# Chiral symmetry in non-Hermitian systems: product rule and Clifford algebra

## TL;DR

This paper explores how chiral symmetry manifests in non-Hermitian systems, providing two general methods—via Clifford algebra and combined symmetry conditions—to identify and generate such symmetries in complex lattice models.

## Contribution

It introduces two novel approaches for recognizing and constructing chiral symmetry in non-Hermitian systems, extending the understanding of symmetry protection in topological phases.

## Key findings

- Clifford algebra approach for chiral symmetry identification
- Symmetry conditions involving non-Hermitian particle-hole and anti-linear symmetries
- Application to lattices with complex on-site potential variations

## Abstract

Chiral symmetry provides the symmetry protection for a large class of topological edge states. It exists in non-Hermitian systems as well, and the same anti-commutation relation between the Hamiltonian and a linear chiral operator, i.e., $\{H,\Pi\}=0$, now warrants a symmetric spectrum about the origin of the complex energy plane. Here we show two general approaches to identify and generate chiral symmetry in non-Hermitian systems, with an emphasis on lattices with detuned on-site potentials that can vary in both their real and imaginary parts. One approach utilizes the Clifford algebra satisfied by the Dirac matrices, while the other relies on the simultaneous satisfaction of non-Hermitian particle-hole symmetry and bosonic anti-linear symmetry, extended beyond simple spatial transformations to include, for example, an imaginary gauge transformation.