Topological symmetry classes for non-Hermitian models and connections to the bosonic Bogoliubov-de Gennes equation

TL;DR

This paper explores the topological classification of non-Hermitian and bosonic BdG models using Bernard-LeClair symmetry classes, revealing that bosonic Hamiltonians inherit non-Hermitian topological properties and analyzing edge instabilities.

Contribution

It establishes a connection between non-Hermitian symmetry classes and bosonic BdG Hamiltonians, extending topological classification to these systems.

Findings

Bosonic BdG spectra relate to non-Hermitian BL classes.

Topological properties are inherited by bosonic Hamiltonians.

Edge instabilities are demonstrated in a 1D model.

Abstract

The Bernard-LeClair (BL) symmetry classes generalize the ten-fold way classes in the absence of Hermiticity. Within the BL scheme, time-reversal and particle-hole come in two flavors, and "pseudo-Hermiticity" generalizes Hermiticity. We propose that these symmetries are relevant for the topological classification of non-Hermitian single-particle Hamiltonians and Hermitian bosonic Bogoliubov-de Gennes (BdG) models. We show that the spectrum of any Hermitian bosonic BdG Hamiltonian is found by solving for the eigenvalues of a non-Hermitian matrix which belongs to one of the BL classes. We therefore suggest that bosonic BdG Hamiltonians inherit the topological properties of a non-Hermitian symmetry class and explore the consequences by studying symmetry-protected edge instabilities in a simple 1D system.