# Topological phases without crystalline counterparts

## TL;DR

This paper demonstrates the theoretical existence of higher-order topological phases protected by quasicrystalline symmetries, featuring Majorana zero modes in a superconductor on an Ammann-Beenker tiling, beyond crystalline classifications.

## Contribution

It introduces a new class of topological phases protected by quasicrystalline symmetries, with a model hosting Majorana modes outside existing classification schemes.

## Key findings

- Majorana zero modes localized at octagonal corners
- Protection by particle-hole, 8-fold rotation, and reflection symmetries
- Robustness of zero modes against symmetry-preserving deformations

## Abstract

Recent years saw the complete classification of topological band structures, revealing an abundance of topological crystalline insulators. Here we theoretically demonstrate the existence of topological materials beyond this framework, protected by quasicrystalline symmetries. We construct a higher-order topological phase protected by a point group symmetry that is impossible in any crystalline system. Our tight-binding model describes a superconductor on a quasicrystalline Ammann-Beenker tiling which hosts localized Majorana zero modes at the corners of an octagonal sample. The Majorana modes are protected by particle-hole symmetry and by the combination of an 8-fold rotation and in-plane reflection symmetry. We find a bulk topological invariant associated with the presence of these zero modes, and show that they are robust against large symmetry preserving deformations, as long as the bulk remains gapped. The nontrivial bulk topology of this phase falls outside all currently known classification schemes.