Topology of ordered and amorphous chiral matter

Building on direct analogies with electronic matter, the concepts of topological insulator and topological protection have been successfully applied to a host of different physical systems as diverse as photonic metamaterials, geophysical fluids and mechanical structures. In this work we focus on materials having a chiral symmetry such as crystals with a sub-lattice symmetry and all mechanical systems assembled from beads and springs.

Firstly, we show that the topology of waves in chiral materials is naturally encoded in their chiral polarization, which quantifies the spatial imbalance of localized Wannier states between the two sub-lattices. Secondly, we demonstrate that the chiral polarization originates both from topological properties of the Hamiltonian and of the frame on which this Hamiltonian is defined. In crystals, we use it to redefine the concept of bulk-boundary correspondence. In amorphous metamaterials, we use it to lay out generic geometrical rules to locate topologically distinct phases, and explain how to engineer localized zero-mode wave guides even more robust than in periodic structures.