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
In unconventional high temperature superconductors, supercurrent vortices are known to spoil the Landau levels. In this thesis the emergence of Landau levels is studied in different types of... Show moreIn unconventional high temperature superconductors, supercurrent vortices are known to spoil the Landau levels. In this thesis the emergence of Landau levels is studied in different types of superconductors: Weyl superconductors, and the Fu-Kane heterostructure. It is shown that in those materials the zeroth Landau level can withstand the scattering off vortices. Show less
Electrons in a crystal lattice have properties that may differ from those of a free electron in vacuum. The effective mass can be different from the bare electron mass, and it may even vanish, ... Show moreElectrons in a crystal lattice have properties that may differ from those of a free electron in vacuum. The effective mass can be different from the bare electron mass, and it may even vanish, so that the electron behaves in some respects as a relativistic massless particle such as a photon. The magnetic moment of the intrinsic angular momentum, the electron spin, may be also different from that of an elementary particle. Spin-like degrees of freedom, referred to as "pseudospin" or "valley isospin", can also arise in the effective low-energy description of electrons in the lattice fields. These various degrees of freedom are of interest as ways to store and transport information: one speaks of "spintronics" and "valleytronics" as alternatives to "electronics". For these purposes it is of interest to study the interplay between the orbital motion of electrons and their spin (spin-like) degrees of freedom, the so-called "spin-orbit coupling". In some systems where this interaction is strong, it causes the electron spin to be tied to the direction of motion. This thesis contains results about the effects of this "spin-momentum locking" on two classes of materials, oxide interfaces and Weyl semimetals, with a focus on their electrical transport properties. Show less
In this thesis we study quantum transport phenomena on the nanometer scale, in two classes of materials: topological insulators with induced superconductivity and graphene superlattices. Both... Show moreIn this thesis we study quantum transport phenomena on the nanometer scale, in two classes of materials: topological insulators with induced superconductivity and graphene superlattices. Both topics are motivated by recent experimental developments: the first topic arose from the search for Majorana fermions in a quantum spin Hall insulator, the second topic arose from the search for massive Dirac fermions in the Kekulé band structure of graphene on a copper substrate. We focus on lattice models, solving them both numerically and analytically. 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
