Topological Phases and Phase Transitions in Clean and Disordered Systems

Abstract

Topological systems exhibit a variety of unique characteristics including robust edge states, an effective relativistic Dirac theory in the vicinity of the critical point and quantisation linked to the topological properties of the wavefunction. In this Thesis we explore the phase diagram of the Haldane model and its topological phase transitions. We present the scaling of the topological marker and set up transfer matrix calculations in order to investigate the critical behaviour of the clean Haldane model. We further apply these methods to the Haldane model in the presence of on-site disorder. We extract the correlation length critical exponent ν for the disorder-driven and the mass-driven topological phase transitions. In the case of the disorder-driven transition the value of the exponent is compatible with previous results found in the context of the Integer Quantum Hall Effect (IQHE). In the case of the mass-driven transitions the results are compatible with the existence of a continuously varying critical exponent in the disordered Haldane model. The exponent interpolates between that of the clean Haldane model with ν = 1 and the value obtained for the disorder-driven transition with ν ∼ 5/2. We discuss potential scenarios and limitations including the presence of marginal perturbations. Finally, we examine the sample-to-sample fluctuations of the topological marker in the vicinity of topological phase transitions in the disordered Haldane model. The fluctuations are compatible with a power-law divergence with increasing systems size. The exponent associated with the divergence of fluctuations exhibits similar variation as the correlation length exponent. We conclude with a discussion on the potential interpretations of the obtained results as well as further work and research directions.

Date of Award	1 Nov 2024
Original language	English
Awarding Institution	King's College London
Supervisor	Joe Bhaseen (Supervisor)