[论文解读] Critical properties of disordered XY model on sparse random graphs

Cosimo Lupo|arXiv (Cornell University)|Jun 27, 2017

Theoretical and Computational Physics参考文献 137被引用 4

一句话总结

本论文通过腔方法与信念传播研究了稀疏随机图上无序XY模型的临界行为。结果表明，Q-state时钟模型可对连续XY模型实现指数级精确逼近，揭示了稀释XY自旋玻璃中比离散模型更强的玻璃态行为，甚至在随机场铁磁相中也存在自旋玻璃的副本对称性破缺。

ABSTRACT

This thesis focuses on the XY model, the simplest vector spin model, used for describing numerous physical systems. It is studied for different sources of quenched disorder: random couplings, random fields, or both them. The belief propagation algorithm and the cavity method are exploited to solve the model on the sparse topology provided by Bethe lattices. It is found that the discretized version of the XY model, the so-called $Q$-state clock model, provides a reliable and efficient proxy for the continuous model with an error going to zero exponentially in $Q$, so implying a remarkable speedup in numerical simulations. Interesting results regard the low-temperature solution of the spin glass XY model, which is by far more unstable toward the replica symmetry broken phase with respect to what happens in discrete models. Moreover, even the random field XY model with ferromagnetic couplings exhibits a replica symmetry broken phase, at variance with both the fully connected version of the same model and the diluted random field Ising model, as a further evidence of a more pronounced glassiness of the diluted XY model. Then, the instabilities of the spin glass XY model in an external field are characterized, recognizing different critical lines according to the different symmetries of the external field. Finally, the inherent structures in the energy landscape of the spin glass XY model in a random field are described, exploiting the capability of the zero-temperature belief propagation algorithm to actually reach the ground state of the system. Remarkably, the density of soft modes in the Hessian matrix shows a non-mean-field behaviour, typical of glasses in finite dimension, while the critical point of replica symmetry instability predicted by the belief propagation algorithm seems to correspond to a delocalization of such soft modes.

研究动机与目标

• 分析无序XY模型在稀疏随机图上的临界性质。
• 评估Q-state时钟模型作为连续XY模型代理的准确性。
• 研究自旋玻璃与随机场XY模型中副本对称解的稳定性。
• 表征随机场下自旋玻璃相的能量景观与软模。
• 探讨外场对称性在诱导XY自旋玻璃模型中不同临界线中的作用。

提出的方法

• 在Bethe格点上应用腔方法与信念传播求解无序XY模型。
• 使用零温信念传播精确达到自旋玻璃模型的基态。
• 应用副本方法分析副本对称性破缺的不稳定性。
• 通过计算Hessian矩阵并分析其软模来映射能量景观。
• 探索不同外场对称性下解的收敛性与稳定性。
• 评估Q-state时钟模型近似的误差，显示其随Q指数衰减。

实验结果

研究问题

• RQ1 Q-state时钟模型作为稀疏图上连续XY模型的有限逼近，其准确性如何？
• RQ2 与离散模型相比，稀释XY自旋玻璃模型中副本对称解的稳定性如何？
• RQ3 具有铁磁耦合的随机场XY模型是否表现出副本对称性破缺，与完全连接版本或伊辛模型类比时有何差异？
• RQ4 外场的不同对称性如何影响XY自旋玻璃中临界行为与相变？
• RQ5 Hessian矩阵中自旋玻璃基态的软模性质为何，其与副本对称性破缺有何关联？

主要发现

• Q-state时钟模型可对连续XY模型实现指数级精确逼近，误差按 e^(-cQ) 衰减（c > 0）。
• 稀释XY自旋玻璃模型表现出显著强于离散模型的副本对称性破缺不稳定性，表明玻璃态行为增强。
• 即使在具有铁磁耦合的随机场XY模型中，也存在副本对称性破缺相，这与完全连接版本及稀释随机场伊辛模型相反。
• 外场的不同对称性导致相图中出现不同的临界线，表明玻璃态行为具有对称性依赖性。
• Hessian矩阵中软模的密度表现出非平均场行为，典型于有限维玻璃态，暗示偏离标准平均场理论。
• 信念传播预测的副本对称性破缺不稳定性对应于软模的非局域化，表明能量景观中存在结构相变。