds to a $G$-torsor $Y\rightarrow X$, see for example, [Mil80, Proposition 4.6]. The torsors $Y$ were studied by Ekedahl:
Proposition 1.6 ([Eke88, p 106-107][PW17, Theorem 2.11]).
Let $X$ be a normal variety over an algebraically closed field $k$ of characteristic $p>0$. Let $\mathcal{L}\in Pic(X)$. A non-trivial element of the kernel under the Frobenius action:
$H^{1}(X,\mathcal{L})\rightarrow H^{1}(X,\mathcal{L}^{p}),$ gives rise to a non-trivial $\alpha_{\mathcal{L}}$-torsor $\beta:Y\rightarrow X$. Locally over $U=Spec(A)$ the $\alpha_{L}$- torsor $\beta^{-1}U\rightarrow U$ is given by $A\rightarrow A[x]/x^{p}-a$ for some element $a\in A$ which is not a $p$-power. Therefore, $\beta$ is purely inseparable of degree $p$. If $X$ is a $G_{1}$ and $S_{2}$ variety, then $Y$ is also a $G_{1}$ and $S_{2}$ variety which satisfies:
$\omega_{Y}=\beta^{*}(\omega_{X}\otimes\mathcal{L}^{p-1})$.
Here $S_{2}$ denotes Serre’s condition two and $G_{1}$ denotes Gorenstein in codimension one.
Remark 1.7.
Y is in general not normal.
1.4. Geometric construction for Weil-divisors
Let $X$ be a projective normal surface over an algebraically closed field of charcteristic $p>0$ and let $D$ a $\mathbb{Q}$-Cartier ample Weil divisor on $X$ such that $H^{1}(X,\mathcal{O}_{X}(-D))\neq 0$ and $H^{1}(X,\mathcal{O}_{X}(-pD))=0.$ Let $X_{0}$ denote the smooth locus of $X$ and let $D_{0}=D\cap X_{0}$. Let $\alpha_{0}$ be a non-trivial element of $H^{1}_{\operatorname{fppf}}(X,\alpha_{\mathcal{O}_{X_{0}}(D_{0})})$ coming from a non-trivial element $\alpha\in H^{1}(X,\mathcal{O}_{X}(D))$ as described in point 1.3. Let $\pi_{0}:Y_{0}\rightarrow X_{0}$ be the corresponding non-trivial $\alpha_{\mathcal{O}_{X_{0}}(D_{0})}$-torsor. Then ${\pi_{0}}_{*}\mathcal{O}_{Y_{0}}$ is a locally free $\mathcal{O}_{X_{0}}$-algebra. Let $i:X_{0}\hookrightarrow X$ denote the inclusion of the regular locus.
The multiplication on $X_{0}$ extends (because $i_{*}{\pi_{0}}_{*}\mathcal{O}_{Y_{0}}$ is reflexive) to make $i_{*}{\pi_{0}}_{*}\mathcal{O}_{Y_{0}}$ into a sheaf of $\mathcal{O}_{X}$-algebras. We may therefore define $Y\mathrel{\mathop{:}}=\operatorname{Spec}_{X}(i_{*}{\pi_{0}}_{*}\mathcal{O}_{Y%
_{0}})$. Let $\pi:Y\rightarrow X$ denote the natural morphism.
Lemma 1.8.
In the situation above $\pi:Y\rightarrow X$ is a finite degree $p$ morphism and $Y$ is a projective $S_{2}$ and $G_{1}$ surface that satisfies:
$$K_{Y}=\pi^{*}(K_{X}-(p-1)D).$$
Proof.
Since $\pi_{*}\mathcal{O}_{Y}=i_{*}{\pi_{0}}_{*}\mathcal{O}_{Y_{0}}$ is $S_{2}$ we have by [KM98, Proposition 5.4] that $Y$ is $S_{2}.$ Since $X$ and $X_{0}$ agree in codimension one, $Y$ is $G_{1}.$ Since everything is $S_{2}$ and $G_{1}$ we may replace $X$ with $X_{0}$ by [Har94, Theorem 1.12], therefore $\pi^{*}(K_{X})$ and $\pi^{*}(D)$ are well-defined and the formula $K_{Y}=\pi^{*}(K_{X}-(p-1)D)$ follows from Proposition 1.6. ∎
Lemma 1.9.
With the notation as above, let $\eta:\overline{Y}\rightarrow Y$ be the normalization of $Y$ and let $\overline{\pi}:\overline{Y}\rightarrow X$ denotes the induced morphism. There exists an effective $\mathbb{Z}$-divisor $\Delta\geq 0$ on $\overline{Y}$ such that $K_{\overline{Y}}+\Delta=\overline{\pi}^{*}(K_{X}-(p-1)D).$
Proof.
