Motivated by the recent discovery of superconductivity in a quasicrystal , we investigate the occurrence of topological superconductivity (TSC) in two-dimensional (2D) quasicrystals. Quasicrystals are characterized by long-range order without periodicity. The most fundamental examples are Penrose (Fig. 1(a)) and Ammann-Beenker tilings (Fig. 1(b)). The Penrose tiling is composed of two rhombuses, colored by red and blue in Fig. 1(a). Similarly, the Ammann-Beenker tiling consists of a square and a rhombus, colored by red and blue, respectively, in Fig. 1(b). Quasicrystals are inherently fractal, characterized by self-similarity. Theoretically, so far TSC has been studied mainly in periodic crystals such as square lattice systems. Although there is a previous study about TSC in two-dimensional quasicrystals, the uniform order parameter was forcely assumed . Therefore, it is not trivial whether a stable TSC phase can exist in quasicrystals, despite their aperiodic and fractal structure.
We solve the Bogoliubov-de Gennes equations on the tight-binding model for 2D TSC of Sato, Takahashi, and Fujimoto  generalized for quasicrystals self-consistently . This model  describes 2D TSC with broken time-reversal symmetry as experimentally realized . According to the symmetry classification by Schnyder et al. , this model is in class D, which has an integer classification given by the TKNN number.
By solving for the superconducting order parameter self-consistently, we examine the stability of TSC in quasicrystals. Furthermore, in terms of our self-consistent solutions that properly reflect the local environment of each lattice site (vertex) in a quasicrystal and hence its self-similarity, we explore possible fractal structure in the distribution of order parameters.
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Fig. 1. Two of the most fundamental examples of quasicrystals: (a) Penrose tiling, (b) Ammann-Beenker tiling
Keywords: Topological superconductivity, Quasicrystals, Penrose tiling, Ammann-Beenker tiling