Band structures under non-Hermitian periodic potentials: Connecting nearly-free and bi-orthogonal tight-binding models

TL;DR

This paper investigates the band structures of one-dimensional non-Hermitian periodic systems, revealing how imaginary potentials influence band gaps and degeneracies, and introduces bi-orthogonal tight-binding models that accurately reproduce continuum model dispersions.

Contribution

It demonstrates that imaginary scalar potentials do not open gaps but create exceptional points, and develops bi-orthogonal tight-binding models for non-Hermitian systems based on Wannier functions.

Findings

Imaginary scalar potentials lead to exceptional points without opening gaps.

Imaginary vector potentials prevent low-energy band separation.

Bi-orthogonal tight-binding models accurately reproduce continuum dispersions.

Abstract

We explore band structures of one-dimensional open systems described by periodic non-Hermitian operators, based on continuum models and tight-binding models. We show that imaginary scalar potentials do not open band gaps but instead lead to the formation of exceptional points as long as the strength of the potential exceeds a threshold value, which is contrast to closed systems where real potentials open a gap with infinitesimally small strength. The imaginary vector potentials hinder the separation of low energy bands because of the lifting of degeneracy in the free system. In addition, we construct tight-binding models through bi-orthogonal Wannier functions based on Bloch wavefunctions of the non-Hermitian operator and its Hermitian conjugate. We show that the bi-orthogonal tight-binding model well reproduces the dispersion relations of the continuum model when the complex scalar potential is sufficiently large.