Geometry as a novel source for topological bound states

Topological phases of matter represent a relatively new and highly active branch of condensed matter. Evading the usual Landau classification, topological insulators and superconductors manifest robust states at their boundaries, whose existence is related to a non trivial topology of the bulk bands. The interest in these systems is both fundamental and technological. Indeed, they are regarded as promising assets for the development of resilient quantum technologies. Their potential applications include low-power electronic devices and fault-tolerant quantum computation, with 0-dimensional topological bound states being prime candidates for robust qubits due to their topological protection. This thesis addresses some conceptual aspects, so far partially overlooked, that are expected to assume crucial relevance as the nanostructuration of topological materials progresses towards the realization of topologically protected quantum technologies. Specifically, it investigates the interplay between geometry and topology, examining the effects of geometric confinement and lattice specifics on topological bound states in three different scenarios: finite size Chern insulating phases, finite size nodal superconducting topological phases, and higher-order topological phases. The analysis is carried out on honeycomb lattice based topological phases, which are attracting renewed interest due to the recent experimental results on bismuthene and germanene. Overall, the reported findings emphasize the need for effective theories that account for lattice geometry in topological systems. Moreover, they imply that geometric confinement can be a general mechanism for engineering effectively 1-dimensional topological phases- and thus topologically protected bound states -through the nanostructuration of 2-dimensional topological counterparts.