Topological phases of matter with disorder and quasiperiodicity

The topological materials have been theoretically and experimentally studied intensively in the past decades. The topological properties’ robustness against distortion makes them a natural playground to explore novel phase transitions. In this thesis, we focus on topological insulators and semimetals to study the phase transitions driven by disorder and quasiperiodicity. The main focus of this thesis is on the non-perturbative effects that demand the application of state-of-the-art numerical methods. We first consider models of semimetals under quasiperiodic modulation. We show that the non-perturbative incommensurate effect can drive semimetals through a quantum phase transition into a diffusive phase. Such phase transition will be referred to as the “magic-angle” effect, which will be interpreted to be central behind the magic-angle twisted bilayer graphene. The phase transition is shown to present universally in many models of semimetals. Meanwhile, the transitions have different characters based on the symmetry, in a way analogous to but fundamentally different from the 10-fold classification of Anderson transition. On top of quasiperiodicity, we study the effect of adding disorder to models of “magic-angle” semimetals. For both the experimentally realized twisted bilayer graphene and the simpler models that emulate the same universal physics, we analyze a special type of disorder that is native to the physics of quasiperiodicity - the inhomogeneity of the modulation. Such disorder effects correspond to the varying twist angle in twisted bilayer graphene which has significant experimental relevance. We then naturally generalize the considerations of semimetal to a 2D topological insulator. The topological mass intertwines with the strength of quasiperiodicity to create a fascinatingly rich phase diagram with interesting eigenstate criticality. In such criticality, we captured topological flat bands, which are of interest because of the potential to host strongly correlated topological phases. Lastly, we study the effect of disorder in the 3D topological insulator. We revisit the question of the stability of the semimetal line as a phase boundary between topological insulator phases. Because of the rare-region effect that is strictly non-perturbative, we find the semimetal line is destabilized which prevents the occurrence of a quantum critical point.Ph.D.Includes bibliographical reference