The Uhlmann connection in fermionic systems undergoing phase transitions
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 1D, as well as the BCS theory of superconductivity in 3D. We show that the Uhlmann connection signals phase transitions during which the eigenbasis of the state of the system changes. Moreover, using the established fidelity approach, 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. Finally, the analysis of the different behaviours of the BCS model and the Kitaev chain suggests that in realistic scenarios the Uhlmann connection could be used to probe the finite-temperature behaviour of superconducting systems (topological and trivial).