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∼ c(∂Γ,∂Ω)L^d-1log⁡L as the size of a subsystem L→∞, where ∂Γ is the Fermi surface and ∂Ω is the boundary of the region in real space. The expression for the constant c(∂Γ,∂Ω) 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.