Machine learning of the Ising model on a spherical Fibonacci lattice

TL;DR

This paper explores the Ising model on a spherical Fibonacci lattice using simulations and machine learning to analyze phase transitions, highlighting how lattice irregularities affect magnetic behavior in curved geometries.

Contribution

It introduces a novel approach combining Monte Carlo, PCA, and GCNs to study the Ising model on a Fibonacci lattice, revealing the impact of geometric irregularities on phase transitions.

Findings

Higher phase transition temperature in ferromagnetic case due to irregular sites.

Lattice irregularities cause geometric frustration in antiferromagnetic case.

Phase transition temperatures are accurately predicted using specific heat, susceptibility, and GCNs.

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 (PCA), 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.