Singularities and symmetry breaking in swarms.

A large-scale system consisting of self-propelled particles, moving under the directional alignment rule (DAR), can often self-organize to an ordered state that emerges from an initially rotationally symmetric configuration. It is commonly accepted that the DAR, which leads to effective long-range interactions, is the underlying mechanism contributing to the collective motion. However, in this paper, we demonstrate that a swarm under the DAR has unperceived and inherent singularities. Furthermore, we show that the compelled symmetry-breaking effects at or near the singularities, as well as the topological connectivity of the swarm in the evolution process, contribute fundamentally to the emergence of the collective behavior; and the elimination or weakening of singularities in the DAR will induce an unexpected sharp transition from coherent movement to isotropic dispersion. These results provide some insights into the fundamental issue of collective dynamics: What is the underlying mechanism causing the spontaneous symmetry breaking and leading to eventual coherent motion?