Connections between the Open-boundary Spectrum and Generalized Brillouin Zone in Non-Hermitian Systems

TL;DR

This paper reveals deep connections between the open-boundary spectrum, generalized Brillouin zone, and periodic-boundary spectrum in non-Hermitian systems, introducing a new numerical method and extending the theory to symplectic symmetry classes.

Contribution

It demonstrates that the relationships among these spectral properties can be understood through special similarity transformations, providing a novel numerical approach and extending the framework to symplectic symmetry classes.

Findings

Deep connections between spectra and GBZ via similarity transformations

A new efficient numerical method for calculating spectra and GBZ

Extension of the theory to symplectic symmetry class systems

Abstract

Periodic-boundary spectrum, open-boundary spectrum, as well as the generalized Brillouin zone (GBZ) are three essential properties of a one-dimensional non-Hermitian system. In this paper we illustrate that the deep connections between them can be revealed by a series of special similar transformations. This viewpoint closely connects the topological geometry of the open-boundary spectrum with the GBZ and provides a new efficient numerical method of calculating them accurately. We further extend these connections to non-Hermitian systems in the symplectic symmetry class. We show that if just the open-boundary features of a non-Hermitian system such as the spectrum and the GBZ, are concerned, the relevant symmetry we should consider is not that of the original system itself, but that of one which has higher symmetry and is related to the original system by a similarity transformation.