Topological mechanics in quasicrystals

TL;DR

This paper extends the concept of topological mechanics from periodic lattices to quasicrystals, demonstrating the existence of topologically protected boundary floppy modes in quasicrystalline tilings like Penrose tilings, revealing new physics and design possibilities.

Contribution

It introduces the application of topological mechanics to quasicrystals, proving the existence of boundary floppy modes using a duality theorem and exploring unique polarization orientations.

Findings

Topological boundary floppy modes exist in quasicrystalline parallelogram tilings.

The duality theorem relates floppy modes and states of self stress in these tilings.

Quasicrystals exhibit unique topological polarization orientations not seen in periodic lattices.

Abstract

We study topological mechanics in two-dimensional quasicrystalline parallelogram tilings. 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 quasicrystalline parallelogram tillings and we use the Penrose tiling as our example to demonstrate how these topological boundary floppy 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 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. 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.