Information geometry in the analysis of phase transitions

The Uhlmann connection is a mixed state generalization of the Berry connection. The latter has a very important role in the study of topological phases at zero temperature. Closely related, the fidelity is an information theoretical measure of distinguishability between quantum states. We show how one can use the fidelity and the Uhlmann connection to study phase transitions at finite temperature. We apply the analysis to free fermion Hamiltonians in 1D exhibiting symmetry protected topological order at zero temperature and also to the BCS theory of superconductivity. We show how one can study finite-temperature dynamical phase transitions by means of the fidelity and interferometric Loschmidt echoes. Moreover, we explain the physical and mathematical origin of the different behaviour of the two Loschmdit echoes by means of the associated susceptibilities.