# Effective field theories for disordered systems from the logarithmic derivative of the wave-function

Van Biljon, Andrew (2001-12)

Thesis (PhD)--Stellenbosch University, 2001.

Thesis

ENGLISH ABSTRACT: In this dissertation, we give an overview of disordered systems, where we concentrate on the theoretical calculation techniques used in this field. We first discuss the general properties of disordered systems and the different models and quantities used in the study of these systems, before describing calculation techniques used to investigate the quantities introduced. These calculation techniques include the phase formalism method used one dimension, as well as the scaling approach and field theoretic approaches leading to non-linear c-models in higher dimensions. We then introduce a complementary effective field theoretic approach based on the logarithmic derivative of the wave-function, and show how the quantities of interest are calculated using this method. As an example, the effective field theory is applied to one dimensional systems with Gaussian disorder. The average density of states, the average 2-point correlator and the conductivity are calculated in a weak disorder saddle-point approximation and in strong disorder duality approximation. These results are then calculated numerically and in the case of the density of states compared to the exact result.

AFRIKAANSE OPSOMMING: In hierdie tesis, gee ons 'n oorsig van sisteme met wanorde, waar ons konsentreer op teoretiese berekeningsmetodes wat in die veld gebruik word. Eerstens bespreek ons die algemene eienskappe van sisteme met wanorde en verskillende modelle en hoeveelhede wat gebruik word in die studie van hierdie sisteme, voordat ons die berekeningsmetodes beskryf wat gebruik word om die bogenoemde hoeveelhede te ondersoek. Hierdie berekeningstegnieke sluit in die fase formalisme wat in een dimensie gebruik word, asook die skalingsbenadering en veldteoretiese metodes wat lei tot nie-lineêr u-modelle in hoër dimensies. Ons voer in 'n komplementere effektiewe veldeteorie gebaseer op die logaritmiese afgeleide van die golffunksie, en wys hoe hoeveelhede van belang met hierdie metode bereken word. As 'n voorbeeld, word die effektiewe veldetoerie toegepas op 'n een dimensionele sisteem met 'n Gauss verdeling. Die gemiddelde digtheid van toestande, die gemiddelde 2-punt korrelator en die gemiddelde geleidingsvermoë word bereken in 'n swak wanorde saalpunt benadering en in 'n sterk wanorde duale benadering. Hierdie resultate word dan numeries bereken, en in die geval van die digtheid van toestande vergelyk met die eksakte resultaat.

Please refer to this item in SUNScholar by using the following persistent URL: http://hdl.handle.net/10019.1/52253

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