Odd elastic response of colloidal spinner crystals

The elastic response of two-dimensional solids assembled from passive units that interact isotropically is completely determined by their bulk and shear moduli. Driving these units out of equilibrium enables the addition of isotropic transverse interactions. This generically expands the allowed mechanics to include intrinsic chiral stresses and moduli unconstrained by energy conservation. To each of these additions are associated new phenomena, from the self-propulsion of dislocations driven by odd stress to the anomalous coupling of crystalline deformations enabled by odd elasticity. We ground our discussion using coarse-grained microscopic models, molecular dynamics simulations, and colloidal experiments.