Self-consistent study of topological superconductivity in quasicrystals

We examine s-wave topological superconductivity (TSC) in two-dimensional Penrose and Ammann-Beenker quasicrystals (QCs) with Rashba spin-orbit coupling and Zeeman field by solving the Bogoliubov-de Gennes equations. The mean-field approximation is applied and the superconducting order parameter as well as the spin-dependent Hartree potential are obtained self-consistently. We find that the self-consistently obtained mean fields are spatially inhomogeneous in both QCs. We demonstrate how the underlying aperiodic structure of a QC is reflected in the superconducting order parameter. We also 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. Our results confirm the existence of a stable TSC state in QCs and the appearance of a Majorana zero mode along the edges of a QC, despite the lack of translational symmetry.