Studies of a novel renormalization group technique for strongly correlated many body systems
Author(s)
Edwards, Khan
Type
Thesis
Abstract
The unusual low temperature behaviour of metals and metal alloys, such
as heavy fermion systems, Mott insulators and unconventional superconductors,
has been a central topic of scientific investigation for over half a
century. Due to strong correlations between the many particles, the solution
to most theoretical models of these systems is difficult because standard
perturbation methods break down. In certain cases, non-perturbative
renormalization group approaches can be used to access the low energy behaviour
and take into account the effects of strong correlations. However,
these methods do not work for several important and interesting classes
of model. These include lattice models with finite dimension, models with
impurities of high degeneracy and steady state transport through quantum
dots. The aim of this thesis is therefore to develop a new and general renormalization
group approach that will lend itself to wide application in models
where existing techniques fail. The method explored here utilises renormalized
perturbation theory (RPT) in conjunction with scaling equations and
collective excitations. It is tested on the single impurity Anderson model,
the paradigmatic model used to understand magnetic impurities within a
non-magnetic host metal. The Anderson model is simple yet exhibits interesting
features due to strong correlation, such as a single renormalized
low energy scale called the Kondo temperature. This model is also well understood
as it has been solved using the numerical renormalization group
(NRG), which provides ample numerical data against which this new technique
is compared. Finally, a preliminary study is given for this method
when it is applied to a wider class of models.
as heavy fermion systems, Mott insulators and unconventional superconductors,
has been a central topic of scientific investigation for over half a
century. Due to strong correlations between the many particles, the solution
to most theoretical models of these systems is difficult because standard
perturbation methods break down. In certain cases, non-perturbative
renormalization group approaches can be used to access the low energy behaviour
and take into account the effects of strong correlations. However,
these methods do not work for several important and interesting classes
of model. These include lattice models with finite dimension, models with
impurities of high degeneracy and steady state transport through quantum
dots. The aim of this thesis is therefore to develop a new and general renormalization
group approach that will lend itself to wide application in models
where existing techniques fail. The method explored here utilises renormalized
perturbation theory (RPT) in conjunction with scaling equations and
collective excitations. It is tested on the single impurity Anderson model,
the paradigmatic model used to understand magnetic impurities within a
non-magnetic host metal. The Anderson model is simple yet exhibits interesting
features due to strong correlation, such as a single renormalized
low energy scale called the Kondo temperature. This model is also well understood
as it has been solved using the numerical renormalization group
(NRG), which provides ample numerical data against which this new technique
is compared. Finally, a preliminary study is given for this method
when it is applied to a wider class of models.
Date Issued
2011-09
Date Awarded
2012-06
Copyright Statement
Attribution NoDerivatives 4.0 International Licence (CC BY-ND)
Advisor
Hewson, Alex
Publisher Department
Mathematics
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)