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We experimentally and theoretically study the dynamics of a one-dimensional array of pendula with a mild spatial gradient in their self-frequency and where neighboring pendula are connected with weak and alternating coupling. We map their dynamics to the topological Su-Schrieffer-Heeger model of charged quantum particles on a lattice with alternating hopping rates in an external electric field. By directly tracking the dynamics of a wave-packet in the bulk of the lattice, we observe Bloch oscillations, Landau-Zener transitions, and coupling between the isospin (i.e., the inner wave function distribution within the unit cell) and the spatial degrees of freedom (the distribution between unit cells). We then use Bloch oscillations in the bulk to directly measure the nontrivial global topological phase winding and local geometric phase of the band. We measure an overall evolution of 3.1 ± 0.2 radians for the geometrical phase during the Bloch period, consistent with the expected Zak phase of π. Our results demonstrate the power of classical analogs of quantum models to directly observe the topological properties of the band structure and shed light on the similarities and the differences between quantum and classical topological effects. [ABSTRACT FROM AUTHOR]

We introduce a method to design topological mechanical metamaterials that are not constrained by Newtonian dynamics. The unit cells in a mechanical lattice are subjected to active feedback forces that are processed through autonomous controllers preprogrammed to generate the desired local response in real time. As an example, we focus on the quantum Haldane model, which is a two-band system with nonreciprocal coupling terms, the implementation of which in mechanical systems requires violating Newton's third law. We demonstrate that the required topological phase characterized by chiral edge modes can be achieved in an analogous mechanical system only with closed-loop control. We then show that our approach enables us to realize, a modified version of the Haldane model in a mechanical metamaterial. Here, the complex-valued couplings are polarized in a way that modes on opposite edges of a lattice propagate in the same direction, and are balanced by counterpropagating bulk modes. The proposed method is general and flexible, and could be used to realize arbitrary lattice parameters, such as nonlocal or nonlinear couplings, time-dependent potentials, non-Hermitian dynamics, and more, on a single platform. [ABSTRACT FROM AUTHOR]

• Metamaterial with embedded feedback control mechanism for autonomous sound navigation. • Real-time sound pressure field shaping, mimicking quantum topological wave phenomena. • Two-dimensional acoustic beam guiding along artificial programmed curved trajectories. • Topological protection keeps beams localized on the trajectories despite sharp turns. • A tool to realize non-local, nonlinear, time dependent or non-reciprocal interactions. We present the concept of a feedback-based topological acoustic metamaterial as a tool for realizing autonomous and active guiding of sound beams along arbitrary curved paths in free two-dimensional space. The metamaterial building blocks are acoustic transducers, embedded in a slab waveguide. The transducers generate a desired dispersion profile in closed-loop by processing real-time pressure field measurements through preprogrammed controllers. In particular, the metamaterial can be programmed to exhibit analogies of quantum topological wave phenomena, which enables unconventional and exceptionally robust sound beam guiding. As an example, we realize the quantum valley Hall effect by creating, using a collocated pressure feedback, an alternating acoustic impedance pattern across the waveguide. The pattern is traversed by artificial trajectories of different shapes, which are reconfigurable in real-time. Due to topological protection, the sound waves between the plates remain localized on the trajectories, and do not back-scatter by the sharp corners or imperfections in the design. The feedback-based design can be used to realize arbitrary physical interactions in the metamaterial, including non-local, nonlinear, time-dependent, or non-reciprocal couplings, paving the way to unconventional acoustic wave guiding on the same reprogrammable platform. We then present a non-collocated control algorithm, which mimics another quantum effect, rendering the sound beams uni-directional. [ABSTRACT FROM AUTHOR]