Braiding a Flock: Winding Statistics of Interacting Flying Spins

When animal groups move coherently in the form of a flock, their trajectories are not all parallel, the individuals exchange their position in the group. In this Letter, we introduce a measure of this mixing dynamics, which we quantify as the winding of the braid formed from the particle trajectories. Building on a paradigmatic flocking model we numerically and theoretically explain the winding statistics and show that it is predominantly set by the global twist of the trajectories as a consequence of a spontaneous symmetry breaking.