Coulomb Pseudogaps and Their Role at Metal-Insulator Transitions

We investigate various fundamental aspects of the metal insulator transition (MIT) by employing several theoretical approaches. Describing the novel features of interacting electronic systems remains one of the principle challenges to theoretical condensed matter physics. Therefore, any effort to provide a deep insight into understanding the nature of MITs opens a new avenue for development of modern technology. First, we present a complete analytical and numerical solution of the Typical Medium Theory (TMT) for the metal-insulator transition. In this theory, we self-consistently calculate the typical amplitude of the electron wave-functions, representing the conceptually simplest order-parameter for the Anderson transition. We classify all possible universality classes for the critical behavior, that can be found within such a mean-field approach. This provides insights into how interaction-induced renormalizations of the disorder potential may produce qualitative modifications of critical behavior. We also formulate a simple description of the leading critical behavior of varies quantities, and then obtain an effective Landau theory for Anderson localization. We also develop an efficient numerical algorithm, "Cluster Typical Medium Theory" (CTMT) to capture both non-local effects and localization in disordered electronic systems. Our formalism utilizes the momentum-resolved typical density of states to characterize the localization transition. We apply this approach to the Anderson model of localization in one and two-dimensions. In one dimension, we find that the critical disorder strength scales inversely with the linear cluster size with a power-law, W[subscript c] ∼ (1/L[subscript c])[superscript 1/v]; whereas in two dimensions, the critical disorder strength decreases logarithmically with the linear cluster size. Our results are in agreement with the one-parameter scaling theory. Furthermore, we show how spatial correlations can also be captured analytically within such a self-consistent theory, by utilizing the standard Landau method of allowing for (slow) spatial fluctuations of the order parameter, and performing an appropriate gradient expansion. Our theoretical results provide insight into recent STM experiments, which were used to visualize the spatially fluctuating electronic wave functions near the metal insulator transition in Ga[subscript 1-x]Mn[subscript x]As.$ We show that, within our theory, all features of the experiment can be accounted for by considering a model of disorder renormalized by long-range Coulomb interactions. This includes the pseudogap formation, the C(R) ∼ 1/R form of the LDOS autocorrelations function, and the ζ ∼ 1/E energy dependence of the correlation length at criticality. In the second part of my Thesis, we show that introducing long-range Coulomb interactions immediately lifts the massive ground state degeneracy induced by geometric frustration for electrons on quarter-filled triangular lattices in the classical limit. Important consequences include the stabilization of a stripe-ordered crystalline (global) ground state, but also the emergence of very many low-lying metastable states with amorphous "stripe-glass" spatial structures. Melting of the stripe order thus leads to a frustrated Coulomb liquid at intermediate temperatures, showing remarkably slow (viscous) dynamics, with very long relaxation times growing in Arrhenius fashion upon cooling, as typical of strong glass formers. On shorter time scales, the system falls out of equilibrium and displays the aging phenomena characteristic of supercooled liquids above the glass transition. Our results show remarkable similarity with the recent observations of charge-glass behavior in ultra-clean triangular organic materials of the θ-(BEDT-TTF)₂ family.