Topological classifications of quadratic bosonic excitations in closed and open systems with examples

TL;DR

This paper classifies quadratic bosonic systems' topological phases in closed and open settings, revealing how system-reservoir interactions influence their symmetry classes and topological properties, with specific models illustrating these effects.

Contribution

It provides a comprehensive topological classification framework for quadratic bosonic systems in both closed and open regimes, highlighting the impact of reservoir coupling on symmetry classes.

Findings

The non-Hermitian dynamic matrix and effective Hamiltonian follow a ten-fold way classification.

System-reservoir coupling can change the topological class of a system.

Certain models like the 2D Chern insulator are insensitive to classification changes.

Abstract

The topological classifications of quadratic bosonic systems according to the symmetries of the dynamic matrices from the equations of motion of closed systems and the effective Hamiltonians from the Lindblad equations of open systems are analyzed. While the non-Hermitian dynamic matrix and effective Hamiltonian both lead to a ten-fold way table, the system-reservoir coupling may cause a system with or without coupling to a reservoir to fall into different classes. A 2D Chern insulator is shown to be insensitive to the different classifications. In contrast, we present a 1D bosonic Su-Schrieffer-Heeger model with chiral symmetry and a 2D bosonic topological insulator with time-reversal symmetry to show the corresponding open systems may fall into different classes.