## Designing mechanical complexity

The quantum properties of topological insulators translate to mechanical systems governed by Newton's equations of motion. Many-body interactions and the multiple degrees of freedom available to charge carriers give electronic systems a range of exotic behaviors. Süsstrunk and Huber show that this extends to mechanical systems made up of a large lattice of coupled pendula. Mechanical excitations can be eliminated from the inner part of the lattice and confined to the edges, much like topological insulators. In addition to presenting a tractable toy system in which to study complex phenomena, the approach has potential uses in vibration isolation.

*Science*, this issue p. 47

## Abstract

A topological insulator, as originally proposed for electrons governed by quantum mechanics, is characterized by a dichotomy between the interior and the edge of a finite system: The bulk has an energy gap, and the edges sustain excitations traversing this gap. However, it has remained an open question whether the same physics can be observed for systems obeying Newton’s equations of motion. We conducted experiments to characterize the collective behavior of mechanical oscillators exhibiting the phenomenology of the quantum spin Hall effect. The phononic edge modes are shown to be helical, and we demonstrate their topological protection via the stability of the edge states against imperfections. Our results may enable the design of topological acoustic metamaterials that can capitalize on the stability of the surface phonons as reliable wave guides.

The experimental hallmarks of the quantum spin Hall effect (QSHE) in semiconductor quantum wells (*1*–*5*) are two counterpropagating edge modes that differ by their spin degree of freedom. As long as time reversal symmetry is preserved, these two modes are independent and do not scatter into each other (*6*, *7*). Much of the interest in condensed matter research involving topological states is driven by the use of these protected edge modes for technological applications such as spintronics (*8*, *9*), magnetic devices (*10*), or quantum information processing (*11*). The transfer of the phenomenology of the QSHE from the quantum mechanical realm to classical mechanical systems is therefore of fundamental interest, and its accomplishment would offer a gateway to new design principles in mechanical metamaterials.

Several key problems in the engineering of acoustic materials can potentially be addressed by capitalizing on the physics of the QSHE. The edge channels are robust counterparts to the well-known whispering gallery modes (*12*, *13*). Any application that requires energy to be confined to the surfaces of some device—for example, vibration insulators—can potentially make use of such edge states. In contrast to the whispering gallery modes, which are extremely sensitive to the shape of the surface (*14*), the topological edge modes are stable under a variety of perturbations. Moreover, because of the stability of these modes, scattering-free phonon waveguides of almost arbitrary shape can be realized. This in turn enables the engineering of robust acoustic delay lines (*15*–*17*), which are useful for purposes such as acoustic lensing (*18*).

How a mechanical system described by Newton’s equations can reproduce the phenomenology of a quantum mechanical model such as the QSHE has remained an open question (*19*–*23*). We sought to derive a mapping of the physics of the QSHE to a general mechanical system and to provide a specific experimental verification of our proposal. With a view to potential applications, it is essential to demonstrate that the physics of the noninteracting QSHE can be observed in a real mechanical system, which necessarily suffers from losses, disorder, and nonlinearities inherent to coupled mechanical oscillators.

A quantum mechanical lattice problem is described by a Schrödinger equation of the form (1)where are the wave function amplitudes for an electron with spin α on lattice site *i*, and is the hermitian Hamiltonian matrix describing the QSHE (*24*, *25*). On the other hand, the dynamics of a collection of undamped classical harmonic oscillators is described by Newton’s equation of motion (2)where *x _{i}* are the coordinates of

*N*pendula and

*is the real, symmetric, and positive semidefinite dynamical matrix containing the couplings between them. For either system, the existence and properties of edge modes are features of the eigenstates of ℋ or alone and depend neither on the interpretation of versus*

_{ij}*x*nor on the nature of the dynamics (i.e.,

_{i}*i*∂

*versus ). Hence, we mean to design a dynamical matrix incorporating the properties of the QSHE.*

_{t}We start from two independent copies of the Hofstadter model (*26*) on the square lattice with flux Φ = ±2π/3 per plaquette, (3)or, in matrix notation, (4)Here, (*r*, *s*) denotes the location on the lattice of size *L _{r}* ×

*L*, ϕ

_{s}*= Φ*

_{s}*s*, and

*f*

_{0}represents the hopping amplitude. The two copies are labeled by a pseudo-spin index α = ±, which also represents the two blocks ℋ

_{Φ}= ℋ

_{+,Φ}and = ℋ

_{−,Φ}in Eq. 4.

*H*is symmetric under time reversal

*T*and has three doubly degenerate bands, which are separated by nonzero gaps with a nontrivial topological

**Z**

_{2}index (

*27*). Therefore, in a finite system, we expect one chiral edge state per pseudo-spin in both gaps.

