  A system of fermions forming a Fermi surface exhibits a large degree of
quantum entanglement, even in the absence of interactions. In particular, the
usual case of a codimension one Fermi surface leads to a logarithmic violation
of the area law for entanglement entropy, as dictated by the Widom formula. We
here generalize this formula to the case of arbitrary codimension, which is of
particular interest for nodal lines in three dimensions. We first rederive the
standard Widom formula by calculating an entanglement Hamiltonian for Fermi
surface systems, obtained by repurposing a trick commonly applied to
relativistic theories. The entanglement Hamiltonian will take a local form in
terms of a low-energy patch theory for the Fermi surface, though it is nonlocal
with respect to the microscopic fermions. This entanglement Hamiltonian can
then be used to derive the entanglement entropy, yielding a result in agreement
with the Widom formula. The method is then generalized to arbitrary
codimension. For nodal lines, the area law is obeyed, and the magnitude of the
coefficient for a particular partition is non-universal. However, the
coefficient has a universal dependence on the shape and orientation of the
nodal line relative to the partitioning surface. By comparing the relative
magnitude of the area law for different partitioning cuts, entanglement entropy
can be used as a tool for diagnosing the presence and shape of a nodal line in
a ground state wavefunction.
