# Bell-Type Correlation at Quantum Phase Transitions in Spin-1 Chain

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## Abstract

**:**

## 1. Introduction

## 2. The 1d Spin-1 XXZ Model with the On-Site Anisotropy

## 3. Bell-Type Correlation from the Generalized Nonlocality Criteria

## 4. Quantum Criticality through the Correlation for the Nonlocality Tests

#### 4.1. The CGLMP Correlation at D = 0

#### 4.2. The CGLMP Correlation at ${J}_{z}$ = 1

#### 4.3. The CGLMP Correlation at ${J}_{z}=-$0.1

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CHSH | Clauser–Horne–Shimony–Holt |

CGLMP | Collins–Gisin–Linden–Massar–Popescu |

SLK | Son–Lee–Kim |

iDMRG | infinite-size Density-Matrix Renormalization Group |

MPS | Matrix Product States |

QPTs | Quantum Phase Transitions |

AFM | Antiferromagnetic |

FM | Ferromagnetic |

BKT | Berezinskii, Kosterlitz, Thouless |

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**Figure 1.**(Color online) (

**a**) the CGLMP correlation $\langle {\widehat{\mathcal{B}}}_{r}\rangle $ and (

**b**) the spin–spin correlations ${C}_{r}^{\left(2\right)}$ and (

**c**) ${C}_{r}^{\left(1\right)}$ as varying the parameter ${J}_{z}$ at $D=0$ for odd distances.

**Figure 2.**(

**a**) CGLMP correlation $\langle {\widehat{\mathcal{B}}}_{r}\rangle $ and the first derivative of $\langle {\widehat{\mathcal{B}}}_{r}\rangle $ with respect to D near (

**b**) $D=-0.31$ and (

**c**) $D=0.97$ as varying the parameter D at ${J}_{z}=1$ for odd distances.

**Figure 3.**(

**a**) CGLMP correlation $\langle {\widehat{\mathcal{B}}}_{r}\rangle $, (

**b**) spin–spin correlations ${C}_{r}^{\left(1\right)}$ and ${C}_{r}^{\left(2\right)}$ as varying the parameter D, and (

**c**) first derivative of $\langle {\widehat{\mathcal{B}}}_{r}\rangle $ with respect to D at ${J}_{z}=-0.1$ for odd distances r.

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Lee, D.; Son, W. Bell-Type Correlation at Quantum Phase Transitions in Spin-1 Chain. *Entropy* **2020**, *22*, 1282.
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**AMA Style**

Lee D, Son W. Bell-Type Correlation at Quantum Phase Transitions in Spin-1 Chain. *Entropy*. 2020; 22(11):1282.
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**Chicago/Turabian Style**

Lee, Dongkeun, and Wonmin Son. 2020. "Bell-Type Correlation at Quantum Phase Transitions in Spin-1 Chain" *Entropy* 22, no. 11: 1282.
https://doi.org/10.3390/e22111282