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
Flór, I.M.; Donís Vela, A.; Beenakker, C.W.J.; Lemut, G. 2023
The symmetries that protect massless Dirac fermions from a gap opening may become ineffective if the Dirac equation is discretized in space and time, either because of scattering between multiple... Show moreThe symmetries that protect massless Dirac fermions from a gap opening may become ineffective if the Dirac equation is discretized in space and time, either because of scattering between multiple Dirac cones in the Brillouin zone (fermion doubling) or because of singularities at zone boundaries. Here an implementation of Dirac fermions on a space-time lattice that removes both obstructions is introduced. The quasi-energy band structure has a tangent dispersion with a single Dirac cone that cannot be gapped without breaking both time-reversal and chiral symmetries. It is shown that this topological protection is absent in the familiar single-cone discretization with a linear sawtooth dispersion, as a consequence of the fact that there the time-evolution operator is discontinuous at Brillouin zone boundaries. Show less