The above matrix ℋ is complex and hence does not comply with our design goals of a real matrix. However, owing to the time reversal symmetry, we can transform to a real basis by combining the local Kramers pairs α = ±, (5)This transformation yields (6)which is real, symmetric, and can easily be made positive definite. The four blocks of the matrix encode *x*-*x* and *y*-*y* couplings through Re ℋ_{Φ} and the couplings between *x* and *y* via Im ℋ_{Φ}. Both *x _{r}*

_{,}

*and*

_{s}*y*

_{r}_{,}

*describe a one-dimensional oscillator at lattice site (*

_{s}*r*,

*s*). In the following, we interpret the combined coordinate (

*x*

_{r}_{,}

*,*

_{s}*y*

_{r}_{,}

*) as an effective two-dimensional oscillator. It is then evident from the structure of*

_{s}*u*that the eigenmodes of the system characterized by α = ± correspond to left and right circularly polarized motions. Hence, these two polarizations replace the notion of pseudo-spin up and down of the quantum mechanical problem.

We implement using pendula, as shown in Fig. 1. inherits its structure from the explicit Landau gauge in Eq. 3: In the *s*-direction, there is no cross-coupling between *x _{r}*

_{,}

*and*

_{s}*y*

_{r}_{,}

_{s}_{±1}, as there is no imaginary part to the coupling

*f*=

*f*

_{0}. Along the

*r*-direction, mixes the

*x*- with

*x*-, the

*y*- with

*y*-, as well as the

*x*- with

*y*-degrees of freedom (

*28*) (Fig. 1). The total system consists of 270 pendula in a lattice of

*L*×

_{r}*L*= 9 × 15 sites. The bare eigenfrequency of each pendulum due to gravity is ω

_{s}_{0}/2π ≈ 0.75 Hz, which changes with the restoring forces of the spring couplings to ω/2π ≈ 2.34 Hz (

*28*).

To analyze the system, we harmonically excite one lattice site with a well-defined polarization by forcing the position of the two pendula (see also movies S1 and S2). By tracking the position of all pendula, one can extract the two-dimensional traces [*x _{r}*

_{,}

*(*

_{s}*t*),

*y*

_{r}_{,}

*(*

_{s}*t*)]. From these we obtain the mean deflection

*A*

_{r}_{,}

*as well as the polarization (left-handed, linear, or right-handed) related to the lag between the*

_{s}*x*and

*y*pendula (

*28*) (Fig. 2A).

To establish the existence of the edge states, we scan the frequency at which one edge site is excited. For every frequency, we wait until the system reaches a steady state before we extract the total response χ = ∑_{r}_{,}_{s}A_{r}_{,}* _{s}/N*, where

*N*is the number of sites. Figure 2E shows that the system responds appreciably between 1.7 Hz and 2.9 Hz, as expected (

*28*). Moreover, there are two regions with sequences of pronounced peaks. The width of these peaks is Γ ~ 0.04 Hz, indicating the damping of the oscillators and a quality factor

*Q*= ω/2πΓ ≈ 60. To make the connection to edge modes, we separate the system into the outermost line of lattice sites χ

_{e}= ∑

_{edge}

*A*

_{r}_{,}

_{s}/N_{edge}and the rest χ

_{b}= ∑

_{bulk}

*A*

_{r}_{,}

_{s}/N_{bulk}. The relative weight χ

*(χ*

_{e}/*+ χ*

_{b}*) is shown in Fig. 2F. There are three bands where the response lies mainly in the bulk; these are separated by two frequency regions where the response is dominated by the edge. To illustrate this further, we show in Fig. 2, B and C, the recorded mode structure at a bulk and an edge frequency, respectively.*

_{e}Given the analogy to the QSHE effect, we expect the edge states to be helical. The edge spectrum ω(*k*), where *k* is the wave vector along the edge, can be extracted from the steady states shown in Fig. 2C: Beyond *A _{r}*

_{,}

*and the polarization, we determine the position of each two-dimensional oscillator at a given time [*

_{s}*x*

_{r}_{,}

*(*

_{s}*t*

_{0}),

*y*

_{r}_{,}

*(*

_{s}*t*

_{0})]. The angle of this vector with respect to the positive

*x*-direction defines a local phase ϕ

_{r}_{,}

*(Fig. 2A). The change(7)defines the wave vector in units of the inverse lattice constant*

_{s}*a*. The resulting edge dispersion ω(

*k*) is shown in Fig. 2G. For each polarization, there is a chiral (that is, unidirectional) mode per gap, as expected.

