Local probes of higher order topology in chiral matter: theory and experiments

The standard description of chiral topological insulators relies on winding and sub winding numbers of eigenstates in periodic bipartite lattices. Generalised bulk-boundary correspondences then relate these topological bulk properties to a zero mode counting at the surfaces, hinges, and corners of finite samples.  

This description, while insightful in theory, poses a number of  difficulties. The first of them is very practical: how can one measure a winding number in an actual experiment? The second is more fundamental and is rooted in the very definition of the topological winding  numbers: they are not material properties but depend on the arbitrariness of the Bloch description, questioning further their relation to measurable quantities.

Building on the chiral polarization, a genuine material property of chiral insulators, we rectify these two unsatisfactory situations. We introduce real-space probes of higher-order topology in  chiral matter and demonstrate their relevance in mechanical experiments conducted both on periodic and amorphous lattices.