Mathematical foundations of the non-Hermitian skin effect

TL;DR

This paper investigates the non-Hermitian skin effect in one-dimensional resonator systems, proving eigenmode condensation at edges, and distinguishes between different sources of non-Hermiticity using Toeplitz matrix theory and complex band analysis.

Contribution

It introduces a mathematical framework for analyzing the non-Hermitian skin effect, including a generalized complex Brillouin zone, and compares different non-Hermitian mechanisms.

Findings

Eigenmodes condense at system edges

Spectral bands of infinite and finite systems match in the large limit

Distinction between imaginary gauge potentials and complex material parameters

Abstract

We study the skin effect in a one-dimensional system of finitely many subwavelength resonators with a non-Hermitian imaginary gauge potential. Using Toeplitz matrix theory, we prove the condensation of bulk eigenmodes at one of the edges of the system. By introducing a generalised (complex) Brillouin zone, we can compute spectral bands of the associated infinitely periodic structure and prove that this is the limit of the spectra of the finite structures with arbitrarily large size. Finally, we contrast the non-Hermitian systems with imaginary gauge potentials considered here with systems where the non-Hermiticity arises due to complex material parameters, showing that the two systems are fundamentally distinct.