  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.
