Nonreciprocal yet Symmetric Multi-Species Active Matter: Emergence of Chirality and Species Separation

Nonreciprocal active matter systems typically feature an asymmetric role among interacting agents, such as a pursuer-evader relationship. We propose a multi-species nonreciprocal active matter model that is invariant under permutations of the particle species. The nonreciprocal, yet symmetric, interactions emerge from a constant phase shift in the velocity alignment interactions, rather than from an asymmetric coupling matrix. This system possessing permutation symmetry displays rich collective behaviors, including a species-mixed chiral phase with quasi-long-range polar order and a species separation phase characterized by vortex cells. The system also displays a coexistence phase of the chiral and the species separation phases, in which intriguing dynamic patterns emerge. These rich collective behaviors are a consequence of the interplay between nonreciprocity and permutation symmetry.

💡 Research Summary

The paper introduces a multi‑species Vicsek‑type model in which particles of different species align their velocities with a constant phase shift α, while the coupling matrix remains symmetric (J_{nm}=J_{mn}). This construction yields nonreciprocal interactions without breaking permutation (S_Q) symmetry, a feature that distinguishes it from previously studied nonreciprocal active matter systems that rely on asymmetric coupling matrices.

Model definition: each particle n carries a position r_n, a heading angle θ_n∈(−π,π], and a species label s_n∈{1,…,Q}. The discrete‑time update rules are

θ_n(t+Δt)=Arg