Floating Wigner crystal and periodic jellium configurations

Abstract

Extending on ideas of Lewin, Lieb, and Seiringer [Phys. Rev. B 100, 035127 (2019)], we present a modified “floating crystal” trial state for jellium (also known as the classical homogeneous electron gas) with density equal to a characteristic function. This allows us to show that three definitions of the jellium energy coincide in dimensions d ≥ 2, thus extending the result of Cotar and Petrache [“Equality of the Jellium and uniform electron gas next-order asymptotic terms for Coulomb and Riesz potentials,” arXiv: 1707.07664 (2019)] and Lewin, Lieb, and Seiringer [Phys. Rev. B 100, 035127 (2019)] that the three definitions coincide in dimension d ≥ 3. We show that the jellium energy is also equivalent to a “renormalized energy” studied in a series of papers by Serfaty and others, and thus, by the work of Bétermin and Sandier [Constr. Approximation 47, 39–74 (2018)], we relate the jellium energy to the order n term in the logarithmic energy of n points on the unit 2-sphere. We improve upon known lower bounds for this renormalized energy. Additionally, we derive formulas for the jellium energy of periodic configurations.

Coulomb and Riesz gases: The known and the unknown

“…In dimension d = 2, we have c 1 0.6612, which is also remarkably close to the expected best constant −ζ trg (0) 0.6606 (see Ref. 285, Prop. B.1).…”

“…19) is the interaction of each of the N points with its periodic copies, which is not contained in the first sum. For N = 1, only the Madelung term remains in Lemma 31 and we conclude that the Jellium energy per unit volume of the infinite lattice X = L with uniform background ρ b = |Q| −1 is just ρ b M L (s)/2, for the mentioned values of s 285 .…”

“…64-66, which only dealt with periodic systems (see also Ref. 285). The spirit is again that each point owns a small piece of the background, of constant volume.…”

“…The convergence of the energy holds in a larger range of s than for the potential. This is because the energy can be expressed in terms of the doubly-screened potential (IV.18) 285 . Physically, each cell z + Q contains N points and a uniform background of density ρ b = N/|Q|, hence is neutral.…”

“…The new function f in the sum now has a Fourier transform proportional to|k| s−d (1 − (2π) d/2 |Q| −1 1 Q (k))M and the arguments are the same as before. The proof of Lemma 29 can be adapted to handle the new terms and this is how one can get the stated analytic continuation to the whole of C \ {d}, increasing M step by step.For instance, for M = 2 and d − 4 < s < d we get the doubly-screened potential54,285,298…”