Curvature-driven shifts of the Potts transition on spherical Fibonacci graphs: a graph-convolutional transfer-learning study

TL;DR

This study uses graph convolutional networks to analyze the Potts model on spherical Fibonacci graphs, revealing how curvature and irregularities slightly shift phase transition temperatures, with transfer learning enabling efficient classification across different graph types.

Contribution

It introduces a transfer-learning approach with GCNs for phase classification of the Potts model on irregular spherical graphs, demonstrating minimal curvature effects on transition temperatures.

Findings

GCNs can transfer from Ising to Potts models without retraining.

Curvature causes modest shifts in transition temperatures.

Shift magnitude decreases as q increases, aligning with phase transition theory.

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 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 pretrained 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 produce only modest shifts in the inferred transition temperatures relative to planar baselines. 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 of the correlation length.