Topological indices for open and thermal systems via Uhlmann's phase.

Two-dimensional topological phases are characterized by Thouless-Kohmoto-Nightingale-den Nijs integers, which classify Bloch energy bands or groups of Bloch bands. However, quantization does not survive thermal averaging or dephasing to mixed states. We show that using Uhlmann's parallel transport for density matrices [Rep. Math. Phys. 24, 229 (1986).