Chern number and Berry curvature for Gaussian mixed states of fermions

TL;DR

This paper extends the concept of topological invariants to mixed states in two dimensions, defining a proper Chern number based on the ensemble geometric phase for Gaussian states, which can be expressed via Berry curvature.

Contribution

It introduces a new topological invariant for mixed states in two dimensions, generalizing the Chern number using the ensemble geometric phase for Gaussian states.

Findings

The Chern number is expressed as an integral of Berry curvature over the Brillouin zone.

The Chern number is non-zero only if the fictitious Hamiltonian breaks time-reversal symmetry.

The approach applies to finite-temperature and non-equilibrium steady states.

Abstract

We generalize the concept of topological invariants for mixed states based on the ensemble geometric phase (EGP) introduced for one-dimensional lattice models to two dimensions. In contrast to the geometric phase for density matrices suggested by Uhlmann, the EGP leads a proper Chern number for Gaussian, finite-temperature or non-equilibrium steady states. The Chern number can be expressed as an integral of the Berry curvature of the so-called fictitious Hamiltonian, constructed from single-particle correlations, over the two-dimensional Brillouin zone. For the Chern number to be non-zero the fictitious Hamiltonian has to break time-reversal symmetry.