Abstract

Ferromagnetic and spin-glass-like transitions in nonequilibrium spin models in contact with two thermal baths with different temperatures are investigated. The models comprise the Sherrington-Kirkpatrick model and the dilute spin glass model which are the Ising models on complete and random graphs, respectively, with edges corresponding, with certain probability, to positive and negative exchange integrals. The spin flip rates are combinations of two Glauber rates at the two temperatures, and by varying the coefficients of this combination probabilities of contact of the model with each thermal bath and thus the level of thermal noise in the model are changed. Particular attention is devoted to the majority vote model in which one of the two above-mentioned temperatures is zero and the other one tends to infinity. Only in rare cases such nonequilibrium models can be mapped onto equilibrium ones at certain effective temperature. Nevertheless, Monte Carlo simulations show that transitions from the paramagnetic to the ferromagnetic and spin-glass-like phases occur in all cases under study as the level of thermal noise is varied, and the phase diagrams resemble qualitatively those for the corresponding equilibrium models obtained with varying temperature. Theoretical investigation of the model on complete and random graphs is performed using the TAP equations as well as mean-field and pair approximations, respectively. In all cases theoretical calculations yield reasonably correct predictions concerning location of the phase border between the paramagnetic and ferromagnetic phases. In the case of the spin-glass-like transition only qualitative agreement between theoretical and numerical results is achieved using the TAP equations, and the mean-field and pair approximations are not suitable for the study of this transition. The obtained results can be interesting for modeling opinion formation by means of the majority-vote and related models and suggest that in the presence of negative interactions between agents, apart from the ferromagnetic phase corresponding to consensus formation, spin-glass-like phase can occur in the society characterized by local rather than long-range ordering.

Graphical abstract

中文翻译：

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完整和随机图中多数投票模型中的铁磁和类自旋玻璃跃迁

摘要

研究了与两个温度不同的热浴接触的非平衡自旋模型中的铁磁和类似自旋玻璃的转变。这些模型包括Sherrington-Kirkpatrick模型和稀纺玻璃模型，它们分别是完整图和随机图上的Ising模型，其边沿（以一定概率）对应于正负交换积分。自旋翻转速率是在两个温度下两个格劳伯速率的组合，并且通过改变该模型与每个热浴的接触概率的组合系数，从而改变了模型中的热噪声水平。特别关注多数投票模型，其中上述两个温度之一为零，而另一个则趋于无穷大。只有在极少数情况下，此类非平衡模型才能在特定有效温度下映射到平衡模型。然而，蒙特卡洛模拟显示，在研究的所有情况下，随着热噪声水平的变化，从顺磁性到铁磁性和类自旋玻璃的相变都会发生，并且相图在定性上类似于由温度变化。分别使用TAP方程以及均场和对近似，对模型在完整图和随机图上进行了理论研究。在所有情况下，理论计算都可以得出有关顺磁性和铁磁性相之间的相边界位置的合理正确的预测。在类似自旋玻璃的转变的情况下，使用TAP方程只能在理论和数值结果之间达成定性一致，并且均值场和对近似不适用于该转变的研究。获得的结果对于通过多数投票和相关模型对观点形成进行建模可能是有趣的，并且表明在代理之间存在负相互作用的情况下，除了对应于共识形成的铁磁相之外，还可能发生自旋玻璃状相在以当地秩序而非长期秩序为特征的社会中。

图形概要