  I compute the entanglement entropy of a strongly coupled 2+1d quantum field
theory containing fermions at finite density using gauge/gravity duality. The
dual geometry is an extremal black hole in 3+1d Einstein-Maxwell theory. This
system was recently shown to exhibit non-Fermi liquid behavior, but the leading
geometrical contribution to the entanglement entropy does not produce an
expected violation of the boundary law. I discuss this negative result in the
context of attempts to find highly entangled states of quantum matter.
