Periodic orbits, pair nucleation, and unbinding of active nematic defects on cones

Geometric confinement and topological constraints present promising means of controlling active materials. By combining analytical arguments derived from the Born-Oppenheimer approximation with numerical simulations, we investigate the simultaneous impact of confinement together with curvature singularity by characterizing the dynamics of an active nematic on a cone. Here, the Born-Oppenheimer approximation means that textures can follow defect positions rapidly on the time scales of interest. Upon imposing strong anchoring boundary conditions at the base of a cone, we find a a rich phase diagram of multi-defect dynamics including exotic periodic orbits of one or two $+1/2$ flank defects, depending on activity and non-quantized geometric charge at the cone apex. By characterizing the transitions between these ordered dynamical states, we can understand (i) defect unbinding, (ii) defect absorption and (iii) defect pair nucleation at the apex. Numerical simulations confirm theoretical predictions of not only the nature of the circular orbits but also defect unbinding from the apex.