Machine learning of the XY model on a spherical Fibonacci lattice

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.