In recent years, topology has been used to predict and harness the propagation of waves of electrons, light, and sound in materials. Most of these advances, however, were realized in idealized scenarios, where waves do not attenuate. The recent advent of non-Hermitian topological systems —wherein energy is not conserved— suggests the possibility of topological control also in more generic settings.
We will first introduce a system realising a purely non-Hermitian type of topology, and show that a change in the bulk non-Hermitian topological invariant leads to a change of topological edge-mode localization. We use a quantum-to-classical map to create a mechanical metamaterial with nonreciprocal interactions, in which the predicted bulk–edge correspondence can be experimentally observed.
Switching to a closely related but non-topological system, we then show that the non-Hermitian skin effect, which forces all bulk modes to one side of a finite system with broken energy conservation and reciprocity, can similarly be tuned to switch its direction of localisation. Moreover, we find that it can display a split skin effect, in which an extensive part of the usual skin effect modes change direction and localize on the side of the system opposite to the driving. We present a physical understanding of this behaviour based on the mathematical properties of Toeplitz matrices, and show that although the skin effect is not by itself topological in nature, the skin modes can be interpreted as topological edge modes of a hypothetical higher-dimensional system.
