Universität Potsdam Institut für Physik KarlLiebknechtStr. 24/25 14476 PotsdamGolm 



AbstractWe 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 noncritical harmonic lattice system of arbitrary spatial dimension, for general finiteranged harmonic interactions, regions of arbitrary shape and states of nonzero temperature. For nearestneighbor interactions  corresponding to the KleinGordon 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 onedimensional case show that under general conditions, one can expect an area law for the entanglement in noncritical harmonic manybody 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 twopoint 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. file generated: 18 Apr 2007


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