  We provide a natural generalization of a Riemannian structure, i.e., a
metric, recently introduced by Sj\"{o}qvist for the space of non degenerate
density matrices, to the degenerate case, i.e., the case in which the
eigenspaces have dimension greater than or equal to 1. We present a physical
interpretation of the metric in terms of an interferometric measurement. We
apply 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 behaviors 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 the system's density matrix (i.e., the ranks of its
spectral projectors).
