Scaling and nonlinear field studies of the jellium model for Wigner electron systems in $d$ dimensions

A jellium model of interacting electrons has been investigated using scaling arguments on the kinetic and potential energy (KE and PE, respectively) in $d$ spatial dimensions. We find that the model exhibits no natural length scale in one dimension (1D), but in 2D and 3D, finite lengths appear indicating a tendency to form periodic structures. This confirms qualitatively the ideas of Wigner, who many years ago [E. P. Wigner, Phys. Rev. 46, 1002 (1934)] realized the possibility, in three dimensions, below a certain critical electron density, ${\rho }_c,$ that the effects of the PE due to Coulomb interactions would outweigh those of the KE and that the PE would be minimized by electrons localizing about sites on a body-centered-cubic lattice. In 4D we find a critical length for periodicity that is infinite, indicating the impossibility of a stable periodic structure. We have also cast the model into Landau-Ginzburg functional form with an appropriate order parameter. A minimization procedure is shown to lead to criteria for lattice formation in terms of electron density and screening length. In the continuum limit, the problem has been mapped into two coupled nonlinear field equations whose 1D versions are found to be exactly integrable. A perturbative treatment of these field equations in 2D, at absolute zero temperature, reveals the emergence of a stable triangular lattice structure.