Floating Wigner Crystal and Periodic Jellium Configurations

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.