The helicity of the edge dispersion indicates that the classical system faithfully implements the QSHE. Hence, for a clean system, a linearly polarized excitation on a boundary site is split into two counterpropagating circularly polarized modes. Consequently, these edge states act as a polarizing beam splitter.

The QSHE is protected by the quantum mechanical time reversal symmetry. In the classical case, the corresponding symmetry is the combination of time reversal together with the local exchange (*x _{r}*

_{,}

*,*

_{s}*y*

_{r}_{,}

*) → (*

_{s}*y*

_{r}_{,}

*, –*

_{s}*x*

_{r}_{,}

*) (*

_{s}*28*). Disorder on the local couplings typically breaks this symmetry. To assess the effect of the symmetry breaking due to disordered spring constants, we characterized our edge states in terms of their efficiency as a beam splitter.

We measured the ratio of excitation between the left and the right long edge after exciting in the middle of a short edge (Fig. 3A). By changing the relative amplitude and time lag between the two local pendula, we scanned all possible polarizations. The resulting splitting ratios are represented in Fig. 3B as a color map on a Poincaré sphere, where the north and south poles correspond to right and left circular polarization of the drive, respectively. The maximal imbalance between the left and right edge is not reached on the north or south pole, but on two approximately antipodal points rotated by ~15° away from the poles. However, as shown in Fig. 3C, we reach a splitting fidelity of 99.80 ± 0.04% at the optimal points. From this we conclude that on the length scale of our system, only symmetry preserving disorder is relevant (*28*).

To demonstrate that our edge states are not mere whispering gallery modes (*12*, *13*) and to highlight the robustness of the phononic edge states described here, we removed a sequence of sites from the dynamical problem, effectively creating a convex boundary. In Fig. 4, A to D, we show the resulting mode structure for a frequency in the lower gap. The results demonstrate that the exact shape of the boundary has no influence on the stability of the edge states.

To further strengthen the point that the edge states are topological rather than imposed by the finite-size geometry, we created a domain wall between two different topological sectors. We inverted the effective flux seen by the two polarizations on six rows of the system. At the boundary between the two sectors, the spin Chern numbers changed their values, which requires the presence of topological in-gap modes (*28*). We illustrate these modes along the sector boundaries by exciting a linearly polarized wave packet on the short edge of the larger sector. As shown in Fig. 4E, the wave packet splits into two circularly polarized packets, each traveling along one edge. At the boundary to the second sector, they are deflected into the interior of the lattice before they each travel independently along the physical edge of the second sector in reversed directions.

Our measured edge spectrum (Fig. 2G), the efficiency of the edge states as a beam splitter (Fig. 3), and their immunity to surface roughness (Fig. 4, A to D) corroborate that the QSHE phenomenology can be implemented in an imperfect mechanical system. Our findings indicate the robustness of our proposal against interactions, in particular against cubic (Duffing) nonlinearities (*29*) inherent to pendula. We indeed find that these nonlinear effects are present in our setup (*28*); however, they do not obstruct our results. In addition, unavoidable damping leads to the dissipation of the mechanical energy injected into the edge channels. However, their chirality would lead one to expect a linear scaling of decay length ξ ~ *Q* with the quality factor of the local oscillators. For applications dependent on the controlled transmission of phonons in one-dimensional channels, this is an improvement over a standard bidirectional wave guide, where one expects in analogy to a random walk (*28*).

Although our experiment was performed with coupled pendula, the mapping from the quantum mechanical problem to a classical dynamical matrix is much more general and can be used in the design of acoustic metamaterials. The key property obtained from a mechanical QSHE is the combination of a phononic band gap with stable helical states on the surface of the material. The guaranteed presence of these states allows the efficiency of a band gap for isolation together with a reduction of reflection, as the surface state can absorb and channel energy around the material (*30*). Moreover, the stability of the surface states allows the specification of reliable phonon wave guides (*31*) or stable acoustic delay lines (*18*), which would be useful for applications such as acoustic lensing. Finally, the helical nature of the surface states provides one-way channels similar to sound and heat diodes (*32*).

## Supplementary Materials

www.sciencemag.org/content/349/6243/47/suppl/DC1

Materials and Methods

Supplementary Text

Fig. S1

Movies S1 to S3

## References and Notes

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**Acknowledgments:**We thank T. Donner, T. Esslinger, P. Maletinsky, E. P. L. van Nieuwenburg, M. Tovmasyan, and O. Zilberberg for discussions. Special thanks go to C. Daraio for pointing out the technological relevance of our work. Supported by the Swiss National Foundation.