Topology by activity in non-Hermitian mechanical metamaterials

A simple view of an active solid is a network of masses interacting via non-conservative bonds. We examine 2D topological lattices equipped with non-reciprocal active bonds that inherit two key properties from their passive counterparts: their interactions conserve linear momentum and depend only on the relative positions of the masses. We characterize the flow of Berry curvature and exceptional points in the resulting non-Hermitian band structure as activity is tuned from the passive limit to the brink of instability. Along this flow, we find a discrete onset of topologically charged bands whose activity threshold can be controlled via zero-mode deformations of the underlying lattice. Simulations and analytical calculations reveal the emergence of persistent edge modes influenced by the interplay of a topologically protected penetration depth and an effective quality factor derived from non-Hermitian gain and loss. Our work sheds light on non-Hermitian band theory and the design of active metamaterials that conserve linear momentum.