Topology in Polar Flocking and Active Nematics
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Active flocking on curved surfaces, such as the 2-sphere and the catenoid, exhibits dynamical symmetry breaking in the form of spontaneous flow, calculable inhomogeneous density patterns and long-wavelength propagating sound modes that get gapped by the curvature of the underlying substrate. Curvature and active flow together result in symmetry-protected topological modes that get localized to special geodesics on the surface. These modes are the analogue of edge states in electronic quantum Hall systems and provide unidirectional channels for information transport in the flock, robust against disorder and backscattering. Active nematics instead exhibit spontaneous motility of strength +1/2 disclinations and active torques that favor the motility-driven unbinding of defects. Despite the directed motion of defects, nematic order is stabilized by rotational noise at low enough activity. Within a perturbative treatment, active forces lower the effective defect-unbinding transition temperature.
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