Integer characterization of topological phases at finite temperature using Uhlmann’s parallel transport
Berry’s geometric phase can be defined for a single Bloch band, or for a group of bands, and allows for integer characterization of topological phases via the TKNN (Chern) numbers. An extension of Berry’s phase to density matrices, first discussed by Uhlmann [Rep. Math. Phys. 24, 229 (1986)], generalizes Berry’s phase in such a way that finite temperature can be accommodated. Topological phases at finite temperature can thereby be characterized by one or more integers in such a way that reduces to the conventional TKNN description at T=0, yet change discretely at one or more critical temperatures. We demonstrate these ideas, and their limitations, in detail, using Haldane’s honeycomb lattice model, and comment on their generalization to multi-band Chern insulators and Z_2 topological insulators.