Synchro-nematic and -antinematic ordering of spheroidal circle swimmers
Chirality gives a microswimmer something a straight-line swimmer lacks: a phase. This variable both modulates, and is affected by, the hydrodynamic interactions between microswimmers. Here we ask what collective order emerges when many such chiral swimmers are free to move, and how the shape and actuation anisotropies of an individual swimmer dictate the outcome. Using a kinetic theory for hydrodynamically interacting circle swimmers, we show that the interplay between intrinsic rotation, stresslet flows, and Jeffery-like reorientation generates effective phase-locking interactions. Asymmetries in the actuation are encoded through a non-axisymmetric stresslet tensor. At the pair level, pusher swimmers select one of two synchronized states depending on particle shape and actuation asymmetry: in-phase/anti-phase locking, or quarter-shifted locking. Extending the analysis to many-body systems, we find that these pair-level synchronization mechanisms drive emergent collective phases. The swimmers develop global \textit{synchro-nematic} order when the hydrodynamic coupling favors parallel or anti-parallel phase locking, and \textit{synchro-antinematic} local order where quarter-shifted locking prevails. A coarse-grained field theory predicts the onset of nematic order through a hydrodynamic instability criterion. In addition, we find that the collective states exhibit crystalline or disordered hyperuniform structure arising from period-averaged hydrodynamic interactions that are effectively repulsive between swimmers. Lattice Boltzmann simulations of chiral oblate squirmers, resolving finite-size and near-field flows, recover the synchro-nematic ordering. Together, these results show how a swimmer's geometric and actuation anisotropies can be leveraged to program synchronization and spatiotemporal order in chiral active matter.