Since everything is $S_{2}$ and $G_{1}$ we may replace $Y$ with a big open set and assume that $Y$ is Gorenstein and $\overline{Y}$ is regular. Affine locally we may assume that $\omega_{Y}\cong\mathcal{O}_{Y}$ and $\omega_{X}\cong\mathcal{O}_{X}$. In this situation [PW17, Lemma 2.14] proves that the natural map $\omega_{\overline{Y}}\rightarrow\eta^{*}\omega_{Y}$ has image equal to the conductor ideal, where $f^{*}\omega_{Y}$ is identified with $\mathcal{O}_{X}$. ∎
1.5. Bend and break
By studying different properties of the variety $\overline{Y}$ appearing in Lemma 1.9 one hopes to arrive at a contradiction to the assumed non-vanishing, $H^{1}(X,\mathcal{O}_{X}(-D))\neq 0$. This strategy was successfully employed by Ekedahl [Eke88] and more recently in [PW17]. The main tool is the use of bend and break, together with the expression for the canonical divisor given by Lemma 1.9. Together, this can be used in order to derive inequalities comparing intersection numbers on $X$ with the characteristic $p$ of the base field. We will repeatedly use the following:
Theorem 1.10 ([Kol96, Theorem 5.8]).
Let $X$ be a projective variety over an algebraically closed field, $C$ a smooth, projective and irreducible curve, $f:C\rightarrow X$ a morphism and $M$ any nef $\mathbb{R}$-divisor. Assume that $X$ is smooth along $f(C)$ and $-K_{X}.C>0.$ Then for every $x\in f(C)$ there is a rational curve $\Gamma_{x}\subset X$ containing $x$ such that:
$$M.\Gamma_{x}\leq 2\dim(X)\frac{M.C}{-K_{X}.C}$$
2. Kodaira vanishing for divisors of big volume
We will use the following terminology: We will say that a curve $C$ on a surface $X$ is big and basepoint free if the linear system $|C|$ defines a birational morphism. If $C$ is big and basepoint free, then for any finite number of points in $X$ there exists $C^{\prime}\in|C|$ such that $C^{\prime}$ avoids all those points.
Lemma 2.1.
Let $X$ be a projective normal surface. Suppose that there exists an effective $\mathbb{Q}$-divisor $E$ such that $-(K_{X}+E)$ is a $\mathbb{Q}$-Cartier ample divisor of Cartier index $I$. If $D$ is a big $\mathbb{Q}$-Cartier semiample $\mathbb{Z}$-divisor such that $D^{2}\geq I^{2}(K_{X}+E)^{2}$ then there exists an ample Cartier divisor $A$ and a big basepoint free curve $C$ on $X$ such that $D.C\geq A.C$.
Proof.
Suppose, in order to arrive at a contradiction, that for every big basepoint free curve $C$ and every ample Cartier divisor $A$ of $X$ we have $D.C<A.C$. For $m$ divisible enough, $-m(K_{X}+E)$ is very ample. Therefore, for $A=-I(K_{X}+E)$ and $C\in|-m(K_{X}+E)|$ we find $-(K_{X}+E).D<I(K_{X}+E)^{2}$. Similarly, for $n$ divisible enough, $nD$ is basepoint free and we may consider a curve $C\in|nD|$ to find that $D^{2}<-I(K_{X}+E).D$. Putting this together, we find that $D^{2}<I^{2}(K_{X}+E)^{2}$.
∎
Theorem 2.2.
Let $X$ be a projective normal surface over an algebraically closed field $k$ of characteristic $p\geq 5$. Suppose that there exists an effective $\mathbb{Q}$-divisor $E$ such that $-(K_{X}+E)$ is a $\mathbb{Q}$-Cartier ample divisor of Cartier index $I$. If $D$ is a $\mathbb{Q}$-Cartier nef and big semiample $\mathbb{Z}$-divisor such that $D^{2}\geq I^{2}(K_{X}+E)^{2}$ then $H^{1}(X,\mathcal{O}_{X}(-D))=0$.
Proof.
Suppose that the theorem is not true. Let $D$ be a nef and big semiample $\mathbb{Z}$-divisor on $X$ such $D^{2}\geq I^{2}(K_{X}+E)^{2}$ and assume that $H^{1}(X,\mathcal{O}_{X}(-D))\neq 0$. By Corollary 1.3 there exists some $j\geq 1$ such that $H^{1}(X,\mathcal{O}_{X}(-p^{j}D))=0$. If we replace $D$ with $p^{j-1}D$ the inequality will be satisfied for $p^{j-1}D$. Suppose we can prove the theorem for $p^{j-1}D$ then $H^{1}(X,\mathcal{O}_{X}(-p^{j-1}D))=0$ and, hence, by descending induction on $j$, we have proven that $H^{1}(X,\mathcal{O}_{X}(-D))=0$. Therefore, there is no loss of generality to assume that $H^{1}(X,\mathcal{O}_{X}(-pD))=0$.
Let $\pi:Y\rightarrow X$ be a degree $p$ cover corresponding to a non-zero element $y\in H^{1}(X,\mathcal{O}_{X}(-D))$ as in subsection 1.3. By replacing $Y$ with its normalisation, we may assume that $Y$ is normal and that
(2.1)
$$K_{Y}=\pi^{*}K_{X}+(1-p)\pi^{*}D-\Delta$$
where $\Delta$ is effective (Lemma 1.9).
