Multipartite nonlocality in the Lipkin-Meshkov-Glick model

Abstract

Lipkin-Meshkov-Glick (LMG) models describe a set of $N$ qubits which are located at the corners of an $N$-dimensional simplex and embedded in an external magnetic field. Because of its high coordinate number, quantum correlations in the models are quite nontrivial. For instance, in the large-$N$ limit, two-site entanglement concurrence vanishes for any magnetic field. Multisite global entanglement and generalized global entanglement also tend to vanish in some quantum regions. We characterize the quantum correlations in the LMG model with multipartite nonlocality. In LMG models with anisotropy in the $x\ensuremath{-}y$ plane, for any fixed magnetic field, the ground-state nonlocality (denoted by $\mathcal{S}$) is found to scale as ${log}_{10}\mathcal{S}\ensuremath{\sim}aN+b$, with $a$ and $b$ the fitting parameters. For LMG models which are isotropic in the $x\ensuremath{-}y$ plane, nevertheless, the nonlocality for each Dicke state scales as ${log}_{10}\mathcal{S}\ensuremath{\sim}a\frac{1}{N}+b$, with $bg0$. Signals for the quantum phase transitions of the models will also be discussed. These results indicate that multipartite nonlocality captures some key ingredient of the ground-state quantum correlations in the models. In addition to the ground states, we also study multipartite nonlocality in the LMG models at finite temperatures. In the anisotropic models, the thermal stability of the nonlocality at low temperatures is determined by the energy gap. For the isotropic models, nevertheless, as the temperature increases, the nonlocality presents a sharp valley which is followed by a round peak. The behavior is far beyond the above-mentioned ``thermal stability vs energy gap'' picture. The underlying mechanics is the ``contribution inversion'' of low-lying energy states in the thermal-state nonlocality $\mathcal{S}({\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{T})$; that is, in a low-temperature region, the first-excited state ``defeats'' the ground state and plays a dominant role in thermal-state nonlocality.