Fidelity spectrum and phase transitions of quantum systems

Quantum fidelity between two density matrices F(rho(1),rho(2)) is usually defined as the trace of the operator F = root root rho(1)rho(2)root rho(1). We study the logarithmic spectrum of this operator, which we denote by the fidelity spectrum, in the cases of the XX spin chain in a magnetic field, a magnetic impurity inserted in a conventional superconductor, and a bulk superconductor at finite temperature. When the density matrices are equal, rho(1) = rho(2), the fidelity spectrum reduces to the entanglement spectrum. We find that the fidelity spectrum can be a useful tool in giving a detailed characterization of the different phases of many-body quantum systems.