Extended Data Fig. 8: Emergent collective order in simulations.

Simulations of square HSLs with the theoretical model developed previously68 is used to explore collective order in 2D. a, b, For appropriate lattice spacings, in-phase ferromagnetic order FM+ (a) and in-phase antiferromagnetic order AFM+ (b) are simulated. Left, schematic; middle, wave field and walker’s trajectories; right, time evolution of the orbital phases.