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

Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved September 27, 2020 (received for review May 28, 2020)

## Significance

In recent years, the mathematical concept of topology has been used to predict and harness the propagation of waves such as light or sound in materials. However, these advances have so far been realized in idealized scenarios, where waves do not attenuate. In this research, we demonstrate that topological properties of a mechanical system can predict the localization of waves in realistic settings where the energy can grow and/or decay. These findings may lead to strategies to manipulate waves in unprecedented fashions, for applications in vibration damping, energy harvesting, and sensing technologies.

## Abstract

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. 1*A*), 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. 1*B*) and create a mechanical metamaterial (Fig. 1*C*) 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.

## Non-Hermitian Winding Number

The one-dimensional model depicted schematically in Fig. 1*A* is described by the quantum mechanical Bloch Hamiltonian**1** has a chiral symmetry,

A non-Hermitian Hamiltonian such as Eq. **1** may host two different types of topological invariants, corresponding either to a winding of the phase of their eigenvectors as the wave vector k is varied across the Brillouin zone (23) (Eq. **A1** in *Materials and Methods*) or to the complex energies winding around one another in the complex energy plane (12, 13) (Eq. **A2** in *Materials and Methods*). The former type of topology exists both for Hermitian and non-Hermitian systems, while the latter is exclusive to non-Hermitian systems, has not been observed yet, and is the focus of the present work.

## Mapping between Non-Hermitian Quantum and Classical Models

The non-Hermitian topology contained in the model of Eq. **1** stems from the nonreciprocity of its hopping parameters. This renders a direct implementation within a quantum material challenging, but recent advances on nonreciprocal mechanical metamaterials (1⇓–3, 24 suggest that such nonreciprocal interactions can be realized within a mechanical platform. In particular, inspired by the works of Kane and Lubensky (27) and Brandenbourger et al. (28), we introduce the one-dimensional mechanical system (Fig. 1*B*), which is described by the dynamical matrix:*B* and *Materials and Methods*). The parameter ε modifies the stiffness of the blue springs in a nonreciprocal way, so that a strain in the spring causes a larger torque on the left rotor than on the right. This nonreciprocal interaction is created locally for each robotic unit cell by an active-control loop: the motor of each unit cell applies a torque that depends on the strain of its neighboring springs (*Materials and Methods*).

The equations of motion imposed by the dynamical matrix **1**, with *Materials and Methods*). This generalizes the formal mapping between the dynamical matrix

## Non-Hermitian Bulk–Edge Correspondence

In the following, we restrict our attention to a particular model with parameter values

We show the full phase diagram of this system in Fig. 3*A*, as a function of the hopping parameters

The hatched pink regions of the phase diagram Fig. 3*A* are based on the behavior of bulk topological invariants, calculated in a system with periodic boundary conditions. The non-Hermitian topology, however, is expected to be most visible experimentally in the emergence or suppression of edge modes localized at the edges of the chain. The edge modes can be found for the quantum (classical) model by solving Schrödinger’s (Newton’s) equation for zero-energy modes (*Materials and Methods*). We focus in the following on a SSH chain with an odd number of sites (Fig. 1*A*) and on the mechanical Kane–Lubensky chain (Fig. 1*B*), for which the bulk–edge correspondences are strictly equivalent. Namely, we investigate an SSH (Kane–Lubensky) chain with N A sites (rotors) and

In the Hermitian limit *A*). In the non-Hermitian case *A*). In all cases, we find that the tails of the edge modes become oscillatory for *B* and *C*), as a consequence of imaginary contributions to their eigenvectors. Other choices of parameters will lead to a qualitatively similar correspondence between edge-mode localization and bulk winding (*Materials and Methods*). Finally, we find that perturbations of the ideal model considered here, such as the inclusion of on-site potentials or mechanical bending interactions, progressively gap the system and suppress the zero modes (*Materials and Methods*).

The coincidence between the change of the non-Hermitian winding number and the change of localization of the zero modes demonstrates that these zero modes are topological and that a change of localization corresponds to a topological transition. These topological zero modes have several peculiar properties that are only possible because Hermiticity is broken: 1) increasing the nonreciprocity at a fixed value of the ratio *A* to cause two consecutive changes in the edge mode location, one of which goes against the direction of the nonreciprocal bias; 2) in the case of the quantum system, the shape of the phase diagram cannot be explained by a simple argument involving the shifting of unit cells, as can be done in the Hermitian limit; and 3) in the mechanical system, as the topological edge mode in the winding region occurs where the mechanical degrees of freedom are constrained—this is a zero-energy mode, and yet it involves stretching of the springs.

The bulk–edge correspondence shown in Fig. 3*A* differs from but is complementary to recent results on even non-Hermitian SSH chains, where the topological modes appear or disappear at the values *Materials and Methods*) (15, 17, 18, 21, 28, 33). Recent results show that taking into account the non-Hermitian skin effect allows the definition of a non-Bloch topological invariant, which switches value at

## Non-Hermitian Bulk–Edge Correspondence in an Active Mechanical Metamaterial

To demonstrate the non-Hermitian bulk–edge correspondence described in *Non-Hermitian Bulk–Edge Correspondence*, we provide an experimental realization. To this end, we build an active mechanical metamaterial (Figs. 1*C* and 4*A*), which consists of nine robotic unit cells and in which a combination of geometry and active control is used to implement **2**. While the geometry allows us to obtain suitable values of *Materials and Methods*). We selectively access properties of the periodic (bulk) or open (edged) system by either including or omitting a rigid connection between the first and last unit cells (Fig. 4*A* and *Materials and Methods*).

