  The entanglement entropy of a noninteracting fermionic system confined to a
two-dimensional honeycomb lattice on a torus is calculated. We find that the
entanglement entropy can characterize Lifshitz phase transitions without a
local order parameter. In the noncritical phase and critical phase with a nodal
Fermi surface, the entanglement entropy satisfies an area law. The leading
subarea term is a constant in the gapped phase rather than a logarithmic
additive term in the gapless regime. The tuning of chemical potential allows
for a nonzero Fermi surface, whose variation along a particular direction
determines a logarithmic violation of the area law. We perform the scaling of
entanglement entropy numerically and find agreement between the analytic and
numerical results. Furthermore, we clearly show that an entanglement spectrum
is equivalent to an edge spectrum.
