In this thesis, I present three of my papers related to the investigation of correlated systems. However, the collaborative nature of this research involves various individuals all of whom I want to acknowledge for the significant contributions they brought to the respective projects.
Chapter 2 revisits the single-impurity Kondo problem and provides a semi-analytical solution thereof. This work was completed in collaboration with S. Kehrein, Th. Pruschke and M. Jarrell and has been published in Physical Review B (24) which holds the copyright thereto.
Chapter 3 introduces a multi-scale extensions to quantum cluster methods for strongly correlated electron systems. Collaborators for this work were M. Jarrell, Th. Maier and J. Deisz. This work has been submitted for publication and is currently in pre-print cond-mat/0603421.
Chapter 4 focuses on the calculation of the spectral properties
of the one-dimensional Holstein and breathing polarons using the self-consistent
Born approximation. The Holstein model electron-phonon coupling is momentum
independent while the breathing coupling increases monotonically with the phonon
momentum. We find that for a linear or tight binding electron dispersion:
i) for the same value of the dimensionless coupling the quasiparticle renormalization
at small momentum in the breathing polaron is much smaller, ii) the quasiparticle
renormalization at small momentum in the breathing polaron increases with phonon
frequency unlike in the Holstein model where it decreases, iii) in the Holstein model
the quasiparticle dispersion displays a kink and a small gap at an excitation energy
equal to the phonon frequency 
while in the breathing model it displays two gaps,
one at excitation energy and another one at 
. I want to acknowledge the
collaboration of A. Macridin, G. A. Sawatzky, M. Jarrell and T. A. Maier in this work which
has been published in Physical Review B (25) which holds the copyright thereto.
One final paper(26) I was involved with during my thesis work is not included here as my contribution was small.
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