By Lemma 2.1 there exists a big basepoint free curve $C$ of $X$ and a ample Cartier divisor $A$ such that $D.C\geq A.C$. Let $\pi:Y\rightarrow X$ be as above. By replacing $C$ with a large enough multiple we see that there exists a curve $C_{Y}$ on $Y$ such that:
•
$C_{Y}$ is contained in the smooth locus of $Y$
•
$C_{Y}$ is not contained in a component of $\Delta$
•
$C=\pi_{*}C_{Y}$ for some curve $C$ on $X$ not contained in a component of $E$ and such that $D.C\geq A.C.$
By Equation 2.1 we have $-K_{Y}.C_{Y}=-K_{X}.C+(p-1)D.C+\Delta.C_{Y}\geq-K_{X}.C$, where the last inequality comes from the assumption that $D$ is nef and that $C_{Y}$ is not contained in a component of $\Delta$. By assumption $-(K_{X}+E)$ is ample, hence $-K_{X}.C>E.C\geq 0$. Therefore $-K_{Y}.C_{Y}>0$. By Theorem 1.10 for $C_{Y}$ and $\pi^{*}A$ on $Y$, there exists a rational curve $\Gamma_{x}$ passing through a point $x\in C_{Y}$ such that:
$$\pi^{*}A.\Gamma_{x}\leq 4\frac{\pi^{*}A.C_{Y}}{-K_{Y}.C_{Y}}.$$
By Equation 2.1, we find:
$$\left(\pi^{*}A.\Gamma_{x}\right)\left((-\pi^{*}K_{X}+(p-1)\pi^{*}D+\Delta).C_{%
Y}\right)\leq 4\pi^{*}A.C_{Y}.$$
However, since $A$ is an ample Cartier divisor and $\pi$ is finite $\pi^{*}A.\Gamma_{x}\geq 1$. Moreover, $-\pi^{*}K_{X}.C_{Y}=-K_{X}.C>0$ and $C_{Y}.\Delta\geq 0$.
By assumption, $\pi^{*}A.C_{Y}\leq\pi^{*}D.C_{Y}$, therefore:
$$(p-1)\pi^{*}A.C_{Y}<\left(\pi^{*}A.\Gamma_{x}\right)\left((-\pi^{*}K_{X}+(p-1)%
\pi^{*}D+\Delta).C_{Y}\right).$$
Putting these inequalities together we find:
$$(p-1)\pi^{*}A.C_{Y}<4\pi^{*}A.C_{Y}$$
and, therefore, $p<5$.
∎
Theorem 2.2 implies, together with results of [Jia13], our main theorem for divisors of big volume:
See C
Proof.
By the Basepoint free theorem [Tan14, Theorem 1.3], $D$ is semiample. Note that a klt surface of index $I$, necessarily, is $\frac{1}{I}$-klt. By [Jia13, Theorem 1.3], we therefore have that $(K_{X}+\Delta)^{2}\leq\textrm{max}\{9,2I+4+\frac{2}{I}\}.$ The result therefore follows from Theorem 2.2.
∎
Corollary 2.3.
Let $X$ be a klt del Pezzo surface over an algebraically closed field of characteristic $p\geq 5$, such that $-K_{X}$ is ample of Cartier index $I$. Let $D$ be a big and nef $\mathbb{Z}$-divisor of Cartier index $c$. Then $H^{1}(X,\mathcal{O}_{X}(D))=0$ if either of the following holds:
•
$K_{X}^{2}\geq(c-1)^{2}D^{2}$ and $D$ is ample
•
$D^{2}\geq(I-1)^{2}K_{X}^{2}.$
Proof.
We have that $D-K_{X}$ is ample and $H^{1}(X,\mathcal{O}_{X}(K_{X}-D))$ is dual to $H^{1}(X,\mathcal{O}_{X}(D))$. Assume that the claimed vanishing does not hold. Since the characteristic of the base field is greater than or equal to five, we may argue as in the proof of Proposition 2.2 to assume that for every big basepoint free curve $C$ and every ample Cartier divisor $A$ we have $(D-K_{X}).C<A.C.$ The first inequality comes from setting $A=cD$ and $C=-mK_{X}$ and $C=lD$ repectively, for large and divisible enough $l$ and $m$. The second inequality comes from setting $A=-IK_{X}$ and $C=-mK_{X}$ and $C=lD$ respectively, for large and divisible enough $l$ and $m$.
∎
3. Kodaira vanishing in large characteristic
In this section we prove Kodaira vanishing in large characteristic for log del Pezzo surfaces of bounded index. That bounded families of log del Pezzo surfaces over $\operatorname{Spec}(\mathbb{Z})$ satisfy Kodaira vanishing in large characteristic (depending only on the family) has been proven by other methods in [CTW17]. Our first Proposition is quite general, it does not demand boundedness of the pair $(X,\Delta)$, it simply demands a very ample line bundle on $X$ of bounded degree and that $X$ is of Fano-type.
Proposition 3.1.
Let $r>0$ be an integer. Let $(X,\Delta)$ be a log pair where $X$ is a projective normal surface over an algebraically closed field $k$ of characteristic $p\geq 4r+1$. Assume that $-(K_{X}+\Delta)$ is $\mathbb{Q}$-Cartier and ample. If there exists a very ample divisor $A$ on $X$ such that $A^{2}\leq r$ then Kodaira vanishing (even for big and nef $\mathbb{Q}$-Cartier $\mathbb{Z}$-divisors) holds on $X$.
Proof.
