Topological edge floppy modes in quasicrystals

Quasicrystals (QCs) are fascinating materials that exhibit physics properties not available in crystals, such as rotational symmetries forbidden in lattices, physics of higher dimensions, and self-similarity. The interplay of these unique features with topological states of matter can offer a rich variety of interesting phenomena. In this talk, we discuss topological mechanics in 2-dimensional quasicrystalline parallelogram tilings. We use the Penrose tiling as our example to demonstrate how these boundary modes arise with a small geometric perturbation to the tiling. The same construction can also be applied to disordered parallelogram tilings to generate topological boundary floppy modes. We find that, due to the unusual rotational symmetry of quasicrystals, the resulting topological polarization can exhibit orientations not allowed in periodic lattices. Our result reveals new physics about the interplay between topological states and quasicrystalline order and leads to novel designs of quasicrystalline topological mechanical metamaterials.