Dynamical defects in a two-dimensional Wigner crystal: Self-doping and kinetic magnetism

We study the quantum dynamics of interstitials and vacancies in a two-dimensional Wigner crystal (WC) using a semiclassical instanton method that is asymptotically exact at low density, i.e., in the $r_s\to \infty$ limit. The dynamics of these point defects mediates magnetism with much higher-energy scales than the exchange energies of the pure WC. Via exact diagonalization of the derived effective Hamiltonians in the single-defect sectors, we find the dynamical corrections to the defect energies. The resulting expression for the interstitial energy extrapolates to 0 at $r_s=r_{\mathrm{mit}}\approx 70$ (at $r_s\approx 30$ for a vacancy), suggestive of a self-doping instability to a partially melted WC for some range of $r_s$ below $r_{\mathrm{mit}}$. We thus propose a "metallic electron crystal" phase of the two-dimensional electron gas at intermediate densities between a low-density insulating WC and a high-density Fermi fluid.