Spectral Curves and Localization in Random Non-Hermitian Tridiagonal Matrices

TL;DR

This paper investigates the spectral properties and localization phenomena in non-Hermitian tridiagonal random matrices using the Hatano-Nelson deformation, revealing annular spectral structures and their relation to eigenstate localization.

Contribution

It introduces a spectral duality and the Argument principle to explain the annular spectrum and localization in non-Hermitian matrices, extending previous Hermitian model insights.

Findings

Spectrum forms an annulus with inner radius from the complex Thouless formula

Inner circle and halo structures correspond to localization features

Spectral duality explains the spectral geometry and eigenstate localization

Abstract

Eigenvalues and eigenvectors of non-Hermitian tridiagonal periodic random matrices are studied by means of the Hatano-Nelson deformation. The deformed spectrum is annular-shaped, with inner radius measured by the complex Thouless formula. The inner bounding circle and the annular halo are stuctures that correspond to the two-arc and wings observed by Hatano and Nelson in deformed Hermitian models, and are explained in terms of localization of eigenstates via a spectral duality and the Argument principle.