A Complex Topological Phase in C-Spin Active Matter

This work explores a theoretical model of self-organization driven by the interplay of positional and orientational order in "complementary-spins" (c-spins), symbolic agents divided into two populations with contrasting positional and orientational interactions. The system, governed by a circular anisotropy parameter that makes it active, exhibits a variety of complex behaviors. For small anisotropy, uniform, stable patterns emerge. As anisotropy increases to a moderate level, the system develops robust, self-repairing topological point defects—vortex complexes characterized by orientational textures and counter-rotating c-spin loops with spin-momentum locking. These novel non-equilibrium dissipative structures are classifiable by a two-valued topological charge. Beyond a local stability threshold, active turbulence (deterministic chaos) occurs, 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 and a novel topological phase transition that activates the vortex complexes. Both analytical and numerical methods were used to evaluate quantitative boundaries in the parameter space, and increasing the system size enhances the complexity of the transport loops. Due to its self-organizational properties, this model provides a new tool for understanding robustness and morphogenesis in living systems.