Spring-loaded: Engineering Dispersion Relations Using Complex Couplings

Motivated by recent theoretical and experimental interest in metamaterials comprising non-local coupling terms, we present an analytic framework to realise materials with arbitrary complex dispersion relations. Building on the inverse design method proposed by Kazemi et al. [Phys. Rev. Lett. 131, 176101 (2023)], which demonstrated arbitrary real dispersion relations with real non-local coupling terms, we show that using complex non-local couplings not only can band structures be drawn, but that arbitrary attenuation of the supported modes can be achieved by tuning their trajectory in the complex frequency plane. This is demonstrated using a canonical mass-spring lattice with higher-order spatial connections; we consider the energy velocity in such systems and show that competing loss and gain of the complex interactions can produce exotic wave dispersion.