Self-Consistent Study of Topological Superconductivity in Two-Dimensional Quasicrystals
In the past several years, there has been a burst of theoretical and experimental activities in the field of topological superconductors. Topological superconductivity (TSC) results in a novel superconducting state characterized by a nonzero topological invariant in the bulk. There is a relation between the bulk and edges or surfaces, which is called the bulk-edge correspondence. The bulk-edge correspondence implies that the topological invariant in the bulk is equivalent to the number of zero-energy excitations per edge or surface. Due to particle-hole symmetry inherent in a superconductor, in the case of TSC, the edge or surface modes in a topological superconductor are zero-energy Majorana fermions. Majorana fermions are their own antiparticles and due to the non-Abelian exchange statistics that they obey, they open the door to new and powerful methods of topological quantum computing. Majorana fermions have been detected, e.g., along the edges of a two-dimensional topological superconductor. Theoretically, so far TSC has only been studied in periodic crystals such as square lattice systems. In such systems with translational symmetry, the superconducting order parameter is uniformly distributed. Motivated by the recent discovery of superconductivity in a quasicrystal (QC), we investigate the occurrence of TSC in two-dimensional QCs. Although QCs present Bragg peaks, they have no periodicity. We generalize a tight-binding model for TSC in two dimensions, which was originally proposed for square lattice systems, for QCs. As the most fundamental examples, the Penrose and Ammann-Beenker QCs are studied. QCs are inherently fractal, and characterized by self-similarity. It is interesting to ask whether a stable TSC phase can exist in QCs, despite their aperiodic and fractal structure. In this thesis, we solve the Bogoliubov-de Gennes (BdG) equations — coupled Schrödinger-like equations for the electron and hole components of quasiparticle excitation — on the tight-binding model for TSC generalized for QCs. This model describes two-dimensional TSC with broken time-reversal symmetry, whose topological nature is governed by the first Chern number in periodic systems. For QCs, we calculate the Bott index as the topological invariant of the system, which is equivalent to the first Chern number in the presence of translational symmetry. The mean-field approximation is applied to the model Hamiltonian of TSC and the superconducting order parameter as well as the spin-dependent Hartree potential are obtained self-consistently. Our numerical results confirm the existence of a stable TSC state in QCs and the appearance of a Majorana zero mode along edges of a QC, despite the lack of translational symmetry. However, we find that the self-consistently obtained mean fields are both spatially inhomogeneous. In particular, we examine how the underlying aperiodic structure of a QC is reflected in the superconducting order parameter.
Topological superconductivity, Quasicrystals, Penrose tiling, Ammann-Beenker tiling
Master of Science (M.Sc.)
Physics and Engineering Physics