Spectral rigidity of non-Hermitian symmetric random matrices near Anderson transition

TL;DR

This paper investigates the spectral rigidity of a non-Hermitian Anderson model near the transition point, revealing eigenvalue repulsion behavior similar to the Ginibre ensemble on the metallic side.

Contribution

It numerically analyzes the eigenvalue statistics of the non-Hermitian Anderson model, demonstrating eigenvalue repulsion and the scaling of the eigenvalue-repelling region near the transition.

Findings

Eigenvalues repel each other strongly in the metallic phase.

The number of eigenvalues in the repelling disk scales with system size.

Eigenvalue behavior resembles the Ginibre ensemble in the metallic regime.

Abstract

We study the spectral rigidity of the non-Hermitian analog of the Anderson model suggested by Tzortzakakis, Makris and Economou (TME). This is a $L\times L \times L$ tightly bound cubic lattice, where both real and imaginary parts of on-site energies are independent random variables uniformly distributed between $-W/2$ and $W/2$. The TME model may be used to describe a random laser. In a recent paper we proved that this model has the Anderson transition at $W= W_c \simeq 6$ in three dimension. Here we numerically diagonalize TME $L \times L \times L$ cubic lattice matrices and calculate the number variance of eigenvalues in a disk of their complex plane. We show that on the metallic side $W < 6$ of the Anderson transition, complex eigenvalues repel each other as strongly as in the complex Ginibre ensemble only in a disk containing $N_c(L,W)$ eigenvalues. We find that $N_c(L,W)$ is proportional to $L$ and grows with decreasing $W$ similarly to the number of energy levels $N_c$ in the Thouless energy band of the Anderson model.