In this setup, we first perform relaxation experiments on the periodic metamaterial to quantify directly the bulk eigenfrequencies for the wave vectors *A*–*D*). We find that the non-Hermitian topological invariant jumps from zero—where the eigenfrequencies are disconnected as in Fig. 4*B*—to one—where the eigenfrequencies wind as in Fig. 4*C*—for a nonreciprocal parameter *D*). Second, we probe the signature of the zero modes of the open chain (Fig. 4*E*) by applying a low-frequency excitation at the central unit cell. We observe a right-to-left (left-to-right) decaying displacement field for small (large) values of the nonreciprocal parameter ε (Fig. 4*F* and Movie S1). We find that the amplification factor *G*). Remarkably, the correspondence

To show more clearly the connection between the topological transition and the behavior of the edge modes, we also create a domain wall in the metamaterial, with the leftmost part remaining reciprocal (*C*) and vice versa (Fig. 5*D*). As expected, beyond the threshold value, the localization of the displacement field changes from the right edge to the domain boundary at the center (Fig. 5*C*) or the displacement field localizes at both edges away from the domain boundary (Fig. 5*D*).

## Discussion and Outlook

To conclude, we discovered and experimentally observed a type of bulk–edge correspondence for the non-Hermitian topological phase of a mechanical metamaterial with nonreciprocal interactions. This particular form of non-Hermitian bulk–edge correspondence, connected to energy winding, exhibits marked differences with the recently proposed non-Hermitian bulk–edge correspondence connected to a biorthogonal expectation value (15, 16). First, the correspondence based on energy winding reported here is unaffected by the non-Hermitian skin effect: despite the complete reorganization of the spectrum between a periodic and an open system, the energy winding of the periodic system predicts changes in the edge modes of the open system. Second, the energy winding and the biorthogonal condition both predict the emergence of zero modes. However, while the biorthogonal condition predicts the existence of edge modes, the energy winding additionally predicts the side of the chain at which the topological mode appears. These differences call for further investigation and generalization of the bulk–edge correspondence based on energy winding, beyond the particular system considered here.

Further, we envision the study of nonlinearity, robustness to disorder, different interactions, higher spatial dimensions, and other strategies to achieve non-Hermiticity to be exciting future research directions. We believe that our work provides conceptual and technological advances, opening up avenues for the topological design of tunable wave phenomena.

## Materials and Methods

### Hermitian and Non-Hermitian Topology of the Nonreciprocal SSH Model.

The Hamiltonian of Eq. **1** may acquire topological character either from the winding of the Berry connection determined by its eigenfunctions **1** is pseudo-Hermitian with respect to a positive definite metric operator (34) and can be diagonalized to find its eigenenergies *A*). In the winding non-Hermitian topological regions (pink hatched regions in Fig. 3*A*), the two bands coalesce into a *A*. Notice that the definition of Eq. **A2** differs by a factor of two from the convention used in some other works (13).

### Edge Modes of the Nonreciprocal SSH Model with an Odd Number of Sites.

In Eq. **1**, a nonreciprocal version of the SSH model is defined in reciprocal space. Here, we use the corresponding real space formulation to identify the edge modes of a finite nonreciprocal SSH chain with open-boundary conditions. Specifically, we consider N sites of type A and *B*, we only plot the right eigenmodes for

### Non-Hermitian Skin Effect.

In the non-Hermitian case, boundary conditions have a significant effect on the shape of the entire spectrum, namely open boundaries shift modes at all energies (frequencies) toward one side of the chain, in what is known as the non-Hermitian skin effect (15, 17, 33, 35). This effect is not related to topology and was recently observed in both a nonreciprocal mechanical metamaterial (28) and a nonreciprocal electronic circuit (18). Theoretically, it has been shown that in a non-Hermitian SSH chain, the closing of the bandgap appears at parameter values that are different for open and periodic boundary conditions, in an apparent breakdown of the bulk–edge correspondence (15⇓–17, 35) (indicated by the gray region in Fig. 6*B*). We find consistent results (indicated by the gray region in Fig. 6*A*), but we report in addition a clear correspondence between the topology of the bulk spectrum computed with closed-boundary conditions and the zero-energy edge modes obtained with open-boundary conditions: 1) a zero-energy edge mode always exists, as calculated analytically in the section above and confirmed numerically (Fig. 6*A*); 2) this edge mode in the chain with open boundaries changes its localization at the exceptional points of the periodic—bulk— model (Fig. 3 *A* and *B*); and 3) surprisingly, this edge mode is unaffected by the gap closing *A* and Fig. 6). These results are consistent with and complementary to recent results in the case of the even non-Hermitian SSH chain (15, 16).

