Ferromagnetic and spin-glass-like transition in the majority vote model on complete and random graphs

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