Generalized Chern numbers based on open system Green's functions

TL;DR

This paper introduces a Green's function-based method to define and compute Chern numbers in open quantum systems, capturing environmental effects directly and revealing novel quantization behaviors.

Contribution

It proposes an energy-dependent Chern number derived from Green's functions, providing a new topological invariant for open systems that does not rely on non-Hermitian Hamiltonians.

Findings

Half-integer quantization of Chern number under specific damping conditions

Loss of quantization and vanishing Chern number away from these conditions

Potential for integer quantization in special cases

Abstract

We present an alternative approach to studying topology in open quantum systems, relying directly on Green's functions and avoiding the need to construct an effective non-Hermitian Hamiltonian. We define an energy-dependent Chern number based on the eigenstates of the inverse Green's function matrix of the system which contains, within the self-energy, all the information about the influence of the environment, interactions, gain or losses. We explicitly calculate this topological invariant for a system consisting of a single 2D Dirac cone and find that it is half-integer quantized when certain assumptions over the damping are made. Away from these conditions, which cannot or are not usually considered within the formalism of non-Hermitian Hamiltonians, we find that such a quantization is usually lost and the Chern number vanishes, and that in special cases, it can change to integer quantization.