Properties of the Vortex Lattice of an Abelian Topological Superconductor

Abstract

Topological materials exhibit behaviour very different from conventional materials. Due to an integer invariant characteristic of the material, topologically-protected zero-energy excitations are guaranteed to exist at a boundary or topological defect -- such as a vortex in a topological superconductor -- of such a material. Topological insulators, materials which are insulating in the bulk but metallic at the surface, host massless Dirac fermions at the surface, excitations with a relativistic, helical nature. In topological superconductors, due to the intrinsic particle-hole symmetry of superconductivity, these surface states become Majorana fermions. The existence of Majorana fermions as condensed matter excitations provides not only a unique opportunity to study properties of this type of excitation, but the possibility of utilizing them for fault-tolerant topological quantum computation.

In this thesis, we study two-dimensional topological superconductivity (TSC) with broken time-reversal symmetry. Extensive research has been done on the non-Abelian phase of TSC, and an index theorem showing the existence of a single Majorana fermion in a vortex core has been derived based on the continuum model of TSC. The Abelian phase, which is allowed in the tight-binding model of TSC and is still topologically distinct from the trivial phase, has been much less studied than the non-Abelian phase. Using the tight-binding model of TSC, we derive an analogous index theorem in the Abelian phase. Our index theorem predicts that a vortex core in the Abelian phase of this system hosts two Majorana fermions as zero-energy bound states, one of which is composed mainly of quasiparticles with momentum near the $\Gamma$ point and the other of quasiparticles with momentum near the $M$ point in the Brillouin zone. We attempt to discern the relation between these two modes, in particular, if they are similar -- and hence will annihilate each other -- or orthogonal.

In contrast with our analytical results, we find no Majorana zero modes numerically in a vortex core in the Abelian phase. Solving the Bogoliubov-de Gennes (BdG) equations self-consistently for the superconducting order parameter, we study how the various ingredients for TSC interact in practice in this system. We use efficient, recently-developed numerical methods designed for large-scale parallel computation, namely, the Chebyshev polynomial method to solve for the mean fields without direct diagonalization of the BdG matrix, and the Sakurai-Sugiura method to find the quasiparticle excitation energies and wave functions within an energy window of one's choice. Calculations were performed on Compute Canada clusters. Our numerical results, lacking any Majorana modes in a vortex core in the Abelian phase, are an indication that the Majorana modes in this phase are not orthogonal to each other, so that when confined in a vortex core they annihilate each other.