Phase transitions of the Swedenborgite antiferromagnet & quasiparticle interference in a topological insulator

In this thesis, we study two separate subjects. First, we examine phase transitions of the antiferromagnetic Heisenberg model on the Swedenborgite lattice. This three-dimensional model contains two different interactions, and there is a critical ratio of these interaction strengths that separates a region with a unique ground state from a region where geometric frustration prevents the existence of such a ground state. First, we explain the possible phase transitions in these different regions by means of phenomenological Landau theory and mean-field theory. Afterwards, a spin wave analysis is performed that shows how thermal fluctuations can destroy the unique ground state. In the vicinity of the critical ratio, we find that the critical temperature that separates the unique ground state from a disordered phase scales linearly with the ratio of interaction strengths. This successfully completes an important section of the phase diagram of this model. The second part covers quasiparticle interference in the Cu doped topological insulator Bi2Te3. The effects of impurity scattering are calculated using an effective low-energy tight-binding model, along with a single local surface impurity. We find two distinct six-fold rotationally symmetric scattering patterns corresponding to two different energy windows. A comparison with the band structure of the system suggests that this distinction is caused by the interplay between edge states among themselves and with the conduction band. We also find that the symmetry and rotational orientation of the model results are in good agreement with experiments. However, this requires a special type of impurity that only couples to the electrons from a single orbital