  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 lambda^2 log(lambda^2). 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.
