Detection of a quantum phase transition in a spin-1 chain through multipartite high-order correlations

We design a Bell inequality that is violated by correlations obtained from the ground states of an XXZ spin-1 chain with on-site anisotropies at the region of phase transition. In order to detect such correlations in spin-1 systems we exploit the formalism of the generalized Bell inequality via the use of multipartite and high-order correlations. We observe a sharp violation in the vicinity of the quantum phase transition between the so-called large-D and AFM phase. Interestingly, the violation of our Bell inequality is manifested by the change in the XXZ spin-1 chain ground state to a Greenberger-Horne-Zeilinger-like state in the critical region. Our results provide a characterization of quantum phase transition via the violation of Bell-type constraint by correlations in the XXZ spin-1 chain with multibody correlations and high-order measurements.