Geometry and Topology of Quantum States in Quasicrystalline Systems

In this thesis we explore how geometrical and topological ideas from band theory can be extended to quasicrystalline systems. We start by studying a model of a shallow quasicrystalline optical lattice. Here we show that due to a natural hierarchy in the spectrum, an external force can be chosen such that the resulting dynamics is simply captured by an effective band structure, albeit with the Brillouin zone replaced by a space referred to as a `pseudo' Brillouin zone. Within a corresponding semiclassical picture, we find the presence of Bloch oscillations, usually synonymous with periodicity, alongside additional anomalous terms due to Berry curvature contributions. Fascinatingly, we also discover a so-called spiral holonomy in the effective band structure, in which circular trajectories result in evolution into an orthogonal state. We show that this feature is a result of the pseudo-Brillouin-zone possessing the topology of a higher genus torus. We then proceed to apply this theory to argue that quantum oscillations can occur in electronic quasicrystals. These were previously observed experimentally but lacked a quantitative theory. We show that due to the spiral holonomy in their band structure, for certain chemical potentials, the quantum oscillations are associated to an exotic ‘spiral Fermi surface’ that is self intersecting and characterised by a turning number—a topological invariant—that is larger than one. Finally, we establish an analytic low-energy theory describing higher-order topological insulator phases in quasicrystalline systems. We find that the localised modes at corners are not associated to conventional mass inversions, but are instead associated to what we dub as ‘fractional mass kinks’. Going beyond the weak coupling limit, we show that a hierarchy of additional gaps occur due to the quasiperiodicity, which also harbour corner-localised modes