平面矩阵量子力学与SYK模型的相变研究

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"平面矩阵量子力学，张量和Sachdev-Ye-Kitaev模型的相图" 在本文中，作者Tatsuo Azeyanagi、Frank Ferrari和Fidel I. Schaposnik Massolo深入研究了平面矩阵量子力学以及张量网络与Sachdev-Ye-Kitaev（SYK）模型的相互作用。这些模型通常被用于理解复杂系统中的量子混沌和强关联现象。 SYK模型是一个特殊的量子多体系统，由随机相互作用的费米子组成，展现出非常规的统计行为。研究的核心是Schwinger-Dyson方程，这是一种在量子场论和统计物理中广泛使用的非线性方程，它描述了系统的平均场性质。在处于先声子阶的费米离子平面矩阵量子力学中，这些方程揭示了两种不同的物理相：一种是高熵状态，类似于SYK黑洞，另一种是低熵状态，其红外行为较为简单。研究发现了一条一阶相变线，这条线在相图中终止于一个新的临界点。这个临界点对于理解系统的行为至关重要，因为它标志着两种相之间的转变变得连续而非跳跃式。关键的指数在相变的两侧表现出非均场特性，这意味着它们不遵循传统的费利曼-金地（Feynman-Ginzburg）理论，该理论假设近似于自由粒子的行为。在相变线上，作者观察到了过渡两侧指数的不同，这表明系统的动力学性质在相变过程中发生了显著变化。此外，他们还在稳定的和不稳定的玻色子模型中发现了有趣的现象，包括Kazakov临界点，这可能预示着系统在特定条件下的特殊对称性破缺模式。同时，他们还指出了一些SYK类似解的不一致性，这对于进一步理解模型的局限性和修正方向提供了线索。这项工作为理解量子混沌、高维量子系统和复杂网络的统计行为提供了重要的理论工具和洞察。通过分析平面矩阵量子力学和张量网络中的相变和临界现象，研究人员可以更好地预测和控制这些系统的行为，这对于未来的量子计算和信息处理技术具有潜在的应用价值。开放获取的特性使得这篇研究对于广大科研人员来说更加可及，有助于推动相关领域的进步。

Phase Diagram of Planar Matrix Quantum Mechanics, Tensor,

and Sachdev-Ye-Kitaev Models

Tatsuo Azeyanagi,

Frank Ferrari,

1,2

and Fidel I. Schaposnik Massolo

Universit´e libre de Bruxelles (ULB) and International Solvay Institutes Service de Physique Th´eorique et Math´ematique

Campus de la Plaine, CP 231, B-10 50 Bruxelles, Belgique

Center for the Theoretical Physics of the Universe Institute for Basic Sciences (IBS), Seoul 08826, Republic of Korea

(Received 14 July 2017; revised manuscript received 19 December 2017; published 6 February 2018)

We study the Schwinger-Dyson equations of a fermionic planar matrix quantum mechanics [or tensor

and Sachdev-Ye-Kitaev (SYK) models] at leading melonic order. We find two solutions describing a high

entropy, SYK black-hole-like phase and a low entropy one with trivial IR behavior. There is a line of first

order phase transitions that terminates at a new critical point. Critical exponents are nonme an field and

differ on the two sides of the transition. Interesting phenomena are also found in unstable and stable

bosonic models, including Kazakov critical points and inconsistency of SYK-like solutions of the IR limit.

DOI: 10.1103/PhysRevLett.120.061602

Introduction.—A series of recent works [1–4] combining

old ideas developed in the condensed matter literature [5],

with new insights in black hole physics [6–9] and tensor

model technology [10–13], is making it possible, for the

first time, to reliably study planar quantum mechanical

matrix models in the fully nonperturbative regime. For a

partial list of interesting recent papers in this area, see, e.g.,

[14]. The aim of the present work is to initiate a systematic

study of the phase diagrams of such models.

We are going to focus on the Hamiltonian

H ¼ NDtr



mψ

†

ﬃﬃﬃﬃ

†



; ð1Þ

where ψ

and ψ

†

are OðDÞvectors of N × N matrices made

of fermionic annihilation and creation operators ðψ

μb

and ðψ

†

¼ðψ

μa

†

satisfying canonical anticommu-

tation relations

fψ

μb

; ðψ

†

g¼

μν

: ð2Þ

After taking the usual planar N → ∞ limit, we are con-

sidering the new D → ∞ limit at fixed λ defined in [4].In

this limit, the model is dominated by melonic planar graphs

and turns out to be equivalent to the model with quenched

disorder investigated by Sachdev in [15]. Our results thus

also immediately apply to models of this type.

We shall also briefly discuss a model of Hermitian

bosonic matrices with an unstable potential, equivalent

at leading order with the Carrozza-Tanasa-Klebanov-

Tarnopolsky theory [12,16], and another model of

Hermitian bosonic matrices with a stable potential, which

can be seen as the bosonic truncation of a natural super-

symmetric extension [17].

In this Letter, we focus on the discussion of the physics

and on the presentation of the main results. Full technical

details, together with a discussion of supersymmetric

models, which can be easily constructed in the present

framework [17], will be given elsewhere [18].

Physical discussion.—Let us start by recalling the three

basic steps that allow us to solve models such as (1) in the

large N and large D limits. We first start from perturbation

theory to compute physical quantities as a power expansion

in the coupling constant λ in terms of Feynman graphs. We

then consider the large N and large D limits, which in our

case, select the planar melonic graphs described in [4]. This

truncation of perturbation theory produces a convergent

series expansion. In step three, we sum the perturbative

series exactly via an appropriate Schwinger-Dyson (SD)

equation. This gives access to the “fully nonperturbative”

regime of the model, by analytic continuation. We thus

obtain a nonperturbative description of the physics, but it is

conceptually crucial to understand that it relies in a

fundamental way on perturbation theory.

A remarkable property of fermionic models like (1) is

that there are two natural but distinct ways to define a

perturbative expansion and thus a priori two distinct paths

to access the nonperturbative physics using the strategy

described in the previous paragraph.

The first perturbative regime corresponds to high temper-

ature T ¼ 1=β ≫ jλj at fixed mass m. The fermions then

have a high effective thermal mass ∼2πT

and, even in the

extreme case m ¼ 0, we get a small effective coupling

constant λ=T. The zeroth order of the resulting perturbative

Published by the American Physical Society under the terms of

the Creative Commons Attribution 4.0 International license.

Further distribution of this work must maintain attribution to

the author(s) and the published article’s title, journal citation,

and DOI. Funded by SCOAP

PHYSICAL REVIEW LETTERS 120, 061602 (2018)

0031-9007=18=120(6)=061602(6) 061602-1 Published by the American Physical Society

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