Signatures of Nonergodicity in Sparse Random Matrices

TL;DR

This paper investigates the spectral properties of sparse random matrices with disorder, revealing signatures of nonergodicity and identifying the Anderson transition through analytical and numerical methods, highlighting a broad nonergodic regime.

Contribution

It analytically derives energy moments and kurtosis to estimate the localization transition in sparse matrices, linking spectral statistics to phase transitions.

Findings

Identification of the Anderson transition via ground state statistics

Estimation of critical sparsity threshold for localization

Discovery of a broad nonergodic regime within the delocalized phase

Abstract

The prevalence of sparsity in interacting many-body systems motivates an investigation into the spectral statistics of sparse random matrices with on-site disorder. We numerically demonstrate that the Anderson transition can be identified through the statistical properties of the ground state. By analytically deriving the energy moments and calculating the shifted kurtosis, we estimate the critical sparsity threshold for this localization-delocalization transition. The short-range energy correlation in the bulk indicates that the Anderson transition at infinite temperature coincides with the quantum phase transition. Furthermore, long-range energy correlations in the bulk spectrum reveal a Thouless energy scale, suggesting a broad nonergodic regime within the delocalized phase.