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Non-Newtonian Topological Mechanical Metamaterials Using Feedback Control

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, pre-programmed 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 directional complex coupling terms, 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 to realize, for the first time, 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 counter-propagating diffusive bulk modes. The proposed method is general and flexible, and could be used to realize arbitrary lattice parameters, such as non-local or nonlinear couplings, time dependent potentials, non-Hermitian dynamics, and more, on a single platform.