Topological Phase Transitions in the Nonlinear Parallel Ising Model

Abstract

We investigate some phase transitions of a nonlinear, parallel version of the Ising model, characterized by an antiferromagnetic linear coupling and a ferromagnetic nonlinear one. This model arises in problems of opinion formation. The mean-field approximation shows chaotic oscillations by changing the linear coupling or the connectivity. The spatial model exhibits bifurcations in the average magnetization, similar to what is seen in the mean-field approximation induced by the change of the topology, after rewiring short-range to long-range connections as predicted by the small-world effect. These coherent periodic and chaotic oscillations of the magnetization reflect a certain degree of synchronization of the spins, induced by long-range couplings. Similar bifurcations may be induced in the randomly connected model by changing the coupling or the connectivity and the synchronism of the updating (dilution of the rule).

Stochastic bifurcations in the nonlinear parallel Ising model

“…In the following we consider multi-spin (plaquette) interactions. We moreover consider only completely asymmetric interactions [49,50], arranged to give a preferred direction that corresponds to time in the standard cellular automata language [24,51].…”

Metastable States in the Parallel Ising Model

“…Thus, the nonlinear voter model can be considered analogous to the Ising model with two-site and multiple-site (plaquette) terms, where the term that couples the given site with all neighbours is infinite (thus producing locally absorbing phases). This model is presented in a companion paper [16], and the absorbing states were used in other opinion models [9], where one can also find a discussion about which model is more representative of a real society.…”