Ashkin-Teller model with antiferromagnetic four-spin interactions: Interference effect between two conflicting issues

Abstract: Spin systems have emerged as powerful tools for understanding collective phenomena in complex systems. In this work, we investigate the Ashkin--Teller (AT) model on random scale-free networks using mean-field theory, which extends the traditional Ising framework by coupling two spin systems via both pairwise and four-spin interactions. We focus on the previously unexplored antiferromagnetic regime of four-spin coupling, in which strong ordering in one layer actively suppresses the formation of order in the other layer. This mechanism captures, for example, scenarios in social or political systems where a dominant viewpoint on one issue (e.g., economic development) can inhibit consensus on another (e.g., environmental conservation). Our analysis reveals a rich phase diagram with four distinct phases -- paramagnetic, Baxter, \langle \sigma \rangle, and antiferromagnetic -- and diverse types of phase transitions. Notably, we find that the upper critical degree exponent extends to \lambda_{c2} \approx 9.237, far exceeding the conventional value of \lambda = 5$ observed in ferromagnetic systems. This dramatic shift underscores the enhanced robustness of hub-mediated spin correlations under competitive coupling, leading to asymmetric order parameters between layers and novel phase transition phenomena. These findings offer fundamental insights into systems with competing order parameters and have direct implications for multilayer biological networks, social media ecosystems, and political debates characterized by competing priorities.