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

Palatini f(R) Gravity and Variants of k-/Constant Roll/Warm Inflation within Variation of Strong Coupling Scenario

We show that upon applying Palatini f(R), characterized by an αR2 term, within a scenario motivated by a temporal variation of strong coupling constant, then one obtains a quadratic kinetic energy. We do not drop this term, but rather study two extreme cases: α<<1 and α>>1. In both cases, one can generate a kinematically-induced inflationary paradigm. In order to fit the Planck 2018 data, the α>>1 case, called k-inflation, requires a fine tuning adjustment with nonvanishing nonminimal coupling to gravity parameter ξ, whereas the α<<1 case, studied in the constant-roll regime, can fit the data for vanishing ξ. The varying strong coupling inflation scenario remains viable when implemented through a warm inflation scenario with or without f(R) gravity

The Emergence of the Hexagonal Lattice in Two-Dimensional Wigner Fragments

International audienceAt very low density, the electrons in a uniform electron gas spontaneously break symmetry and form a crystalline lattice called a Wigner crystal. But which type of crystal will the electrons form? We report a numerical study of the density profiles of fragments of Wigner crystals from first principles. To simulate Wigner fragments, we use Clifford periodic boundary conditions and a renormalized distance in the Coulomb potential. Moreover, we show that high-spin restricted open-shell Hartree–Fock theory becomes exact in the low-density limit. We are thus able to accurately capture the localization in two-dimensional Wigner fragments with many electrons. No assumptions about the positions where the electrons will localize are made. The density profiles we obtain emerge naturally when we minimize the total energy of the system. We clearly observe the emergence of the hexagonal crystal structure, which has been predicted to be the ground-state structure of the two-dimensional Wigner crystal