Second-order topological states in a sixfold symmetric quasicrystal

Recently, higher-order topology has been expanded to encompass aperiodic quasicrystals, including those with eightfold or twelvefold rotational symmetry. The underlying mechanism for these high-order topological phases is generally protected by ${C}_{n}{\mathcal{M}}_{z}$ symmetry, resulting in the presence of $n$ corner states. However, this mechanism is not applicable to other ${C}_{2N}$ quasicrystals when $N$ is an odd number. In this work, we propose the realization of a second-order topological superconductor (SOTSC) within a sixfold symmetric bronze-mean hexagonal quasicrystal with six Majorana zero-energy modes. This SOTSC emerges from the combination of vertical and horizontal mirror symmetries, which flips the mass-term sign along the horizontal mirror-invariant line and produces Majorana zero-energy modes at each corner of the quasicrystal sample. Moreover, this mechanism can extend to quasicrystals with ${C}_{4N+2}$ and ${C}_{4N}$ rotational symmetries, namely encompassing systems with ${C}_{2N}$ symmetry. Our findings provide useful guidance for achieving SOTSC in quasicrystals featuring ${C}_{2N}$ rotational symmetry and introduce bronze-mean hexagonal quasicrystals as a promising platform for exploring quasicrystal SOTSC.