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.