Ashkin-Teller Model on Scale-Free Networks

Abstract

The investigation of various spin models on complex networks has played a crucial role in the comprehension of collective behavior in natural phenomena. In particular, the Ising and the Potts model on networks have received attention in the last decade for such purposes. Recently, multiplex networks have been actively studied in detail, because many real-world networks are networks of networks. Despite these facts, only a few spin models that incorporate interactions between nodes on inter- and intra-networks have been studied yet.

In this paper, we study the Ashkin-Teller (AT) model on scale-free random networks. In the AT model, spins on each site are of two types, and two spins of each type at the nearest neighbors interact with coupling strength $J_2$, and four spins of both types at the nearest neighbors interact with coupling strength $J_4$. In other words, $J_2$ and $J_4$ correspond to the interaction constants of the spins on the intra- and th inter-network, respectively.

As was seen in the mean-field approximation, various phases emerge depending on the ratio $x=J_4/J_2$. Some examples of such phases are the paramagnetic phase, ferromagnetic phase, anti-ferromagnetic phase, the Baxter phase, and the sigma phase. We obtain the phase diagram on scale-free networks. While the phase transition between paramagnetic phase and the Baxter phase is discontinuous in the standard mean-field solution, it can be continuous depending on the degree exponent on scale-free network. In spirit of the Landau theory, we focus on this tricritical point that divides the phase space into regimes of the first- and second-order phase transition. Then we obtain the critical degree exponent as a function of $x$ using the analytical approach. For positive $x$, we can thus determine the type of order of the phase transition on scale-free networks in terms of $x$ and the degree exponent.

In addition, we perform Monte Carlo simulation using the Metropolis algorithm to sketch the schematic phase diagram for $x<0$. Besides the diagram, a variety of thermodynamic quantities, such as magnetizations, heat capacity, susceptibility, the Binder cumulant, can come along for the ride. In this regime, anomalous behavior due to frustration can be observed.

Finally, we examine the analytic and simulation results, and discuss the implications of the difference in the mean-field level between the phase diagrams of scale-free networks and homogeneous space.