Entanglement entropy in the Lipkin-Meshkov-Glick model

Abstract

We analyze the entanglement entropy in the Lipkin-Meshkov-Glick model, which describes mutually interacting spins half embedded in a magnetic field. This entropy displays a singularity at the critical point that we study as a function of the interaction anisotropy, the magnetic field, and the system size. Results emerging from our analysis are surprisingly similar to those found for the one-dimensional XY chain.PACS numbers: 03.65. Ud,03.67.Mn,73.43.Nq Within the last few years, entanglement properties of spin systems have attracted much attention. As initially shown in one-dimensional (1D) spin chains [1,2,3,4], observables measuring this genuine quantum mechanical feature are strongly affected by the existence of a quantum phase transition. For instance, the so-called concurrence [5] that quantifies the two-spin entanglement displays some nontrivial universal scaling properties. Similarly, the von Neumann entropy which rather characterizes the bipartite entanglement between any two subsystems scales logarithmically with the typical size L of these subsystems at the quantum critical point, with a prefactor given by the central charge of the corresponding theory [3,4,6,7]. Note that the role played by the boundaries in these conformal invariant systems has been only recently elucidated [8]. Apart from 1D systems, very few models have been studied so far [9,10,11,12,13], either due to the absence of exact solution or to a difficult numerical treatment. In this context, the LipkinMeshkov-Glick (LMG) model [14,15,16] discussed here has drawn much attention since it allows for very efficient numerical treatment as well as analytical calculations. Introduced by Lipkin, Meshkov and Glick in Nuclear Physics, this model has been the subject of intensive studies during the last two decades because of its relevance for quantum tunneling of bosons between two levels. It is thus of prime interest to describe in particular the Josephson effect in two-mode Bose-Einstein condensates [17,18]. The entanglement properties of this model have been already discussed through the concurrence, which exhibits a cusp-like behavior at the critical point [19,20,21,22] as well as interesting dynamical properties [23]. Note that similar results have also been obtained in the Dicke model [24,25] which can be mapped onto the LMG model in some cases [26], or in the reduced BCS model [27].In this letter, we analyze the von Neumann entropy computed from the ground state of the LMG model. We show that, at the critical point, it behaves logarithmically with the size of the blocks L used in the bipartite decomposition of the density matrix with a prefactor that depends on the anisotropy parameter tuning the universality class. We also discuss the dependence of the entropy with the magnetic field and stress the analogy with 1D systems.The LMG model is defined by the Hamiltonianwhere σ k α is the Pauli matrix at position k in the direction α, and N the total number of spins. This Hamiltonian describes a set of spins half located at the vertices of ...

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