Characterization of quantum phase transition in theXYmodel with multipartite correlations and Bell-type inequalities

Abstract

The ground state of interacting spin chains in external magnetic fields can undergo a quantum phase transition (QPT) characterized by dramatic changes at a critical value of the magnetic field. In this paper, we use Bell-type inequalities to study the multipartite correlations (including multipartite entanglement and multipartite nonlocality in an $n$-spin subsystem) in the QPT of an infinite $XY$ chain. An efficient numerical optimization procedure is proposed to figure out the violation measure ${M}_{n}$ of the inequalities. For $n\ensuremath{\le}7$, the magnetic-field ($\ensuremath{\lambda}$) dependence of ${M}_{n}$ is studied. We find the derivative of ${M}_{n}$ is divergent exactly at the QPT point ${\ensuremath{\lambda}}_{c}=1$ for any $n$. In addition, with the increase of $n$, ${M}_{n}$ converges quickly for $\ensuremath{\lambda}l{\ensuremath{\lambda}}_{c}$ and converges very slowly for $\ensuremath{\lambda}g{\ensuremath{\lambda}}_{c}$, which can be regarded as another signal for the QPT. Furthermore, in the vicinity of ${\ensuremath{\lambda}}_{c}$, high-order Bell-type inequalities will be violated as long as $n$ is large enough. This indicates that high-level multipartite correlation will be present when the system is in the vicinity of the QPT point. Nevertheless, genuine $n$-partite entanglement or genuine $n$-partite nonlocality is not observed in the QPT.