Universal HOM-type S-matrix correlations in chaotic (non-)trivial topological insulators and an improved model for the AFM and STM characteristics in Cu-based quantum corrals

The 2016 Nobel Prize in Physics was awarded to David Thouless, F. Haldane, and J. Kosterlitz for "theoretical discoveries of topological phase transitions and topological phases of matter." Since then, new opportunities have emerged, e.g., the robust edge states in topological insulators could significantly reduce heat generation due to their reduced resistance in electrical devices. In the main part of the dissertation we will focus, moreover, on the application of edge states as waveguides in an electron quantum optical setup. When the waveguides are attached to a chaotic cavity one expects the emergence of universal signatures of chaotic scattering, explored in this thesis for the first time in this topological scenario. We go beyond extensive simulations based on numerical techniques, and present semiclassical results. Together with experimental results, a holistic view of universal scattering correlators in various systems ranging from trivial to nontrivial insulators is presented here. The second part provides a new model to simulate the local density of states of the (un)perturped quantum corral. Furthermore we investigate the bonding characteristics of a AFM tip to the mesoscopic wave function of the quantum corral.