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.

Figure 1

Spectrum of asymmetric Qi-Wu-Zhang model.

Figure 2

The Chern number of a two-dimensional, translational invariant lattice model (a) can be represented as the winding of Zak phases of one-dimensional chains. This can be done in two different ways, either by the winding of the Zak phase ${\varphi }_y={\varphi }_x\left(k_x\right)$ upon cyclic changes in the lattice momentum $k_x$ (b) or by the (negative) winding of the Zak phase ${\varphi }_x={\varphi }_x\left(k_y\right)$ upon cyclic changes in $k_y$ (c).

Figure 3

EGP of the Qi-Wu-Zhang model in the $x$ direction as a function of $k_y$ (top) and in the $y$ direction as a function of $k_x$ (bottom) for different temperatures. One recognizes that in contrast to the Uhlmann case, the windings of both are exactly the same (with the proper sign convention) for all temperatures even much above the single-particle gap.