  We study the behaviour of the Uhlmann connection in systems of fermions
undergoing phase transitions. In particular, we analyse some of the
paradigmatic cases of topological insulators and superconductors in dimension
one, as well as the BCS theory of superconductivity in three dimensions. We
show that the Uhlmann connection signals phase transitions in which the
eigenbasis of the state of the system changes. Moreover, using the established
fidelity approach and the study of the edge states, we show the absence of
thermally driven phase transitions in the case of topological insulators and
superconductors. We clarify what is the relevant parameter space associated
with the Uhlmann connection so that it signals the existence of order in mixed
states.
