Entanglement of collectively interacting harmonic chains: An effective two-dimensional system

We study the ground-state entanglement of one-dimensional harmonic chains that are coupled to each other by a collective interaction as realized, e.g., in an anisotropic ion crystal. Due to the collective type of coupling, where each chain interacts with every other one in the same way, the total system shows critical behavior in the direction orthogonal to the chains, while the isolated harmonic chains can be gapped and noncritical. We derive lower and most importantly upper bounds for the entanglement, quantified by the von Neumann entropy, between a compact block of oscillators and its environment. For sufficiently large size of the subsystems, the bounds coincide and show that the area law for entanglement is violated by a logarithmic correction.