Energy space entanglement spectrum of pairing models with s-wave and p-wave symmetry

TL;DR

This paper analyzes the entanglement spectrum of energy levels in superconducting models with s-wave and p-wave symmetry, revealing how entanglement measures relate to Cooper pairing and quantum phase transitions.

Contribution

It introduces a novel measure of active Cooper pairs based on entanglement spectrum and characterizes entanglement features across different superconducting phases.

Findings

Maximal entanglement entropy scales as 1/2 log(L) with system size.

The entanglement spectrum includes a principal band and higher bands, whose behavior signals phase transitions.

The number of levels with maximal entanglement correlates with the number of active Cooper pairs.

Abstract

Entanglement between blocks of energy-levels is analysed for systems exhibiting s-wave and p-wave superconductivity. We study the entanglement entropy and spectrum of a block of $\ell$ levels around the Fermi point, and also between particles and holes, in the ground state of Richardson-type Hamiltonians. The maximal entropy grows with the number of levels $L$ approximately as $1/2\log(L)$, as suggested by the permutational symmetry of the state at large coupling. The number of levels in the block around the Fermi surface with maximal entanglement is proposed as a measure of the number of {\em active Cooper pairs}, which correlates with standard estimates of this magnitude. The entanglement spectrum is always composed of a principal parabolic band plus {\em higher bands} whose disappearance signals a exact BCS state, e.g. in the Moore-Read line, while the Read-Green quantum phase transition is characterized by a maximum in their weight.