Over the last several decades, many materials exhibiting complex properties have been discovered (1). These materials are characterized by the emergence of competing phenomena in the thermodynamic limit due to the highly correlated nature of their electrons. In most common materials the electrons only weakly interact with each other or their surrounding ions. However, in the so called strongly correlated electrons, the inter-electronic interactions are strong enough as to alter the free electron nature of the system. While all materials are correlated, we tend to classify them in the latter category once the free electron's kinetic energy is out-competed by electron-electron interactions. Beyond this threshold, a multitude of competing phenomena emerge decisively coining the materials properties. Many of these newly discovered features are at odds with the well established Landau Fermi-Liquid theory of metals. However, this intriguing complexity may qualify these materials for possible future technological applications. Continual interest in this area of condensed matter and material science research is manifested by a steady stream of new publications.
Most advances in the field have been driven by experimental discoveries.
During its infancy, most of the research was focused on gaining a better
understanding of the Mott metal-insulator transition and
superconductivity. Recently, the field received a rejuvenated vigor with
discoveries of high-
superconductors and other heavy Fermion systems; system
in which the effective mass of an electron can approach the thousand fold
of its bare mass. In an ongoing theoretical endeavor
a variety of models were introduced to address these novel properties. Amongst these are the
Kondo (2), Anderson (3) and Hubbard (4) models. These models are
intriguingly simple but pose a significant challenge in solving them. While
it is possible to find analytical solutions (5,6) to these models in one dimension,
we are forced to revert to approximations in higher dimensions.
The non-trivial nature of these problems brought about many variations on the original models
with varying degree of approximation. With the emergence and ever increasing
computational resources, we now have added a new generation of tools available
to us which enable us to address the models. These methods include Green's function
techniques, various Quantum Monte Carlo (7) simulations, quantum cluster methods such
as the dynamical mean-field theory and its extensions, and renormalization group
theory (8).