The effect of dualities on elastic moduli

Elasticity describes how a rigid object like a rubber duck goes back to its original form when slightly deformed. Microscopic symmetries impose strong constraints on the elasticity of a crystalline solid. For instance, the elasticity of a 2D crystal with triangular symmetry is isotropic, and hence has only two independent elastic moduli (instead of six without any symmetry). This is because the elastic tensor relating stress and deformation transforms as a tensor under spatial transformations.
Hence, it is surprising to encounter a family of anisotropic microscopic crystals devoid of any symmetry that however exhibits isotropic elasticity. We will show that a duality lies at the root of this riddle.

In addition to usual spatial symmetries, non-spatial symmetries can occur microscopically, that are not captured in the tensorial character of the elastic tensor. Such additional symmetries can emerge in families of microscopic systems where a duality transformation relates pairs of different systems: in self-dual systems (mapped to themselves by the duality), the duality can become an new symmetry different from spatial operations. These relations can constrain the elastic tensor, reducing the number of independent moduli both in self-dual systems and away from the self-dual point.