Describing the critical behavior of the Anderson transition in infinite dimension by random-matrix ensembles: logarithmic multifractality and critical localization
Weitao Chen, Olivier Giraud, Jiangbin Gong, Gabriel Lemarié
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引用次数: 0
Abstract
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.