Perturbed Block Toeplitz matrices and the non-Hermitian skin effect in dimer systems of subwavelength resonators

TL;DR

This paper derives explicit eigenpair formulas for perturbed block Toeplitz matrices, providing a mathematical basis for the non-Hermitian skin effect and its topological origin in dimer systems of subwavelength resonators.

Contribution

It introduces explicit eigenpair formulas for perturbed block Toeplitz matrices and links the non-Hermitian skin effect to topological properties in dimer systems.

Findings

Eigenpairs explicitly derived for perturbed block Toeplitz matrices

Mathematical proof of eigenmode condensation at edges

Topological origin of the non-Hermitian skin effect

Abstract

The aim of this paper is fourfold: (i) to obtain explicit formulas for the eigenpairs of perturbed tridiagonal block Toeplitz matrices; (ii) to make use of such formulas in order to provide a mathematical justification of the non-Hermitian skin effect in dimer systems by proving the condensation of the system's bulk eigenmodes at one of the edges of the system; (iii) to show the topological origin of the non-Hermitian skin effect for dimer systems and (iv) to prove localisation of the interface modes between two dimer structures with non-Hermitian gauge potentials of opposite signs based on new estimates of the decay of the entries of the eigenvectors of block matrices with mirrored blocks.