Entanglement at the quantum phase transition in a harmonic lattice

Elisabeth Rieper,Vlatko Vedral,Janet Anders

doi:10.1088/1367-2630/12/2/025017

Abstract

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.93250404). 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).

Highlights

Νt > νt,crit x νt < νt,crit of entanglement, measured by the negativity and the von Neumann entropy, changes abruptly at the critical point and indicates the occurrence of a QPT, in a manner similar to the way classical correlations indicate standard phase transitions
This is in contrast to models with only nearest neighbour (NN) interaction where ‘area laws’ apply [22] and entanglement does not increase with block size, i.e. volume, as long as the surface of the block is constant
Two measures of entanglement display critical behaviour: the von Neumann entropy of a single site and blocks of two and three sites diverge at the critical point while the negativity is not differentiable

Summary

The model

To calculate the entanglement measures, we approximate the Coulomb potential to second order and expand about the equilibrium positions. Similar to the classical calculation [20], we use a simplified model with equidistant equilibrium position in the x-direction, spaced by the lattice constant a. Such a condition can be realized for the central ions of a long ion chain inside a linear Paul trap [25] or for ions confined in a ring of large radius [18, 26]. The dx,y,xy denote second order Taylor coefficients of the Coulomb potential which are, for the linear and zig-zag configuration, dτx

Calculation of entanglement measures

Diagonalization of the Hamiltonian

Two-site negativity and von Neumann entropy

Behaviour of entanglement at zero temperature

Witnessing entanglement at finite temperature

Conclusions