Extended Data Fig. 7: Emergent order for different vertical bouncing synchronizations and lattice geometries.

a, Oblique view of a 2D spin lattice where the walker in the centre is bouncing vertically in-phase and out-of-phase with its left and right neighbours, respectively. b, Average spin correlation for lattices with the same geometry as those described in Fig. 1 when the walkers all bounce in phase (blue, result presented in text), out of phase (green), or have randomly distributed bouncing phases (red). Vertically in-phase pairs promote in-phase orbital antiferromagnetic order (AFM₊), whereas vertically out-of-phase pairs promote out-of-phase orbital antiferromagnetic order (AFM₋). A random distribution of vertical phases thus leads to competing orbital synchronization modes, which has an effect on the emergent spin correlation. Solid lines are fits resulting from smoothing the piecewise linear plot given by connecting the points. c, Triangular HSLs with a lattice spacing tuned to promote antiferromagnetic order can be used to investigate frustration effects.