用随机矩阵集合描述无限维安德森转变的临界行为:对数多分形和临界定位

Weitao Chen, Olivier Giraud, Jiangbin Gong, Gabriel Lemarié
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引用次数: 0

摘要

随机矩阵集合因其分析上的可操作性,成为探索计算要求高的系统中奇异现象的强大平台。本论文在一封附信[arXiv:2312.17481]的基础上,采用分析技术和大量数值模拟,研究了两种为捕捉无限维安德森转换临界行为而定制的随机矩阵集合。我们的研究揭示了两类临界行为:对数多重分形和批判局部化。与传统的多分形不同,新对数多分形的特征是特征态矩与系统大小的对数呈代数缩放关系。临界局部化的特征是特征状态矩的阶数$q>1/2$收敛到一个有限值,表明局部化,表现出特征对数有限大小或时间效应,这与在有效无限维度的随机正则图和Erd\"os-R\'enyi 图中观察到的临界行为一致。利用微扰方法,我们在模型中建立了对数多分形和批判局部化的存在。此外,我们还探索了时间动力学和空间相关函数中出现的新颖尺度行为。我们的模型为研究无限维量子无序系统提供了一个有价值的框架,我们发现的普遍性使其广泛适用于具有明显有限尺寸效应和缓慢动力学的系统,包括有争议的多体局域化转变,类似于无限维的安德森转变。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Describing the critical behavior of the Anderson transition in infinite dimension by random-matrix ensembles: logarithmic multifractality and critical localization
Due to their analytical tractability, random matrix ensembles serve as robust platforms for exploring exotic phenomena in systems that are computationally demanding. Building on a companion letter [arXiv:2312.17481], this paper investigates two random matrix ensembles tailored to capture the critical behavior of the Anderson transition in infinite dimension, employing both analytical techniques and extensive numerical simulations. Our study unveils two types of critical behaviors: logarithmic multifractality and critical localization. In contrast to conventional multifractality, the novel logarithmic multifractality features eigenstate moments scaling algebraically with the logarithm of the system size. Critical localization, characterized by eigenstate moments of order $q>1/2$ converging to a finite value indicating localization, exhibits characteristic logarithmic finite-size or time effects, consistent with the critical behavior observed in random regular and Erd\"os-R\'enyi graphs of effective infinite dimensionality. Using perturbative methods, we establish the existence of logarithmic multifractality and critical localization in our models. Furthermore, we explore the emergence of novel scaling behaviors in the time dynamics and spatial correlation functions. Our models provide a valuable framework for studying infinite-dimensional quantum disordered systems, and the universality of our findings enables broad applicability to systems with pronounced finite-size effects and slow dynamics, including the contentious many-body localization transition, akin to the Anderson transition in infinite dimension.
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