Momentum-space entanglement for interacting fermions at finite density

We investigate the entanglement between individual field theory modes in finite-density systems of interacting relativistic and non-relativistic fermions in one spatial dimension. We calculate the entanglement entropy for a single field theory mode and the mutual information between any two modes. The calculation is perturbative in the four-fermion (two-body) coupling, with the leading contribution at order λ2 log(λ2). At this leading order, the perturbative expression for the entanglement entropy of a mode diverges logarithmically as the momentum of the mode approaches the Fermi surface from above or below. The mutual information between modes is largest for pairs of modes just above and below the Fermi momentum. The entanglement properties of modes near the Fermi surface are qualitatively the same if the field theory is cut off to eliminate modes away from the Fermi surface.