Exact Numerical Study of Disordered Systems

Abstract

This thesis presents work on the development of new techniques to study the problem of localization in various models of disordered systems with the goal of being able to extend these model calculations to real materials where these various mechanisms of disorder can all be present. I consider the Anderson Model with diagonal, off-diagonal disorder, multiple bands and superconductivity is included at the level of a Bogoliubov - De Gennes mean field (superconductivity is considered by adding the symmetries of the Bogoliubov - De Gennes Hamiltonian on top of the disordered lattice Hamiltonian). The localization of electrons is studied with the transfer matrix method (TMM) in order to compare mobility edge predictions with that of the newly developed Typical Medium Dynamical Cluster Approximation (TMDCA) for systems with both off diagonal disorder and multiple bands. It is verified this method can accurately predict the localization transition in model systems. A model of a disordered superconductor is considered with extended s-wave pairing, but in this case the excitations are no longer electrons but Bogoliubov quasiparticles or bogolons. I study the multifractal properties of the bogolon wavefunction and apply a multifractal analysis similar to what has been applied to the Anderson model and verify the ability to capture the localization of the bogolon quasiparticle excitations with comparison to the TMM.