2023/05/24 by Amer Alrakik, Alrakik, Amer, Miguel Escobar Azor +9 · 1 citation
Chemistry · Materials Science · Physics and Astronomy · #Crystallography and Radiation Phenomena #FOS: Physical sciences #Fullerene Chemistry and Applications #Other Condensed Matter (cond-mat.other) #Quasicrystal Structures and Properties
paper · pdf · doi:10.48550/arxiv.2305.15604
openalex publication_date 2023/05/24 · openalex created_date 2023/05/27 · openalex updated_date 2026/07/28
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 S2) 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., ℝ3. In this work, we consider the analogous problem where the electrons are confined to an n-dimensional flat Clifford torus \cal Tn with n = 1, 2, 3. Since the torus \cal Tn can be embedded in the complex manifold ℂn, we define the distance in the Coulomb law as the Euclidean distance in ℂn, 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.