Solution to the Thomson Problem for Clifford Tori with an Application to Wigner Crystals

In its original version, the Thomson problem consists of the search for the minimum-energy configuration of a set of point-like electrons that are confined to the surface of a two-dimensional sphere ($$) that repel each other according to Coulomb's law, in which the distance is the Euclidean distance in the embedding space of the sphere, i.e., $$. In this work, we consider the analogous problem where the electrons are confined to an n-dimensional flat Clifford torus $$ with n = 1, 2, 3. Since the torus $$ can be embedded in the complex manifold $$, we define the distance in the Coulomb law as the Euclidean distance in $$, in analogy to what is done for the Thomson problem on the sphere. The Thomson problem on a Clifford torus is of interest because supercells with the topology of a Clifford torus can be used to describe periodic systems such as Wigner crystals. In this work, we numerically solve the Thomson problem on a square Clifford torus. To illustrate the usefulness of our approach, we apply it to Wigner crystals. We demonstrate that the equilibrium configurations we obtain for large numbers of electrons are consistent with the predicted structures of Wigner crystals. Finally, in the one-dimensional case, we analytically obtain the energy spectrum and the phonon dispersion law.