Uniform electron gases. I. Electrons on a ring

We introduce a new paradigm for one-dimensional uniform electron gases (UEGs). In this model, $n$ electrons are confined to a ring and interact via a bare Coulomb operator. We use Rayleigh-Schr"odinger perturbation theory to show that, in the high-density regime, the ground-state reduced (i.e. per electron) energy can be expanded as $\eps(r_s,n) = \eps_0(n) r_s^{-2} + \eps_1(n) r_s^{-1} + \eps_2(n) +\eps_3(n) r_s + \ldots$,where$r_s$ is the Seitz radius. We use strong-coupling perturbation theory and show that, in the low-density regime, the reduced energy can be expanded as $\eps(r_s,n) = \eta_0(n) r_s^{-1} + \eta_1(n) r_s^{-3/2} + \eta_2(n) r_s^{-2} + \ldots$. We report explicit expressions for ${\text{eps}}_0(n)$, ${\text{eps}}_1(n)$, ${\text{eps}}_2(n)$, ${\text{eps}}_3(n)$, ${\eta }_0(n)$ and ${\eta }_1(n)$ and derive the thermodynamic (large-$n$) limits of each of these. Finally, we perform numerical studies of UEGs with $n=2,3,\dots ,10$, using Hylleraas-type and quantum Monte Carlo methods, and combine these with the perturbative results to obtain a picture of the behavior of the new model over the full range of $n$ and $r_s$ values.