Self-consistent study of topological superconductivity in two-dimensional quasicrystals

We study two-dimensional $s$-wave topological superconductivity with Rashba spin-orbit coupling and Zeeman field in Penrose and Ammann-Beenker quasicrystals. By solving the Bogoliubov--de Gennes equations self-consistently for not only the superconducting order parameter, but also the spin-dependent Hartree potential, we show the stable occurrence of topological superconductivity with broken time-reversal symmetry in both Penrose and Ammann-Beenker quasicrystals. The topological nature of the quasicrystalline system is signified by the Bott index $B$. Topological phase transitions are found to occur, where $B$ changes between 0 and $\ifmmode\pm\else\textpm\fi{}1$, as the chemical potential or Zeeman field is varied. In terms of self-consistent solutions, we demonstrate the existence of a Majorana zero mode per edge or vortex when $B=\ifmmode\pm\else\textpm\fi{}1$, consistently with the bulk-edge/defect correspondence for periodic systems.