Interferometric geometry from symmetry-broken Uhlman gauge group and applications to topological phase transitions

We provide a natural generalization of a Riemannian structure, i.e., a metric, recently introduced by Sjoqvist for the space of non degenerate density matrices, to the degenerate case, i.e., in which the eigenspaces have dimension greater or equal to one. We present a physical interpretation of the metric in terms of an interferometric measurement. We study this metric, physically interpreted as an interferometric susceptibility, to the study of topological phase transitions at finite temperatures for band insulators. We compare the behaviors of this susceptibility and the one coming from the well-known Bures metric, showing them to be dramatically different. While both infer zero temperature phase transitions, only the former predicts finite temperature phase transitions as well. The difference in behaviours can be traced back to a symmetry breaking mechanism, akin to Landau-Ginzburg theory, by which the Uhlmann gauge group is broken down to a subgroup determined by the type of system's density matrix (i.e., the ranks of its spectral projectors).