Floating Wigner Crystal and Periodic Jellium Configurations

TL;DR

This paper introduces a modified trial state for Jellium, proving the equivalence of multiple energy definitions in higher dimensions and relating Jellium energy to logarithmic energy, with improved bounds and formulas for periodic configurations.

Contribution

It extends the equivalence of Jellium energy definitions to dimensions d≥2 and connects it to renormalized and logarithmic energies, providing new bounds and formulas.

Findings

Proves the equivalence of three Jellium energy definitions in dimensions d≥2.

Relates Jellium energy to the renormalized energy studied by Serfaty and others.

Provides formulas for the Jellium energy of periodic 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 $d\geq 2$, thus extending the result of Cotar and Petrache (arXiv: 1707.07664) and Lewin, Lieb and Seiringer (Phys Rev B, 100, 035127, (2019)) that the three definitions coincide in dimension $d \geq 3$. 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 work of B\'etermin and Sandier (Constr Approx, 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.