Disorder, Interactions, and the Fate of the Two-Dimensional Electron System: Wigner Crystal, Metal-Insulator Transition, and Amorphous Solid

Abstract

We present a unified theoretical study of the interplay between Coulomb disorder and Coulomb interactions in two-dimensional electron systems, covering the Wigner crystallization transition, the metal-insulator transition, and the crossover from a Wigner crystal to an amorphous localized solid in both zero magnetic field and the lowest Landau level. Using Hartree-Fock theory, variational Monte Carlo, and percolation and strong-localization arguments, we show that quenched charged disorder destroys true long-range Wigner crystalline order in the thermodynamic limit at any nonzero charged-impurity density, within the Imry-Ma-Larkin framework; the finite-system density image is that of an amorphous solid with crystalline coherence over at most a few lattice spacings. For the two-dimensional metal-insulator transition we show, in a controlled nonlinear-Thomas-Fermi model of long-range Coulomb disorder at setback d, that the half-depletion density of the semiclassical landscape and the Ioffe-Regel-Mott density of Born-RPA transport track each other to within a factor of order unity over the physically relevant range of impurity density and setback, with a slow n_i^(-1/10) d^(-1/5) spread; the agreement reflects the common energy scale E_dis ~ E_F at the transition rather than a theorem, and modest independent short-range transport channels can shift the ratio by factors of order two. The article is a theoretical study under well-defined approximations; it takes the recent scanning tunneling microscopy imaging experiments on two-dimensional electron systems and the longstanding disorder-versus-interaction debate on the two-dimensional metal-insulator transition as motivation, and leaves to the reader the comparison between its calculated signatures and the published experimental analyses.