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 $d$, $S\sim c(\partial \Gamma ,\partial \Omega )L^{d-1}\log L$ 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 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 $S$.

Figure 1

The Fermi sea $\Gamma$ in momentum space, and a region $\Omega$ in real space.