# Topological band theory for non-Hermitian systems from the Dirac equation

## TL;DR

This paper explores topological properties of non-Hermitian systems derived from the Dirac equation, analyzing how Lorentz-symmetry violation and complex mass influence bulk-boundary correspondence and skin effects.

## Contribution

It introduces a unified framework for understanding topological phases in non-Hermitian Dirac systems with Lorentz-symmetry violation and complex mass, clarifying conditions for skin effects and bulk-boundary correspondence.

## Key findings

- Non-Hermitian skin effect occurs in systems with Lorentz-symmetry violation.
- Bulk-boundary correspondence is restored via gauge transformation in LSV systems.
- No skin effect and conventional bulk-boundary correspondence in systems with complex mass.

## Abstract

We identify and investigate two classes of non-Hermitian systems, i.e., one resulting from Lorentz-symmetry violation (LSV) and the other from a complex mass (CM) with Lorentz invariance, from the perspective of quantum field theory. The mechanisms to break, and approaches to restore, the bulk-boundary correspondence in these two types of non-Hermitian systems are clarified. The non-Hermitian system with LSV shows a non-Hermitian skin effect, and its topological phase can be characterized by mapping it to the Hermitian system via a non-compact $U(1)$ gauge transformation. In contrast, there exists no non-Hermitian skin effect for the non-Hermitian system with CM. Moreover, the conventional bulk-boundary correspondence holds in this (CM) system. We also consider a general non-Hermitian system in the presence of both LSV and CM, and we generalize its bulk-boundary correspondence.