Notice that in our system, a signature of the non-Hermitian skin effect can be seen in the response to a local excitation. While the localization of the response at low frequency (Fig. 7*A*) is essentially the same as that of the zero mode (Fig. 3), changing localization edge at the topological phase boundaries, the localization at large frequencies (Fig. 7*B*) solely depends on the nonreciprocal parameter, which is a direct signature of the non-Hermitian skin effect. For extensive portions of parameter space, the localization of the topological zero mode dominating the low-frequency response is opposite to that induced by the non-Hermitian skin effect at high frequencies.

### Nonreciprocal Kane–Lubensky Chain.

The classical analog of the non-Hermitian SSH chain is a nonreciprocal version of the Kane–Lubensky chain (27), as shown in Fig. 1*B*. In this system, N rotors of length r, with an initial tilt angle θ and a staggered orientation, are connected by **2**, we assume that the ratio **A5** becomes **2**. This factored mathematical expression allows us to construct the mapping between quantum and classical systems, following refs. 27 and 29⇓–31, where the quantum Hamiltonian is written as in Eq. **1**. Notice that the physical meaning of the Fourier equilibrium and compatibility matrices

### Role of Perturbations.

The computational model introduced in Eq. **A5** and discussed in *Non-Hermitian Bulk–Edge Correspondence* is necessarily an idealization. In the actual mechanical, integrated system in the experimental setup, there are unavoidable small effects of bending in each of the elastomeric bands, on top of other essential effects from frictional forces, geometrical and electromechanical nonlinearities in the chain, time delays and noise from microcontrollers, and geometric irregularities. That we nevertheless see the theoretically predicted bulk–edge correspondence in our experimental results is thus witness to the robustness of the non-Hermitian topology described by the numerical predictions. To test the limits of the topological robustness, we explicitly probe the role of two types of perturbations in the numerical model: 1) bending interactions inherently present in the elastomeric bands connecting the nearest-neighbor rotors and 2) an on-site potential. With these, the Fourier-transformed dynamical matrix becomes *A* and *B* shows that the presence of bending gaps the spectrum of the model with periodic boundary conditions around *C* and *D*). Both of these observations are a testament to the robustness of the non-Hermitian topology and its bulk–boundary correspondence, which remain intact even in the presence of perturbations, for sufficiently large values of the nonreciprocity. These considerations have been taken into account in the design of the experiments described in *Experimental Platform*, wherein a specific shape of the rubber band is chosen to minimize the bending.

### Experimental Platform.

To perform the experiments, we followed Brandenbourger et al. (28) and created a one-dimensional, nonreciprocal active mechanical metamaterial (Fig. 1*C*) consisting of nine unit cells, each of which has a single rotational degree of freedom *B*). Each unit cell consists of a rigid rotor of 36 mm in length and initial angle *C*). The lattice spacing between subsequent rotors is **A5**, and therefore its mechanical response exhibits the bulk–edge correspondence shown by the right zero modes of D. We record the rotors’ instantaneous positions

### Measurements of the Non-Hermitian Winding Number.

In order to measure the winding of the bands in the spectrum of the system with periodic boundary conditions, we connect the first and the last rotors with a rigid bar and ball-bearing hinges. To ensure homogeneity of the moments of inertia throughout the system, we add small masses at the end of each rotor. A rigid pin is attached to each rotor. We impose the initial position of each rotor *B* and *C*). From there, we use a discretized version of Eq. **A2***D*.

### Measurements of the Edges Modes.

We excite the metamaterial at the center rotor (*F*–*E*.

## Data Availability.

Automation codes and raw and postprocessed data for the experiments have been deposited in Github (https://github.com/corentincoulais/nHtopo_1DactiveMM.git).

## Acknowledgments

We thank D. Giesen, T. Walstra, and T. Weijers for their skillful technical assistance. We are grateful to C. Bender, L. Fu, T. Lubensky, D. Z. Rocklin, R. Thomale, and Z. Wang for insightful discussions and to M.S. Golden for critical review of the manuscript. J.v.W. acknowledges funding from a Netherlands Organization for Scientific Research Vidi grant. C.C. acknowledges funding from European Research Council Grant ERC-StG-Coulais-852587-Extr3Me.

## Footnotes

↵

^{1}A.G. and M.B. contributed equally to this work.- ↵
^{2}To whom correspondence may be addressed. Email: vanwezel{at}uva.nl or coulais{at}uva.nl.

Author contributions: J.v.W. and C.C. designed the research; A.G., J.v.W., and C.C. performed the theoretical calculations and the numerical simulations; M.B. and C.C. designed the experiments; A.G. and M.B. performed the experiments; A.G. and M.B. analyzed the data; and A.G., M.B., J.v.W., and C.C. wrote the paper.

The authors declare no competing interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at https://www.pnas.org/lookup/suppl/doi:10.1073/pnas.2010580117/-/DCSupplemental.

- Copyright © 2020 the Author(s). Published by PNAS.

This open access article is distributed under Creative Commons Attribution-NonCommercial-NoDerivatives License 4.0 (CC BY-NC-ND).

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