  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.
