Integer characterization of 2D topological insulators at finite temperature

2D band topological insulators (TI) are characterized by the TKNN number and its variants. However, this only works for zero temperature as the TKNN number is no longer quantized for $T>0$. We show that using Uhlmann's parallel transport for density matrices, TI at finite temperature can still be characterized by an integer, which (1) reduces to the corresponding TKNN number at $T = 0$, and (2) exhibits a phase transition, i.e. drops to zero, at a critical temperature. Prototypical models such as Haldane's honeycomb lattice model and the Bernevig-Hughs-Zhang model will be discussed.