Topological mechanical metamaterials are artificial structures whose unusual properties are protected very much like their electronic and optical counterparts. Here, we present an experimental and theoretical study of an active metamaterial-composed of coupled gyroscopes on a lattice-that breaks time-reversal symmetry. The vibrational spectrum displays a sonic gap populated by topologically protected edge modes that propagate in only one direction and are unaffected by disorder. We present a mathematical model that explains how the edge mode chirality can be switched via controlled distortions of the underlying lattice. This effect allows the direction of the edge current to be determined on demand. We demonstrate this functionality in experiment and envision applications of these edge modes to the design of oneway acoustic waveguides.topological mechanics | gyroscopic metamaterial | metamaterial A vast range of mechanical structures, including bridges, covalent glasses, and conventional metamaterials, can be ultimately modeled as networks of masses connected by springs (1-6). Recent studies have revealed that despite its apparent simplicity, this minimal setup is sufficient to construct topologically protected mechanical states (7-11) that mimic the properties of their quantum analogs (12). This follows from the fact that, irrespective of its classic or quantum nature, a periodic material with a gapped spectrum of excitations can display topological behavior as a result of the nontrivial topology of its band structure (13-21).All such mechanical systems, however, are invariant under time reversal because their dynamics are governed by Newton's second law, which, unlike the Schrödinger equation, is second order in time. If time-reversal symmetry is broken, as in recently suggested acoustic structures containing circulating fluids (16), theoretical work (13) has suggested that phononic chiral topological edge states that act as unidirectional waveguides resistant to scattering off impurities could be supported. In this paper, we show that by creating a coupled system of gyroscopes, a "gyroscopic metamaterial," we can produce an effective material with intrinsic time-reversal symmetry breaking. As a result, our gyroscopic metamaterials support topological mechanical modes analogous to quantum Hall systems, which have robust chiral edge states (22)(23)(24). We demonstrate these effects by building a real system of gyroscopes coupled in a honeycomb lattice. Our experiments show long-lived, unidirectional transport along the edge, even in the presence of significant defects. Moreover, our theoretical analysis indicates that direction of edge propagation is controlled both by the gyroscope spin and the geometry of the underlying lattice. As a result, deforming the lattice of gyroscopes allows one to control the edge mode direction, offering unique opportunities for engineering novel materials.Much of the counterintuitive behavior of rapidly spinning objects originates from their large angular momentum, which endows th...

The discovery that the band structure of electronic insulators may be topologically non-trivial has revealed distinct phases of electronic matter with novel properties 1,2 . Recently, mechanical lattices have been found to have similarly rich structure in their phononic excitations 3,4 , giving rise to protected unidirectional edge modes [5][6][7] . In all of these cases, however, as well as in other topological metamaterials 3,8 , the underlying structure was finely tuned, be it through periodicity, quasi-periodicity or isostaticity. Here we show that amorphous Chern insulators can be readily constructed from arbitrary underlying structures, including hyperuniform, jammed, quasi-crystalline and uniformly random point sets. While our findings apply to mechanical and electronic systems alike, we focus on networks of interacting gyroscopes as a model system. Local decorations control the topology of the vibrational spectrum, endowing amorphous structures with protected edge modes-with a chirality of choice. Using a real-space generalization of the Chern number, we investigate the topology of our structures numerically, analytically and experimentally. The robustness of our approach enables the topological design and self-assembly of non-crystalline topological metamaterials on the micro and macro scale.Condensed-matter science has traditionally focused on systems with underlying spatial order, as many natural systems spontaneously aggregate into crystals. The behaviour of amorphous materials, such as glasses, has remained more challenging 9 . In particular, our understanding of common concepts such as bandgaps and topological behaviour in amorphous materials is still in its infancy when compared with crystalline counterparts. This is not only a fundamental problem; advances in modern engineering, both of metamaterials and of quantum systems, have opened the door for the creation of materials with arbitrary structure, including amorphous materials. This prompts a search for principles that can apply to a wide range of amorphous systems, from interacting atoms to mechanical metamaterials.In the exploration of topological insulators, conceptual advances have proven to carry across between disparate physical realizations, from quantum systems 10 , to photonic waveguides 11 , to acoustical resonators 12,13 , to hinged or geared mechanical structures 3,14 . One promising model system is a class of mechanical insulators consisting of gyroscopes suspended from a plate. Appropriate crystalline arrangements of such gyroscopes break time-reversal symmetry, opening topological phononic bandgaps and supporting robust chiral edge modes 5,6 . Unlike trivial insulators, whose electronic states can be thought of as a sum of independent local insulating states, topological insulators require the existence of delocalized states in each non-trivial band and prevent a description in terms of a basis of localized Wannier states [15][16][17] . It is natural, therefore, to assume that some regularity over long distances may be key to top...

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Topological metamaterials exhibit unusual behaviors at their boundaries, such as unidirectional chiral waves, that are protected by a topological feature of their band structure. The ability to tune such a material through a topological phase transition in real time could enable the use of protected waves for information storage and readout. Here we dynamically tune through a topological phase transition by breaking inversion symmetry in a metamaterial composed of interacting gyroscopes. Through the transition, we track the divergence of the edge modes' localization length and the change in Chern number characterizing the topology of the material's band structure. This work provides a new axis with which to tune the response of mechanical topological metamaterials.A central challenge in physics is understanding and controlling the transport of energy and information. Topological materials have proven an exceptional tool for this purpose, since topological excitations pass around impurities and defects and are immune to back-scattering at sharp corners [1,2]. Furthermore, topological edge modes are robust against weak disorder, such as variations in the pinning energy of each lattice site, in contrast to typical edge waves [3,4].Mechanical topological insulators represent a rapidlygrowing class of materials that exhibit topologicallynontrivial phononic band structure [2,5]. An important subset of these materials send finite-frequency waves around their perimeter, in a direction determined by the band topology [4,[6][7][8][9]. Just as in electronic materials, these edge waves are immune to scattering either into the bulk or in the reverse direction along the edge. Here we present a method for reversibly passing through a topological phase transition, which allows us to switch chiral edge modes on and off in real time.Our system consists of rapidly-spinning gyroscopes hanging from a plate (Figure 1). If displaced from equilibrium, a single gyroscope will precess: its tip moves in a circular orbit about the equilibrium position as a result of the torques from gravity and the spring suspension. To induce repulsive interactions between gyroscopes, we place a magnet in each gyroscope with the dipole moments aligned.A honeycomb lattice of such gyroscopes behaves as a Chern insulator, exhibiting robust chiral edge waves that pass around corners uninhibited. The phononic spectrum has a band gap, and shaking a boundary site at a frequency in the gap generates a wavepacket that travels clockwise along the edge. Figure 1b shows such a wavepacket as seen from below.The origin of these chiral edge modes is broken time reversal symmetry, which arises from a combination of lattice structure and spinning components [7]. As in [7,9], an effective time reversal operation both reverses time (t → −t) and reflects one component of each gyroscope's displacement (ψ → ψ * , where ψ = δx + iδy is the displacement of a gyroscope). Breaking effective time re-motor magnet coil spring suspension FIG. 1. Modulating the magnetic field at each la...

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