Gapless topological phases in two-dimensional quasicrystalline superconductors

We study s-wave superconductivity in two-dimensional Ammann-Beenker quasicrystals at and near half-filling under magnetic field, and examine its real-space topology. Ammann-Beenker quasicrystals have a flat band at the band center due to confined states and the lack of translational symmetry results in nonuniform superconducting order parameter. These two unique features lead to gapless superconductivity that is stabilized under strong magnetic field. Gapless superconductivity can be topologically nontrivial in the presence of Rashba spin-orbit coupling. The nontrivial phase exhibits topological edge states buried in bulk states, which are characterized by a pseudospectrum invariant given by a spectral localizer.