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^d-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, S0ex{0ex}c (, ) L^d-1logL as the size of a subsystem L, where is the Fermi surface and is the boundary of the region in real space. The expression for the constant c (, ) 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.