Flocking on Curved Surfaces and Topological Sound

Flocking, the spontaneous collective motion of a large collection of self-propelled entities, is seen on many scales in nature, and can occur on a curved substrate. Generalizing the continuum Toner-Tu equations of an active polar fluid to a curved surface, we analytically obtain inhomogeneous ordered flocking states that also include the topological defects required by the underlying curved geometry. In addition to frustrating order, the curvature also generates a finite frequency threshold to excite the long-wavelength sound modes that are present in an ordered polar flock. The breaking of time reversal symmetry due to spontaneous flow and the presence of a gap in the sound spectrum together gives rise to topologically protected sound modes that propagate unidirectionally and are localized to special geodesics on the surface (like the equator on a sphere). These excitations are analogous to edge states in electronic quantum Hall systems or well-known equatorial waves in ocean and atmospheric flows, and provide unidirectional channels for information transport in the flock, robust against disorder and backscattering.