Majorana and Weyl Modes in Designer Materials

Topology has recently become one of the main themes of research in condensed-matter physics. Not only does the theory of topological materials predict novel phenomena in already well-established areas of physics, it also suggests intriguing applications in future technology. Topological phases exist in both gapped and gapless systems. In gapped topological systems, the bulk is insulating, but the boundary hosts zero-energy modes. Gapless topological phases, on the other hand, exhibit non-trivial phenomena within the bulk itself.This thesis consists of two parts: in the first part, the focus is on topological superconductors, which are gapped systems where the boundary modes are Majorana zero modes -- condensed-matter analogues of the Majorana fermion. Majorana zero modes are neither fermions nor bosons but obey an exotic form of exchange statistics envisaged to be employable in future quantum-computer architectures. This alone has makes finding implementations of topological superconductivity one of the central objectives for researchers in the field. The second part involves Weyl semimetals, which are three-dimensional gapless topological systems. Much like the Majorana zero modes in topological superconductors, the low-energy modes in Weyl semimetals have a high-energy counterpart, namely the Weyl fermion. The connection between Weyl semimetals and Weyl fermions brings high-energy physics into a low-temperature setting, making relativistic phenomena accessible in table-top experiments. Moreover, the novel transport properties emerging from these effects, combined with non-trivial topology, make Weyl semimetals a prospective building block in future electronics. In Publications I-III, we study one-dimensional chains of magnetic atoms deposited on a conventional superconductor. This setup has previously been studied in a low-energy limit for dilute chains. In Publication I and II, we extend the theory to also incorporate parameter regimes beyond this limit. In Publication III, we study the consequences of two kinds of disorder in these systems: vacancies and disordered coupling between the magnetic atoms and the superconductor. In Publication IV, we apply the machinery developed in Publications I-III to study chains of scalar impurities on top of an intrinsic two-dimensional topological superconductor. In Publication V, we introduce Weyl metamaterials as a platform in which to realize effective curved spaces and gauge fields. A Weyl semimetal owes its existence to the breaking of at least time-reversal or inversion symmetry. By making the symmetry-breaking fields inhomogeneous, the Weyl-like excitations experience an effective curved space and gauge field. We develop a mathematical framework which provides a direct route between the symmetry-breaking fields, and the curved space and gauge field. We also present an example of a geometry with lens-like trajectories.