Spin systems on a spherical lattice

Question

I'm interested in studying spin systems (like the Ising model) on a discretized sphere (say a two-sphere for simplicity). Has this been done before in the literature? I'm finding it difficult to construct a lattice that tiles the sphere in a simple way. Presumably this is a well-understood mathematical problem, so any help would be appreciated.

Answer 1

There is no arbitrarily fine-grained tiling on the sphere, there are only finitely many Platonic solids. The best uniform tiling is the dodecahedron. This is ultimately because there are only finitely many nonabelian finite groups which fit inside SO(3), If there were an arbitrarily fine uniform grid on the sphere, it's symmetry group would act transitively to shuffle these points around.

This makes it difficult to formulate the Ising model directly on the sphere, but you can use the embedding space. Consider a smooth real function f which is zero except he region between R and radius R+a, where it is 1. Then define the couplings in a 3-d Ising model between points at distance r to be J f(r), and take the limit of R getting large, a getting large, and R/a small. The same can be done on an arbitrary manifold, but you need rather high dimension embedding spaces to embed a manifold isometrically.

You might decide to make a random lattice, but this adds a weak noise to the model, it is an Ising spin-glass, because there is always going to be a static difference between the coordination number of different lattice sites, or the couplings, or something or other. Ising spin-glasses are widely studied, but you are presumably not interested in static noise, you want to preserve the symmetry of the sphere.

The basic properties of the system near the phase transition point are described by 2-d conformal field theory on Riemann surfaces, which are also widely studied in string theory, beginning with the famous Belavin Polyakov Zamalodchikov paper.