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We provide a natural generalization of a Riemannian structure, i.e., ametric, recently introduced by Sj\"{o}qvist for the space of non degeneratedensity matrices, to the degenerate case, i.e., the case in which theeigenspaces have dimension greater than or equal to 1. We present a physicalinterpretation of the metric in terms of an interferometric measurement. Weapply this metric, physically interpreted as an interferometric susceptibility,to the study of topological phase transitions at finite temperatures for bandinsulators. We compare the behaviors of this susceptibility and the one comingfrom the well-known Bures metric, showing them to be dramatically different.While both infer zero temperature phase transitions, only the former predictsfinite temperature phase transitions as well. The difference in behaviors canbe traced back to a symmetry breaking mechanism, akin to Landau-Ginzburgtheory, by which the Uhlmann gauge group is broken down to a subgroupdetermined by the type of the system's density matrix (i.e., the ranks of itsspectral projectors).

Interferometric geometry from symmetry-broken Uhlmann gauge group with applications to topological phase transitions