Chern number and Berry curvature for Gaussian mixed states of fermions

We generalize the concept of topological invariants for mixed states based on the ensemble geometric phase (EGP) to two-dimensional band structures. In contrast to the geometric Uhlmann phase for density matrices, the EGP leads to a proper Chern number for Gaussian, finite-temperature, or nonequilibrium steady states. The Chern number can be expressed as an integral of the ground-state Berry curvature of a fictitious lattice Hamiltonian, constructed from single-particle correlations. For the Chern number to be nonzero the fictitious Hamiltonian has to break time-reversal symmetry.

Automated summary

The study extends the concept of ensemble geometric phase to mixed states in two-dimensional band structures, introducing a proper Chern number calculation method for various types of states. The Chern number can be calculated through the ground-state Berry curvature of a fictitious Hamiltonian that breaks time-reversal symmetry.