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## Quantum phase transition and topological order scaling in spin-1 bond-alternating Heisenberg model with Dzyaloshinskii-Moriya interaction

Chen Ai-Min, Liu Dong-Chang, Duan Jia, Wang Hong-Lei, Xiang Chun-Huan, Su Yao-Heng
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• #### 摘要

利用张量网络表示的无限矩阵乘积态算法研究了含有Dzyaloshinskii-Moriya (DM)相互作用的键交替海森伯模型的量子相变和临界标度行为. 基于矩阵乘积态的基态波函数计算了系统的量子纠缠熵及非局域拓扑序. 数据表明, 随着键交替强度变化, 系统从拓扑有序的Haldane相转变为局域有序的二聚化相. 同时DM相互作用抑制了系统的二聚化, 并最终打破系统的完全二聚化. 另外, 通过对相变点附近二聚化序的一阶导数和长程弦序的数值拟合, 分别得到了此模型相变的特征临界指数αβ的值. 结果表明, 随着DM相互作用强度的增强, α逐渐减小, 同时β逐渐增大. DM相互作用强度影响着此模型的临界行为. 针对此模型的临界性质的研究, 揭示了量子自旋相互作用的彼此竞争机制, 对今后研究含有DM相互作用的自旋多体系统中拓扑量子相变临界行为提供一定的借鉴与参考.

#### Abstract

Quantum phase transitions are driven by quantum fluctuations due to the uncertainty principle in many-body physics. In quantum phase transitions, the ground-state changes dramatically. The quantum entanglement, specific heat, magnetization and other physical quantities diverge according to certain functions, and show specific scaling behaviors. In addition, there is a topological quantum phase transition beyond the conventional Landau-Ginzburg-Wilson paradigm, which is relevant to emergent phenomena in strongly correlated electron systems, with topological nonlocal order parameters as a salient feature. Thus, topological order is a new paradigm in the study of topological quantum phase transitions.To investigate competition mechanism of the different quantum spin interactions, in this paper, the one-dimensional spin-1 bond-alternating Heisenberg model with Dzyaloshinskii-Moriya (DM) interaction is considered. The DM interaction drives the quantum fluctuations resulting in a phase transition. By using the one-dimensional infinite matrix product state algorithm in tensor network representation, the quantum entanglement entropy and order parameters are calculated from the ground-state function. The numerical result shows that with the change of bond alternating strength, there is a quantum phase transition from the topological ordered Haldane phase to the local ordered dimer phase. Based on the von Neumann entropy and order parameter, the phase diagram of this model is obtained. There is a critical line that separates the Haldane and the dimer phase. The DM interaction inhibits the dimerization of the quantum spin system and finally breaks the fully dimerization. Due to the fact that the structurally symmetry of system is broken, the local dimer order exists in the whole parameter range when the bond-alternative strength parameter changes. The first derivative of the local dimer order behaves as a peak corresponding to the critical point. Furthermore, from the numerical scaling of the first derivative of dimer order and the non-local string order near the phase transition point, the characteristic critical exponents α and β are obtained, respectively. It shows that the characteristic critical exponent α decreases, and β increases gradually with the interaction strength of DM increasing. The resulting state i.e. the anti-symmetric anisotropic DM interaction produced by spin-orbit coupling, affects the critical properties of the system in the phase transition. This reveals that the competition mechanism of the quantum spin interaction, also provides some guidance for the future study of the critical behavior in topological quantum phase transition with the DM interaction.

#### 作者及机构信息

###### 通信作者: 陈爱民, chenaimin_xa@163.com
• 基金项目: 国家自然科学基金(批准号: 11504283)和陕西省自然科学基金(批准号: 2019JM-017)资助的课题

#### Authors and contacts

###### Corresponding author: Chen Ai-Min, chenaimin_xa@163.com
• Funds: Project supported by the National Natural Science Foundation of China (Grant No. 11504283) and the Natural Science Foundation of Shaanxi Province, China (Grant No. 2019JM-017)

#### 施引文献

• 图 1  D = 0.5和D = 1时(a)奇键von Neumann熵${S_{\rm{o}}}$和(b)偶键von Neumann熵${S_{\rm{e}}}$$\delta 的变化 Fig. 1. (a) Odd-bond von Neumann entropy {S_{\rm{o}}} and (b) even-bond von Neumann entropy {S_{\rm{e}}} as a function of \delta for D = 0.5 and D = 1. 图 2 二聚化序{O_{\rm{D}}}$$\delta$的变化(内插图为${O_{\rm{D}}}$$\delta 的一阶导数) (a) D = 0.5, (b) D = 1 Fig. 2. Dimer order parameter {O_{\rm{D}}} as a function of \delta for (a) D = 0.5 and (b) D = 1 (first derivation of {O_{\rm{D}}} in the insert). 图 3 二聚化序{O_{\rm{D}}}随DM相互作用强度D的变化, 其中键交替参数选取为\delta = 1 Fig. 3. Dimer order parameter {O_{\rm{D}}} as a function of DM interaction D for \delta =1 . 图 4 非局域拓扑弦序{O_{\rm{S}}}$$\delta$的变化

Fig. 4.  Non-local topological string order parameter as a function of $\delta$.

图 5  含有DM相互作用的自旋1键交替反铁磁海森伯模型相图

Fig. 5.  Phase diagram of spin-1 bond-alternating Heisenberg model with DM interaction.

图 6  相互作用参数(a) D = 0, (b) D = 0.5和(c) D = 1时在对应相变点$\delta _{\rm{c}}^{D = 0} \approx 0.260$, $\delta _{\rm{c}}^{D = 0.5} \approx 0.305$$\delta _{\rm{c}}^{D = 1} \approx0.418附近二聚化序的一阶导数\partial {O_{\rm{D}}}/\partial \delta 的特征临界指数\alpha 拟合 Fig. 6. Characteristic critical exponent \alpha from the first derivative of the dimer order \partial {O_{\rm{D}}}/\partial \delta for (a) D = 0, (b) D = 0.5, and (c) D = 1 in the vicinity of the critical points \delta _{\rm{c}}^{D = 0} \approx 0.260, \delta _{\rm{c}}^{D = 0.5} \approx 0.305, and \delta _{\rm{c}}^{D = 1} \approx 0.418, respectively. 图 7 相互作用参数(a) D = 0.5和(b) D = 1时在对应相变点\delta _{\rm{c}} ^{D = 0.5} \approx 0.305$$\delta _{\rm{c}}^{D = 1} \approx 0.418$附近非局域拓扑弦序${O_{\rm{S}}}$的特征临界指数$\beta$拟合

Fig. 7.  Characteristic critical exponent $\beta$ from the nonlocal topological string order${O_{\rm{S}}}$ for (a) D = 0.5 and (b) D = 1 in the vicinity of the critical points $\delta _{\rm{c}} ^{D = 0.5} \approx 0.305$ and $\delta _{\rm{c}}^{D = 1} \approx 0.418$, respectively.

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##### 出版历程
• 收稿日期:  2019-11-21
• 修回日期:  2020-02-16
• 刊出日期:  2020-05-05

## 含有Dzyaloshinskii-Moriya相互作用的自旋1键交替海森伯模型的量子相变和拓扑序标度

• 1. 西安工程大学理学院, 西安　710048
• 2. 重庆医科大学医学信息学院, 重庆　400016
• 3. 重庆医科大学公共卫生与管理学院, 重庆　400016
• ###### 通信作者: 陈爱民, chenaimin_xa@163.com
基金项目: 国家自然科学基金(批准号: 11504283)和陕西省自然科学基金(批准号: 2019JM-017)资助的课题

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