Topological phase transitions at finite temperature

The ground states of noninteracting fermions in one-dimension with chiral symmetry form a class topological band insulators, described by invariant that can be related to the Zak phase. Recently, generalization this quantity open systems---known as ensemble geometric phase (EGP)---has emerged robust way describe topology at nonzero temperature. By using quantity, we explore nature allowed for dissipation beyond Lindblad description, allow coupling external baths finite temperatures. We introduce two main aspects theory open-system topology. First, discover transitions function temperature $T$, manifesting changes differences EGP accumulated over closed loop parameter space. characterize these and reveal corresponding nonequilibrium steady state exhibit nontrivial structure---contrary previous studies where it was found fully mixed state. Second, demonstrate itself becomes quantized when key symmetries are present, allowing viewed marker which undergo equilibrium