Chern number and Berry curvature for Gaussian mixed states of fermions

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