Curvature-driven topological protection

Active systems on curved geometries are ubiquitous in the living world. Ordered polar flocks in the presence of spatial curvature are forced to be inhomogeneous and must also exhibit topological defects, even in the steady state, when the curved manifold is topologically non-trivial. I will discuss how dynamical symmetry breaking via spontaneous flow (flocking), together with spatial curvature, are the only ingredients necessary to produce long-wavelength symmetry-protected gapped sound modes. The steady state profile of an active polar flock on a two-sphere and a catenoid will be presented. The topologically-protected sound modes are localized to special geodesics on the surface (the equator or the neck respectively). 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.