Dislocation Loops in Smectic Liquid Crystals from Topology

Dislocation loops play a critical role in the theory of melting smectic and crystalline order. Given de Gennes' analogy between smectics and superconductors, one might presuppose that defects in smectics enjoy the same kinds of excitations as superconductors. However, broken translational symmetry in smectics yields extra subtlety for the theory of topological defects. For example, edge and screw dislocations in smectics are topologically distinct, and disclination charge is limited to q≤1. One is required to adhere to these and all other topological rules when constructing defects in smectics, making looped, linked, or knotted dislocations highly nontrivial. By starting from the purely topological fact that dislocations can be split into disclination pairs, we analyze how edge and screw dislocations can form loops in smectic liquid crystals. We discuss the implications of this construction on the Gaussian curvature of the smectic layers as well as the prospect of knotted defect configurations.