Complex Band Structure and localisation transition for tridiagonal non-Hermitian k-Toeplitz operators with defects

TL;DR

This paper introduces a novel Bloch-Floquet based method to analyze eigenvectors of non-Hermitian k-Toeplitz operators, revealing detailed localization phenomena and broad applicability in complex band structure analysis.

Contribution

It presents an innovative technique for eigenvector computation of non-Hermitian tridiagonal k-Toeplitz operators, extending understanding of localization beyond traditional zones.

Findings

Validated method through numerical simulations of resonator chains.

Identified non-Hermitian skin localization and tunneling effects.

Demonstrated broad applicability to non-Hermitian Hamiltonians.

Abstract

Using the Bloch-Floquet theory, we propose an innovative technique to obtain the eigenvectors of tridiagonal k-Toeplitz operators. This method offers a more extensive and quantitative basis for describing localised eigenvectors beyond the non-trivial winding zone, yielding sharp decay bounds. The validity of our results is confirmed numerically in one-dimensional resonator chains, showcasing non-Hermitian skin localisation, bulk localisation, and tunnelling effects. We conclude the paper by analysing non-Hermitian tight binding Hamiltonians, illustrating the broad applicability of the complex band structure.