Exchange Frequencies in the 2D Wigner Crystal

Using path integral Monte Carlo we have calculated exchange frequencies as electrons undergo ring exchanges in a “clean” 2D Wigner crystal as a function of density. The results show agreement with WKB calculations at very low density, but show a more rapid increase with density near melting. Remarkably, the exchange Hamiltonian closely resembles the measured exchanges in 2D ${}^3\mathrm{He}$. Using the resulting multispin exchange model we find the spin Hamiltonian for $r_s\le 175\pm 10$ is a frustrated antiferromagnetic; its likely ground state is a spin liquid. For lower density the ground state will be ferromagnetic.