Suppose that $D$ is a big and nef $\mathbb{Z}$-divisor and suppose that
$H^{1}(X,\mathcal{O}_{X}(-D))\neq 0$. By replacing $D$ with $p^{j}D$ for some $j\geq 1$ we may assume that $H^{1}(X,\mathcal{O}_{X}(-pD))=0$ (Corollary 1.3). By Ekedahl’s construction we may assume that there exists a normal variety $Y$ and a purely inseparable degree $p$ morphism $\pi:Y\rightarrow X$ such that
(3.1)
$$K_{Y}=\pi^{*}K_{X}-(p-1)D-E$$
for $E\geq 0$. Let $A$ be a very ample general curve on $X$ as above. There exists an integer $l>0$ such that $l\pi^{*}A$ is very ample on $Y$. Therefore there exists a curve $A_{Y}$ on $Y$ such that:
•
$A_{Y}$ is contained in the smooth locus of $Y$
•
$A_{Y}$ is not contained in a component of $E$
•
$A^{\prime}=\pi_{*}A_{Y}$ for some $A^{\prime}\in|nA|$ where $n>0$ is an integer.
By Equation 3.1 we have
$$-K_{Y}.A_{Y}=n(-K_{X}.A+(p-1)D.A)+E.A_{Y}\geq-nK_{X}.A.$$
By assumption $-(K_{X}+\Delta)$ is ample, hence $-K_{X}.A>\Delta.A\geq 0$. Therefore $-K_{Y}.A_{Y}>0$. By Theorem 1.10 for $A_{Y}$ and $\pi^{*}(A)$ on $Y$, there exists for every point $x\in A_{Y}$ a rational curve $\Gamma_{x}$ passing through $x$ such that
$$\pi^{*}(A).\Gamma_{x}\leq 4\frac{\pi^{*}(A).A_{Y}}{-K_{Y}.A_{Y}}.$$
However, since $A$ is an ample Cartier divisor and $\pi$ is finite, $\pi^{*}A.\Gamma_{x}\geq 1$. By Equation 3.1 and the assumption $A^{2}\leq r$, we therefore have:
$$(-\pi^{*}K_{X}+(p-1)\pi^{*}D+E)A_{Y}\leq 4rn.$$
Since $-K_{X}.A>0$ and $E.A_{Y}\geq 0$, we find that:
$$(p-1)\pi^{*}D.A_{Y}<4rn.$$
Since $\pi^{*}D.A_{Y}\geq n$, the result follows.
∎
See A
Proof.
By [Wit17] we have that $(13-45I)IK_{X}$ is very ample. By [Jia13, Theorem 1.3] we have that $K_{X}^{2}\leq\textrm{max}\{9,2I+4+\frac{2}{I}\}.$ The result therefore follows from Proposition 3.1 and Theorem C.
∎
For the boundary version of the theorem above we need to use a common bound for the Cartier indices of $K_{X}$ and $K_{X}+\Delta$.
Theorem 3.2.
Let $(X,\Delta)$ be a log del Pezzo surface over an algebraically closed field $k$ of characteristic $p>0$. Let $m\geq 2$ be such that $m(K_{X}+\Delta)$ and $mK_{X}$ are both Cartier. Let $D$ be a big and nef $\mathbb{Z}$-divisor on $X$. If
$$p\geq 173056m^{8}-86528m^{7}+16200m^{5}+32400m^{4}+16200m^{3}+1$$
then $H^{1}(X,\mathcal{O}_{X}(-D))=0$.
Proof.
Let $H=13mK_{X}-45m^{2}(K_{X}+\Delta)$. According to [Wit17], $H$ is a very ample Cartier divisor. By [Wit17, Lemma 6.2], we have inequalities $(K_{X}+\Delta).K_{X}\geq 0$ and $K_{X}^{2}\leq 128m^{5}(2m-1)$. By [Jia13, Theorem 1.3], we have that $(K_{X}+\Delta)^{2}\leq 2m+4+\frac{2}{m}$ . Therefore:
$$H^{2}\leq(13)^{2}128m^{7}(2m-1)+(45)^{2}(2m^{5}+4m^{4}+2m^{3}).$$
We conclude by Proposition 3.1.∎
Remark 3.3.
A klt log del Pezzo surface $(X,\Delta)$ where $(K_{X}+\Delta)$ has Cartier index $I$ is necessarily $\epsilon$-klt for $\epsilon=\frac{1}{I}$. As in the proof below, the Cartier index of $K_{X}$ can therefore be bounded in terms of the Cartier index of $(K_{X}+\Delta)$.
See B
Proof.
By assumption $X$ is $\epsilon$-klt, therefore the $\mathbb{Q}$-factorial index $q$ at any point $x\in X$ satisfies
$$q\leq 2\left(\frac{2}{\epsilon}\right)^{\frac{128}{\epsilon^{5}}}$$
by [Wit17, Prop 6.1]. Moreover, by the same proposition, the Picard rank of a minimal resolution of $X$ is bounded by $\frac{128}{\epsilon^{5}}$. Hence, for every $\mathbb{Z}$-divisor $D$, there exists some $c\leq 2^{\frac{128}{\epsilon^{5}}}\left(\frac{2}{\epsilon}\right)^{\frac{(128)%
^{2}}{\epsilon^{25}}},$ depending on $D$, such that $cD$ is Cartier.
