  We extend the study of finite-entanglement scaling from one-dimensional
gapless models to two-dimensional systems with a Fermi surface. In particular,
we show that the entanglement entropy of a contractible spatial region with
linear size $L$ scales as $S\sim L\log[\xi f(L/\xi)]$ in the optimal tensor
network, and hence area-law entangled, state approximation to a metallic state,
where $f(x)$ is a scaling function which depends on the shape of the Fermi
surface and $\xi$ is a finite correlation length induced by the restricted
entanglement. Crucially, the scaling regime can be realized with numerically
tractable bond dimensions. We also discuss the implications of the
Lieb-Schultz-Mattis theorem at fractional filling for tensor network state
approximations of metallic states.
