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 ${\cal F}=\sqrt{\sqrt{\rho_1} \rho_2 \sqrt{\rho_1}}$. We study the logarithmic spectrum of this operator, which we denote by {\it 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 different phases of many-body quantum systems.