Suppose that $H^{1}(X,\mathcal{O}_{X}(-D))\neq 0$. By replacing $D$ with $p^{k-1}D$ for some $k>1$, we may assume that $H^{1}(X,\mathcal{O}_{X}(-pD))=0$. We may, therefore, assume that there exists a normal variety $Y$ and a degree $p$ inseparable morphism $Y\rightarrow X$ such that $K_{Y}=\pi^{*}(K_{X}+(1-p)D)-E$, for some $E\geq 0$. Let $C_{Y}$ be a general complete intersection curve on $Y$, not contained in the support of $E$, such that $\pi_{*}C_{Y}=C$ is not contained in the support of $\Delta$. Then $-K_{Y}.C_{Y}>0$. By Theorem 1.10 applied to $C_{Y}$ and
$$\pi^{*}(-(K_{X}+\Delta)+(p-1)D)\equiv-(K_{Y}+E)-\pi^{*}(\Delta),$$
there exists a rational curve $\Gamma$ through a point on $C_{Y}$ such that:
$$\pi^{*}(-(K_{X}+\Delta)+(p-1)D).\Gamma\leq 4\frac{(-(K_{Y}+E)-\pi^{*}(\Delta))%
.C_{Y}}{-K_{Y}.C_{Y}}.$$
Since $-\pi^{*}(K_{X}+\Delta).\Gamma>0$, $E.C_{Y}\geq 0$ and $\pi^{*}(\Delta).C_{Y}\geq 0$, we find that
$$(p-1)D.\Gamma<4.$$
We have that $D.\Gamma\geq\frac{1}{c}.$
Therefore:
$$p<4c+1\leq 2^{\frac{128}{\epsilon^{5}}+2}\left(\frac{2}{\epsilon}\right)^{%
\frac{(128)^{2}}{\epsilon^{25}}}+1.$$
∎
Remark 3.4.
From the proof of Theorem B we see that on a log del Pezzo surface $(X,\Delta)$ over an algebraically closed field $k$ of characteristic $p\geq 4c+1$ we have $H^{1}(X,\mathcal{O}_{X}(-D))=0$, for every ample $\mathbb{Z}$-divisor $D$ of Cartier index $\leq c$.
4. Kodaira vanishing in char $p>5$ using the liftability of log del Pezzo surfaces of Picard rank one
Proposition 4.1.
Let $X$ be a normal surface birational to a surface $Y$ via a birational morphism $g:X\rightarrow Y$, such that $Y$ admits a fibration $f:Y\rightarrow B$ to a curve $B$ with general fiber $F$ isomorphic to $\mathbb{P}^{1}$. Let $D$ be a $\mathbb{Q}$-Cartier big and nef $\mathbb{Z}$-divisor on $X$. Then $H^{1}(X,\mathcal{O}_{X}(-D))=0$.
Proof.
According to Corollary 1.3 by replacing $D$ with $p^{m}D$, we may assume that $H^{1}(X,\mathcal{O}_{X}(-pD))=0.$ Suppose that
$H^{1}(X,\mathcal{O}_{X}(-D))\neq 0$. We apply Ekedahl’s construction 1.3 to get a degree $p$ finite morphism $\pi:Z\rightarrow X$, where $Z$ is normal, and
$$K_{Z}+E=\pi^{*}K_{X}+(1-p)\pi^{*}D$$
for an effective divisor $E\geq 0.$ We illustrate the situation with a diagram:
$$\tikzcd Z\arrow{d}{\pi}\\
X\arrow{r}{g}&Y\arrow{d}{f}\\
&B$$
Let $F$ be a fiber of $f$ not containing any point which is the image of a curve contracted by $g$. Then $F$ can be identified with a fiber of $f\circ g:X\rightarrow B$. By abuse of notation we denote a general fiber of $f\circ g$ by $F$. A general fiber of the composition $f\circ g$ avoids the exceptional locus of $g$ since the curves contracted by $g$ map to a finite number of points on $Y$. We compute:
$$K_{Z}.\pi^{*}F=\pi^{*}((K_{X}-(p-1)D).F)-E.\pi^{*}F.$$
The morphism $\pi$ is finite of order $p$, therefore, we have:
$$K_{Z}.\pi^{*}F=p((K_{X}-(p-1)D).F-E.\pi^{*}F.$$
Since two general fibers $F$ do not intersect we have $F^{2}=0$ and therefore $K_{Y}.F=-2$. Since $D$ is big and $F$ is general, we have $D.F>0$.
Moreover, since $E$ is effective, we see (upon varying $F$) that the intersection number $E.\pi^{*}F\geq 0$. We conclude that:
$$K_{Z}.\pi^{*}F\leq p(K_{X}-(p-1)D).F=$$
$$p(deg(K_{F})-(p-1)D.F)=$$
$$-2p-p(p-1)D.F.$$
Denote by $G=\pi^{*}F$. Let $F^{\prime}$ denote the reduction of a general fiber of $f\circ g\circ\pi$.
If $G$ is reduced, then $G=F^{\prime}$, otherwise $G=pF^{\prime}.$
In either case, we have:
$$-2p\leq K_{Z}.G\leq-2p-p(p-1)D.F,$$
but $-p(p-1)D.F<0.$ This is a contradiction.
∎
Definition 4.2 ([EV92, Definition 8.11]).
Let $k$ be a perfect field of positive characteristic and let $X$ be a scheme smooth over $\operatorname{Spec}(k)$. Let $D=\sum_{i}^{n}D_{i}$ be a reduced simple normal crossing divisor on $X$.
