Topological phases in quasicrystalline and amorphous matter

_{Recent years saw the complete classification of topological band structures, revealing an abundance of topological crystalline insulators. Here we investigate whether the classification can be further extended to materials that inherently lack translation invariance, but possess average spatial symmetries. We develop new theoretical and numerical methods based on local topological markers and effective momentum-space Hamiltonians to study topological phases in these systems.}

_{We demonstrate the possibility of higher-order topological materials beyond the existing classification framework, protected by quasicrystalline symmetries. We construct a higher-order topological phase protected by an eightfold rotation symmetry that is impossible in any crystalline system. We find a bulk topological invariant associated with the presence of Majorana corner modes, and show that they are robust against large symmetry preserving deformations, as long as the bulk remains gapped.}

_{Next we turn to amorphous materials, topological phases of which are still largely uncharted, even though it is a subset of materials of comparable size to crystals. We introduce the topological Weaire-Thorpe class of models, whose short-range properties allow us to analytically predict spectral gaps and the topological phase diagram using symmetry indicators for the first time in an amorphous system.}

Join URL: https://ucph-ku.zoom.us/j/69590299926