  We demonstrate that the entropy of entanglement and the distillable
entanglement of regions with respect to the rest of a general harmonic lattice
system in the ground or a thermal state scale at most as the boundary area of
the region. This area law is rigorously proven to hold true in non-critical
harmonic lattice system of arbitrary spatial dimension, for general
finite-ranged harmonic interactions, regions of arbitrary shape and states of
nonzero temperature. For nearest-neighbor interactions - corresponding to the
Klein-Gordon case - upper and lower bounds to the degree of entanglement can be
stated explicitly for arbitrarily shaped regions, generalizing the findings of
[Phys. Rev. Lett. 94, 060503 (2005)]. These higher dimensional analogues of the
analysis of block entropies in the one-dimensional case show that under general
conditions, one can expect an area law for the entanglement in non-critical
harmonic many-body systems. The proofs make use of methods from entanglement
theory, as well as of results on matrix functions of block banded matrices.
Disordered systems are also considered. We moreover construct a class of
examples for which the two-point correlation length diverges, yet still an area
law can be proven to hold. We finally consider the scaling of classical
correlations in a classical harmonic system and relate it to a quantum lattice
system with a modified interaction. We briefly comment on a general
relationship between criticality and area laws for the entropy of entanglement.
