Effect of a lattice upon an interacting system of electrons in two dimensions: Breakdown of scaling and decay of persistent currents

Abstract

The ground state of an electron gas is characterized by the interparticle spacing to the effective Bohr radius ratio r_s=a/a_B*. For polarized electrons on a two dimensional square lattice with Coulomb repulsion, we study the threshold value r_s* below which the lattice spacing s becomes a relevant scale and r_s ceases to be the scaling parameter. For systems of small ratios s/a_B*, s becomes only relevant at small r_s (large densities) where one has a quantum fluid with a deformed Fermi surface. For systems of large s/a_B*, s plays also a role at large r_s (small densities) where one has a Wigner solid, the lattice limiting its harmonic vibrations. The thermodynamic limit of physical systems of different a_B* is qualitatively discussed, before quantitatively studying the lattice effects occurring at large r_s. Using a few particle system, we compare exact numerical results obtained with a lattice and analytical perturbative expansions obtained in the continuum limit. Three criteria giving similar values for the lattice threshold $r_s^*$ are proposed. The first one is a delocalization criterion in the Fock basis of lattice site orbitals. The second one uses the persistent current which can depend on the interaction in a lattice, while it becomes independent of the interaction in the continuum limit. The third one takes into account the limit imposed by the lattice to the harmonic vibrations of the electron solid.