Cálculo dos Elementos de Matriz para Determinação da Densidade Espectral do Modelo de Anderson de Dois Canais

Abstract

Non-magnetic metals hosting magnetic atoms, present in a highly diluted concentration, are known as systems with diluted magnetic impurities. Within a certain range of low temperatures, they exhibit physical properties that differ from those expected for an ideal metal (without impurities). Below this temperature range, they return to the expected behavior of an ideal metal, described as the Fermi liquid regime.

The cause of this difference is attributed to a physical phenomenon known as the Kondo effect, named after Jun Kondo. This researcher proposed a model that partially described these results. According to this model, the physical properties at low temperatures can be characterized by a parameter called the Kondo temperature.

For some time, the Kondo model was able to explain the behavior of systems with magnetic impurities. It was found that another model, the Anderson model, was equivalent and also capable of explaining the Kondo effect. However, experiments with new materials (metal alloys) showed the limitations of this theoretical description. Subsequent works, particularly by Nozières and Blandin, gave rise to several new models in which a system with impurities can exhibit characteristics different from those indicated by Kondo.

In these new models, the physical properties at low temperatures always display behavior different from that of an ideal metal, and this situation is referred to as the non-Fermi liquid regime. One of these models, called the Two-Channel Anderson model, is used in this work. This model describes an "anomalous" Kondo effect, where the conduction band electrons of the metal can interact with the magnetic impurity through two resonances.

The objective of this dissertation is to calculate the matrix elements for determining the spectral density of the Two-Channel Anderson model. Depending on the assigned values of physical parameters, this model is capable of reproducing the results of Jun Kondo or describing physical properties in the non-Fermi liquid regime.

We start with the analytical definition of the spectral density found in the literature. We adapt this expression to be used with the Numerical Renormalization Group method, a technique that computationally diagonalizes the Anderson model. We calculate matrix elements that will assist in a future computational implementation of the spectral density calculation.