Crossover of Level Statistics between Strong and Weak Localization in Two Dimensions

TL;DR

This paper studies the spectral statistics of two-dimensional disordered systems, revealing a crossover from Wigner to Poisson distributions, governed by a single parameter related to localization length, without critical behavior.

Contribution

It introduces a finite-size scaling analysis of spectral statistics in 2D disordered systems, identifying a universal crossover function and relating it to localization length.

Findings

Spectral statistics exhibit a crossover from Wigner to Poisson distributions.

A single-parameter scaling function $\gamma(L/\xi)$ describes the crossover.

No critical behavior observed in the scaling function.

Abstract

We investigate numerically the statistical properties of spectra of two-dimensional disordered systems by using the exact diagonalization and decimation method applied to the Anderson model. Statistics of spacings calculated for system sizes up to 1024 $\times$ 1024 lattice sites exhibits a crossover between Wigner and Poisson distributions. We perform a self-contained finite-size scaling analysis to find a single-valued one-parameter function $\gamma (L/\xi)$ which governs the crossover. The scaling parameter $\xi(W)$ is deduced and compared with the localization length. $\gamma ( L/\xi)$ does {\em not} show critical behavior and has two asymptotic regimes corresponding to weakly and strongly localized states.