Defect Loops in Three-Dimensional Active Nematics as Active Multipoles

We develop a description of defect loops in three-dimensional active nematics based on a multipole expansion of the far-field director and show how this leads to a self-dynamics dependent on the loop's geometric type. The dipole term leads to active stresses that generate a global self-propulsion for splay and bend loops. The quadrupole moment is non-zero only for non-planar loops and generates a net `active torque', such that defect loops are both self-motile and self-orienting. Our analysis identifies right- and left-handed twist loops as the only force and torque free geometries, suggesting a mechanism for generating an excess of twist loops. Finally, we determine the Stokesian flows created by defect loops and describe qualitatively their hydrodynamics.