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.

Figure 1

In the $J_z\text{−}D$ plane the ratio of Bell correlation to the local realistic bound $\mathcal{B}/{\beta }_{\mathrm{LR}}$ maximized over $\tilde{f}$ for $N=8$. Inset: weight ${\tilde{f}}_{\max }$ choices in a specific region. The violation of a Bell inequality $\mathcal{B}\ge {\beta }_{\mathrm{LR}}$ arises (in purple). This violation signifies the transition of quantum phase from the large-$D$ phase to AFM phase.

Figure 2

Value of $\mathcal{B}/{\beta }_{\mathrm{LR}}$ with $f_{max}^v$ for $J_z=\{6,12\}$. The maximum value of $\mathcal{B}/{\beta }_{\mathrm{LR}}$ occurs at the first-order phase transition for large $J_z$. At this criticality, $\mathcal{B}/{\beta }_{\mathrm{LR}}$ increases as $N$ grows large and behaves as an exponential function of $N, \mathcal{B}/{\beta }_{\mathrm{LR}}\sim {\gamma }^N$.

Figure 3

The fidelity $F$ between the ground state and the state $|{\psi }_{max}\rangle$ defined in Eq. (7) for $J_z=12$ near the first-order phase transitions up to the particle number $N\le 10$. We can see that the maximum of the fidelity $F$ occurs at the criticality.

Figure 4

Entanglement entropy $S\left({\widehat{\rho }}_{N/2}\right)$ in the vicinity of the first-order phase transition for exchange anisotropies $J_z=6$ and $J_z=12$.

Figure 5

Scaling behaviors of the ratio of the Bell correlation to the local realistic bound $\mathcal{B}/{\beta }_{\mathrm{LR}}$ (left) and the entanglement entropy $S$ (right) at the quantum criticality. The green line is the exponential fit $\mathcal{B}/{\beta }_{\mathrm{LR}}=a{\gamma }^N+c$, and the red line is the logarithmic fit $S=alogN+c$.