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.
An entanglement-area law for general bosonic harmonic lattice systems
我们表明，熵的纠缠和纠缠的区域相对于其余的一个一般的谐波晶格系统在地面或热状态的规模最多作为该地区的边界面积。严格证明了该面积定律在任意空间维度的非临界调和晶格系统中，对于一般的有限范围调和相互作用、任意形状的区域和非零温度的状态都是成立的。对于最近邻相互作用-对应于Klein-Gordon情况-纠缠度的上界和下界可以明确地为任意形状的区域陈述，概括了[Phys. Rev. Lett. 94，060503（2005）]。在一维情况下，块熵分析的这些高维类似物表明，在一般条件下，可以预期的面积律的纠缠在非临界谐波多体系统。证明利用纠缠理论的方法，以及块带状矩阵的矩阵函数的结果。还考虑了无序系统。此外，我们构造了一类例子，两点相关长度发散，但仍然可以证明面积法成立。最后，我们考虑了经典谐振子系统中经典关联的标度性，并将其与具有修正相互作用的量子晶格系统联系起来。我们简要地评论临界性和面积定律之间的一般关系的熵纠缠。
