Localization challenges quantum chaos in the finite two-dimensional Anderson model

TL;DR

This paper investigates the transition from quantum chaos to localization in the finite 2D Anderson model, revealing that quantum chaos breaks down at any nonzero disorder in the thermodynamic limit, with universal scaling behaviors observed.

Contribution

The study provides a detailed numerical analysis demonstrating the breakdown of quantum chaos in the 2D Anderson model at nonzero disorder, supporting the single-parameter scaling theory.

Findings

Indicators suggest emergence of quantum chaos at weak disorder

Universal behavior of spectral form factor at weak disorder

Scaling of indicators with parameter u = (W ln V)^(-1)

Abstract

It is believed that the two-dimensional (2D) Anderson model exhibits localization for any nonzero disorder in the thermodynamic limit and it is also well known that the finite-size effects are considerable in the weak disorder limit. Here we numerically study the quantum-chaos to localization transition in the finite 2D Anderson model using standard indicators used in the modern literature such as the level spacing ratio, spectral form factor, variances of observable matrix elements, participation entropy and the eigenstate entanglement entropy. We show that many features of these indicators may indicate emergence of robust single-particle quantum chaos at weak disorder. However, we argue that a careful numerical analysis is consistent with the single-parameter scaling theory and predicts the breakdown of quantum chaos at any nonzero disorder value in the thermodynamic limit. Among the hallmarks of this breakdown are the universal behavior of the spectral form factor at weak disorder, and the universal scaling of various indicators as a function of the parameter $u = \left(W \ln V\right)^{-1}$ where $W$ is the disorder strength and $V$ is the number of lattice sites.