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We generalize the concept of topological invariants for mixed states based onthe ensemble geometric phase (EGP) introduced for one-dimensional latticemodels to two dimensions. In contrast to the geometric phase for densitymatrices suggested by Uhlmann, the EGP leads a proper Chern number forGaussian, finite-temperature or non-equilibrium steady states. The Chern numbercan be expressed as an integral of the Berry curvature of the so-calledfictitious Hamiltonian, constructed from single-particle correlations, over thetwo-dimensional Brillouin zone. For the Chern number to be non-zero thefictitious Hamiltonian has to break time-reversal symmetry.

Chern number and Berry curvature for Gaussian mixed states of fermions