Spectra and pseudo-spectra of tridiagonal $k$-Toeplitz matrices and the topological origin of the non-Hermitian skin effect

TL;DR

This paper links the spectral properties of tridiagonal $k$-Toeplitz matrices to topological phenomena, explaining the non-Hermitian skin effect through eigenvalue winding numbers and eigenvector decay, supported by numerical verification.

Contribution

It establishes a theoretical connection between eigenvalue winding numbers and eigenvector decay in $k$-Toeplitz matrices, revealing the topological origin of the non-Hermitian skin effect.

Findings

Eigenvalue winding number determines eigenvector decay

Topological origin of the non-Hermitian skin effect confirmed

Numerical verification supports theoretical results

Abstract

We establish new results on the spectra and pseudo-spectra of tridiagonal $k$-Toeplitz operators and matrices. In particular, we prove the connection between the winding number of the eigenvalues of the symbol function and the exponential decay of the associated eigenvectors (or pseudo-eigenvectors). Our results elucidate the topological origin of the non-Hermitian skin effect in general one-dimensional polymer systems of subwavelength resonators with imaginary gauge potentials, proving the observation and conjecture in arXiv:2307.13551. We also numerically verify our theory for these systems.