  We discuss the relation between entanglement and criticality in
translationally invariant harmonic lattice systems with non-randon,
finite-range interactions. We show that the criticality of the system as well
as validity or break-down of the entanglement area law are solely determined by
the analytic properties of the spectral function of the oscillator system,
which can easily be computed. In particular for finite-range couplings we find
a one-to-one correspondence between an area-law scaling of the bi-partite
entanglement and a finite correlation length. This relation is strict in the
one-dimensional case and there is strog evidence for the multi-dimensional
case. We also discuss generalizations to couplings with infinite range.
Finally, to illustrate our results, a specific 1D example with nearest and
next-nearest neighbor coupling is analyzed.
