Non-Hermitian bulk-boundary correspondence and auxiliary generalized Brillouin zone theory

TL;DR

This paper introduces an analytical method to determine the generalized Brillouin Zone in one-dimensional non-Hermitian systems, clarifying the bulk-boundary correspondence and enabling spectral analysis of open systems.

Contribution

It presents a systematic, algebraic approach to calculate the GBZ and auxiliary GBZ, advancing understanding of non-Hermitian topological phenomena.

Findings

Analytic characterization of GBZ via algebraic equations

Method to derive GBZ from auxiliary GBZ

Application to spectral analysis of open boundary systems

Abstract

We provide a systematic and self-consistent method to calculate the generalized Brillouin Zone (GBZ) analytically in one dimensional non-Hermitian systems, which helps us to understand the non-Hermitian bulk-boundary correspondence. In general, a n-band non-Hermitian Hamiltonian is constituted by n distinct sub-GBZs, each of which is a piecewise analytic closed loop. Based on the concept of resultant, we can show that all the analytic properties of the GBZ can be characterized by an algebraic equation, the solution of which in the complex plane is dubbed as auxiliary GBZ (aGBZ). We also provide a systematic method to obtain the GBZ from aGBZ. Two physical applications are also discussed. Our method provides an analytic approach to the spectral problem of open boundary non-Hermitian systems in the thermodynamic limit.