Banded non-Hermitian random matrices, neural networks, and eigenvalue degeneracies

TL;DR

This paper investigates the spectral properties of two-banded non-Hermitian random matrices inspired by neural networks, revealing how disorder and directional bias influence eigenstate localization, delocalization, and degeneracies.

Contribution

It introduces and analyzes two models, the SSH chain and ladder, demonstrating novel delocalization phenomena, eigenvalue degeneracies, and spectral structures driven by disorder and bias.

Findings

Eigenstates localize due to sign disorder following Dale's Law.

Directional bias causes a delocalization transition in eigenstates.

Distinct band structures lead to different delocalization behaviors in models.

Abstract

We study two-banded, non-Hermitian random matrices inspired by sparse neural networks with a circular, 1d topology. We focus on two paradigmatic models, an SSH chain and a ladder model, which have both non-Hermitian directional bias and random sign disorder in the hoppings. The random sign disorder, which follows Dale's Law, leads to localization of the eigenstates, while the directional bias drives a delocalization transition in these states. The competition between disorder and directional bias results in rich eigenspectra with loops of extended states in the complex plane surrounded by regions of localized ones, and the eigenvalues are all confined to an annular region. Furthermore, the distinct band structures of the SSH chain and ladder model lead to different delocalization phenomena. Even in the absence of disorder, tuning the directional bias can lead to an eigenvalue degeneracy, which is an exceptional point for the SSH chain but a diabolic point for the ladder. In the presence of the disorder, these special eigenvalue degeneracies are preserved and also highlight key stages in the delocalization process. For both models, increasing the directional bias initially delocalizes states starting from within the bands. For the SSH chain, for large enough directional bias, the delocalized states open up a hole in the spectrum in the complex plane, similar to prior results for single band systems. But for the ladder model, as the directional bias is increased, the states delocalize in two stages, leading to two separate loops of extended states with localized states in between. The precise contours on which the extended states reside can be predicted from the Lyapunov exponents associated with products of random transfer matrices, in agreement with direct numerical diagonalization. Although we focus on periodic boundaries, results are discussed for open boundaries as well.