Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2405.10975

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
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

cond-mat.dis-nnquant-ph

Keep this discovery

BibTeXRIS

Weitao Chen, Olivier Giraud, Jiangbin Gong, Gabriel Lemarié. 2024-05-12. Describing the critical behavior of the Anderson transition in infinite dimension by random-matrix ensembles: logarithmic multifractality and critical localization. https://doi.org/10.1103/physrevb.110.014210

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Irrationality Measure Controls Long-Wavelength Charge Fluctuations in Quasiperiodic Systems

Quasiperiodic order is characterized by irrational frequencies whose rational approximability known as irrationality measure defines distinct arithmetic classes. We establish that this arithmetic classification has direct physical consequences for long-wavelength charge fluctuations. In translation-covariant quasiperiodic systems, the infrared scaling of charge fluctuation is governed by the interplay between the irrationality exponent of irrational frequency and the large-harmonic decay of the hull charge profile: the latter determines the available charge weight, while the former controls how efficiently that weight is transferred to the infrared. Consequently, all algebraic irrational frequencies share the same arithmetic scaling, whereas exceptionally well-approximable transcendental frequencies can exhibit strongly enhanced infrared fluctuation scaling. We further prove that occupied states separated from the Fermi level by a gap stable throughout the hull contribute only an analytic infrared background, leaving the nontrivial scaling to near-Fermi states. Our results extend to general translation-covariant multi-frequency quasiperiodic systems.

cond-mat.dis-nn↗
arXiv2026

Curvature-Induced Geometric Universality in Non-Hermitian Anderson Transitions

In Euclidean space, universality classes of Anderson transitions are primarily determined by symmetry and spatial dimensionality. Here, we present evidence for a geometry-controlled universality class of non-Hermitian Anderson transitions on hyperbolic-like lattices. In this setting, critical behavior is influenced by the large-scale hyperbolic geometry, characterized by negative curvature, exponential volume growth, and a non-Euclidean notion of spatial scaling. Finite-size scaling of participation ratios across several distinct \( \{p,q\} \) tilings reveals one-parameter scaling collapses with a common critical exponent \( \nu\simeq1 \) within numerical accuracy. A complementary phenomenological coarse-grained Landau-Ginzburg analysis shows how exponential correlation-volume growth suppresses critical fluctuations, offering a rationale for the observed mean-field-like scaling. Our results suggest that spatial curvature can act as an additional organizing principle for Anderson-transition universality beyond the conventional dimensionality- and symmetry-based classification.

cond-mat.dis-nn↗
arXiv2026

Finite-rank multiplicative perturbations of rotationally invariant non-Hermitian random matrices

We study finite-rank multiplicative deformations of rotationally invariant non-Hermitian random matrices. More precisely, we consider models of the form $\mathbf{A}(\mathbf{I}+\mathbf{T})$, where $\mathbf{A}$ is a large rotationally invariant non-Hermitian random matrix, $\mathbf{T}$ is a finite-rank normal perturbation, and $\mathbf{I}$ denotes the identity matrix. We characterize the emergence of outlier eigenvalues, their fluctuations, and the associated eigenvector overlaps. Our results provide a multiplicative non-Hermitian counterpart to the classical Baik--Ben Arous--P\'ech\'e framework.

cond-mat.dis-nn↗