GUE随机矩阵中的RSB： wormhole网络模型的微观解析
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本文主要探讨了在高斯单位整体（Gaussian Unitary Ensemble, GUE）随机矩阵模型中观察到的副本对称性破坏（Replica Symmetry Breaking, RSB）现象。在统计力学和量子场论中，随机矩阵模型常被用于模拟复杂系统的统计性质，特别是在研究许多物理系统中的级联效应时，如量子混沌和量子重正化群行为。
在高维随机矩阵的N趋近于无穷大（large-N limit）的极限下，作者发现RSB的临界温度TRSB遵循一个幂律关系，即TRSB ~ N^(-2/3)。这一结果表明，随着系统的规模增加，矩阵模型中的对称性结构变得更为复杂，非平凡的对称性破缺模式开始显现。这种转变可以理解为自由能的微观景观发生了显著变化，其中包含了一系列几乎退化的局部极小值，这些极小值由Young图（一种描述分形排列的数学工具）进行标记。
文章进一步将这个理论框架与"ER = EPR"猜想联系起来。该猜想是弦理论中的一种假设，主张在某些极端条件下，额外维度的“连接”（ER）可以等同于量子纠缠的“实体”（EPR）。在随机矩阵模型的背景下，作者推测多重耦合的Sachdev-Ye-Kitaev (SYK) 模型，这是一种具有随机交互的量子系统，其量子力学特性与黑洞的信息悖论有关，可能受到RSB现象的启发。同时，作者还探讨了这种RSB现象在全息双Jackiw-Teitelboim (JT) 引力理论中的潜在意义，JT引力是黑洞物理学的一个简化模型，它与量子信息有着深刻的关联。
这篇论文提供了一个深入理解随机矩阵模型中RSB行为的新视角，并可能为理解量子信息的复杂性、黑洞物理学以及理论物理学中的其他未解问题提供关键线索。通过结合高维随机矩阵理论、统计物理方法和弦理论的概念，作者提出了一个关于量子系统和引力理论之间潜在联系的理论框架。开放获取的文章形式使得这一研究成果可以广泛传播，促进跨领域的学术交流和进一步探索。
Physics Letters B 803 (2020) 135280
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Replica symmetry breaking in random matrix model: A toy model of
wormhole networks
Kazumi Okuyama
Department of Physics, Shinshu University, 3-1-1 Asahi, Matsumoto 390-8621, Japan
a r t i c l e i n f o a b s t r a c t
Article history:
Received
26 July 2019
Received
in revised form 20 January 2020
Accepted
4 February 2020
Available
online 7 February 2020
Editor:
N. Lambert
We study the replica symmetry breaking (RSB) in the Gaussian Unitary Ensemble (GUE) random matrix
model. In the large N limit the RSB occurs at the transition temperature T
RSB
∼ N
−2/3
. We argue that
this transition originates from a landscape of almost degenerate local minima of free energy labeled by
Young diagrams. We also discuss a possible implication of our findings for the multiply-coupled Sachdev-
Ye-Kitaev
models and their holographic dual Jackiw-Teitelboim gravity from the viewpoint of ER=EPR
conjecture.
© 2020 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP
3
.
1. Introduction
The replica symmetry breaking (RSB) in disordered system is
an important concept in the study of glass transition. In particular,
the spin-glass model of Sherrington and Kirkpatrick [1]was solved
by Parisi [2,3]based on the RSB ansatz (see [4]for a review).
In
the disordered system a natural quantity to compute is the
quenched average of the free energy log Z(β), where Z (β) =
Tr e
−β H
is the partition function at temperature T = β
−1
. The av-
erage
··· in the disordered system is defined by integrating over
the distribution of random couplings in the Hamiltonian H. The
replica method is commonly used to compute the quenched aver-
age
log Z(β) by preparing n copies of the system (replicas) and
taking the limit n →0at the end of calculation
log Z (β)=lim
n→0
∂
∂n
Z(β)
n
.
(1)
RSB is characterized by the condition that the quenched average
and the annealed average of n replicas do not agree
Z(β)
n
=Z (β)
n
.
(2)
In this paper, we consider the RSB in the Gaussian Unitary En-
semble
(GUE) random matrix model, where the random couplings
in the Hamiltonian are modeled by treating the Hamiltonian H
itself
as a random N × N hermitian matrix with the Gaussian dis-
tribution
E-mail address: kazumi@azusa.shinshu-u.ac.jp.
···=
dHe
−
N
2
Tr H
2
(···)
dHe
−
N
2
Tr H
2
. (3)
Note that the RSB in the GUE random matrix model has been con-
sidered
in [5], where the correlator of resolvent Tr(E − H)
−1
was
studied by the replica method. In this paper we consider the cor-
relator
of Z (β) = Tr e
−β H
and study the RSB characterized by (2).
In this paper, we will not consider the limit n → 0in (1); we study
Z(β)
n
with positive integer n in its own right.
1
In the large N limit Z(β)
n
in the GUE random matrix model
indeed exhibits the RSB at the transition temperature T
RSB
Z(β)
n
≈
Z(β)
n
,(T > T
RSB
),
Z(β)
n
c
,(T < T
RSB
),
(4)
where the subscript c in Z(β)
n
c
denotes the connected part of
the correlator.
2
It turns out that transition temperature scales as
T
RSB
∼ N
−2/3
, which follows from the known large N behavior of
correlators in the GUE random matrix model (see e.g. [7]). We will
argue that the transition at T = T
RSB
originates from the existence
of a landscape of almost degenerate local minima of free energy.
This landscape of free energy arises as follows: When we expand
1
In the high temperature regime T > T
RSB
, taking the limit in (1)we find
log Z(β) ≈ logZ(β). On the other hand, the n → 0 limit in the low tempera-
ture
regime T < T
RSB
is rather subtle and the analytic continuation to n = 0is not
straightforward [6]. As emphasized in [6], the correct free energy is not reproduced
by simply taking the n →0 limit of Z(β)
n
c
in (4).
2
A similar behavior was observed in the so-called random-energy model, where
each energy levels are assumed to be Gaussian distributed but the correlations be-
tween
different energy levels are neglected [6].
https://doi.org/10.1016/j.physletb.2020.135280
0370-2693/
© 2020 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by
SCOAP
3
.
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