  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 $S$.
