Observation of non-Hermitian topology and its bulk–edge correspondence in an active mechanical metamaterial

Topological edge modes are excitations that are localized at the materials’ edges and yet are characterized by a topological invariant defined in the bulk. Such bulk–edge correspondence has enabled the creation of robust electronic, electromagnetic, and mechanical transport properties across a wide range of systems, from cold atoms to metamaterials, active matter, and geophysical flows. Recently, the advent of non-Hermitian topological systems—wherein energy is not conserved—has sparked considerable theoretical advances. In particular, novel topological phases that can only exist in non-Hermitian systems have been introduced. However, whether such phases can be experimentally observed, and what their properties are, have remained open questions. Here, we identify and observe a form of bulk–edge correspondence for a particular non-Hermitian topological phase. We find that a change in the bulk non-Hermitian topological invariant leads to a change of topological edge-mode localization together with peculiar purely non-Hermitian properties. Using a quantum-to-classical analogy, we create a mechanical metamaterial with nonreciprocal interactions, in which we observe experimentally the predicted bulk–edge correspondence, demonstrating its robustness. Our results open avenues for the field of non-Hermitian topology and for manipulating waves in unprecedented fashions.

The inclusion of non-Hermitian features in topological insulators has recently seen an explosion of activity. Exciting developments include tunable wave guides that are robust to disorder (1–3), structure-free systems (4, 5), and topological lasers and pumping (6–10). In these systems, active components are introduced to typically 1) break time-reversal symmetry to create topological insulators with unidirectional edge modes (1–5) and 2) pump topologically protected edge modes, thus harnessing Hermitian topology in non-Hermitian settings (6–9, 11). In parallel, extensive theoretical efforts have generalized the concept of a topological insulator to truly non-Hermitian phases that cannot be realized in Hermitian materials (12–14). However, such non-Hermitian topology and its bulk–edge correspondence remain a matter of intense debate. On the one hand, it has been argued that the usual bulk–edge correspondence breaks down in non-Hermitian settings, while on the other hand, new topological invariants specific to non-Hermitian systems have been proposed to capture particular properties of their edge modes (15–20).Here, focusing on a non-Hermitian version of the Su–Schrieffer–Heeger (SSH) model (15–17, 21) with an odd number of sites (Fig. 1A), we find that a change in the bulk non-Hermitian topological invariant is accompanied by a localization change in the zero-energy edge modes. This finding suggests the existence of a bulk–edge correspondence for this type of truly non-Hermitian topology. We further construct a mechanical analogue of the non-Hermitian quantum model (Fig. 1B) and create a mechanical metamaterial (Fig. 1C) in which we observe the predicted correspondence between the non-Hermitian topological invariant and the topological edge mode. In particular, we report that the edge mode in the non-Hermitian topological phase has a peculiar nature, as it is localized on the rigid rather than the floppy side of the mechanical metamaterial.Open in a separate windowFig. 1.Quantum-to-classical mapping of a chain with non-Hermitian topology. (A) An SSH chain with two sublattices, A (in red) and B (in blue), augmented with nonreciprocal variations in the hopping amplitudes (indicated by ±ε). (B) The nonreciprocal classical analog of the augmented SSH chain, in which the classical masses (in red) correspond to the A sites in the quantum model, while the nonreciprocal springs (in blue) are analogous to the B sites. (C) Picture of the mechanical metamaterial realizing the nonreciprocal classical analogue of the augmented SSH model.