Entanglement Entropy of Fermions in Any Dimension and the Widom Conjecture

We show that entanglement entropy of free fermions scales faster than area law, as opposed to the scaling ${L}^{densuremath{-}1}$ for the harmonic lattice, for example. We also suggest and provide evidence in support of an explicit formula for the entanglement entropy of free fermions in any dimension $d$, $Sensuremath{sim}phantom{rule{0ex}{0ex}}c(ensuremath{partial}ensuremath{Gamma},ensuremath{partial}ensuremath{Omega}){L}^{densuremath{-}1}mathrm{log}L$ as the size of a subsystem $Lensuremath{rightarrow}ensuremath{infty}$, where $ensuremath{partial}ensuremath{Gamma}$ is the Fermi surface and $ensuremath{partial}ensuremath{Omega}$ is the boundary of the region in real space. The expression for the constant $c(ensuremath{partial}ensuremath{Gamma},ensuremath{partial}ensuremath{Omega})$ is based on a conjecture due to Widom. We prove that a similar expression holds for the particle number fluctuations and use it to prove a two sided estimate on the entropy $S$.