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

TL;DR

This work investigates the ferromagnetic $q$-state Potts model on curved spherical Fibonacci graphs to quantify curvature- and connectivity-induced shifts in the critical temperature $T_c$. It combines Swendsen–Wang cluster Monte Carlo sampling with graph convolutional networks (GCNs) and introduces a binarization-based transfer learning strategy that allows an Ising-trained GCN to infer $T_c$ for different $q$ without retraining. Planar benchmarks validate the transfer, matching the exact planar result $T_c^{2\mathrm{D}}(q)/J=1/\ln(1+\sqrt{q})$; on the sphere, curvature effects yield percent-level shifts at small $q$ that vanish as $q$ grows, consistent with the weakly first-order nature of the Potts transition for $q>4$. The results demonstrate a robust, geometry-aware GCN framework that can generalize to other curved manifolds and multi-state spin models, highlighting a clear separation between geometric effects and spin multiplicity in determining critical behavior.

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.

Figures (12)

• Figure 1: Spherical Fibonacci sites for $N=1000$ ($R=10$), shown in orthographic front (left) and top (right) views.
• Figure 2: Interaction graph constructed from the Fibonacci sites in Fig. \ref{['fig:lattice']} using $r_c/R=0.1298$: edges connect pairs with $d_{ij}<r_c$ (orthographic front/top views).
• Figure 3: Bond activation in the Swendsen--Wang algorithm ($N=1000$, $R=10$, and $r_c/R=0.1298$). Left: a representative spin configuration at $T/J=1.0$ for $q=3$. Right: the same configuration overlaid with the activated FK bonds, which connect equal-spin neighbors into FK clusters.
• Figure 4: GCN confidence for the 2D $q=3$ Potts model ($L=128$).
• Figure 5: Planar Potts configurations (q=3) and their binarized versions at $T/J=1.0$. Left: Potts spins $\{\sigma_i\}$; right: binarized Ising spins $\{s_i=\pm1\}$ (see Eq. \ref{['eq:binarize_rule']}).