  We study the fidelity approach to quantum phase transitions (QPTs) and apply
it to general thermal phase transitions (PTs). We analyze two particular cases:
the Stoner-Hubbard itinerant electron model of magnetism and the BCS theory of
superconductivity. In both cases we show that the sudden drop of the mixed
state fidelity marks the line of the phase transition. We conduct a detailed
analysis of the general case of systems given by mutually commuting
Hamiltonians, where the non-analyticity of the fidelity is directly related to
the non-analyticity of the relevant response functions (susceptibility and heat
capacity), for the case of symmetry-breaking transitions. Further, on the case
of BCS theory of superconductivity, given by mutually non-commuting
Hamiltonians, we analyze the structure of the system's eigenvectors in the
vicinity of the line of the phase transition showing that their sudden change
is quantified by the emergence of a generically non-trivial Uhlmann mixed state
geometric phase.
