Topology of chiral insulators

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, or metamaterials, 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. Our results elucidate a long-standing ambiguity on the topological characterization of the simplest, and probably oldest example of a topological state: the Su-Schrieffer-Hegger (SSH) chain. Finally, we establish a generalized bulk-boundary correspondence for chiral systems and show how to effectively use the concept of chiral polarization to predict the existence of interfacial states at the boundary of chiral insulators.