Electric and magnetic signatures of boundary states in topological insulators and superconductors

This thesis is based on original research that was carried out in the field of topological phases in quantum condensed matter physics. The results have been published in five papers in Physical Review B, which form the core of the thesis. Three of these papers are concerned with magnetic order at the surface of a three-dimensional topological insulator, more precisely phenomena related to the so-called topological magnetoeletric effect (TME). This effect causes an electric field to induce a magnetic polarization. The first paper explores the role of Coulomb interaction at an interface of a ferromagnetic insulator with a topological insulator. The main finding is that the long-range interaction causes a nonlocal contribution to the TME. This is further investigated by deriving the Landau-Lifshitz equation describing the magnetization dynamics at the interface. In the second paper, the system is extended to a multilayer heterostructure. It is shown that the effect of the Coulomb interaction can effectively be described as a topological magnetic dipole-dipole interaction. Furthermore a setup is proposed where a magnetic texture can be manipulated nonlocally by means of an electric voltage. The third paper represents a generalization of the bilayer system to bipartite magnetic insulators. The main finding is a topological staggered-field-electric effect in ferrimagnetic insulators. All these three papers are based on completely analytical calculations within effective continuum models. The other two papers deal with topological superconducting systems. The first one is about a so-called Majorana wire, a spin-orbit coupled semiconductor nanowire with proximity-induced superconductivity in an external magnetic field. Such a system can host localized Majorana zero modes at the wire ends under certain conditions, when it is in a topologically nontrivial phase. One aspect of this condition, namely the allowed range of magnetic field orientations, is analytically derived in the paper. This is supplemented by calculating the conductance spectra of a Majorana wire-trivial conductor junction, which show the decay of the prominent zero-bias peak at the predicted critical angle. The last paper resulted from an external collaboration, where nodal noncentrosymmetric superconductors are studied. For certain crystallographic orientations, the surface of such a material features Majorana zero modes as flat-band states that are restricted to bounded regions in the surface Brillouin zone. By self-consistent numerical computation of the superconducting gaps, it is shown that the surface spontaneously breaks time-reversal symmetry for sufficiently low temperatures. The surface states become weakly dispersive and are no longer Majorana modes