Topological Boundary Floppy Modes in Quasicrystals

Topological mechanics has been studied intensively in periodic lattices in the past a few years, leading to the discovery of topologically protected boundary floppy modes in Maxwell lattices. In this paper, we extend this concept to two-dimensional quasicrystalline parallelogram tilings, and we use the Penrose tiling as our example to demonstrate that topological boundary floppy modes can arise from 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 a topological polarization can be defined for quasicrystalline structures as a bulk topological invariant. Remarkably, due to the unusual orientational symmetry of quasicrystals, the resulting topological polarization can exhibit orientational symmetries not allowed in periodic lattices. We prove the existence of these topological boundary floppy modes using a duality theorem which relates floppy modes and states of self-stress in parallelogram tilings and fiber networks, which are Maxwell reciprocal diagrams to one another. Our result reveals new physics about the interplay between topological states and quasicrystalline order and leads to novel designs of quasicrystalline topological mechanical metamaterials.