| Since 60s in the last century, Kondo effect and its related physical problem have been received great amount of interest from researchers in condensed matter physics. The early research into the Kondo problem not only helped physicists to build up the important theoretical framework of the renormalization group theory, it also motivates the development of a series of numerical technique based on numerical renormalization group. On the other hand, in the last decade, due to the underlying application in quan-tum transport and topological quantum computation, the topological state of matter has grown up into a field that draws a lot of attention in condensed matter physics. Besides, a number of different materials have been experimentally realized and were proved to be topologically nontrivial, such as the topological insulators and the topological semimetals. One naturally interesting question then arises as:in the exotic quantum topological phases, will the Kondo effect manifests itself by new physical phenomena? Does any difference should be observed between the conventional Kondo effect and that in the topological states, in terms of quantities such as the Kondo resonance width or the Kondo temperature. Moreover, is it possible for the Kondo-related physics to reveal more undiscovered topological state of matter? Once being combined with topology, the Kondo problem will leave us a lot of interesting questions to answer. The aim of this thesis is to introduce my preliminary attempt to explore this field.In the first chapter, I will give a general introduction to two main topics that will be discussed in detail in this thesis, one is the Kondo effect and the other is the topo-logical quantum systems. As to the former, I will review the main breakthrough made in the history. As to the latter topic, I mainly try to answer what is the topology in solid state physics and how to describe the topological phases. The second chapter is an introduction to the perturbative theory of the Kondo effect. Firstly, I discuss the non-magnetic impurity model and the T-matrxi method. Then, using the T-matrix, I solved the non-interacting Anderson model. As to the Anderson model and the s-d exchange model, I follow the work by Kondo, discussing his perturbative theory on this prob-lem. However, we can at the same time find out that the physical quantities such as the resistance will become divergent in the low temperature regime, which is obviously in contradiction with the experimental results. This inconsistency forces one to search for the unperturbed solution of the Anderson model or s-d exchange model. Hence, in the third chapter, I mainly devote to introduce several unperturbation approaches to the Kondo problem. Firstly, the Abrikosov fermion representation and the perturba-tive renormalization group theory is discussed in detail, since it serves as a basis of the following method. Then the numerical renormalization group theory and the exact Bethe ansatz approach is introduced. In the fourth chapter, an effective theory is con-structed to describe the strong coupling regime at low temperature. It is found that a weakly-coupling Fermi liquid theory can be used to depict the strong-coupling fixed point. The emergent effective theory in a certain energy window is actually the essence of the renormalization group theory. Starting from Chapter five, I discuss the applica-tion of the Kondo effect into topologically nontrivial phases. The fifth chapter focuses on the current research in this field. The sixth to the eighth chapter show us in detail three different specific examples. We can find out from the three examples that, even though the Kondo problem is an odd topic that enjoys a history of more than half a cen-tury, it is brought to new life when it combines with topology in the sense that, it only only shows us different physical behavior compared to the conventional Kondo physics but also induces an amount of exotic topological states of matter which are beyond the previous research. |
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