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arXiv:2601.07346 · Chinese Physics B

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

Zheng Zhou, Xu-Yang Hou, Hao 昊 Guo 郭

Published Apr 29, 2026• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1674-1056/ae6637

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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.
Abstract We investigate the Ising model on a spherical surface, utilizing a Fibonacci lattice to approximate uniform coverage. This setup poses challenges in achieving consistent lattice distribution across the sphere for comparison with planar models. We employ Monte Carlo simulations, principal component analysis, graph convolutional networks (GCNs) to study spin configurations across a range of temperatures and to determine phase transition temperatures. The Fibonacci lattice, despite its uniformity, contains irregular sites that influence spin behavior. In the ferromagnetic case, sites with fewer neighbors exhibit a higher tendency for spin flips at low temperatures, though this effect weakens as temperature increases, leading to a higher phase transition temperature than in the planar Ising model. In the antiferromagnetic case, lattice irregularities induce geometric frustration, resulting in highly degenerate ground states and the phase transition temperature lower than the planar square lattice. Phase transition temperatures are derived through specific heat, magnetic susceptibility analysis and GCNs predictions, yielding T c values for both ferromagnetic and antiferromagnetic scenarios. This work emphasizes the impact of the Fibonacci lattice’s geometric properties-namely curvature and connectivity-on spin interactions in non-planar systems, with relevance to microgravity environments.
The XY model on a spherical surface, inspired by recently realized atomic gases trapped in a spherical shell, is analyzed here. Instead of a traditional latitude-longitude lattice, we introduce the much more homogeneous Fibonacci lattice and use classical Monte Carlo (MC) simulations to generate spin configurations. The results clearly show that topological defects, in the form of vortices, must exist in the stable configuration on a sphere but vanish in a plane due to a mathematical theorem. Using these spin configurations as training samples, a graph-convolutional network-based method is implemented to recognize different phases and successfully predict the phase-transition temperature. We also apply the density-based spatial clustering of applications with noise, a powerful machine-learning algorithm, to monitor the path of two vortices with topological charges on the sphere during MC simulations. Our results provide reliable predictions for quantum simulators using polaritons or future space-based experiments on ultracold atoms in microgravity.
Abstract We study third-order transitions in the Ising and Potts models on regular two-dimensional lattices and on Watts—Strogatz small-world networks. Cluster observables are used to track post-critical boundary reorganization and pre-critical cluster breakup. For the Ising model, the critical temperature TcT_cTc​ is calibrated independently from Binder-cumulant crossings and susceptibility peaks, whereas for the Potts model on small-world networks it is identified operationally from the dominant critical peak of d⟨P⟩/dT\mathrm d\langle P\rangle/\mathrm dTd⟨P⟩/dT and independently checked against the specific-heat peak. The independent and dependent third-order transitions are identified from the isolated-spin peak and the post-critical structural extremum, respectively. For both lattice and small-world topologies, we find the robust ordering Tind＜Tc＜TdepT_{\mathrm{ind}}＜T_c＜T_{\mathrm{dep}}Tind​＜Tc​＜Tdep​. Increasing the rewiring probability shifts all three characteristic temperatures upward and enhances the visibility of the post-critical transition. The effect is especially clear in the Potts model, where perimeter-based observables are more sensitive to multistate boundary fluctuations. The systematic persistence of the characteristic temperature hierarchy across topologies and finite sizes argues against interpreting these features as incidental finite-size irregularities. Instead, our results support their interpretation as robust third-order precursor transitions whose structural detectability can be amplified by network topology.
The main question raised in the article is whether a neural network trained on a spin lattice model in one universality class can be used to test a model in another universality class. The quantities of interest are the critical phase transition temperature and the correlation length exponent. In other words, the question of transfer learning is how "universal" the trained network is and under what conditions. For this purpose, we applied a supervised learning procedure to three two-dimensional models for which critical properties are precisely known: The Ising model, the four-state Potts model, and the Baxter-Wu model. We consider two datasets: One with spin configurations and one with binding energy configurations. We find that estimates of the critical temperature agree well with the known results for both datasets, but not with the results of cross-testing using the energy datasets of the two models: The four-state Potts model and the Ising model. Estimates of the critical length exponent are less regular, and appear to be more accurate for energy datasets. A good example is the cross-testing using the energy dataset between the Ising model and the Baxter-Wu model in both training and testing directions.
The classical nnn-vector model on the face-centered cubic lattice with ferromagnetic and antiferromagnetic interactions is studied by using the framework of an effective field theory approach in cluster with two spins. For n=1n=1n=1 (Ising), n=2n=2n=2 (XYXYXY-planar rotator), n=3n=3n=3 (Heisenberg), and $n\ensuremath{\rightarrow}\ensuremath{\infty}$ (spherical) models the ferromagnetic system undergoes a second-order phase transition while the antiferromagnetic system presents a first-order behavior. The first-order character for n=3n=3n=3, which is inconclusive in the literature, seems actually to be the case for the isotropic Heisenberg antiferromagnet, and the same is expected to hold for the less studied XYXYXY and spherical models. In addition, the transition temperatures are quantitatively comparable to the exact one (when available) and to those from Monte Carlo simulations and series expansion.
An approach based on a graph neural network has been developed to recognize phase states and to estimate the critical temperature of finite-size systems within the Ising model. The advantage of the new approach is the explicit specification of the lattice geometry and bond structure using a graph. This allows a unified transfer of equilibrium spin configurations of lattices of different spatial dimensions and geometries to the input of the neural network. This method has been applied to seven different lattices: two-dimensional square, triangular, hexagonal, and two-dimensional Apamea lattice, as well as three-dimensional simple cubic, bcc, and fcc lattices. It has been shown that the critical temperature can be extracted by crossing the output of the neural network with the 1/2 level. Feeding a small sample of equilibrium configurations to the input of a graph neural network allows achieving the mean absolute error 9.7 × 10–3 relative to the reference value TcrefT_{c}^{{{\text{ref}}}}Tcref​ . An additional scaling analysis has been performed for the Apamea lattice, which has shown that the neural network estimate of the critical temperature is consistent with the intersection point of the Binder cumulant curves for systems of different sizes.