INTERPLAY OF GEOMETRY WITH IMPURITIES AND DEFECTS IN TOPOLOGICAL STATES OF MATTER

The discovery of topological quantum states of matter has required physicists to look beyond Landau’s theory of symmetry-breaking, previously the main paradigm for
studying states of matter. This has led also to the development of new topological theories for describing the novel properties. In this dissertation an investigation in this
frontier research area is presented, which looks at the interplay between the quantum geometry of these states, defects and disorder. After a brief introduction to the topological quantum states of matter considered herein, some aspects of my work in this area are described. First, the disorder-induced band structure engineering of topological insulator surface states is considered, which is possible due to their resilience from Anderson localization, and believed to be a consequence of their topological origin.
Next, the idiosyncratic behavior of these same surface states is considered, as observed in experiments on thin film topological insulators, in response to competition between
hybridization effects and an in-plane magnetic field. Then moving in a very different direction, the uncovering of topological ‘gravitational’ response is explained: the
topologically-protected charge response of two dimensional gapped electronic topological states to a special kind of 0-dimensional boundary – a disclination – that encodes spatial curvature. Finally, an intriguing relation between the gravitational response of quantum Hall states, and their response to an apparently unrelated perturbation – nonuniform electric fields is reported.