  Discrete models provide concise descriptions of complex physical phenomena,
such as negative refraction, topological insulators, and Anderson localization.
While there are multiple tools to obtain discrete models that demonstrate
particular phenomena, it remains a challenge to find metamaterial designs that
replicate the behavior of desired nontrivial discrete models. Here we solve
this problem by introducing a new class of metamaterial, which we term
'perturbative metamaterial', consisting of weakly interacting unit cells. The
weak interaction allows us to associate each element of the discrete model
(individual masses and springs) to individual geometric features of the
metamaterial, thereby enabling a systematic design process. We demonstrate our
approach by designing 2D mechanical metamaterials that realize Veselago lenses,
zero-dispersion bands, and topological insulators. While our selected examples
are within the mechanical domain, the same design principle can be applied to
acoustic, thermal, and photonic metamaterials composed of weakly interacting
unit cells.
