Entanglement properties and quantum phases for a fermionic disordered one dimensional wire with attractive interactions

A fermionic disordered one dimensional wire in the presence of attractive interactions is known to have two distinct phases: A localized and a superconducting one depending on the strength of interaction and disorder. The localized region may also exhibit a metallic behaviour if the system size is shorter than the localization length. Here we show that the superconducting phase has a distinct distribution of the entanglement entropy and entanglement spectrum distinct from the metallic regime. The entanglement entropy distribution is strongly asymmetric with L\'evy alpha stable distribution (compared to the Gaussian metallic distribution), and the entanglement level spacing distribution is unitary (compared to orthogonal). Thus, entanglement properties may reveal properties which can not be detected by other methods.