Projective approach to the entanglement entropy of 1-$d$ fermions

TL;DR

This paper introduces a novel projective approach to compute the entanglement entropy of 1D fermions without using replicas, by leveraging a BCS analogy and solving a gap equation to find entropy scaling.

Contribution

It presents a new method that approximates entanglement entropy in 1D fermions using a BCS-inspired framework, avoiding traditional replica techniques.

Findings

Entropy scales as (w^2/t^2) log L for large L

The approach links entanglement spectrum to number fluctuations

Provides an approximate solution to the BCS gap equation for entropy

Abstract

The entanglement entropy of two gapless non-interacting fermion subsystems is computed approximately in a way that avoids the introduction of replicas and a geometric interpretation of the reduced density matrix. We exploit the similarity between the Schmidt basis wavefunction and superfluid BCS wavefunction and compute the entropy using the BCS approximation. Within this analogy, the Cooper pairs are particle-hole pairs straddling the boundary and the effective interaction between them is induced by the projection of the Hilbert space onto the incomplete Schmidt basis. The resulting singular interaction may be thought of as "lifting" the degeneracy of the single particle distribution function. For two coupled fermion systems of linear size $L$, we solve the BCS gap equation approximately to find the entropy $S \approx (w^2/t^2)\log{L}$ where $w$ is the hopping amplitude at the boundary of the subsystem and $2t$ is the bandwidth. We further interpret this result based upon the relationship between entanglement spectrum, entropy and number fluctuations.