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 Ld-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 similar to c(partial derivative Gamma,partial derivative Omega)L(d-1)logL as the size of a subsystem L ->infinity, where partial derivative Gamma is the Fermi surface and partial derivative Omega is the boundary of the region in real space. The expression for the constant c(partial derivative Gamma,partial derivative 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.