Fluctuation-induced phases and localisation in quasicrystalline systems

Abstract

In physics, quasicrystals are a unique kind of condensed matter system that lie in between the usual paradigms of periodic and disordered matter. While they lack any form of short-range translational invariance, they will retain long-range order, which can allow for the observation of distinct physical properties. Different kinds of quasicrystals will manifest their quasiperiodic order to model parameters in distinct ways, including the geometry of a discrete lattice or a separate energy potential. In this thesis, we will show how different quasicrystals can lead to the formation of fluctuation-induced phases and localisation across a lattice. The models we consider will concern that of interacting systems composed of bosonic atoms, allowing for close analogies to be made to the field of ultracold gases. Indeed, ultracold atom experiments with optical lattices offer the possibility to realise highly controllable lattice environments, including those that are quasicrystalline. By first considering quasicrystalline lattices, we show that non-uniform coordination numbers can introduce off-site disorder. If the interactions between atoms are then non-local, the system can exhibit fluctuation-induced insulators and structures which have no crystalline counterpart. On the other hand, when considering quasicrystals that contain on-site disorder, such as the 2D Aubry-Andr´e model, long-range order can also affect the percolation of observables. For this scenario, the physics of a small region in a quasicrystal can play a vital role in the underlying quantum phase transitions. Finally, if a magnetic field interacts with atoms confined to a quasiperiodic lattice, the induced cyclical motion can also lead to localisation. In particular, we will show how off-site disorder from a magnetic field can introduce localised, incompressible phases, i.e. current-carrying states that are robust against particle number fluctuations. The study of interacting, quasicrystalline systems is at an early stage of its development, especially in regards to the numerical methods that can solve these models. By studying the different quasicrystals across this thesis, we will be able to uncover the rich potential of these fascinating systems and their possible realisation in experiments.