  Exactly solving a spinless fermionic system in two and three dimensions, we
investigate the scaling behavior of the block entropy in critical and
non-critical phases. The scaling of the block entropy crucially depends on the
nature of the excitation spectrum of the system and on the topology of the
Fermi surface. Noticeably, in the critical phases the scaling violates the area
law and acquires a logarithmic correction \emph{only} when a well defined Fermi
surface exists in the system. When the area law is violated, we accurately
verify a conjecture for the prefactor of the logarithmic correction, proposed
by D. Gioev and I. Klich [quant-ph/0504151].
