Nonlinear conduction via solitons in a topological mechanical insulator

Networks of rigid bars connected by joints, termed linkages, provide a minimal framework to design robotic arms and mechanical metamaterials built of folding components. Here, we investigate a chain-like linkage that, according to linear elasticity, behaves like a topological mechanical insulator whose zero-energy modes are localized at the edge. Simple experiments we performed using prototypes of the chain vividly illustrate how the soft motion, initially localized at the edge, can in fact propagate unobstructed all of the way to the opposite end. Using real prototypes, simulations, and analytical models, we demonstrate that the chain is a mechanical conductor, whose carriers are nonlinear solitary waves, not captured within linear elasticity. Indeed, the linkage prototype can be regarded as the simplest example of a topological metamaterial whose protected mechanical excitations are solitons, moving domain walls between distinct topological mechanical phases. More practically, we have built a topologically protected mechanism that can perform basic tasks such as transporting a mechanical state from one location to another. Our work paves the way toward adopting the principle of topological robustness in the design of robots assembled from activated linkages as well as in the fabrication of complex molecular nanostructures.Mechanical structures composed of folding components, such as bars or plates rotating around pivots or hinges, are ubiquitous in engineering, materials science, and biology (1). For example, complex origami-like structures can be created by folding a paper sheet along suitably chosen creases around which two nearby faces can freely rotate (2–4). Similarly, linkages can be viewed as 1D versions of origami where rigid bars (links) are joined at their ends by joints (vertices) that permit full rotation of the bars (Fig. 1 A–C). Some of the joints can be pinned to the plane while the remaining ones rotate relative to each other under the constraints imposed by the network structure of the linkage (5). Familiar examples include the windshield wiper, robotic arms, biological linkages in the jaw and knee, and toys like the Jacob’s ladder (6) and the Hoberman sphere. Moreover, linkages and origami can be used in the design of microscopic and structural metamaterials whose peculiar properties are controlled by the geometry of the unit cell (7, 8).Open in a separate windowFig. 1.The chain of rotors in the flipper phase. (A) The translation symmetric system with θ=θ¯ constant. We show a linkage made from plastic and metal screws. (B) A computer sketch of the elastic chain (11): The masses are blue, rigid rotors are black, and springs are dashed red lines. The green arrows depict the amplitude of displacement of each mass of the edge-localized zero mode of the system. (C) A configuration of the linkage showing a soliton as a domain wall between right-leaning and left-leaning states. (D) A computer-simulated static configuration. The arrows beneath show the x projections of each rotor.Many of these examples are instances of what mechanical engineers call mechanisms: structures where the degrees of freedom are nearly balanced by carefully chosen constraints so that the allowed free motions encode a desired mechanical function. However, as the number of components increases, more can go wrong: lack of precision machining or undesired perturbations. Robustness in this sense is a concern relevant to the design of complex mechanical structures from the microscopic to the architectural scale, typically addressed at the cost of higher manufacturing tolerances or active feedback.Here, we take an alternative approach inspired by recent developments in the design of fault tolerant quantum devices (9). Consider, as an example, the quantized Hall conductivity of a 2D electron gas that is topologically protected in the sense that it cannot change when the Hamiltonian is smoothly varied (10). In this article, we present a topologically protected classical mechanism that can transport a mechanical state across a chain-like linkage without being affected by changes in material parameters or smooth deformations of the underlying structure, very much like its quantum counterparts.Kane and Lubensky (11) recently took an important step toward establishing a dictionary between the quantum and classical problems. Their starting point, which seems at first disconnected from the linkages we study here, was to analyze the phonons in elastic systems composed of stretchable springs. In particular they derived a mathematical mapping between electronic states in topological insulators and superconductors (10) and the mechanical zero modes in certain elastic lattices (12). The simplest is the 1D elastic chain, shown in Fig. 1B, inspired by the Su–Schrieffer–Heeger (SSH) model for polyacetylene (13), a linear polymer chain with topologically protected electronic states at its free boundaries. In the mechanical chain, the electronic modes map onto zero-energy vibrational modes with a nontrivial topological index, whose eigenvectors represented as green arrows in Fig. 1B are localized at one of the edges (11). An intriguing question then arises: Could these zero-energy edge modes propagate through the system in the form of finite deformations?We address this question by building and analyzing a linkage of rigid bars as an extreme limit of the 1D lattice of springs. This linkage allows no stretching deformations, yet it still displays the distinctive zero-energy mode localized at the edges (Fig. 1 and Movie S1). By nudging the rotors along the direction of the zero-energy mode (Fig. 1B and Movie S2), we provide a vivid demonstration of how the initially localized edge mode can indeed propagate and be moved around the chain at an arbitrarily small energy cost. We then show analytically and numerically that the mechanism underlying the mechanical conduction is in fact an evolution of the edge mode into a nonlinear topological soliton, which is the only mode of propagation in the chain of linkages that costs zero potential energy. The soliton or domain wall interpolates between two distinct topological mechanical phases of the chain and derives its robustness from the presence of a band gap within linear elasticity and the boundary conditions imposed at the edges of the chain. Although the topological protection ensures the existence of a domain wall, the dynamical nature of the soliton falls into two distinct classes that can ultimately be traced to the geometry of the unit cell. The prototypes we built therefore provide simple examples of structures that we dub topological metamaterials whose excitations are topologically protected zero-energy solitons (9).