In some condensed matter systems, such as the surface of a 3D topological insulator, the electrons are effectively massless and we must necessarily use the massless Dirac equation to describe them... Show moreIn some condensed matter systems, such as the surface of a 3D topological insulator, the electrons are effectively massless and we must necessarily use the massless Dirac equation to describe them.A very convenient way to numerically solve this equation is to discretise them. However, the Nielsen-Ninomiya theorem proves that if we try to do it naively, extra unphysical massless fermion species appear, giving rise to a number of undesired artefacts. This is known as fermion doubling, and the main focus of this thesis is to tackle this problem via the discretisation method of tangent fermions.Chapters 2,3 and 4 are devoted to developing various aspects of this method. Chapters 5 and 6 are not directly related to the method of tangent fermions but still describe processes that arise in materials with a Dirac-like dispersion relation. In chapter 5, we study the effect a non-zero net supercurrent parallel to the edges of a topological superconductor. We find that the supercurrent can induce a "chirality inversion'' of the Majorana edge modes.In the last chapter, we simulate the injection of "edge-vortices'' into a topological superconductor. These are a type of quasiparticles that can theoretically be used to realise a quantum computer. Show less
This thesis is devoted to the effects of disorder on two-dimensional systems of Dirac fermions. Disorder localizes the usual electron system governed by the Schroedinger equation. The influence of... Show moreThis thesis is devoted to the effects of disorder on two-dimensional systems of Dirac fermions. Disorder localizes the usual electron system governed by the Schroedinger equation. The influence of disorder on Dirac fermions is qualitevely different. We concentrate on a random mass term in the Dirac equation. We have discovered that Dirac fermions in graphene are localized by a random mass, without any transition into metallic state. The situation is entirely different for Dirac fermions in a p-wave superconductor. There electrostatic disorder appears in the Dirac equation as a random mass, which localizes the excitation, but only if the disorder is relatively weak. For large mass fluctuations a transition into metallic state appears. This qualitatively different response to disorder in graphene and in p-wave superconductors is explained by the appearance of Majorana bound states, which allow for resonant tunneling and metallic state. Electrostatic disorder in a d-wave superconductor represented as random vector potential in the Dirac equation. We look at the transmission of Dirac fermions for electrostatic potential with long- and short-range fluctuations. We study the interplay of electrical and mechanical properties of suspended graphene by calculating the correction to the conductivity due to its deformation by a gate electrode. Show less
This thesis addresses a variety of systems in which the diffusion is anomalous, mainly motivated by recent experimental developments. The main topics discussed are: * We look at the consequences... Show moreThis thesis addresses a variety of systems in which the diffusion is anomalous, mainly motivated by recent experimental developments. The main topics discussed are: * We look at the consequences that subdiffusion on fractals has for shot noise. * The effects correlations have on superdiffusion in one dimension are examined. * We develop and demonstrate the usefulness of a method to simulate the anomalous diffusion of Dirac fermions in a computer. * A spin precession experiment in topological insulators is proposed and analyzed. * We present the mechanism for the conversion of an ordinary insulator into a topological insulator by disorder which was reported in the literature, on the basis of computer simulations, but had remained unexplained. Show less
