Topological Phases without Crystalline Counterparts

We construct a two-dimensional higher-order topological phase protected by a quasicrystalline eightfold rotation symmetry. Our tight-binding model describes a superconductor on the Ammann-Beenker tiling hosting localized Majorana zero modes at the corners of an octagonal sample. In order to analyze this model, we introduce Hamiltonians generated by a local rule, and use this concept to identify the bulk topological properties. We find a Z₂ bulk topological invariant protecting the corner modes. Our work establishes that there exist topological phases protected by symmetries impossible in a crystal.