Entanglement at the quantum phase transition in a harmonic lattice

TL;DR

This paper investigates how entanglement measures signal quantum phase transitions in a two-dimensional harmonic lattice, showing abrupt changes at critical points and comparing thermal entanglement with witness predictions.

Contribution

It demonstrates that entanglement measures can indicate quantum phase transitions in harmonic lattices, extending previous finite-dimensional results to continuous-variable systems.

Findings

Entanglement measures change abruptly at the critical trapping potential.

Entanglement signals quantum phase transitions in harmonic lattices.

Thermal entanglement is compared with an entanglement witness, showing consistency.

Abstract

The entanglement properties of the phase transition in a two dimensional harmonic lattice, similar to the one observed in recent ion trap experiments, are discussed both, for 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 [Phys. Rev. Lett. {\bf 93}, 250404 (2004)]. Finally, we consider the thermal state and compare its exact entanglement with a temperature entanglement witness introduced in [Phys. Rev. A {\bf 77} 062102 (2008)].