A Complex Topological Phase in C-Spin Active Matter

This work unveils complex topological properties within a recent theoretical model concerning the interplay of positional and orientational order. The model features "complementary-spins" (c-spins), symbolic agents having two-dimensional spatial mobility and rotational freedom, divided into two populations with contrasting positional and orientational interactions. The model is governed by the system size and a control parameter that splits their natural rotational frequencies, a form of circular anisotropy. For a given system size and for small anisotropy, equilibrium patterns showing both positional and orientational regularity emerge, consistent with local stability predictions. For moderate anisotropy, the system develops complex topological point defects, driven by phase singularity and bistable with the uniform patterns. The defects are constituted by curled orientational textures embedding two c-spin loop trains that counter-rotate around the same center, exhibiting regular spacing, spin-momentum locking, and dissipationless flow. These emergent non-local defect complexes are extremely robust to noise and capable of self-repair, and constitute a whole new class of non-equilibrium dissipative states. These are, in fact, topological vortex states, classifiable by a two-valued topological charge. For anisotropy exceeding the local stability threshold, active turbulence (deterministic chaos) takes place and order is lost. A statistical analysis revealed the coexistence of a double phase transition at a critical parameter value: an "ordinary" symmetry-breaking transition associated with standard collective synchronization and a novel topological phase transition activating the vortex complexes. Quantitative boundaries have been evaluated, either analytically or numerically, in the parameter space. Increasing system size enhances organizational complexity, developing spin-momentum locked transport networks of increasing complexity. Thanks to its self-organizational properties, this work provides a new tool to understand robustness and morphogenesis in living systems.