Engineering low-dimensional topological superconductivity in magnetic heterostructures

Topological matter, which derives its properties from the overall topology of the bulk material, has in the past few decades emerged as one of the most important topics in condensed-matter physics. As a consequence of the non-local character of these systems, their exotic properties -- such as conducting edges on bulk insulators, or fractally-charged quasiparticles -- are remarkably robust to perturbations, potentially enabling a multitude of applications. Topological superconductors, in particular, have attracted significant interest, as they are predicted to support zero-energy Majorana bound states, quasiparticles that arise as equal superpositions of electrons and holes. A characteristic feature of Majorana bound states are their non-Abelian exchange statistics, which could see future use in, for example, topological quantum computers. It is known that embedded point-like magnetic impurities in a superconducting substrate support slowly-decaying subgap bound states known as Yu-Shiba-Rusinov states. The bands created when these Yu-Shiba-Rusinov states hybridise have symmetry properties which make them especially suitable as a platform for topological superconductivity in both one and two dimensions. The main contribution of this dissertation lies in studying this type of system in mean-field BCS approach, using both analytical and numerical methods with focus on topological properties. The overarching goal is to contribute to the ongoing search for systems in which topological superconductivity can be realised and experimentally observed.  In Publications I-III we study the topological properties of systems based on one-dimensional chains of magnetic impurities, focusing on analytically solving for the topological phase transitions and spectral properties of the systems. In Publication I the focus is on coupled chains with nearest-neighbour hopping, while in Publications II-III, we move on to consider systems with infinite hopping, developing a general framework for approaching Yu-Shiba-Rusinov-based topological superconductors. In Publications V-VI we consider generalisations of the basic Yu-Shiba-Rusinov lattice; in the former, by making use of non-pointlike nanomagnets, and in the latter, by abandoning lattice structure altogether and showing that topologically nontrivial phases can arise even in amorphous systems with sufficiently dense impurities. In Publication IV, we instead examine a magnetic skyrmion coupled to an intrinsic two-dimensional topological superconductor. We find that the character of the bound states depends on the type of skyrmion and the form of the superconducting pairing terms, allowing the bound-state properties to act as a probe of the underlying system.