Conformal soft modes at elastic instabilities

In case a crystal is close to a continuous elastic phase transition, pronounced critical fluctuations emerge. We show that on the level of linear elasticity theory the displacement vector for these critical modes obeys a condition reminiscent of the conformal Killing equation. For an isostructural instability, four critical modes arise corresponding to one dilation and three special conformal transformations. At crystal symmetry-breaking instabilities, the critical fluctuations include a continuum of modes even for crystals with finite size. For certain cases, these comprise the conformal transformations associated with two-dimensional Minkowski space. These results have implications for critical elastic phenomena as well as for extremal mechanical metamaterials.