Machine learning of the Ising model on a spherical Fibonacci lattice

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 and 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 prevent the total energy from reaching its minimum at zero temperature, highlighting the role of curvature and connectivity in shaping interactions. Phase transition temperatures are derived through specific heat, magnetic susceptibility analysis and GCN 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.