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

Two-dimensional topological phases are characterized by TKNN 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)), an integer classification of topological phases can be defined for a finite generalized temperature $T$ or dephasing Lindbladian. This scheme reduces to the familiar TKNN classification for $T<T_{{\rm c},1}$, becomes trivial for $T>T_{{\rm c},2}$, and exhibits a `gapless' intermediate regime where topological indices are not well-defined. We demonstrate these ideas in detail, applying them to Haldane's honeycomb lattice model and the Bernevig-Hughes-Zhang model, and we comment on their generalization to multi-band Chern insulators.