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The entanglement entropy of two gapless non-interacting fermion subsystems iscomputed approximately in a way that avoids the introduction of replicas and ageometric interpretation of the reduced density matrix. We exploit thesimilarity between the Schmidt basis wavefunction and superfluid BCSwavefunction and compute the entropy using the BCS approximation. Within thisanalogy, the Cooper pairs are particle-hole pairs straddling the boundary andthe effective interaction between them is induced by the projection of theHilbert space onto the incomplete Schmidt basis. The resulting singularinteraction may be thought of as "lifting" the degeneracy of the singleparticle 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 thesubsystem and $2t$ is the bandwidth. We further interpret this result basedupon the relationship between entanglement spectrum, entropy and numberfluctuations.

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