Spectra and pseudospectra of non-Hermitian Toeplitz operators: Eigenvector decay transitions in banded and dense matrices

TL;DR

This paper develops a mathematical framework using Floquet-Bloch theory to analyze eigenvector decay in non-Hermitian Toeplitz operators, revealing decay transitions and applying these insights to model the non-Hermitian skin effect in 3D resonator systems.

Contribution

It introduces a generalized method for constructing eigenvectors of non-Hermitian Toeplitz operators, including dense matrices, and provides decay estimates and insights into the non-Hermitian skin effect.

Findings

Eigenvector decay estimates for banded and dense operators

Poor reconstruction of dense operators by banded approximations

Mechanism for transition between skin effect and localization

Abstract

Using a generalised Floquet-Bloch theory, we present a mathematical method to construct eigenvectors for non-Hermitian Toeplitz operators. We extend the method to both banded Toeplitz operators and those with algebraically decaying, fully dense off-diagonal structure. We present sharp decay estimates for the amplitude of bulk eigenmodes as well as eigenmodes associated with defect eigenfrequencies inside the spectral band gap. The validity of those results is illustrated numerically and we show that banded approximations give poor reconstructions of the dense operators, due to the slow algebraic decay. We apply the insights gained to model the non-Hermitian skin effect in a three-dimensional system of subwavelength resonators, where the corresponding operator exhibits only algebraic decay of off-diagonal entries. We use our approach to demonstrate the fundamental mechanism responsible for the transition between the non-Hermitian skin effect and defect-induced localisation in the bulk.