Topology: Impurities in the Hofstadter model & Surface behaviour in Weyl semimetals
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In this thesis we consider two specific models in which topology plays an important role. In the first part we study the Hofstadter model. This is a two dimensional lattice model exhibiting the quantum Hall effect. We investigate the role of impurities in the emergence of bound states in the material. Every lattice impurity causes a bound state to occur around the impurity. These bound states only occur in the band gaps. In the second part we investigate Weyl semimetals. These materials have linear band touching points to which a topological charge can be attributed. This gives rise to exotic boundary effects like the emergence of Fermi arcs on the surface between two band touching points and the anomalous Hall effect. The surface cannot be described by a two dimensional analytic theory, but instead the whole bulk has to be taken into account. We study the effect on the polarization function on the surface. This also turns out to be highly dependent on the full bulk.