General theory for geometry-dependent non-Hermitian bands

TL;DR

This paper develops a higher-dimensional non-Bloch band theory using strip generalized Brillouin zones to explain the phenomenon of geometry-dependent bands in non-Hermitian lattices, revealing their origin and criteria.

Contribution

It introduces a novel theoretical framework based on SGBZs to understand geometry-dependent non-Hermitian bands and derives a general criterion for their occurrence.

Findings

Geometry-dependent bands arise from incompatibility of SGBZs.

A criterion based on the complex energy spectrum's area explains band dependence.

The theory advances understanding of non-Hermitian band topology.

Abstract

In two- and higher-dimensional non-Hermitian lattices, systems can exhibit geometry-dependent bands, where the spectrum and eigenstates under open boundary conditions depend on the bulk geometry even in the thermodynamic limit. Although geometry-dependent bands are widely observed, the underlying mechanism for this phenomenon remains unclear. In this work, we address this problem by establishing a higher-dimensional non-Bloch band theory based on the concept of "strip generalized Brillouin zones" (SGBZs), which describe the asymptotic behavior of non-Hermitian bands when a lattice is extended sequentially along its linearly independent axes. Within this framework, we demonstrate that geometry-dependent bands arise from the incompatibility of SGBZs and, for the first time, derive a general criterion for the geometry dependence of non-Hermitian bands: non-zero area of the complex energy spectrum or the imaginary momentum spectrum. Our work opens an avenue for future studies on the interplay between geometric effects and non-Hermitian physics, such as non-Hermitian band topology.