Topological invariant for gaussian mixed states of fermion systems with time-reversal symmetry

Topological properties of Gaussian mixed states of fermions can be fully classified by the single-particle correlation matrix, which defines the ficticious Hamiltonian 

    $\hat H_\textrm{fict}$. 

    If the ficticious Hamiltonian breaks time-reversal (TR) symmetry, a topological invariant can be constructed based on the ensemble geometric phase or mixed-state many-body polarization \cite{PRX}, which

    is identical to the Chern number of the ground state of  $\hat H_\textrm{fict}$.      For a TR

    invariant ficticious Hamiltonian this invariant vanishes. We here introduce a generalization of the time-reversal polarization to mixed states of fermions and show that

    it defines a $Z_2$ topological invariant. Specifically we discuss the one-dimensional Fu-Kane model at finite temperature to illustrate the invariant.