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, S approximately c(deltagamma, deltaomega)L(d-1) logL as the size of a subsystem L-->infinity, where deltagamma is the Fermi surface and is the boundary of the region in real space. The expression for the constant c(deltagamma, deltaomega) 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.