Machine learning of the Ising model on a spherical Fibonacci lattice

TL;DR

Ising spins on a spherical surface are studied using a Fibonacci lattice to approximate uniform coverage, with Monte Carlo simulations, PCA, and graph convolutional networks used to analyze spin configurations across temperatures and identify the phase transition temperature $T_c$ for both ferromagnetic and antiferromagnetic cases. The curvature and irregular connectivity of the spherical lattice elevate or suppress $T_c$ relative to planar lattices: ferromagnetic $T_c$ is raised due to heterogeneous neighbor counts, while antiferromagnetic $T_c$ is lowered due to geometric frustration. The study demonstrates that ML tools, particularly GCNs and PCA-based clustering, can robustly detect phase boundaries on non-Euclidean lattices and that results align with, or complement, traditional thermodynamic indicators. The findings underscore the importance of lattice geometry in spin dynamics and offer a framework for exploring spin systems on curved surfaces with potential applications in microgravity contexts.

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.

Figures (15)

• Figure 1: A comparison of the uniformity between two types of lattices of $N=1000$: the latitude-longitude lattice on the left and the Fibonacci lattice on the right.
• Figure 2: Perspectives of a $N=1000$ Fibonacci lattice from different directions.
• Figure 3: Spin configuration of the spherical ferromagnetic Ising Model at $T/J=2.0$ (Top panel) and $T/J=8.0$ (Bottom panel): the left panel shows the front view, while the right panel presents the top view. Up and down spins are white and black pixels.
• Figure 4: (Top panel) A perspective of the Fibonacci lattice at $T/J=2.0$, in which the spins at 6 sites with three neighbours change direction (Left). The ranking of the statistical spin-flip probabilities $P_3$, $P_4$ and $P_5$ at different temperatures (Right). (Bottom panel) Statistical spin-flip probabilities as functions of temperature.
• Figure 5: (Top panel) Total energy as a function of temperature for the spherical ferromagnetic Ising model with $N=1000$. (Middle panel) Specific heat as a function of temperature, calculated using $C_V=\left(\frac{\partial E}{\partial T}\right)_V$. (Bottom panel) Specific heat as a function of temperature, calculated using Eq.(\ref{['Cveqn']}). The black dashed lines represent the raw Monte Carlo data points, and the red solid line shows the Gaussian fit around the peak.