Entanglement at the quantum phase transition in a harmonic lattice

The entanglement properties of phase transition in a two-dimensional harmonic lattice, similar to the one observed in recent ion trap experiments, are discussed for both finite number of particles and thermodynamical limit. We show that for the ground state at the critical value of the trapping potential, two entanglement measures, the negativity between two neighbouring sites and the block entropy for blocks of size 1, 2 and 3, change abruptly. Entanglement thus indicates quantum phase transitions in general, not only in the finite-dimensional case considered in Wu et al (2004 Phys. Rev. Lett. 93 250404). Finally, we consider the thermal state and compare its exact entanglement with a temperature entanglement witness introduced in Anders (2008 Phys. Rev. A 77 062102).