关键词: $q$-state Potts model, spherical fibonacci graphs, graph convolutional networks (GCNs), critical temperature

Abstract: We investigate the ferromagnetic $q$-state Potts model on spherical Fibonacci graphs. These graphs are constructed by embedding quasi-uniform sites on a sphere and defining interactions via a chord-distance cutoff chosen so as to yield a network approximating four-neighbor connectivity. By combining Swendsen-Wang cluster Monte Carlo simulations with graph convolutional networks (GCNs), which operate directly on the adjacency structure and node spins, we develop a unified phase-classification framework applicable to both regular planar lattices and curved, irregular spherical graphs. Benchmarks on planar lattices demonstrate an efficient transfer strategy: after a fixed binarization of Potts spins into an effective Ising variable, a single GCN pre-trained on the Ising model can localize the transition region for different $q$ values without retraining. Applying this strategy to spherical graphs, we find that curvature- and defect-induced connectivity irregularities induce only modest shifts in the inferred transition temperatures relative to their planar counterparts. Further analysis shows that the curvature-induced shift of the critical temperature is most pronounced at small $q$ and diminishes rapidly as $q$ increases. This trend is consistent with the physical picture that, in two dimensions, the Potts model undergoes a transition from a continuous phase transition to a weakly first-order one for $q$>4, accompanied by a pronounced reduction in the correlation length.

Key words: $q$-state Potts model, spherical fibonacci graphs, graph convolutional networks (GCNs), critical temperature