A lifting of $D=\sum_{i=1}^{n}D_{i}\subset X$ to $\operatorname{W_{2}(k)}_{2}(k)$ consits of a scheme $X^{\prime}$ and subschemes $D^{\prime}_{i}$ all defined and flat over $\operatorname{W_{2}(k)}_{2}(k)$ such that:
$X=X^{\prime}\times_{\operatorname{Spec}(\operatorname{W_{2}(k)}_{2}(k))}%
\operatorname{Spec}(k)$ and $D_{i}=D_{i}^{\prime}\times_{\operatorname{Spec}(\operatorname{W_{2}(k)}_{2}(k)%
)}\operatorname{Spec}(k)$,
for $1\leq i\leq n$.
For a pair $(X,D)$ consisting of a smooth surface and a reduced simple normal crossing divisor as above, [CTW17] defines the notion of liftability to characteristic zero over a smooth base.
We refer the reader to [CTW17, Definition 2.15] for the definition. If the pair $(X,D)$ lifts to characteristic zero over a smooth base then the pair $(X,D)$ lifts to $\operatorname{W_{2}(k)}_{2}(k)$ [CTW17, Remark 2.16].
Theorem 4.3 ([Lac, Theorem 7.2]).
Let $S$ be a log del Pezzo surface of Picard rank one over an algebraically closed field of characteristic $p>5$. Then there exists a log resolution $\mu:V\rightarrow S$ such that $(V,\operatorname{Exc}(\mu))$ lifts to characteristic zero over a smooth base.
We believe the following lemma to be well-known to experts, however since we did not find a reference we include it here.
Lemma 4.4.
Let $X$ be a smooth variety and $D=\sum_{i=1}^{n}D_{i}$ be a reduced simple normal crossing divisor. Suppose that $(X,D)$ lifts to $\operatorname{W_{2}(k)}_{2}(k)$. Let $x\in X$ be a closed point. If $\pi:Y\rightarrow X$ is the blow up of $X$ at $x$ then $(Y,Exc(\pi)+\pi_{*}^{-1}D)$ lifts to $\operatorname{W_{2}(k)}_{2}(k)$.
Proof.
Let $(X^{\prime},D^{\prime})$ be a lift of $(X,D)$ to $\operatorname{W_{2}(k)}_{2}(k)$. Let $x\in X$ be a closed point. By formal smoothness there exists a lift $x^{\prime}\in X^{\prime}$ of $x$. We claim the following:
Claim.
If $J\subset\{1,2,\dots,n\}$ is an index set such that $x\in\bigcap_{i\in J}D_{i}$ but $x\notin D_{k}$ for $k\notin J$, then, there exists a lift $x^{\prime}\in X^{\prime}$, such that $x^{\prime}\in\bigcap_{i\in J}D^{\prime}_{i}$, but $x^{\prime}\notin D^{\prime}_{k}$ for $k\notin J$.
By [EV92, Lemma 8.13 d)], we may assume that we have a diagram with all squares being Cartesian:
$$\tikzcd U\arrow{rrrrr}\arrow{rrd}{e}\arrow{dd}&&&&&U^{\prime}\arrow{lld}[swap]%
{e^{\prime}}\arrow{dd}\\
&&\mathbb{A}_{k}\arrow{r}\arrow{lld}&\mathbb{A}_{\operatorname{W_{2}(k)}_{2}(k%
)}\arrow{rrd}\\
\operatorname{Spec}(k)\arrow{rrrrr}&&&&&\operatorname{Spec}(\operatorname{W_{2%
}(k)}_{2}(k))$$
where $\mathbb{A}_{k}=\operatorname{Spec}(k[t_{1},t_{2}])$, $\mathbb{A}_{\operatorname{W_{2}(k)}_{2}(k)}=\operatorname{Spec}(\operatorname{%
W_{2}(k)}_{2}(k)[t_{1},t_{2}])$, $U$ and $U^{\prime}$ are open subsets of $X$ and $X^{\prime}$ respectively and the morphisms $e$ and $e^{\prime}$ are étale. By [EV92, Lemma 8.14 e)], we may assume that we have chosen local parameters:
$\phi_{i}=e^{*}(t_{i})\in\mathcal{O}_{U}$ and $\phi^{\prime}={e^{\prime}}^{*}(t_{i})\in\mathcal{O}_{U^{\prime}}$
such that $x=\operatorname{V}((\phi_{1},\phi_{2}))$ and $x^{\prime}=\operatorname{V}(({\phi^{\prime}}_{1},{\phi^{\prime}}_{2}))$ and such that there exists a subset $I$ of $\{\emptyset,1,2\}$, such that $D\cap U=\operatorname{V}(\prod_{i\in I}\phi_{i})$ and $D^{\prime}\cap U^{\prime}=\operatorname{V}(\prod_{i\in I}{\phi^{\prime}}_{i})$. This proves the claim.
