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 (${\cal S}^2$) that repel each other according to Coulomb's law, in which the distance is the Euclidean distance in the embedding space of the sphere, {\em i.e.}, $\mathbb{R}^3$. In this work, we consider the analogous problem where the electrons are confined to an $n$-dimensional flat Clifford torus ${\cal T}^n$ with $n = 1, 2, 3$. Since the torus ${\cal T}^n$ can be embedded in the complex manifold $\mathbb{C}^n$, we define the distance in the Coulomb law as the Euclidean distance in $\mathbb{C}^n$, in analogy to what is done for the Thomson problem on the sphere. The Thomson problem on a Clifford torus is of interest because super-cells with the topology of 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 a 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.