Entanglement entropy of fermions in any dimension and the Widom conjecture

Abstract

We show that entanglement entropy of free fermions scales faster then 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 $d$, $S\sim c(\partial \Gamma ,\partial \Omega )\cdot L^{d-1}\log L$ as the size of a subsystem $L\to \infty$, where $\partial \Gamma$ is the Fermi surface and $\partial \Omega$ is the boundary of the region in real space. The expression for the constant $c(\partial \Gamma ,\partial \Omega )$ is based on a conjecture due to H. Widom. We prove that a similar expression holds for the particle number fluctuations and use it to prove a two sided estimates on the entropy $S$.

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