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

TL;DR

This work develops a mean-field framework for the antiferromagnetic Ashkin–Teller model on random scale-free networks, revealing how hub-dominated connectivity reshapes competing order parameters. By deriving the free-energy density and self-consistency equations under an annealed-network approximation, it uncovers four phases (PM, Baxter, $\langle \sigma\rangle$, AF) and a complex phase diagram that depends sensitively on the degree exponent $\lambda$ and inter-layer coupling ratio $x$. A key finding is the extension of the upper critical degree exponent to $\lambda_{c2} \approx 9.237$, indicating robust hub-mediated correlations even in strongly heterogeneous networks, plus the emergence of a distinct $\langle \sigma\rangle$ phase where one layer orders while the other remains disordered. These results illuminate how network topology and competing interactions drive polarization and consensus dynamics in systems with multiple interacting orders, with potential applications to social, biological, and political contexts.

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 σ\rangle, and antiferromagnetic -- and diverse types of phase transitions. Notably, we find that the upper critical degree exponent extends to λ_{c2} \approx 9.237, far exceeding the conventional value of λ= 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.

Figures (8)

• Figure 1: In (a), the Ashkin-Teller (AT) model on a single-layer network shows nodes containing two types of Ising spins (red and blue), with interactions depicted by solid lines. Panel (b) illustrates the equivalent system as a two-layer multiplex network, where each layer contains a single spin type. Solid and dashed lines represent intra- and inter-layer interactions, respectively.
• Figure 2: Distinct spin alignments characterizing ground states in each phase: paramagnetic (a), Baxter (b), $\langle \sigma \rangle$ (c), and antiferromagnetic (d). Each phase exhibits unique order parameter values, as indicated below each configuration.
• Figure 3: Phase behavior of the AF-AT model across different network types. Panels (a-e) display phase diagrams for random scale-free networks with varying degree exponents $\lambda$, while (f) shows the homogeneous network case. The critical temperature for continuous phase transitions is indicated by a red line, while the temperature for discontinuous phase transitions is marked by a black line. We examine the order parameters and free energy ratios for $x<1$ (g-l) and $x\geq1$ (m-r), where the critical temperatures are represented by red vertical lines.
• Figure 4: (Color online) Temperature dependence of order parameters in Regime III when $\lambda > \lambda_{c1}$. Panel (a) at $x=1^-$ shows the $\langle \sigma \rangle$-PM phase transition with order parameters $m$, $m_{s}$, and $M$. Panel (b) at $x=1^+$ depicts the AF-PM transition characterized by $M_{AF}$, $m$, and $M$. Red dotted lines indicate critical temperatures.
• Figure 5: (Color online) Temperature dependence of order parameters in Regime II at $(\lambda, x)=(6, 0.8)$. As temperature increases, the system undergoes successive transitions through Baxter, $\langle \sigma \rangle$, and PM phases. This regime's presence becomes more pronounced in the $\lambda > \lambda_{c1}$ phase diagram. A red dotted line marks the critical temperature.