Let $\pi^{\prime}:Y^{\prime}\rightarrow X^{\prime}$ be the blow up of $x^{\prime}=\operatorname{V}(({\phi^{\prime}}_{1},{\phi^{\prime}}_{2})),$ as above. We now prove that $(Y^{\prime},{\pi^{\prime}}^{-1}_{*}D^{\prime}+\operatorname{Exc}(\pi^{\prime}))$ is a lift of $(Y,\operatorname{Exc}(\pi)+\pi_{*}^{-1}D)$ to $\operatorname{W_{2}(k)}_{2}(k)$. To this end, let $S$ be equal to $k$ or $\operatorname{W_{2}(k)}_{2}(k)$ and $\mathbb{A}_{S}=\operatorname{Spec}(S[t_{1},t_{2}])$ and $e:U\rightarrow\mathbb{A}^{2}_{S}$ an étale morphism. Then by the commutativity of blowing up with flat base change [Sta20, Lemma 085S] the following natural diagram is Cartesian:
$$\tikzcd\operatorname{Bl}_{V((e^{*}(t_{1}),e^{*}(t_{2})))}(U)\arrow{d}\arrow{r}%
&\operatorname{Bl}_{V((t_{1},t_{2}))}(\mathbb{A}^{2}_{S})\arrow{d}\\
U\arrow{r}{e}&\mathbb{A}^{2}_{S}$$
Moreover, the strict transform of $\operatorname{V}(\prod_{i\in I}e^{*}(t_{i}))$ is the base change to $U$ of the strict transform of $\operatorname{V}(\prod_{i\in I}t_{i})$ for any subset $I$ of $\{\emptyset,1,2\}$. Therefore:
$$\operatorname{Bl}_{x^{\prime}}(U^{\prime})=\operatorname{Bl}_{\operatorname{V}%
(t_{1},t_{2})}(\mathbb{A}^{2}_{\operatorname{W_{2}(k)}_{2}(k)})\times_{\mathbb%
{A}^{2}_{\operatorname{W_{2}(k)}_{2}(k)}}U^{\prime}$$
and
$$\operatorname{Bl}_{x}(U)=\operatorname{Bl}_{\operatorname{V}(t_{1},t_{2})}(%
\mathbb{A}^{2}_{k})\times_{\mathbb{A}^{2}_{k}}U$$
and the strict transform of the subschemes defined by the local parameters correspond in the adequate way.
We are therefore reduced to considering the following situation:
$$\tikzcd\operatorname{Bl}_{\operatorname{V}(t_{1},t_{2})}\mathbb{A}^{2}_{k}%
\arrow{r}\arrow{d}&\operatorname{Bl}_{V(t_{1},t_{2})}\mathbb{A}^{2}_{%
\operatorname{W_{2}(k)}_{2}(k)}\arrow{d}\\
\mathbb{A}^{2}_{k}\arrow{r}\arrow{d}&\mathbb{A}^{2}_{\operatorname{W_{2}(k)}_{%
2}(k)}\arrow{d}\\
\operatorname{Spec}(k)\arrow{r}&\operatorname{Spec}(\operatorname{W_{2}(k)}_{2%
}(k))$$
The top square, and therefore the whole diagram, is clearly Cartesian. In this situation it is evident that $\operatorname{Bl}_{\operatorname{V}(t_{1},t_{2})}\mathbb{A}^{2}_{\operatorname%
{W_{2}(k)}_{2}(k)}$ is smooth over $W_{2}(k)$ and that the strict transform of each of the coordinate axes, as well as the exceptional divisor of the blow up, are flat and hence smooth over $\operatorname{W_{2}(k)}_{2}(k)$. Since the base change of an étale morphism is étale this shows that $Y^{\prime},{\pi^{\prime}}^{-1}_{*}D^{\prime}_{i}$, for $1\leq i\leq n$, and $\operatorname{Exc}(\pi^{\prime})$ are all flat over $\operatorname{W_{2}(k)}_{2}(k)$. Consequently $(Y^{\prime},{\pi^{\prime}}^{-1}_{*}D^{\prime}+\operatorname{Exc}(\pi^{\prime}))$ is a lift of $(Y,\operatorname{Exc}(\pi)+\pi_{*}^{-1}D)$ to $\operatorname{W_{2}(k)}_{2}(k)$.
∎
See D
Proof.
Let $D$ be an ample $\mathbb{Q}$-Cartier $\mathbb{Z}$-divisor. It is sufficient to prove that $0=H^{1}(X,\mathcal{O}_{X}(-D))=H^{1}(X,\mathcal{O}_{X}(K_{X}+D))$. We run a $K_{X}$-MMP to get a birational morphism $g:X\rightarrow Y$ where either $f:Y\rightarrow B$ is a MFS onto a curve $B$ or $Y$ is a klt del Pezzo surface of Picard rank one.
(1)
If $f:Y\rightarrow B$ is a $K_{X}$-MFS onto a curve, then the result follows from Proposition 4.1 .
