Fidelity, Fidelity Spectrum and Geometric Phases as Signatures of Phase Transitions

We show that the fidelity, a measure of state distinguishability, used
in quantum information and computation, can be efficiently used as a
tool to detect some macroscopic phase transitions, and we establish
its relation to standard many-body properties. In particular, we make
the analytical study of the anisotropic XY spin chain in a magnetic
field, the Stoner-Hubbard model of itinerant magnetism and the BCS
model of superconductivity, and make the numerical study of an
impurity in a superconductor film. We show that the sudden drop of the
fidelity marks the line of the phase transition. We study the
fidelity-induced metric on the space of parameters that define
system's Hamiltonian and its connection to geometric Berry and Uhlmann
phases. We also establish the relation between the fidelity-induced
metric, often referred to as fidelity susceptibility, and
thermodynamical susceptibility associated to a given phase transition.
Finally, we study the logarithmic spectrum of the operator whose trace
defines the quantum fidelity between two density operators, and denote
it by the fidelity spectrum. We find that the fidelity spectrum can be
a useful tool in giving a more detailed characterization of different
phases of many-body quantum systems and of the particular modes that
drive the phase transition.