Fig. 4: Orientational interactions between adjacent streamlines and topological-defect fractionalization.
a, Close-up view of the flow field in two adjacent streamlines separated by a channel that supports no net flux. Below ϵ*, the central channel hosts a single vortex. Above ϵ*, the central channel hosts two vortices. When two vortices rotate in the same direction, the flow field includes one topological charge of −1 and two charges of +1. When the two vortices rotate in opposite directions, the −1 charge is fractionalized into two −1/2 defects bound to the channel walls. b, Fraction of parallel contacts across the channels supporting no net flow. The fraction of parallel contacts is given by \(\frac{1}{2}(1+{\sigma }_{1}{\sigma }_{2})\). The orange hexagons indicate honeycomb lattices. The dark rectangles and bow-tie symbols indicate experiments performed in the brick and deformed honeycomb lattices, respectively (Supplementary Fig. 1). The flow couplings are mostly antiparallel below ϵ* and mostly parallel above. Each experiment has been repeated ten times. c, Numerical resolution of the Toner–Tu equations in cigar-shaped boxes (Supplementary Information). We define the aspect ratio of the boxes as ϵ = ℓ/w, where w is the width of the simulation box and ℓ is the length of the straight line joining the half-circles. Increasing the aspect ratio results in a transition from a flow field with a single +1 defect to flow fields hosting two vortices either separated by a +1 defect or two −1/2 defects. Each simulation has been repeated 50 times. As shown in b, the error bars correspond to the 95% confidence interval. d, Steady states where the right-most (v2) and left-most (v1) flows are parallel and prevail in boxes with large aspect ratios. The transition from antiferromagnetic to ferromagnetic contacts are not specific to colloidal rollers but generic to flocking matter (Toner–Tu fluids). e, Close-up view of the experimental streamlines measured in a channel where ϵ < ϵ*. The green arrow indicates the local orientation of the flows. The antiferromagnetic couplings between adjacent streamlines promote crumpled and segregated streamlines. f, Same data as in e, but for ϵ > ϵ*. The ferromagnetic couplings promote persistent and nested streamlines.