(2)
If there exists a birational morphism $X\rightarrow Y$ where $Y$ is a log del Pezzo surface of Picard rank one then by Theorem 4.3 there exists a log resolution $\mu:(V,\operatorname{Exc}(\mu))\rightarrow Y$ such that $(V,\operatorname{Exc}(\mu))$ lifts to characteristic zero over a smooth base where $\operatorname{Exc}(\mu)$ denotes the reduced divisor supported at the exceptional locus of $\mu$. There is an induced rational map $\mu^{\prime}:V\dashedrightarrow X$. Let $\pi:V^{\prime}\rightarrow V$ be a resolution of the indeterminacy locus of $\mu^{\prime}$. The morphism
$\pi:V^{\prime}\rightarrow V$ can be facorised as a composition of blowups at points. By Lemma 4.4, $(V^{\prime},\pi_{*}^{-1}\operatorname{Exc}(\mu)+\operatorname{Exc}(\pi))$ lifts to $\operatorname{W_{2}(k)}_{2}(k)$. The diagram below illustrates the situation:
$$\tikzcd X\arrow{d}{g}&(V^{\prime},\pi_{*}^{-1}\operatorname{Exc}(\mu)+%
\operatorname{Exc}(\pi))\arrow{l}{\tilde{\mu}}\arrow{d}{\pi}\\
Y&(V,\operatorname{Exc}(\mu))\arrow{l}{\mu}$$
We have that the support of $\operatorname{Exc}(\tilde{\mu})$ is contained in the support of $\pi_{*}^{-1}\operatorname{Exc}(\mu)+\operatorname{Exc}(\pi)$. Therefore $\tilde{\mu}:(V^{\prime},\operatorname{Exc}(\tilde{\mu}))\rightarrow X$ is a log-resolution that lifts to $\operatorname{W_{2}(k)}_{2}(k)$. By [CTW17, Lemma 6.1] the result follows.
∎
References
[Ber17]
Fabio Bernasconi.
Kawamata-Viehweg vanishing fails for log del Pezzo surfaces in
characteristic 3.
arXiv e-prints, page arXiv:1709.09238, Sep 2017.
[Ber19]
Fabio Bernasconi.
On the base point free theorem for klt threefolds in large
characteristic.
arXiv e-prints, page arXiv:1907.10396, Jul 2019.
[CT18]
Paolo Cascini and Hiromu Tanaka.
Smooth rational surfaces violating Kawamata-Viehweg vanishing.
Eur. J. Math., 4(1):162–176, 2018.
[CT19]
Paolo Cascini and Hiromu Tanaka.
Purely log terminal threefolds with non-normal centres in
characteristic two.
Amer. J. Math., 141(4):941–979, 2019.
[CTW17]
Paolo Cascini, Hiromu Tanaka, and Jakub Witaszek.
On log del Pezzo surfaces in large characteristic.
Compos. Math., 153(4):820–850, 2017.
[Eke88]
Torsten Ekedahl.
Canonical models of surfaces of general type in positive
characteristic.
Inst. Hautes Études Sci. Publ. Math., (67):97–144, 1988.
[EV92]
Hélène Esnault and Eckart Viehweg.
Lectures on vanishing theorems, volume 20 of DMV Seminar.
Birkhäuser Verlag, Basel, 1992.
[Fuj83]
Takao Fujita.
Vanishing theorems for semipositive line bundles.
In Algebraic geometry (Tokyo/Kyoto, 1982), volume 1016 of
Lecture Notes in Math., pages 519–528. Springer, Berlin, 1983.
[Har80]
Robin Hartshorne.
Stable reflexive sheaves.
Math. Ann., 254(2):121–176, 1980.
[Har94]
Robin Hartshorne.
Generalized divisors on Gorenstein schemes.
In Proceedings of Conference on Algebraic Geometry and
Ring Theory in honor of Michael Artin, Part III (Antwerp,
1992), volume 8, pages 287–339, 1994.
[HW19]
Christopher Hacon and Jakub Witaszek.
On the rationality of Kawamata log terminal singularities in
positive characteristic.
Algebr. Geom., 6(5):516–529, 2019.
[Jia13]
Chen Jiang.
Bounding the volumes of singular weak log del Pezzo surfaces.
Internat. J. Math., 24(13):1350110, 27, 2013.
[KM98]
János Kollár and Shigefumi Mori.
Birational geometry of algebraic varieties, volume 134 of Cambridge Tracts in Mathematics.
Cambridge University Press, Cambridge, 1998.
With the collaboration of C. H. Clemens and A. Corti, Translated from
the 1998 Japanese original.
[Kol96]
János Kollár.
Rational curves on algebraic varieties, volume 32 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of
Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd
Series. A Series of Modern Surveys in Mathematics].
Springer-Verlag, Berlin, 1996.
[Lac]
Justin Lacini.
On rank one log del pezzo surfaces in characteristic different from
two, three and five.
Preprint, available at http://www.math.ucsd.edu/ jlacini/.
[Mil80]
James S. Milne.
Étale cohomology, volume 33 of Princeton Mathematical
Series.
Princeton University Press, Princeton, N.J., 1980.
[PW17]
Zsolt Patakfalvi and Joe Waldron.
Singularities of General Fibers and the LMMP.
arXiv e-prints, page arXiv:1708.04268, Aug 2017.
[Ray78]
Michel Raynaud.
Contre-exemple au “vanishing theorem” en caractéristique
$p>0$.
In C. P. Ramanujam—a tribute, volume 8 of Tata Inst.
Fund. Res. Studies in Math., pages 273–278. Springer, Berlin-New York,
1978.
[Sta20]
The Stacks project authors.
The stacks project.
https://stacks.math.columbia.edu, 2020.
[Tan14]
Hiromu Tanaka.
Minimal models and abundance for positive characteristic log
surfaces.
Nagoya Math. J., 216:1–70, 2014.
[Tan18]
Hiromu Tanaka.
Minimal model program for excellent surfaces.
Ann. Inst. Fourier (Grenoble), 68(1):345–376, 2018.
[Wit17]
Jakub Witaszek.
Effective bounds on singular surfaces in positive characteristic.
Michigan Math. J., 66(2):367–388, 2017. |