Electronic structure and dynamics in quasi-disordered systems, from quasiperiodic potentials to quantum spin ice

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Condensed matter physics is traditionally concerned with long-range ordered systems living on crystalline lattices, and this is the domain where the central tools of the field, such as Bloch's theorem, are applicable. This picture was changed by the work of Anderson and Mott who realised that disorder and strong interactions can result in localisation, contradicting the Bloch's theorem picture of extended electronic eigenstates. Understanding the connection between disorder, strong interactions, and localisation has since led to important developments in condensed matter physics, most recently to the discovery of many-body localisation. My project is concerned with the rich physics of "quasi-disordered" systems, that is, phases of matter between the limits of perfect order and complete, uncorrelated disorder. I focus on two examples:1. Quasicrystals are, in fact, long-range ordered; unlike crystals, however, this order is aperiodic, and so Bloch's theorem does not apply. As such, they emerge as a middle ground between periodic crystals and disordered systems, with unique localisation and topological properties. I investigate Anderson localisation transitions of one- and two-dimensional quasicrystals which display novel kinds of critical behaviour.2. Frustrated magnets have massively degenerate ground states: although these are disordered, the form of interactions result in subtle and unusual long-range correlated disorder and unique spin liquid phases. In particular, I investigate novel methods to understand the spin dynamics and electronic structure of spin ice systems.