Fracton-elasticity duality on curved manifolds

The mechanical properties of crystals on curved substrates mix elastic, geometric, and topological degrees of freedom. In order to elucidate the properties of such crystals, we formulate the low-energy effective action that combines metric degrees of freedom with displacement fields and defects. We propose dualities for elasticity coupled to curved geometry formulated in terms of tensor gauge theories. We show that the metric degrees of freedom, evolving akin to linearized gravity, are mapped to tensors with three indices. When coupled to crystals, these degrees of freedom become gapped and, in the presence of dislocations and disclinations, multivalued. The elastic degrees of freedom remain gapless and mapped to symmetric gauge fields with two indices. In analogy with elasticity on flat-space formulation, we assume that the trace of the total quadrupole moment is conserved. In the dual formulation, topological defects, which act as sources for the gauge fields, are fractons or excitations with restricted mobility. This leads to generalized glide constraints that restrict both displacement and gravitational defects.

Figure 1

(a) As an example of a crystal on a curved manifold, we start with a sphere. (b) We allow atoms to break the rotation symmetry spontaneously by creating a crystal on its surface. (c) Curvature is visible by the necessity of including pentagons in a hexagonal lattice, which unfolded on a plane shows angle deficiency.