Anderson transition in three-dimensional systems with non-Hermitian disorder

TL;DR

This paper investigates the Anderson transition in three-dimensional non-Hermitian systems with complex disorder, identifying a clear transition point through eigenvalue statistics, which differs from the crossover observed in 2D models.

Contribution

The study demonstrates the existence of a true Anderson transition in 3D non-Hermitian systems using eigenvalue statistics, contrasting with the crossover behavior in 2D models.

Findings

Transition point at W_c = 6.0 ± 0.1 identified

Eigenvalue ratio r(W) crosses at W_c for different N

3D non-Hermitian systems exhibit a true Anderson transition

Abstract

We study the Anderson transition for three-dimensional (3D) $N \times N \times N$ tightly bound cubic lattices where both real and imaginary parts of onsite energies are independent random variables distributed uniformly between $-W/2$ and $W/2$. Such a non-Hermitian analog of the Anderson model is used to describe random-laser medium with local loss and amplification. We employ eigenvalue statistics to search for the Anderson transition. For 25\% smallest-modulus complex eigenvalues we find the average ratio $r$ of distances to the first and the second nearest neighbor as a function of $W$. For a given $N$ the function $r(W)$ crosses from $0.72$ to 2/3 with a growing $W$ demonstrating a transition from delocalized to localized states. When plotted at different $N$ all $r(W)$ cross at $W_c = 6.0 \pm 0.1$ (in units of nearest neighbor overlap integral) clearly demonstrating the 3D Anderson transition. We find that in the non-Hermitian 2D Anderson model, the transition is replaced by a crossover.